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Math Grade 6 Instructional Guide 2011-2012
1
Subject: Math Grade 6
Benchmark Assessments and Instructional Guide
Instructional Guides are provided as resource for Alliance classroom teachers. They identify high priority grade-level standards to be taught during each quarter of
instruction in the context of proposed units with a suggested amount of time. High priority standards are assessed on quarterly benchmark exams.
The secondary curriculum begins in the sixth grade, where the concept of the unknown is solidified through the study of expressions and equations in a wide range
of settings. The rational numbers are thoroughly explored in a variety of forms including fraction, decimal, and percent. Examples of rational numbers in a real
world context involve the concepts of ratio, proportion, percentage, and rate. The concepts of shape and angle are studied, and formulas for area and volume are
discovered. Various methods of representing data are then used, and probabilities of events are calculated.
Unit
Unit 1: Exploring Integers
This course begins with a discussion
of the relationship between positive
and negative numbers, and the concept of an opposite. Examples of
positive and negative numbers in the
real world are explored. The set of
integers is then defined as the set of
whole numbers,
, and
their opposites. This leads to a discussion of the relationship between
the number line and the set of integers, including the meaning of addition and subtraction in terms of the
number line.
The addition and subtraction of integers is then performed, away from
the number line, and rules for adding
and subtracting two integers are discussed. The multiplication and division of two integers is also studied.
Rules for multiplying or dividing integers are then developed. The concept of positive and negative is revisited in terms of the rules for operations on integers. It is emphasized
High Priority Standards
And
Learning Targets*
Number Sense:
2.3 Solve addition, subtraction, multiplication, and division
problems, including
those arising in concrete situations that
use positive and negative integers and
combinations of these
operations.
Learning Targets
1C Explain how numbers are added or subtracted
1D Add or subtract one positive
and one negative integer.
1F Add or subtract more than two
integers and explain the process
1G Explain how to add or subtract
integers without using a
number line and perform
these operations.
1H Solve real world problems by
adding or subtracting integers.
1I Multiply or divide two positive
integers and explain the process
1J Explain the rules of multiplication and division of integers
and perform these opera-
# CST
Items
# Q1
Items
6
4
Supporting Medium/Low
Priority Standards
& Learning Targets
Number Sense:
1.1 Students compare and
order positive and negative fractions, decimals,
and mixed numbers and
place them on a number
line.
# CST Items
4.1
3
4.2
4.2 & 4.3
4.4
Learning Targets
1A Compare, explain, and give examples of positive and negative integers.
1B Identify and order positive and
negative integers on a number
line.
Algebra and Functions:
1.0 Students write verbal
expressions and sentences as algebraic expressions and equations; they evaluate algebraic expressions,
solve simple linear equations, and graph and interpret their results.
Textbook
McDougal Littell**
4.2-4.4
4.2-4.4
4.2-4.4
17
(all 1.0)
Learning Targets
1H Solve real world problems by adding or subtracting integers.
1Q Write and evaluate an algebraic
expression involving adding, subtracting, multiplying and dividing
4.5 & 4.6
4.5 & 4.6
4.5 & 4.6
4.5 & 4.6
4.5 & 4.6
4.5 & 4.6
4.5 & 4.6
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
Unit
that while multiplying or dividing two
negative integers gives a positive
integer, this is not true for addition
and subtraction. A variety of examples are given to justify this conclusion.
The end of this unit begins a theme
that is studied throughout the course,
namely the concepts of variable and
unknown. Variables are used to
write expressions, and when values
are assigned to those variables, the
expressions can be evaluated. The
meaning of variable in this context is
emphasized, namely that these expressions can be evaluated for a variety of numbers (at this level, usually
any number) and thus can vary when
different numbers are used. On the
other hand, an equation is a statement where one or both sides may
have an unknown number, and when
a number is used for the unknown,
the truth of the statement is evaluated. The process of solving an equation is introduced as the process of
finding numbers (there may be more
than one) that make the equation a
true statement. Throughout this discussion the difference between variable and unknown is emphasized,
namely that an unknown is a particular value that makes an equation
true, while a variable can be a choice
of values that make an equation true,
or a choice of values that are used to
evaluate an expression.
Unit 2: Order of Operations
High Priority Standards
And
Learning Targets*
# CST
Items
# Q1
Items
tions.
1K Multiply or divide one positive
and one negative integer.
1L Multiply or divide two negative
integers.
1M Explain the difference between
adding and subtracting two
negative integers and multiplying/dividing two negative
integers.
1N Multiply or divide more than
two integers.
1O Solve a real world problem by
multiplying or dividing integers.
Algebra and Functions:
1.1 Write and solve onestep linear equations
in one variable.
6
4
Learning Targets
1R Write and solve one-step linear
equations, explaining each
step in the process
1S Explain the difference between
an expression and an equation and the difference between evaluating and solving.
1.2 Write and evaluate an
algebraic expression
for a given situation,
using up to three variables.
Learning Targets
1P Explain the difference between
a variable and an unknown.
1Q Write and evaluate an algebraic expression involving adding, subtracting, multiplying
and dividing positive and
negative integers.
Algebra and Functions:
1
2
Math Grade 6 Instructional Guide 2011-2012
Supporting Medium/Low
Textbook
Priority Standards
# CST Items
McDougal Littell**
& Learning Targets
positive and negative integers.
5.3 & 5.4
1.3 Apply algebraic order of
1
operations and the
5.3 & 5.4
commutative, associative, and distributive
5.3 & 5.4
properties to evaluate
expressions; and justify
5.3 & 5.4
each step in the process.
Learning Targets
1C Explain how numbers are added or
subtracted
1F Add or subtract more than two
integers and explain the process
1G Explain how to add or subtract
integers without using a number
line and perform these operations.
1R Write and solve one-step linear
equations, explaining each step
in the process
Mathematical Reasoning:
1.0 Students make decisions
about how to approach
problems
2.0 Students use strategies,
skills, and concepts in
finding solutions.
3.0 Students move beyond a
particular problem by
generalizing to other situations.
All are embedded
Number Sense:
1.3-3.6
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
2
Unit
This unit expands upon the concept
of operations on integers from the
previous unit, to include multiple operations in one problem. Students
explore the correct order of operations, and perform error analysis to
solidify this understanding of the correct order. The importance of the
organization of steps in solving problems with more than one operation is
highlighted. Continuing the theme of
a variable, algebraic expressions are
written using variables and more
than one operation, and they are
evaluated for given values of a variable.
High Priority Standards
And
Learning Targets*
1.3 Apply algebraic order
of operations and the
commutative, associative, and distributive
properties to evaluate
expressions; and justify each step in the
process.
# CST
Items
# Q1
Items
1
2
Learning Targets
2A State the correct order of operations and explain why this
rule is necessary
2B Read the solution steps to an
order of operations problem,
find and explain the error,
and rework the problem correctly, justifying each step
2D Evaluate an algebraic expression using order of operations.
1.4 Solve problems manually by using the
correct order of operations or by using a
scientific calculator.
Learning Targets
2E Use the order of operations in
solving real world problems
that involve more than one
operation, including finding
the mean of a group of integers.
Math Grade 6 Instructional Guide 2011-2012
Supporting Medium/Low
Textbook
Priority Standards
# CST Items
McDougal Littell**
& Learning Targets
2.3 Solve addition, subtrac6
tion, multiplication, and
3.6& 4.7
division problems, including those arising in
10.1
concrete situations, that
use positive and negative integers and combinations of these operations.
Learning Targets
2C Evaluate numerical expressions
with more than one operation and
explain each step in the process.
2E Use the order of operations in solving real world problems that involve more than one operation,
including finding the mean of a
group of integers.
1
2
Algebra and Functions:
1.2 Write and evaluate an
algebraic expression for
a given situation, using
up to three variables.
1
Learning Targets
2D Evaluate an algebraic expression
using order of operations.
Statistics, Data Analysis,
and Probability:
1.1 Compute the range,
mean, median, and mode
of data sets.
1
Learning Targets
2E Use the order of operations in solving real world problems that involve more than one operation,
including finding the mean of a
group of integers.
Mathematical Reasoning:
1.0 Students make decisions
about how to approach
problems
2.0 Students use strategies,
skills, and concepts in
finding solutions.
Embedded
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
3
Unit
Unit 3: Factors and Multiples of
Positive Integers
Multiplication plays a critical role in
connecting the concepts of the integers and the rational numbers (fractions and decimals). In this unit, multiplication of positive integers is explored in two contrasting ways: writing a positive integer as the product
of other, smaller, positive integers
(factors), and multiplying a single
positive integer by a set of other
positive integers (multiples).
The concept of a factor is explored
first, and used to introduce prime and
composite numbers. Factoring is
used to write a positive integer as the
product of prime factors (i.e. the
prime factorization). Common prime
factors are then identified amongst a
group of positive integers. The product of the common prime factors is
defined as the greatest common factor, abbreviated the GCF. This concept is connected with the notion of
the greatest common divisor, learned
in elementary school. Prime factorization is then used to find the GCF of
a group of positive integers. Fractions are defined as an expression
for the division of two integers, and
the need for studying the GCF is
then illustrated through the process
of reducing fractions to lowest terms.
The GCF is then used to reduce frac-
High Priority Standards
And
Learning Targets*
Number Sense:
1.1 Compare and order
positive and negative
fractions, decimals,
and mixed numbers
and place them on a
number line.
# CST
Items
# Q1
Items
3
2
Learning Targets
3N Apply GCF and LCM to graph
a set of fractions on a number
line and to order a set of fractions from least to greatest or
vice versa.
2.4 Determine the least
common multiple and
the greatest common
divisor of whole numbers; use them to
solve problems with
fractions (e.g., to find
a common denominator to add two fractions or to find the reduced form of a fraction).
Learning Targets
3A Explain the concept of a factor
3B Explain the difference between
prime and composite numbers, and give examples.
3C Find the greatest common
factor of a number using
prime factorization.
3D Identify whether a fraction is in
lowest terms using the GCF.
3E Reduce a fraction to lowest
terms using the prime factorization of the numerator and
denominator.
3F Explain how the GCF is used in
3
2
Math Grade 6 Instructional Guide 2011-2012
Supporting Medium/Low
Textbook
Priority Standards
# CST Items
McDougal Littell**
& Learning Targets
3.0 Students move beyond a
particular problem by
generalizing to other situations.
Number Sense:
1.0 Students compare and
order positive and negative fractions, decimals,
and mixed numbers.
Students solve problems
involving fractions, ratios, proportions, and
percentages.
15
(all of 1.0)
Learning Targets
3C Find the greatest common factor of
a number using prime factorization.
3F Explain how the GCF is used in
reducing fractions.
3K Explain how the LCM is used in
ordering fractions.
Mathematical Reasoning:
1.0 Students make decisions
about how to approach
problems
2.0 Students use strategies,
skills, and concepts in
finding solutions.
3.0 Students move beyond a
particular problem by
generalizing to other situations.
Embedded
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
4
Unit
tions to lowest terms, and write
statements showing the equivalence
of the two fractions.
Another notion of the multiplication of
positive integers is then introduced,
namely the concept of a multiple.
Lists of multiples are written for given
positive integers and using these
lists, students identify common multiples from those lists. The smallest of
these common multiples is defined
as the least common multiple, abbreviated LCM. The need for the least
common multiple is also related to
fractions by finding the least common
denominator (LCD). Students then
find the LCD of a group of fractions,
rewrite the fractions using the LCD,
and then write the fractions in order
from least to greatest.
High Priority Standards
And
Learning Targets*
# CST
Items
# Q1
Items
Math Grade 6 Instructional Guide 2011-2012
Supporting Medium/Low
Textbook
Priority Standards
# CST Items
McDougal Littell**
& Learning Targets
reducing fractions.
3G Find the least common multiple
of a number using the lists of
multiples.
3H Find the least common multiple
of a number using prime factorization.
3I Explain the situations you would
use each method of finding
the LCM.
3J Use the least common multiple
to write a set of fractions so
that each fraction has the
same denominator.
3K Explain how the LCM is used in
ordering fractions.
3L Explain the similarities and
differences of the least common multiple, and the greatest common factor.
3M Compare and contrast methods for finding the least
common multiple, and the
greatest common factor.
This unit finishes with a careful discussion of the similarities and differences in the concepts of a factor of a
positive integer, and a multiple of a
positive integer. It is emphasized
that there are a finite number of
prime factors of a positive integer
and that those factors cannot be
larger than the original number. Furthermore, it is emphasized that there
are an infinite number of multiples of
a given positive integer, and that
those multiples cannot be smaller
than the original number. The methods for finding the GCF and the LCM
are also compared and contrasted,
as well as the relationship of each
value to fractions.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
5
Math Grade 6 Instructional Guide 2011-2012
Unit
Unit 4: Fractions & Decimals
The study of the concept of a fraction
is continued in this unit, with a discussion of the relationship between
fractions and decimals, and how to
perform operations on both kinds of
numbers. The discussion begins
with the addition and subtraction of
fractions with like denominators. The
notion of adding and subtracting fractions, and why we need to be able to
perform those operations on fractions, is illustrated through a variety
of real world examples. Connections
are then made to the previous unit,
as the LCD is used in adding and
subtracting fractions with unlike denominators.
The connection between multiplication and repeated addition is explored, and used to introduce the
concept of the multiplication of two
fractions. The relationship between
multiplication and division is then
highlighted, as the division of two
integers is interpreted as multiplication by the reciprocal. The concept
of a reciprocal is then used to divide
two fractions. The four arithmetic
operations on fractions are mastered,
and used to solve problems in a real
world context.
The concept of a fraction as division,
from the previous unit, leads into a
discussion of the decimal form of a
rational number. This discussion
begins with fractions whose numera-
High Priority Standards
And
Learning Targets*
Number Sense:
2.1 Solve problems involving addition, subtraction, multiplication, and division of
positive fractions and
explain why a particular operation was
used for a given situation.
# CST
Items
# Q2
Items
1
2
1
2
Textbook
McDougal Littell**
10
(all 2.0)
2.3 Solve addition, subtraction, multiplication, and
division problems, including those arising in
concrete situations, that
use positive and negative integers and combinations of these operations.
6
Learning Targets
4K Convert between the fraction and
decimal form of the number.
4L Add or subtract decimals and explain the algorithm
4M Multiply or divide decimals and
explain the algorithm
4N Solve real world problems involving
operations on decimals.
Learning Targets
4G Explain the reciprocal, and use
it to divide fractions.
4H Explain the meaning of dividing
by a fraction.
4J Solve the real world problems
involving multiplication and
division of fractions.
2.4 Determine the least
common multiple and
the greatest common
divisor of whole numbers; use them to
solve problems with
fractions (e.g., to find
a common denominator to add two factions or to find the re-
# CST Items
Learning Targets
4B Solve real world problems involving
the addition and subtraction of
fractions.
4L Add or subtract decimals and explain the algorithm
4M Multiply or divide decimals and
explain the algorithm
Learning Targets
4A Add or subtract fractions with
like denominators.
4B Solve real world problems
involving the addition and
subtraction of fractions.
2.2 Explain the meaning
of multiplication and
division of positive
fractions and perform
the calculations. (e.g.,
5/8 divided by 15/16=
5/8 x 16/15 = 2/3).
Supporting Medium/Low
Priority Standards
& Learning Targets
Number Sense:
2.0 Students calculate and
solve problems involving addition, subtraction,
multiplication, and division.
3
2
Mathematical Reasoning:
1.0 Students make decisions
about how to approach
problems
2.0 Students use strategies,
skills, and concepts in
finding solutions.
3.0 Students move beyond a
particular problem by
generalizing to other sit-
Embedded
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
6
Unit
tor is a multiple of the denominator.
The value of these fractions upon
division is an integer. The concept of
a decimal naturally arises in discussing fractions where the numerator is
not a multiple of the denominator.
The division process is then used to
write the decimal expansion of a fraction of this type. Connections are
made between the place value of a
decimal and powers of ten in the denominator of a fraction. Using this
concept, decimals are written in fraction form. Processes for the four
operations on decimals are mastered, and used to solve problems in
a real world context.
The theme of variable and unknown
is once again revisited at the end of
this unit. Simple equations involving
fraction and decimal coefficients are
solved using similar procedures as in
earlier units. Simple equations with
fraction and decimal solutions are
also studied.
High Priority Standards
And
Learning Targets*
duced form for a fraction).
# CST
Items
# Q2
Items
6
4
1
2
Learning Targets
4C Explain how to find a least
common denominator (LCD).
4D Add or subtract fractions with
unlike denominators and explain the process.
4E Explain the differences in adding or subtracting fractions
with like denominators, and
fractions with unlike denominators.
4F Multiply two or more fractions.
4J Solve the real world problems
involving multiplication and
division of fractions.
Algebra and Functions:
1.1 Write and solve onestep linear equations
in one variable.
Math Grade 6 Instructional Guide 2011-2012
Supporting Medium/Low
Textbook
Priority Standards
# CST Items
McDougal Littell**
& Learning Targets
uations.
Algebra and Functions:
1.0 Students write verbal
17
expressions and sen(all 1.0)
tences as algebraic expressions and equations; they evaluate algebraic expressions,
solve simple linear equations, and graph and interpret their results.
Learning Targets
4O Solve simple equations with
fraction and decimal coefficients.
4P Solve simple equations with
fraction and decimal solutions.
1.3 Apply algebraic order
of operations and the
commutative, associative, and distributive
properties to evaluate
expressions; and justify each step in the
process.
Learning Targets
4I Use order of operations for
problems with more than one
operation on fractions.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
7
Unit
Unit 5: Ratio and Proportion
In this unit, the concept of a fraction
from previous units is used to establish a relationship between two numbers or quantities. This discussion
begins by defining a ratio as a way of
writing the relationship between two
quantities using a fraction. Other
representations of a ratio are also
discussed (i.e. a:b, a to b). Ratios
are then used to explicitly write the
relationships between given quantities, including those arising from a
real world context. Variables are
also used to write ratios that represent a given relationship. These expressions are evaluated for given
values of the variables.
Just as a ratio relates two quantities
together using a fraction, a proportion also establishes a relationship
between two quantities, namely the
equivalence of two ratios. The difference between a ratio and proportion is highlighted and compared to
the relationship between an expression and an equation. Now that we
have established the proportion as
an equation, it is natural to discuss a
method for solving proportions,
namely cross-multiplication. Proportions are then solved using this
method, and their solutions are interpreted in a real world context. Finally, ratios and proportions are compared and contrasted, and linked to
an understanding of the difference
between expressions and equations
from previous units (i.e. ratios  ex-
High Priority Standards
And
Learning Targets*
Number Sense:
1.2 Interpret and use ratios in different contexts ( e.g., batting
averages, miles per
hour) to show the relative sizes of two
quantities, using appropriate notations (
a/b, a to b, a:b)
# CST
Items
# Q2
Items
1
2
Learning Targets
5H Compare ratios and proportions to
expressions and equations.
2.1 Convert one unit of
measurement to another
(e.g., from feet to miles,
from centimeters to
inches).
Learning Targets
5A Describe a ratio in your own
words.
5B Give a variety of examples of
ratios written in different ways
(a/b, a to b, a:b)
5C Represent a real life situation
using a ratio and explain the
connection between the situation and the mathematical
model
1.3 Use proportions to
solve problems (e.g.,
determine the value of
N if 4/7 = N/21, find
the length of a side of
a polygon similar to a
known polygon). Use
cross-multiplication
as a method for solving such problems,
understanding it as
the multiplication of
both sides of an equation by a multiplicative inverse.
Learning Targets
5D Relate two ratios using a proportion.
5E Explain the difference between
a ratio and a proportion.
5F Solve proportions by cross
multiplication.
5G Solve real world problems
using proportions and justify
Math Grade 6 Instructional Guide 2011-2012
Supporting Medium/Low
Textbook
Priority Standards
# CST Items
McDougal Littell**
& Learning Targets
Algebra and Functions:
1.2 Write and evaluate an
1
algebraic expression for
a given situation, using
up to three variables.
1
Learning Targets
5A Describe a ratio in your own words.
5D Relate two ratios using a proportion.
6
4
2.2 Demonstrate an understanding that rate is
measure of one quantity
per unit value of another
quantity.
6
Learning Targets
5C Represent a real life situation using
a ratio and explain the connection
between the situation and the
mathematical model
2.3 Solve problems involving
rates, average speed,
distance, and time.
1
Learning Targets
5G Solve real world problems using
proportions and justify the mathematical model used to solve the
problem.
Mathematical Reasoning:
1.0 Students make decisions
about how to approach
problems
2.0 Students use strategies,
skills, and concepts in
finding solutions.
3.0 Students move beyond a
particular problem by
Embedded
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
8
High Priority Standards
And
Learning Targets*
Unit
pressions and proportions  equations)
# CST
Items
# Q2
Items
6
4
6
4
the mathematical model used
to solve the problem.
Algebra and Functions:
1.1 Write and solve onestep linear equations
in one variable.
Math Grade 6 Instructional Guide 2011-2012
Supporting Medium/Low
Textbook
Priority Standards
# CST Items
McDougal Littell**
& Learning Targets
generalizing to other situations.
Learning Targets
5H Compare ratios and proportions to expressions and
equations.
Unit 6: Percentages
The concepts of ratio and proportion
from the previous unit are critical for
the study of percentage, as well as
the concepts of variable and unknown. A percentage, p%, is first
represented as the ratio
, and
then compared with another ratio
using the proportion
. Us-
ing this relationship, methods of conversion between the fraction form,
decimal form and percent form of a
number are developed. From here,
the focus of the unit shifts to a discussion of two different interpretations of percentage: the percent
equation (
) and percent
change.
The percent equation is studied and
used to answer the following types of
questions: what is p% of a number,
what number is p% of another number, and a number is what percent of
another number? These types of
questions are also posed in a real
world context. Percent change is
then introduced, and used to find the
Number Sense:
1.3 Use proportions to
solve problems (e.g.,
determine the value of
N if 4/7 = N/21, find
the length of a side of
a polygon similar to a
known polygon). Use
cross-multiplication
as a method for solving such problems,
understanding it as
the multiplication of
both sides of an equation by a multiplicative inverse.
Learning Targets
6A Explain the meaning of per-
1
Learning Targets
6B Explain how percent is related to
ratios and proportion.
Algebra and Functions:
1.1 Write and solve one-step
linear equations in one
variable.
6
Learning Targets
6G Use the percent equation to find
the percent of a number.
6I Use an algebraic equation to find
the base.
6L Increase or decrease a number by
a given percent, including
sales/taxes, tips, etc.
Learning Targets
6B Explain how percent is related
to ratios and proportion.
6C Explain how to convert between fractions and percents.
6D Convert between decimal,
fraction and percent form.
6E Solve real world problems by
converting between fractions,
decimals, and percents.
1.4 Calculate given percentages of quantities
and solve problems
involving discounts at
sales, interests
earned, and tips.
Number Sense:
1.2 Interpret and use ratios
in different contexts
(e.g., batting averages,
miles per hour) to show
the relative sizes of two
quantities, using appropriate notations (a/b, a to
b, a:b).
5
3
1.2 Write and evaluate an
algebraic expression for
a given situation, using
up to three variables.
Learning Targets
6J Find the percent decrease.
6K Find the percent increase.
Mathematical Reasoning:
1.0 Students make decisions
about how to approach
problems
1
Embedded
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
9
Unit
percent increase or decrease of two
numbers or quantities. Percent
change is then used to increase or
decrease a given quantity by a given
percent (e.g. increase 10 by 20%).
Percent change is also studied in a
real world context by solving problems involving growth, discounts, and
markup.
High Priority Standards
And
Learning Targets*
cents.
6F Find the percentage of a number and explain the connection to 100.
6G Use the percent equation to
find the percent of a number.
6H Use the percent equation to
find the part of a base and
explain what the base represents
6I Use an algebraic equation to
find the base.
6J Find the percent decrease.
6K Find the percent increase.
6L Increase or decrease a number
by a given percent, including
sales/taxes, tips, etc.
6M Explain the difference between
using the percent equation,
and finding a percent change.
# CST
Items
# Q2
Items
Math Grade 6 Instructional Guide 2011-2012 10
Supporting Medium/Low
Textbook
Priority Standards
# CST Items
McDougal Littell**
& Learning Targets
2.0 Students use strategies,
skills, and concepts in
finding solutions.
3.0 Students move beyond a
particular problem by
generalizing to other situations.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
Math Grade 6 Instructional Guide 2011-2012 11
Unit
Unit 7: Rates
Continuing the discussion of how
quantities are related, and how to
represent those relationships, the
focus of this unit is to relate two
quantities that have different units. A
rate is then defined as a special type
of ratio that compares two such
quantities. A variety of examples of
rates are introduced in a real world
context. Given that a rate is a special kind of ratio, a proportion is used
to show equivalent rates. It is emphasized that equivalent rates must
have the same type of units (e.g.
miles/hours can never equal gallons/hours). Using proportion, given
rates are converted to other, equivalent rates, with the same type of units
(e.g. convert miles/hour to feet per
second). Very often, rates are written so that the denominator represents one unit of measure, allowing
us to write statements such as “per
hour”. A method for finding a unit
rate from a given rate is discussed,
and used in a real world context.
Throughout this unit, solutions to
problems involving rates are checked
using dimensional analysis.
High Priority Standards
And
Learning Targets*
Algebra and Functions:
2.1 Convert one unit of
measurement to another ( e.g., from feet
to miles, from centimeters to inches).
# CST
Items
# Q3
Items
1
2
6
4
Learning Targets
7D Use proportion to find equivalent rates.
7F Use proportion to find unit
rates.
2.2 Demonstrate an understanding that rate
is a measure of one
quantity per unit value of another quantity.
Learning Targets
7A Compare the meaning of a rate
and a ratio.
7B Write a rate to represent a real
world situation.
7C Explain the meaning of equivalent rates.
7E Explain the meaning of a unit
rate, and give examples.
2.3 Solve problems involving rates, average
speed, distance, and
time.
Learning Targets
7G Solve real world problems
involving rates.
1
2
Supporting Medium/Low
Priority Standards
& Learning Targets
Number Sense:
1.2 Interpret and use ratios
in different contexts
(e.g., batting averages,
miles per hour) to show
the relative sizes of two
quantities, using appropriate notations (a/b, a to
b, a:b).
# CST Items
Textbook
McDougal Littell**
1
Learning Targets
7A Compare the meaning of a rate
and a ratio.
1.3 Use proportions to solve
problems (e.g., determine the value of N if 4/7
= N/21, find the length of
a side of a polygon similar to a known polygon).
Use cross-multiplication
as a method for solving
such problems, understanding it as the multiplication of both sides of
an equation by a multiplicative inverse.
6
Learning Targets
7C Explain the meaning of equivalent
rates.
7D Use proportion to find equivalent
rates.
1.4 Calculate given percentages of quantities and
solve problems involving discounts at sales,
interests earned, and
tips.
5
Learning Targets
7E Explain the meaning of a unit rate,
and give examples.
Algebra and Functions:
1.1 Write and solve one-step
6
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
Unit
High Priority Standards
And
Learning Targets*
# CST
Items
# Q3
Items
Math Grade 6 Instructional Guide 2011-2012 12
Supporting Medium/Low
Textbook
Priority Standards
# CST Items
McDougal Littell**
& Learning Targets
linear equations in one
variable.
1.2 Write and evaluate an
algebraic expression for
1
a given situation, using
up to three variables.
Learning Targets
7G Solve real world problems involving rates.
Unit 8: Measurement and Geometry
Continuing the study of how quantities are related even further, this unit
focuses on using variables and equations to relate geometric objects and
concepts. The ratio continues to be
a tool used to describe such a relationship. This unit begins with the
definition of the number
as the
ratio of a circle’s circumference to its
diameter. This ratio is explored for a
variety of circles, and it is shown that
these ratios are all equivalent. A
decimal and fractional estimate of the
number  (3.14, 22/7) is then developed, and it is emphasized that
is
simply a symbol that represents a
certain ratio, and that it is neither a
variable nor an unknown. Formulas
for area and circumference of a circle
Algebra and Functions:
3.0 Students investigate
geometric patterns
and describe them algebraically.
1
2
Learning Targets
8I Find the perimeter of a triangle
or quadrilateral.
8J Find the area of a triangle and a
quadrilateral and explain how
the formulas work in finding
the solution.
8K Solve real world problems
involving the area and perimeter of a triangle or quadrilateral.
3.1 Use variables in expressions describing
geometric quantities
(e.g., P = 2w + 2l, A =
1/2bh, C = d).
Mathematical Reasoning:
1.0 Students make decisions
about how to approach
problems
2.0 Students use strategies,
skills, and concepts in
finding solutions.
3.0 Students move beyond a
particular problem by
generalizing to other situations.
Number Sense:
1.2 Interpret and use ratios
in different contexts
(e.g., batting averages,
miles per hour) to show
the relative sizes of two
quantities, using appropriate notations (a/b, a to
b, a:b).
Embedded
1
Learning Targets
8K Solve real world problems involving
the area and perimeter of a triangle or quadrilateral.
1
2
Algebra and Functions:
1.1 Write and solve one-step
linear equations in one
variable.
6
Learning Targets
8N Describe different types of angle
pairs, including vertical, adjacent,
complementary and supplementary and use these properties to
find missing values of angles.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
Unit
are explored in terms of , and the
different estimates of
are used in
the formulas. Estimates of the circumference are then compared with
direct measurements.
The idea of an equation as a formula
is continued in a discussion of triangles and quadrilaterals. Properties of
these objects are explored, and the
formulas for the area of a triangle
and rectangle are discussed and
used in a real-world context. The
concept of a two-dimensional shape
is extended further to threedimensional shapes, or solids. Some
types of solids are given special
names, including a cube, a prism,
and a cylinder. Formulas for the volume of a cube, a triangular prism and
a cylinder are found and connected
back to the concept of area in two
dimensions.
The focus of the unit then shifts back
to two dimensions with the study of
angles. Different types of angles
(vertical, adjacent, complementary,
supplementary) are defined. The
theme of an unknown is then explored geometrically as problems are
solved involving finding a missing
angle. Angles within a shape are
also identified. The measures of the
angles of a triangle are explored, and
the triangle sum theorem is developed. This theorem is then used to
find the missing angle of a triangle,
once again solidifying the theme of
the unknown.
High Priority Standards
And
Learning Targets*
Measurement and
Geometry:
1.1 Understand the concept of a constant
such as ; know the
formulas for the circumference and area
of a circle.
# CST
Items
# Q3
Items
3
2
Learning Targets
8A Explain the meaning of pi, and
how it relates to the study of
ratios.
8C Find the area of a circle.
8D Find the circumference of a
circle.
1.2 Know common estimates of  (3.14; 22/7)
and use these values
to estimate and calculate the circumference
and the area of circles; compare with
actual measurements.
1
2
Learning Targets
8J Find the area of a triangle and a
quadrilateral and explain how the
formulas work in finding the solution.
8L Use the triangle sum theorem to
find the missing angle of a triangle.
8P Explain the relationship between a
cylinder and a circle.
Mathematical Reasoning:
1.0 Students make decisions
about how to approach
problems
2.0 Students use strategies,
skills, and concepts in
finding solutions.
3.0 Students move beyond a
particular problem by
generalizing to other situations.
Learning Targets
8B Use different estimates of pi
including 3.14 and 22/7 and
explain how these connect to
the exact value of pi
8E Directly measure the circumference of a circle and compare these measurements to
measurements found using
the formula for circumference.
1.3 Know and use the
formulas for the volume of triangular
prisms and cylinders
(area of base x
height); compare
these formulas and
explain the similarity
between them and the
formula for the vol-
Math Grade 6 Instructional Guide 2011-2012 13
Supporting Medium/Low
Textbook
Priority Standards
# CST Items
McDougal Littell**
& Learning Targets
1.4 Solve problems manually
by using the correct or1
der of operations or by
using a scientific calculator.
3.2 Express in symbolic
form simple relation1
ships arising from geometry.
1
Embedded
2
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
Unit
This unit closes by once again comparing and contrasting the different
ways a variable and an unknown are
used in a geometric context. This is
then connected to the notion of variable and unknown from previous
units.
High Priority Standards
And
Learning Targets*
ume of a rectangular
solid.
# CST
Items
# Q3
Items
1
2
4
3
1
2
Math Grade 6 Instructional Guide 2011-2012 14
Supporting Medium/Low
Textbook
Priority Standards
# CST Items
McDougal Littell**
& Learning Targets
Learning Targets
8O Identify three-dimensional
solids, including a prism and
cylinder.
8P Explain the relationship between a cylinder and a circle.
8Q Find the volume of a prism and
cylinder.
2.1 Identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions of
these terms.
Learning Targets
8N Describe different types of
angle pairs, including vertical,
adjacent, complementary and
supplementary and use these
properties to find missing values of angles.
2.2 Use the properties of
complementary and
supplementary angles
and the sum of the
angles of a triangle to
solve problems involving an unknown
angle.
Learning Targets
8L Use the triangle sum theorem
to find the missing angle of a
triangle.
8M Explain why the triangle sum
theorem is valid.
8N Describe different types of
angle pairs, including vertical,
adjacent, complementary and
supplementary and use these
properties to find missing values of angles.
2.3 Draw quadrilaterals
and triangles from
given information
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
Unit
High Priority Standards
And
Learning Targets*
about them (e.g. a
quadrilateral having
equal sides but no
right angles, a right
isosceles triangle).
# CST
Items
# Q3
Items
Math Grade 6 Instructional Guide 2011-2012 15
Supporting Medium/Low
Textbook
Priority Standards
# CST Items
McDougal Littell**
& Learning Targets
Learning Targets
8F Identify triangles and quadrilaterals.
8G Classify triangles by angle or
by side length.
8H Classify quadrilaterals by angle
or by side length.
Unit 9: Probability and Statistics
This unit begins with a discussion of
the concept of a data set. Different
statistical measures (range, mean,
median, mode) are used to analyze
different data sets. Tables and
graphs are used to represent data
that has been collected through a
variety of sampling methods. The
focus then shifts to probability, with a
discussion of theoretical and experimental probability. Using real world
examples, the experimental probability of an event is found. Given a
compound event, all possible outcomes are determined and the theoretical probability is calculated for
each outcome. This concept is
framed in a real world context and
used to solve problems involving
proportion and probability, continuing
the study of the unknown.
Statistics, Data Analysis,
and Probability:
1.1 Compute the range,
mean, median, and
mode of data sets.
1
2
3
2
Learning Targets
9A Find the range, mean, median,
and mode of a set of different
data sets and explain the results
2.2 Identify different ways
of selecting a sample
(e.g., convenience
sampling, responses
to a survey, random
sampling) and which
method makes a
sample more representative for a population.
Learning Targets
9A Find the range, mean, median, and
mode of a set of different data
sets and explain the results
1
1
1.3 Understand how the inclusion or exclusion of
outliers affect measures
of central tendency.
Learning Targets
9A Find the range, mean, median, and
mode of a set of different data
sets and explain the results
Learning Targets
9C Collect data using different
sampling methods.
2.3 Analyze data displays
and explain why the
way in which the
question was asked
might have influenced
the results obtained
and why the way in
which the results
were displayed might
Statistics, Data Analysis,
and Probability:
1.2 Understand how additional data added to data
sets may affect these
computations of
measures of central tendency.
1
2
1.4 Know why a specific
measure of central tendency (mean, median,
mode) provides the most
useful information in a
given context.
Learning Targets
9B Use tables and graphs to represent
data sets and how representation
of the data sets can influence
conclusions reached.
2.1 Compare different samples of population with
1
1
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
Unit
High Priority Standards
And
Learning Targets*
have influenced the
conclusions reached.
# CST
Items
# Q3
Items
Learning Targets
9B Use tables and graphs to represent data sets and how representation of the data sets
can influence conclusions
reached.
3.0 Students determine
theoretical and experimental probabilities
and use these to
make predictions
about events: SDP
3.1, SDP 3.2 and SDP
3.3
Learning Targets
9D Solve real world problems
involving experimental probability.
9E Find all possible outcomes of a
compound event.
9F Find the theoretical probability
of the outcome of a compound event.
9G Explain the difference between
theoretical and experimental
probability.
9H Solve real world problems
involving proportion and
probability.
9H Solve real world problems
involving proportion and
probability.
3
2
Math Grade 6 Instructional Guide 2011-2012 16
Supporting Medium/Low
Textbook
Priority Standards
# CST Items
McDougal Littell**
& Learning Targets
the data from the entire
population and identify a
situation in which it
makes sense to use a
sample.
Learning Targets
9C Collect data using different sampling methods.
2.4 Identify data that represent sampling errors and
explain why the sample
(and the display) might
be biased.
Learning Targets
9B Use tables and graphs to represent
data sets and how representation
of the data sets can influence
conclusions reached.
9C Collect data using different sampling methods.
1
1
3.2 Use data to estimate the
probability of future
events (e.g., batting averages or number of accidents per mile driven).
Learning Targets
9F Find the theoretical probability of
the outcome of a compound
event.
1
3.4 Understand that the
probably of either of two
disjoint events occurring
is the sum of the two individual probabilities
and that the probability
of one event following
another, in independent
trials, is the product of
the two probabilities.
Learning Targets
9G Explain the difference between
theoretical and experimental
probability.
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell
Unit
High Priority Standards
And
Learning Targets*
# CST
Items
# Q3
Items
Math Grade 6 Instructional Guide 2011-2012 17
Supporting Medium/Low
Textbook
Priority Standards
# CST Items
McDougal Littell**
& Learning Targets
Mathematical Reasoning:
1.0 Students make decisions
about how to approach
problems
2.0 Students use strategies,
skills, and concepts in
finding solutions.
3.0 Students move beyond a
particular problem by
generalizing to other situations.
Embedded
Unit 10: CST Review Unit
In this final unit, the connections that
have been formed from unit to unit
are solidified. The relationships between all the skill sets and concepts
that have been learned throughout
the course are communicated. As a
way of differentiating instruction, understanding of skills and concepts
from the course is self-assessed, and
an individualized plan is developed to
address any deficiencies.
Instruction Continues After CST- Q4
Math Culminating Projects
* Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A.
**Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell