Catherine Shelfer

Ratio, Proportion, and Percent

A Sixth Grade Mathematics Unit Plan

Created by Catherine Shelfer

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Catherine ShelferTable of Contents

I. Introduction and General Information 2

II. Depth of Content Knowledge and Unit Content 5Content Standards 5Unit Summary 5Unit Plan Outline 6

III. Subject Matter Content 7

IV. Unit Goals 8Enduring Understandings 8Essential Questions 8Students Will Know… 8Students Will Be Able To… 9

V. Content 10Unit Summary 10Graphic Organizer 10Unit Planning Calendar 11

VI. Acceptable Evidence 12Examples of Evidence 12Methods of Assessment 12Grading Outline 13

VII. Lesson Plans 14Day 1—Ratio and Rates 14Day 2—Proportions 16Day 3—Proportions and Measurement 18Day 4—Similar Figures and Indirect Measures 19Day 5—Mid-Unit Quiz and Scale Drawings and Maps 21Days 6—Percents, Decimals, and Fractions 22Day 7—Using Percents 24Day 8—Unit Review 26Day 9—Unit Test 26

VIII. Appendix 27

IX. Summative Reflection 45

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Catherine ShelferIntroduction

The following unit plan is designed for a sixth grade mathematics class. The unit will take place over

nine days and it will cover ratio, proportion, and percent.  Specifically, we will look at:  ratios and rates;

proportions and customary measurements; similar figures; indirect measurements; scale drawings and maps;

percent, decimals, and fractions; and percent applications.  Students need to learn about ratio, proportion, and

rate because these topics are useful in everyday life, for example, computing test scores or lottery odds, tipping

in a restaurant, converting measurements, reading maps, and comparing statistics. Also, ratio, proportion, and

percent are a basic concepts in understanding more advanced mathematics principles and applications.

The unit is designed to reach all learners. The lessons are paced and "scaffolded" so that students of any

level can master the skills, and then apply the skill with critical thinking.  There are many places for students to

ask questions and questions are encouraged. In addition, there are multiple opportunities to reach all learners

through journals, examples, models, group activities, pair projects, graphic organizers, etc.   Throughout the

unit, a variety of instructional strategies are used to accommodate various learning styles or special needs. 

The curriculum to be covered in this unit is in-line with the North Carolina Standard Course of Study for

6th grade mathematics. The first competency goal the unit will cover outlines the understanding and

computation of rational numbers. Students will develop meaning for percents which involves connecting the

variety of representations as well as making estimates in appropriate situations. In addition, students will

compare and order rational numbers and should develop fluency in addition, subtraction, multiplication and

division of nonnegative rational numbers. Students will also estimate the results of computations and be able to

judge the reasonableness of solutions. Finally, the students will develop flexibility in solving problems by

selecting strategies and using mental computation, estimation, calculators or computer, and paper and pencil.

The second goal the unit covers will teach students to demonstrate an understanding of simple algebraic

expressions. Students will use graphs, tables and symbols to model and solve problems involving rate and

ratios.

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Catherine ShelferThe goal of the unit is to provide students with the lessons needed for a full understanding of percents.

The students will be able to write ratios and rates, find unit rates, write and solve proportions, use ratios to

identify similar figures and use proportion and similar figures to find unknown measures. Students will also be

able to read and use map scales and scale drawing. They will learn to write percents as decimals and fractions

and write decimals and fractions as percents. Students will be able to apply their knowledge of percents, ratios

and proportions to solve a variety of complex problems by selecting the appropriate strategy. They will also be

able to relate ratio, proportion, and percent to previous topics and to future problems. Finally, the unit will teach

students to compare, analyze, and interpret data using ratios, proportion and percent, create related charts and

graphs, and use their knowledge to identify situations in the real world or classroom where ratio, proportion or

percent is used to solve or simplify.

After the implementation of this unit plan students will have clear knowledge of ratio, proportion, and

percent and understand that they are a means of simplifying and organizing for comparison. They will know

that percent is a ratio compared to 100, that a ratio is a comparison of two quantities using division, a proportion

is an equation that shows two equivalent ratios, and that proportions make conversions in measurements and

identify similar figures. Lastly, students will have knowledge that proportions and similar figures are used to

find unknown measures of items we cannot measure directly and acknowledge that a scale drawing is of a real

object that is proportionally smaller or larger than the real object.

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Catherine ShelferStep 1: General Information

Unit Title: Proportional Reasoning Unit Topic: Ratio, Proportion, and Percent Course Content: Mathematics Grade Level: 6 Length of Class Time: 85 minutes Length of Unit: 9 Days

Student Population, Characteristics, and Accommodations: Based on Clinical Observation, Community House Middle School, 6th grade population

In the two classes that I observed, there are three LEP students who work with a specialist outside of class and have access to additional notes, if needed. However, they are expected to take their own notes and participate. There are three students with a 504 plan. Two students have ADD and one has Dyslexia. All of the plans allow for extra time on tests, preferential seating, and access to notes.

Average Class Size: 28 students RACE % ENROLLED

African American 11.5

White 66.7

Asian 9

Hispanic 10

American Indian less than 1

Multi-Racial 2.7

The unit will be taught to a sixth grade mathematics class of mixed academic levels. The unit contains seven different lessons regarding ratio, proportion, and percent, and will take nine days to complete.

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Catherine ShelferStep 2: Demonstrating Depth of Content Knowledge and Defining Unit Content

Link to Content Standards:

North Carolina Standard Course of Study

Competency Goal 1 : The learner will understand and compute with rational numbers.

1.02 Develop meaning for percents. a) Connect the model, number word, and number using a variety of representations. b) Make estimates in appropriate situations. 1.03 Compare and order rational numbers. 1.04 Develop fluency in addition, subtraction, multiplication, and division of nonnegative rational numbers.

a) Estimate the results of computations. b) Judge the reasonableness of solutions.

1.07 Develop flexibility in solving problems by selecting strategies and using mental computation, estimation, calculators or computers, and paper and pencil.

Competency Goal 5 : The learner will demonstrate an understanding of simple algebraic expressions.

5.04 Use graphs, tables, and symbols to model and solve problems involving rates of change and ratios.

Summary of Unit: The unit will cover ratio, proportion, and percent. Specifically, we will look at: ratios and rates; proportions and customary measurements; similar figures; indirect measurements; scale drawings and maps; percent, decimals, and fractions; and percent applications.

Students will be able to…1. Write ratios and rates and find unit rates2. Write and solve proportions3. Use ratios to identify similar figures4. Use proportion and similar figures to find unknown measures5. Read and use map scales and scale drawings6. Write percents as decimals and fractions7. Write decimals and fractions as percents8. Find the missing value in a percent problem9. Apply their knowledge of percents, ratios, and proportions to solve a variety complex problems by

selecting appropriate strategies10. Estimate, when appropriate, using ratio, proportion, and percent11. Relate ratio, proportion, and percent to previous topics and to future problems12. Compare, analyze, and interpret data using ratio, proportion, and percent13. Identify situations in the real world or class room where ratio, proportion, or percent is used to solve

or simplify14. Read and create graphs that involve ratio, proportion, and percent

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Catherine ShelferRequired Prior Knowledge:

1. Division/multiplication2. Number line3. Decimals, rounding decimals4. Fractions, reducing fractions5. Base 10 number system

Important Subject Matter Elements1. Percent means “out of 100”2. Fractions, decimals, and percents can express the same number in different forms3. A ratio is a comparison of two numbers; a percent is a number compared to 1004. A proportion makes two ratios equal5. Charts, graphs, and diagrams display data 6. Similar figures have the same shape but not necessarily the same size7. A scale drawing is of a real object that is proportionally smaller or larger than the real object

Unit Plan OutlineDay 1—Ratio and RatesStudents will write ratios and rates and find unit rates

Day 2—ProportionsStudents write and solve proportions using ratios

Day 3—Proportions and MeasurementStudents use ratios and proportions to find measurements

Day 4—Similar Figures and Indirect MeasuresStudents use ratios to identify similar figuresStudents use proportions and similar figures to find unknown measures

Day 5—Short Mid-Unit Quiz and Scale Drawings and Maps8 question quiz from days 1 to 4Students read about and use map scales and scale drawings

Days 6—Percents, Decimals, and FractionsStudents use ratio and proportion to write percent as decimals and fractions Students use ratio and proportion to write decimals and fractions as percent

Day 7—Using Percents Students find missing values in real-world percents problems

Day 8—Unit ReviewReview study guide and answer questions about unit

Day 9—Unit Test

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Catherine ShelferStep 3: Subject Matter Content

Textbook: Middle School Math: Course 1 Worksheets from Middle School Math: Course 1, Resource Book Graphic Organizers and Charts Activity from www.LearnNC.org Activity from National Council of Teachers of Mathematics website http://illuminations.nctm.org Teacher resources include worksheets, activities, models, word problems, and diagrams. PowerPoint and Excel

Bibliography

Holt Middle School Math Course 1, Chapter 8, Resource Book. New York: Holt Rinehart and Winston, 2002.

Illuminations: Welcome to Illuminations. The National Council of Teachers of Mathematics. 12 October 2009 <http://illuminations.nctm.org>.

LEARN NC. 12 October 2009 <http://www.learnnc.org>.

Middle School Math Course 1 Algebra Readiness with CD ROM (Middle School Math). New York: Holt, 2004.

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Catherine ShelferStep 4: Indentifying Unit Goals

Enduring Understandings: Students will understand…1. Math allows us to organize data for easier comparison2. Basic mathematic principles can simplify situations, so we can often solve a more complex problem3. Math concepts translate into real world scenarios; a variety of applications and adaptations must be

employed to solve problems according to their constraints4. Estimations are helpful in everyday life, especially as a consumer5. Mathematics is a process; what we learn today builds upon previous lessons and serves a base for future

lessons 

Essential Questions1. How does percent relate to our previous lesson on ratio? How do we use ratios when solving percent

problems?2. How are ratios related to proportions?3. How are equivalent ratios like equivalent fractions?4. Why does 55% rest between 0 and 1 on the number line?5. Tell of a time (other than assignments and tests) when you will use your knowledge of ratio, proportion,

and percent. 6. How and when can we use our math knowledge in the world of sports?7. As a consumer, when is our knowledge of estimation useful?8. When would you use scale drawings?9. When would you use indirect measure?10. Pretend you are an English teacher, when would you use ratio, proportion, and percent in your daily

activities? 11. Why do I need to know ratio, proportion, and percent?

Students will know…1. Ratio, proportion, and percent are a means of simplifying and organizing data for comparison2. A variety of representations for numbers between 0 and 1 (decimal, fraction, percent)3. Percent is a ratio compared to 1004. A ratio is a comparison of two quantities with division5. A proportion is an equation that shows two equivalent ratios6. Proportions make conversions in measurements and identify similar figures7. Proportions and similar figures are used to find unknown measures of items we cannot measure directly8. A scale drawing is of a real object that is proportionally smaller or larger than the real object

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Catherine ShelferStudents will be able to…

1. Write ratios and rates and find unit rates2. Write and solve proportions3. Use ratios to identify similar figures4. Use proportion and similar figures to find unknown measures5. Read and use map scales and scale drawings6. Write percents as decimals and fractions7. Write decimals and fractions as percents8. Find the missing value in a percent problem9. Apply their knowledge of percents, ratios, and proportions to solve a variety complex problems by

selecting appropriate strategies10. Estimate, when appropriate, using ratio, proportion, and percent11. Relate ratio, proportion, and percent to previous topics and to future problems12. Compare, analyze, and interpret data using ratio, proportion, and percent13. Identify situations in the real world or class room where ratio, proportion, or percent is used to solve or

simplify14. Read and create graphs that involve ratio, proportion, and percent

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Catherine ShelferStep 5: Scope and Sequencing of Content

Unit Summary

The unit will cover ratio, proportion, and percent. First, students will learn about ratios and rates. Students will learn how to find a unit rate and write ratios. Then, students will extend their knowledge of ratios by writing and solving proportions. Students will learn that a proportion is made up of two equivalent ratios. Also, students will apply their knowledge of proportions to convert measurements, identify similar figures, calculate indirect measures, and explore scale drawings. Next, the unit moves into percents. Students learn the definition of percent and how fractions and decimals relate to percents. Then, the students use their knowledge of ratios and proportions to calculate percents. Finally, the students will investigate real-world applications of percents.

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Catherine ShelferUnit Planning Calendar

Day 1Ratio and Rates

Skill and Concepts : Students will write ratios and rates and find unit ratesImportant Subject Matter:A ratio is a comparison of two quantities with divisionEssential Questions:How are equivalent ratios like equivalent fractions?Vocabulary: rate, ratio, unit rate

Day 2Proportions

Skill and Concepts : Students write and solve proportions Important Subject Matter:A proportion makes two ratios equalEssential Questions:How are ratios related to proportions?Vocabulary: proportion

Day 3Proportion & Measurements

Skill and Concepts:Students use proportions to find measurementsImportant Subject Matter:Proportions make conversions in measurements Essential Questions:Tell of time when knowledge of proportions and measurements might be helpful.

Day 4Similar Figures and Indirect

Measures

Skill and Concepts : Students use ratios to identify similar figuresStudents use proportions and similar figures to find unknown measuresImportant Subject Matter:Similar figures have the same shape but not necessarily the same sizeProportions and similar figures are used to find unknown measures of items we cannot measure directlyEssential Questions:When would you use indirect measure?Vocabulary: congruent, similar figure

Day 5Mid-Unit Quiz

Scale Drawings and Maps

Skill and Concepts : Students read and use map scales and scale drawingsImportant Subject Matter:A scale drawing is of a real object that is proportionally smaller or larger than the real objectEssential Questions:When would you use scale drawings?Vocabulary: scale

Day 6Percents, Decimals, & Fractions

Skill and Concepts : Students write percent as decimals and fractionsStudents write decimals and fractions as percentImportant Subject Matter:Percent means “out of 100”Fractions, decimals, and percents can express the same number in different formsA percent is a number compared to 100Essential Questions:How does percent relate to our previous lesson on ratio? How do we use ratios when solving percent problems?Why does 55% rest between 0 and 1 on the number line?Vocabulary: percent

Day 7Using Percents

Skill and Concepts : Students find missing values in percents problemsStudents solve percents problems that relate to real lifeImportant Subject Matter:Math concepts translate into real world scenarios; a variety of applications and adaptations must be employed to solve problems according to their constraintsCharts, graphs, and diagrams display and compare data Essential Questions:Tell of a time when you will use your knowledge of ratio, proportion, and percent. How and when can we use our math knowledge in the world of sports?As a consumer, when is our knowledge of estimation useful?

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Catherine ShelferStep 6: Determining Acceptable Evidence

Examples of acceptable evidence1. Explains how to represent percentages by either moving the decimal two spaces to the right or by

multiplying by 1002. Reveals understanding that a percent in less than 1 when asked to mark a percent on the number line3. Estimates the tip at a restaurant4. Constructs and/or interprets a pie chart that reflects the results of a class election5. Calculates net income and creates a monthly budget6. Analyzes and compares the same products at different stores, considering discounts and tax7. Recognizes and records uses of percent in day-to-to activities8. Solves percent problem using ratio and proportion9. Converts a group of fractions, decimals, and percents to the same form and organizes from greatest to

least10. Uses proportion and similar figures to find the measure of something that could not be directly measured11. Identifies instances when indirect measure is important12. Determines the unit rate of items in a store and identifies the importance of unit rate as a consumer13. Sets up and solves proportions14. Applies proportion knowledge to various situations, such as identifying similar figures, finding indirect

measures, converting measurements, determining percent, etc

Methods of Assessment1. Diagnostic

“Are You Ready?” Worksheet will be assigned the week before we begin the unit on ratio, proportion, and percent. The worksheet will assess: vocabulary, simplifying fractions, writing equivalent fractions, writing fractions as decimals, writing decimals as fractions, and multiplying decimals.

2. Formative Formative assessments are included in the essential questions and checks for understanding. I will observe and interact with the students during their group activities in selected lessons; therefore, I will see their thought processes and possible misconceptions. Also, students will work examples (guided by me) throughout each lesson, either in their notes or on the board (if they volunteer). During the examples we will discuss each step and why we are taking that step. Students self-assess their learning and understanding during the math journals, reflections, KWL, and vocabulary chart.

3. SummativeDaily homework, in class activities, warm-ups, a mid-unit quiz, lesson reflections, math journal, class participation, and a unit test will be used as summative assessments.

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Catherine ShelferGrading Outline

Assessment Points/Assignment Total Points5 homework assignments 10 50Reflection 10 10Vocabulary Self-Awareness Chart 40 404 Warm-Ups 10 40Currency Activity 20 20Shadow Activity 30 30Basketball Challenge 20 202 Math Journal Entries 10 20Mid-Unit Quiz 30 30GIST 10 10Think-Pair-Share 20 20Sale, Sale, Sale Activity 50 50Unit Test 100 100Class participation 10 10Total 450

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Catherine ShelferStep 7: Unit Strategies and Activities

Day 1—Ratio and Rates Objective: Students will be able to write ratios and rates and find unit rates

Materials: KWL Chart, overhead projector/Smart Board, Vocabulary Chart

Procedureo Attention Grabber with KWL:  Have you ever heard the term rate or ratios? When? How was the word

used in a sentence? Allow the entire class to brainstorm and share what they KNOW about rate and ratio; write

information in the "K" column of the KWL chart on overhead projector or Smart Board. After brainstorming, ask students what they WANT TO KNOW about rate and ratio. Record

questions in the "W" column of the KWL chart. After lesson, come back to the KWL chart. Ask students what they LEARNED about rate and

ratio and fill in the "L" column of the chart. See if any student can answer the questions posed in the "W" column.

o Ratio Comparison of two quantities using division Compare two groups using division Example:

Compare the number of pugs with the number of poodles. Can be written three ways

8 to 9 8:989

Order is very important! Pug to Poodles is different than Poodles to Pugs Ratios can compare:

A part to a part A part to the whole The whole to a part

Practice writing ratios using dog breed chart. Maltese to Labradors

5 to 12 5:12512

Mixed Breeds to total number of dogs at the kennel

10 to 48 10:481048

Total number of dogs at the kennel to Boston Terriers

48 to 4 48:4484

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Number of Dogs at Kennel, by BreedPugs 8Mixed Breeds 10Poodles 9Boston Terriers 4Labradors 12Maltese 5

Catherine Shelfero Equivalent Ratios

Ratios that name the same comparison. Found by multiplying or dividing both terms of a ratio by the same number. Example:

Write three equivalent ratios to compare the number of stars to the number of hearts.

Number of starsNumber of hearts

= 46

There are 4 stars and 6 hearts

4 ÷ 26 ÷ 2

= 23

There are two stars for every 3 hearts

4 × 36 ×3

= 1218

If you triple the ratio, there will be 12 stars for every 18 hearts

o Rate Compares two quantities that have different units of measure Example: A 15 ounce can of soup costs $1.35.

Rate = price

number of ounces =

$ 1.3515

Unit Rate When the comparison is to one unit Divide both terms by the second term

Unit Rate = $ 1.35 ÷ 15

15 ÷ 15 = $ 0.09

1 $0.09 for 1 ounce of soup

Compare unit rates of two or more items to find the better deal Which is a better deal? A 2 liter bottle of soda that costs $2.02 or a 3 liter bottle of soda

that costs $2.79?

$2.022liters =

$ 1.011 liter

$2.793liters =

$ 0.931liter

o Reflection: Define ratio and give examples of ratios.  Explain how to find equivalent ratios. Why is the ratio 2 cats: 6 dogs different from the ratio 6 dogs: 2 cats?

Add terms to vocabulary self-awareness chart: rate, ratio, unit rate

o Homework: Exercises from text book

Closure: Return to the KWL chart. Ask students what they LEARNED about rate and ratio and fill in the "L" column of the chart. See if any student can answer the questions posed in the "W" column

 

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Catherine ShelferDay 2—Proportions Objective: Students will be able to write and solve proportions 

Materials: Vocabulary Chart, “Patriotic Proportion” worksheets

Procedureo Warm up with ratios

Brown Bears 3 Write each ratio: giraffes to monkeys = 2:17polar bears to all bears = 4:7monkeys to all animals = 17:26 all animals to all bears = 26:7

Giraffes 2Monkeys 17

Polar Bears 4

o Proportion An equations that shows two equivalent ratios

21

= 42

42

= 126

21

= 63

Cross Multiplication to find missing values in proportions

Example:

34

= n

16 Cross Multiply

3×16 = 4×n Products are equal

48 = 4n 4 is multiplied by n

484

= 4 n4

Divide both sides by 4 to undo multiplication

n = 12

Write a proportion for the following:

Find the missing values in the proportions:

86

= n3

(n = 4)

t5

= 2820

(t = 7)

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Catherine Shelferp

40 =

38

(p = 15)

o Small group activity: Currency Exercise from Holt’s Middle School Math: Course 1

Europe (euro) Canada (dollar) Israel (shekel) Mexico (peso)0

1

2

3

4

5

6

7

8

9

10

0.88 0.620000000000001

0.23 0.11

1.131.6

4.4

9.1

Currency Conversions

Value of Foreign Currency Value of $1 US, in foreign currency

“The value of the US Dollar as compared to the values of currencies from other countries changes every day. The graph shows the recent value of various currencies compared to the US Dollar. Use the graph to answer the following questions.1. What is the value of 9.72 European Euros in US Dollars?2. You have $100 US Dollars. Determine how much money this is in Euros, Canadian Dollars, and

Mexican pesos. 3. A watch in Israel costs 82 shekels. In the US, the watch costs $25. In what country does it cost

less?4. Would you prefer to have five US Dollars or five Canadian Dollars? Why?”

Add terms to vocabulary self-awareness chart: proportion

“Patriotic Proportion” challenge worksheet for homework 

Closure: Give an example of a proportion. Tell how you know it is proportion.

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Catherine ShelferDay 3—Proportions and Measurement Objective: Students will be able to make conversions within the customary system.

Materials: “Basketball Challenge” worksheets, Math Journals

Procedureo Warm up with missing values in proportion

41

= 24x

(x=6)

x

30 =

56

(x=25)

1416

= x

40 (x=35)

8x

= 1218

(x=12)

o Attention Grabber: How many centimeters in 1 meter? (100) How many centimeters are in 4 meters? (400) Demonstrate proportion with this info. (This shows that proportions convert measurements.)100 cm

1 m =

400 cm4 m

 

o Review common customary measurements

Length Weight Time Capacity1 foot = 12 inches1 yard = 3 feet1 mile = 5280 feet

1 pound = 16 ounces1 ton = 2000 pounds

1 minute = 60 seconds1 hour = 60 minutes1 day = 24 hours1 week = 7 days1 year = 365 days1 year = 12 months

1 cup = 8 fluid ounces1 pint = 2 cups1 quart = 2 pints1 gallon = 16 cups

o Examples: The units must be in the same order in both ratios1 yard3 feet

= 185 yards

x feet (x = 555 feet)

1 Hour60 minutes

= x Hour

360 minutes (x = 6 hours)

1ton2000 pounds

= x tons

55,000 pounds (x = 27.5 tons)

o “Basketball Challenge” worksheet in pairs in class

Students write about proportion and measurement in Math Journal for homework

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Catherine Shelfer Closure: Describe a situation in which you would need to convert measurements.  

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Catherine ShelferDay 4—Similar Figures and Indirect Measures Objective: Students will be able to use ratios to identify similar figures; Students will be able to use

proportions and similar figures to find unknown measures

Materials: Protractors, Overhead Projector or Smart Board, Shadow Activity sheets, Vocabulary Chart, Discussion Web, yard sticks, string, scissors, calculators

Procedureo Warm up: Students draw two congruent angles with protractors.  Ask how they know they are

congruent.  (Same measure)  Then draw and label two pairs of line segments: 1 inch and 2 inch, 3 inch

and 6 inch.   Ask students if the measurements of these lines can form a proportion.  Yes, 12

= 36

o Attention Grabber with Discussion Web: On the board, make a list of things that are too tall for us to measure. 

Pick one really tall item from the list, let's say The Empire State Building Display the Discussion Web on a projector or Smart Board. Write the question in the center of

the discussion web: Can we find the height of The Empire State Building using our mathematical knowledge?

Let students answer the question with yes or no, but they MUST give a reason. Write the reasons in the corresponding columns on the web.

Return to the chart after the Shadow Activity to fill in and reflect on the conclusion. o Similar Figures

Two or more figures are similar if they have exactly the same shape. They may be different sizes. Corresponding sides have proportional lengths Corresponding angles are congruent Example

Draw and label two similar rectangles on the board.

Locate and discuss the corresponding sides and corresponding angles. Sides: Angles:AB corresponds to WX A corresponds to WBC corresponds to XY B corresponds to XCD corresponds to YZ C corresponds to YAD corresponds to WZ D corresponds Z

Set up proportions with the data.

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Catherine ShelferABWX

= ADWZ

or 26

= 39

If you cannot use corresponding side lengths to write proportions, or if the corresponding angles are not congruent, then the figures are not similar.

Example: Missing Values with Similar Figures The two triangles are similar. Find the missing length x and the measure of angle b.

812

= 6x

(x = 9 cm)

Angle A is congruent to angle B, so angle B must equal 65°o Indirect measure

One way to find height that you cannot measure directly is to use similar figures and proportions On a sunny day, a tree cast a shadow that was 228 feet long. A 6 foot tall man standing near the

tree casts a 12 foot long shadow.

Both the person and the tree form right angles with the ground and their shadows are cast at the angle. What does this mean?

We can form two similar right triangles and use proportions to find the missing height. 6h

= 12

228 h = 114 feet

o Shadow Activity in pairsFrom National Council of Teachers of Mathematics http://illuminations.nctm.org

Add terms to vocabulary self-awareness chart: congruent, similar figure

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Catherine Shelfer

Answer Shadow Activity questions for homework

Closure: Return to the discussion web after the Shadow Activity to fill in and reflect on the conclusion

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Catherine ShelferDay 5—Short Mid-Unit Quiz and Scale Drawings and Maps Materials: Quizzes, GIST sheets, Math Journals, Vocabulary Charts

Procedureo Administer 8 question quiz from days 1 to 4        o Early finishers begin reading lesson on scale drawings with GIST strategy , then write about scale maps

in Math Journal

Add terms to vocabulary self-awareness chart: scale

Math Journal for homework: Write about scale maps in Math Journal.

Students only need to be familiar with the scale drawing and map section.  They should know that it is related to proportion. 

Closure: How is the scale on a map useful?  If the scale is 1 inch = 5 inches, what does this mean? How can this info be used? How is this related to our previous lessons?

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Catherine ShelferDays 6—Percents, Decimals, and Fractions Objective: Students will be able to express percent as decimals and fractions and express decimals and

fractions as percent

Materials: Common Equivalents Charts sheets, Vocabulary Charts, “Problem Solving” worksheets

Procedureo Warm-Up: Review fractions and decimals

Write the following fraction as a decimal: ¾, 9/10 Write the following decimal as a fraction: 0.375, 0.05 Reduce the following fractions: 8/10,  21/63

o Attention Grabber: Where and when have you seen/heard percentages in everyday life?  Does anyone know anything about percent?

o Percent a percent is a ratio whose second term is always 100 means per 100 Using ratio terms of “part to whole”, the percent is the part and the whole is 100 100% means “the whole thing” Example:

8% means 8 out of 100 or 8

10010x10 grid has 100 squares 8 out of 100 squares are shaded to show 8%

Percent is also a fraction whose denominator is 100 Example: Changing percent to fraction and reducing

Given 40%, express as a ratio 40

10040

100 is also a fraction which can be reduced to

25

A percent is less than 1 Draw number line on board.

Who can show me where 50% lies on the number line? Why does it lie there? Percent can be represented as a decimal Example: Moving decimals using base 10 knowledge to show less than 1

Water frozen in glaciers makes up 75% of the worlds fresh water supply. Write 75% as a decimal.

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Catherine Shelfer

75% = 75

10075

100 means 75.00 ÷ 100.00

We move decimal two places to the left 00.75 ÷ 1.00 = .75.75 is less than 1

Fractions decimals and percents appear in real life. To understand the data, we should be able to change from one form to another.

Examples:Write the decimal as percent using place value0.3

0.3 = 3

10

3× 1010× 10

= 30

100 = 30%

Write the decimal as percent using multiplication by 1000.74310.7431 x 100= 74.31% (move decimal two place to the left)

Write the fraction as a percent—denominator is factor of 10045

4 × 205 ×20

= 80

100= 80%

Write the fraction as a percent—denominator is not a factor of 10038

3 ÷ 8 = 0.375 (use division or calculator)0.375 = 37.5% (move decimal two places to the left)

o Students complete common fractions, percents, and decimals chart in pairs

Fraction15

14

13

25

12

35

23

34

45

Decimal 0.2 0.25 0.333 0.4 0.5 0.6 0.666 0.75 0.8

Percent20%

25% 33.3%40%

50% 60% 66.6%75%

80%

Add terms to vocabulary self-awareness chart: percent

"Problem Solving" worksheet for homework

Closure: Which method do you prefer for converting decimals to percent? Why?

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Catherine ShelferDay 7—Using Percents Objective: Students will be able to calculate missing values in real-world percents problems   

Materials: “Sale, Sale, Sale” instructions, calculators, “Unit Review” worksheets, construction paper, rulers

Procedure o Review of percent and ratio and proportion

What do you remember from yesterday’s lesson about percent? What is a percent? Can anyone tell or show me what a ratio is? Can anyone tell or show me how a proportion is related to ratio?

o Attention Grabber: Does anyone know how to tip a waiter?

o Percents of numbers other than 100 can be found with ratios and proportion and multiplication Examples:

What is 80% of 40? Set up proportion with given info 80

100 =

x40

Cross multiply and solve for x. 100x = 3200x = 32

Find 20% of 150. 20% = 0.20 write the percent as a decimal0.20 x 150 = 30 multiply using the decimal

A shirt is $20 and sales tax is 8%. How much will the sales tax be?

Set up proportion with provided info 8

100 =

x20

Cross multiply and solve. 160=100xx = $1.60

Mary is downloading a file from the internet. So far 75% of the file has downloaded. If 30 minutes has passed since she began, how long will it take to download the rest of the file?75

100 =

30m

30 minutes is a part (75%) of the entire time needed to download

3,000= 75m 40 = mThe time needed to download the entire file is 40 minutes. So far, the file has been downloading 30 minutes. Because 30-40 = 10, the remainder of the file will download in 10 minutes.

o Percents can be represented in charts and graphs and percent can be calculated from info on charts and graphs.

o Pie chart displays percent very effectively… the entire pie represents 100 and the pieces represent the percents.

Example:Results of a vote for 6th graders t-shirt color 60% want blue

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60%16%20%4%

If 10 kids voted for green shirts, how many kids are in the class?

Since 10 kids represent “part” of the class, we are looking for the “whole” this time.

Set up proportion 20

100 =

10x

Cross multiply and solve for x. 20x = 1000x = 50

o Estimation of percent is appropriate in some instances, round numbers for easier calculation

o Calculating mentally: 10% of a number move decimal one place to the left

o 1% of a number move decimal two places to the left

o 5% of a number, half of 10%

o Estimation can serve as method to check our work Example:

A store sign reads “10% off the regular price.” If Nicole wants to buy a CD whose regular price is $14.99, about how much will she pay for it after discount?

Round $14.99 to $1510% of $15 = $1.50 move decimal one place to the leftDiscount = $1.50$15 - $1.50 = $13.50 Subtract discount from original price

o "Sale, Sale, Sale!!" Activity in groups of 4 From LearnNC.org

o If students finish early, then they can start on their homework.

"Unit Review" worksheet for homework

Closure: What do you think is important to know about percent? Why do you need to understand percent?

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Catherine ShelferDay 8—Unit Review Materials: “Sale, Sale, Sale” instructions and handouts, calculators, Vocabulary Charts, completed “Unit

Review” worksheets, Math Journals

Procedureo Finish “Sale, Sale, Sale” activity, if neededo Review, complete, correct, revise vocabulary self-awareness chartso Review study guide and answer questions about unito Share reflections and math journals

 Day 9—Unit Test

 

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Vocabulary Self Awareness ChartWord + - Example Definition

Procedure:1. Examine the list of words you have written in the first column.2. Put a “+” next to each word you know well, and give an accurate example and definition of the word. Your

definition and example must relate to what we are studying.3. Put a “ “ next to any words for which you can write only a definition or example, but not both.4. Put a “-“ next to words that are new to you.

This chart will be used throughout our unit. By the end of the unit you should have the entire chart completed. Since you will be revising this chart, write in pencil.

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Length Weight1 foot = ____ inches

1 yard = ____ feet

1 yard = ____ inches

1 mile = _____ feet

1 mile = ____ yards

1 pound = ____ ounces

1 ton = _____ pounds

Time Capacity1 minute = ____ seconds

1 hour = ____ minutes

1 day = ____ hours

1 week = ____ days

1 year = ____ days

1 year = ____ months

1 cup = ____ fluid ounces

1 pint = ____ cups

1 quart = ____ pints

1 gallon = ____ cups

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Fraction14

12

35

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Decimal 0.2 0.33… 0.4 0.6 0.66… 0.75

Percent 20% 25% 33.3…% 40% 50% 75% 80%

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“Sale, Sale, Sale!”Activity (from LearnNC.org)

1. Students work in groups of 4.

2. The teacher models a scenario, buying pizza. As guided practice, the class works on the problem together to find the new sale price offered by each restaurant, and then applies the NC 6% sales tax for total cost.

Pizza Hut: $13.99 at 20% off + tax = $11.86 Dominos: $14.00 at 1/4 off + tax = $11.13 Franks: Pay only 3/5 of $16.50 + tax = $10.49

3. Students are given the sale ads of retailers and their competitors. Students are directed to use the example on the board as a model for organization and layout of information. Each product to purchase has three retailers for which to compare and this could take a whole side of a piece of paper (8.5 X 11) divided into 3 sections using a ruler.

4. The goal is to find the best deal.

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Summative Reflection

Before this mathematics education course, I had studied lesson planning and pedagogy,

but not specifically for mathematics. Of course teachers across the curriculum use similar

planning tactics but mathematics has special requirements. And, to be honest, until this course, I

am not sure that I even considered mathematics’ unique needs.

Throughout the course, we examined an array of factors that affect student learning in

mathematics, factors such as classroom management, motivation, manipulatives/technology,

standards, questioning/communication, homework, cooperative learning, assessments, etc. Prior

to this course, I had not thought about ALL of these factors and the effects they have on each

other. All of these factors create a tangled web that we, teachers, must consider when planning

for our students. For instance, we must plan lessons to reach as many students as possible with

thoughtful use of manipulatives, while managing student behavior, asking meaningful questions,

and meeting standards! On top of that, we must keep track of grades, communicate with parents,

and keep students motivated. In general, I learned that there are many aspects that enhance or

hinder student learning in the mathematics’ classroom. And we must consider these aspects

during our planning in order to provide an effective learning experience for our students.

In addition to the aforementioned factors, we must also consider the combination of

characteristics that our students possess. These combinations make each student unique and

contribute to their learning needs or preferences. Thus, it is essential that we know our students,

so we can plan appropriate learning experiences for them. I believe that it is important to know

our students as individuals. We must challenge our students and encourage them; show them that

we believe in them. We must acknowledge and embrace the students’ differences but treat all of

them fairly and as an individual.

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Catherine ShelferMany instances in the course served as reflection-points for me and forced me to ponder

how I would address the many factors effecting mathematics education. For example, until

recently, I never thought about how I would use manipulatives in my classroom, or if my way

was the most beneficial to my students. Additionally, I learned a lot from the ideas of my

classmates. The discussion boards challenged my opinions and showed me other people’s

perspectives. After reading about and discussing the variety of topics in this course, I feel like I

am knowledgeable on many aspects that effect mathematical learning and know how to plan or

respond properly.

Now, I understand that an effective mathematics educator must consider a combination of

factors and the effects of those factors on students. Furthermore, to effectively teach to a diverse

group of students, we must know and embrace the characteristics that make our students unique.

Our students’ needs and the world of education are ever-changing. It is our responsibility to stay

abreast of the issues in mathematics education and constantly communicate with our peers and

students in order to be the most effective teachers we can be.

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