BIG IDEA 3: Write, interpret, and use mathematical expressions and equations.
BENCHMARK CODE
BENCHMARK
MA.6.A.3.1
Write and evaluate mathematical expressions that correspond to given situations.
MA.6.A.3.2
Write, solve, and graph one- and two- step linear equations and inequalities.
MA.6.A.3.3
Works backward with two-step function rules to undo expressions.
MA.6.A.3.4
Solve problems given a formula.
MA.6.A.3.5
Apply the Commutative, Associative, and Distributive Properties to show that two expressions are equivalent.
MA.6.A.3.6
Construct and analyze tables, graphs and equations to describe linear functions and other simple relations using
both common language and algebraic notation.
Access Points for Students with Significant Cognitive Disabilities
Independent:
MA.6.A.3.In.a Write and solve number
sentences (equations) that correspond to realworld problem situations involving addition
and subtraction with two-digit numbers.
MA.6.A.3.In.b Use models and diagrams to
solve problems with inequalities, including the
> and < signs.
MA.6.A.3.In.c Identify rules that use addition
and subtraction for functions represented in
number pairs.
MA.6.A.3.In.d Use the commutative and
associative properties of addition to show that
two number sentences (equations) are equal.
MA.6.A.3.In.e Solve addition and subtraction
number sentences (equations) using
information from physical models, diagrams,
and tables.
MA.6.A.3.In.f Use visual models, such as
tables, to show the relationship between
addition and subtraction problems.
Supported:
MA.6.A.3.Su.a Write and solve number
sentences (equations) that correspond to
real-world problem situations involving
addition and subtraction with one-digit
numbers.
MA.6.A.3.Su.b Use physical models and
diagrams to solve problems with
inequalities, including the terms more than
and less than.
MA.6.A.3.Su.c Use the commutative
property of addition to show that two
number sentences represented by physical
and visual models are equal.
MA.6.A.3.Su.d Use visual models, such as
two-column tables, to represent the
relationship of addition and subtraction
basic facts.
MA.6.A.3.Su.e Use information from
physical models, diagrams, tables, and
pictographs to solve number sentences
(equations) involving addition and
subtraction with one-digit numbers.
Participatory:
MA.6.A.3.Pa.a Follow a prompt to identify
a missing item from a familiar set.
MA.6.A.3.Pa.b Indicate the next step in a
sequence for two or more activities.
MA.6.A.3.Pa.c Match a repeating pattern
with two elements in two or more routines
or activities.
MA.6.A.3.Pa.d Indicate 1 and 1 more by
imitating a model in more than one activity.
FLORIDA GRADE 6 CONTENT MODULE MA.6.A.3.5
MA.6.A.3.5:
Apply the Commutative, Associative, and Distributive
Properties to show that two expressions are equivalent.
Table of Contents:
A. Vertical Alignment of prerequisite and subsequent benchmarks…………….4
B. Content description and glossary of terms used……………..………………..5
C. Description of student developmental level……………………………………11
Literacy strategies
Numeracy strategies
Brain-compatible learning strategies
E. Rigor/relevance Chart………………………………………………………...…..13
F. Gold Seal Lesson: a Mathematical Private Eye……………………………….14
G. Gold Seal Lesson: Arithmetic Made Easy ……………………………………..16
H. Gold Seal Lesson: Game Show Formats for Review …………………………22
I. Differentiation………………………………………………………………………..25
Strategies for SWD……………
Strategies for ELL…………
Strategies for enrichment…………
J. Sample questions from FCAT ……………………………..……………………..26
K. Links to web-based instructional materials……………………………….…….27
MA.6.A.3.5 – VERTICAL ALIGNMENT
1st Grade:
MA.1.A.1.2 Identify, describe, and apply addition and
subtraction as inverse operations.
MA.1.A.1.3 Create and use increasingly sophisticated
strategies, and use properties such as Commutative,
Associative and Additive Identity, to add whole numbers.
3rd Grade:
MA.3.A.1.2
Solve multiplication and division fact problems by
using strategies that result from applying number properties.
MA.6.A.3.5 Apply the Commutative, Associative, and Distributive
Properties to show that two expressions are equivalent.
7th Grade:
MA.7.A.3.1
Use and justify the rules for adding,
subtracting, multiplying, dividing, and finding the absolute value
of integers.
CONTENT DESCRIPTION : MA.6.A.3.5
Apply the Commutative, Associative, and Distributive Properties to show
that two expressions are equivalent.
Supports Big Idea 3: Write, interpret and use mathematical expressions and
equations.
Students will recognize equivalent expressions based on the application of the
commutative, associative, identity, inverse and/or distributive property.
Students will identify and name the following properties: commutative,
associative, identity, inverse, distributive
The FCAT will assess this benchmark using multiple choice questions. No more
than two properties will be needed to convert from one expressions to an
equivalent expression. The maximum number of variables is two. The questions
can be real-world or mathematical in context.
Students can consider many examples of actions in real life to see if they are
commutative. After considering these, variables can be assigned to each action
and expressions can be written with = or ≠.
 Getting dressed:
o Does the result change if the order of putting on underwear and
jeans is reversed?
o Does the result change if the order of putting on a ring and a watch
is reversed?
 Cooking:
o Does the result change if the order of putting milk and cereal in a
bowl is reversed?
o Does the result change if you heat oil first, then sauté the onions or
is it the same if you sauté the onions, then put in the oil?
An interesting discussion can ensue if getting dressed or cooking is considered
to be an operation. It seems that sometimes that operation is commutative and
sometimes it is not! Can an operation be considered commutative if sometimes
the result changes and sometimes it doesn’t?
The associative property can be illustrated by thinking of the items purchased at
the cafeteria. One student buys a sandwich ($1) and an apple ($.50) the first
time through the line, and then goes back to buy chips ($.75). Another student
just buys a sandwich ($1) on the first time through the line, and then goes back to
buy an apple ($.50) and chips ($.75). Does the amount of money spent change?
The teacher can pose the following problem to students to introduce the
distributive property. Sara and Brie are best friends and when they go shopping
together they always buy two of the same thing. Last Saturday, they bought two
bracelets priced at $5.50 each and two necklaces priced at $7.50 each. How
much did the two of them spend together? Ask many students to come up to the
board simultaneously and write down their work and answer. Some students will
inevitably add $5.50 and $7.50 first and then multiply by 2. Other students will
double $5.50 and double $7.50 then add the results. All of this work will be
visible on the board and can lead to a great discussion of the power of the
distributive property.
GLOSSARY OF RELATED MATHEMATICAL LANGUAGE
algebra The branch of mathematics that uses letters, symbols, and/or
characters to represent numbers and express mathematical relationships.
algebraic Making use of or referring to concepts or methods of algebra.
algebraic expression A mathematical phrase that is written using one or more
variables and constants, but which does not contain a relation symbol
( , , , , ,  ) (e.g., 3y + 6).
algebraic inequality An algebraic statement that is written using one or more
variables and constants that shows a greater than or less than relationship (e.g.,
3x + 7 > 22).
associative property A property of real numbers that states that the sum or
product of a set of numbers is the same, regardless of how the numbers are
grouped.
Examples: Addition:
2 + (3.5 + 1.3) = (2 + 3.5) + 1.3
Multiplication:
–6  (18  7) = (–6  18)  7
coefficient A constant that multiplies a variable (e.g., 3x + 4y = 14, 3 is the
coefficient of x and 4 is the coefficient of y).
commutative property of addition A property of real numbers that states that
the sum of two terms is unaffected by the order in which the terms are added;
i.e., the sum remains the same (e.g., 2 + 3.5 = 3.5 +  2 ).
commutative property of multiplication A property of real numbers that states
that the product of two factors is unaffected by the order in which they are
multiplied; i.e., the product remains the same (e.g., 3  5 = 5  3 and 5 • x = x •
5).
coordinate axes The two intersecting perpendicular lines in a plane that form
the four quadrants for locating points, given the ordered pair of the points; the
axes are referred to as the x-axis and the y-axis.
coordinate plane A plane containing a set of coordinate axes in which each
point is located by a set of coordinates (x,y); the point of intersection of the axes
is called the origin and has coordinates (0,0).
Example: On
below, point A has
the coordinate plane
coordinates ( 3,4) .
y
A
+
4
+
2
-
4
-
+
2
2
-
2
-
4
+
4
x
coordinate geometry The study of geometry using a coordinate plane.
coordinate system (also called rectangular coordinate system) A method of
representing points in the plane or in space by means of numbers; a point in a
plane can be located by its distance from both a horizontal and a vertical line
called the axes; the horizontal line is called the x-axis; the vertical line is called
the y-axis; the pairs of numbers are called ordered pairs; the first number, called
the x-coordinate, designates the distance along the horizontal axis; the second
number, called the y-coordinate, designates the distance along the vertical axis;
the point at which the two axes intersect has the coordinates (0,0) and is called
the origin.
coordinates An ordered pair of numbers that identifies a point on a coordinate
plane, written as (x,y).
distributive property A property of real numbers that states that the product of
the sum or difference of two numbers is the same as the sum or difference of
their products.
Example:
Multiplication over addition:
Multiplication over subtraction:
2(15 + 4) = 2  15 + 2  4
4(12  8) = 4  12  4  8
equal to ( = ) A symbol that means two things have the same amount, size,
number, or value.
equation A mathematical sentence stating that two expressions are equal.
equivalent Equal in value.
3 + 3 is equivalent to 2  3
(equivalent numerical expressions)
2.9 is equivalent to 2.90
(equivalent decimals)
1 yard is equivalent to 3 feet (equivalent lengths)
Examples:
function A mathematical relationship between two variables, an independent
variable and a dependent variable, where every value of the independent
variable corresponds to exactly one value of the dependent value.
greater than (>) A relationship showing that the first term or expression has a
value larger than the second term or expression (e.g., 5 + 3 > 5 – 2 and 2 x – 4 <
18).
inequality A mathematical statement containing one of the symbols:
 ,  ,  ,  , or  to indicate the relationship between two quantities (e.g., 6 – 2
 4  2, 5z  25, and
7  2 + 2).
inverse operation An operation that is the opposite of, or undoes, another
operation; addition and subtraction are inverse operations; multiplication and
division are inverse operations.
less than (<) A relationship showing that the first term or expression has a value
smaller than the second term or expression (e.g., 2 < 3 or –5 < –1).
linear equation An equation of the first degree that contains one or more
variables; a linear equation with one or two variables yields a straight line when
represented graphically (e.g., 3x – 1 = –7 or x + 2y = 12).
linear inequality An inequality of the first degree that contains one or more
variables (e.g., 3x > 2 or y < 2x – 1).
line graph A graph that uses line segments to show changes in data; the data
usually represents a quantity changing over time.
Example:
A value omitted from an equation
missing value
)).
that is needed to make the equation true (e.g., (2  3)  5 = 2  (3 
multiplier The number that one is multiplying by in a multiplication problem.
numeric expression Any combination of words, variables, constants, and/or
operators that result in a number; also known as an arithmetic expression.
numeric pattern An arrangement of numbers that repeat or that follow a
specified rule.
one-step equation or inequality An equation or inequality that can be solved in
one step.
operations Procedures used to combine numbers, expressions, or polynomials
into a single result (e.g., addition, subtraction, multiplication, division, exponents).
order of operations A specified sequence in which mathematical operations
are expected to be performed; an arithmetic expression is evaluated by following
these ordered steps: (1) simplify within grouping symbols such as parentheses or
brackets, starting with the innermost; (2) apply exponents – powers and roots; (3)
perform all multiplications and divisions in order from left to right; (4) perform all
additions and subtractions in order from left to right.
ordered pair A set of two numbers named in an order that matters; represented
by (x,y) such that the first number, x, represents the x-coordinate and the second
number, y, represents the y-coordinate when the ordered pair is graphed on the
coordinate plane; each point on the coordinate plane has a unique ordered pair
associated with it.
properties of real numbers Rules that apply to the operations with real
numbers.
Examples:
Commutative Property
Associative Property
Distributive Property
a+b=b+a
a + (b + c) = (a + b) + c
a(b + c) = ab + ac
ab = ba
a(bc) = (ab)c
rate of change The amount the function's output increases or decreases for
each unit of change in the input.
table A systematic or orderly list of values, usually in rows and columns.
table of values An organized list of values from a function/relation.
two-step algebraic equation An algebraic equation that requires two different
steps such as multiplication/division and addition/subtraction to solve.
value of a digit How much a digit is worth according to its place in a number.
variable A symbol used to represent a number or group of numbers in an
expression or an equation.
verbal expression A phrase stating a relationship; can be translated into a
mathematical/algebraic expression (e.g., twice a number can be represented as
2n).
verbal form A mathematical expression or relationship using words rather than
symbols.
verbally Expressed in written form or spoken in words.
written symbol A sign used to represent something such as an operation ( +, –,
 ,  ), a relationship (  ,  ,  ,  , or  ), or a special quantity (π).
DESCRIPTION OF STUDENT DEVELOPMENTAL LEVEL
LITERACY STRATEGIES
To master the vocabulary, students should make their own booklet of properties.
Each page will be about a single property and will include a Frayer model.
Outside the model, students can write cues to themselves to help them
remember each property.
COMMUTATIVE PROPERTY
DEFINITION
CHARACTERISTICS
EXAMPLES
NON-EXAMPLES
CUES TO
REMEMBER
THIS WORD
NUMERACY STRATEGIES
Numeracy, or quantitative literacy, is the ability to reason with numbers and other
mathematical concepts. Numeracy includes a comfort with using logic and
reasoning and the ability to apply mathematical knowledge to the solution of realworld problems and everyday tasks.
It should be automatic that students reverse the order of addends or factors to
ease computation. Be diligent in looking for students who missed this concept in
earlier grades and always think, for instance, that the “bigger” number always
goes on the top line.
Other students may already be very set in their ways and may only feel confident
when an addition or multiplication problem is set up vertically. Questions about
the properties are almost always set up horizontally! Be deliberate in designing
instruction so that students relate the two formats.
BRAIN-COMPATIBLE LEARNING STRATEGIES
Even though the properties mean very different things, they are often taught
together. This can exacerbate transfer problems.
The brain absorbs new learning into working memory for a short time and then
seeks to store that new learning into long-term memory. It does this, in part, by
searching long-term memory for similar concepts that allow the brain to make
meaning out of the new concept. The new concept is associated withsensory
memory cues.
When the brain tries to retrieve this knowledge, it searches for the memory cue.
If that cue is very distinct, it is easier to locate. If the cue is too similar to another
one, retrieval is less sure.
Consider teaching the properties at different times – separated by days or even
weeks. Revisit the learned properties repeatedly over time. The properties will
be more likely to be understood with their individual characteristics instead of
being confused with each other.
RIGOR/RELEVANCE CHART
QUADRANT C
(assimilation)
1. Students validate a given set of
instructions in “Guess a number” and
write the algebraic code for the
questions.
2. Students explain properties and
give examples of each one.
QUADRANT A
(acquisition)
QUADRANT D
(adaptation)
1. Students use the distributive
property to create a list of questions
for “guess a number”.
2. Students create questions for
game-show format situations.
QUADRANT B
(application)
GOLD
SEAL
LESSON
Subject(s)
Mathematics
Grade Level
6
Instructional
Focus
Student
Learning
Performance
Task
A MATHEMATICAL PRIVATE EYE
Rigor/Relevance
Framework
K
n
o
w
l
e
d
g
e
6
5
4
3
2
1
1
C
D
A
B
2 3 4
Application
5
Number Operation and Concepts
Students use number, number sense, and number relationships in a
problem-solving situation. Students communicate the reasoning
used in solving these problems.
Algebraic Concepts and Relationships
Students use algebraic methods to investigate, model, and interpret
patterns and functions involving numbers, shapes, data, and graphs
in a problem-solving situation. Students evaluate and communicate
the reasoning used in solving these problems.
Problem-Solving and Mathematical Reasoning
Students apply a variety of problem-solving strategies to investigate
and solve problems from across the curriculum as well as from
practical applications.
 Students manipulate algebraic expressions that involve the
distributive property into equivalent expressions.
 Students translate verbal expressions into algebraic
expressions.
 Students evaluate algebraic expressions for different values
of the variable.
 Students verify the reasonableness of the results obtained by
working backwards.
This lesson is adapted from “A Mathematical Private Eye” by Ji-Eun
Lee and Kyoung-Tae Kim which was published in Mathematics
Teaching in the Middle School, Vol. 12 No. 7 March, 2007.
Teachers should search www.nctm.org to find an archived copy of
this article.
Ms. Lee and Mr. Kim have come up with a clever take on “Guess
My Number” – adding a role play on the part of the students.
Working in pairs, one student plays a “private eye” and the other
student plays the “client”. The private eye asks questions of the
client based on personal numerical data. For instance, the private
eye tells the client, “I know your favorite number” or “I know your
house number” or “I know your dog’s age.” The private eye has
previously worked out some algebraic manipulations, broken down
into three or four steps, that will result in the original number.
Ms. Lee and Mr. Kim suggest that students keep track of both the
numerical value and its algebraic equivalent in table form. Here is
an example that might be shared in whole-group instruction so that
students get an idea of how to construct the question sequence prior
to the role-play.
The private eye says:
The client
writes down
the numbers:
The private
eye writes
down the
algebra:
I know your house
number.
Write it down, don't show
it to me.
Add 7 to your number.
12
x
19
Multiply the result by 3.
57
Subtract 21.
36
Divide by 3.
12
x+7
3(x + 7) = 3x
+ 21
3x + 21 - 21
= 3x
3x/3 = x
Students should make up more problems based on similar algebraic
manipulations – especially utilizing the distributive property. Pairs
of students should switch roles after each turn. After each turn, the
private eye should explain to the client the “secret” formula and how
it was developed. Verbalizing the algebra helps students deepen
their understanding of the processes.
Scoring Guide
Depending upon the class objectives, students can make up algebraic
manipulations based on other properties such as combining like
terms, adding the opposite, multiplying by a fraction, etc.
Students can be given a blank spreadsheet similar to the ones
used during the lesson. Students should make up a list of
instructions, fill in the algebraic justification and test three
numbers to prove by example that the algebra is accurate.
Attachments/
Resources
“A Mathematical Private Eye” by Ji-Eun Lee and Kyoung-Tae Kim
published in Mathematics Teaching in the Middle School, Vol. 12
No. 7 March, 2007.
GOLD
SEAL
LESSON
Subject(s)
Mathematics
Grade Level
6
Instructional
Focus
Student
Learning
Performance
Task
ARITHMATIC MADE EASY
Rigor/Relevance
Framework
K
n
o
w
l
e
d
g
e
6
5
4
3
2
1
1
C
D
A
B
2 3 4
Application
5
Number Operation and Concepts
Students use number, number sense, and number relationships in a
problem-solving situation. Students communicate the reasoning
used in solving these problems.
Problem-Solving and Mathematical Reasoning
Students apply a variety of problem-solving strategies to
investigate and solve problems from across the curriculum as well
as from practical applications.
Students will use their knowledge of properties to simplify
arithmetic problems.
This is a pretty traditional lesson that presents a review of the
properties in terms of how they help make it easier to do
arithmetic.
Students read through examples of the properties presented by two
cartoon characters.
The key to the lesson is that after each set of examples, the student
is asked to describe the property in their own words and to write an
example of the property. Verbalizing their understanding of the
properties will increase student understanding of how they apply to
arithmetic operations.
Scoring Guide
Teachers should read the student explanations and use Post-It
notes to give students feedback about their descriptions.
Attachments/
Resources
Worksheet: Simplifying Arithmetic With Properties
SIMPLIFYING ARITHEMETIC WITH PROPERTIES
I don’t really see why
we have to learn about
these properties. They
are all big, long words
that don’t help me do
anything!
That’s because you
don’t know the tricks
that you can use with
the properties to make
it easier to compute
the answer!
THE IDENTITY PROPERTY
This property is great! Anything that is multiplied by the number 1 keeps its
same identity. You use this all the time with fractions when you change to a
different denominator. You’re really multiplying by another name for 1.
4
3
+
You can’t add these without changing them both to a common
9
5
denominator of 45.
4
5
20
5
9
x =
When you multiplied by and
you were really multiplying by
9
5
25
5
9
another name for 1.
That means that the new fraction had the same value as the
original fraction.
3 9
27
x =
5 9
45
In algebra, they just use a variable in place of the original number and show that
when they multiply that variable by 1, it stays the same.
w ·1 = w
or
1 ·c2 = c2
For addition, you can always add zero to any number and the number keeps its
same identity.
0 + 12 = 12
67.3 + 0 = 67.3
p +0 = p
0+j=j
In your own words, describe the identity property rules:
THE COMMUTATIVE PROPERTY
By the way, there is no letter N in this
word!! C O M M U T A T I V E !!
This property can really make adding and multiplying easier because you can
change the order of two numbers without changing the answer.
6 + 75
If you start thinking of being 6 years old and trying to figure
out how old you’ll be in 75 years, it’s hard to do.
= 81
75 + 6
It’s much easier to think of being 75 years old and figuring
out how hold you’ll be in 6 years!
81 x 5
It’s tricky to figure out in your head how much 81 groups of 5
is.
= 405
5 x 81
It’s much easier to figure out the amount for 5 groups of 81.
Not all operations are commutative.
In your own words, explain how you know that subtraction is not commutative.
Show at least two examples.
In your own words, explain how you know that division is not commutative. Show at least
two examples.
THE ASSOCIATIVE PROPERTY
The only thing I
know about this
property is that it
always has three
numbers and
parentheses.
That’s true but also . . .
The order of the numbers
does not change, only the
grouping changes.
When you want to add three numbers, you have the freedom to group them
differently and the answer will not be affected.
8 + ( 2 + 75 )
Grouped this way, you’d have to add 8 + 77.
= 85
( 8 + 2 ) + 75
Grouped this way, it’s easier to add 10 + 75
5 x ( 4 x 11)
Grouped this way, you’d have to multiply 5 x 44.
= 220
( 5 x 4 ) x 11
Grouped this way, it’s easier to multiply 20 x 11.
The Associative Property doesn’t work for all operations – it only works for
addition and multiplication.
In your own words, explain how you know that division is not associative.
Use 20 ÷ 10 ÷ 2 in your explanation.
In your own words, explain how you know that subtraction is not associative.
Use 50 – 40 – 5 in your explanation.
DISTRIBUTIVE PROPERTY
This is my favorite property because it can help you do so many
computations way easier!
I know somebody who calls this the “paper-boy’ property, because
it reminds her of throwing the paper at each house.
7( 2 + 5 + 8 ) = 14 + 35 + 56
Let’s say that you have to multiply 8 x 521. You can do it the long way or use the
distributive property.
8( 500 + 21 ) = (8 x 500) + (8 x 21)
4000
The answer is
+
168
4,168
You could even do it this way: 8( 500 + 20 + 1 ) = (8 x 500) + (8 x 20) + (8 x
1)
4,000
+
160
+
4,168
This works great with mixed fractions, too. Do it the long way or use the
distributive property.
9x5
1
3
144
Simplify: 48
3
The long way is to write
9 16
9 x16
x
then multiply.
=
1
3
1x3
8
= 48
9x5
1
3
Distribute: 9( 5 +
1
1
) = (9 x 5) + ( 9 x )
3
3
=
45 + 3
= 48
Write a description of the distributive property in your own words. Write down
some cues you will be able to use in order to remember how it works.
GOLD
SEAL
LESSON
Subject(s)
Mathematics
Grade Level
6
Instructional
Focus
Student
Learning
Performance
Task
GAME SHOW FORMATS FOR REVIEW
Rigor/Relevance
Framework
K
n
o
w
l
e
d
g
e
6
5
4
3
2
1
1
C
D
A
B
2 3 4
Application
5
Number Operation and Concepts
Students use number, number sense, and number relationships in a
problem-solving situation. Students communicate the reasoning used
in solving these problems.
Algebraic Concepts and Relationships
Students use algebraic methods to investigate, model, and interpret
patterns and functions involving numbers, shapes, data, and graphs in
a problem-solving situation. Students evaluate and communicate the
reasoning used in solving these problems.
Statistics and Probability
Students use statistics and probability to analyze given situations and
the results of experiments. Students communicate the reasoning used
in arriving at a conclusion.
The student will identify the commutative, associative, distributive,
identity and inverse properties.
This activity uses a game-show format to provide distributed practice
with the properties of arithmetic and algebra.
What’s My Line? works like this: Three students walk before the
class and one after the other, they say, “I am the Commutative
Property”, “No, I am the Commutative Property”, then “No, I am the
Commutative Property” then sit down facing the class. (Prior to the
students walking on stage, the teacher has identified only one of the
students as being the “true” commutative property.)
The students in the class begin to present questions with yes or no
answers to each of the three contestants. The person who is the “true”
commutative property must always answer truthfully but the other two
people have the option to tell the truth or a lie.
Questions might be something like, “Are you a valid property for
division?” or “Can 2(x + y) be written as 2x + 2y using your
property? After five or six questions, the teacher says, “Will the true
commutative property please stand up!”
To prepare for the activity, students could be given a few minutes
before the game to write down some questions that they plan to ask. It
might work well for groups of two students to write these questions
together.
The next day, this game can be played again, this time with another
property. It is an activity that need not take more than 10 minutes but
provides a review activity in which all students are participants.
Another old game show was Concentration. This matching game can
work well for identifying equivalent expressions. The teacher
provides twelve index cards to pairs of students. An algebraic
expression is written on six of them and the student pairs write an
equivalent expression on the other six. The teacher should verify that
the student-written expressions are correct! This preparatory task can
take place in about 10 minutes on the first day.
Scoring Guide
Before class meets the next day, the teacher should rearrange the pairs
of cards into sets of 24 cards. Pairs of students place the cards
facedown on the desk, mixing up the order. The students take turns
turning over two cards to try to find equivalent expressions. When a
pair is uncovered, that student removes the cards and retains them.
The winner is the student who ends up with the most pairs.
Scoring Guide Attached
Scoring Guide:
After several days of review, teachers can use journal entry-type questions to
assess student learning. For example:
Write two ways that you
know that the expressions
below are equivalent.
4(x + 6)
4x + 24
Write your answer in complete sentences.
DIFFERENTIATION: MA.6.A.3.5
Students with disabilities
Although understanding is the goal, students with disabilities can also benefit
from gaining skill in alternative methods for answering multiple choice questions.
SAT-prep books call this the “substitute numbers” strategy.
When a question and all of the answer choices are algebraic expressions,
students can be taught to pick a value for each variable. They then can
substitute numbers for each variable to find the choice whose value matches the
expression in the question.
English Language Learners
Imagine that your native language is not English and you have to memorize
words like commutative, associative, distributive, identity and inverse! An ELL
student will certainly need additional time to conquer the terminology.
Splitting the content up into one property at a time will give ELL students a better
chance at conquering the vocabulary.
Extended Activities for Enrichment
Accelerated students should delve further into the distributive property with
division. Realizing that division is the same as multiplying by the reciprocal, can
the distributive work with division?
( 20 + 12 ) ÷ 4
32 ÷ 4
8
= ( 20 ÷ 4) + ( 12 ÷ 4 )
5
+
8
3
SAMPLE QUESTIONS FROM THE FCAT: MA.6.A.3.5
LINKS TO WEB-BASED INSTRUCTIONAL RESOURCES: MA.6.A.3.5
1. http://www.youtube.com/watch?v=u3dexXyyb3c This is a video clip where a
teacher teaches the distributive property to a student. It progresses from
numerical examples to algebraic examples.
2. http://videos.howstuffworks.com/hsw/22881-discovering-math-the-distributiveproperty-video.htm This short video relates a researcher mixing solutions to the
distributive property.
3. http://www.youtube.com/watch?v=iSuteidOkD4 This is kind of a cute
PowerPoint set to music that illustrates properties. Check the associative
property part to see if you agree with the presentation . . .
4.
http://www.education.vic.gov.au/studentlearning/teachingresources/maths/maths
continuum/structure/st275ip.htm This is a website for teachers to use. It gives
some good background ideas about how to teach these properties in a
meaningful way.
5. http://www.quia.com/hm/26073.html This is a Hang-Man game in which
students enter letters from a keyboard to guess a mathematical term. There are
a variety of words, not just properties, but it’s a nice little interactive game.
6. http://www.quia.com/cm/61114.html?AP_rand=1110244373 This is a game
that matches expressions with the names of properties.