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Accelerated Math 6-8 CC3 Packet
CC3 Chapter 2 – Simplifying with Variables
CC3 2.1.1 Exploring Variables and Combining Like Terms
2-4. Build each collection of tiles represented below. Then name the collection using a simpler algebraic expression,
if possible. If it is not possible to simplify the expression, explain why not.
a. 3x + 5 + x2 + y + 3x2 + 2
b. 2x2 + 1 + xy + x2 + 2xy + 5
c. 2 + x2 + 3x + y2 + 4y + xy
d. 3y + 2 + 2xy + 4x + y2 + 4y + 1
Math Notes: Algebraic Vocabulary
Variable: A letter or symbol that represents one or more numbers.
Work Space and Examples:
Expression: A combination of numbers, variables, and operation symbols. For example,
2x + 3(5 – 2x) + 8. Also, 5 – 2x is a smaller expression within the larger expression.
Term: Parts of the expression separated by addition and subtraction. For example, in the
expression 2x + 3(5 – 2x) + 8., the three terms are 2x, 3(5 – 2x), and 8. The
expression 5 – 2x has two terms, 5 and –2x.
Coefficient: The numerical part of a term. In the expression 2x + 3(5 – 2x) + 8, for
example, 2 is the coefficient of 2x. In the expression 7x – 15x2, both 7 and 15 are
coefficients.
Constant term: A number that is not multiplied by a variable. In the expression 2x +
3(5 – 2x) + 8, the number 8 is a constant term. The number 3 is not a constant term,
because it is multiplied by a variable inside the parentheses.
Factor: Part of a multiplication expression. In the expression 3(5 – 2x), 3 and 5 – 2x are
factors.
CC3 2.1.2 Simplifying Expressions by Combining Like Terms
2-13. For each of the shapes formed by algebra tiles below:
Use tiles to build the shape.
Sketch and label the shape on your paper. Then write an expression that represents the perimeter.
Simplify your perimeter expression as much as possible. This process of writing the expression more simply by
collecting together the parts of the expression that are the same is called combining like terms.
a.
b.
c.
d.
2-14. Calculate the perimeter of the shapes in problem 2-13 if the length of each x-tile is 3 units and the length of
each y-tile is 8 units. Show all work. Use the above space at the right of the images.
Work Space and Examples:
Math Notes: Combining Like Terms
Combining Like Terms
Combining tiles that have the same area to write a simpler
expression is called combining like terms. See the example
shown at the right.
When you are not working with actual tiles, it can help to picture the tiles in your
mind. You can use these images to combine the terms that are the same. Here are two
examples:
Example 1: 2x2 + xy + y2 + x + 3 + x2 + 3xy + 2
⇒
3x2 + 4xy +y2 + x + 5
Example 2: 3x2 − 2x + 7 − 5x2 + 3x − 2
⇒
−2x2 + x + 5
A term is an algebraic expression that is a single number, a single variable, or the product
of numbers and variables. The simplified algebraic expression in Example 2 above
contains three terms. The first term is −2x2, the second term is x, and the third term is 5.
CC3 2.1.3 What does “minus” mean? Writing Algebraic Expressions.
2-22. LEARNING LOG
What does “ – ” mean? Find as many ways as
you can to describe this symbol and discuss how
these descriptions differ from one another.
Meanings of Minus
2-25. BUILDING EXPRESSIONS
Use the Expression Mat (below) to create two of the following expressions with algebra tiles. Find at least two
different representations for each expression. How do these relate to the “Meaning of Minus?”
a.
−3x + 4
b. −(y − 2)
c. −y − 3
d. 5x − (3 − 2x)
2-26. In problem 2-25, you represented algebraic expressions with algebra tiles. In this problem, you will need to
reverse your thinking to write an expression from a diagram of algebra tiles. Write algebraic
expressions for each diagram below.
Math Notes: Evaluating Expressions and Order of Ops
a.
b.
Evaluating Expressions and the Order of Operations
To evaluate an algebraic expression for particular values of the
variables, replace the variables in the expression with their known
numerical values and simplify. Replacing variables with their known
values is called substitution. An example is provided below.
Evaluate 4x – 3y + 7 for x = 2 and y = 1.
Replace x and y with their known values of 2 and 1, respectively,
and simplify.
4(2) − 3(1) + 7
=8−3+7
= 12
When evaluating a complex expression, you must remember to use
the Order of Operations that mathematicians have agreed
upon. Create an example of Order of Ops to the right.
c.
Work Space and Examples:
CC3 2.1.4 What makes zero? Using Zero to Simplify Algebraic Expressions
2-35. How can you represent zero with tiles on an Expression Mat? With your team, try to find at least two
different ways to do this (and more if you can). Be ready to share your
ideas with the class.
2-36. Gretchen used seven algebra tiles to build the expression shown at
right. Represent this same value three different ways, each time using a different
number of tiles. Be ready to share your representations with the class.
2-37. Build each expression below so that your representation does not match those of your teammates. Once
your team is convinced that together you have found at least two different, valid representations, sketch your
representations for the letter of your choice in multiple ways.
a.
b.
c.
−3x + 5 + y
−(−2y + 1)
2x − (x − 4)
2-38. Write the algebraic expression shown on each Expression Mat below. Use the model and then simplify the
expression by removing as many tiles as you can without changing the value of the expression. Finally, write the
simplified algebraic expression.
a.
b.
Closure: Practice the skill without tiles. Simplify the following expressions using your knowledge of combining like
terms and the ability to eliminate zero pairs.
Example: 3𝑥 − (5𝑥 + 1) + 4 = 3𝑥 − 5𝑥 − 1 + 4 = 𝟑𝒙 − 𝟑𝒙 − 2𝑥 − 𝟏 + 𝟏 + 3 = −2𝑥 + 3
a.
3x − (2x + 4) = ________________________________________________________________________
b. 7 − (4y − 3) + 2y – 4 = __________________________________________________________________
Math Notes: Commutative Property
The Commutative Property of Addition states that when adding two or
more numbers or terms together, order is not important. That is:
a+b=b+a
For example, 2 + 7 = 7 + 2
The Commutative Property of Multiplication states that when
multiplying two or more numbers or terms together, order is not
important. That is:
a·b=b·a
For example 3 · 5 = 5 · 3
However, subtraction and division are not commutative, as shown below.
7 − 2 ≠ 2 − 7 since 5 ≠ − 5
50 ÷ 10 ≠ 10 ÷ 50 since 5 ≠ 0.2
CC3 2.1.5 How can I simplify the expression? Using Algebra Tiles to
Simplify Algebraic Expressions
2-47. COMPARING EXPRESSIONS
Two expressions can be represented at the same time using an Expression
Comparison Mat. The Expression Comparison Mat puts two Expression
Mats side-by-side so you can compare them and see which one is
greater. For example, in the picture at right, the expression on the left
represents –3, while the expression on the right represents –2. Since −2 >
−3, the expression on the right is greater.
a.
Can you simplify each of the expressions so that fewer tiles are
used? Develop a method to simplify both sides of the Expression
Comparison Mats. Why does it work?
b. Which side of the Expression Comparison Mat do you think is greater
(has the largest value)? Agree on an answer as a team. Make sure each
person in your team is ready to justify your conclusion to the class.
2-48. As Karl simplified some
algebraic expressions, he recorded his
work on the diagrams below.
What move did he make in each example?
A. _______________________________________
B. _______________________________________
C. _______________________________________
Determine which is greater. Circle the greater mat
and write the appropriate inequality (<, >, or =)
between the words “Left” and “Right” for each.
Work Space and Examples:
2-49. Use Karl’s “legal” simplification moves to determine which side of each Expression Comparison Mat below
is greater. After each expression is simplified, state which side is greater (has the largest value). Circle the greater
mat and write the appropriate inequality (<, >, or =) between the words “Left” and “Right” for each.
Math Notes: : Legal Moves for Simplifying
Expressions
Three common ways to simplify or alter expressions on an Expression Mat
are illustrated below.
Work Space and Examples:
Removing an equal number of opposite tiles that are in the
same region. For example, the positive and negative tiles in
the same region at right combine to make zero.
Flipping a tile to move it out of one region into the opposite
region (i.e., finding its opposite). For example, the tiles in the
“−” region at right can be flipped into the “+” region.
Removing an equal number of identical tiles from both the
“−“ and the “+” regions. This strategy can be seen as a
combination of the two methods above, since you could first
flip the tiles from one region to another and then remove the
opposite pairs.
CC3 2.1.6 How can I simplify the expression? Using Algebra Tiles to Simplify Algebraic Expressions
2-56. WHICH IS GREATER? A little extra practice simplifying. Circle the greater mat and write the appropriate
inequality (<, >, or =) between the words “Left” and “Right” for each.
The Associative Property of Addition states that when adding three or more
numbers or terms together, grouping is not important. That is:
Math Notes: Associative Property
(a + b) + c = a + (b + c)
Work Space and Examples:
For example, (5 + 2) + 6 = 5 + (2 + 6)
The Associative Property of Multiplication states that when
multiplying three or more numbers or terms together, grouping is not
important. That is:
(a · b) · c = a · (b · c)
For example, (5 · 2) · 6 = 5 · (2 · 6)
However, subtraction and division are not associative, as shown below.
(5 − 2) −3 ≠ 5 −(2 −3) since 0 ≠ 6
(20 ÷ 4) ÷ 2 ≠ 20 ÷ (4 ÷ 2) since 2.5 ≠ 10
The Identity Property of Addition states that adding zero to any expression
gives the same expression. That is:
a+0=a
For example, 6 + 0 = 6
The Identity Property of Multiplication states that multiplying any
expression by one gives the same expression. That is:
1·a=a
For example, 1 · 6 = 6
Math Notes: Connecting algebra
steps to tile moves.
CC3 2.1.7 How can I write it? Simplifying and Recording Work.
While Athena was comparing the expressions shown
below, she was called out of the classroom. When her
teammates needed help, they looked at her paper and
saw the work shown to the right. Can you see how
Athena justified each step and connected it to her
“legal” tile
moves?
2-63. For each pair of expressions below, determine which is greater, carefully recording your steps as you go. If you
cannot tell which expression is greater, state, “Not enough information.” Make sure that you record your result after
each type of simplification. For example, if you flip all of the tiles from the “–” region to the “+” region, record the
resulting expression and indicate what you did using either words or symbols.
A. Which is greater?
Explanation or Justification
Left
3x − (2 − x) + 1
Right
−5 + 4x + 4?
B. Which is greater?
Left
2x2 − 2x + 6 − (−3x)
Right
−(3 − 2x2) + 5 + 2x
C. Which is greater?
Left
−1 + 6y − 2 + 4x − 2y
Right
x + 5y − (−2 + y) + 3x − 6
2-66. Practice the skill without tiles. Simplify the following expressions using your knowledge of combining like terms
and the ability to eliminate zero pairs. For each pair of expressions below, determine which is greater, carefully
recording your steps as you go. You do not need to justify each step with legal moves but try to make connections
between the tile experience and the non-tile experience. Be ready to share your process with the class.
a.
b.
c. Which is greater: 5 − (2y − 4) − 2 or −y − (1 + y) + 4?
d. Which is greater: 3xy + 9 − 4x − 7 + x or −2x + 3xy − (x − 2)?