2012-2013 6th grade math curriculum pacing...

Curriculum Pacing Guide

6th grade

2011 - 2012

First Nine Weeks at a Glance:

• SOL 6.15 Measures of central

tendency

• SOL 6.14 Graphing data and

data analysis

• SOL 6.17 Geometric and

arithmetic sequences

• SOL 6.19 Properties of real

numbers

• SOL 6.16 Probability

Second Nine Weeks at a Glance:

• SOL 6.18 Linear equations

• SOL 6.20 Graphing inequalities

• SOL 6.11 Coordinate Planes

• SOL 6.10c Area and Perimeter

• SOL 6.10 d Volume and Surface

area

• SOL 6.10 a, b Circumference and

area of circle

Third Nine Weeks at a Glance:

• SOL 6.9 Customary and Metric

measurement systems

• SOL 6.2 Fractions, decimals,

percents

• SOL 6.1 Ratios

• SOL 6.4 Number sense: Fractions

• SOL 6.6 Problem solving

Fractions

• SOL 6.7 Problem solving

decimals

• SOL 6.3 Integers

• SOL 6.5 Positive exponents and

perfect squares (*Scientific

notation)

Fourth Nine Weeks at a Glance:

• SOL 6.8 Order of operations

• SOL 6.12 Congruence and

similarities

• SOL 6.13 Properties of

Quadrilaterals

Time Strand, Big Idea, & Student Objectives

Essential Knowledge, Skills, Processes

Instructional Strategies and Model Lessons Assessment Items

Quarter

1

Week 1

Pretesting of Grade 6 SOLs, Fact Drills, Basic computation skills Instructional Strategies:

• Teach students to add, subtract, multiply, and divide whole numbers

• Drill students on addition, subtraction, multiplication, and division facts

(Continue throughout the year to give children several chances to

master facts).

Resources:

Glencoe Course 1: Prerequisite skills

Virginia SOL Mathematics Coach Grade 6 Pretest

Virginia SOL Mathematics Coach Grade 6

Pretest, Pretest of Math Facts

Quarter

1 Week

2

Probability and Statistics

SOL 6.15

The student will

a) describe mean as balance point; and

b) decide which measure of center is appropriate for a given

purpose.

Essential Understandings

• What does the phrase “measure of center” mean?

This is a collective term for the 3 types of averages for a set of data – mean,

Instructional Strategies:

• Review of division (concept and method) will be helpful prior to teaching

mean.

• Have students keep track of their grades on tests, quizzes, and

homework. Have them find the mean, median, mode, and range of the

data. Have the students look for trends in their grades.

• Pass around a bucket of color cubes and have each student pick one.

Students, who have a particular color cube, are asked to link their cubes

together. In order for the students to count the number o cubes, of each

color, place the columns of cubes on a desk. Next, the students organize

Open response:

Explain how to find the median of a set

of

data that has an even number of data in

the set. List six numbers such that the

mean is 14, the median is 14, the modes

are 12 and 14, and the range is 5.

Describe a real-life situation in which the

mean is lower than the median.

Writing prompts:


of


6th grade

2011 - 2012 median, and mode.

• What is meant by mean as balance point? Mean can be defined as the

point on a number line where the data distribution is balanced. This

means that the sum of the distances from the mean of all the points above

the mean is equal to the sum of the distances of all the data points below

the mean. This is the concept of mean as the balance point.

• Understand that measures of central tendency are types of averages for a

set of data.

• Understand that mean, median, and mode are measures of central

tendency that are useful for describing data in different situations.

• Understand that the range indicates how data is spread out or dispersed.

• Understand and appropriately use measures of central tendency for a data

set.

Essential Knowledge and Skills

• Find the mean for a set of data.

• Describe the three measures of center and a situation in which each

would best represent a set of data.

• Identify and draw a number line that demonstrates the concept of mean

as balance point for a set of data.

• Solve problems by finding the mean of a set of no more than 20 numbers.

• Solve problems by finding the median of a set of data of no more than 20

numbers when the numbers are arranged from least to greatest, including

data sets that have one middle number and data sets that have two

middle numbers.

• Solve problems by finding the mode of a set of data of no more than 20

numbers.

• Identify the mode in a set of data, given that there may be one, more than

one, or no mode.

• Examine the range to understand spread or dispersion of the data.

• Solve problems by finding the range of a set of data of no more than 20

numbers.

the data in a table and determine the mean, median, mode, and range

of the data.

• Have students use real-life data from various sources and then

determine the mean, median, mode, and range of the data.

• Students collect the scores for their favorite team or player in ten games

and find the average score, mode, and the median.

Model Lessons:

Enhanced Scope and Sequence, Grade 6

“Balancing Act,”

Resources:

Course 1: 2-6, 2-7

Glencoe Teacher ‘s Resource Kit

Understanding Math



the set.

Explain how you can use a stem-and-leaf

plot to determine the median of a set of

data.

SOL-like Multiple Choice:

VDOE Released SOL items (6.15)

Pretests, Posttest, Formative

assessments, Bellwork, Classwork, and

Homework


6th grade

2011 - 2012

Teacher Notes

• Measures of center are types of averages for a data set. They represent

numbers that describe a data set. Mean, median, and mode are measures

of center that are useful for describing the average for different situations.

– Mean works well for sets of data with no very high or low numbers.

– Median is a good choice when data sets have a couple of values

much higher or lower than most of the others.

– Mode is a good descriptor to use when the set of data has some

identical values or when data are not conducive to computation

of other measures of central tendency, as when working with

data in a yes or no survey.

• The mean is the numerical average of the data set and is found by adding

the numbers in the data set together and dividing the sum by the number

of data pieces in the set.

• In grade 5 mathematics, mean is defined as fair- share.

• Mean can be defined as the point on a number line where the data

distribution is balanced. This means that the sum of the distances from

the mean of all the points above the mean is equal to the sum of the

distances of all the data points below the mean. This is the concept of

mean as the balance point.

• Defining mean as balance point is a prerequisite for understanding

standard deviation.

• The range is the difference between the greatest and least values in a set

of data and shows the spread in a set of data.

Quarter

1

Week 3

& 4


SOL 6.14

The student, given a problem situation, will

a) construct circle graphs;

b) draw conclusions and make predictions, using circle

graphs; and

c) compare and contrast graphs that present information

from the same data set.


• What types of data are best presented in a circle graph? Circle graphs are

best used for data showing a relationship of the parts to the whole.


• Review of division (concept and method) will be helpful prior to teaching

mean.

• Have students keep track of their grades on tests, quizzes, and

homework. Have them find the mean, median, mode, and range of the

data. Have the students look for trends in their grades.

• Pass around a bucket of color cubes and have each student pick one.

Students, who have a particular color cube, are asked to link their cubes

together. In order for the students to count the number o cubes, of each

color, place the columns of cubes on a desk. Next, the students organize

the data in a table and determine the mean, median, mode, and range

of the data.

Open response:


of data that has an even number of data

in the set. List six numbers such that the

mean is 14, the median is 14, the modes

are 12 and 14, and the range is 5.

Describe a real-life situation in which the

mean is lower than the median.

Writing prompts:


of


the set. Explain how you can use a stem-


6th grade

2011 - 2012

• Understand that data can be displayed in a variety of graphical

representations.

• Select and use appropriate statistical methods to analyze data.

• Understand that different types of representations can tell different things

about the same data.

• Understand that graphs tell a story.


• Collect, organize and display data in circle graphs by depicting information

as fractional.

• Draw conclusions and make predictions about data presented in a circle

graph.

• Compare and contrast data presented in a circle graph with the same data

represented in other graphical forms.

• Collect data sets of no more than 20 items by using tally sheets, surveys,

observations, questionnaires, interviews, and polls.

• Organize data by using lists, charts, tables, frequency distributions, and

line plots.

• Organize and display data in bar and line graphs, displaying the

information as clearly as possible by using increments of whole numbers,

fractions, and decimals rounded to the nearest tenth.

• Decide which type of graph is appropriate for a given situation.

• Bar graphs are used to display categorical (discrete) data.

• Line graphs are used to display continuous data.

• Circle graphs are used to show a relationship of the parts to a whole.

• Interpret data from line, bar, and circle graphs and from stem-and-leaf

and box and-whisker plots.

• Collect, analyze, display, and interpret a data set of no more than 20

items, using stem-and-leaf plots where the stem is listed in ascending

order and the leaves are in ascending order with or without commas

between leaves.

• Have students use real-life data from various sources and then

determine the mean, median, mode, and range of the data.

• Students collect the scores for their favorite team or player in ten games

and find the average score, mode, and the median.

Model Lessons:


“May I Have Fries with That?”


“It’s In the Bag” SOL 5.14

“Enough Room?” SOL 5.6

Resources:

Course 1:2-1, 2-2, 2-3, 2-5, 2-6, 2-7, 2-7b, 2-7


Understanding Math


and-leaf plot to determine the median of

a set of data.





Homework


6th grade

2011 - 2012

• Organize, analyze, display, and interpret data sets of no more than 20

numbers in box-and-whisker plots, identifying the lower extreme

(minimum), lower quartile, median, upper quartile, and upper extreme

(maximum). Use the critical points in a box-and-whisker plot to determine

the range and the interquartile range (IQR).

• Determine patterns and relationships within data sets (e.g., trends).

• Add charts, graphs, and tables into a project from another file on the

computer, from the internet, or from a software gallery.

• Format and insert a table with data into a document.

• Utilize short cut keys to manipulate and format data.

• Evaluate data and draw conclusions from tables and graphs.

• Manipulate an existing spreadsheet template to add content

• Identify cells, row, and columns by their alpha/numeric label.

• Convert data in a spreadsheet into various graphs.

• Export graphs, charts, and data to other applications to present

information about a concept or skill.

• Analyze collected information in a spreadsheet to draw conclusions.

Teacher Notes

• To collect data for any problem situation, an experiment can be designed,

a survey can be conducted, or other data-gathering strategies can be used.

The data can be organized, displayed, analyzed, and interpreted to answer

the problem.

• Different types of graphs are used to display different types of data.

– Bar graphs use categorical (discrete) data (e.g., months or eye color).

– Line graphs use continuous data (e.g., temperature and time).

– Circle graphs show a relationship of the parts to a whole.

• All graphs include a title, and data categories should have labels.

• A scale should be chosen that is appropriate for the data.

• A key is essential to explain how to read the graph.


6th grade

2011 - 2012

• A title is essential to explain what the graph represents.

• Data are analyzed by describing the various features and elements of a

graph.

Quarter

1

Week 5

Patterns, Functions, and Algebra

SOL 6.17

The student will identify and extend geometric and arithmetic sequences.


• What is the difference between arithmetic and a geometric sequence?

While both are numerical patterns, arithmetic sequences are additive and

geometric sequences are multiplicative.

• Understand that mathematical patterns can be represented in various

forms, geometrically or numerically.

• Understand that patterns regularly occur in everyday life.

• Understand that patterns can be recognized, extended, or generalized.

• Understand that numerical patterns may involve adding or multiplying by

the same number.

• Understand that geometric patterns may involve shape, size, angles,

transformations of shapes, and growth.

• Understand that patterns in mathematics are often represented by using a

rule that relates elements in one set to elements in another set.

Essential knowledge and Skills

• Investigate and apply strategies to recognize and describe the change

between terms in arithmetic patterns.

• Investigate and apply strategies to recognize and describe geometric

patterns.

• Describe verbally and in writing the relationships between consecutive

terms in an arithmetic or geometric sequence.

• Extend and apply arithmetic and geometric sequences to similar


• Given a list of numbers or a table of values, have the students determine

the nth value(s). Have them find the rule that applies to the particular

list or table. Explain and develop strategies to solve problems involving

numerical patterns, then create an algebraic equation. Given a series of

blocks, have the students create the next 5 figures. Have the students

explain orally and in writing how the pattern works.

• Have students write number sequences for other students to complete,

by finding and continuing a pattern. Ask the problem solvers to state the

rule used.

• Have students work in pairs to create a pattern using centimeter grid

paper. Have them challenge their classmates to find the next 2 figures in

the pattern.

• Suggested manipulatives include pattern blocks, grid paper, and quilting

patterns.

Model Lessons:


“Growing Patterns and Sequences”

ARI Curriculum Companion

“ Patterns, Functions, and Algebra”, p. 33-37

Resources:

Course 1: 7-6a, 7-6

Course 2: 1-7, 1-7b, 3-6a

NCTM Illuminations 6-8

“Chairs”, “Limits”

MathScience Innovation Center

“Playing with Patterns”

http://www.mathinscience.info/public/playing_with_patterns/PlayingWithPatt

erns.htm

Open response:

Consider the following sequence: 1, 4, 7,

10, 13… Is 100 a member of this

sequence? Explain your reasoning. If you

saved $2.00 on January 1, $4.00 on

February 1, $6.00 on March 1, and $8.00

on April 1 and so on, how much money

would you save in one year?

Explain how you would find the missing

terms in the following geometric

sequence:

_, _, 3, _, _, 1/9

Writing prompts:

Create a pattern in which the fifth

picture in the series would be the picture

shown below. Describe your pattern.

Groups of campers were going to an

island. On the first day, 10 went over and

2 came back. On the second day, 12

went over and 3 came back. If this

pattern continues, how many would be

on the island at the end of a week? Draw

a table. Write a minimum of 3 sentences

to explain the pattern and how you

found out the number of campers on the

island at the end of the week.





Homework


6th grade

2011 - 2012 situations.

• Extend arithmetic and geometric sequences in a table by using a given rule

or mathematical relationship.

• Compare and contrast arithmetic and geometric sequences.

• Identify the common difference for a given arithmetic sequence.

• Identify the common ratio for a given geometric sequence.

• Analyze numeric and geometric sequences to discover a variety of

patterns.

• Create numerical and geometric patterns by using a given rule or

mathematical relationship.

• Describe numerical and geometric patterns, including triangular numbers.

Teacher Notes:

• Integrate patterns in the curriculum throughout the entire year.

• Numerical patterns may include linear and exponential growth, perfect

squares, triangular and other polygonal numbers, or Fibonacci numbers.

• Arithmetic and geometric sequences are types of numerical patterns.

• In the numerical pattern of an arithmetic sequence, students must

determine the difference, called the common difference, between each

succeeding number in order to determine what is added to each previous

number to obtain the next number. Sample numerical patterns are 6, 9,

12, 15, 18… and 5, 7, 9, 11, 13…

• In geometric number patterns, student must determine what each

number is multiplied by to obtain the next number in the geometric

sequence. This multiplier is called the common ratio. Sample geometric

number patterns include 2, 4, 8, 16, 32… 1, 5, 25, 125, 625… and 80, 20, 5,

1.25…

• Strategies to recognize and describe the differences between terms in

numerical patterns include, but are not limited to, examining the change

between consecutive terms, and finding common factors. An example is

the pattern 1, 2, 4, 7, 11, 16…

Glencoe Teacher’s Resource Kit

Understanding Math



6th grade

2011 - 2012 Quarter

1 Week

6


SOL 6.19

The student will investigate and recognize

a) the identity properties for addition and multiplication;

b) the multiplicative property of zero; and

c) the inverse property for multiplication.

Esssential Understandings

• How are the identity properties for multiplication and addition the same?

Different? For each operation the identity elements are numbers that

combine with other numbers without changing the value of the other

numbers. The additive identity is zero (0). The multiplicative identity is one

(1).

• What is the result of multiplying any real number by zero? The product is

always zero.

• Do all real numbers have a multiplicative inverse? No. Zero has no

multiplicative inverse because there is no real number that can be

multiplied by zero resulting in a product of one.


• Identify a real number equation that represents each property of

operations with real numbers, when given several real number equations.

• Test the validity of properties by using examples of the properties of

operations on real numbers.

• Identify the property of operations with real numbers that is illustrated by

a real number equation.

• NOTE: The commutative, associative and distributive properties are taught

in previous grades.

Teacher Notes

• Identity elements are numbers that combine with other numbers without

changing the other numbers. The additive identity is zero (0). The

multiplicative identity is one (1). There are no identity elements for

subtraction and division.

Instructional Strategies

• Model 2 x 4 = 4 x 2 by showing 2 groups of 4 and 4 groups of 2 both

equal to 8.

• Use index cards and markers to create flash cards for properties.

Students will illustrate the property on one side and the property name

on the other side.

• Students can match cards naming properties to examples of each

property.

• Use Venn Diagrams to compare and contrast Identity property of

addition and multiplication

• Use Inverse properties when solving one step equations.

Model Lessons:


“Pick and Choose”


“Modeling Properties”, p. 9 – 26

Resources:

Course 1: 9-1

Course 2: 1-6, 3-4, 3-6, 6-5


Understanding Math


Open Response:

Explain how properties are used when

solving equations like

x + 5 = 12.

Provide students with information for

one side of a statement and ask students

to apply a particular property and

demonstrate the result.

(i.e. “(2 + 3) + 7 =” use the associative

property of addition to write an

equivalent statement.)

Writing prompts:

You have to buy 3 video games and the

prices are $37, $42, and $23. Explain

how these numbers can be added

mentally and what property you would

use and why.





Homework


6th grade

2011 - 2012

• The additive identity property states that the sum of any real number and

zero is equal to the given real number (e.g., 5 + 0 = 5).

• The multiplicative identity property states that the product of any real

number and one is equal to the given real number (e.g., 8 · 1 = 8).

• Inverses are numbers that combine with other numbers and result in

identity elements.

• The multiplicative inverse property states that the product of a number

and its multiplicative inverse (or reciprocal) always equals one (e.g., 4 · 1

4 =

1).

• Zero has no multiplicative inverse.

• The multiplicative property of zero states that the product of any real

number and zero is zero.

• Division by zero is not a possible arithmetic operation. Division by zero is

undefined.

Quarter

1

Week 7

& 8


SOL 6.16

The student will

a) compare and contrast dependent and independent

events; and

b) determine probabilities for dependent and independent

events.


• How can you determine if a situation involves dependent or independent

events? Events are independent when the outcome of one has no effect

on the outcome of the other. Events are dependent when the outcome of

one event is influenced by the outcome of the other.

• Understand how to use and interpret information given a sample space.

• Understand that a probability can be expressed as a ratio, decimal, or

percent.

• What is the Fundamental (Basic) Counting Principle? The Fundamental

(Basic) Counting Principle is a computational procedure used to determine

the number of possible outcomes of several events.


• Students work in pairs with two number cubes. Each pair brainstorm to

list all the possible outcomes of rolling two number cubes. When the

pair of students have completed their list they will get the teacher to

check it. Afterwards, the students will give, in ratio form:

- favorable number of outcomes/total number of outcomes

- P(two 3’s)

- P(same number on both cubes)

- P(one or two 6’s)

- P(the sum of numbers showing 7)

• Use real-life examples involving coins, dice, clothing, and food.

• Students study the chances of winning in the Virginia Lottery Pick 3 and

Pick 4 daily events using the Basic

• Counting Principle. Compare the chances of winning with the size of the

prize.





Homework


6th grade

2011 - 2012

• What is the role of the Fundamental (Basic) Counting Principle in

determining the probability of compound events? The Fundamental

(Basic) Counting Principle is used to determine the number of outcomes of

several events. It is the product of the number of outcomes for each event

that can be chosen individually.

• What is the difference between the theoretical and experimental

probability of an event? Theoretical probability of an event is the expected

probability and can be found with a formula. The experimental probability

of an event is determined by carrying out a simulation or an experiment.

In experimental probability, as the number of trials increases,


• Determine whether two events are dependent or independent.

• Compare and contrast dependent and independent events.

• Determine the probability of two dependent events.

• Plan and carry out experiments that use concrete materials to find a

sample space.

• Determine the number of possible arrangements for selected experiments

and represent the sample space for no more than three types of objects as

a list, chart, picture, and tree diagram.

• Compute the number of possible arrangements of no more than three

types of objects by using the Fundamental (Basic) Counting Principle.

• Given a sample space, determine the probability of a simple event.

Represent the probability as a ratio, fraction, decimal, or percent where

the fraction’s denominator does not exceed 20, decimals are rounded to

tenths, and percent is rounded to 1/10 of a percent.

• Determine the theoretical probability of an event.

• Determine the experimental probability of an event.

• Describe changes in the experimental probability as the number of trials

increases.

• Investigate and describe the difference between the probability of an

event found through experiment or simulation versus the theoretical

probability of that same event.

Model Lessons:


“It Could Happen”


“What are the Chances?” SOL 7.9


“ Probability”, p. 8-14, 25-32

Resources:

Course 1: 11-1, 11-1b, 11-2, 11-5

Course 2: 9-1, 9-2, 9-3, 9-6, 9-6b, 9-7

Understanding Math Software




6th grade

2011 - 2012

• Compute the number of possible outcomes by using the Fundamental

(Basic) Counting Principle.

Teacher Notes

• The probability of an event occurring is equal to the ratio of desired

outcomes to the total number of possible outcomes (sample space).

• The probability of an event occurring can be represented as a ratio or the

equivalent fraction, decimal, or percent.

• The probability of an event occurring is a ratio between 0 and 1.

• A probability of 0 means the event will never occur.

• A probability of 1 means the event will always occur.

• A simple event is one event (e.g., pulling one sock out of a drawer and

examining the probability of getting one color).

• Events are independent when the outcome of one has no effect on the

outcome of the other. For example, rolling a number cube and flipping a

coin are independent events.

• The probability of two independent events is found by using the following

formula: ( ) ( ) ( )P Aand B P A P B= ⋅

Ex: When rolling two number cubes simultaneously, what is the

probability of rolling a 3 on one

cube and a 4 on the other?

1 1 1(3 4) (3) (4)

6 6 36P and P P= ⋅ = ⋅ =

• Events are dependent when the outcome of one event is influenced by the

outcome of the other. For example, when drawing two marbles from a

bag, not replacing the first after it is drawn affects the outcome of the

second draw.

• The probability of two dependent events is found by using the following

formula:

( ) ( ) ( )P Aand B P A P B after A= ⋅


6th grade

2011 - 2012

Ex: You have a bag holding a blue ball, a red ball, and a yellow ball.

What is the probability of

picking a blue ball out of the bag on the first pick and then without

replacing the blue ball in the

bag, picking a red ball on the second pick?

1 1 1(blue red) (blue) (red blue)

3 2 6P and P P after= ⋅ = ⋅ =

• A sample space is the set of all possible outcomes of an experiment.

• A sample space can be organized by using a list, chart, picture, or tree

diagram.

• The sample space for tossing two coins is (H,H), (H,T), (T,H), and (T,T).

• The probability of an event occurring is equal to the ratio of desired

outcomes to the total number of possible outcomes (sample space).

• The probability of an event occurring can be represented as a ratio or the

equivalent fraction, decimal, or percent.

• The probability of an event occurring is a ratio between 0 and 1.

– A probability of 0 means the event will never occur.

– A probability of 1 means the event will always occur.

• A simple event is one event (e.g., pulling one sock out of a drawer and

examining the probability of getting one color).

Quarter

1

Week 9

Review for Benchmarks Use old release test items from past years revised to higher level of thinking

Use Virginia SOL Coach

Benchmark 1 Assessment

(Oct. 29 – Nov. 5)

Quarter

2

Week 1


SOL 6.18

The student will solve one-step linear equations in one variable involving

whole number coefficients and positive rational solutions.



• Stress vocabulary and use continuously throughout the course.

• Use only whole numbers in your expressions and equations.

• Have students complete an equation magic square. Once they have

solved the equation in each square correctly, their sum across, down and

Open response:

Give students balance-scale pictures and

have them record the steps taken along

with the resulting equations.

Writing prompts:

Have students write a description how


6th grade

2011 - 2012

• When solving an equation, why is it necessary to perform the same

operation on both sides of an equal sign?

• To maintain equality, an operation performed on one side of an equation

must be performed on the other side.

• Understand that physical objects can be used to represent and solve

algebraic equations.

• Understand that in an equation, the equal sign indicates that the value on

the left side of the sign is the same as the value on the right side.

• Understand that to maintain equality an operation performed on one side

of an equation must be performed on the other side.


• Represent and solve a one-step equation, using a variety of concrete

materials such as colored chips, algebra tiles, or weights on a balance

scale.

• Solve a one-step equation by demonstrating the steps algebraically.

• Identify and use the following algebraic terms appropriately: equation,

variable, expression, term, and coefficient.

• Write verbal expressions as algebraic expressions. Expressions will be

limited to no more than two operations.

• Write verbal sentences as algebraic equations. Equations will contain no

more than one variable term.

• Translate algebraic expressions and equations to verbal expressions and

sentences. Expressions will be limited to no more than two operations.

Teacher Notes

• A one-step linear equation is an equation that requires one operation to

solve.

• A mathematical expression contains a variable or a combination of

variables, numbers, and/or operation symbols and represents a

mathematical relationship. An expression cannot be solved.

on the diagonal will yield the magic number.

• Use flashcards with equations on one side and the solution on the other.

• Have students circle or highlight the variable they are solving for.

• Use manipulatives such as Algebra Tiles and balance scales to help

students build a conceptual understanding of solving linear equations.

Model Lessons:


“Equation Vocabulary,”

“Balanced,”


“Variables and Open Sentence.” SOL 5.18


“Using Balance Mats and Counters to Solve One-step Equations”, p. 49-53

“Solving One-step Equations using Equation Mats”, p. 81-88

“Solving One-step Equations and Word Problems”, p. 89-92

Resources:

Course 1: 1-7, 9-2, 9-3, 9-4, 9-5




they would solve an addition/subtraction

or multiplication/division equation for a

student who was absent and missed the

lesson on solving equations.

Have students describe the process of

solving equations using a balance scale

or mat. Students write a story/narrative

using key mathematical vocabulary and

giving an example of each vocabulary

term.





Homework


6th grade

2011 - 2012

• A term is a number, variable, product, or quotient in an expression of

sums and/or differences. In 7x2 + 5x – 3, there are three terms, 7x

2, 5x,

and 3.

• A coefficient is the numerical factor in a term. For example, in the term

3xy2, 3 is the coefficient; in the term z, 1 is the coefficient.

• Positive rational solutions are limited to whole numbers and positive

fractions and decimals.

• An equation is a mathematical sentence stating that two expressions are

equal.

• A variable is a symbol (placeholder) used to represent an unspecified

member of a set.

• A verbal expression is a word phrase (e.g., “the sum of two consecutive

integers”).

• A verbal sentence is a complete word statement (e.g., “The sum of two

consecutive integers is five.”).

• An expression is a name for a number.

• An algebraic expression is a variable expression that contains at least one

variable (e.g., 2x – 5).

• An algebraic equation is a mathematical statement that says that two

expressions are equal (e.g., 2x + 1 = 5).

• Key words in translating verbal expressions/ sentences to algebraic

expressions/equations may include words and their translations such as: is

to =, of to multiplication, more than to +, less than to –, increased by to +,

and decreased by to -.

• An expression that contains only numbers is called a numerical expression.

• An expression that contains a variable is called a variable expression.

Quarter

2

Week 2


SOL 6.20

The student will graph inequalities on a number line.



• Match inequality with graph of inequality cards.

• Put inequalities on the board and give students a card with a number on

it. Students stand if their car is a solution to the given inequality.

Open response:

Write inequality statements given a

representation on a number line.

Use real life examples – What time

would you need to get to beat the

school’s 400 m record? Long jump

record? Graph your results on a number


6th grade

2011 - 2012

• In an inequality, does the order of the elements matter?

Yes, the order does matter. For example, x > 5 is not the same relationship

as 5 > x. However, x > 5 is the same relationship as 5 < x.


• Given a simple inequality with integers, graph the relationship on a

number line.

• Given the graph of a simple inequality with integers, represent the

inequality two different ways using symbols (<, >, <, >).

Teacher Notes

• Inequalities using the < or > symbols are represented on a number line

with an open circle on the number and a shaded line over the solution set.

Ex: x < 4

• When graphing x ≤ 4 fill in the circle above the 4 to indicate that the 4 is

included.

• Inequalities using the or≤ ≥ symbols are represented on a number line

with a closed circle on the number and shaded line in the direction of the

solution set.

• The solution set to an inequality is the set of all numbers that make the

inequality true.

• It is important for students to see inequalities written with the variable

before the inequality symbol and after. For example x > -6 and 7 > y.

• Emphasize the definition of inequality: A mathematical sentence that

contains (<, >, ≥ , ≤ ).

• Only graph inequalities that are already solved (the variable is isolated on

one side of the inequality).

• Look at Guinness Book of World Records website and choose a record.

Write an inequality to beat that record.

Model Lessons:


“Give or Take a Few,”

Resources:

Course 2: 4-2a, 4-2, 4-3, 4-5




line.

Writing prompts:

Explain the difference between x > 7 and

7 > x. Explain how to verify a solution to

an inequality.


Which number is a solution to the

inequality 6 >x?

A 6

B 8

C 5

D 12





Homework


6th grade

2011 - 2012 Quarter

2 Week

3

Geometry

SOL 6.11

The student will

a) identify the coordinates of a point in a coordinate plane;

and

b) graph ordered pairs in a coordinate plane.


• Can any given point be represented by more than one ordered pair?

The coordinates of a point define its unique location in a coordinate plane.

Any given point is defined by only one ordered pair.

• In naming a point in the plane, does the order of the two coordinates

matter? Yes. The first coordinate tells the location of

the point to the left or right of the y-axis and the second point tells the

location of the point above or below the x-axis. Point (0, 0) is at the origin.

• Understand that the coordinates of a point define its location in a

coordinate plane.


• Identify and label the axes of a coordinate plane.

• Identify and label the quadrants of a coordinate plane.

• Identify the quadrant or the axis on which a point is positioned by

examining the coordinates (ordered pair) of the point.

• Graph ordered pairs in the four quadrants and on the axes of a coordinate

plane.

• Identify ordered pairs represented by points in the four quadrants and on

the axes of the coordinate plane.

• Relate the coordinate of a point to the distance from each axis and relate

the coordinates of a single point to another point on the same horizontal

or vertical line.

Teacher Notes

• In a coordinate plane, the coordinates of a point are typically represented


• Give students graph paper and ordered pairs which result in a picture

(i.e., rocket, fish, etc.) or have students draw a picture on a coordinate

graph and label the coordinates of specific points and list the ordered

pairs.

• Create a foldable showing the axes, quadrants and coordinates.

• Give students cards with an ordered pair or the graph of the ordered

pair and have them find their match.

• Use two large coordinate plans displayed in the front of the room. Pick

two students to come up at time and graph an ordered pair. Winner is

the one who plots the point correctly first..

• Teach vocabulary associated with graphing points on a coordinate plane:

Axes, origin, ordered pair, coordinate plane and quadrant.

Model Lessons:


“What’s the Point?”


“Patterns, Functions, and Algebra”, p. 63-67

Resources:

Course 1: 8-6

Course 2: 3-3




Open response:

Explain how the points (7, -10) and

(-10, 7) are different on a coordinate

plane.

Writing prompts:

Explain how to graph the point (-3, 6).

Have pairs of students draw a coordinate

grid complete with positive and negative

numbers. Ask them to plot any point in

quadrant II, III, and IV. Have them

explain

the meaning of any negative sign

associated with the ordered pair that

they plotted.





Homework


6th grade

2011 - 2012 by the ordered pair (x, y), where x is the first coordinate and y is the

second coordinate. However, any letters may be used to label the axes

and the corresponding ordered pairs.

• The quadrants of a coordinate plane are the four regions created by the

two intersecting perpendicular number lines. Quadrants are named in

counterclockwise order. The signs on the ordered pairs for quadrant I are

(+,+); for quadrant II, (–,+); for quadrant III, (–, –); and for quadrant IV, (+,–

).

• In a coordinate plane, the origin is the point at the intersection of the x-

axis and y-axis; the coordinates of this point are (0,0).

• For all points on the x-axis, the y-coordinate is 0. For all points on the y-

axis, the x-coordinate is 0.

• The coordinates may be used to name the point. (e.g., the point (2,7)). It is

not necessary to say “the point whose coordinates are (2,7)”.

Quarter

2

Week 4

Measurement

SOL 6.10 (c only)

The student will

c) solve practical problems involving area and perimeter


• What is the difference between area and perimeter? Perimeter is the

distance around the outside of a figure while area is the measure of the

amount of space enclosed by the perimeter.

• Understand the attributes of polygons and the use of measures to

determine area and perimeter.

• Understand the derivation of formulas related to area and perimeter of

polygons; and how to determine which is used in problem situations.

• Understand how to apply area or perimeter in real-life situations.


• Apply formulas to solve practical problems involving area and perimeter of

triangles and rectangles.

• Determine if a problem situation involving polygons of four or fewer sides


• Draw rectangles on coordinate plane and determine area and perimeter.

Use ordered pairs in ALL quadrants.

• Have students use manipulatives such as tiles, one-inch cubes, adding

machine tape, graph paper, geo-boards, or tracing paper with real world

applications to develop a deeper understand of the formulas for area

and perimeter.

• Give students blocks. Ask them to create shapes with a certain area and

perimeter (ex. An area of 8 and a perimeter of 12).

• For real world application of area and perimeter, use tools to measure

items in and around classroom.

• Give students situations like building a fence, putting in new carpet,

painting a wall adding baseboard to the floor of a room, etc… and have

them state whether they would find area or perimeter.

• Use the Shade Game, Build Perimeter , Area/Perimeter, activity to

engage students.

Model Lessons:


“Out of the Box,”

Open response:

Draw and label two different triangles

that each have an area of 24 square feet.

Find an object that has an area of ____.

Explain how you know.

Area and Perimeter: Word Problems

Writing prompts:

Describe the relationship between the

area of a parallelogram and the area of a

triangle with the same base and height.

Explain. If you were given the area of a

square, explain how you would find the

length of one side and the perimeter.


VDOE Released SOL items (6.10.c)



Homework


6th grade

2011 - 2012 represents the application of perimeter or area.

• Subdivide a polygon into rectangles and right triangles, estimate the area

of the rectangles and/or right triangles to estimate the area of the

polygon, and find the area of the rectangles and/or right triangles to

determine the area of the polygon.

Teacher Notes

• Experiences in deriving the formulas for area, perimeter, and volume using

manipulatives such as tiles, one-inch cubes, adding machine tape, graph

paper, geoboards, or tracing paper, promote an understanding of the

formulas and facility in their use.†

• The perimeter of a polygon is the measure of the distance around the

polygon.

• The perimeter of a square whose side measures s is 4 times s (P = 4s), and

its area is side times side (A = s2).

• The perimeter of a rectangle is the sum of twice the length and twice the

width [P = 2l + 2w, or

P = 2(l + w)], and its area is the product of the length and the width (A =

lw).

• An estimate of the area of a polygon can be made by subdividing the

polygon into rectangles and right triangles, estimating their areas, and

adding the areas together.

• The area of a triangle is one half of the measure of the base times the

height: A = ½ bh, or A = bh ÷ 2.

• Experiences in deriving the formulas for area and perimeter, using

manipulatives such as tiles, one inch cubes, adding machine tape, graph


formulas and facility in their use.


“2-D Measurement,” p. 10-21

Resources:

Course 1 : 1-8, 4-5, 12-1b, 14-1, 14-2

Course 2 : 6-8, 11-4, 11-5



“Area Triangles”




6th grade

2011 - 2012 Quarter

2

Week 5

& 6

Measurement

SOL 6.10 (D only)

The student will

d) describe and determine the volume and surface area of a

rectangular prism.


• What is the relationship between area and surface area?

Surface area is calculated for a three-dimensional figure. It is the sum of

the areas of the two-dimensional surfaces that make up the three-

dimensional figure.

• How are volume and surface area related? Volume is the measure of the

amount a container holds while surface area is the sum of the areas of the

surfaces on the container.

• How does the volume of a rectangular prism change when one of the

attributes is increased? There is a direct relationship between the volume

of a rectangular prism increasing when the length of one of the attributes

of the prism is changed by a scale factor.

• Understand the attributes of polygons and the use of measures to

determine area and perimeter.

• Understand the derivation of formulas related to area and perimeter of

polygons and how to determine which is used in problem situations.

• Understand how to apply volume and surface area in real-life situations.

• Understand the derivation of formulas related to volume and surface area

of polygons.


• Solve problems that require finding the surface area of a rectangular

prism, given a diagram of the prism with the necessary dimensions

labeled.

• Determine if a practical problem involving a rectangular prism represents

the application of volume or surface area.

• Apply formulas to solve problems involving area and perimeter for

triangles and rectangles.


• Use nets to develop and understand formula application.

• Review and practice algebraic, fraction, and decimal skills by using

formulas.

• Draw a 7 x 10 cm rectangle on a sheet of paper and find the perimeter.

Determine how the perimeter is affected when: length and width are

doubled; length and width are halved.

• Draw as many rectangles as you can with a perimeter of 24.

• Give students an object (i.e., school playground, number of tiles on the

floor, etc.) and a description of an attribute to be measured. Students

should determine whether area or perimeter should be measured. For

example, fence around a playground (perimeter), grass for playground

(area), and number of tiles covering the floor (area).

• Suggested manipulatives: 3-dimensional hollow figures with rice or sand

to fill, wooden cubes, sugar cubes, linker or snap cubes, polydron

shapes, card stock paper, rulers, and scissors.

• Use 3-dimensional figures that are hollow; then fill with sand, rice,

water, etc. to determine the volume.

• Have students bring in assorted 3-dimensional figures to determine

surface area by covering the figures with centimeter paper.

Model Lessons:


“Out of the Box”


“Volume of a Rectangular Prism,” SOL 7.5

“Surface Area of a Rectangular Prism,” SOL 7.5

“Attributes of a Rectangular Prism,” SOL 7.5

Resources:

Course 1: 14-5, 14-6

Course 2: 12-2, 12-4a, 12-4,



Writing prompts:

Write a paragraph explaining real-life

situations, which apply to area and

perimeter.

Have students discuss the difference

between surface area and volume or

explain in written form how to calculate

surface area and volume.


VDOE Released SOL items (6.10d)



Homework


6th grade

2011 - 2012

• Develop a procedure and formula for finding the surface area and volume

of a rectangular prism.

• Solve problems that require finding the volume of the rectangular prism

given a diagram of the prism with the necessary dimensions labeled.

• Describe how the volume or surface area of a rectangular prism is affected

when one measured attribute is multiplied by a scale factor. Problems will

be limited to changing attributes by scale factors only.

Teacher Notes

• The surface area of a rectangular prism is the sum of the areas of all six

faces ( 2 2 2SA lw lh wh= + + ).

• The volume of a rectangular prism is computed by multiplying the area of

the base, B, (length x width) by the height of the prism (V lwh Bh= = ).

• There is a direct relationship between changing one measured attribute of

a rectangular prism by a scale factor and its volume. For example,

doubling the length of the prism will double its volume. This direct

relationship does not hold true for surface area.

• Experiences in using a variety of measuring devices and making real

measurements promote an understanding of measurements and the

formula associated with measurements.

• Experiences in deriving the formulas for area and perimeter, using

manipulatives such as tiles, one-inch cubes, adding machine tape, graph


formulas and facility in their use.


Quarter

2

Week 7

& 8

Measurement

SOL 6.10 (a & b only)

The student will

a) define pi (π) as the ratio of the circumference of a circle

to its diameter;

b) solve practical problems involving circumference and

area of a circle, given the diameter or radius;


• What is the relationship between the circumference and diameter of a


• Give students a circular object (i.e., plastic lids, cups, etc.) two different

colors of strings, and a centimeter ruler. Students will work in groups.

The students will measure the distance around the object and the

diameter using the strings and then measuring the string with the ruler.

Regardless of the lid size used, if the students have been careful

wrapping the string, they should get three diameters out of the string

length with a little string left over. After all groups have finished their

measurements and calculations for the circles, have them write the ratio

of circumference/diameter, for their circle and use a calculator to

convert the ratio into decimal form. Record decimal forms on the

Open response:

Compare and contrast the circumference

and area of a circle. What is the

relationship between the radius and

diameter of a circle? If you know the

circumference of a circle, can you figure

out the area? Explain why or why not.

Circle Word Problems: Circles

Writing prompts:

Ask students to write a description of a


6th grade

2011 - 2012 circle?

The circumference of a circle is about 3 times the measure of the diameter.

• Select the approximation for pi (π) when solving problems.

• Understand the derivation of pi and formulas for finding circumference

and area of a circle.


• Derive an approximation for pi (3.14 or 22

7 ) by gathering data and

comparing the circumference to the diameter of various circles, using

concrete materials or computer models.

• Find the circumference of a circle by substituting a value for the diameter

or the radius into the formula C = πd or C = 2πr.

• Find the area of a circle by using the formula

A = πr2.

• Determine the circumference and/or area of a circle, using various tools.

• Create and solve problems that involve finding the circumference and area

of a circle when given the diameter or radius.

Teacher Notes

• Circumference is the distance around or perimeter of a circle.

• The area of a closed curve is the number of non-overlapping square units

required to fill the region enclosed by the curve.

• The value of pi (π) is the ratio of the circumference of a circle to its

diameter.

• The ratio of the circumference to the diameter of a circle is a constant

value, pi (π), which can be approximated by measuring various sizes of

circles.

• The fractional approximation of pi generally used is 22

7 .

• The decimal approximation of pi generally used is 3.14.

overhead or chalkboard. Ask the student to find the average of all ratios

found. When students have measured carefully, the average is usually

close to 3.14.

• Using a compass, have students construct a circle and then draw the

diameter and the radius. With a ruler have students measure the

diameter and radius and then find the circumference and area of the

circle they drew using the appropriate formula.

• Use the Bubble Mania, Circle Investigation, Parallelogram, and Pi

Investigation to engage students.

• Connect to literacy with “Sir Cumference and The Dragon of Pi” story

helping students understand how pi was derived.

Model Lessons:


“Going the Distance,”


“Circles”, p. 10-21

Resources:

Course 1: 4-6, 14-3, 11-6

Course 2: 11-6




diameter’s relationship to its

circumference (i.e. a circumference of a

circular object equals about 3.14 as long

as its diameter).

If the radius of the larger circle is twice

that of the smaller circle;

Explain how you can find the area

of the larger circle when you are

given the radius of the smaller circle.

Find the area of the larger circle.


VDOE Released SOL items (6.10a, b)



Homework

5cm


6th grade

2011 - 2012

• The circumference of a circle is computed using C dπ= or 2C rπ= ,

where d is the diameter and r is the radius of the circle.

• The area of a circle is computed using the formula2

A rπ= , where r is

the radius of the circle.

Quarter

2

Week 9

Review for Benchmarks Use old release test from past years revised to higher level of thinking

Use Virginia SOL Coach

Benchmark 2 Assessment

(Jan. 22 – 28)

Quarter

3

Week 1

Measurement

SOL 6.9

The student will make ballpark comparisons between measurements in the

U.S. Customary System of measurement and measurements in the metric

system.


• What is the difference between weight and mass? Weight and mass are

different. Mass is the amount of matter in an object. Weight is the pull of

gravity on the mass of an object. The mass of an object remains the same

regardless of its location. The weight of an object changes dependent on

the gravitational pull at its location.

• How do you determine which units to use at different times? Units of

measure are determined by the attributes of the object being measured.

Measures of length are expressed in linear units, measures of area are

expressed in square units, and measures of volume are expressed in cubic

units.

• Why are there two different measurement systems? Measurement

systems are conventions invented by different cultures to meet their

needs. The U.S. Customary System is the preferred method in the United

States. The metric system is the preferred system worldwide.

• Understand that there is a structured relationship between and among

units of measure for length, area, weigh/mass, and volume in the metric

and U.S. Customary systems.

• Understand that weight and mass are different.


• The metric system and conversions between the systems will be

taught/reviewed in the third nine weeks.

• Use the Real World Intro to get students excited about the concept.

• Give students real life objects to measure using tools like measuring

tape, pan balance scale, measuring cups…

• Have them convert the measurements they find into other compatible

units. (i.e. 1 cup of Kool-aide could be 8 oz of Kool-aide or ½ pint).

• Give students pictures of items and have them match the picture to it

appropriate measurement unit. (i.e. Gorilla would go with ton or feet…)

• Use graphic organizers like G-man and Big G, to help students convert

liquid measures.

• Use Bingo to engage students.

• Take students outside to measure and compare objects and/or items

around the school. Take rulers, yard sticks, and meter sticks to use for

measuring.

• Students measure height, width, and length of classroom. Find the

number of cubic yards of air in the room. Make a model using cubes to

show that 27 cubic feet = one cubic yard. Suggested manipulatives:

straws, sticks, cubes, string, rulers, yardsticks, trundle wheel, measuring

cups, meter sticks, and scales.

• Students will use a variety of objects for measuring parts of the body

(fingers, arms, legs, waist to top of head, etc.) using things such as

Open response:

A recipe for macaroni and cheese calls

for

2 quarts of milk. I only have a pint

measuring tool. How can I use this tool

to make sure I used 2 quarts?

Fuji apples are $1.49 a pound. If the

apple I buy weighs 10 oz, how would I

figure the price of my apple?

Writing prompts:

Detrick is from Germany and has only

used the metric system. Explain to

Detrick

how the U.S. Customary system is similar

but different than the metric system.





Homework


6th grade

2011 - 2012

• Understand that measures are determined by quantitative comparison to

a standard unit.

• Understand that units of measure are determined by the attributes of the

object being measured.

• Understand that measures of length are expressed in linear units,

measures of area are expressed in square units, and measures of volume

are expressed in cubic units.


• Estimate the conversion of units of length, weight/mass, volume, and

temperature between the U.S. Customary system and the metric system

by using ballpark comparisons. Ex: 1 L ≈ 1qt. Ex: 4L ≈ 4 qts.

• Estimate measurements by comparing the object to be measured against

a benchmark.

• Compare and convert units of measure for length, area, weight/mass, and

volume within the U.S. Customary and metric system.

• Determine the most appropriate unit of measure for a given situation.

• Estimate measurements by comparing the object to be measured against

a benchmark.

• Solve measurement problems by estimating and determining length, using

standard and nonstandard units of measure.

• Solve measurement problems by estimating and determining

weight/mass, using standard and nonstandard units of measure.

• Solve measurement problems by estimating and determining area, using

standard and nonstandard units of measure.

• Solve measurement problems by estimating and determining liquid

volume/capacity, using standard and nonstandard units of measure.

Teacher Notes

• Making sense of various units of measure is an essential life skill, requiring

reasonable estimates of what measurements mean, particularly in relation

to other units of measure.

straws, cubes, books, and whatever else is readily available.

• Solve measurement problems by estimating and determining length,

using standard and nonstandard units of measure (i.e., rulers, tape

measures, trundle wheels, etc,).

Model Lessons:


“Measuring Mania,”


“Working with Units”, p. 2-18

Resources:

Course 1: 12-1, 12-2, 12-4, 12-5





6th grade

2011 - 2012 – 1 inch is about 2.5 centimeters.

– 1 foot is about 30 centimeters.

– 1 meter is a little longer than a yard, or about 40 inches.

– 1 mile is slightly farther than 1.5 kilometers.

– 1 kilometer is slightly farther than half a mile.

– 1 ounce is about 28 grams.

– 1 nickel has the mass of about 5 grams.

– 1 kilogram is a little more than 2 pounds.

– 1 quart is a little less than 1 liter.

– 1 liter is a little more than 1 quart.

– Water freezes at 0°C and 32°F.

– Water boils at 100°C and 212°F.

– Normal body temperature is about 37°C and 98°F.

– Room temperature is about 20°C and 70°F.

• Mass is the amount of matter in an object. Weight is the pull of gravity on

the mass of an object. The mass of an object remains the same regardless

of its location. The weight of an object changes dependent on the

gravitational pull at its location. In everyday life, most people are actually

interested in determining an object’s mass, although they use the term

weight, as shown by the questions: “How much does it weigh?” versus

“What is its mass?”

• The degree of accuracy of measurement required is determined by the

situation.

• Whether to use an underestimate or an overestimate is determined by the

situation.

• Physically measuring objects along with using visual and symbolic

representations improves student understanding of both the concepts and

processes of measurement.

• Multiple experiences with using nonstandard and standard units of

measure to measure physical objects help students develop an intuitive

understanding of size.

• Chunking or benchmarks are strategies used to make measurement

estimates.

• Chunks of length such as a window’s length can be used to estimate the

length of classroom wall.

• Benchmarks such as the two-meter height of a standard doorway can be

used to estimate height.


6th grade

2011 - 2012 Quarter

3

Week 2

Number and Number Sense

SOL 6.2

The student will

a) investigate and describe fractions, decimals and percents

as ratios;

b) identify a given fraction, decimal or percent from a

representation;

c) demonstrate equivalent relationships among fractions,

decimals, and percents; and

d) compare and order fractions, decimals, and percents.


• What is the relationship among fractions, decimals and percents?

Fractions, decimals, and percents are three different ways to express the

same number. A ratio can be written using fraction form ( 2

3 ), a colon

(2:3), or the word to (2 to 3). Any number that can be written as a fraction

can be expressed as a terminating or repeating decimal or a percent.

• Understand how the magnitude of a fraction compares to another number

represented by a fraction.

• Understand how to represent the same fraction in multiple ways, using

concrete material, a drawing, a symbol, or a statement.

• Develop strategies to compare, order, and determine equivalency among

fractions and decimals.

• Understand that a percent is a way of representing fractions and decimals.

• Understand that a number can be written as a fraction, decimal, or

percent.

• Understand that percent is a method of standardization that is efficient

because each number is always based on 100ths.

• Understand that percents are used in real-life applications to compare or

describe data.

• Select appropriate methods for computing with rational numbers

according to the context of the problem.



• At this point students are working with fractions only – decimals and

percents will come later.

• Comparisons using pictorial representations.

• Construct as many true fractions sentences as possible using 4 digits.

Using the digits 5, 6, 7, and 8:

• Suggested manipulatives: base ten blocks, fraction bars, fraction circles,

decimal squares.

• Examples of real-life situations using estimation strategies include

shopping for groceries, buying school supplies, budgeting allowance,

deciding what time to leave for school or the movies, and sharing a pizza

or the prize money from a contest.

• Have students solve problems involving doubling recipes and/or figuring

out the amount of an ingredient needed in more than one recipe calling

for the same ingredient.

• Students will write problems using manipulative (like compare 5/8 and

0.6) then switch problems with a neighbor to solve.

• Change decimals to fractions by putting the decimal part over the

corresponding place value and reducing.

• Use newspapers and magazines to find equivalent fractions and

decimals.

• Divide students into groups of about 8. Give each student a card with a

decimal on it. Have them order themselves from least to greatest.

Teacher should check. Results can be displayed on a string by attaching

the cards with clothes pins. After they order decimals repeat with just

fractions. Then do with both decimals and fractions together.

• Create a card game. Students should match the equivalent fraction and

decimal.

• Compare equivalent fraction method and proportion method of

conversion.

• Use spreadsheet template to represent fractions on a 10x10 grid and

express fractions as equivalent decimals.

• Use newspapers and magazines to find equivalent fractions, decimals,

and percents.

Open response:

Jarred has two cakes that are the same

size. The first cake was chocolate, which

he cut into 12 equal parts. The second

cake was marble, which he cut into 6

equal parts. His family eats 5 slices of

chocolate cake and 3 slices of marble

cake. Did they eat more chocolate or

marble cake?

Show the decimal 0.8 as a picture

representation and write in fraction

form.

Explain your steps.

Order the numbers 4/5, 0.4, and 3/4

from least to greatest. Explain how you

got your answer. My mom said I could

eat 3/8 of my holiday candy or 0.4 of it.

Which option should I choose and why?

Writing prompts:

John said 5/8 was closer to a half and

Jenny said it was closer to a whole. Who

is correct? Draw a picture and write 3

sentences to support your answer.

For each situation, decide whether the

best estimate is more or less than one

half.

Record your conclusions and reasoning.

1. When pitching, Alan struck out 9 of

the 19 batters.

2. Sidney made 7 out of 17 free throws

3. Steven made 8 field goals out of 11

attempts

4. Kyle made 8 hits in 15 times at bat.

5. Maria only responded to 4 of her 35

text messages.

Have students explain in writing their

process for representing a fraction on a

10x10 grid and how they represent the

same fraction as an equivalent decimal.

Have students explain how to find the

sales tax and tip on a restaurant meal.


6th grade

2011 - 2012

• Identify the decimal and percent equivalents for numbers written in

fraction form including repeating decimals.

• Represent fractions, decimals, and percents on a number line.

• Describe orally and in writing the equivalent relationships among

decimals, percents, and fractions that have denominators that are factors

of 100.

• Represent, by shading a grid, a fraction, decimal, and percent.

• Compare two decimals through thousandths using manipulatives, pictorial

representations, number lines, and symbols (<, ,, ≥≤ >, or =).

• Compare two fractions with denominators of 12 or less using

manipulatives, pictorial representations, number lines, and symbols

(<, ,, ≥≤ >, or =).

• Compare two percents using pictorial representations and symbols

(<, ,, ≥≤ >, =).

• Order no more than 3 fractions, decimals, and percents (decimals through

thousandths, fractions with denominators of 12 or less), in ascending or

descending order.

• Identify, compare, order, and determine equivalent relationships among

fractions and decimals. Decimals are limited to the thousandths place.

• Recognize that percent means “out of 100” or hundredths, using the

percent symbol (%).decimals, and percents. Decimals are limited to the

thousandths place.

• Draw a shaded region on a 10-by-10 grid to represent a given percent.

• Represent in decimal, fraction, and/or percent form a given shaded region

of a 10-by-10 grid.

• Represent a number in fraction, decimal, and percent forms. Fractions will

have denominators of 12 or less.

Teacher Notes

• Percent means “per 100” or how many “out of 100”; percent is another

name for hundredths.

• Use statistics from baseball cards to make comparisons.

• Have students calculate sales tax, tips, and discounts.

• Students bring to class newspaper and magazine clippings which express

the discount on sale items in a variety of ways, including percent off,

fraction off, and dollar amount off. For items chosen from the clippings,

the students discuss which form is the easiest form of expression of

discount, which is most understandable to the consumer, and which

makes the sale seem the biggest bargain.

• Students obtain menus from their favorite restaurants. In groups,

students record what they would like to order and the cost of each item.

Afterwards they are to determine the tax and the tip that they should

leave (using 20%) and the total cost of their meal.

Model Lessons:


“Rational Speed Matching”


“Order Up!” SOL 5.2


“Comparing and ordering”, p. 9-13

Resources:

Course 1: 3-1, 3-2, 5-5, 5-6, 5-7, 10-4, 10-5, 10-6, 10-7

Course 2: 5-4, 5-5, 5-6, 5-8, 7-5



“Fraction Model” I, II, III







Homework


6th grade

2011 - 2012

• A number followed by a percent symbol (%) is equivalent to that number

with a denominator of 100 (e.g., 30% = 30

100 =

3

10 = 0.3).

• Percents can be expressed as fractions with a denominator of 100 (e.g.,

75% = 75

100 =

3

4 ).

• Percents can be expressed as decimal (e.g., 38% = 38

100 = 0.38).

• Some fractions can be rewritten as equivalent fractions with

denominators of powers of 10, and can be represented as decimals or

percents

(e.g., 3

5 =

610

= 60

100 = 0.60 = 60%).

• Decimals, fractions, and percents can be represented using concrete

materials (e.g., Base-10 blocks, number lines, decimal squares, or grid

paper).

• Percents can be represented by drawing shaded regions on grids or by

finding a location on number lines.

• Percents are used in real life for taxes, sales, data description, and data

comparison.

• Fractions, decimals and percents are equivalent forms representing a

given number.

• The decimal point is a symbol that separates the whole number part from

the fractional part of a number.

• The decimal point separates the whole number amount from the part of a

number that is less than one.

• The symbol • can be used in Grade 6 in place of “x” to indicate

multiplication.

• Fractions may be represented and compared by using fraction

manipulatives, drawings, pictures, or symbols.

• Rational number is the set of numbers that can be written as a ratio or

fraction. Percents, and numbers represented in scientific notation with

positive exponents.


6th grade

2011 - 2012

• Equivalent relationships among fractions and decimals can be determined

by using manipulatives (e.g., fraction bars, base-ten blocks, fraction circles,

graph paper, and calculators).

• Strategies using 0, 1

2 and 1 as benchmarks can be used to compare

fractions.

• When comparing two fractions, use 1

2 as a benchmark. Example: Which

is greater, 4

7 or

3

9?

4

7 is greater than

1

2 because 4, the numerator, represents more than

half of 7, the denominator. The denominator tells the number of parts that

make the whole. 3

9 is less than

1

2 because 3, the numerator, is less than

half of 9, the denominator, which tells the number of parts that make the

whole. Therefore,

4

7 >

3

9.

• When comparing two fractions close to 1, use distance from 1 as your

benchmark. Example: Which is greater, 6 8

?7 9

or 6

7 is

1

7away from 1

whole. 8 1

9 9is away from 1 whole. Since

1 1

7 9> , then

6

7 is a greater

distance away from 1 whole than 8

9so

8 6

9 7> .

• Students should have experience with fractions such as

1

8 , whose

decimal representation is a terminating decimal (e. g.,

1

8 = 0.125) and


6th grade

2011 - 2012

with fractions such as

2

9 , whose decimal representation does not end but

continues to repeat (e. g.,

2

9 = 0.222…). The repeating decimal can be

written with ellipses (three dots) as in 0.222… or denoted with a bar above

the digits that repeat as in 0.2 .

Quarter

3

Week 3


SOL 6.1

The student will describe and compare data, using ratios, and will use

appropriate notations, such as a

b , a to b, and a:b.


• What is a ratio?

A ratio is a comparison of any two quantities. A ratio is used to represent

relationships within a set and between two sets. A ratio can be written

using fraction form

( 2

3 ), a colon (2:3), or the word to (2 to 3).

• What is a ratio? A ratio is a comparison of any two quantities. A ratio is

used to represent relationships within a set and between two sets. A ratio

can be written using fraction form, (2/3), a colon (2:3), or the word to (2 to

3).

• What makes two quantities proportional? Two quantities are proportional

when one quantity is a constant multiple of the other.

• Understand that a ratio is a comparison of two quantities.

• Understand that ratios can be represented in more than one way.


• Describe a relationship within a set by comparing part of the set to the

entire set.


• Include rates and unit rates.

• Have students in the classroom collect and compare data. For example,

boys to girls, students to teacher, colors of shoes, hair color, eye color,

etc.

• Survey classmates on favorite color and write ratios from results.

• Use ratio tables to solve word problems.

• Working in pairs, have students draw four cards. Use the numbers on

cards to write two different ratios.

• Students will exchange ratios and find equivalent ratios.

• Repeat the activity using different cards.

• Relate proportions to cooking recipes and other everyday things

(doubling, halving, etc.)

Model Lessons:


“Field Goals, Balls, and Nets”


“ Ratios and Proportions”, p. 21 – 26

Resources:

Course 1: 10-1, 10-2








Homework


6th grade

2011 - 2012

• Describe a relationship between two sets by comparing part of one set to

a corresponding part of the other set.

• Describe a relationship between two sets by comparing all of one set to all

of the other set.

• Describe a relationship within a set by comparing one part of the set to

another part of the same set.

• Represent a relationship in words that makes a comparison by using the

notations a

b, a:b, and a to b.

• Create a relationship in words for a given ratio expressed symbolically.

• Write proportions that represent equivalent

Teacher Notes

• A ratio is a comparison of any two quantities. A ratio is used to represent

relationships within and between sets.

• A ratio can compare part of a set to the entire set (part-whole

comparison).

• A ratio can compare part of a set to another part of the same set (part-

part comparison).

• A ratio can compare part of a set to a corresponding part of another set

(part-part comparison).

• A ratio can compare all of a set to all of another set (whole-whole

comparison).

• The order of the quantities in a ratio is directly related to the order of the

quantities expressed in the relationship. For example, if asked for the ratio

of the number of cats to dogs in a park, the ratio must be expressed as the

number of cats to the number of dogs, in that order.

• A ratio is a multiplicative comparison of two numbers, measures, or

quantities.

• All fractions are ratios and vice versa.

• Ratios may or may not be written in simplest form.


6th grade

2011 - 2012

• Ratios can compare two parts of a whole.

• Rates can be expressed as ratios.

• A ratio is used to represent a variety of relationships within a set and

between two sets.

• A ratio can be written using a fraction form (2/3), a colon 2:3, or the word

to (2 to 3).

• A proportion can be written as a/c is to b/d, a:b = c:d, or a is to b as c is to

d.

• A proportion can be solved by finding the product of the means and the

product of the extremes. For example, in the proportion a:b = c:d. a and d

are the extremes and b an c are the means. If values are substituted for a,

b, c, and d such as 5:12= 10:24, then the product of extremes (5 x 24) is

equal to the product of the means (12 x 10).

• In a proportional situation, both quantities increase or decrease together.

• In a proportional situation, two quantities increase multiplicatively. Both

are multiplied by the same factor.

• A proportion can be solved by finding equivalent fractions.

Quarter

3

Week 4


SOL 6.4

The student will demonstrate multiple representations of multiplication and

division of fractions.


• When multiplying fractions, what is the meaning of the operation?

When multiplying a whole by a fraction such as 3 x 1

2 , the meaning is the

same as with multiplication of whole numbers: 3 groups the size of 1

2 of

the whole. When multiplying a fraction by a fraction such as 2 3

3 4⋅ , we are

asking for part of a part. When multiplying a fraction by a whole number


• Non-calculator section of Math 6 SOL test includes questions from all

standards within the Computation and Estimation reporting category.

• Students, working in groups, explore fraction multiplication and division.

They use fraction circles and fraction bars to solve problems, such as,

“What is the fairest way to divide four cakes among five people?” Next,

they write in their math journal about the methods they used and the

reasons they believe their answers to be correct.

Model Lessons:


“Modeling Multiplication of Fractions,”

“Modeling Division of Fractions.”

Resources:

Course 1: 7-2a, 7-4a


Open response:

A factory used 3/5 of a barrel of almonds

to make 6 batches of granola bars. How

many barrels of almonds did the factory

put in each batch? Sara’s lemon cookie

recipe calls for 2 2/3 cups of sugar. How

much sugar would Sara need to make

1/3 of a batch of cookies? Jan is baking.

She needs 4 cups of sugar. Her problem

is that she only has a ½ measuring cup

and a ¾ measuring cup. What is the least

number of scoops that she could make in

order to get 4 cups?

Writing prompts:

Explain why the product of two fractions,

such as ½ x ½ is less than each fraction.

Draw a picture and use at least 3

sentences to explain your answer.



6th grade

2011 - 2012

such as 1

2 x 6, we are trying to find a part of the whole.

• What does it mean to divide with fractions?

For measurement division, the divisor is the number of groups and the

quotient will be the number of groups in the dividend. Division of fractions

can be explained as how many of a given divisor are needed to equal the

given dividend. In other words, for 1 2

4 3÷ the question is, “How many

2

3 make

1

4?”

For partition division the divisor is the size of the group, so the quotient

answers the question, “How much is the whole?” or “How much for one?”

• How are multiplication and division of fractions and multiplication and

division of whole numbers alike? Fraction computation can be approached

in the same way as whole number computation, applying those concepts

to fractional parts.

• What is the role of estimation in solving problems? Estimation helps

determine the reasonableness of answers.

• Understand that fraction computation uses the same ideas a whole-

number computation, applying those concepts to fractional parts.


• Demonstrate multiplication and division of fractions using multiple

representations.

• Model algorithms for multiplying and dividing with fractions using

appropriate representations.

• Multiply and divide with fractions and mixed numbers. Answers are

expressed in simplest form.

• Solve single-step and multistep practical problems that involve

multiplication and division with fractions and mixed numbers that include

denominators of 12 or less. Answers are expressed in simplest form.

Teacher Notes






Homework


6th grade

2011 - 2012

• Using manipulatives to build conceptual understanding and using pictures

and sketches to link concrete examples to the symbolic enhance students’

understanding of operations with fractions and help students connect the

meaning of whole number computation to fraction computation.

• Multiplication and division of fractions can be represented with arrays,

paper folding, repeated addition, repeated subtraction, fraction strips,

pattern blocks and area models.

• When multiplying a whole by a fraction such as 3 x 1

2 , the meaning is

the same as with multiplication of whole numbers: 3 groups the size of 1

2

of the whole.

• When multiplying a fraction by a fraction such as 2 3

3 4⋅ , we are asking for

part of a part.

• When multiplying a fraction by a whole number such as 1

2 x 6, we are

trying to find a part of the whole.

• For measurement division, the divisor is the number of groups. You want

to know how many are in each of those groups. Division of fractions can

be explained as how many of a given divisor are needed to equal the given

dividend. In other words, for 1 2

4 3÷ , the question is, “How many

2

3

make1

4?”

• For partition division the divisor is the size of the group, so the quotient

answers the question, “How much is the whole?” or “How much for one?”

• Using an area model assists with students’ developing understanding of

multiplication and division of fractions.

• Simplifying fractions to simplest form assists with uniformity of answers.

• It is helpful for students to simplify before they multiply fractions, using

the commutative property of multiplication to reduce fractions to simplest

form before multiplying.


6th grade

2011 - 2012 Quarter

3

Week 5

Computation and Estimation

SOL 6.6

The student will

a) multiply and divide fractions and mixed numbers; and

b) estimate solutions and then solve single-step and

multistep practical problems involving addition,

subtraction, multiplication, and division of fractions.


• How are multiplication and division of fractions and multiplication and

division of whole numbers alike? Fraction computation can be approached

in the same way as whole number computation, applying those concepts

to fractional parts.

• What is the role of estimation in solving problems? Estimation helps

determine the reasonableness of answers.

• Understand that fraction computation uses the same ideas as whole-

number computation, apply those concepts to fractional parts.


• Multiply and divide with fractions and mixed numbers. Answers are

expressed in simplest form.

• Solve single-step and multistep practical problems that involve addition

and subtraction with fractions and mixed numbers, with and without

regrouping, that include like and unlike denominators of 12 or less.

Answers are expressed in simplest form.

• Solve single-step and multistep practical problems that involve

multiplication and division with fractions and mixed numbers that include

denominators of 12 or less. Answers are expressed in simplest form.

• Convert fractions to equivalent forms to perform the operations of

addition and subtraction.

• Simplify fractional answers to simplest form.

Teacher Notes


• Have students solve problems involving doubling recipes and/or figuring

out the amount of an ingredient needed in more than one recipe calling

for the same ingredient. (Example: How much flour will be needed to

make to batches of chocolate chip cookies and one batch of oatmeal

cookies?)

• Give students a recipe and have them make enough for different

number of servings. (Ex. The recipe makes 4 servings. Ask them to make

enough for 12 people.)


standards with the Computation and Estimation reporting category.

• Use the area model to introduce concept.

• Work with fraction computation is divided into two weeks to allow more

time for mastery.

• Using manipulatives such as circle diagrams, pattern blocks grid paper,

and fraction bars help to build conceptual understanding and using

pictures and sketches to link concrete examples to the symbolic enhance

students’ understanding of operations with fractions and help students

connect the meaning of whole-number computation to fraction

computation.

Model Lessons:


“Modeling Multiplication of Fractions,”

“Modeling Division of Fractions.”

Resources:

Course 1: 5-3, 6-3, 6-4, 6-5, 6-6, 7-2, 7-3, 7-4, 7-5




Open response:

List 3 different fraction pairs that can be

combined to produce a whole number.

Explain how you know they will result in

a whole number. To design an outfit, you

need 5 ½ yards of fabric, but you only

have 2 2/3 yards. How do you know how

much fabric to buy, and what special

considerations should you take into

account? Compare adding and

subtracting fractions to adding and

subtracting whole numbers. Give several

examples to support your comparison.

A recipe for cookies calls for 2/3 cups of

sugar. If I want to make three batches of

cookies, should I double or triple the

recipe? Explain.

Writing prompts:

What are some instances that an exact

sum or difference is needed rather than

an estimate?

On the 2007 SOL test, 12 year olds were

given the following question:

Estimate the answer to 14/15 + 9/10.

A 1

B 2

C 23

D 25

27% of the students answered correctly,

with approximately equal numbers of

students choosing each of the wrong

answers. Explain what the thinking was

behind each of the choices.

I have a board that is 2 ¾ feet long. I

need to cut it so it will make a shelf that

is 1 4/5 feet long. I cut off 1 foot. Did I

cut off too much or too little? Explain

and include the correct amount to cut

off.




6th grade

2011 - 2012

• Simplifying fractions to simplest form assists with uniformity of answers.

• Addition and subtraction are inverse operations as are multiplication and

division.

• It is helpful to use estimation to develop computational strategies. For

example,

7 32

8 4⋅ is about

3

4 of 3, so the answer is between 2 and 3.

• When multiplying a whole by a fraction such as 1

32

⋅ , the meaning is the

same as with multiplication of whole numbers: 3 groups the size of 1

2 of

the whole.

• When multiplying a fraction by a fraction such as 2 3

3 4⋅ , we are asking for

part of a part.

• When multiplying a fraction by a whole number such as 1

62

⋅ , we are

trying to find a part of the whole.

• Equivalent forms are needed to perform the operations of addition and

subtraction of fractions.

• Rewriting an improper fraction as a mixed numeral assists with uniformity

of answers and concepts.

• There is an implied addition of the whole number part and the fractional

part in mixed numerals.

• Using manipulatives to build conceptual understanding and using pictures

and sketches to link concrete examples to the symbolic enhances

students’ understanding of operations with fractions and helps students

connect the meaning of whole-number computation of fraction

computation.



Homework


6th grade

2011 - 2012 Quarter

3

Week 6


SOL 6.7

The student will solve single-step and multistep practical problems involving

addition, subtraction, multiplication, and division of decimals.


• What is the role of estimation in problem solving? Estimation gives a

reasonable solution to a problem when an exact answer is not required. If

an exact answer is required, estimation allows you to know if the

calculated answer is reasonable.

• Understand how mathematics relates to problems in daily life.

• Understand how to represent problems within various contexts.

• Understand the importance of planning and maintaining a budget.


• Solve single-step and multistep practical problems involving addition,

subtraction, multiplication and division with decimals expressed to

thousandths with no more than two operations.

• Determine essential information necessary, including the operations

required to solve consumer application problems.

• Solve one step and multi-step consumer application problems involving

whole numbers, fractions with denominators not greater than 12 and

decimals.

• Represent the solution as a data table or graph.

• Present and justify the solution orally or in writing.

Teacher Notes

• Estimation and checking the reasonableness of a result enhances

computational proficiency.

• Various estimation strategies, such as front-end, compatible numbers, or

rounding, are effective for various operations and situations.

• Understanding the placement of the decimal point is very important when


• Teach students to look for compatible numbers as an estimation

strategy. Using a grocery store ad, find two items whose total is about

$2, $3, or $5.

• Students imagine they are in line at a checkout counter at a grocery

store. Give students a $25.00 spending limit and a list of items which has

a cost that is over the limit. Students estimate the total cost and, then,

determine which products to drop from the list to get below the limit by

rounding the prices to the nearest dollar.

• Use grid paper to make decimal models to show division of decimals.

• Use grid paper when solving decimal division to help students line up

decimal.

• Using catalogs or newspaper ads have students estimate the cost of

buying a certain number of items, including some multiple items (i.e. 3

gallons of milk, 2 boxes of cereal, and a half gallon of orange juice).

• Solving multi-step problems in the context of all real-life situations

enhances interconnectedness and proficiency with estimation strategies.

• Use simple fractions (1/2, 1/3, 1/4) in consumer applications, i.e., ½ off

of a $90.00 coat, ¼ of a budgeted amount, etc.

• Examples of practical situations solved by using estimation strategies

include shopping for groceries, buying school supplies, budgeting an

allowance, deciding what time to leave for school or the movies, and

sharing a pizza or the prize money from a contest.

• Include real-life application problems involving money, travel, work,

recreation, and home life.

• Give students catalogs and get them to make their own two-step

decimal problem. (Ex. Choose school supplies that you need for the

semester for a $20 budget.)

• Make sure to include real world problems displayed in charts graphs and

tables.

• Students should display their answers in many different forms (charts,

graphs, tables, and paragraph).

Model Lessons:


Open response:

You want to purchase three items at a

store for $4.43, $5.42, and $6.45. You

have $16. Estimate the total cost by

rounding the cost of each item to the

nearest dollar. Will you have a problem

making this purchase? Explain. Compare

to the actual cost of the three items.

How are the steps similar/different when

dividing by a whole number verses a

decimal?

How much money will you need to bring

to the movies if you plan to buy a ticket

for you and your friend, drinks, and

snacks? Research online to find current

prices.

You and two friends sold lemonade

during a yard sale and made $47.35.

Your expenses were $12.85. About how

much money will each friend get if you

share your earnings equally? Explain

your results.

Writing prompts:

Sally has $7.45 to spend on crayons for

his class. If each box of crayons cost 25

cents, how many boxes can he buy?

Justify your answer.

When are estimates more useful or

descriptive than exact answers?

Use grocery store advertisements to

estimate a budget for food for a birthday

party. Spend less than $5 per person.

Explain your choices.


VDOE Released SOL items (SOL 6.7)



Homework


6th grade

2011 - 2012 finding quotients of decimals. Examining patterns with successive decimals

provides meaning, such as dividing the dividend by 6, by 0.6, by 0.06, and

by 0.006.

• Solving multistep problems in the context of real-life situations enhances

interconnectedness and proficiency with estimation strategies.

• Examples of practical situations solved by using estimation strategies

include shopping for groceries, buying school supplies, budgeting an

allowance, deciding what time to leave for school or the movies, and

sharing a pizza or the prize money from a contest.

• A consumer application problem is defined as the type of problem that is

normally encountered in daily living, such as, but not limited to, money,

travel, work, recreation, and home life.

• A budget may be kept for short or long periods of time. Students may

keep a short-term budget to enable the purchase of an expensive item or

a long-term budget to facilitate a long-term spending plan.

“Practical Problems Involving Decimals,”


Resources:

Glencoe Course 1: 1-1, 1-7a, 3-4, 3-5, 3-5b, 4-2, 4-4



Quarter

3

Week 7


SOL 6.3

The student will

a) identify and represent integers;

b) order and compare integers; and

c) identify and describe absolute value of integers.


• What role do negative integers play in practical situations?

Some examples of the use of negative integers are found in temperature

(below 0), finance (owing money), below sea level. There are many other

examples.

• How does the absolute value of an integer compare to the absolute value

of its opposite? They are the same because an integer and its opposite are

the same distance from zero on a number line.

• Understand how to identify, represent, order and compare integers.

• Understand that an integer and its opposite are the same distance from

zero.


• Whole numbers may be represented and compared by using whole

number manipulatives, drawings, pictures, and symbols.

• Suggested manipulatives: laminated number lines showing -20 to +20,

game markers

• Have students work in groups to investigate integers. Give each group a

number line showing -20 to +20 and a deck of cards with the face cards

removed. Each student places a different color marker on zero. As the

student is dealt a card face up, the student moves that number of

places: red is negative, black is positive. The first student to reach -20 or

+20 wins.

• Students create a Celsius thermometer naming the temperature as

positive and negative integers. Give students diagrams of thermometers

with missing negative and positive integers, and have them fill it in.

• Suggestions for curriculum integration are: Health- body temperatures,

Geography- longitude and latitude, elevation, Physical Science- electrical

charges.

• Students are assigned an integer value from -20 to +20 and create a

human number line. Ask questions about absolute value, opposites,

greater than/less than or equal to, etc.

Open response:

Explain why any negative integer is less

than any positive integer.

Writing prompts:

Dan went on a diving trip. He dove 20

feet below sea level to the top of a

sunken ship. He then floated up towards

a fish that was five feet above the

sunken ship. Finally he dove down to the

floor of the lake which was fifteen feet

below the fish. Draw a picture, using

integers, to display his dive. Extension:

How many feet below sea level is the

bottom of the lake?

Compare a number line to a

thermometer. How are they alike? How

are they different?






6th grade

2011 - 2012 Essential Knowledge and Skills

• Identify an integer represented by a point on a number line.

• Represent integers on a number line.

• Order and compare integers using a number line.

• Compare integers, using mathematical symbols (<, >, =).

• Identify and describe the absolute value of an integer.

• Demonstrate absolute value using a number line.

• Determine the absolute value of a rational number.

Teacher Notes

• Integers are the set of whole numbers, their opposites, and zero.

• Positive integers are greater than zero.

• Negative integers are less than zero.

• Zero is an integer that is neither positive nor negative.

• A negative integer is always less than a positive integer.

• When comparing two negative numbers, the negative number that is

closer to zero is greater.

• An integer and its opposite are the same distance from zero on a number

line. For example, the opposite of 3 is -3.

• On a conventional number line, a smaller number is always located to the

left of a larger number (e.g., –7 lies to the left of –3; thus –7 < –3; 5 lies to

the left of 8 thus 5 is less than 8).

• Comparison between integers can be made by using the mathematical

symbols: < (less than), > (greater than), or = (equal to).

• The absolute value of a number is the distance of a number from zero on

the number line regardless of direction. Absolute value is represented as

|-6| = 6.

Model Lessons:


“Ground Zero”


“Organizing Numbers”, SOL 8.2


“Integers”, p. 13-19

Resources:

Course 1: 8-1

Course 2: 3-1

Venn Diagram Example


“Pan Balance Shapes”




Homework

Integers

Whole

Natural


6th grade

2011 - 2012 Quarter

3

Week 8


SOL 6.5

The student will investigate and describe concepts of positive exponents and

perfect squares.


• What does exponential form represent?

Exponential form is a short way to write repeated multiplication of a

common factor such as

5 x 5 x 5 x 5 = 54

.

• What is the relationship between perfect squares and a geometric square?

A perfect square is the area of a geometric square whose side length is a

whole number.

• How is taking a square root different from squaring a number? Squaring a

number and taking a square root are inverse operations.

• Understand that a power of a number is repeated multiplication of that

number by itself.


• Recognize and describe patterns with exponents that are natural numbers

by using a calculator.

• Recognize and describe patterns of perfect squares, not to exceed 202, by

using grid paper, square tiles, tables, and calculators.

• Recognize and describe patterns with square roots and squares by using

squares, grid paper, and calculators.

• Recognize powers of ten by examining patterns in a place-value chart: 104

= 10,000, 103 = 1000, 10

2 = 100, 10

1 = 10, 100 = 1.

• Determine the square root of a perfect square less than or equal to 400.

• Write scientific notation for a number greater than 10.

Teacher Notes

Instructional Strategies

• Have students explore the exponent key, the x^ 2 key, and the square

root key on a calculator. Students are encouraged to define the function

of each key.

• Students will work in groups using a number cube, and a 0-9 spinner to

express numbers in exponent form. One student will toss the cube to get

a number that all students will use as an exponent. Another student will

spin the spinner to get three different numbers. Each student will then

use any combination of the numbers with the exponent to write the

largest possible number. The student(s) to write the largest possible

number wins a point.

• Create a 9 or 16 square puzzle. The matching edges of the individual

squares should represent exponent form, standard form, and square

roots, and perfect squares. Give students the pieces of the puzzle cut

apart and ask them to arrange the pieces so that edges are equivalent.

• Apply prior knowledge of patterns to: powers of 10, exponents, and

square roots.

• Play BINGO. Students find numbers on the card that are equal to the

power called by the teacher.

• Write square and square root problems on a beach ball. Students catch

the ball and evaluate the problem under their right thumb.

• Have students use an almanac, astronomical chart, internet, or other

source to find and create a chart listing numbers, in both standard form

and scientific notation, such as, population, distances to planets, etc.

Model Lessons:


“Perfecting Squares,”


“Patterns, Functions, and Algebra”, p. 38-42(powers of

10, exponents)

“Patterns, Functions, and Algebra”, p. 43-48(squares and

square roots)

“Patterns, Functions, and Algebra”, p. 54-58(scientific notation)

Resources:

Course 1: 1-4, 4-1

Course 2: 1-9, 11-1

Open Response

Evaluate 10³, 104, 10

5, and 10

6. Explain

how you can evaluate 1020 without

using a calculator.

Explain how finding the square of a

number is like finding the area of a

square.

Writing prompts:

Write a short paragraph to explain why

expressions like 107 are written with

exponents.

Explain how finding the square of a

number is like finding the area of a

square.





Homework


6th grade

2011 - 2012

• In exponential notation, the base is the number that is multiplied; the

exponent represents the number of times the base is used as a factor. In

83, 8 is the base and 3 is the exponent.

• A power of a number represents repeated multiplication of the number

(e.g., 83 = 8 · 8 · 8 and is read “8 to the third power”).

• Any real number (other than zero) raised to the zero power is 1. Zero to

the zero power (0) is undefined.

• Perfect squares are numbers that result from multiplying any whole

number by itself, i.e. 36 = 6 x 6.

• Perfect squares may be represented geometrically as the areas of squares

the length of whose sides are whole numbers, i.e., 1 x 1, 2 x 2, or 3 x 3.

This can be modeled with grid paper, tiles, geoboards, and virtual

manipulatives.

• A square root of a number is a number which, when multiplied by itself,

produces the given number, i.e., the square root of 49 is 7 since 7 x 7 = 49.

• The square root of a number can be represented geometrically as the

length of a side of the square.

• Patterns in place-value charts provide visual meaning of exponents: 103 =

1000, 102 = 100, 10

1 = 10.

• Scientific notation for a number is expressed by writing the number as a

number greater than or equal to 1 but less than 10 times a power of 10,

e.g., 3.2 x 103 is scientific notation for 3,200.




Quarter

3

Week 9

Review SOLs for Benchmark Use old Release tests from past years Benchmark 3 (SIMS) Assessment

(March 22 – 28)

Quarter

4

Week 1


SOL 6.8

The student will evaluate whole number numerical expressions, using the order

of operations.


• What is the significance of the order of operations? The order of



standards within the Computation and Estimation reporting category.

• Use whole numbers and benchmark decimals only at this point.

• Mnemonic devices (i.e. PEMDAS)

• Create a list of real-life situations where order matters (i.e., cooking, car

Open response:

Using each of the numbers 1, 2, 3, and 4

only once, any or all of the four

operations and grouping symbols if

necessary, write expressions that equal

1, 2, 3 and 4.




6th grade

2011 - 2012 operations prescribes the order to use to simplify expressions containing

more than one operation. It ensures that there is only one correct answer.

• Understand that the order of operations describes the order to use to

simplify expressions containing more than one operation.

• Select appropriate strategies and tools to simplify expressions.

• Understand that whole numbers, fractions, and decimals are rational

numbers.

• Simplify expressions with whole numbers by using the order of operations

in a demonstrated sep-by-step approach. The expressions should be

limited to positive values and not include braces { } or absolute value | |.

• Find the value of numerical expressions, using order of operations, mental

mathematics, and appropriate tools. Exponents are limited to positive

values.


• Simplify expressions by using the order of operations in a demonstrated

step-by-step approach. The expressions should be limited to positive

values and not include braces { } or absolute value | |.

• Find the value of numerical expressions, using order of operations, mental

mathematics, and appropriate tools. Exponents are limited to positive

values.

Teacher Notes

• An expression, like a phrase, has no equal sign.

• Expressions are simplified by using the order of operations.

• The order of operations is a convention that defines the computation

order to follow in simplifying an expression.

• The order of operations is as follows:

� First, complete all operations within grouping symbols. If there are

grouping symbols within other grouping symbols, do the innermost

operation first.

� Second, evaluate all exponential expressions.

repair work, structural designs, getting ready for school).

Model Lessons:


“Order Up!”


“Order of operations and Properties”, p. 2

Resources:

Course 2: 1-3





Homework


6th grade

2011 - 2012

� Third, multiply and/or divide in order from left to right.

� Fourth, add and/or subtract in order from left to right.

• Parentheses ( ), brackets [ ], braces {}, and the division bar – as in

3 4

5 6

+

+

should be treated as grouping symbols.

• The power of a number represents repeated multiplication of the number

(e.g., 83 = 8 · 8 · 8). The base is the number that is multiplied, and the

exponent represents the number of times the base is used as a factor. In

the example, 8 is the base, and 3 is the exponent.

• Any number, except 0, raised to the zero power is 1. Zero to the zero

power is undefined

Quarter

4 Week

2

Geometry

SOL 6.12

The student will determine congruence of segments, angles, and polygons.


• Given two congruent figures, what inferences can be drawn about how the

figures are related? The congruent figures will have exactly the same size

and shape.

• Given two congruent polygons, what inferences can be drawn about how

the polygons are related? Corresponding angles of congruent polygons

will have the same measure. Corresponding sides of congruent polygons

will have the same measure.

• How do polygons that are similar compare to polygons that are

congruent?

• Congruent polygons are the same size and shape.

• Similar polygons have the same shape, and corresponding angles between

the similar figures are congruent. However, the lengths of the

corresponding sides are proportional. All congruent polygons are

considered similar with the ratio of corresponding sides being 1:1.

• Understand the meaning of congruence.

• Understand that similar geometric figures have the same shape but have


• Review proportions and ratios.

• Indirect measurement should be incorporated.

• Suggested manipulatives: rulers, cards stock paper for cutting out similar

shapes, grid paper for dilations, tracing paper, compass, and protractors.

• Using magazines have students look for examples of congruent figures

that are the same size and shape.

• Students use a straightedge to draw a picture (e.g., houses, books,

boxes, etc.). Only straight line segments that connect to form angles may

be used. Next, students exchange pictures with each other and, then,

attempt to duplicate their partner’s drawing. The compass is used to

construct line segments and angles that are congruent to those in their

partner’s drawing. Each student tries to make the drawing as close to the

original as they can. Finally students compare the copy with the original

to see how well they did.

• Create and compare scale drawings.

• Give each student two rectangular cards of different size to see if they

are similar. Have the students measure the cards in inches and compare

the two ratios to see if they are equal. If they are not similar, have

students cut one of the cards so they are similar. This can also be done

with triangles.

Writing prompts:

Compare and contrast various segments.

Compare and contrast various angles.





Homework


6th grade

2011 - 2012 different sizes.

• Understand how ratios and proportions can be used to determine the

length of something that cannot be measured directly.


• Characterize polygons as congruent and noncongruent according to the

measures of their sides and angles.

• Determine the congruence of segments, angles, and polygons given their

attributes.

• Draw polygons in the coordinate plane given coordinates for the vertices;

use coordinates to find the length of a side joining points with the same

first coordinate or the same second coordinate. Apply these techniques in

the context of solving practical and mathematical problems.

• Identify corresponding sides and corresponding angles of similar figures

using the traditional notation of curved lines for the angles.

• Determine if quadrilaterals or triangles are similar by examining the

congruence of corresponding angles and proportionality of corresponding

sides.

• Write proportions to express the relationships between the lengths of

corresponding sides of similar figures.

• Given two similar figures, write similarity statements using symbols such

as ΔABC ≈ ΔDEF, angle A corresponds to angle D, and AB corresponds to

DE.

Teacher Notes

• Congruent figures have exactly the same size and the same shape.

• Noncongruent figures may have the same shape but not the same size.

• The symbol for congruency is ≅ .

• The corresponding angles of congruent polygons have the same measure,

and the corresponding sides of congruent polygons have the same

measure.

• The determination of the congruence or noncongruence of two figures can

Model Lessons:


“Side to Side”

Resources:

Course 1: 13-3a, 13-6





6th grade

2011 - 2012 be accomplished by placing one figure on top of the other or by comparing

the measurements of all sides and angles.

• Construction of congruent line segments, angles, and polygons helps

students understand congruency.

• Two polygons are similar if corresponding (matching) angles are congruent

and the lengths of corresponding sides are proportional.

• Congruent polygons are a special type of similar polygons; the ratio of the

corresponding sides is 1:1.


6th grade

2011 - 2012 Quarter

4

Week 3

Geometry

SOL 6.13

The student will describe and identify properties of quadrilaterals.


• Can a figure belong to more than one subset of quadrilaterals?

• Any figure that has the attributes of more than one subset of

quadrilaterals can belong to more than one subset. For example,

rectangles have opposite sides of equal length. Squares have all 4 sides of

equal length thereby meeting the attributes of both subsets.

• Understand that plane figures are identified and described by their

similarities, differences, and defining properties.

• Understand that plane figures are classified by their defining properties.

• Understand that quadrilaterals can be classified according to the attributes

of their sides and/or angles.

• Understand that a quadrilateral can belong to one or more subsets of the

set of quadrilaterals.

• Understand that every quadrilateral in a subset has all of the defining

attributes of the subset. (If a quadrilateral is a rhombus, it has all the

attributes of a rhombus.)

• Understand the meaning of prefixes associated with the number of sides

of a polygon.


• Sort and classify polygons as quadrilaterals, parallelograms, rectangles,

trapezoids, kites, rhombi, and squares based on their properties.

Properties include number of parallel sides, angle measures and number

of congruent sides.

• Identify the sum of the measures of the angles of a quadrilateral as 360°.

• Classify and draw triangles, quadrilaterals, pentagons, hexagons,

heptagons, octagons, nonagons, and decagons, using a variety of tools.

• Identify by the number of sides or number of angles the following

polygons: pentagon, hexagon, heptagon, octagon, nonagon, and decagon.


• Emphasize key vocabulary.

• Emphasize deductive reasoning. (Given what you know, what does that

tell you?)

• Use Venn Diagrams and concept maps for classification.

• Have students work with a partner. Each partner describes a triangle or

quadrilateral according to its characteristics. The other partner must

model the figure described using a rubber band and a geo-board.

• Students take turns describing and modeling.

• Suggested manipulatives: A selection of quadrilateral shapes that can be

sorted by characteristics, geo-strips, geo-boards.

• Use a variety of manipulatives to construct examples of polygons.

• Have students make mobiles displaying polygons from 3 to 10 sides.

They may use any resources that they want including poster board,

wood, string, etc.

Model Lessons:


“Exploring Quadrilaterals,”


“Polygons” p. 2-11, 80-92

“Classifying Angles” p. 41-50

Resources:

Course 1: 13-4




Writing prompts:

Compare and contrast various

quadrilaterals.

Compare and contrast various plane

figures.





Homework


6th grade

2011 - 2012

• Classify and describe the similarities and differences in sets of triangles by

sorting.

• Classify a triangle based on the size of its angles and/or its sides.

• Determine that the sum of the measures of the angles of a triangle is 180°.

• Identify the classification(s) to which a quadrilateral belongs. (Include

classification by pairs of parallel sides.)

• Classify quadrilaterals, using deductive reasoning and inference.

Teacher Notes

• A quadrilateral is a closed planar (two-dimensional) figure with four sides

that are line segments.

• A parallelogram is a quadrilateral whose opposite sides are parallel and

opposite angles are congruent.

• A rectangle is a parallelogram with four right angles.

• Rectangles have special characteristics (such as diagonals are bisectors)

that are true for any rectangle.

• To bisect means to divide into two equal parts.

• A square is a rectangle with four congruent sides or a rhombus with four

right angles.

• A rhombus is a parallelogram with four congruent sides.

• A trapezoid is a quadrilateral with exactly one pair of parallel sides. The

parallel sides are called bases, and the nonparallel sides are called legs. If

the legs have the same length, then the trapezoid is an isosceles

trapezoid.

• A kite is a quadrilateral with two pairs of adjacent congruent sides. One

pair of opposite angles is congruent.

• Quadrilaterals can be sorted according to common attributes, using a

variety of materials.

• Quadrilaterals can be classified by the number of parallel sides: a

parallelogram, rectangle, rhombus, and square each have two pairs of

parallel sides; a trapezoid has only one pair of parallel sides; other


6th grade

2011 - 2012 quadrilaterals have no parallel sides.

• Quadrilaterals can be classified by the measures of their angles: a

rectangle has four 90° angles; a trapezoid may have zero or two 90°

angles.

• Quadrilaterals can be classified by the number of congruent sides: a

rhombus has four congruent sides; a square, which is a rhombus with four

right angles, also has four congruent sides; a parallelogram and a rectangle

each have two pairs of congruent sides.

• A square is a special type of both a rectangle and a rhombus, which are

special types of parallelograms, which are special types of quadrilaterals.

• The sum of the measures of the angles of a quadrilateral is 360°.

• A chart, graphic organizer, or Venn Diagram can be made to organize

quadrilaterals according to attributes such as sides and/or angles.


6th grade

2011 - 2012 Quarter

4

Week 4-

8

Review SOLs for end of year SOL Testing Instructional Strategies:

• Give students a recently past release test to complete. Identify areas of

weakness (include those from past benchmark tests). Review all class

weaknesses in whole group. Address individual weaknesses in small

groups.

• Utilize Glencoe Virginia Standards of Learning Assessment Practice for

SOL skills using old SOL numbers and crosswalk.

• Utilize Virginia SOL Mathematics Coach for practice of weak skills in

small groups by identified students based on past performance on

Benchmarks.

Posttest, 2009/2010 Release test,

Formative assessments, Bellwork,

Homework, Classwork

Final SOL Test in May

2012-2013 6th grade math curriculum pacing...

Documents