south carolina support systems instructional...

SOUTH CAROLINA SUPPORT SYSTEM INSTRUCTIONAL GUIDEContent Area Seventh Grade MathSeventh Grade Math

Recommended Days of Instruction First Nine WeeksFirst Nine Weeks

Standards/Indicators Addressed:Standard: 7-2: The student will demonstrate through the mathematical processes an understanding of the representation of

rational numbers, percentages, and square roots of perfect squares; the application of ratios, rates, and proportions to solve problems; accurate, efficient, and generalizable methods for operations with integers; the multiplication and division of fractions and decimals; and the inverse relationship between squaring and finding the square roots of perfect squares.

7-2.1 Understand fractional percentages and percentages greater than one hundred7-2.2 Represent the location of rational numbers and square roots of perfect squares on a number line.7-2.3 Compare rational numbers, percentages, and square roots of perfect squares by using the symbols

≤, ≥, <, >, and =. 7-2.4 Understand the meaning of absolute value. 7-2.6 Translate between standard form and exponential form.7-2.7 Translate between standard form and scientific notation7-2.9 Apply an algorithm to multiply and divide fractions and decimals.7-2.10 Understand the inverse relationship between squaring and finding the square roots of perfect squares.

* These indicators are covered in the following 3 Modules for this Nine Weeks Period.

Module 1-1 Rational NumbersModule 1-1 Rational Numbers

South Carolina Curriculum ProjectSouth Carolina Curriculum Project DRAFTDRAFT 12-15-200812-15-2008 1

Indicator Recommended Resources Suggested Instructional Strategies

Assessment Guidelines

Module 1-1 Lesson A:

7-2.1 Understand fractional percentages and percentages greater than one hundred.

NCTM's Online Illuminations http://illuminations.nctm.org

NCTM's Navigations Series

SC Mathematics Support Document

Teaching Student-Centered Mathematics Grades 5-8 and Teaching Elementary and Middle School Mathematics Developmentally 6th Edition, John Van de Walle

NCTM’s Principals and Standards for School Mathematics (PSSM)

Textbook Correlations – See Appendix A

See Instructional Planning Guide Module 1-1 Introductory Lesson A

See Module 1-1, Lesson A Additional Instructional Strategies

See Instructional Planning Guide Module 1-1 Lesson A Assessment

Module 1-1 Lesson B:

7-2.10 Understand the inverse relationship between squaring and finding the square roots of perfect squares.

See Instructional Planning Guide Module 1-1, Introductory Lesson B

See Instructional Planning Guide Module 1-1, Lesson B Additional Instructional Strategies

See Instructional Planning Guide Module 1-1 Lesson B Assessment

Module 1-1 Lesson C

7-2.2 Represent the location of rational numbers and square roots of perfect squares on a number line.

See Instructional Planning Guide Module 1-1 Introductory Lesson C

See Instructional Planning Guide Module 1-1, Lesson C Additional In-structional Strategies

See Instructional Planning Guide Module 1-1 Lesson C Assessment

Module 1-1 Lesson D

7-2.3 Compare rational numbers, percentages, and square roots of perfect squares by using the symbols ≤, ≥, <, >, and =.

See Instructional Planning Guide Module 1-1, Introductory Lesson D

See Instructional Planning Guide Module 1-1, Lesson D Additional Instructional Strategies

See Instructional Planning Guide Module 1-1 Lesson D Assessment

Module 1-1 ContinuedIndicator Recommended Resources Suggested Instructional Assessment


http://illuminations.nctm.org/

Strategies GuidelinesModule 1-1 Lesson E:

7-2.4 Understand the meaning of absolute value.



SC Mathematics Support DocumentTeaching Student-Centered Mathematics Grades 5-8 and Teaching Elementary and Middle School Mathematics Developmentally 6th Edition, John Van de Walle



See Instructional Planning Guide Module 1-1 Introductory Lesson E

See Instructional Planning Guide Module 1-1, Lesson E Additional Instructional Strategies

See Instructional Planning Guide Module 1-1 Lesson E Assessment

Module 1-2 Number StructureIndicator Recommended Resources Suggested Instructional

StrategiesAssessment Guidelines



7-2.6 Translate between standard form and exponential form.







See Instructional Planning Guide Module 1-2, Lesson A Additional Instructional Strategies



7-2.7 Translate between standard form and scientific notation




Module 1-3 Operations on Fractions/DecimalsModule 1-3 Lesson A:

7-2.9 Apply an algorithm to multiply and divide fractions and decimals.

NCTM's Online Illuminations http://illuminations.nctm.org NCTM's Navigations SeriesSC Mathematics Support DocumentTeaching Student-Centered Mathematics Grades 5-8 and Teaching Elementary and Middle School Mathematics Developmentally 6th Edition, John Van de WalleNCTM’s Principals and Standards for School Mathematics (PSSM)Textbook Correlations – See Appendix A





7-2.9 Apply an algorithm to multiply and divide fractions and decimals

See Instructional Planning Guide Module 1-3,Introductory Lesson B



Module 1-3 ContinuedIndicator Recommended Resources Suggested Instructional


Module 1-3 Lesson C: NCTM's Online Illuminations See Instructional Planning Guide See Instructional




http://illuminations.nctm.org





Module 1-3 Introductory Lesson C

See Instructional Planning Guide Module 1-3, Lesson C Additional Instructional Strategies

Planning Guide Module 1-3 Lesson C Assessment

Module 1-3 Lesson D:


See Instructional Planning Guide Module 1-3, Introductory Lesson D

See Instructional Planning Guide Module 1-3, Lesson D Additional Instructional Strategies



MODULE1-1

Rational Numbers

I. Background for the Module


South Carolina Curriculum ProjectSouth Carolina Curriculum Project DRAFTDRAFT 12-15-200812-15-2008

This module addresses the following indicators:

7-2.1 Understand fractional percentages and percentages greater than one hundred. (B2) 2 days

7-2.10 Understand the inverse relationship between squaring and finding the square roots of perfect squares. (B2) 2 days

7-2.2 Represent the location of rational numbers and square roots of perfect squares on a number line. (B2) 2 days

7-2.3 Compare rational numbers, percentages, and square roots of perfect squares by using the symbols ≤, ≥, <, >, and =. (B2) 5 days

7-2.4 Understand the meaning of absolute value. (B2) 1 day

Module 1–1 consists of 5 introductory lessons. Teaching time should be adjusted to allow for sufficient learning experiences in each of the modules.

6

1. Learning Continuum

In sixth grade, students were introduced to whole number percents of one hundred or less.

In third grade, students had their first experiences with perfect squares as they learned basic multiplication facts such as 4 x 4, 7 x 7, etc. In fifth grade, students were exposed to the concept of squares when they determined the area of geometric squares. Students have been exposed to inverse relationships for addition and subtraction in first grade and multiplication and division in third grade. These relationships can be used to help explain these inverse relationships.

In fifth grade students compared whole numbers, decimals, and fractions. In sixth grade the comparisons were continued and whole number percents were included.

2. Key Vocabulary

Percent CubedBaseExponentPowerSquareSquare rootRational NumberPerfect squareAbsolute valueFractional percentages

3. Content Overview

Seventh grade students should extend this knowledge to percentages less than one and percentages greater than one hundred. Using concrete models with enable the students to connect the new learning to prior knowledge. Since students worked with fractions that are less than or greater than one in third grade, they should now be given opportunities to line that to fractional percents and percents greater than 100%. Students should be given opportunities to develop models using materials such as base-ten blocks, 10 x 10 grid paper, or graph paper to help students visualize the connection of fractions to percents less than one percent and mixed number to percents greater than 100%.

When the terms squares and square roots are introduced, it is essential that the connection is made between the squared number and the corresponding geometric square. In other words, students should understand that “find the length of a side of a square with area equal to 25 units” and “find the square root of 25” are basically the same question. Students have been exposed to


inverse relationships for addition and subtraction in first grade and multiplication and division in third grade. In seventh grade, the concept of inverse relationships is expanded to include squaring and finding square roots of perfect squares. Learning opportunities should include both models and numbers.

In seventh grade the expectation is to locate rational numbers and square roots of perfect squares on a number line. Students should explore the location of fractions, decimals, percents, and square roots of perfect squares on a number line. It is equally important that students justify the placement of these representations on a number line, as well as understand the relationship to the numbers between which a given value lies. Being able to justify the placement on a number line will enable students to compare and order rational numbers, percentages, and square roots of perfect squares using the symbols ≤, ≥, <, >, and =.

Students new to seventh grade will be an understanding of the meaning of absolute value. Student instruction should focus on the fact that the absolute value of a number is the distance of the number from zero. The absolute value of any number except zero is a positive value. An understanding that distance is always a positive value is essential to develop a solid understanding.

II. Teaching the Lesson

1. Teaching Lesson A

a. Indicators with Taxonomy

7-2.1 Understand fractional percentages and percentages greater than one hundred.

Cognitive Process Dimension: UnderstandKnowledge Dimension: Conceptual Knowledge

b. Introductory Lesson –

Adapted from a CEEMM lessonEssential Question: What are real-life examples that you would express as a percent greater than 100 or less than one?

1. Ask students to give several examples that show how percents are used in everyday life. Ask students for a definition of “percent”. Remind students that the word percent means “per 100” or “out of 100”. Using 100-block models will help students understand the concept of “percent” meaning “for each 100.” In 6th grade students had experiences in shading 10 × 10 grids to represent given percents. Distribute the Grid Worksheet to all students. (NCTM suggests these sheets be laminated so that students can shade the grids with dry-erase, water-based, or grease markers.)

2. Model percents using 100-block models to help students understand the

concept of “percent” meaning “for each 100.” Assess the students


http://illuminations.nctm.org/Lessons/Percent/Percent-AS-10x10Grids.pdf

understanding of whole number percents by asking them to shade the percents shown below on the grid worksheet.

3. Show students how to change a fraction to a percent by finding an equivalent fraction using 100 as the denominator then use the numerator as the percent. (Because 7-2.9 has not been introduced all denominators must divide into 100 without a remainder, unless they are benchmark fractions.)

4. Model changing a percent to a fraction by writing the percent over 100 and

simplifying the fraction. For example, to change 72% to a fraction, place 72 over 100 to equal 72/100 and then simplify the fraction. 72/100 =18/25.

5. Before using percents to solve problems, students should have experiences in shading 10 × 10 grids to shade in percentages that include decimals and fractions, such as 63.25% and 18½%. (Because 7-2.9 has not been introduced any fractions used here must be benchmark fractions.) Using money may help them understand why you would have less than one percent. If you present $100, $1 would be 1%, any change, like 75 cents would be less than 1%. The figures below show that when ¼ of a square is shaded, this represents ¼% (or 25 cents), and when an entire unit square (100 small squares= $100) plus another 34 small squares are shaded; this represents 134% (representing $134).

6. To change a decimal to a percent, you will move the decimal to the right. Examples should include decimals such as:

a.) 0.0054 = 0.54% b.) 15.5 = 1550%c.) 2 = 200%


7. Write a decimal less than one such as 0.135 on the board. Ask them to write it as a percent (13.5%). Explain that it can also be written as 13½% since .5 = ½. When it is written as 72 ½% or 17.5% the number can be referred to as a “fractional percentage.” Give several more examples using only benchmark fractions.

8. Show students how to change a percent to a decimal by moving the decimal point two places to the left. (Remind students of decimal placement in a whole number.) Show several examples of percents greater than 100%, less than one, and percents that also contain a decimal. Examples should include percents such as:

a.) 150% = 1.5. b.) 0.95% = 0.0095c.) 12.8% = 0.128.

Point out to students that mixed numbers will always be percents greater than 100. Because 1 is 100% any mixed number must be greater than 100%. Throughout the lesson remind students that the decimal is always moved two places because percent means “per hundredth”.

9. Distribute copies of a table similar to the one below for pairs of students to complete.

PercentDecimal Fractio

n PercentDecimal

Fraction

1 2/5 7 2 9/10

2 0.75 8 0.03

3 50% 9 19%4 0.6 10 0.275

5 1/3 11 588%6 0.05 12 0.003

c. Misconceptions/Common Errors –

Students often will memorize rules for decimal movement when changing a decimal to a percent or a percent to a decimal. This may result in incorrect movement of decimal in changing to percent. Students may leave the decimal in its original position and simply add a percent sign such as 0.25 ≠ 0.25% or they may move the decimal to right instead of to the left, such as .25 ≠ .0025%. Students may also not understand decimal placement in any whole number, such as 25% = 25.0%.

d. Additional Instructional Strategies –


Teaching Student-Centered Mathematics Grades 5-8 , John A. Van de Walle and LouAnn H. Lovin, Pearson, 2006. Pages 119 – 122 provide additional opportunities for instruction and/or practice.

Glencoe Mathematics Course 2, pages 71-72, 159, 162, 338 and 340 provide additional opportunities for instruction and/or practice.

Enrichment:

CEEMM: The following NCTM introductory lesson allows students to apply that knowledge to problem solving.

Materials: centimeter grid paper

There are two other preliminary activities that can set the stage for solving percent problems. 1. Let the unit square (10 × 10 grid) represent given amounts, and then determine the

value of one of the small squares (1 percent). To help students determine the value of 1 percent of the unit square, they can think of sharing the given amount equally among the 100 parts of the unit square. Dividing by 100 can be done by mental computation. As examples, if the unit square represents 400 people (that is, if 400 people represents 100% in some situation), then each small square represents 4 people. Similarly, if the unit square represents 85 pounds, then each small square has a value of 0.85 pounds; and, if the unit square represents 162 days, then the value of each small square is 1.62 days. Each of these can easily be related to mental division by 100. (Relate division of money - $1 divided into 10 parts would be 10 cents or 0.10 which moves the decimal 1 place. $1 divided into 100 parts would be 1 cent or 0.01 which moves the decimal 2 places. Several examples of this type should help.)

Successfully determining the value of one square (1%) is key to solving percent problems. To prepare students for this task, it may be helpful to pose questions that involve convenient fractional parts of the grid. For example, if the unit square represents 400 people:

How many people would be represented by half of the unit square? … by one-fourth of the unit square? … by twenty small squares? … by ten small squares?

What part of the unit square would represent 200 people? …100 people? …40 people?


…4 people?

The following percent problems can be solved using the unit square grids. The first three problems use a two-part approach: first, students are asked to use a percent grid to represent the given information; then, students use the sketch to answer a question related to that information. This approach emphasizes the importance of thinking through the given information before attempting to obtain an answer.

Problems 1-3 are typical percent problems. Problem 4 involves percents greater than 100. Problems 5-6 deal with percent increase, a situation that often causes anxiety in students. Finally, problem 7 uses the grid method to solve a problem involving a percent discount.

1.Twenty percent of a company's 240 employees are classified as minorities. a. Use a percent grid to show that 20 percent of a business's 240 employees are

classified as minorities. b. How many employees are classified as minorities?

[The given information can be represented by letting the unit square represent 240 employees and shading 20 percent of the square.]

Students may notice immediately from the shaded grid that one shaded column represents 1/10 of 240, or 24, so two shaded columns—which is 2/10, or 20%—represent 48.

[Only if 7.2.9 has been introduced…Alternatively, if the whole square represents 240 employees, then one small square represents 2.4 employees, so 20 squares represent 20 × 2.4 = 48. Therefore, 48 employees are classified as minorities.]

2.Twenty-five acres of land are donated to a community, but the donor stipulates that six acres of this land should be developed as a playground. a. Use a percent grid to represent this situation. b. What percent of the land is to be used for playground?

[If the unit square represents 25 acres, we need to determine how many small squares represent 6 acres. Each small square represents 0.25 = ¼ acres, so 4 small squares represent 1 acre, and 24 small squares represent 6 acres. Students have not been exposed to division of decimals except with models so they will not understand since each


small square has a value of 0.25 acres, then 6 ÷ 0.25 = 24 squares are needed to represent 6 acres.

The playground is represented by 24 shaded squares, so 24 percent of the land is to be used for the playground.]

3.In Hazzard County, 57 of the schools have a teacher-to-student ratio that meets or exceeds the requirements for accreditation. These 57 schools represent 38 percent of the schools in the county. a. Use a unit square sketch to represent this situation. b. How many schools are in Hazzard County?

[The unit square represents the number of schools in the county, and 38 small squares (38 percent) represent 57 schools.

Because 38 small shaded squares represent 57 schools, each small shaded square represents 57 ÷ 38 = 1.5 schools. So the unit square represents 100 × 1.5 = 150, which is the number of schools in the county.] 4.The school population in Provincetown is 135 percent of the school's population from the

preceding year. The school population in Provincetown is 135 percent of the school's population from the preceding year. The new student population is 270. How many students did the school have the previous year?

[The current population can be represented by 135 small squares, so each small square


represents 270 ÷ 135 = 2 students.

The school population for the preceding year is the value of one unit square, so the number of students the previous year was 100 × 2 = 200.]


2

270

14

e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activites, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)Stream Line Videos Math Mansion: 27.Percentigimole (09:16)http://player.discoveryeducation.com/index.cfm?guidAssetId=0355D4BC-C944-46B4-A502-B4CBF687AB10&blnFromSearch=1&productcode=USPromethean Board – See Prometheanplanet.com lesson “Match the Percent” - Sharon Willard, for a review of whole number percentshttp://www.prometheanplanet.com/server.php?show=ConResource.9886“Percentages” – Pete Lambert (slides 1 – 5)http://www.prometheanplanet.com/server.php?show=ConResource.8891Percent, Decimal, and Fraction, Cheryle Trupphttp://www.prometheanplanet.com/server.php?show=ConResource.8584www.brainpop.com/math/ratioproportionandpercent/percents/preview.wemlwww.yourteacher.com/prealgebra/mathpercentage.phpA tutorial on percents: www.math.com/school/subject1/lessons/S1U1L6GL.html

f. Assessing the Lesson

The following examples of possible assessment strategies may be modified as necessary to meet student/teacher needs. These examples are not derived from nor associated with any standardized test.

1. Exit Slip: Answer today’s essential question.

2. What is 0.9% expressed as a decimal?

a. .009b. .900c. 9.00d. 90.0

3. What is 0.237 written as a percent?

a. 0.00237%b. 0.237%c. 23.7%d. 237%


http://www.math.com/school/subject1/lessons/S1U1L6GL.html

http://www.yourteacher.com/prealgebra/mathpercentage.php

http://www.brainpop.com/math/ratioproportionandpercent/percents/preview.weml

http://www.prometheanplanet.com/server.php?show=ConResource.8584



http://player.discoveryeducation.com/index.cfm?guidAssetId=0355D4BC-C944-46B4-A502-B4CBF687AB10&blnFromSearch=1&productcode=US

http://player.discoveryeducation.com/index.cfm?guidAssetId=0355D4BC-C944-46B4-A502-B4CBF687AB10&blnFromSearch=1&productcode=US

4. Use these figures to answer the question.

What percent of the figures is shaded?a. 0.134%b. 1.34% c. 13.4%d. 134%

5. What is 1/500 written as a percent?a. 0.002%b. 0.02%c. 0.2%d. 2%

6. Many times athletes are said to “give 110%” to their sport. What is 110% expressed as a mixed number?a. 100 1/100 b. 100 1/10c. 1 1/10d. 1 1/100

7. What kind of fraction would represent a percent greater than 100?

8. How do you convert a _____to a _____?

9. Draw a model to represent 0.25 %.

10. Draw a model to represent 225%.

II. Teaching Lesson B


7-2.10 Understand the inverse relationship between squaring and finding the square roots of perfect squares.


b. Introductory Lesson –Materials Needed:


Essential Question: What is the relationship between square root and area?

Students will need centimeter grid paper.

1. Using the Kagan strategy, “Think-Pair-Share” have students discuss the answer(s) to the following question(s):What is the inverse operation of addition?What is the inverse operation of multiplication?What does it mean to “square” a number?

2. Explain that squaring a number also has an inverse; it is called finding the “square root” of a number.

3. Students should work together in teams of 3-4 to investigate squares. Divide

the numbers from 1 to 100 among the teams. Ask them to try to make a square on the grid paper with area equal to each of the numbers they received. They should also record the dimensions of the square and keep a record of the numbers that can be used to make a square and the ones that cannot. Have the students demonstrate their successes on the board or overhead. Discuss the dimensions of the squares and the inverse relationship between the squares and the square roots.

4. Explain that a number whose square root is a whole number is called a “perfect square”. Go over the list of perfect squares from 1 to 144, explaining that the square roots and the perfect squares are a fact family.


No typical student misconceptions noted at this time.Misconceptions may be noted in learning continuum.


Teaching Student-Centered Mathematics Grades 5-8 , John A. Van de Walle and LouAnn H. Lovin, Pearson, 2006. See page 150.

See the resources listed below.

e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activites, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)Stream Line Videos Promethean Board – See Prometheanplanet.comhttp://www.math.com/school/subject1/lessons/S1U1L9GL.htmlhttp://www.mathsisfun.com/square-root.html


http://www.mathsisfun.com/square-root.html


http://math-and-reading-help-for-kids.org/articles/Middle_school_math_help:_Sixth_grade_square_roots.htmlwww.brainpop.com/math/numbersandoperations/squareroots/preview.weml




2. What is the difference between squaring the number 16 and finding the square root of the number 16?

3. The model below represents √36 = 6.

Which arrangement of squares can be used to model a large square that represents √225?

a. 3 rows of 75 squaresb. 5 rows of 45 squaresc. 9 rows of 225 squaresd. 25 rows of 25 squares

4. Which integer is a perfect square?a. 88b. 164c. 196d. 250

5. The length of a square can be represented by √36. What is another way to represent the length of this square?

a. √6b. 6c. √9d. 9

6. Are all numbers perfect squares?

7. What is the opposite of squaring a number?

8. When would a number be called a perfect square?

9. Explain the relationship between squaring a number and finding the area of a square.

3. Teaching Lesson C



http://www.brainpop.com/math/numbersandoperations/squareroots/preview.weml

http://math-and-reading-help-for-kids.org/articles/Middle_school_math_help:_Sixth_grade_square_roots.html

http://math-and-reading-help-for-kids.org/articles/Middle_school_math_help:_Sixth_grade_square_roots.html

7-2.2 Represent the location of rational numbers and square roots of perfect squares on a number line.



Essential Question: How do you locate rational numbers and square roots of perfect squares on a number line?

Materials:

Number lineStraight edgeCompassPencilPaper.

As students walk into the classroom, give each a post-it note or index card with a rational or a perfect square recorded on it. A number line that includes positive and negative integers should be drawn on the board. Students may need to review the placement of integers on the number line.

Introduce the term “rational number” as a number that can be written as the ratio of two integers and give examples. Using the number line on the board or overhead tell students it is often easier to convert all numbers involved to a common form first. ( such as all in decimal form) Then write the number in its original form when placing it on the number line. If the rational number is an improper fraction, it can be converted into a mixed number to easily identify which two integers this rational number can be located between. For fractional parts use 0.5 or ½ as a guide to determine where each number should go. Give them practice examples using rational numbers held by several students from the class.

Ask students to discuss with their partner the location of their number on the number line. When the students agree, the pair should place their numbers on the number line. Once all rational numbers have been posted, the class will check for correctness and discuss what the reasoning was for each location.

Adapted from a lesson by: Winebrenner, William Dunbar Vocationalhttp://www.iit.edu/~smile/mathinde.html

“The following is a collection of almost 200 single concept lessons. These lessons may be freely copied and used in a classroom but they remain the copyright property of the author.”



http://www.iit.edu/~smile/mathinde.html

Positioning numbers on a number line is a common error. When given a blank number line, a student may order the numbers on the line by placing the negative on the right and the positive on the left. Improper or mixed numbers may be incorrectly ordered because students do not understand fractions or their value. Students will often not solve square roots before placement. Stress to students that often it is easier to change each of the given numbers to a common form to order then place on the number line in its original form.


Additional Practice:Mathematics: Applications and Connections, Course 2, Glencoe, 2001 Point Search

Students should work in pairs to complete this lesson. Give each pair one sentence strip and the following list of numbers. Have them draw a number line on the sentence strip, plot 12 of the points, and label them on index cards with at least two equivalent forms of the numbers. You might assign the evens to the girl teams and the odd to the boy teams, for example. They should be expected to discuss the locations of the points and be prepared to justify their reasoning.

1. 2. - 3. 0.4 4. 20%

4. 5. 6. 2 8. -1.6

9. 125% 10. 11. - 12. -413. 3.85 14. 450% 15. 16. -0.9617. 18. 19. 0.01 20. -2.521. 22 22. 9 * 101 23. 0.5% 24. 3.9

Additional ResourcesA Grain of Rice, Helean Clare Pitman, Bantam Skylark, 1986The King’s Chessboard, David Burch, Dial Books, 1988


e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activites, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)Promethean Board – See Prometheanplanet.com http://www.brainpop.com/math/numbersandoperations/squareroots/preview.wemlhttp://www.bced.gov.bc.ca/irp/math89/ma8nnc.htmhttp://www.iit.edu/~smile/ma8720.htmlhttp://www.lessonplanspage.com/MathCILAIdentifyingAndOrderingRationalAndIrrationalNumbers8.htm



1. Exit Slip: Answer the essential question.

2. Which number on the number line represents √‾144?

A B C D E

7 8 9 10 11 12 13 14

a) A b) B c) C d) D e) E

3. Which point represents 2 ¼?

A B C D E

1 2 3 4

a) A b) B c) C d) D e) E


http://www.lessonplanspage.com/MathCILAIdentifyingAndOrderingRationalAndIrrationalNumbers8.htm

http://www.lessonplanspage.com/MathCILAIdentifyingAndOrderingRationalAndIrrationalNumbers8.htm

http://www.iit.edu/~smile/ma8720.html

http://www.bced.gov.bc.ca/irp/math89/ma8nnc.htm



4. Give each student a 8 ½ by 11 sheet of paper or a large index card. Have them put any rational number on their card. Then have the class stand in a circle. Pass around a card that has a > on it and an = on the back. (Students flip the > symbol card when needed to become a < symbol.) When a student gets the symbol card, they must manipulate it to make it the symbol that will make the statement true, comparing the 2 rational numbers of the people on both sides. Everyone should have their cards facing in to the circle for the class to see. The class will give thumbs-up or a thumbs-down for agree or disagree. Pass the symbol card around for everyone to get a turn. (Can also be used with 7-2.3)

Use the number line below to answer numbers 5-8.

A B C D E F G H I J K L

5. What letter identifies ? _______

6. What letter identifies 3² ? _________

7. What letter identifies ? ________

8. What letter identifies 4² ? __________

9. What is a rational number?

10. Can a decimal be a rational number?

11. Using the number line, how do you know which number is greater?

4. Teaching Lesson D


7-2.3 Compare rational numbers, percentages, and square roots of perfect squares by using the symbols ≤, ≥, <, >, and=.



Essential Question: How do you compare and order rational numbers?

Materials:


Post-it notes with rational and perfect squares written on them.White boards, markers, and erasers

A common activity to compare rational numbers, percentages, and perfect squares is to play “Who Am I?” As the students enter the room, each student is given a label to be placed on their back. A post-it note is often used for this activity. The label will have a rational number, a percentage or a perfect square written on it. The students will circulate among each other asking questions of their classmates to find out “who” they are. They will assemble themselves in order from least to greatest. Ordering the students according to the numbers that were on their backs simulates locating rational and perfect squares on a number line.

Continue the lesson with a review of the symbols ≤, ≥, <, >, and = and their meanings. Make sure students are reading the inequalities from left to right. Have the students write each symbol at the top of their note page, along with how each is read. Ask each student to write at least one true statement using each symbol then share with their neighbor or group.

Ask for 2 student volunteers to come to the front of the room. Question the class which symbol should be used to compare the 2 rational numbers. (Be sure to question students which sign would be used if the students switched sides.)

Ask 2 other student volunteers to come to the front and show the class their numbers. Have the rest of the class use the whiteboards to write the symbol they would use to compare the two numbers. Discuss the correct answer with the class. Continue by alternating 1 and 2 until comfortable with your students understanding.

Have students number their paper from 1 to 10. Create problems for the students to complete. Write these on the overhead or promethean. (Students may also be asked to draw a number line during this lesson to again illustrate their understanding of 7-2.2).


If students do not understand “rational” they may incorrectly think repeating decimals cannot be rational. Students may not understand the inequality symbols. They often confuse the > and the < symbols. In using the ≤ or ≥ symbol they may not include the “…or equal to” causing the solution set to be incomplete and/or the inequality to be graphed incorrectly. Students may not understand comparison of 2 negative rational numbers. For ex., in comparing -2.5 and -7.5 students may incorrectly think -7.5 is larger than -2.5 because of the digits used. Other common errors include the student may not convert improper fractions before comparing them or not finding the square root before ordering it.

d. Additional Instructional Strategies


Give each student a 8 ½ by 11 sheet of paper or a large index card. Have them put any rational number on their card. Then have the class stand in a circle. Pass around a card that has a >, <, ≥, ≤ and an = symbol. (Students may flip the > symbol card when needed to become a < symbol.) When a student gets the symbol card, they must manipulate it to make a true statement by comparing the 2 rational numbers of the people on both sides. Everyone should have their cards facing in to the circle for the class to see. The class will give thumbs-up or a thumbs-down for agree or disagree. Pass the symbol cards around for everyone to get a turn discussing any misconceptions as you progress.

e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activites, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)




2. Which statement is true?a) √‾36 > 25b) 8 < √‾64c) √‾64 < √‾36d) √‾100 > √‾144

5. Teaching Lesson E


7-2.4 Understand the meaning of absolute value.



Essential Question: What is the relationship between absolute value and distance?

Ask students for several real-life examples of opposites. Discuss with students a real-life example of driving to work and then driving home. If you leave home to come to school and then go back home, did you travel zero miles?


What if you drove to work and then drove the car back home in reverse? Even if you backed all the way home, the odometer in the car still moves forward because distance is always positive. What if you walk two miles to the park? Does walking the two miles back mean you walked negative two miles making the distance you traveled equal zero? You cannot walk a negative two miles. You just walk two miles in the opposite direction.

Use a number line and a piece of string to demonstrate the meaning of

absolute value and stress that it is the distance from zero. Measure the string from 0 to positive 10. Place a knot in the string at each number interval. Measure the string from 0 to negative 10. Use an overhead sheet and the knotted string to demonstrate that 10 and -10 are the same distance from zero and a distance (or the piece of string) cannot be negative.

Use the following questions to engage students in discussion:

If four red chips represent a negative four, what would its opposite look like? What changed? (The direction to move from zero) Where would that be on the number line? How far are both numbers from zero? Do you see a pattern with regard to opposite integers and their distance from

zero?

Write the following on the board, “The absolute value of an integer is its relative distance from zero”. Show and explain the notation for absolute value. (Examples might include | 4 | = 4 and | -4 | = 4.) Ask what they notice about the absolute value of each? (They are both the same distance from zero.) Ask students to explain in writing what this means to them. Collect explanations to assess for understanding.

Divide students into groups of two to play Absolutely! from the Glencoe Mathematics: Applications and Connections, Course 3.

Materials:

index cards

* Copy onto cards the integers on the back of this card, one integer per card. Shuffle the cards and divide them equally. Each partner writes “absolute value” on two other cards and puts these cards aside.

* Each partner places the top card of his or her pile face up. The partner with the greater card takes both cards and puts them in a separate pile. When there are no cards left in your original pile, shuffle the cards in the second pile and use them.

* Twice during the game you can use an “absolute value” card after the other two cards have been played. When this card is played, partners compare the absolute value of the integers on the cards. When there is a tie, continue play, the first partner with a greater integer takes all the unclaimed cards. The winner is the partner who takes all the cards.



Students often have the misconception that distance can be a negative number when determining absolute value of negative numbers. Students need to be reminded that absolute value provides a distance and not a direction. Students may incorrectly think the absolute value of a number is its opposite. For ex., incorrectly thinking | 4 | = -4 and |-4 |= 4.


* “Using, Seeing, Feeling, and Doing Absolute Value for Deeper Understanding”, Gregorio A. Ponce, Mathematics Teaching in the Middle School, Vol.14, No. 4, November 2008, pages 234-240.* Enrichment: Play “Touchdown!”

e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activities, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)

Promethean Board – See Prometheanplanet.com lesson“Absolute Value” by Coach Brunsonhttp://www.prometheanplanet.com/server.php?show=ConResource.15005“Absolute Value” by Katrina Campbellhttp://www.prometheanplanet.com/server.php?show=ConResource.12234“Comparing and Ordering Integers” by Megan Colehttp://www.prometheanplanet.com/server.php?show=ConResource.11538http://www.yourteacher.com/prealgebra/absolutevalues.phphttp://www.brainpop.com/math/numbersandoperations/absolutevalue/preview.weml


http://www.brainpop.com/math/numbersandoperations/absolutevalue/preview.weml

http://www.brainpop.com/math/numbersandoperations/absolutevalue/preview.weml

http://www.yourteacher.com/prealgebra/absolutevalues.php






1. Exit Slip: Answer today’s essential question.2. Does a negative number always have a positive absolute value?3. Does a positive number always have a negative absolute value?4. Does absolute value just change the sign of the number inside the bars to its

opposite?5. If -10 is less than -1, is the absolute value of -10 also less than the absolute

value of -1?

III. Assessing the Module

At the end of this module summative assessment is necessary to determine student understanding of the connections among and between the indicators addressed in this module.


MODULE 1-2

Number Structure





7-2.6 Translate between standard form and exponential form. (B2)

7-2.7 Translate between standard form and scientific notation. (B2)


29


In sixth grade, students applied strategies and procedures to determine values of powers of 10 up to 106. In sixth grade, students represented whole numbers in exponential form.

2. Key Vocabulary

Factor Prime CompositeBaseExponentPowerStandard formScientific notationExponential formSquaredCubed

3. Content Overview

In seventh grade, students build on strategies to determine values of powers of 10 by translating between standard form to exponential form and to scientific notation. Seventh grade is the first time students transfer numbers between standard form and exponential form and scientific notation. Students need to understand that in scientific notation the first number should be greater than or equal to one and less than ten. Students need to work with a variety of numbers, both very large and very small, as well as decimal and whole numbers.




7-2.6 Translate between standard form and exponential form.




EQ:

1. Bell work may be to review students concerning factors, prime numbers and composite numbers. Remind students that factors are the numbers that are multiplied together to give a product. Prime numbers are numbers that have only two factors, one and itself, whereas composite numbers are numbers that have more than two factors. Tell students one way to determine if numbers are prime or composite are to create a tree shaped figure, called a factor tree.

2. Students could sing “Oh Factor Tree” to the tune of “Oh Christmas Tree.” A

powerpoint can be found at www.mrsoregan.net/MathPowerPoints2008/Ch4/ Oh _ Factor _ Tree .ppt

3. Show and explain how to factor the number 50 using the factor tree. 50 ^ 2•25 / ^ 2• 5•5

Explain to the students that the prime factorization of a number is the product of its prime factors (ex. 2•5•5). A shorter way to write the number is using a “base” and “exponents” (ex. 2• 5²). Tell the students that when you write the factorization of a number using exponents, such as 2•5², it is called “exponential notation”.

4. Have the students factor the number 30 using a factor tree. Give the

students ample time to complete this problem. Ask students to share with the class their solutions and discuss each answer stressing the use of the terms base and exponent. Answer any questions about writing a number in exponential notation.

5. Pair students and assign the following problems before moving on to the next activity.1) 152) 403) 604) 1005) 35Review the answers to the five problems and answer any questions.

6. Continue the class by giving examples of numbers written in exponential form in which the base is not a prime number, such as 102 = 100.

7. Explain to students that a base raised to the zero power is 1(with the exception of 00) and a base raised to the first power is the number itself. Give several examples, such as 71 = 7, 70 = 1, 251 = 25, 250 = 1, etc.


http://www.mrsoregan.net/MathPowerPoints2008/Ch4/Oh_Factor_Tree.ppt

As a culminating activity groups or the class could play a game of EXPO Bingo found in Mathematics Applications and Connections, Glencoe, Course 2, 2001. This game can be for 2 to 4 players or may be used with whole class.

*Each player copies the EXPO Bingo playing card onto their paper.*Copy at least 20 possible powers for students to use on paper. Cut apart and

place these inside an envelope. Each player selects 16 different powers from those listed and writes them in any of the upper right hand corner boxes on the EXPO card.

*Choose a person to be the bingo caller. The caller draws one expression from the envelope to read aloud. (To accommodate visual learners I would have the caller write the expression as they are read.) Each player marks the space by writing the equivalent number in the larger box.

*The first player with four powers listed in any row, column, or diagonal wins. E X P O


Students may multiply the base by the exponent. Ex. 23 = 2 x 3 = 6. Students may not understand the terminology, such as “squared” or “cubed”. Student may incorrectly think that a base with an exponent of zero equals zero. (such as 40 = 0). The student may incorrectly think every number raised to the zero power is zero forgetting that there is the exception of 00.


Read aloud the book G is for Googol; A Math Alphabet Book. David Schwartz, Tricycle Press, 1998, pgs. 12-13.

e. Technology


OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activites, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)Promethean Board – See Prometheanplanet.com lesson“Exponents” – Tami Scruggshttp://www.prometheanplanet.com/server.php?show=ConResource.11886“Exponents” by Sandie Fusillohttp://www.prometheanplanet.com/server.php?show=ConResource.11414“Exponents” by Mike Kinghttp://www.prometheanplanet.com/server. php?show=ConResource.169



1. The distance between Johnsonville and Macadenville is 53 miles. Written in standard form, what is the distance between the two towns?a.) 15 milesb.) 25 milesc.) 75 milesd.) 125 miles

2. Explain why 34 ● 4 is not written in exponential form.

2. Teaching Lesson B


7-2.7 Translate between standard form and scientific notation.



Essential Question: Why is scientific notation used to write numbers?CEEMM-

Materials:

25 m of string per group or 25 m of cash register paper per groupMasking tapeIndex cards – 10 per group (can use 5 per group and have students cut in half)Solar System Chart below – one per group


http://www.prometheanplanet.com/server.%20php?show=ConResource.169



To introduce the concept of translating very large numbers into standard form only the solar system chart is necessary. However, creating a scale model will help students begin to understand the relative magnitude of those very large numbers. In this lesson students will create a scale model of distances from the sun of various planets with the solar system. Give each group of students the materials listed above. Instruct the students to wrap a piece of masking tape around the string at one end and attach an index card labeled Sun. At the other end of attach an index card labeled Pluto. Use the solar system chart to discuss the relationship of the distance between the Sun and Pluto. Instruct student groups to use the scale distance information (they are familiar with comparing decimals) on the solar system chart to estimate the location for the other planets and then attach index cards with the planets’ names.

Example:

Allow student groups to compare their estimations. Call on a couple of groups to share the strategies they used to estimate the location of planets. Next, use Pluto’s information (5.9 09; 5.9 x 109; 5,900,000,000) to introduce the students to proper calculator notation and scientific notation. DO NOT go into a lot of detail, simply point the information out on the chart. Instruct student groups to discuss how they might use the given information to complete the chart. DO NOT complete the chart, simply discuss how the calculator and scientific notation can be used to determine actual distance.

Allow student groups to share their strategies. Instruct groups to complete the chart.


SunPluto095.9 x 109

5,900,000,000

34

SOLAR SYSTEM CHART

Planet Scale distance from the sun (meters) *

Distance to the planet in calculator notation

Distance to the planet in Scientific Notation

Actual distance to the planet -km (Standard form)

Mercury .25 5.8 07 5.8 x 107 58,000,000Venus .45 1.08 08 1.08 x 108

Earth .63 1.5 08 1.5 x 108

Mars .96 2.28 08 2.28 x 108

Jupiter 3.3 7.78 08 7.78 x 108

Saturn 6 1.43 09 1.43 x 109 1,430,000,000Uranus 12.1 2.87 09 2.87 x 109

Neptune 19 4.5 09 4.5 x 109

Pluto 25 5.9 09 5.9 x 109 5,900,000,000*Note to teacher: This scale distance was calculated by dividing the useable distance (25 meters) by the distance each planet is from the sun in astronomical units to get a scaling factor. Once the scaling factor of .63 was determined, it was multiplied by the actual distance a planet is from the sun in kilometers to get this scale distance.Ask, “What is the relationship between the calculator notation and the scientific notation? … the scientific notation and the standard form? …the standard form and the calculator notation?”


The student may not understand prime and composite numbers. The student may not understand the correct placement of a decimal in a whole number. The student may incorrectly think 7.69 x 105 simply means add 5 zeroes and not move the decimal. (as in 7.6900000) The student may add the zeroes and move the decimal to the end without regard to the original placement of the decimal. (as in 7.69 x 105 = 76,900,000)


G is for Googol; A Math Alphabet Book. David Schwartz, Tricycle Press, 1998, pgs. 16-17.

e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activites, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)Promethean Board – See Prometheanplanet.com lesson “Scientific Notation” – Megan Colehttp://www.prometheanplanet.com/server.php?show=ConResource.11553“Scientific Notation” – Heather Wisehttp://www.prometheanplanet.com/server.php?show=ConResource.17903http://www.brainpop.com/math/numbersandoperations/standardandscientificnotation/preview.wemlhttp://www.yourteacher.com/prealgebra/scientificnotation.php


http://www.yourteacher.com/prealgebra/scientificnotation.php

http://www.brainpop.com/math/numbersandoperations/standardandscientificnotation/preview.weml

http://www.brainpop.com/math/numbersandoperations/standardandscientificnotation/preview.weml



http://www.yourteacher.com/prealgebra/converttoscientificnotation.php



1. A number is written in standard form. To express it in scientific notation, ________________________________________________________.

2. A number is written in scientific notation. To express it in standard form, ________________________________________________________.

3. If writing a one digit number in scientific notation, what would be the exponent of the 10?

4. Does the value of a number change when it is written in scientific notation?

5. What are some real-world examples where people use scientific notation? III. Assessing the Module



http://www.yourteacher.com/prealgebra/converttoscientificnotation.php

MODULE 1-3Operations on Fractions/Decimals


I. Background for the Modules



7-2.9 Apply an algorithm to multiply and divide fractions and decimals. (C3)


37


In sixth grade, students generated strategies to build conceptual understanding of multiplying and dividing fractions and decimals.

2. Key Vocabulary

Product QuotientDivisorDividend

3. Content Overview

This is a statement of what is expected by this major concept. It may be derived from the Support Document. Seventh grade is the first time students are required to multiply and divide fractions and decimals symbolically (numerals only). As a result, students should be given opportunities to relate their prior concrete and pictorial experiences to the new symbolic operations. In addition to building on those previous experiences, students should estimate the products and quotients of problems involving fractions and decimals and use those estimations as the basis for explaining the reasonableness of results after actually solving. Furthermore, students should be given opportunities to apply multiplication and division of fractions and decimals in context – not merely perform the operations for the sake of multiplying or dividing. Teachers should be alert to the misconception that all multiplication results in “a bigger number”. Multiplication with fractions and decimals may result in a smaller product.





Cognitive Process Dimension: ApplyKnowledge Dimension: Procedural Knowledge


EQ: How does the process of multiplying fractions differ from adding fractions?

1. Have students fold a piece of paper in four sections horizontally, unfold it, and shade one fourth using a colored pencil. Then have them fold the sheet in half the other way and shade one half using a different colored pencil. Ask them to write the multiplication problem shown with the


shading. Be sure students understand the product is represented by the region shaded in both colors.

2. Tell students that today they will be talking about multiplying fractions. Ask students for real-world examples of multiplying fractions. A common example is in recipes. Show students a recipe for cookies. The following makes 4 dozen “Crocodiles” (Minneapolis Star Tribune, December 3, 2008).

• 2 c. flour• 1 tsp. baking soda• 1 1/2 tsp. ground cinnamon• 1/2 tsp. ground ginger• 1/2 tsp. salt• 1 c. (2 sticks) butter, at room temperature• 1 1/3 c. packed brown sugar• 1 egg• 1 tsp. vanilla extract• 2 c. chocolate chips• 1 c. chopped walnuts• Powdered sugar for rolling cookies

Explain that this recipe makes 4 dozen cookies. Because these are petite cookies most men eat 2 at a time! So they may need to double the recipe. Using models guide students to multiply these measurements by 2. Encourage students to try the multiplication in their head. Show how to multiply a fraction by a whole number using the algorithm. Have students write an explanation of multiplying fractions by a whole number in their notes. Students should share these with a partner or group. Have a volunteer read their explanation aloud to the class.

3. Lead the discussion into what happens when you multiply two fractions,

including several examples using mixed fractions. Show students several examples, stressing products must be expressed in simplest form. Have students write an explanation for multiplying 2 fractions in their notes. Students should share these with a partner or group. Have a volunteer read their explanation aloud to the class.

4. Assign students several problems to work with their group or partner. Circulate the room to help where necessary. Have them compare answers. If they agree then they continue. If they disagree, they need to compare their processes until satisfied with a common answer. (The Kagan strategy “Pairs Check” works great here.) Bring the class back together to discuss answers and misconceptions.


In a problem such as 4 ½ ● 3 1/4, the student multiplies the whole numbers then multiplies the fractions. Ex., A student mistakenly thinks 4 ½ * 3 ¼ means 4*3 + ½ * ¼ to find a product of 12 1/8. Teachers should be alert to the misconception that all multiplication results in “a bigger number”.



Teaching Student-Centered Mathematics Grades 5-8 , John A. Van de Walle and LouAnn H. Lovin, Pearson, 2006. See pages 96-98 for multiplication of fractions algorithm, pages 102 – 104 for division of fractions algorithm, and 126 -128 for multiplication and division of decimals algorithms.

Active Learning in the Mathematics Classroom, Grades 5-8 , Hope Martin, Corwin Press, 2007. See pages 22-23 for a real-world application of multiplying fractions.

Active Learning in the Mathematics Classroom, Grades 5-8, Hope Martin, Corwin Press, 2007. See pages 22-23 for a real-world application of multiplying fractions and decimals.

An enrichment activity: “Multiplication” by Virginia Muhammad. http://www.iit.edu/~smile/ma9708.html

Remediation: “Math Technique in Multiplication and Division of Fractions”, Frieda Garth. http://www.iit.edu/~smile/ma8805.html

e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activites, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)Stream Line Videos *Multiplying Fractions (01:05)*Example 2: Multiplying Fractions—Vacation Time (01:37)*Lesson 3: Multiplying Fractions (03:40)*Math Mastery: Fractions (30:00)*Lesson 4: More About Fractions (07:44)*Lesson 11: Working with Mixed Numbers (03:21) *The Zany World of Basic Math, Module 9: Multiplying and Dividing Fractions (09:26)Promethean Board – See Prometheanplanet.com

Multiplying Fractions – Shannon DeLaRosahttp://www.prometheanplanet.com/server.php?show=ConResource.13457Multiplying Fractions – Ruth Pitthttp://www.prometheanplanet.com/server.php?show=ConResource.10218Multiplying Fractions – Joyce Doddhttp://www.prometheanplanet.com/server.php?show=ConResource.13231Your Teacher Video http://www.yourteacher.com/prealgebra/multiplyfractions.phpBrain Pop http://brainpop.com/math/numbersandoperations/multiplyinganddividingfractions/Your Teacher Videohttp://www.yourteacher.com/prealgebra/multiplyingmixednumbers.php



http://www.yourteacher.com/prealgebra/multiplyingmixednumbers.php

http://brainpop.com/math/numbersandoperations/multiplyinganddividingfractions/

http://brainpop.com/math/numbersandoperations/multiplyinganddividingfractions/

http://www.yourteacher.com/prealgebra/multiplyfractions.php







1. When you multiply a whole number by a fraction does your answer get bigger? Give an example.

2. Does ½ ● 4 mean the same thing as 4 ● ½? Explain.

3. In multiplying fractions do you have to find a common denominator? 4. A possible exit slip may be for a student to write a word problem in which it is

necessary to multiply fractions to solve.

5. A box of Frosted Flakes cereal contains 15 servings of ¾ cup per serving. How many cups does the box contain?

6. If a bicycle tire has a 26 inch diameter, what is the circumference of the bicycle tire? Use 22/7 for ∏.






EQ: Why does it work to invert and multiply when dividing fractions? Teaching Student-Centered Mathematics Grades 5-8 , John A. Van de Walle and

LouAnn H. Lovin, Pearson, 2006. See pages 96-98 for multiplication of fractions algorithm, pages 102 – 104 for division of fractions algorithm, and 126 -128 for multiplication and division of decimals algorithms.


Teachers should be alert to the misconception that all division results in “a smaller number”.



While additional learning opportunities are needed no suggestions are included at this time.

e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activities, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)Stream Line Videos *The Zany World of Basic Math, Module 9: Multiplying And Dividing Fractions (09:26)*Lesson 10: Dividing Fractions (03:03)Promethean Board – See Prometheanplanet.comMultiplying Fractions – Shannon DeLaRosahttp://www.prometheanplanet.com/server.php?show=ConResource.13457http://www.brainpop.com/math/numbersandoperations/multiplyinganddividingfractions/preview.weml



1. What is the reciprocal of 2/3?

2. What is the product of two reciprocals?3. A recipe calls for 1 2/3 cup of milk. Terri wants to make half of the recipe.

How much milk does she need?

4. In a long-distance bike race, a rider is 4 ½ miles from the finish line. If he continues riding at a constant rate of ½ mile per minute, in how many minutes will the rider finish?

5. Richard can build a cabinet in 3 ¼ hours. How many completed cabinets can he build in 40 hours?

III. Teaching Lesson C





http://www.brainpop.com/math/numbersandoperations/multiplyinganddividingfractions/preview.weml

http://www.brainpop.com/math/numbersandoperations/multiplyinganddividingfractions/preview.weml



EQ: How do we know where to place the decimal in the product when multiplying decimals?

Begin with questioning students simply multiplication problems such as 10 x 1 = ?, 14 x 1 = ?, 1/3 x 1 = ?, etc. What happens when a number is multiplied by one? Can someone tell me what a decimal is? If I multiply a decimal by one, such as 0.5 x 1, will the result be larger or smaller than 1? OR If I multiply 4.375 x 1, will the result be larger or smaller than 1? Why?

Let the students know that today they will be talking about multiplying decimals. Ask students for real-world examples of multiplying decimals. The most obvious is money. Show students a sale flyer from a paper. Find an item students would enjoy and ask how many of the students wanted one. Use the price to demonstrate how much it would cost (without tax) to purchase the item for one, two, or ten students. Ask if they see a pattern in the multiplication. Explain to multiply decimals just as you would whole numbers. Then in the product, beginning at the right, count off as many decimal places as there are in the multiplier and the multiplicand together. Place the decimal point that many places in

Lead the discussion into what happens when you multiply two decimals. Show students several examples of multiplying decimals. Have students turn their notebook paper sideways to write their multiplication and division problems with each digit of the numbers in its own column. Stress counting the decimal places in both numbers before placing the decimal in the product. Have students write an explanation for multiplying 2 decimals in their notes, share these with a partner or group, then have several read aloud their writing.

Assign students several problems to work with their group or partner. Circulate the room to help where necessary. Have them compare answers. If they agree then they continue. If they disagree, they need to compare their processes until satisfied with a common answer. (The Kagan strategy “Pairs Check” works great here.) Bring the class back together to discuss answers and misconceptions.

If time, give students cards with the digits 1, 6, 8, and 9. Again in pairs, have the students draw the 2 diagrams below on a sheet of paper.

. x .

Which digits should go where to make the indicated products the largest possible? Moving the cards allows for visual reordering. As a variation, you could have students try to find the smallest possible products.



A student may divide decimal by a decimal without removing the decimal point in the divisor. Or they may move the decimal in the divisor but incorrectly move the decimal in the dividend. Teachers should be alert to the misconception that all multiplication results in “a bigger number”.



Active Learning in the Mathematics Classroom, Grades 5-8 , Hope Martin, Corwin Press, 2007. See pages 22-23 for a real-world application of multiplying fractions.

Active Learning in the Mathematics Classroom, Grades 5-8 , Hope Martin, Corwin Press, 2007. See pages 22-23 for a real-world application of

multiplying fractions and decimals. A multiplying decimal lesson is found at

http://www.shodor.org/interactive/lessons/MultiplyingDecimals

e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activities, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)Stream Line Videos *Math Mansion: 30. Another Sad Hair Day (09:18)*Math Mansion: The Return of the Big Red Hen (09:26)*The Zany World of Basic Math, Module 6: Working with Decimals (15:15)*Math Mastery: Decimals and Percents (30:00)*Lesson 5: Multiplying Decimals (03:38)Prometheanplanet.com“Multiplying Decimals by a Power of Ten”, Holli Cooper, http://www.prometheanplanet.com/server.php?show=ConResource.10294“Multiplication by 2-digit Multipliers”, Holli Cooper, http://www.prometheanplanet.com/server.php?show=ConResource.10673“Multiplying Decimals by Whole Numbers”, Jaclyn Lavigne, http://www.prometheanplanet.com/server.php?show=ConResource.12513“Multiplying and Dividing Decimals”, Crissie Fowler, http://www.prometheanplanet.com/server.php?show=ConResource.15745








http://www.shodor.org/interactive/lessons/MultiplyingDecimals

1. It takes 23 minutes per pound to cook a turkey. Mrs. Hinson bought a 9.8 pound turkey. How many minutes will it take to cook?a. 205.4 minutesb. 225.4 minutesc. 2054minutesd. 2254 minutes

2. At the Bay City Restaurant Jonnie and her 3 friends each orders Shrimp Caesar salad. If each salad costs $9.95, what is the total bill for the meals, excluding tax?

3. Jose’s family is going to tile their home’s entryway. If each tile costs $2.45, what will be the total cost of 45 tiles, excluding tax?

4. Holiday Cruise Company charges $0.55 per minute for calls made from their cruise ship. What is the cost of a 16 minute call?

5. Mike built a rectangular shaped counter top. The dimensions of the top are

6.625 ft by 2.25 ft. Use the formula P = 2 l + 2 w to find the perimeter of the top. Show your work.a. 8.875 ftb. 9.75 ftc. 10.75 ftd. 17.75 ft

IV. Teaching Lesson D





EQ: Is it possible to divide by a decimal?

Dividing decimals is the most challenging operation of this indicator. Because most problems don’t come out “perfect” and require thought to decide what should be done next, many students have trouble with division. Explain to students that even though we have calculators that will do the division for us, it does not mean we shouldn't learn how to divide decimals! As with any calculations number sense must be used to check answers to see if they are reasonable.

The skills needed to divide decimals are almost the same as dividing whole numbers. The difference is the positioning of the decimal point in the dividend to determine the decimal places in the quotient. Begin modeling by using examples that divide a decimal by a whole number. (such as 7.65 ÷ 5)


Refer to the essential question. Explain that division by a decimal is impossible. You must first change the divisor to a whole number by moving the decimal to the right. Explain this is the same as multiplying the divisor by a multiple of ten. Tell students if they move the decimal in the divisor you must also move it the same number of places in the dividend. Explain this is the same as multiplying the dividend by a multiple of ten. Then place the decimal in the quotient directly above the decimal in the dividend. After several examples ask students to write the steps to dividing decimals in their notebook.

To divide decimals

a) Is the divisor a whole number? If no, move the decimal point in the divisor to the right to make the divisor a whole number. Move the decimal point in the dividend the same number of places as you moved the decimal in the divisor.

b) Divide as if both numbers were whole numbers. If the divisor doesn't go into the dividend evenly, add zeroes to the right of the last digit in the dividend and keep dividing until it comes out evenly or starts repeating.

c) Place the decimal point in the quotient straight above the decimal point in the dividend. [Use a highlighter to show this.]

d) Check your answer by multiplying the quotient by the divisor. Does it equal the dividend? (Using a calculator here is optional.)

Additional problems should be assigned for practice.


The student divides decimal by a decimal and does not move the decimal point in the divisor or the dividend. Stress to students to check to make sure the decimal point in the quotient is directly above the decimal point in the dividend.



*Active Learning in the Mathematics Classroom, Grades 5-8, Hope Martin, Corwin Press, 2007. See pages 22-23 for a real-world application of multiplying fractions.

Active Learning in the Mathematics Classroom, Grades 5-8 , Hope Martin, Corwin Press, 2007. See pages 22-23 for a real-world application of multiplying fractions and decimals.

e. Technology

OneplaceSC.orgIlluminations by NCTM.org


Interactive at shodor.org (searches for lessons/activities, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)Stream Line Videos *Math Mansion: The Return of the Big Red Hen (09:26)*The Zany World of Basic Math, Module 6: Working with Decimals (15:15)vi. Promethean Board – See Prometheanplanet.com“Dividing Decimals by a Power of Ten”, Holli Cooper, http://www.prometheanplanet.com/server.php?show=ConResource.10323“Decimal Division”, n/a, http://www.prometheanplanet.com/server.php?show=ConResource.525vii. A tutorial on dividing decimals: http://www.math.com/school/subject1/lessons/S1U1L6GL.html



1. A gallon of gasoline costs $3.98 per gallon. If the total cost to fill a car is $47.76, how many gallons were purchased?

2. Rakeem wants to buy a gift for his grandfather that costs $68.25. If he wants to save the same amount each week, how much money must Rakeem save each week? Give your answer in dollars and cents.

3. At the lumber yard, Harold makes $9.75 an hour. Last week he worked 38.5 hours. What was his gross salary?

4. EXIT TICKET: Write an explanation to an absent student explaining how to divide decimals.



From Prometheanplanet.com, an ACTIVote quiz covering multiplication, division, rounding, estimating, and comparing decimals can be found at http://www.prometheanplanet.com/server.php?show=ConResource.11671







Recommended Days of Instruction Second Nine WeeksSecond Nine Weeks

Standard/Indicators Addressed:Standard: 7-2: The student will demonstrate through the mathematical processes an understanding of the

representation of rational numbers, percentages, and square roots of perfect squares; the application of ratios, rates, and proportions to solve problems; accurate, efficient, and generalizable methods for operations with integers; the multiplication and division of fractions and decimals; and the inverse relationship between squaring and finding the square roots of perfect squares.

7-2.8 Generate strategies to add, subtract, multiply, and divide integers

Standard: 7-3: The student will demonstrate through the mathematical processes an understanding of proportional relationships.

7-3.1 Analyze geometric patterns and pattern relationships. 7-3.2 Analyze tables and graphs to describe the rate of change between and among quantities.7-3.3 Understand slope as a constant rate of change.7-3.4 Use inverse operations to solve two-step equations and two-step inequalities. 7-3.5 Represent on a number line the solution of a two-step inequality.7-3.6 Represent proportional relationships with graphs, tables, and equations.7-3.7 Classify relationships as either directly proportional, inversely proportional, or nonproportional.



Module 2-1 IntegersIndicator Recommended Resources Suggested Instructional



7-2.8 Generate strategies to add, subtract, multiply, and divide integers.






Textbook Correlations –See Appendix A

See Instructional Planning Guide Module 2-1 Introductory Lesson A See Instructional Planning Guide Module 2-1, Lesson A Additional In-structional Strategies




See Instructional Planning Guide Module 2-1, Introductory Lesson B Appendix Two

See Instructional Planning Guide Module 2-1, Lesson B Additional Instructional Strategies Appendix Two


Module 2-1 Lesson C:


See Instructional Planning Guide Module 2-1 Introductory Lesson C Appendix Two

See Instructional Planning Guide Module 2-1, Lesson C Additional In-structional Strategies Appendix Two


Module 2-1 Lesson D:


See Instructional Planning Guide Module 2-1, Introductory Lesson D Appendix Two

See Instructional Planning Guide Module 2-1, Lesson D Additional Instructional Strategies Appendix Two



Module 2-2 Solve Mathematical SituationsIndicator Recommended Resources Suggested Instructional



7-3.4 Use inverse operations to solve two-step equations and two-step inequalities.

NCTM's Online Illuminations http://illuminations.nctm.org/







See Instructional Planning Guide Module 2-2, Lesson A Additional In-structional Strategies







Module 2-2 Lesson C:

7-3.5 Represent on a number line the solution of a two-step inequality.

See Instructional Planning Guide Module 2-2 Introductory Lesson C

See Instructional Planning Guide Module 2-2, Lesson C Additional In-structional Strategies


Module 2-3 Equivalencies





7-3.1 Analyze geometric patterns and pattern relationships.








See Instructional Planning Guide Module 2-3, Lesson A Additional In-structional Strategies



7-3.6 Represent proportional relationships with graphs, tables, and equations.

7-3.7 Classify relationships as either directly proportional, inversely proportional, or nonproportional.

See Instructional Planning Guide Module 2-3 Introductory Lesson B

See Instructional Planning Guide Module 2-3, Lesson B Additional In-structional Strategies


Module 2-4 Change in Various Contexts


https://spike.scsu.edu/exchweb/bin/redir.asp?URL=http://illuminations.nctm.org/




7-3.2 Analyze tables and graphs to describe the rate of change between and among quantities.

7-3.3 Understand slope as a constant rate of change.

.







See Instructional Planning Guide Module 2-4 Introductory Lesson A See Instructional Planning Guide Module 2-4, Lesson A Additional In-structional Strategies




MODULE 2-1Integers


I. Background for the Modules



7-2.8 Generate strategies to add, subtract, multiply, and divide integers. (C3)



In sixth grade, students developed a conceptual understanding of an integer. Spending the time to fully explore strategies to perform integer operations will pay off in future mathematics courses for students. Students should work with concrete models and pictorial representations to build the foundation needed for eighth grade when abstract/symbolic integer operations are performed. Students should be allowed to generate algorithms for addition, subtraction, multiplication and division before introduction to traditional algorithms in eighth grade.

2. Key Vocabulary

IntegerZero pair

3. Content Overview

Seventh grade students generate strategies to add, subtract, multiply and divide integers. Students should not be expected to perform symbolic operations with integers. To support and promote conceptual understanding of operations with integers, manipulatives should be used.






b. Introductory Lesson – Invitation to Integers (Grade 5-7)Master Teacher: Deb Childs South Carolina ETV


Materials:

overhead counters of 2 different colors (If available, the counters should be the same two colors as the student counters. If not write + and - signs on the overhead counters). Timer or clock with second hand (optional)marker (1 per student) 2-color counters (13 counters per student) sentence strip or a copy of the number line (1 per student) 3x5 or 4x6 note cards (at least 5 cards per group)

Learning Activities

Step 1: Give out counters. Agree on which color will represent positive

and which will represent negative. (Note to Teacher: Usually 2-color counters have a red side that is often modeled as negative.) Model each problem in the video clip on the overhead as students manipulate their counters. Place 3 negative counters on the desk or overhead. Place 5 positive counters on the surface. Remove zero pairs (1 positive and 1negative).

You will be left with 2 positive counters as the answer. Repeat with -9 + 4 = -5 and -3 +3 = 0. Ask students "When would you see -3 + 3 = 0, outside of a math classroom or textbook?" (Borrow and pay back $3."Mother may I" taking 3 steps forward, 3 steps backward. Temperature dropping 3 degrees below zero and then rising 3 degrees. Football team losing 3 yards and then gaining it back. A dieter losing 3 pounds and gaining it back. Diving 3 feet below sea level and then coming back up.)

Ask students to show the following problems and solutions with their counters. Solve the equations and discuss the answers one at a time. -8 + 5 = -3. 5 + (-3) = +2. 8 +(-2) = +6. -6 + 6 = 0. Write each problem down after the student has modeled it. You may wish to use raised signs or parentheses on the "plus a negative" problems as was demonstrated in the previous four problems. Remind students when there is no sign in front of a number, it is assumed to be positive.

Step 2: FAST FORWARD the video until you hear "She's a self-assured

little tyke" and see the cartoon hostess and parrot disappear. Then you will see and hear "Pauline's Perilous Pyramid." Provide a Focus for Media Interaction by asking students to look for another way to add integers. PAUSE when you hear "I'll get you next time" and see a box that says UNTIL NEXT TIME. Ask how this clip demonstrated adding positive and negative integers in a different way from the counters used previously. (Number line).

Step 3: Either give students the number lines you have made, or make number lines from -25 to 25 using sentence strips. Label by 5's


but put hash marks for each integer between the multiples of 5. (See Activity Sheet 1). Ask them if there is any way you can get to the top without using the zapper. Remember you must stay in the range between -25 and +25. REWIND the video until you see the game board for Pauline's Perilous Pyramid. FREEZE FRAME at that spot. Guide the students through one example. Start with 5 and jump to -15. Then jump somewhere besides -10.

After you finish that one, let the students work individually or in groups to come up with other paths Pauline could take to win the game. You may wish to add the following limits to the rules. You may go left to right but not back "down." You may move to the left or right only one space per (left to right) row. Allow students about 5-10 minutes to complete this activity. Let groups or individuals describe other possible solutions. (Note to Teacher: See Activity Sheets 2 and 3 for some possible solutions.)

Step 4: FAST FORWARD through the cartoon hostess and parrot until you see a black and white maze that says "MATH MAN." You will hear "Math Man, no he's as sick as a pig." Provide a Focus for Media Interaction by asking students to listen for why 0 does not work in the game. STOP when you see GAME OVER, and hear Math Man. Listen for the students' responses to why zero does not work? (7 + 0 = 7, seven is not less than seven.) If your students are familiar with the symbol for less than or equal to, ask them what symbol could have replaced the < to make the statement true. If they are not familiar with this symbol, show them the symbol for less than or equal to < and greater than or equal to >. Ask the students what the "p" represented. (Some number or the unknown). Show the other numbers listed on the clip (5 No. -3 Yes.), and let the students decide which ones would make the inequality true. Ask the students for other integers that would make the inequality true? (Any negative number)

Culminating Activity

Step 1: Go the Quia Web site (http://www.quia.com/custom/66762). If you have an AverKey, you can guide the students through the matching game the first time. Provide a Focus for Media Interaction by asking students to write down any equations they miss as they play the game. Now, have the students go the Web site that has been bookmarked. Allow the students to play a few rounds of the matching game. Ask about problems they missed. Work those on the overhead. (Note to Teacher: All equations and solutions used on the Internet activity are included on Activity Sheet 4).


Step 2: Arrange the students in groups of two. Ask them to make up 5 integer problems that can be solved using the number line from +25 to -25. Each group will write the problems on one side and the answers on the other side, like a flash card. Ask groups to exchange cards with another group and check their flash cards. Put flash cards that both groups agree to be correct in one pile. Make another pile of flash cards with possible errors. (Note to Teacher: Groups that finish early can copy some of the equations from Activity Sheet 4 to include in the upcoming game.) Check all cards and destroy any with errors.

Step 3: Play "Around the World" with the flash cards. Ask the student in seat number 1 to stand beside the student in seat number 2. Show a flash card. The first one to answer correctly gets to move to seat number 3 and challenge that student. The student who did not move sits in seat number 2. If there is a tie, the two students get another card, until the tie is broken. If they both miss, they get another card, until a student answers correctly and moves to the next seat. Each student gets one guess at a card. If the student misses, he/she must remain silent while the other student has 20 seconds to answer the question correctly. The goal is to be the first student to go completely around the room and return to his/her seat. (Note to Teacher: If your desks are not numbered, designate a pattern around the room before play begins).

Step 4: Divide the flash cards into three piles: a negative plus a negative, a positive plus a positive and a third pile in which one addend is negative and the other is positive. Ask students to hold up those cards at the front of the room. Ask the students to notice any patterns they observe. They will probably respond that a negative plus a negative is always negative, a positive plus a positive is always positive. They may need some guidance to see that when the signs are different, the answer will match the sign of the number that is farthest from zero. You may wish to write on the

-5 + 3 = -2 5 + -3 = +2

board or overhead. Negative 5 is farther from zero in the first problem so the answer is negative. Positive 5 is farther from zero in the second problem so the answer is positive. At this point they will probably notice they can find the number that is the answer by subtracting instead of walking the number line or finding zero pairs. Ask them to explain why the answer is sometimes positive and sometimes negative in these cases. They may talk about going past the zero on the number line or they may notice the answer is always the same as the addend with the larger absolute value (farthest from zero in either direction).


Step 5: In class or as homework, ask students to write a paragraph to a friend who was absent from school that day (a real student or a pretend one if all students are present). In that note they will explain how to add positive and negative numbers. Ask them to include examples of simple problems in their paragraph that illustrate adding two positive numbers, two negative numbers, and a problem where one number is positive and the other is negative. They may use the rules just discussed, the number line, or the 2-color counters demonstrated earlier, to explain the process to the absent classmate. They may wish to write about all the methods.


Students may not understand zero pairs. Students may misinterpret direction on a number line.


*Throughout the modules stress to students there are multiply ways to represent a number using chips. For example, 6 can be represented with 6 yellow chips, 10 yellow and 4 red chips, etc. Remind students that one red and one yellow chip equal a zero pair.

*Play “I Have…Who Has”. You will need to make enough cards so each of your students has at least one card. If you have less students than cards, give the stronger students two cards. If you have more students than cards, pair a weaker student with a stronger one. Focus on situations using integers such as 100 feet above sea level, debt of $20 , rise of 5 degrees in temperature, etc.

CEEMM

Adapted from: Activities from Kennedy, Tipps, Steve, (2000). Children’s Learning of Mathematics Ninth Edition, Belmont, CA: Wadsworth/Thompson Learning, page 238.

Materials:

ThermometerResearch materials / Internet access

1. Present a problem situation: “The lowest temperature ever recorded in Florida was minus 2 degrees. The lowest temperature in Georgia was minus 17 degrees. Which temperature was the lower, or colder, of the two?

2. Display a thermometer. Turn it horizontally to show how it is a number line with positive and negative numbers.

3. On the horizontal thermometer, ask “What is happening if you move toward the left on the thermometer?” “What is happening if you move to


the right on the thermometer?” “Where would our temperature today fall on this number line?”

4. Have students research the highest and lowest temperatures and discuss where they would fall on a number line.

5. Give the students an integer and ask for a real world example of that integer. For instance +23 would be the increases of the price of gas in the last week, or -20 the drop in the temperature since yesterday.

6. Give the students real world situations and have them represent the situation with an integer. For instance, last night our quarterback was sacked and we lost 13 yards, or, because of all of the rain the water level has risen 8 inches in the last month.GLENCOE, 7th grade, 2001

Materials:

Two color Counters – about 15 per studentInteger Cards (made with index cards) numbered from –15 to +15

Introduce the modeling of integers with the use of two color counters. The red side of the counter could represent a negative number and the yellow a positive number. Explain that to represent a positive 2, two chips would be turned over to the yellow side. A negative 2 would be represented with two red chips, relating back to sixth grade where a company’s loss was seen as “being in the red”. Go through several examples having students represent different numbers until it is certain that all students understand this concept. As the class is modeling various numbers, locate each number modeled on the number line in the front of the room. Note the position of the number as it relates to zero, that is, is it to the right or the left of zero?

Give each pair two of the integer cards. Students use the counters to model their integer and then locate their integers on a number line. Students should take turns placing their team’s integers at the appropriate place on the number line in front of the room and describing how they determined each integer’s location. As integers are placed on the number line, elicit strategies for determining the order of two or more integers. Follow this discussion up with questions that include: “How do negative numbers compare to a positive number? If you are comparing two positive numbers which one is always bigger? When comparing two negative numbers which one is bigger? Is there a way to tell which integer is bigger in any situation?”

Ask students to select numbers that were not used on the index cards but that fit following criteria: One negative, one positive – not equal Two negatives – not equal Two positives – not equal

Glencoe Mathematics Applications and Connections, Course 3


Number of players: 4Materials: Spinners "Sum"-times

Label the sections of one spinner with the following integers: -3, 12, 7, -9, 2, -8. Label the sections of a second spinner with the following integers: -5, -10, -1, 15, 6, 4. Label the sections of a third spinner with the following integers: 13, -16, 5, -2, 18, -11. Label the sections of a fourth spinner with the following integers: -14, 20, -17, 19, -7, 8.

All four group members spin a spinner at the same time. Try to be the first to find the sum of the four resulting integers.Play several rounds.Glencoe Mathematics Applications and Connections, Course 3Number of players: 2Materials: Counters, spinners It Evens Out

Label the sections of one spinner with the integers 1 through 6. Label the sections of a second spinner with the integers -1 through -6. Decide which color of counters will represent positive integers and which color will represent negative integers. One partner takes twenty counters of one color, and the other partner takes twenty counters of the other color.

The partner with the counters that represent positive integers spins the positive integer spinner; the partner with the counters that represent negative integers spins the negative integer spinner. Both partners then place between them the number of counters indicated on their spinners. Zero pairs are removed from play. Leftover counters are returned to the partner with that color. Before the next round, partners write an equation that describes their actions and solve for the variable. Play continues until one partner runs out of counters.

Glencoe Mathematics

Number of players: 2Materials:

Index cards

Copy onto cards the integers shown below, one integer per card. Shuffle the cards and place them face down in a pile. Decide which partner will go first.

The first partner selects two cards from the pile and places them face up, with the greater integer to the right. The second partner then selects two cards and arranges all four cards in order from least to greatest. Each partner takes the two cards he or she selected unless two of one partner's cards surround one or both of the other partner's cards. If you can place the card to the left and to the right of one or both of your partner's cards, you "capture"


the card or cards. Continue in the same way, taking turns drawing the first two cards of each round, until no card remains in the pile. The partner with more cards wins.Numbers to put on cards: -14

1-7

-11

-18

-13

1-6

-12

-19

-12

1-5

-13

-10

-11

1-4

-14

-11

-10

1-3

-15

-12

-9

-2

-6

13

-8

-1

-7

14

Additional Resources

Teaching Student-Centered Mathematics Grades 5-8 , John A. Van de Walle and LouAnn H. Lovin, Pearson, 2006. See pages 138 – 146 for discussion and models.

Students may make a 5-tab foldable as integer operations are introduced.

e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activites, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)Stream Line Videos :Zany World of Basic Math, The, Module 2: Subtracting IntegersPromethean Board – See Prometheanplanet.com lesson“Integer Addition and Subtraction” - Anna Mikahttp://www.prometheanplanet.com/server.php?show=ConResource.7820“Adding and Subtracting Integers” – Laurie Morrishttp://www.prometheanplanet.com/server.php?show=ConResource.11073“Adding Integers” – Hanoko Owenhttp://www.prometheanplanet.com/server.php?show=ConResource.14008“Modeling Integer Addition” – Elizabeth Herringhttp://www.prometheanplanet.com/server.php?show=ConResource.15744“Adding and Subtracting Integers” – Kim Benziehttp://www.prometheanplanet.com/server.php?show=ConResource.13498“An Introduction to Adding Integers” – Shannon Browninghttp://www.prometheanplanet.com/server.php?show=ConResource.11073Quia, pronounced Key-uh and short for Quintessential Instructional Archive, provides a variety of educational services. This page on the site,













Quia.com, looks at "Adding and Subtracting Positive and Negative Numbers." Click on the game that says "matching." There is an index to the terms. Students can play several rounds of an adding positive and negative numbers game.Brain Pop lesson http://www.brainpop.com/math/numbersandoperations/addingandsubtractingintegers/preview.weml



1. Use the number line below to show a method to solve -6 + 9 = x.


http://www.brainpop.com/math/numbersandoperations/addingandsubtractingintegers/preview.weml


http://www.quia.com/custom/66762

2. The model below demonstrates which equation?

a. – 2 – 7 = 5b. – 2 – 7 = 9c. – 2 + 7 = 5d. – 2 + 7 = 9

3. Have students work with a partner to write 4 addition problems that have a sum of 7. Have students use tiles to show the solution to the first two problems and draw a number line to show the sum of the third and fourth problem. Each pair should choose one problem and its solution to share with the class.

4. What are some words that represent a positive integer?

5. If there are more red chips than yellow chips when adding two integers, will the answer be a positive or a negative?

6. If there are 10 reds and 6 yellows, how many zero pairs can be created?

7. Can your answer be in two colors?

8. If your answer is in two colors, what should be done to complete the problem?

9. Complete the first section of the 5 tab foldable described at the end of the module.







+ ++

+ ++

+

-_

-_

+ ++

+ ++

+

-_

-_

63

Essential Question: Why is subtraction considered the opposite operation of addition?

Adapted from: ROVER, Richland One Virtual Educational Resources, “Zero”-ing in on Integers, designed by David K. Blackwell, Gibbes Middle School.

Begin the class with an overview of the lesson which includes a statement of the objective.

Ask the focus question and allow a few minutes for reflection and discussion. Begin the lesson on zero pairs by modeling the concept with an overhead mat and overhead red and yellow counters.

Write the problem 6 – (-6) = ? on the overhead or board. Ask students how you can subtract 6 negative counters from 6 positive counters? To model subtraction, you need to start by representing the first number, 6, then take away from that. Remember the phrase “create the possibility with zero pairs.” Remind students they cannot just add negative counters for no reason. You must keep the same value that you started with, and the way to do that is add zero pairs. So you must add 6 sets of zero pairs. Pair the positive and negative integers together. Remove as many pairs as you can.

Ask questions, such as: * How many counters are left on the overhead? * What is the difference +6 and -6? * How are +6 and -6 related? * What is the difference of any pair of opposites? * What is the value of zero pair? * What happens to the value of the counters when you remove a zero

pair from the mat?

Monitor the questions and clarify any misunderstandings.

Pair the students by counting 1 - 2 or by letting them turn to a person next to them. Make sure everyone has a partner.

Pass out Zip-lock bags that contain a laminated mat, red and yellow counters and two index cards with a negative sign on one and a positive sign on the other. Tell the students to take out the contents of their bags. Tell them they are going to use the index cards to answer some questions.

Ask the students, "What type of integer is represented by the yellow counters? The red counters?" This is a quick way to make sure all students understand which color represents what.

Pass out ATTACHMENT 1. Tell the students to use their counters to solve the problems. Tell them that this is a practice session. Monitor the room while the students are completing the activity. One student should record the answers to the problems.


Have the pairs share the answers to each problem. Clarify any misunderstandings at this time. Hand out an ATTACHMENT 2 to each student and let them complete the problems on their own. Provide additional bags of counters and mats at this time. (See Attachments 3, 4, & 5 at the end of the module)

After students complete this activity, have students trade papers with their partners. Let the partners correct the papers as you review the answers. Have volunteers give the answers.

SUBTRACTION OF INTEGERSATTACHMENT 1 (Cooperative Activity)

Use the mat and counters to solve the following problems involving addition of integers.

1. +4 AND -2 Answer: ____________

2. -6 and +3 Answer: ____________

3. -7 and +5 Answer: ____________

4. -2 and +2 Answer: ____________

5. -8 and +4 Answer: ____________

6. +5 and -10 Answer: ____________

7. +9 and -6 Answer: ____________

8. -8 and +8 Answer: ____________

9. +3 and -3 Answer: ____________

10. -6 and -4 Answer: ____________

11. -1 and +4 Answer: ____________

12. +1 and -1 Answer: ____________

13. +4 and +5 Answer: ____________

14. -3 and -2 Answer: ____________

15. -8 and -6 Answer: ____________

Be prepared to explain your answers during the class sharing. You can write down anything that you see occurring when you complete each task to help you in the class discussion.

SUBTRACTION OF INTEGERS ATTACHMENT 2 (Individual Activity)

Use the mat and counters to solve the following problems involving addition of integers.

1. +9 and -2 Answer:____________

2. +6 and -1 Answer:____________

3. +5 and -3 Answer:____________


4. +4 and -1 Answer:____________

5. -8 and -4 Answer:____________

6. +2 and +2 Answer:____________

7. -3 and -1 Answer:____________

8. +6 and -7 Answer:____________

9. +9 and -7 Answer:____________

10. -5 and -3 Answer:____________

11. +5 and -5 Answer:____________

12. -7 and +4 Answer:____________

13. +5 and -6 Answer:____________

14. +3 and +8 Answer:____________

15. -6 and -2 Answer:____________

Be prepared to explain your answers during the class sharing. You can write down anything that you see occurring when you complete each task to help you in the class discussion.


Students may simply remove the 2nd number (subtrahend) from the mat without regard to the sign.

Students may confuse a positive number following a minus sign with a negative number.

Students may not understand or be able to visualize some of the terminology, such as “ground level”, “below zero”, “par” in golf, “debt”, “credit”, “net gain”, “net loss”, etc. Be aware that some students may not have heard of the Stock Market.




Glencoe Mathematics: Applications and Concepts, Course 2

Use groups of 2.Materials: Playing cards 1-10 Subtraction War

In this game, black cards 1 through 10 are positive integers, and red cards 1 through 10 are negative integers. Shuffle and deal all of the cards.


Each player turns over his or her top two cards, laying the first card down and then the second card to the right of the first card. Each player then finds the difference between the two cards, first card minus the second card. The player with the greater total gets all of the cards. Play continues until all cards are used. The winner is the one with the most cards at the end of the round. Play several rounds.

e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activities, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)Promethean Board – See Prometheanplanet.com“Adding and Subtracting Integers” – Kim Benziehttp://www.prometheanplanet.com/server.php?show=ConResource.13498 “Integer Addition and Subtraction” - Anna Mikahttp://www.prometheanplanet.com/server.php?show=ConResource.7820“Adding and Subtracting Integers” – Laurie Morrishttp://www.prometheanplanet.com/server.php?show=ConResource.11073“Subtracting Integers” – Nicole Beckstead http://www.prometheanplanet.com/server.php?show=ConResource.11447Brain Pop lessonhttp://www.brainpop.com/math/numbersandoperations/addingandsubtractingintegers/preview.wemlRemediation: “Positive and Negative Charge for Integers”, Barbara Williams.http://www.iit.edu/~smile/ma8718.html



1. Use models to show 5 – 8 =______.












2. Which one of the following number sentences does the model represent?

● represents -1

●●●●● ●●● ●●●●● ооо

a. -10 - + 3 = -13b. -10 - + 3 = +7c. +3 – (-10) = -7d. -3 – (-10) = 13

3. Have students choose a positive integer less than 5. They then subtract 3, -4,-3, and -2 from the number they chose. After the class has finished, have them find another student who began with the same integer. The group members then compare answers.

4. Why is the -7 in parentheses in 19 – (-7)?

5. Complete the subtraction tab of the 5 tab foldable described at the end of this module.






Essential Question: How does multiplication of integers compare to addition and subtraction of integers?

Explain to your students that multiplication is different, because you start with nothing. The beginning value is zero, and from there you build groups. Ask a student what 4 • 1 means: four groups with one item in each group. Focus on the language: gaining or losing, positives or negatives.

When you begin to use positive and negative signs, you have to add more language. + 2 • + 3 means you are gaining two groups of positive 3. Since the sign of the first factor is positive, they will be gaining groups. Gaining positives gives you a positive answer.

+ 2 • -3 means you are gaining two groups of negative 3. Gaining negatives gives you a negative answer. −2 • + 3 means you are losing two groups of positive 3. Since the sign of the first factor is negative,


they will be losing groups. Losing positives leaves you with negatives. −2 • 3 means you are losing two groups of negative 3. Losing negatives leaves you with positives.

Remember if you need to lose something that is not there, create the possibility with zero pairs and start with that mat at zero! Model each problem with the integer chips and with r’s and y’s.


Students may confuse the process for addition with the process of multiplication.




e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activites, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)



1. Use counters and a number line to illustrate 7 x -9 = _______.2. What does the 7 represent in 7 x -9?3. What does the -9 represent in 7 x -9?4. What does -7 x -9 mean?5. Complete the next tab of the foldable shown at the end of this

module.

4. Teaching Lesson D





b. Introductory Lesson

Essential Question: How does division of integers compare to multiplication of integers?

1. Modeling is designed to help build conceptual understanding. Division of integers using models can often confuse students. In fact, problems involving dividing a positive by a negative do not have a model, because it is not a concrete, physical reality. Guide students to the discovery that multiplication and division have the same rules.

2. Model the following problems using the division model of dividing into equal groups.

+ 6 ÷ + 2 = ____ and −6 ÷ −2 =____.

You can ask “How many groups of +2 are in +6” and, “How many groups of −2 are in −6?” You can then split the 6 items into groups of 2 and count the groups for the answer. Split the −6 counters into groups of 2 and count the groups for the answer.

3. There is no model for 6 ÷−2. Six does not contain groups of −2, and six cannot be divided into two negative groups since there is no such thing as a negative group. If your students need more focus on the relationship between multiplication and division, ask: for +6 ÷2, “What do you multiply −2 by to get a +6?”

4. Continue examples until assured student understanding. Then give groups of student’s exercises to complete.


No typical student misconceptions noted at this time.





e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activities, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)



1. What does -6 ÷ 2 mean?2. Complete the next tab of the foldable described at the end of this

module.



These additional activities may be used at the end of the module to check for student understanding.

Once you have completed all four operations with your students, you can let students practice working with integers on the following Internet site. http://funbrain.com/linejump/index.html

Students may make a 5-tab foldable as integer operations are introduced similar to the one shown below.


http://funbrain.com/linejump/index.html

ATTACHMENT 1MAT

Make a transparency and copies of this for your students. ATTACHMENT 2

Run copies of this in yellow for the positive counters.


ATTACHMENT 3Run copies of this in red for the negative counters.


The 5-tab foldable below can be created as you teach each lesson. It is just an example of how the foldable could be arranged.


Integers----------------------Addition----------------------Subtraction----------------------Multiplication----------------------Division

75


Define zero pair

Yellow = positive (+)

Red = negative (-)

-5 + 3 = R R R R R = -2 Y Y Y

* represent both #’s* push together* ask, “Is my answer in one color?”* if needed, remove zero pairs

-5 - 2 = R R R R R RR = -2 Y Y

* represent the 1st #* remove the 2nd # (If necessary create the possibility)

-3 * -2 = loose 3 groups of -2 items Begin with nothing…Add 3 sets of zero pairs…Now can I loose 3 groups of negative 2? = 6

* Begin with a zero square * Think “Can I gain/loose ____ groups of ____ items?”* If loosing, you need to create the possibility using zero pairs.

1 8 ÷ 4 = 2 YYYY YYYY 1 22. -4 ÷ -2 = 2 R R R R 1 23. -8 ÷ 4 = -2 RR RR RR RR 1 2 3 4

1. How many groups of 4 are in 8?

2. How many groups of -2 are in -4?

3. What is in each of 4 groups using -8 items?

YR

= Zero pair

RR RR RRY Y Y Y Y Y

76

MODULE 2-2 Solve Mathematical Situations





7-3.4 Use inverse operations to solve two-step equations and two-step inequalities. (C3)

7-3.5 Represent on a number line the solution of a two-step inequality. (B2)

Module 2-2 consists of 3 introductory lessons. Teaching time should be adjusted to allow for sufficient learning experiences in each of the modules.

77


In grade six, students use inverse operations to solve one-step equations that involve only whole numbers. Although sixth grade students do represent algebraic relationships with variables in simple inequalities, they have not yet had any instruction in solving inequalities. As grade eight students prepare for Algebra I, a strong foundation in solving equations is a necessity. The foundation begins to be built in grade six with simple one step equations (with whole numbers only), transitions to one and two-step equations (with rational numbers) and inequalities in grade seven, and the process continues into grade eight with the focus on solving inequalities and multi-step equations.

See sixth grade Algebra Indicator 6-3.2 (Apply order of operations to simplify whole-number expressions) for information on prior knowledge for order of operations.

A more in-depth look at the concept of equality/inequality began in the 6th

grade and continues throughout seventh and eighth grade.

2. Key Vocabulary

EquationExpressionInequalityInverseInverse operations

3. Content Overview

Seventh grade will be the first time that solving any type of inequality (both one-step and two-step) is introduced and the first time to be exposed to solving two-step equations.

Note that students in grade seven are to use inverse operations to solve equations and inequalities. A connection can be made here to order of operations in that when solving equations or inequalities (particularly two steps), we proceed in isolating the variable by doing the order of operations in reverse order.

Caution should be exercised when introducing solving inequalities that include negative numbers. The tendency is to simply tell students to reverse the inequality symbol when multiplying or dividing by a negative number. However, without understanding why, a student will soon forget that “rule” of inequalities. It is important that students understand "why" the sign is reversed when multiplying or dividing by a negative number.

If students have a solid foundation with the concepts of equality/inequality and that understanding is applied to solving equations and inequalities, the


notion of “balancing” both sides of an equation or inequality should not present a problem for students.

In seventh grade, students should also understand that solutions to inequalities can be written as an inequality, in set notation, or graphed on a number line. It is important to distinguish between the meaning of < verses ≤ and > verses ≥, particularly in regards to their graphs.

II. Teaching the Lesson(s)



7-3.4 Use inverse operations to solve two -step equations and two-step inequalities.



EQ: How does one and two-step equations differ?

Warm-up exercises: Give several problems for students to solve that involves solving linear one step equations. (e.g.) m + 17 = 95, 3n = 21, 16 = 4p, and x/3 = 18

1. Review inverse operations. The inverse operation of addition is subtraction, etc.

2. Review the order of operations involved when using inverse operations.Addition and/or subtraction are completed first, and then multiplication and/or division are completed next.

3. Demonstrate the value of algebra tiles using a set of overhead algebra tiles.

(e.g.) = x =1 = -1

NOTE: If algebra tiles are not available, a small plastic cup works well for the variable and chips may be used for the integers.


Have students draw these at the top of their paper for reference during class work.Model several two step linear equations using the algebra tiles. For example:

2x + 3 = 9 = =

2x + 3 - 3 = 9 - 3 = Subtract three from both sides

2x = 6 =

2x = 6 = Divide each side into 2 groups 2 2

x = 3 = , x = 3 items in each of 2 groups

4. Show another example such as 2 m - 3 = 11.

2 m - 3 = 11 =

2x - 3 +3 = 11+3 = Add three to both sides

2x = 14 =

2x = 14 2 2 = Divide each side into 2 equal groups

x = 7 x = 7 items in each of the 2 groups


5. You will not be able to model every single type of two step equation. Just use the cups and chips to help explain why you add and subtract first, and then multiply and divide. You can help your students see that in equations you use Order of Operations in reverse when trying to solve for the variable. You are trying to undo a finished equation, so start with addition and subtraction and move up.

6. Give student groups at least five two step linear equations to solve using a drawing representation of the algebra tiles and also show the algorithm for solving the equations. Before students begin ask them to state which operation they should undo first when solving each equation.

7. As students complete the assignment, give each group a problem to demonstrate how to solve on the overhead using algebra tiles.

8. Tell the students you are now going to ask them to solve the equations without manipulatives. Using the algorithm demonstrate how to solve two-step equations.

9. Using a form of a Kagan strategy called Rally Table, students play “Pass the Folder”. Cooperative teams of 4 are given one piece of paper and one pencil. Give each team a problem to solve. A time limit is set. A student is chosen to begin. This student writes the first step to solving the 2-step equation and passes the folder to the student on their left. That student checks the work of the first student and adds the next step to the problem. The paper continues to go around the team as each student adds a step to the team’s problem. When the teacher calls time, all pencils are placed on the team table. The teams take turns sharing their steps/solutions with the rest of the class.


Student may perform inverse operation to only one side.Student may use incorrect inverse operation.Students may become confused when given problems with the variable on the right side of the equals sign such as 25 = 3x + 7.Students may undo the multiplication or division first, then undo the addition or subtraction.Some students may try to simplify across the equal sign. An example may be in x + 7 = 7, the student may add 7 to the right side to get x = 14.


Active Learning in the Mathematics Classroom, Grades 5-8 , Hope Martin, Corwin Press, 2007. See pages 54-55 for solving equations with one variable.

The Hands-On Equations Learning Kit, Borenson and Associate, Inc. Balance the Beans – 2-7, Glencoe Mathematics Applications and

Connections, Course 3. Use groups of 2.


Materials:

Balance, beans, envelopes

* Each partner secretly places an equal number of beans in each of several envelopes and then seals the envelopes.

* Decide which partner will go first. The first partner places his or her sealed envelopes plus several loose beans on one pan of the balance. The second partner places beans on the other pan until the two pans balance. Working together, both partners write an equation that describes this situation.

* The second partner solves for the variable in the equation, beginning by removing the loose beans from the pan with the envelopes and an equal number from the other pan. The next step is dividing the remaining loose beans so that an equal number can be placed in the same number of envelopes as there are on the other pan.

* Trade roles and repeat the activity.

Glencoe Mathematics: Applications and Connections, Interactive Mathematics Tools. This multimedia software provides an interactive lesson tied to this lesson. Students will click and drag cups and counters to explore two-step equations.

Solving for Variables The original game was introduced by - Andy FieldGrade level - 7/8/Algebra 1/Math Tech

Materials:For each group of three students: 8 small containers, 80 countable objects, code sheet prepared by the teacher

Set:Explain to students that today the class will play a game in which they try to solve for the other teams' "Secret Numbers".

Review:Go over solving equations. Ensure everyone remembers how to solve equations. Explain that today they will learn how to write equations. This can be used at any time during instruction. (I have found it the most useful as I introduce solving algebraic equations.)

Procedure:1. Divide the class into groups of two or three. Instruct each group to

select a recorder to keep an account of the events beginning in step 6. Distribute 8 containers and 80 counters to each group.

2. Each group is assigned a different letter of the alphabet and each group's 8 containers are labeled with the lowercase form of that letter. If the same lesson is taught repeatedly, the same containers can be used over and over.


3. Each group chooses a "secret number" between one and ten and informs the teacher of their choice. The teacher keeps a record of all "secret numbers" on his code sheet.

4. Have each group place the "secret number" of counters in each of their containers.

5. Each group will now have 8 containers, each of which contains the same number of counters and the same letter of the alphabet. Discuss ways to express the total number of counters in all 8 containers. For example: m+m+m+m+m+m+m+m or x+x+x+x+x+x+x+x. Build on that idea: 8m or 8x.

6. Have each group exchange some containers with one other group. For example, 3 m's are exchanged for 3 x's. Each group records its holdings in the following manner: m+m+m+m+m+x+x+x or 5m + 3x and x+x+x+x+x+m+m+m or 5x + 3m.

7. Each group confers with the teacher who checks the code sheet to tell them the total number of counters their group is holding. For example, the first group has 5m + 3x counters. The teacher tells them they have 22 counters.

8. Discuss if necessary how to write an equation to express the total number of counters. For example, 5m + 3x = 22.

9. Each group solves the equation they have developed to solve for the unknown variable.

10.Students continue to trade until they have discovered each group's "secret number" or until time has run out.

11.Encourage students to keep solutions within their group so each group can make the discoveries on their own.

Closure:Have students return to their desks. Explain that they now should be familiar with how to write algebraic expressions and solve algebraic equations that represent real objects.

Assessment:Students will be assessed through informal observation and formal written evaluation on a written test.

e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activities, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)Promethean Board – See Prometheanplanet.comSolving Two-Step Equations – Founders’ Hall Math Dept.http://www.prometheanplanet.com/server.php?show=ConResource.8251Solving Two-Step Equations – Cathy Durhamwww.prometheanplanet.com/server.php?show=ConResource.15017Solving Two-Step Equations – Sandi Twetenhttp://www.prometheanplanet.com/server.php?show=ConResource.11794





Solving Inequalities – Danielle Rabidashttp://www.brainpop.com/math/algebra/twostepequations/preview.wemlhttp://www.youtube.com/watch?v=jW7qko-bBY4http://www.yourteacher.com/prealgebra/2stepequations.php



1. Ask students to identify and correct the error in the following solution of the equation 3 x - 7 = 22.

3 x – 7 = 213 x – 7 = 21 3 3x – 7 = 7x = 14

2. Assign a two step equation for students to trade with a partner. They should show their solution by using models and by showing their work step-by-step.

3. Have students write a step-by-step explanation for solving an equation such as 4x – 7 = 13.

4. Solve the equation 3y + 6 = 42.a. -12b. 12c. -16d. 16

5. Solve 60 = n/4 – 20. a. -10b. 10c. -20d. 20

6. Explain the steps you would use to solve the equation 5y + 2 = 17.

7. In a two step equation which operation do you get rid of first?

8. What does it mean to “isolate the variable”?

9. How do you check an equation to see if your solution is correct?

10.Why do you apply the order of operations in reverse when solving equations?



http://www.yourteacher.com/prealgebra/2stepequations.php

http://www.youtube.com/watch?v=jW7qko-bBY4

http://www.brainpop.com/math/algebra/twostepequations/preview.weml


7-3.4 Use inverse operations to solve two -step equations and two-step inequalities.



EQ: What are some real-life situations that could be represented using inequalities?

Complete a lesson on inequalities modeled after Teaching Lesson A. Additionally, ask students to name several numbers that are possible solutions. Have them substitute these in the inequality to check.


1. Student may solve inequality but fail to graph solution correctly.


Active Learning in the Mathematics Classroom, Grades 5-8 , hope Martin, Corwin Press, 2007. See pages 67-70 for an introductory activity to graphing inequality expressions.

Move On 2-10, Glencoe Mathematics: Applications and Connections, Course 3.

Number of players: 2

Materials:

Index cards, spinner, counters

* Copy onto cards the digits 0 through 30, one digit per card. Shuffle the cards and place them face down in a pile. Label the sections of a spinner with the following inequalities: 3x - 5 > 11, 17 + c < 25, 4p > 16, d/2 + 15 < 20, 2f + 60 > 75, m – 42 > 21. On a large sheet of paper (or several sheets taped together) copy the game board below.

* Both partners place a counter in the first square of a row. Decide which partner will go first. In turn, each partner selects a card and spins the spinner. You may advance one square on each turn only if the number on the card is part of the inequality’s solution set.

* The winner is the first partner to reach the tenth square.


e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activites, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)Promethean Board – See Prometheanplanet.comhttp://www.yourteacher.com/prealgebra/solveinequality.php



Solve the inequality. 1. 0.6 + 3p > 1.8

a. p > 0.4b. p > 0.8c. p > 4d. p > 40

2. m/2 – 1 < 6a. m < 10b. m > 10c. m < 14d. m > 14

III. Teaching Lesson C





EQ: How do you graph the solutions to a two-step inequality using a number line?

1. Now that students understand how to solve two step inequalities they should be ready to begin graphing solutions on a number line. Remind students that in the sixth grade and in previous lesson students were reminded of the positioning of integers on a number line.


http://www.yourteacher.com/prealgebra/solveinequality.php

2. In this lesson they are introduced to “open” and “closed” notations when graphing solutions. Have students write “open” and “closed” at the top of their notes and place the appropriate signs beneath each. Draw examples of graphs for students. Remind students also that the number line goes on indefinitely in each direction. Remember to question students to name several integers that will make the solution true. For example, if x < 5 is the solution students may substitute 4, 0, or -3 into the inequality to prove the solution true.

3. Have students work in cooperative pairs to solve and graph the solutions

to two step inequalities. They must agree on a solution before moving to the next problem.


Students may solve the inequality but fail to graph its solution correctly. If students have trouble with two step inequalities, it is usually because they do not understand two step equations.


Active Learning in the Mathematics Classroom, Grades 5-8, Hope Martin, Corwin Press, 2007. See pages 67-70 for an introductory activity to graphing inequality expressions.

e. Technology

OneplaceSC.orgIlluminations by NCTM.orgInteractive at shodor.org (searches for lessons/activities, etc.)NLVM.usu.edu/en/nav/vlibrary.html (searches for grade level activities, lessons, etc.)Promethean Board – See Prometheanplanet.comhttp://www.yourteacher.com/prealgebra/solveinequalities.php


http://www.yourteacher.com/prealgebra/solveinequalities.php



1. EXIT SLIP: What part of solving and graphing inequalities do you find the most difficult? Why?

2. Sketch several graphs on the overhead that show the solution sets of several inequalities. Have students state the corresponding inequalities.

3. Solve each inequality. Then show the solution on a number line. x + 5 ≤ 21 5c – 21 < 9 z/4 + 8 < 16 4f – 17 ≥ 25

1. Explain to your partner the difference in using an open and a closed circle when graphing the solution of an inequality.




MODULE

2-3Patterns, Relationships, and Functions

andRepresentations, Properties, and

Proportional Reasoning




7-3.1 Analyze geometric patterns and pattern relationships7-3.7 Classify relationships as either directly proportional,

inversely proportional, or nonproportional. 7-3.6 Represent proportional relationships with graphs, tables,

and equations. (B2)

Module 2-3 consists of 2 introductory lessons. Additional lessons may be required.

89


Students have used patterns all their lives and began to learn about and study patterns as early as kindergarten. In kindergarten through second grade, students progress from identifying patterns to translating patterns into rules. The emphasis on the creation of numeric patterns begins in third grade, and in fourth and fifth grades, students transition to analyzing patterns, then representing these patterns in words, expressions, and equations. Middle school continues this study of patterns by placing emphasis on numeric and algebraic patterns in the sixth grade, with the seventh grade focusing on geometric patterns.

In sixth grade students determined whether two ratios were equivalent and used proportions to determine units rates. Seventh grade students will classify relationships as directly proportional, inversely proportional, or nonproportional.

Beginning in fourth grade students interpreted data in tables, line graphs, bar graphs, and double bar graphs with scale increments greater than one. Building on prior knowledge, seventh grade students will represent relationships through graphs, tables, and equations.

2. Key Vocabulary

Directly ProportionalInversely ProportionalNonproportional

3. Content Overview

From the study of patterns seventh grade students should advance to classifying relationships as directly proportional (when one quantity always changes by the same factor as another, = k or y=kx, where k is a constant), inversely proportional (when one quantity decreases by the same factor as the other increases, xy=k or y= where k is a constant), or nonproportional. Please note that directly proportional is also known as direct variation and inversely proportional as inverse variation. Because seventh grade also includes the introduction of slope, students should be led to discover the connection between slope and relationships that are directly proportional, the constant k being the same as the slope. (Slope will be dealt with in Module 2-4.) This is the first time that students have been introduced to the terms directly and inversely proportional, and instruction should enable them to differentiate between the two, both numerically and graphically. Students should also be able to identify nonproportional relationships.

Teacher Note: When using two order pairs (x1, y1) and (x2, y2), teachers may refer to these alternative forms:


Directly proportional is = . Inversely proportional is .

An overall curriculum theme seen throughout the seventh grade Algebra strand is the concept of a constant rate of change (slope) and the tables, equations, and graphs that result from a relationship that has a constant rate of change. It is important that students be given ample opportunities to discover the connection between direct proportionality and the table, equation, and graph this relationship produces as it leads to the gentle introduction of slope and then linear functions in the eighth grade. As students graph directly proportional relationships, they should be able to identify the unit rate as the slope of the related line.

Representing inversely proportional relationships in tables, equations, and graphs allows students to understand that not all tables and equations produce similar graphs and that slope only exists when there is a "constant" rate of change. This understanding is important as the focus in eight grade will be on the table, equations, and graphs derived from linear functions.

II. Teaching the Lessons



7-3.1 Analyze geometric patterns and pattern relationships. (B4)

Cognitive Process Dimension: AnalyzeKnowledge Dimension: Conceptual Knowledge


Teacher Information:

Some texts define geometric sequences and “figurate numbers” differently. Because the focus of this indicator is geometric patterns, we will make no distinction other than to explain how texts see them differently. The explanation is provided to alert you as a teacher that student learning opportunities should include all types without students having to identify the various types. Also, some texts label geometric patterns as growing patterns.


The distinction:Geometric sequence – Each successive term is determined by multiplying the preceding term by a fixed number, the ratio. Take a family tree for example. You start with one couple who each has two parents. Each of those have two more parents (4 grandparents for the original couple), etc. In table form the sequence becomes:Step/Number of the term Term1 (original couple) 2 = 21

2 (4 parents) 4 = 22

3 (8 grandparents) 8 = 23

4 (16 great grandparents) 16 = 24

So, following this pattern/sequence the nth term would be 2n.

Figurate numbers – Typically we think of we think of triangular or square numbers – representations can be arranged in the shape of triangles or squares. There are figurate numbers for each polygon and many interesting patterns can be found by exploring the relationships. Take square numbers for example:

* * * * * * * * * * * * * * 1 star 4 stars 9 stars

As with the geometric sequence, this is not arithmetic because the difference between the number of stars is not the same each time. In table form the pattern becomes:Step/Number of the term Term

1 12

2 22

3 32

So, following this pattern/sequence the nth term would be n2 . The number of stars does not depend on the previous number of stars and, therefore not geometric and the difference between the number of stars is not constant and, therefore not arithmetic.

Lesson –As a quick review, have students draw or build examples of growing patterns.

Circulate the class and ask questions to determine students’ justification. Also, look for patterns that were developed based on adding versus multiplication to find the next term. Allow a couple of students to share their growing patterns – select students who used addition and multiplication. As students share, point out vocabulary such as “number of term or step” and “term”. Ask students what is different about how the next term/step was found (addition versus multiplication). If no student used multiplication be prepared to give an example before asking about the difference.

Next show the students three or four steps of a pattern (you may decide to use the star pattern shown above, give them material (grid paper, blocks,


toothpicks, etc.) and ask them to extend the pattern and to justify the extension in writing. It will be best if students work in small groups to share their thinking. When deemed appropriate, allow student groups to share how they extended the pattern and to provide their justification.

Explain to students that just as in earlier grades it is not always possible to draw out an entire pattern to determine the number of items/shapes that will be in say, the 100th term. Therefore, it is critical to begin to look at how the steps fit together or are related to form the next step. Further explain that in this grade they will begin to work with geometric patterns – patterns that rely on multiplication – including exponents they used in the first nine weeks.

Tell students that this summer you visited an art exhibit and saw a sculpture entitled “Stairs to the Stars”. The artist created steps by placing a series of concrete blocks in a pattern like:

After viewing the exhibit you wondered how many blocks would it take if the artist made 50 rather than 4 parts to the sculpture. Challenge the students to determine how many blocks would be needed in the 50 th part and to do so without drawing out all the parts between 4 and 50. If students are struggling suggest that they make a table or chart and look for relationships/patterns. (If students need cubes to begin their thinking make that manipulative available.)

Additional learning opportunities will be needed by students.


Students are accustomed to looking for repeating or growing patterns that involve simple addition – either perform the operation of addition to get the next step or add another shape/figure, etc. When observing the class look for students who may not have a solid understanding of those simple relationships and are thus not able to move to more complex thinking.


While additional learning opportunities are needed, no suggestions are included at this time.


e. Technology

The following Web sites are possible resources:

Web site for teacher content knowledge:http://www.mathematische-basteleien.de/triangularnumber.htm

Good Web site for teachers and students:http://milan.milanovic.org/math/english/triangular/triangular.htmlhttp://mathforum.org/workshops/usi/pascal/pascal_triangular.html

SCETVhttp://www.Oneplacesc.org

Interactivatehttp://www.showdor.org/interactivate/standards/organization/6/92

Virtual Manipulativeshttp://NLVM.usu.edu/en/nav/vlibrary.html


The following examples of possible assessment strategies may be modified as necessary to meet student/teacher needs. These examples are not derived from nor associated with any standardized testing. Use teacher observation and questioning during the introductory lesson. Use a ticket out check method - Show students a geometric pattern with at

least three terms. Ask students to extend the pattern and explain their reasoning. Their response is a ticket out. Note: The verb in the indicator is “analyze” and the verb in this assessment is “extend”. Students will need to “analyze” prior to extending. Therefore the strategy is appropriate for the indicator.

2. Teaching Lesson B:


7-3.7 Classify relationships as either directly proportional, inversely proportional, or nonproportional. (B2)


7-3.6 Represent proportional relationships with graphs, tables, and equations. (B2)




http://NLVM.usu.edu/en/nav/vlibrary.html

http://www.showdor.org/interactivate/standards/organization/6/92

http://www.Oneplacesc.org/

http://mathforum.org/workshops/usi/pascal/pascal_triangular.html

http://milan.milanovic.org/math/english/triangular/triangular.html

http://www.mathematische-basteleien.de/triangularnumber.htm

NOTE: In this lesson students will be asked to create T-Charts and Graphs. These should be done individually because the charts and graphs will be used in the next module of this nine weeks. They may compare and share but each student will need his/her own T-Chart and Graph for future use. If students are absent, they should be required to make up this portion of the lesson so they will have the necessary material for later use.

Introductory lesson for Directly Proportional

Tell the students that David, the son of a friend of yours, is getting a part-time job to earn money during the holidays. He is making $7.50 per hour. Ask students to make a T-Chart showing how much David will make for each hour earned if he works up to ten hours. Remind students to label the T-Chart and title it “David’s Earnings”. Ask students to compare their charts for possible errors.

Next give students grid paper and ask them to graph the data with each line on the X axis valued at increments of one for hours worked and each line on the Y axis valued at increments of $7.50 for money earned. Remind them to label the graph and title it “David’s Earnings”. Circulate the room and ask probing questions as students work.

Ask students questions about the graph such as “Can the dots be connected on the graph? If so, why is that possible? If not, why not?” (Yes they can be connected because David may work part of an hour and the points not actually shown on the graph would tell how much he would make if he worked less than whole hours.) “Describe the pattern on either the T-Chart or the graph.” (The more hours worked, the money is earned. So, there is a relationship between hours worked and money earned. As one increases so does the other.) “Could there ever be information on this graph below zero on the X or Y axis – why or why not?” (No, because you cannot work negative hours nor be paid negative amounts). Ask students to set the chart and graph aside for just a moment.

Introductory Lesson for Indirectly Proportional

Next tell students that David is saving money to take a train ride to Washington, D.C. to visit his friend, Cameron, who moved away over the summer. David knows that it is 240 miles from his home to D.C. If the train travels at a constant rate of speed and goes 20 mph it will take 12 hours, at 60 mph it will take 4 hours, at 80 mph it will take 3 hours, at 120 mph it will take 2 hours, etc. Ask students to again make a T-Chart showing the mph and time to travel the 240 miles. Remind students to label the T-Chart and title it “David’s 240 Miles Trip”.

Next give students grid paper and ask them to graph the data with each line on the X axis valued at increments of one for hour needed to travel 240 miles and each line on the Y axis valued at increments of 20 for miles per hour. Remind them to label the graph and title it “David’s 240 Miles Trip”. Circulate the room and ask probing questions as students work.


Ask students questions about the graph such as “Can the dots be connected on the graph? If so, why is that possible? If not, why not?” (Yes they can be connected because time continues as miles are covered.) “Describe the pattern on either the T-Chart or the graph.” (The faster the train travels the less hours are needed to go 240 miles. So, there is a relationship between speed and distance traveled. One quantity increases by the same factor that the other decreases.) “Could there ever be information on this graph below zero on the X or Y axis – why or why not?” (No, because you cannot go back in time nor travel negative miles.)

Working in small groups ask students to compare the T-Charts and the graphs and to record their observations. Allow time for whole class sharing. Next put the definitions for “Directly Proportional” and “Indirectly Proportional” on the board/overhead. You may use:

Directly Proportional – As one quantity increases, the other quantity increases by the same constant factor. The two quantities must have the same constant ratio to be directly proportional.

Inversely Proportional – As one quantity increases, the other quantity decreases by the same constant factor. Note: Some definitions may say the quantities change by reciprocal factors. That definition is correct also because for one quantity to decrease proportionally, the reciprocal factor must be applied.

Ask students to read the two definitions, discuss in small group which T-Chart and graph matches which definition and why.

Introductory Lesson for Nonproportional

Tell students that not all relationships are either directly nor inversely proportional – even if it at first it appears they fit the definition. For example:(These examples were adapted from Algebra To Go, Great Source Publishing, 2000.)

Example A: You are 14 and your sister is half your age – 7. Next year you will be 15 and your sister will be 8 – more than half your age. While both ages increase, the factor does not remain constant. Thus, there is no direct proportionality. (NOTE: It would be best to state the first sentence of this example and have student make a T-Chart and a graph of the ages for the next six years. Remind students to label and title “Ages”. Next ask them how is this example like and different from the other T-Charts and graphs. Both factors increase but not by a constant factor. Also, you may want to ask students which definition does this appear to fit at first glance and why does it not meet the requirements?)

Example B: The longer a candle burns, the shorter it gets. Does it get shorter at the same rate? No, after one hour of burning, a 10-inch candle may be only 8-inches tall. However, after two hours of burning it may be 6 inches tall. While the


burning time increased and the height decreased, the height did not decrease by a constant factor. Therefore, there is no inverse proportional relationship.

Explain to students that the two examples they just discussed are called nonproportional. Add a third definition to the board –

Nonproportional – A relationship exists between quantities but the relationship is neither directly proportional nor inversely proportional.

Ask students to provide examples of each of the three types of relationships and to justify their reasoning.

IMPORTANT: You may either ask students to save the T-Charts and graphs they made OR you may collect and hold for when you move to the introductory lesson on indicators 7-3.2 and 7-3.3 which follows this module.

Before moving to representing proportional relationships with equations, students will need multiple opportunities to classify, make T-Charts, and graph proportional and nonproportional relationships.


A common misconception is to think two quantities are directly proportional simply because they increase at a constant rate. In fact, it is the ratio of the quantities that must remain constant.


While additional learning opportunities are needed, no suggestions are available at this time.

e. Technology

These are suggestions for resources

A good Web site for teachers and students on proportionality is:http://www.wikihow.com/Determine-Whether-Two-Variables-Are-Directly-Proportional

The following site has a variety of lessons and ideas:http://mathforum.org/library/drmath/sets/select/dm_direct_indirect.html

SCETVhttp://oneplacesc.org

Interactivatehttp://showdor.orf/interactivate/standards/organization/6/92




http://showdor.orf/interactivate/standards/organization/6/92

http://oneplacesc.org/

http://mathforum.org/library/drmath/sets/select/dm_direct_indirect.html

http://www.wikihow.com/Determine-Whether-Two-Variables-Are-Directly-Proportional

http://www.wikihow.com/Determine-Whether-Two-Variables-Are-Directly-Proportional


The following examples of possible assessment strategies may be modified as necessary to meet student/teacher needs. These examples are not derived from nor associated with any standardized testing.

Use a ticket out check method - Show students graphs depicting directly proportional, inversely proportional and nonproportional relationships, labeled A,B,C. Have students put their name on a sheet of paper, identify graphs, and turn in their response as a ticket out of the class.

Use a ticket out check method - Show students a graph depicting directly proportional, inversely proportional or nonproportional relationships. Ask students to select a graph, identify the relationship and justify their identification. Have students turn in their response as a ticket out of the class.

Use either of the above mentioned strategies but substitute a chart or table for the graph.




MODULE2-4

Change in Various Contexts



7-3.2 Analyze tables and graphs to describe the rate of change between and among quantities. (B4)

7-3.3 Understand slope as a constant rate of change. (A2)

Module 2-4 consists of 1 introductory lesson. Additional lessons may be required.

99



Beginning as early as first grade students begin to classify change over time as quantitative or qualitative. Then they move to illustrate situations that show change over time as increasing or decreasing in grades 3 and 4 respectively. By the end of the fifth grade, students are analyzing change over time. In seventh grade students should analyze tables and graphs to describe rate of change and understand slope as a constant rate of change.

2. Key Vocabulary

Rate of ChangeSlope

3. Content Overview

As stated earlier, patterns continue to be explored in the seventh grade, but the focus becomes more symbolic. Seventh grade students should examine patterns in tables and graphs, describe the change among quantities, and connect their observations to a rate of change to determine if the rate of change is constant or not. Students should be provided with opportunities to discover that the rate of change and slope are one in the same. Once this observation is made students can use this understanding to solve problems as they analyze tables and graphs.




7-3.2.1 Analyze tables and graphs to describe the rate of change between and among quantities. (B4)


7-3.3 Understand slope as a constant rate of change. (A2)

Cognitive Process Dimension: UnderstandKnowledge Dimension: Remember



In the previous module students created T-Charts and Graphs based on David, the son of your friend and on ages. If you took these up, pass them back to students. If students saved the material ask them to retrieve it.

Ask students to refresh your memory about the T-Charts and Graphs. Ask what was the most important point that distinguished proportional relationships from nonproportional relationships? (Changes are based on a constant factor.)

Tell students that David got a new job and is now earning $15.00 per hour. Ask students to make a T-Chart showing how much David will make for each hour earned if he works up to ten hours. Remind students to label the T-Chart and title it “David’s New Earnings”. Ask students to compare the new charts with each other for possible errors.

Next give students grid paper and ask them to graph the data with each line on the X axis valued at increments of one for hours worked and each line on the Y axis valued at increments of $15.00 for money earned. Remind them to label the graph and title it “David’s New Earnings”. Circulate the room and ask probing questions as students work.

Ask students to compare the graph of “David’s Earnings” with “David’s New Earnings”. Again, ask can the dots on the graph be connected? Why or why not? Ask students to connect the dots on both graphs. What do they notice? If a student does not mention the steepness of the line on the graph ask “What do you notice about the “slant” of the line on the graph? (The slant is steeper on the graph where he makes more money.)

Tell students to pretend that the graphs of David’s Earnings” and David’s New Earnings” reflect millions of dollars earned by a corporation over a number of years. Which graph do you think stockholders/owners would rather see and why? (The steeper the line, the greater the earnings over time.)

Another way to refer to steepness is “slope”. Ask students what they think of when they hear the word “slope”. Allow time for answers.

(If students did not give stairs as examples of what they think of for the word slope, then point that out. Draw a few stairs on the board. Ask students when they think about how steep stairs are do they think about the “dip” for each step or do they think about the total slant of the stairs – draw a straight line touching the steps to illustrate your point. Typically we think of the slant or slope of the total stairs. So, even though hills and stairs may have “dips” the steepness is determined by the degree of slant of a straight line, not the “dips”.

Ask students to compare “David’s Earning”, “David’s New Earnings” and “Ages” graphs. What to they notice about the lines in those graphs? (Ages is not a straight line.)

Again say, that in mathematics the steepness of a line is referred to as “slope”. Next put this or a similar definition on the board:


Slope – A constant rate of change

Put this or a similar definition on the board:

Graph – A picture of the change in one quantity in terms of the other quantity.

Ask students to mentally relate that to David’s graphs they made and then to share their thinking with a partner. Next allow students to share whole class. (The point is to have students focus on the fact that a graph represents relationships and when there is a constant change in the relationship the graph of the line has slope.)

Ask students to again pretend that David’s graphs are a picture of business earnings in the millions of dollars. Would stockholders/owners want to know that a graph of their earnings had slope or would they want to know how much slope and why? (How much slope because that is an indication of the amount of earnings.) Slope is determined by how much one quantity changes compared to the other quantity. In the case of David’s graphs, his initial rate of change was $7.50 – in other words for each hour worked he made $7.50 – each time a new point was added to the graph his earnings changed by $7.50. For his new earnings the rate of change was $15.00 – each time a new point was added to the graph his earnings changed by $15.00. Let’s look at some other graphs and talk about the slope or rate of change.

Show students examples of other graphs and guide them through a rise over run discussion. Follow this by independent student work on determining slope. Be certain to include examples of graphs that do not have a constant rate of change. Next move to work with tables.

c. Misconceptions/Common Errors

A common mistake that students make is to put the horizontal (run) quantity over the vertical (rise) quantity when computing rate of change. This is an indication that they lack conceptual understanding.

d. Additional Instructional Strategies

Giving students pictures of different shaped bottles and pictures of what the graphs would look like if the bottles were filled with water at a constant rate and asking the students to match the graphs and bottles will provide insight into student understanding of graphical and contextual representations. Such an activity can be found on page 288 of Elementary and Middle School Mathematics, Sixth Edition, John A Van De Walle.

Enrichment student lessons involving slope can be found at: (Note: Teachers should explore the lessons first to determine student academic level appropriateness.)http://www.terragon.com/tkobrien/algebra/topics/slope/slope.html


http://www.terragon.com/tkobrien/algebra/topics/slope/slope.html

http://www.analyzemath.com/Slope/Slope.html

e. Technology

These are suggestions for resources:

A tool such as a CBR (Calculator Based Ranger by Texas Instrument) is a great way to generate a lot of interest and make learning about slope fun experience for students. Activities for the CBR at the middle level can be found at:http://education.ti.com/educationportal/sites/US/nonProductSingle/activitybook_cbr_cbl_73_datacollection.html

While this site contains information beyond the level of these 7th grade indicators, it is a good site for teacher reference:http://mathforum.org/library/drmath/sets/select/dm_slope.html




Note: The first part of each of the Web based lessons listed under “Additional Instructional Strategies” above would be appropriate resources for introducing the concept of slope.



During the lesson student observation and questioning is a way to determine if students are on target.

Since indicator 7-3.3 is at the remember level of Blooms, students could simply complete the sentence “Slope is a _____________”. Students written response could serve as a ticket out.

Since indicator 7-3.2 is at the analyze level of Blooms a simple graph or table could be provided to students and they could be required to identify the depicted rate of change.







http://mathforum.org/library/drmath/sets/select/dm_slope.html

http://education.ti.com/educationportal/sites/US/nonProductSingle/activitybook_cbr_cbl_73_datacollection.html

http://education.ti.com/educationportal/sites/US/nonProductSingle/activitybook_cbr_cbl_73_datacollection.html

http://www.analyzemath.com/Slope/Slope.html

Recommended Days of Instruction Third Nine WeeksThird Nine Weeks

Standards/Indicators Addressed:Standard 7-4: The student will demonstrate through the mathematical processes an understanding of proportional

reasoning, tessellations, the use of geometric properties to make deductive arguments, the results of the intersection of geometric shapes in a plane, and the relationships among angles formed when a transversal intersects two parallel lines.

7-4.1 Analyze geometric properties and the relationships among the properties of triangles, congruence, similarity, and transformations to make deductive arguments. (B4)

7-4.2 Explain the results of the intersection of two or more geometric shapes in a plane. (B2)7-4.3 Illustrate the cross section of a solid. (C2)7-4.4 Translate between two- and three-dimensional representations of compound figures. (B2)7-4.5 Analyze the congruent and supplementary relationships—specifically, alternate interior, alternate exterior,

corresponding, and adjacent—of the angles formed by parallel lines and a transversal. (B4)7-4.6 Compare the areas of similar shapes and the areas of congruent shapes. (B2)7-4.7 Explain the proportional relationship among attributes of similar shapes. (B2)7-4.8 Apply proportional reasoning to find missing attributes of similar shapes. (C3)7-4.9 Create tessellations with transformations. (B6)7-4.10 Explain the relationship of the angle measurements among shapes that tessellate. (B2)

Standard 7-5: The student will demonstrate through the mathematical processes an understanding of how to use ratio and proportion to solve problems involving scale factors and rates and how to use one-step unit analysis to convert between and within the U.S. Customary System and the metric system.

7-5.4 Recall equivalencies associated with length, mass and weight, and liquid volume: 1 square yard = 9 square feet, 1 cubic meter = 1 million cubic centimeters, 1 kilometer = mile, 1 inch = 2.54 centimeters; 1 kg = 2.2 pounds; and 1.06 quarts = 1 liter. (A1)

7-5.5 Use one step unit analysis to convert between and within the US Customary System and the metric system. (C3)



Module 3-1 EquivalenciesIndicator Recommended Resources Suggested Instructional



7-5.4 Recall equivalencies associated with length, mass and weight, and liquid volume.


NCTM's Navigations SeriesSC Mathematics Support DocumentTeaching Student-Centered Mathematics Grades 5-8 and Teaching Elementary and Middle School Mathematics Developmentally 6th Edition, John Van de Walle

NCTM’s Principals and Standards for School Mathematics (PSSM)Textbook Correlations – See Appendix A


See Instructional Planning Guide Module 3-1 Lesson A Assessing the Lesson

Module 3-2 ConversionsModule 3-2 Lesson A:

7-5.5 Use one step unit analysis to convert between and within the US Customary System and the metric system. (C3)




NCTM’s Principals and Standards for School Mathematics (PSSM)Textbook Correlations – See Appendix A

See Instructional Planning Guide Module 3-2, Introductory Lesson A


See Instructional Planning Guide Module 3-2Lesson A Assessment




Module 3-3 Plane GeometryIndicator Recommended Resources Suggested Instructional



7-4.2 Explain the results of the intersection of two or more geometric shapes in a plane. (B2)

7-4.3 Illustrate the cross section of a solid. (C2)

7-4.4 Translate between two- and three-dimensional representations of compound figures. (B2)





Textbook Correlations -See Appendix A


See Module 3-3, Lesson AAdditional Instructional Strategies



7-4.5 Analyze the congruent and supplementary relationships—specifically, alternate interior, alternate exterior, corresponding, and adjacent—of the angles formed by parallel lines and a transversal. (B4)


See Module 3-3, Lesson B Additional Instructional Strategies


Module 3-4 Plane and Transformational Geometry






7-4.9 Create tessellations with transformations. (B6)

7-4.10 Explain the relationship of the angle measurements among shapes that tessellate. (B2)





Textbook Correlations - SeeAppendix A

See Instructional Planning Guide Module 3-4,Introductory Lesson A



Module 3-4 Lesson B





Module 3-5 Proportional Reasoning






7-4.6 Compare the areas of similar shapes and the areas of congruent shapes. (B2)

7-4.7 Explain the proportional relationship among attributes of similar shapes. (B2)

7-4.8 Apply proportional reasoning to find missing attributes of similar shapes. (C3)







See Module 3-5 Lesson A Additional Instructional Strategies




MODULE 3-1 Equivalencies




7-5.4 Recall equivalencies associated with length, mass and weight, and liquid volume: 1 square yard = 9 square feet, 1 cubic meter = 1 million cubic centimeters, 1 kilometer = mile, 1 inch = 2.54 centimeters; 1 kg = 2.2 pounds; and 1.06 quarts = 1 liter.


110


In grades two through four, students have been asked to recall equivalencies with length, time, liquid volume, and weight within the U.S. Customary System. In grade five, students recalled equivalencies within the metric system associated with length, liquid volume, and mass.

Seventh grade students extend their learning of equivalencies to include equivalencies between the U.S. Customary System and the metric system.

2. Key Vocabulary

Cubic meterSquare Yard

3. Content Overview

As stated earlier, seventh grade students extend their learning of equivalencies to include equivalencies between the U.S. Customary System and the metric system. The specific equivalencies students should recall are cited in the indicator.




7-5.4 Recall equivalencies associated with length, mass and weight, and liquid volume: 1 square yard = 9 square feet, 1 cubic meter = 1 million cubic

centimeters, 1 kilometer = mile, 1 inch = 2.54 centimeters; 1 kg = 2.2 pounds; and 1.06 quarts = 1 liter. (A1)

Cognitive Process Dimension: RecallKnowledge Dimension: Remember


Since the verb for this indicator is “Recall” then all that is expected of students is to remember the specific equivalencies cited in the indicator. This should be accomplished over time by giving students fun engaging activities such as a game of Concentration. Concentration is a simple game to make and play and the instructions follow. It is essential to recall or memorization that students follow the instructions and verbalize equivalencies as they play. Verbalization not only helps the student who is stating the equivalency but addresses the auditory needs of other players as well. Thus, all learning modalities are addressed. Since games of this nature should be played multiple


times in order to achieve recall, students could be allowed to play a game such as Concentration prior to the start of class, if there is unstructured time at the end of class, etc.

Materials:

4 x 6 (or smaller) index cards Note: It is recommended that more than one set of cards be made so that multiple groups of students can play at the same time. If so, then colored index cards should be used. If cards are accidentally mixed, then it will be easy to see that all “yellow” go together, all “green” go together, etc. without taking time to sort the cards.

Permanent MarkerLaminate to preserveGallon size zip lock storage bag or similar storage container

Write one-half of each equivalency on an index card with a permanent marker. Laminate all index cards to preserve. Place the entire set of cards into a gallon size zip lock bag. Write “Equivalencies Concentration” on the front of the bag. NOTE: It would be a good idea to include the equivalencies required by other “Equivalencies” indicators in previous grades. This makes the game more interesting and challenging and helps with recall.

To Play:One or more players

Shuffle/mix all “equivalency cards” and place them face down. The cards can be placed face down in a random, square, triangle, rectangle, etc. pattern. The players make that decision. Once all cards are face down, player one turns over any one card of their choice, reads the card, and announces what they need for a match. For example, if a card stating “1 square yard” is turned over, then the player announces “1 square yard = 9 square feet. I am looking for 9 square feet”. That player then turns over a second card of their choice. If the two cards form an equivalency, the player takes those two cards, and has earned another turn. That player continues to turn over cards, read and announce until no match is made. When no match is made, the two revealed cards are turned back over and the next player takes a turn. If any player neglects/forgets to read and announce they loose a turn. The players who are waiting for a turn must “concentrate” on which cards were revealed, where the cards are located face down, and then use that information to make matches when it is their turn to play. After all cards are revealed or time has been called, the player with the most matches wins. Note: If through observation it is determined that students are not verbalizing the equivalencies prior to turning over a card, then students could be required to write the equivalencies in order to earn play privileges.





Connections as well as associations should be made with familiar equivalencies such as from linear units of measure to square and cubic units of measure. For example, there are 3 feet in 1 yard so to find the number of square feet in 1 square yard, have students draw a square with side lengths of 3 feet (1 yard), then find the area of the square. The area will be 9 square feet which is equivalent to 1 square yard.

The use of pictures (visual images) to recall equivalencies is another strategy to teach this indicator. For example, rulers with both customary and metric units can help with the relationship between cm and inches. By examining the ruler students will be able to see that 2.54 cm = 1 inch.

The strategies above are used to provide visual images to develop measurement equivalencies. These visual images will help student be able to recall the required equivalencies.

e. Technology


This site has lesson ideas for the teacher http://desktoppub.about.com/cs/intermediate/a/basicmetric_2.htm

This site has conversion charts that may be printed out for studentshttp://www.sawmillsoftware.com/activeserverpages/worksheets/measurementconversions.asp

Conversionshttp://www.bartleby.com/61/charts/M0182500.html

Lesson on Conversionshttp://www.eduref.org/Virtual/Lessons/Mathematics/Measurement/MEA0007.html

Good conversions http://www.convert-me.com/en/

Good Table http://www.mathleague.com/help/metric/metric.htm

SCETVhttp://www.oneplacesc.orgInteractivatehttp://www.showdor.org/interactivate/standards/organization/6/92


Since the verb for this indicator is “recall” then assessment can be addressed through questioning, “observation” as students play games such as Concentration, or through a formal “fill in the blank” type of assessment.



http://www.oneplacesc.org/

http://www.mathleague.com/help/metric/metric.htm

http://www.convert-me.com/en/

http://www.eduref.org/Virtual/Lessons/Mathematics/Measurement/MEA0007.html

http://www.bartleby.com/61/charts/M0182500.html

http://www.sawmillsoftware.com/activeserverpages/worksheets/measurementconversions.asp

http://www.sawmillsoftware.com/activeserverpages/worksheets/measurementconversions.asp

http://desktoppub.about.com/cs/intermediate/a/basicmetric_2.htm

MODULE 3-2 Conversions



7-5.5 Use one step unit analysis to convert between and within the U.S. Customary System and the metric system. (C3)


115



In fourth grade students began to make conversions within the U.S. Customary system. In fifth grade students made conversions within the metric system. In seventh grade students should now convert between the U.S. Customary and the metric systems.

2. Key Vocabulary

Unit AnalysisDimensional AnalysisOne-step Unit Analysis

3. Content Overview

In seventh grade students are asked to convert a quantity expressed in one set of units to another equivalent quantity expressed in a different set of units. In other words, using unit (dimensional) analysis means that you will keep the units of measure throughout the problem. For example, if there are 2.54 cm to 1 inch, how many centimeters are in a foot? By writing each in fraction format (horizontally), students can see that all units cancel except the final ones in the answer.

2.54 cm x 12 in. = 30.48 cm = 30.48 cm in a foot. 1 in. 1 ft. 1 ft.

Another example: 10 yd to feet

10 yd x 3 ft = 10 x 3 ft = 30 ft 1 1 yd 1

In other words, you can cancel units just like you do factors when dealing with fractions. This is called “unit cancellation”.

Teacher Notes: In science unit analysis is typically called dimensional analysis. Therefore, students should be familiar with both unit and dimensional analysis terms.




7-5.5 Use one step unit analysis to convert between and within the U.S. Customary System and the metric system. (C3)

Cognitive Process Dimension: Apply


Knowledge Dimension: Procedural Knowledge


Note: This indicator should be based on indicator 7-5.4. The equivalencies which students should recall in 7-5.4 could serve as the basis for beginning a discussion on one-step unit analysis.

Tell the students the following story or give them a copy to read:After school while waiting for the bus you are watching some repairmen begin to patch a small crack in the concrete sidewalk. You hear one say they need six quarts of water for the small amount of concrete mix that is needed. However, the only measuring tool they have is a 1 liter Pepsi bottle. You can tell they need help determining how many 1 liter bottles of water they should use. They look over at you and ask for your help. How many 1 liter bottles of water should they use to equal 6 quarts of water?

Allow the students to work in pairs or small groups to find an answer. Note: As you move around the room if groups are struggling, remind them of the equivalencies they are memorizing for indicator 7-5.4.

Ask students to share strategies. Make certain that students express their answer in terms of liters and indicate that it will take 5 whole liters and seven tenths of another OR almost 6 liters.

Ask students to keep the strategies in their mind because you will come back to them shortly.

Next write the following on the board: A. 8 x 1 = 8

B. 6 x 3 1 3 =

2 6 x 3 1 3 1 =

6 1

Ask how are equations A and B above similar. (They both use the multiplicative identity element – students may respond that in both instances something is multiplied times one.)

Remind students that any time something is multiplied by one the result is equal to what you started with.

Leave the above examples on the board and now add the following example:How many square feet are in 3 square yards?


3 sq yds X 9 sq ft1 1 sq yd

3 sq yds X 9 sq ft Tell students units can be cancelled just like 1 1 sq yd when working with fractions.

27 sq ft

Ask students to think about the strategies they heard for solving the concrete/water mixture problem and challenge them to find another way to solve the problem using the “anything times one = anything” and cancellation rules you just demonstrated on the board.

Move about the room and provide prompting questions. Allow students to share correct strategies.

6 quarts 1 liter 1 x 1.06 quarts =

6 liters1.06 =

5.7 liters

Tell students that this strategy is called one-step unit analysis or dimensional analysis and can be used to convert both within and between the US Customary System and metric units of measurement – time, mass and weight, length, liquid volume, etc. Next year they will build on this strategy by using two-steps or more than two different units.

It is important that students understand that unit analysis is related to other mathematics they have previously learned and that it is not something entirely new they have to accommodate in their memory bank. By introducing the concept in this exploratory manner, students will have a basis for understanding rather than simply repeating a process that appears to be random.

Multiple opportunities to use one-step unit analysis will be necessary. Opportunities should be in context and should include conversions within and between the US Customary System and the metric system.


A common mistake students make is to put the units in such a fashion that they cannot be cancelled.



While additional learning opportunities are necessary no additional instructional strategies are included at this time.

e. Technology


Good tutorial sites that can be used for absent or struggling students:http://hotmath.com/hotmath_help/topics/unit-analysis.html

http://www.purplemath.com/modules/units.htm

http://mathforum.org/library/drmath/view/62176.html






During classroom instruction teacher questioning is a critical assessment strategy.

Ask students to compare in writing one-step unit analysis and multiplying fractions.

Ask students what is the typical mistake most students make when setting up unit analysis problems and why it is important to set up problems a certain way.



MODULE





http://mathforum.org/library/drmath/view/62176.html

http://www.purplemath.com/modules/units.htm

http://hotmath.com/hotmath_help/topics/unit-analysis.html

3-3 Plane Geometry



In fifth grade students explored methods for translating between two-dimensional representations and three-dimensional objects. They learned to sketch the front, top, and side views of a three-dimensional object built with cubes. They also learned to draw a net for a given three-dimensional shape and




7-4.3 Illustrate the cross section of a solid. (C2)7-4.4 Translate between two- and three-dimensional representations of

compound figures. (B2)7-4.5 Analyze the congruent and supplementary relationships—specifically,

alternate interior, alternate exterior, corresponding, and adjacent—of the angles formed by parallel lines and a transversal. (B4)


120

construct and/or state the three-dimensional shape when given its two-dimensional representation (net).

The general attributes and names of parallel, perpendicular, and intersecting lines were discussed in prior grades.

2. Key Vocabulary

Transversal lineAlternate exterior anglesAlternate interior anglesCorresponding anglesAdjacent anglesCross SectionCompound Figure

3. Content Overview

Seventh grade students are required to explain the results of the intersection of two or more geometric shapes in a plane. For example, if a line intersects a circle, the result of the intersection is two points. This can be modeled by drawing a picture of a circle and the line going through it. Using models and pictures will aid students as they transition to attaining the ability to visualize the intersection of two or more geometric shapes. An example of a hands-on model would be using a pen and a piece of paper to demonstrate what happens when a plane and line intersect, the intersection being a point. Poking the pen through the paper gives the students a visual demonstration of this concept. Encourage students to think of everyday objects that can represent points, lines, planes, etc. in an effort to help them visualize the intersection.

In seventh grade students should acquire the ability to translate between two- and three-dimensional representations of compound (when two or more, two-dimensional or three-dimensional figures are joined together) figures and to illustrate the cross section of a solid. Extensive modeling with concrete objects needs to be done in order for the students to develop a mental picture of compound three-dimensional shapes and the two-dimensional view points that give the figure it’s overall shape and vice versa. Student work with cross sections at seventh grade should be limited to “deconstructing” the layers of a three-dimensional object. For example, to illustrate the cross section of a rectangle, students may build a rectangle using interlocking cubes. Then the students might illustrate on isometric dot paper the original rectangle and a view of one of the “layers” or cross sections that make up the rectangle. The ability to do this will enable students to develop, justify, and understand formulas (such as area, surface area, volume, etc.) that are used in regards to two- and three-dimensional figures. Because seventh grade is the first time students are introduced to the concept of volume, illustrating the cross section of three-dimensional shapes is an appropriate prerequisite to generating strategies for finding volume. It should be noted that in the example just cited, students should illustrate a horizontal and a vertical cross section. Also, it would be sound educational practice to do the same with a cube and discuss the relationship


between the horizontal and vertical cross sections. When engaging in such a discussion, it is important for students to understand that the concrete models and pictorial illustrations are the result of the intersection of a plane in a segment of the geometric shape. (This stays true to the definition of a cross section – the intersection of a plane and a geometric solid.)

In seventh grade the specific angle relationships formed between parallel lines and a transversal (a line that intersects two or more lines in different points) are explored and identified. Students in seventh grade should analyze the congruent and supplementary relationships of the angles formed by two parallel lines and a transversal. Students should have the ability to understand, identify, and use in mathematical communication the terms alternate exterior, alternate interior, corresponding, and adjacent in regards to the angles formed by parallel lines and a transversal.

Students need to formulate their own conclusions in regards to the angles formed by two parallel lines and a transversal through guided investigation before the formal definitions are introduced. They should have the opportunity to measure the angles formed by these lines, and from their findings, make conjectures and draw conclusions about which angles are congruent (a concept taught in fifth grade) and which angles are supplementary (a concept taught in sixth grade). Once the students have a good understanding of which angles are congruent and/or supplementary, then the formal definitions can be introduced.

II. Teaching the Lessons





7-4.3 Illustrate the cross section of a solid. (C2)


Cognitive Process Dimension: UnderstandKnowledge Dimension: Procedural Knowledge

7-4.4 Translate between two- and three-dimensional representations of compound figures. (B2)



NCTM’s Navigating through Geometry, Grades 6-8, pp. 67-72, 115, 116


The students may have difficulty visualizing the results of the intersections.


1. Tell students that you are going to “deconstruct” the layers of a three-dimensional object. Build a rectangle using interlocking cubes. Have students illustrate on isometric dot paper the original rectangle and a view of a horizontal or cross section that makes up the rectangle. Then, have them illustrate a vertical cross section.

2. Ask students how formulas (such as area, surface area, volume, etc.) that are used in regards to two- and three-dimensional figures relate to their figure.

3. Discuss the relationship between the horizontal and vertical cross sections. When engaging in such a discussion, it is important for students to understand that the concrete models and pictorial illustrations are the result of the intersection of a plane in a segment of the geometric shape. (This stays true to the definition of a cross section – the intersection of a plane and a geometric solid.)

4. Have students perform the following activity. First, have them make a cube using modeling clay. Then, have them slice the cube using dental floss or fishing line. Once they are comfortable with “slicing” have them try to make the following cross sections by slicing a cube. A couple of the shapes are impossible to make. Discuss what makes them impossible.

Square Equilateral triangle Rectangle (not a square)Triangle (not equilateral) Pentagon HexagonOctagon Parallelogram (not

rectangle)Circle

5. For classwork or homework have students describe the cross sections they can get from a sphere, cylinder, and cone. To help them in their descriptions, they may want to create each using modeling clay. For more information, see


http://www.learner.org/channel/courses/learningmath/geometry/session9/part_c/index.html

e. Technology

The applets Spinning and Slicing Polyhedra and Cube Challenge on the Navigating Through Geometry: Grades 6-8 cd-rom can be used with these activities. http://mason.gmu.edu/~mmankus/Handson/pbinfo.htm

http://www.richlandone.org/learningcenter/Webquests/IntroGeometry/pointslinesandplanes.html


Students should demonstrate conceptual knowledge of the concepts and procedures of these indicators.



7-4.5 Analyze the congruent and supplementary relationships—specifically, alternate interior, alternate exterior, corresponding, and adjacent—of the angles formed by parallel lines and a transversal. (B4)



Use three thin strips of cardboard to make a physical model of a pair of lines crossed by a transversal. Use brads to connect the strips in a way that allows them to move. Have students to measure the angles and form conjectures. Then, have them to use the brads to change the angles measuring again to test their conjectures.






http://mason.gmu.edu/~mmankus/Handson/pbinfo.htm




While additional learning opportunities are necessary no additional instructional strategies are available at this time.

e. Technology

Graphing CalculatorsMaterials Needed:TI-83+ or higher graphing calculatorsCalculator Application: Cabri Geometry Lab Worksheet

Make sure that each student or team of students has a graphing calculator with the Cabri Geometry application loaded on it. Provide the lab activity worksheet for each student or team of students. Follow the lab activity procedures on the lab worksheet. Discuss the findings as a class.

* This is a modified version of the activity “Angles Formed by Parallel Lines” by Karen Droga Campe found on the Texas Instruments Activity Exchange web site. http://education.ti.com/educationportal/activityexchange

Teacher Tube: http://www.teachertube.com/view_video.php?viewkey=53269bb8882b81b54430


Summative – The following example of a possible assessment strategy may be modified as necessary to meet student/teacher needs. This examples is not derived from nor associated with any standardized testing.

Lines m and n do not intersect. Line p intersects line m to form a right angle. Which of the following is NOT true?

a. Lines p and n are parallel.b. Lines m and n are parallel.c. Lines p and m are perpendicular.d. All three statements are true.

(Source: NCTM, Mathematics Assessment Sampler: Grades 6-8)




http://www.teachertube.com/view_video.php?viewkey=53269bb8882b81b54430

http://www.teachertube.com/view_video.php?viewkey=53269bb8882b81b54430

http://education.ti.com/educationportal/activityexchange

MODULE 3-4

Transformational Geometry



7-4.9 Create tessellations with transformations. (B6)7-4.10 Explain the relationship of the angle measurements among shapes that

tessellate. (B2)7-4.1 Analyze geometric properties and the relationships among the

properties of triangles, congruence, similarity, and transformations to make deductive arguments. (B4)


126



Students studied transformations in fifth and sixth grades. Prior to seventh grade, students have had experience with triangles and transformations. In fifth grade they classified shapes and congruent and in sixth grade they classified shapes as similar.

2. Key Vocabulary

TessellationTransformation

3. Content Overview

Seventh grade students should use transformations to create tessellations, but more importantly should be able to explain the relationship of the angle measurements among shapes that tessellate. This is the students’ first introduction to tessellations. Tessellations (the covering of a plane without overlaps or gaps using congruent figures or a combination of congruent figures) are formed by transformations such as translations, reflections, and rotations (students studied transformations in both the fifth and sixth grades). The emphasis should not be on creating a tessellation. The emphasis should be on the transformations used to create the tessellation and the relationship among the polygons that can be used to tessellate (or tile) a plane (piece of paper).

Discovering the relationship of angle measures among shapes that tessellate will require guidance by the teacher. The objective is for students to discover that in order for regular (all sides are the same length) polygons to tessellate, the sum of the measures of the angles surrounding a point (at a vertex) must be 360°. This sum can be attained by using a combination of different regular polygons or just several of the same polygon. Using regular polygon manipulatives and the following chart may be helpful.

Shape Triangle Square Hexagon OctagonMeasure of one interior angle

60° 90° 120° 135° Combinations that total 360° and therefore will tessellate.

Number 6 60 (6) = 360of 4 1 60 (4) + 120 = 360

each 1 2 90 (1)+ 135 (2) = 360shape 3 120 (3) = 360used 1 2 1 60(1)+90(2) +120(1) = 360

Seventh grade students should build on those experiences by analyzing the geometric properties of congruent and similar triangles. After doing so, students should come to the conclusion that congruent triangles can be matched by


simply placing one on top of the other. On the other hand, similar triangles have congruent corresponding angles and sides that match by a constant scale factor. Student experiences should enable them to come to those conclusions and be able to defend their thinking – not merely memorize the factual relationships. The goal is for students to begin to think more formally about the concepts of congruency and similarity and how transformations might be used when proving relationships. “Investigations into the properties of, and relationships among, similar “triangles” can afford students many opportunities to develop and evaluate conjectures inductively and deductively. For example, an investigation of the perimeters, areas, and side lengths of the similar and congruent triangles. . .could reveal relationships and lead to generalizations. Teachers might encourage students to formulate conjectures about the ratios of the side lengths, of the perimeters, and of the areas of the . . .triangles. (Students) might conjecture that the ratio of the perimeters is the same as the scale factor relating the side lengths and that the ratio of the atreas is the square of that scale factor.” (Principles and Standards for School Mathematics, 2000, page 234-234) All of this could then be linked to the Indicators listed in the following “Proportional Reasoning” module.




7-4.9 Create tessellations with transformations. (B6)

Cognitive Process Dimension: CreateKnowledge Dimension: Conceptual Knowledge

7-4.10 Explain the relationship of the angle measurements among shapes that tessellate. (B2)



Materials: pattern blocks, graph paper, scissors, tape, markersThis activity lays the foundation for understanding how geometric figures tessellate, and easily translates to real-world applications of flooring and tiling found in buildings.Let students arrange a variety of regular polygons on graph paper using lines to help keep form. (Polydrons can also be used as a manipulative, as well as traced figures reproduced and cut from paper.)

Make a chart and record the interior angle degrees of regular polygons (pattern blocks). The object is for students to discover that in order for regular polygons to tessellate the sum of the angles must be 360 degrees (using a combination of different regular polygons or just several of the same regular polygon). When an arrangement of polygons are placed side by side to cover a


surface without overlaps or gaps, creating a convergence of angles meeting at one vertex that equals 360 degrees, a tessellation will occur. Use this information to help create a tessellation.

(Examples)Triangle Rectangl

eHexagon Octagon

60 degrees

90 degrees

120 degrees

135 degrees

4 1 60 + 60 + 60 + 60 + 120 = 360

6 60 x 6 = 3601 2 90 + 135 + 135 = 360

When a combination is discovered that will tessellate, the student should sketch or trace it on graph paper. Next, record the angle measures that created the tessellation. Have students create their own portfolio of ‘discovered’ patterns of regular polygons that tessellate. Move about the class asking if students notice anything about the polygons that will tessellate compared to the ones that will not.




NCTM, Navigating through Geometry Grades 6-8: pp. 46-47NCTM, Navigating through Geometry Grades 6-8: pp. 68-70, 113-114

e. Technology-

No technology resources noted at this time to address the intent of this indicator.


Summative – The following examples of possible assessment strategies may be modified as necessary to meet student/teacher needs. These examples are not derived from nor associated with any standardized testing.

Which of the following two shapes can be combined to form a tessellation?

Pentagon Hexagon


Octagon Square

e. the pentagon and hexagonf. the octagon and the squareg. the pentagon and the octagonh. the hexagon and the square

(Source: NCTM, Mathematics Assessment Sampler: Grades 6-8)






Give students triangles that are congruent and have then compare the side lengths and angles. Have them to discuss and come up with the properties of


congruent triangles and similar triangles. Make sure to conclude the discussion with a teacher led demonstration/discussion of SAS, SSS, ASA, AAS HL. Also make sure students understand that the corresponding angles are congruent in similar triangles. (proportional sides will be addressed in the next unit)- ask students which transformations you could use to prove that 2 triangles are congruent or similar.-use different triangles to perform transformations (translations, reflections, and rotations). Discuss what happens to the angles and sides when the transformation occurs. Have students form conjectures (such as the angle measure will always stay the same) and test them.




While additional learning opportunities are necessary no additional instructional strategies are available at this time.

e. Technology

http://standards.nctm.org/document/eexamples/chap6/6.4/index.htm


Assessment opportunities should allow for analyzing at the conceptual level.



MODULE South Carolina Curriculum ProjectSouth Carolina Curriculum Project DRAFTDRAFT 12-15-200812-15-2008 131


3-5Proportional Reasoning

b. Background for the Module



In sixth grade, the concept of similarity was introduced and students compared the angles, side lengths, and perimeters of similar shapes and also







132

classified shapes as similar. Fifth grade is when the concept of congruency was introduced and students compared the angles, side lengths, and perimeters of congruent shapes and also classified shapes as congruent.

2. Key Vocabulary

Similar shapesCongruent shapes

3. Content Overview

Chapter 19 in Elementary and Middle School Mathematics: Teaching Developmentally (Sixth Edition) by John A. Van de Walle provides valuable information about proportional reasoning for teachers.

In seventh grade, students will now compare the areas of similar shapes and the areas of congruent shapes. Students should be given the opportunity to discover that the areas of congruent shapes are equal whereas the areas of similar shapes are not. This indicator may be extended to allow students to discover and understand that the area of the larger of two similar shapes will be the area of the smaller shape multiplied by the square of the scale factor needed to create the larger similar shape. This may be difficult for students to initially conclude; therefore numerous examples should be done to help students see this relationship. It is sound educational practice to start with Indicator 7-4.1 explained under the “Plane and Transformational - Transformational” section above. Once students are comfortable making deductive arguments with triangles, they are ready to move on to a variety of congruent or similar shapes.

A natural link can be made to the major concept of ratio and proportion when students begin to compare attributes of similar and/or congruent figures such as ratios of angles to corresponding angles. Seventh grade students should explore how similar shapes compare proportionally and be able to explain the proportional relationship among attributes of similar shapes. For example, corresponding angle measures are congruent while corresponding sides are proportional.

Seventh grade students should extend their understanding of ratio and proportion as they explore similarity at a more in-depth level. This understanding will provide students the prerequisites needed to find missing attributes of similar shapes. Once again, it is extremely important that students have made the connections in regards to similar figures that corresponding angle measures are congruent while corresponding sides are proportional.

Seventh grade students must have an in-depth understanding of when and how to apply proportional reasoning to solve various types of problems. Teachers should provide problems that encourage students to use these concepts. Students should be encouraged to not only find solutions to these problems, but also justify their solutions and explain the method they chose to derive their solution.










Cognitive Process Dimension: Apply Knowledge Dimension: Procedural Knowledge


Have students find the areas of shapes that are congruent. Keep a table of the areas and discuss the results. Have students find the areas of shapes that are similar. Keep a table of the areas and discuss the results.




Re-teach: Have students to build shapes using pattern blocks. Have them to build a similar shape using the same type of pattern blocks and then find the proportional relationship between the number of pattern blocks used (or the area). Have students to draw similar shapes on graph paper and find the areas. Have them to find the proportional relationships between the areas. Have students use different size manipulatives such as straws, or tiles to build figures. Measure sides and predict the measures of other sides using proportions before measuring

e. Technology

http://standards.nctm.org/document/eexamples/chap6/6.3/index.htmhttp://findarticles.com/p/articles/mi_m0STR/is_n8_v107/ai_20322829


http://findarticles.com/p/articles/mi_m0STR/is_n8_v107/ai_20322829



Assessments for this lesson should allow students to demonstrate they can apply their conceptual and procedural knowledge.




Recommended Days of Instruction Fourth Nine WeeksFourth Nine Weeks

Standard/Indicators Addressed:Standard 7-5: The student will demonstrate through the mathematical processes an understanding of how to use ratio and

proportion to solve problems involving scale factors and rates and how to use one-step unit analysis to convert between and within the U.S. Customary System and the metric system.

7-5.2 Apply strategies and formulas to determine the surface area and volume of the three-dimensional shapes prism, pyramid, and cylinder. (C3)

7-5.3 Generate strategies to determine the perimeters and areas of trapezoids. (B6)

Standard 7-6: Through the process standards students will demonstrate an understanding of relationships between two populations or samples through data analysis and probability.

7-6.1 Predict the characteristics of two populations based on the analysis of sample data. (B2)7-6.2 Organize data in box plots or circle graphs as appropriate. (B4)7-6.3 Apply procedures to calculate the interquartile range. (C3)7-6.4 Interpret the interquartile range for data. (B2)7-6.5 Apply procedures to calculate the probability of mutually exclusive simple or compound events. (C3)7-6.6 Interpret the probability of mutually exclusive simple or compound events. (B2)7-6.7 Differentiate between experimental and theoretical probability of the same event. (A4)7-6.8 Use the fundamental counting principle to determine the number of possible outcomes for a multistage event. (C3)7-6.1 Predict the characteristics of two populations based on the analysis of sample data. (B2)

These indicators are covered in the following 3 Modules for this Nine Weeks Period.


Module 4-1 Perimeter, Area, and VolumeIndicator Recommended Resources Suggested Instructional







NCTM’s Principals and Standards for School Mathematics (PSSM)Textbook Correlations - See Appendix A






NCTM's Online Illuminations http://illuminations.nctm.org/ NCTM's Navigations SeriesSC Mathematics Support DocumentTeaching Student-Centered Mathematics Grades 5-8 and Teaching Elementary and Middle School Mathematics Developmentally 6th Edition, John Van de WalleNCTM’s Principals and Standards for School Mathematics (PSSM)

Textbook Correlations -See Appendix A




Module 4-2 Data Collection, Representation, and Analysis







7-6.2 Organize data in box plots or circle graphs as appropriate. (B4)

7-6.3 Apply procedures to calculate the interquartile range. (C3)

7-6.4 Interpret the interquartile range for data. (B2)





Textbook Correlations - See Appendix A





7-6.1 Predict the characteristics of two populations based on the analysis of sample data. (B2)



See Instructional Planning Guide Module 4-2Lesson B Assessment



Module 4-3 ProbabilityIndicator Recommended Resources Suggested Instructional



7-6.5 Apply procedures to calculate the probability of mutually exclusive simple or compound events. (C3)

7-6.6 Interpret the probability of mutually exclusive simple or compound events. (B2)

7-6.7 Differentiate between experimental and theoretical probability of the same event. (A4)










7-6.8 Use the fundamental counting principle to determine the number of possible outcomes for a multistage event. (C3)






MODULE4-1

Perimeter, Surface Area, and Volume






140



Fifth grade students applied strategies and formulas to determine the volume of rectangular prisms and worked with nets – the two-dimensional representations of both rectangular prisms and cylinders. Sixth grade students generated strategies to determine the surface area of a rectangular prism and a cylinder.

Fifth grade students applied formulas to determine the perimeters and areas of triangles, rectangles, and parallelograms. Sixth grade students focused on the perimeter and area of irregular shapes. In fourth grade one of the quadrilaterals students analyzed was trapezoid. As a result they should have familiarity with that geometric shape. However, a quick review of the characteristics of trapezoids may be necessary prior to work on perimeter and area, especially the concept of “height” of a trapezoid.

2. Key Vocabulary

Surface areaVolumeThree dimensionalPrismPyramidCylinderPerimeterTrapezoid

3. Content Overview

Seventh grade students should apply formulas to determine the surface area and volume of prisms and cylinders. Now for the first time they should apply strategies and formulas to determine the surface area and volume of pyramids. That means seventh grade students should be fluent applying formulas to find the surface area and volume of prisms, pyramids, and cylinders.

Seventh grade students should use that knowledge to generate strategies to determine the perimeters and areas of trapezoids.

When generating strategies to determine the perimeters and areas of trapezoids, students should analyze and discuss the formula for finding the area of a rectangle, A = l x w. When area formula is changed into an equivalent form of A = b x h, this form can be useful in developing the area formula for trapezoids. One strategy to determine area of a trapezoid is to have students cut out two identical trapezoids, put them together to form a parallelogram, and relate the area of the parallelogram formed to the area of the trapezoid.

base 2


base 1 base = base 1 + base 2A = height x (base 1 + base 2)

Two trapezoids make a parallelogram with the same height and a base equal to the sum of bases of the trapezoid. So the formula for are of trapezoid is:

A = ½ h(base 1 + base 2)

Another strategy is to use the area formulas for a rectangle and a triangle to see why the formula for a trapezoid works.

Students should spend time investigating problem situations involving perimeters of trapezoids and be given the opportunity to discover the formulas for themselves using concrete materials and computer models. The Pythagorean Theorem is not introduced until the eighth grade. Therefore, when finding height of a trapezoid other strategies should be used.

As stated in sixth grade; “memorizing” measurement formulas becomes unnecessary when the mathematics makes sense to students and they understand the concepts.

Measurement is closely tied to many topics in geometry and algebra and should not be taught in isolation.





Cognitive Process Dimension: CreateKnowledge Dimension: Conceptual Knowledge


Have students cut out two identical trapezoids, put them together to form a parallelogram, and relate the area of the parallelogram formed to the area of the trapezoid.


A common error made by students when using formulas comes from no conceptual understanding of the meaning of height in geometric figures. Before using formulas involving height, students should discuss the meaning of height of a geometric figure and be able to identify where a height could be measured.



Additional instructional strategies may be needed, but there are none available at this time.

e. Technology

No technology resources available at this time.


Generating strategies means that the concept is being introduced for first time. Instructional and assessment activities should foster conceptual understanding with concrete and pictorial models only.






Construct rectangles with unifix or multi-link cubes. Calculate area. Construct rectangular prisms using these rectangles as bases. Calculate volume. Students develop relationship between area of base and volume of prisms.




Additional instructional strategies may be necessary but none are available at this time.

e. Technology



As the indicator indicates, apply strategies and formulas means the concept is introduced for first time. The goal must be to progress to fluency

III. Assessing the ModuleSouth Carolina Curriculum ProjectSouth Carolina Curriculum Project DRAFTDRAFT 12-15-200812-15-2008 143


MODULE 4-2

Data Collection, Representation, and Analysis




7-6.2 Organize data in box plots or circle graphs as appropriate. (B4)7-6.3 Apply procedures to calculate the interquartile range. (C3)7-6.4 Interpret the interquartile range for data. (B2)


145



Since kindergarten, students have been collecting and representing data. The representations used include:

Kindergarten drawings and picturesFirst Grade picture, object, bar graphs and tablesSecond Grade charts, pictographs, and tablesThird Grade tables, bar graphs, and dot plotsFourth Grade tables, line graphs, bar graphs, and double

bar graphs (scale increments greater than or equal to one)

Sixth grade frequency tables, histograms, and stem-and-leaf plots

Third grade students were asked to find the range of a data set as well as analyze dot plots and bar graphs to make predictions about populations. Fourth grade students interpreted data in tables, line graphs, bar graphs, and double bar graphs with scale increments greater than or equal to 1. Fifth grade students applied procedures to calculate to calculate the measures of central tendency as well as interpreted the meaning and applications of the measures. Sixth grade students predicted the characteristics of one population based on the analysis of sample data and analyzed which measure of central tendency (mean, median, or mode) was the most appropriate for a given purpose.

2. Key Vocabulary

Box-and-whisker plot (box plot) Upper QuartileUpper Extreme Lower QuartileLower Extreme Inter-quartile Range

3. Content Overview

In seventh grade, students should learn to use box-and-whisker plots as well as circle graphs to organize data. Students extend their predictions of characteristics of one population to two populations. Emphasis should be placed on linearity and proportionality.

Students should plan and design experiments to collect and compare relevant two population data. Students should be expected to gather reliable information with well-written questions. Experiments should be used to make inferences and predictions. For example, based on data regarding sneaker prices at three different stores, “What would you expect to pay for a pair of sneakers?” Box plots are useful when making comparisons between populations.

Students should describe observed relationships mathematically and discuss whether the conjectures that they drew from the sample data might apply to a larger population. In deciding whether two data sets are similar or different consists essentially in deciding whether the distributions of the data sets are similar or different. Students need to understand that differences in the variability, or spread,


in two data sets are an important part of deciding whether the data sets are similar or different. Students should describe the variability by characterizing the shape(s) of the distributions: bell curve, u-shape distribution, rectangle-shaped (flat or uniform), and J-shaped or backward J-shaped.

In the middle grades, not only is the range a measure of the spread in a data set, but the use of quartile is another measure. Quartiles are the three values that divide an ordered set of data into four equal-sized subsets. Of the data, 25% fall between two successive quartiles. The number of data elements between successive quartiles depends on the size of the data set, but the percent of the data between the successive quartiles is always 25%.

Comparing quartiles involves comparing concentrations of data in two data sets. This comparison is an example of multiplicative reasoning (i.e., ratios of numbers of elements).

To find the interquartile range, students need to find the difference between the upper and lower quartiles (third and first or Q3 – Q1). Half of the data is in that range. The “box” of the box plot denotes the middle 50% of the data and the difference between the two ends of the box is the interquartile range.

Students need to understand that data analysis is a sophisticated process.






7-6.3 Apply procedures to calculate the interquartile range. (C3)


7-6.4 Interpret the interquartile range for data. (B2)



Create a human box-and-whisker plot by lining up from shortest to tallest. Beginning at the extremes, pair off until the midpoint (median) is reached. Repeat this process to find the upper and lower quartiles.



Be certain that students understand that the quartile refers to ¼ of the number of items and not ¼ of the range. The indicators focus on analyzing and interpreting data, not just constructing a box plot and memorizing vocabulary.


Practice—Teacher Ages Lesson

Materials:

Centimeter Grid PaperData set: For example, some ages of teachers such as:24,62,30,40,31,32,33,36,37,38,39,40,46,48,55,57,37,31,26,32,54,55,35,25,41, 35,60Transparency of Number line on gridBlank Transparencies

Any set of data the students collect can be used for this exercise. Test grades for each class or comparing one class to another (no one has to know how each student scored, simply the list of test scores) would be of interest to students. The answers to the questions for this lesson are based on the teacher’s age data set above. NOTE: Box plots can be drawn vertically or horizontally.

Tell students that today they are going to learn about a new way to represent data – a box-and-whisker plot; some people call it a box plot for short. A box plot is a visual way to easily see the median and range of a data set.

Ask students what they think about when they hear the terms box and whiskers? Allow a student to come to the overhead/board and draw what comes to mind. (It might be a silly box with whiskers but that is a visual to help students begin to make a connection to the formal name.) Leave that up and tell students you’ll come back to that drawing at the end of today’s lesson.

Show the teacher’s ages data set. Either gather real data or make up a story about the data set.

First, have students make a number line on grid paper long enough to record teacher ages (the grids ensure even intervals).

Since students have had previous experience with stem-and-leaf plots, ask them to display the data in that format on a separate sheet of notebook paper or in their journal. (A QUICK review may be necessary.)

Stem-and-Leaf Plot Answer:Teachers’ Ages

2 4, 5, 63 0, 1, 1, 2, 2, 3, 5, 5, 6, 7, 7, 8, 94 0, 0, 1, 6, 8, 5 4, 5, 5, 7 6 0, 2


Tell the students to find the median three times on the data set - first the median of all the data and second the median of each “half” above and below the central median, to mark the medians on the data set and on the number line. (The median of all the data is 37; the median of the lower half is 32 and the median of the upper half is 48 – NOTE the use of upper half and lower half refer to the position of the data in numerical order NOT the upper part of the stem-and-leaf plot.) Allow time for students to mark and record the medians. Allow students to compare with a partner.

Next have students make two rectangles “the box” that encloses all three medians. (See sample below. Box is around 32, 37, and 48.) Ask students to talk with a partner about how many parts the number line has now been divided into. (Four parts/quartiles.) Allow students to share their reasoning and demonstrate on the overhead. (Cover your teacher made grid transparency with a clear transparency so several students can explain and no time is wasted cleaning your original.)

Explain to students that the data is now divided into quartiles. Ask what the word quartiles makes them think of mathematically. (quarters, fourths, etc.) Say, “Since you know the data is divided into quarters, what can you tell me about the data contained in the box?” (The box contains the middle half of the data.)

Point to the lower extreme (24) and tell students that is what it is called. Label your box plot and tell students to do the same. Ask students, “Why is it called the lower extreme when it is at the upper part of the stem and leaf plot?” (Because lower extreme refers to the data set values not the location.) Do the same for the upper extreme (62).

24

32

37

48

62

Point to 32 and tell students that is called the “first quartile” or “lower quartile”. Label your graph and have students do the same. Point to 37 and remind students that is the median of the set of data. Tell them it is sometimes referred to as the “second quartile”. Label your graph with both “median” and “second quartile” and tell students to do the same. Point to 48 and ask students what do they think it is called. (Third quartile or Upper quartile) Label your graph and tell students to do the same. To help students remember the vocabulary, ask them to look at the labels and discuss with a partner how the labels are related. (Lower, Median, Upper OR First, Median, Third) Allow students to share their thinking.

Refer students to the drawing on the board (when a student drew their interpretation of box and whiskers). Ask students to look at their box plot and talk with a partner about what is missing (whiskers) and where they think the missing pieces should go? (Extend from the box to the extremes.) Say, “We said the data is divided into quarters/quartiles. How are the whiskers related to that?” (The whiskers


cover the upper and lower fourths of the graph.) (Common Student error – Students may think that the longer the half of the box and/or the longer the whiskers, the more data points that are included. Each half of the box and each whisker represent one-fourth of the data, no matter how the lengths compare. The length of the box and/or whiskers is an indicator of the spread of the data, not the quantity of data points.)

Ask students for the range of the data. (60-24 = 36)

Say, “While the whole data set has a range, it is sometimes useful to know the range of the data within the “box”. This is the last feature to our graph. What is the range of the data within the box? (48-32 = 16) This is called the inter-quartile range.” Label your graph and ask students to do the same.

Ask students, “What does 16 represent?” (half of the teachers’ ages fall within a 16 year range)

Practice: Navigating Through Data Analysis in Grades 6-8, pp. 53-55 (teacher information), 63-65, 97-99

e. Technology

http://www.prometheanplanet.com/server.php?show=ConResource.10502http://illuminations.nctm.org/ActivityDetail.aspx?ID=77



The box and whisker plot represents the quiz scores of students.

Which of the statements could be true of the test scores earned by the class?

a. Three-fourths of the class had a test score lower than 12 points.

b. Half of the class scored between 8.5 and 12.

c. Three fourths of the class had a test score of 8.5 or above.

d. Half of the class scored between 14 and 20.


http://illuminations.nctm.org/ActivityDetail.aspx?ID=77


(Source: NCTM Assessment Sampler)






Edible Circles Activity

Materials:

Bag of M&M’s or Skittles (or any candy with multiple color pieces)CompassStraight-edge

Have the students sort all the M&M’s in the bag by color. Create a circle using all the M&M’s from the bag. Use the compass to draw a circle the same size as the one created using the M&M’s. On the circle, make a tick mark to indicate the separation of colors. Draw a radius between each new color of M&M to the center of the circle. To determine the percentage of each color of M&M, have the students divide the number of M&M’s of a specific color by the total number of M&M’s in their bag. Ask the students, “What does the percentage mean about your M&M’s? What is the relationship between the number of M&M’s and the percentages?” Also ask, “How can we determine a good estimate of the number of each color in a regular bag of M&M’s?”


Students should have practice using a protractor and compass to create circles and measure angles. The knowledge of equivalent ratios is crucial.


No additional instruction strategies are available at this time.

e. Technology

Create a Graph—http://nces.ed.gov/nceskids/createagraph/default.aspxhttp://illuminations.nctm.org/ActivityDetail.aspx?ID=60http://illuminations.nctm.org/LessonDetail.aspx?id=L737



http://illuminations.nctm.org/LessonDetail.aspx?id=L737

http://illuminations.nctm.org/ActivityDetail.aspx?ID=60

http://nces.ed.gov/nceskids/createagraph/default.aspx


Sue found this data in an almanac.

Crop Area planted (in acres)Wheat 55,000Corn 32,500Potatoes 30,000Lettuce 40,000Carrots 22,500Total 180,000

She needs to use it to make a circle graph. What is the best estimate of the number of degrees that should be in the central angle of the section that represents wheat?

60º

65º

80º

110º

(Source: Released PACT items)






Once students are comfortable using box plots, it would be good practice to compare two data sets. For example, the above introductory lesson could be saved, teacher ages from another school gathered, a stem-and-leaf plot and a box plot for the new data created, and the data set for two populations compared.





PracticeStudents collect, analyze, and interpret data involving males and females, such as back-pack weight. Students weigh their back-packs and investigate whether the boys or girls carry the most weight.

Students poll people in their neighborhoods for information. The students predict how their peers will poll on the same information based on their data.

e. Technology

Technology resources for this indicator not available at this time


Assessment opportunities should focus on demonstrating an understanding of conceptual knowledge.



MODULE 4-3

Probability








153

\

c. Background for the Module



Beginning with grade two, students were asked to predict on the basis of data whether events are more likely or less likely to occur.

In grades three through five, students learned how to quantify the likelihood of single-stage events as likely, unlikely, certain, impossible, or equally likely. Third grade students understood when the probability of an event would be 0 or 1. Fourth grade students analyzed possible outcomes for a simple event.

Sixth grade students used theoretical probability to determine the sample space and probability for one-/and two-stage events such as tree diagrams, models, lists, charts, and pictures and applied procedures to calculate the probability of complementary events.

Since sixth students calculated the probability of complementary events, they were introduced to events that were mutually exclusive.

2. Key Vocabulary

Experimental ProbabilityTheoretical Probability


Mutually ExclusiveIndependent EventCompound EventFundamental Counting Principle

3. Content Overview

In seventh grade students will use prior knowledge to calculate the probability of mutually exclusive simple or compound events. Mutually exclusive events can not happen at the same time (the events have no common elements in the sample space). For example, when rolling 2 dice, getting an even sum and getting an odd sum are mutually exclusive. When two outcomes are mutually exclusive, student can use this formula to find the probability; Probability (A or B) = Probability (A) + Probability (B). Students should be given ample opportunities to calculate and interpret the probability of mutually exclusive simple or compound (combination of at least two simple events) events. Students should be able to decide whether or not given events are mutually exclusive and describe how they know.

When finding a probability by tossing a coin 100 times or throwing free throws on the basketball court 50 times are examples of finding an experimental probability (simulation). It is considered experimental because the probability is based on the results of an experiment rather than theoretical analysis. The experimental probability of an event is the ratio of number of observed occurrences of the event to the total number of trials:


Experimental Probability = Number of favorable outcomes Number of Trials

The more trials the more confident one is that the experimental (simulation) probability is close to the actual probability. There are advantages to an experimental approach:

more conceptual and intuitive; eliminates guessing at probability and wondering if it is right; provides experiential background for examining theoretical model; shows how the ratio of a particular outcome to the total number of trials begins

to converge to get closer and closer to a fixed number; develops an appreciation for a simulation approach to solving problems; and it is a lot more fun and interesting.

An experimental (simulation) approach should be used in the classroom whenever possible. (Elementary and middle School Mathematics, Teaching Developmentally 3rd Edition, John A. Van de Walle, p410)

(Adapted from Navigating Through Probability, Grades 6 – 8, NCTM, p.73 – 76)

Students need to know that a simulation is a procedure for answering questions about a real problem by conducting an experiment that closely resembles the real situation. As students think about and discuss the real problem and the factors that make the real problem more complex, remind them a simulation is an approximation of the real problem and that with a simulation some of the factors are eliminated such as any possible danger, complexity of the problem, or length of time necessary to solve the problem.

Have students examine when a simulation’s results appear to be the closest to theoretical probability. (Results of a simulation are more precise when it is run repeatedly. Remember that probability deals more with long term trends than with outcomes of individual events.)

When designing a simulation, students need to keep the following 5 tasks in mind:1. Identify the essential components and assumptions of the problems.

Problem stated clearly.2. Select a random device for the essential components.

Device must be able to generate chance outcomes with probabilities that match those of the problem.

3. Define the trial.One trial must be clearly specified. A trial may require several flips, spins, or draws for example.

4. Conduct a large number of trials and record the information.There is no magic number of trials. Rule of thumb, 25 to 30 trials are appropriate for most of the problems for middle school students.

5. Use the data to draw conclusions.Data must be summarized in some meaningful way such as computing an average, making a graph, or noting trends in data.

A simulation is a model of a problem that occurs in real life. The model is designed so that it reflects the probabilities of the real situation.


Students use simulations to conduct experiments whose results represent experimental probability.

Discuss how results of simulations can be deceptive. (Not enough trials, very few attempts, etc.)

Theoretical probability is the ratio of the number of favorable outcomes to the total number of possible outcomes. Theoretical probability is based on a logical analysis of the experiment. For example, the theoretical probability of a coin landing on tails is ½.

P(heads) = number of sides with heads = ½ number of sides

To use theoretical probability to predict how many times a coin would land on heads if tossed 30 times:

Multiply its probability by the number of attempts.

P(heads) x number of tosses ½ x 30 = 15

You would toss heads 15 times out of 30.

Students should discuss the differences between the probability of an event found through experimental/simulation and the theoretical probability of that same event

Theoretical Probability = Number of favorable outcomes Number of possible outcomes

Experimental Probability = Number of favorable outcomesNumber of Trials

Students should understand that when all outcomes of an experiment are equally likely, the theoretical probability of an event is the fraction of the outcomes in which the events occurs Students should use theoretical probability and proportions to make approximate predictions.

The Fundamental Counting Principle states that if a first event can occur in a ways, and a second event can occur in b ways, then the two events can occur together in a times b ways. For example the school shirt comes in three colors and in short or long sleeves. How many choices of shirts are there? So, 3 x 2 = 6 choices.

In the middle grades, the groundwork is laid to help students know when events are or are not mutually exclusive. This is important for the study of more complex situations in grades 9 – 12.










Cognitive Process Dimension: AnalyzeKnowledge Dimension: Factual Knowledge


Use “Adjustable Spinner” activity (instructions can be found on the NCTM Illuminations website http://illuminations.nctm.org/). This interactive tool allows students to create a spinner and examine the theoretical and experimental outcomes for a specified number of spins.


When a probability experiment has very few attempts or outcomes, the result can be deceptive. Computer simulations may help students avoid or overcome erroneous probabilistic thinking. Simulations afford students access to relatively large samples that can be generated quickly and modified easily.” (NCTM 2000, p254) Using large samples, the distribution is more likely to be close to the actual distribution. When simulations are used, you will need to help students understand what the simulation data represent and how they relate to the problem situation.


There are no additional instructional strategies available at this time.

e. Technology







A basketball player shoots foul shots with a two-thirds accuracy record. He is given a free throw from the foul line and is given a second shot only if he scores a basket on the first attempt. In this one on one situation, he can score 0, 1, or 2 points. Determine the approximate probability that he will score 2 points. Show your work.

(Source: Mathematics Assessment Sampler: Grades 6-8)






http://illuminations.nctm.org/LessonDetail.aspx?ID=U7

Students are encouraged to discover all the combinations for the given situation using problem-solving skills (including elimination and collection of organized data). This unit was adapted from "Ideas: Combinations" by Marcy Cook, which appeared in The Arithmetic Teacher Vol. 36, No. 1 (September, 1988) pp. 31 - 36.


There are no misconceptions or common errors noted at this time.


There are no additional instructional strategies available at this time.

e. Technology


http://www.teachertube.com/view_video.php?viewkey=eb4b91a6c3e5bf35695e



http://www.teachertube.com/view_video.php?viewkey=eb4b91a6c3e5bf35695e


http://my.nctm.org/eresources/journal_home.asp?journal_id=4

http://illuminations.nctm.org/LessonDetail.aspx?ID=U7

Assessment should focus on students applying their procedural knowledge of the Fundamental Counting Principle.




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