Slide 1 / 172content.njctl.org/courses/math/6th-grade-math/... · Slide 5 / 172 Expressions Algebra extends the tools of arithmetic, which were developed to work with numbers, so

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6th Grade

Mathematical Expressions

2015-10-20

www.njctl.org

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Slide 3 / 172

Table of Contents

Click on a topic to go to that section.


Order of Operations

The Distributive Property

Like TermsTranslating Words Into Expressions

Evaluating Expressions

Glossary & Standards

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Table of Contents

Click on a topic to go to that section.


Order of Operations


Like TermsTranslating Words Into Expressions


Glossary & Standards

[This object is a pull tab]

Teac

her N

otes Vocabulary Words are bolded

in the presentation. The text box the word is in is then linked to the page at the end of the presentation with the word defined on it.

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Return to Table of Contents

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Expressions

Algebra extends the tools of arithmetic, which were developed to work with numbers, so they can be used to solve real world problems.

This requires first translating words from your everyday language (i.e. English, Spanish, French) into mathematical expressions.

Then those expressions can be operated on with the tools originally developed for arithmetic.

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Expressions An Expression may contain:

numbers, variables, mathematical operations

Example: 4x + 2 is an algebraic expression.

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There are two terms: 4x; 2

What is a Term?Terms of an expression are the parts of the expression which are separated by addition or subtraction.

Circle the terms of this expression.

Example: 4x + 2

Circle the terms and then click to check.

Slide 8 / 172

What is a Constant?A constant is a fixed value, a number on its own, whose value does not change. A constant may either be positive or negative.

Example: 4x + 2

In this expression 2 is the constant.

Circle the constant and then click to check.

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What is a Variable?

A variable is any letter or symbol that represents a changeable or unknown value.

In this expression x is the variable.

Example: 4x + 2

Circle the variable and then click to check.

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What is a Coefficient?

A coefficient is a number multiplied by a variable. It is located in front of the variable.

In this expression 4 is the coefficient.

Example: 4x + 2

Circle the coefficient and then click to check.

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If a variable contains no visible coefficient, the coefficient is 1.

Example 1: x + 7 is the same as (1)x + 7

Example 2: -x + 7 is the same as (-1)x + 7

Coefficient


If a variable contains no visible coefficient, the coefficient is 1.

Example 1: x + 7 is the same as (1)x + 7

Example 2: -x + 7 is the same as (-1)x + 7

Coefficient


Mat

h Pr

actic

e MP6: Attend to precision.

Continuously emphasize that the coefficient of 1 or -1 exists when when only the variable (or the negative variable) is given.

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1 In 2x - 12, the variable is "x".

True

False


1 In 2x - 12, the variable is "x".

True

False


Ans

wer

True

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2 In 6y + 20, the variable is "y".

True

False


2 In 6y + 20, the variable is "y".

True

False


Ans

wer

True

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3 In 3x + 4, the coefficient is 3.

True

False



True

False


Ans

wer

True

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True

False



True

False


Ans

wer

False

Slide 16 / 172

5 What is the constant in 7x - 3?

A 7B xC 3D -3


5 What is the constant in 7x - 3?

A 7B xC 3D -3


Ans

wer

D

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6 What is the coefficient in - x + 3?

A noneB 1C -1D 3


6 What is the coefficient in - x + 3?

A noneB 1C -1D 3


Ans

wer

C

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7 x has a coefficient.

True

False


7 x has a coefficient.

True

False


Ans

wer

True

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8 19 has a coefficient.

True

False


8 19 has a coefficient.

True

False


Ans

wer

False

Slide 20 / 172

Order of Operations


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Order of Operations

Mathematics has its grammar, just like any language.

Grammar provides the rules that allow us to write down ideas so that a reader can understand them.

A critical set of those rules is called the order of operations.

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Order of OperationsThe order of operations allows us to read an expression and interpret it as intended.

It lets us understand what the author meant.

For instance, the below expression could mean many different things without an agreed upon order of operations.

How would you evaluate this expression?

(5-8)(5)(3)-42÷2+8÷4+(3-2)

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Use ParenthesesParentheses will make your life much easier.

Each time you do an operation, keep the result in parentheses until you use it for the next operation.

You'll be able to read your own work, and avoid mistakes.

When you're done, read each step you did and you should be able to check your work.

Also, when you substitute a value into an expression, put it in parentheses first...that'll save you a lot of trouble.

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Order of Operations

Do all operations in parentheses first.

Then, do all exponents and roots.

(8-5)(5)(3)-42÷2+8÷4+(3-2)

(3)(5)(3)-42÷2+8÷4+(1)

(3)(5)(3)-(16)÷2+8÷4+1Then, do all multiplication and division.

(45)-(8)+(2)+1Then, do all addition and subtraction.

34

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Order of OperationsOne acronym used for the order of operations is PEMDAS which stands for:

ParenthesesExponents/RootsMultiplication/DivisionAddition/Subtraction

This order helps you read an expression...but it also helps you write expressions that others can read.

Since parentheses are always done first, you can always eliminate confusion by putting parentheses around what you want to be done first.

They may not be needed, but they don't ever hurt.


Order of OperationsOne acronym used for the order of operations is PEMDAS which stands for:

ParenthesesExponents/RootsMultiplication/DivisionAddition/Subtraction

This order helps you read an expression...but it also helps you write expressions that others can read.

Since parentheses are always done first, you can always eliminate confusion by putting parentheses around what you want to be done first.

They may not be needed, but they don't ever hurt.


Teac

her N

otes

You may wish to use this to help students remember.

P PleaseE ExcuseM D My DearA S Aunt Sally

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9 Evaluate the expression.

1 + 5 · 7



1 + 5 · 7


Ans

wer

36

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6 - 5 + 2



6 - 5 + 2


Ans

wer

3

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18 ÷ 9 · 2



18 ÷ 9 · 2


Ans

wer

4

Slide 29 / 172


40 ÷ 5 · 9



40 ÷ 5 · 9


Ans

wer

72

Slide 30 / 172


8 + 4 · 3



8 + 4 · 3


Ans

wer

20

Slide 31 / 172


5(3)2



5(3)2


Ans

wer

45

Slide 32 / 172


7 ∙ 9 − (7 − 4)3 ÷ 9 + (14 − 12)



7 ∙ 9 − (7 − 4)3 ÷ 9 + (14 − 12)


Ans

wer

62

Slide 33 / 172


(7 + 3)2 ÷ 25 + 4 ∙ 2 - (1 + 8)



(7 + 3)2 ÷ 25 + 4 ∙ 2 - (1 + 8)


Ans

wer

3

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Order of Operations and FractionsThe simplest way to work with fraction is to imagine that the numerator and the denominator are each in their own set of parentheses.

Before you divide the numerator by the denominator, you must have them both in simplest form.

And, then you must be very careful about what you can do with them.

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Order of Operations and Fractions

How would you evaluate this expression?

45 3(7-2)

45 3(5)

45 15

3

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Order of Operations(4)(3)-32÷3+6÷2+(15-8)

10-8

(4)(3)-32÷3+6÷2+(15-8)(10-8)

First, recognize that terms in a denominator act like they are in parentheses.

Then, do all operations in parentheses first. (Keep all results in parentheses until the next operation.)

Then, do all exponents.

(4)(3)-32÷3+6÷2+(7)(2)

(4)(3)-(9)÷3+6÷2+(7)(2)

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Order of Operations(4)(3)-9÷3+6÷2+(7)

(2)Then, all multiplication and division

Then, do all addition and subtraction.

Then, divide the numerator by the denominator.

(12)-(3)+(3)+(7)(2)

(15)(2)

7.5

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3(5 − 3)3 + 5(7 + 5) − 9 2 ∙ 5 + 5



3(5 − 3)3 + 5(7 + 5) − 9 2 ∙ 5 + 5


Ans

wer 5

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2(9 − 4)2 + 8 ∙ 6 − 3 3 ∙ 42 + 2



2(9 − 4)2 + 8 ∙ 6 − 3 3 ∙ 42 + 2


Ans

wer

1.9 = 1910

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4(10 − 8)2 + 7(3) + 15 25 − 22



4(10 − 8)2 + 7(3) + 15 25 − 22


Ans

wer

5.6

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[ 6 + ( 2 ∙ 8 ) + ( 42 - 9 ) ÷ 7 ] ∙ 3

Let's try another problem. What happens if there is more than one set of grouping symbols?

When there are more than 1 set of grouping symbols, start inside and work out following the Order of Operations.

Grouping Symbols

[ 6 + ( 2 ∙ 8 ) + ( 42 - 9 ) ÷ 7 ] ∙ 3

[ 6 + ( 16) + ( 16 - 9 ) ÷ 7 ] ∙ 3

[ 6 + ( 16) + (7) ÷ 7 ] ∙ 3

[ 6 + ( 16) + 1] ∙ 3

[ 23] ∙ 369

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[(3)(2) + (5)(4)]4-1



[(3)(2) + (5)(4)]4-1


Ans

wer

103

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[(2)(4)]2 - 3(5 + 3)



[(2)(4)]2 - 3(5 + 3)


Ans

wer

40

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[(8-3)(2)]2 ÷ (17 + 3)



[(8-3)(2)]2 ÷ (17 + 3)


Ans

wer

5

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Parentheses

Parentheses can be added to an expression to change the value of the expression.

4 + 6 ÷ 2 - 1 (4 + 6) ÷ 2 - 1 4 + 3 - 1 10 ÷ 2 - 1 7-1 5 - 1 6 4

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Parentheses

Change the value of the expression by adding parentheses. See how many different values your table can come up with.

5(4) + 7 - 22


Parentheses


5(4) + 7 - 22


Ans

wer

5(4) + (7 - 2)2

20 + 52

20 + 25 45

5[(4) + 7] - 22

5[11] - 22

55 - 4 51

Slide 47 / 172

Parentheses


12 - 3 + 9 ÷ 3


Parentheses


12 - 3 + 9 ÷ 3


Ans

wer

12 - (3 + 9) ÷ 312 - 12 ÷ 3

0 ÷ 30

(12 - 3 + 9) ÷ 318 ÷ 3

6

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23 Which expression with parentheses added in changes the value of:

36 ÷ 2 + 7 + 1

A (36 ÷ 2) + 7 + 1

B 36 ÷ (2 + 7) + 1

C (36 ÷ 2 + 7 + 1)

D none of the above change the value



36 ÷ 2 + 7 + 1

A (36 ÷ 2) + 7 + 1

B 36 ÷ (2 + 7) + 1

C (36 ÷ 2 + 7 + 1)

D none of the above change the value[This object is a pull tab]

Ans

wer

B

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5 + 14 - 7

A (5 + 14) - 7

B 5 + (14 - 7)

C (5 + 14 - 7)




5 + 14 - 7

A (5 + 14) - 7

B 5 + (14 - 7)

C (5 + 14 - 7)



Ans

wer

D

Slide 50 / 172


5 + 32 - 1

A (5 + 3)2 - 1

B 5 + (32 - 1)

C (5 + 32 - 1)




5 + 32 - 1

A (5 + 3)2 - 1

B 5 + (32 - 1)

C (5 + 32 - 1)

D none of the above change the value[This object is a pull tab]

Ans

wer

A

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Mat

h Pr

actic

eThis lesson addresses MP1, MP2 & MP4

Additional Q's to address MP standards:What is the problem asking? (MP1)How could you start the problem? (MP1)How could you represent the problem with symbols & numbers? (MP2)What connections do you see between the distributive property and the area of a rectangle? (MP4)

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Area Model

4

x 2

Write an expression for the area of a rectangle whose width is 4 and whose length is x + 2

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Area Model

4

x 2

You can think of this as being two rectangles.

One has an area of (4)(x) and the other has an area of (4)(2)

An expression for the total area would be 4x + 8

Or as one large rectangle of area (4)(x+2).

Slide 54 / 172

Distributive PropertyFinding the area of each rectangle demonstrates the

distributive property.

4(x + 2)4(x) + 4(2)

4x + 8

The 4 is distributed to each term of the sum (x + 2).

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Distributive PropertyNow you try:

6(x + 4) =

5(x + 7) =


Distributive PropertyNow you try:

6(x + 4) =

5(x + 7) =[This object is a pull tab]

Ans

wer

6x + 24

5x + 35

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Write an expression equivalent to:

2(x - 1) 4(x - 8)

Distributive Property


Write an expression equivalent to:

2(x - 1) 4(x - 8)



Ans

wer 2(x - 1)

2x - 24(x - 8)4x - 32

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a(b + c) = ab + ac Example: 2(x + 3) = 2x + 6

(b + c)a = ba + ca Example: (x + 7)3 = 3x + 21

a(b - c) = ab - ac Example: 5(x - 2) = 5x - 10

(b - c)a = ba - ca Example: (x - 3)6 = 6x - 18

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The Distributive Property can be used to eliminate parentheses, so you can then combine like terms.


For example:

3(4x - 6) 3(4x) - 3(6) 12x - 18

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26 Simplify 4(7x + 5) using the distributive property.

A 7x + 20

B 28x + 5

C 28x + 20



A 7x + 20

B 28x + 5

C 28x + 20


Ans

wer

C

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A 12x + 4

B 12x + 24

C 12x + 4

D 8x + 10



A 12x + 4

B 12x + 24

C 12x + 4

D 8x + 10


Ans

wer

B

Slide 61 / 172

28 Simplify 3(5m - 8) using the distributive property.

A 35m - 8

B 15m + 24

C 15m - 24

D 8m - 11


28 Simplify 3(5m - 8) using the distributive property.

A 35m - 8

B 15m + 24

C 15m - 24

D 8m - 11


Ans

wer

C

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29 ) 4(x + 6) is the same as 4 + 4(6).

TrueFalse


29 ) 4(x + 6) is the same as 4 + 4(6).

TrueFalse


Ans

wer

False

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30 Use the distributive property to rewrite the expression without parentheses. 2(x + 5)

A 2x + 5B 2x + 10C x + 10D 7x


30 Use the distributive property to rewrite the expression without parentheses. 2(x + 5)

A 2x + 5B 2x + 10C x + 10D 7x


Ans

wer

B

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31 Use the distributive property to rewrite the expression without parentheses. (x - 6)3

A 3x - 6B 3x - 18C x - 18D 15x


31 Use the distributive property to rewrite the expression without parentheses. (x - 6)3

A 3x - 6B 3x - 18C x - 18D 15x


Ans

wer

B

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32 Use the distributive property to rewrite the expression without parentheses. 0.6(3.1x + 17)

AB

CD

1.86x + 10.2

186x + 1021.86x + 17

.631x + .617


32 Use the distributive property to rewrite the expression without parentheses. 0.6(3.1x + 17)

AB

CD

1.86x + 10.2

186x + 1021.86x + 17

.631x + .617


Ans

wer

C

Slide 66 / 172

33 Use the distributive property to rewrite the expression without parentheses. 0.5(10x - 15)

A

B

C

D

5x - 7.5

5x - 15

10x - 7.5

5x - 75


33 Use the distributive property to rewrite the expression without parentheses. 0.5(10x - 15)

A

B

C

D

5x - 7.5

5x - 15

10x - 7.5

5x - 75


Ans

wer

C

Slide 67 / 172

34 Use the distributive property to rewrite the expression without parentheses. 1.3(6x + 49)

A

B

C

D

7.8x + 63.7

78x + 637

7.8x + 49

1.36x + 1.349


34 Use the distributive property to rewrite the expression without parentheses. 1.3(6x + 49)

A

B

C

D

7.8x + 63.7

78x + 637

7.8x + 49

1.36x + 1.349


Ans

wer

B

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Real Life SituationYou went to the supermarket and bought 4 bottles of orange soda and 5 bottles of purple soda. Each bottle cost $2. How much did you pay in all?

Use the distributive property to show two different ways to solve the problem.

$2 (4 orange sodas + 5 purple sodas)

($2 x 4 orange sodas) + ($2 x 5 purple sodas)

$2 x 9 sodas$18

OR

$8 + $10

$18

Slide 69 / 172

Real Life SituationYou bought 10 packages of gum. Each package has 5 sticks of gum. You gave away 7 packages to each of your friends. How many sticks of gum do you have left?

Use the distributive property to show two different ways to solve the problem.

5 sticks x (10 packages - 7 packages)

5 sticks x 3 packages

15 sticks of gum

(5 sticks x 10 packages) - (5 sticks x 7 packages)

50 sticks of gum - 35 sticks of gum

15 sticks of gum

OR

Slide 70 / 172

35 Canoes rent for $29 per day. Which expression can be used to find the cost in dollars of renting 6 canoes for a day?

A (6 + 20) + (6 + 9)

B (6 + 20) x (6 + 9)

C (6 x 20) + (6 x 9)

D (6 x 20) x (6 x 9)


35 Canoes rent for $29 per day. Which expression can be used to find the cost in dollars of renting 6 canoes for a day?

A (6 + 20) + (6 + 9)

B (6 + 20) x (6 + 9)

C (6 x 20) + (6 x 9)

D (6 x 20) x (6 x 9)


Ans

wer

C

Slide 71 / 172

36 A restaurant owner bought 5 large bags of flour for $45 each and 5 large bags of sugar for $25 each. The expression 5 x 45 + 5 x 25 gives the total cost in dollars of the flour and sugar. Which is another way to write this expression?

A 5 + (45 + 25)B 5 x (45 + 25)C 5 + (45 x 5) + 25D 5 x (45 + 5) x 25


36 A restaurant owner bought 5 large bags of flour for $45 each and 5 large bags of sugar for $25 each. The expression 5 x 45 + 5 x 25 gives the total cost in dollars of the flour and sugar. Which is another way to write this expression?

A 5 + (45 + 25)B 5 x (45 + 25)C 5 + (45 x 5) + 25D 5 x (45 + 5) x 25


Ans

wer

B

Slide 72 / 172

37 Tickets for the amusement park cost $36 each. Which expression can be used to find the cost in dollars of 8 tickets for the amusement park?

A (8 x 30) + (8 x 6)B (8 + 30) + (8 + 6)C (8 x 30) x (8 x 6)D (8 + 30) x (8 + 6)


37 Tickets for the amusement park cost $36 each. Which expression can be used to find the cost in dollars of 8 tickets for the amusement park?

A (8 x 30) + (8 x 6)B (8 + 30) + (8 + 6)C (8 x 30) x (8 x 6)D (8 + 30) x (8 + 6)


Ans

wer

A

Slide 73 / 172

Like Terms


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Like Terms



Mat

h Pr

actic

eThis lesson addresses MP1, MP2 & MP5

Additional Q's to address MP standards:What is the problem asking? (MP1)How could you start the problem? (MP1)How could you represent the problem with symbols & numbers? (MP2)How could you use drawings or manipulatives to show your thinking? (MP5)

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Like Terms: Terms in an expression that have the same variable(s) raised to the same power

Like Terms

6x and 2x

5y and 8y

4z and 7z

NOT Like Terms

6x and y

5y and 8

4y and z

Like Terms

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38 Identify all of the terms like 5y.

A 5B 4zC 18yD 8yE -1y



A 5B 4zC 18yD 8yE -1y


Ans

wer

C, D, E

Slide 76 / 172

39 Identify all of the terms like 8x.

A 5xB 4zC 8yD 8E -10x


39 Identify all of the terms like 8x.

A 5xB 4zC 8yD 8E -10x


Ans

wer

A, E

Slide 77 / 172

40 Identify all of the terms like 8xy.

A 5xB 4zyC 3xyD 8yE -10xy


40 Identify all of the terms like 8xy.

A 5xB 4zyC 3xyD 8yE -10xy


Ans

wer

C, E

Slide 78 / 172


A 51yB 2wC 3yD 2xE -10y



A 51yB 2wC 3yD 2xE -10y


Ans

wer

A, C, E

Slide 79 / 172

42 Identify all of the terms like 14z.

A 5xB 2zC 3yD 2xE -10x


42 Identify all of the terms like 14z.

A 5xB 2zC 3yD 2xE -10x


Ans

wer

B, E

Slide 80 / 172

43 Identify all of the terms like 0.75z.

A 75xB 75zC 3yD 2xE -10z


43 Identify all of the terms like 0.75z.



Ans

wer

B, E

Slide 81 / 172

44 Identify all of the terms like


2 3

x




2 3

x


Ans

wer

A, D

Slide 82 / 172


A 5xB 2xC 3zD 2zE -10x

1 4

z



A 5xB 2xC 3zD 2zE -10x

1 4

z


Ans

wer

C, D

Slide 83 / 172

Simplify by combining like terms

6x + 3x

(6 + 3)x

9x

Notice when combining like terms you add/subtract the coefficients but the variable remains the same.

Combining Like Terms



6x + 3x

(6 + 3)x

9x




Mat

h Pr

actic


Emphasize the addition/subtraction of the coefficients w/ the degree of the variable remaining the same.

Slide 84 / 172


4 + 5(x + 3)

4 + 5(x) + 5(3)

4 + 5x + 15

5x + 19





4 + 5(x + 3)

4 + 5(x) + 5(3)

4 + 5x + 15

5x + 19




Mat

h Pr

actic



Slide 85 / 172


7y - 4y

(7 - 4)y

3y





7y - 4y

(7 - 4)y

3y




Mat

h Pr

actic



Slide 86 / 172

46 Simplify the expression 8x + 9x.

A x

B 17x

C -x

D cannot be simplified


46 Simplify the expression 8x + 9x.

A x

B 17x

C -x



Ans

wer

B

Slide 87 / 172

47 Simplify the expression 7y - 5y.

A 2y

B 12y

C -2y



47 Simplify the expression 7y - 5y.

A 2y

B 12y

C -2y



Ans

wer

A

Slide 88 / 172

48 Simplify the expression 6 + 2x + 12x.

A 6 + 10x

B 20x

C 6 + 14x



48 Simplify the expression 6 + 2x + 12x.

A 6 + 10x

B 20x

C 6 + 14x



Ans

wer

C

Slide 89 / 172

49 Simplify the expression 7x + 7y.

A 14xy

B 14x

C 14y



49 Simplify the expression 7x + 7y.

A 14xy

B 14x

C 14y



Ans

wer

C

Slide 90 / 172

50 ) 8x + 3x is the same as 11x.

TrueFalse


50 ) 8x + 3x is the same as 11x.

TrueFalse


Ans

wer

True

Slide 91 / 172

51 ) 7x + 7y is the same as 14xy.

TrueFalse


51 ) 7x + 7y is the same as 14xy.

TrueFalse


Ans

wer

False

Slide 92 / 172

52 ) -12y + 4y is the same as -8y.

TrueFalse


52 ) -12y + 4y is the same as -8y.

TrueFalse


Ans

wer

True

Slide 93 / 172

53 ) -3 + y + 5 is the same as 2y.

TrueFalse


53 ) -3 + y + 5 is the same as 2y.

TrueFalse


Ans

wer

False

Slide 94 / 172

54 ) 5y - 3y is the same as 2y.

TrueFalse


54 ) 5y - 3y is the same as 2y.

TrueFalse


Ans

wer

True

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55 ) 7x + 3(x - 4) is the same as 10x - 12.

TrueFalse


55 ) 7x + 3(x - 4) is the same as 10x - 12.

TrueFalse


Ans

wer

True

Slide 96 / 172

56 ) 7 + 5(x + 2) is the same as 5x + 9.

TrueFalse


56 ) 7 + 5(x + 2) is the same as 5x + 9.

TrueFalse


Ans

wer

False

Slide 97 / 172

57 ) 4 + 6(x - 3) is the same as 6x -14.

TrueFalse


57 ) 4 + 6(x - 3) is the same as 6x -14.

TrueFalse


Ans

wer

True

Slide 98 / 172

58 ) 3x + 2y + 4x + 12 is the same as 9xy + 12.

TrueFalse


58 ) 3x + 2y + 4x + 12 is the same as 9xy + 12.

TrueFalse


Ans

wer

False

Slide 99 / 172

59 The lengths of the sides of home plate in baseball are represented by the expressions in the accompanying figure.

Which expression represents the perimeter of the home plate?

A 5xyzB x + yzC 2x + 3yzD 2x + 2y + yz

yz

yy

xx


59 The lengths of the sides of home plate in baseball are represented by the expressions in the accompanying figure.

Which expression represents the perimeter of the home plate?

A 5xyzB x + yzC 2x + 3yzD 2x + 2y + yz

yz

yy

xx[This object is a pull tab]

Ans

wer

D

Slide 100 / 172

xx+2

x+3

7

xx+2x+3

7

60 Find an expression for the perimeter of the octagon.

A x +24

B 6x + 24

C 24x

D 30x


xx+2

x+3

7

xx+2x+3

7

60 Find an expression for the perimeter of the octagon.

A x +24

B 6x + 24

C 24x

D 30x


Ans

wer

B

Slide 101 / 172

61 Brianna's teacher asks her which of these three expressions are equivalent to each other.

Brianna says that all three expressions are equivalent because the value of each one is -4 when x = 0. Brianna's thinking is incorrect.

Identify the error in Brianna's thinking. Determine which of the three expressions are equivalent. Explaon of show your process in determining which expressions are equivalent.

A Expression A: 9x - 3x - 4B Expression B: 12x - 4C Expression C: 5x + x - 4

From PARCC PBA sample test calculator #11

http://practice.parcc.testnav.com/#


61 Brianna's teacher asks her which of these three expressions are equivalent to each other.

Brianna says that all three expressions are equivalent because the value of each one is -4 when x = 0. Brianna's thinking is incorrect.

Identify the error in Brianna's thinking. Determine which of the three expressions are equivalent. Explaon of show your process in determining which expressions are equivalent.

A Expression A: 9x - 3x - 4B Expression B: 12x - 4C Expression C: 5x + x - 4



Ans

wer

A & C

Brianna only checked the value of each expression for one

substitution of x. To check which expressions are equivalent, you need to check that they are the

same value for any substitution of x.


Slide 102 / 172

62 Select each expression that is equivalent to 3(n + 6). Select all that apply.

A 3n + 6

B 3n + 18

C 2n + 2 + n + 4

D 2(n + 6) + (n + 6)

E 2(n + 6) + n

From PARCC EOY sample test non-calculator #4



62 Select each expression that is equivalent to 3(n + 6). Select all that apply.

A 3n + 6

B 3n + 18

C 2n + 2 + n + 4

D 2(n + 6) + (n + 6)

E 2(n + 6) + n

From PARCC EOY sample test non-calculator #4


Ans

wer

B & D


Slide 103 / 172

Translating Words Into Expressions


page3svg

Slide 104 / 172

Translating Between Words and Expressions

Key to solving algebra problems is translating words into mathematical expressions.

The two steps to doing this are:

1. Taking English words and converting them to mathematical words.

2. Taking mathematical words and converting them into mathematical symbols.

We're going to practice the second of these skills first, and then the first...and then combine them.

Slide 105 / 172

AdditionList words that indicate addition.


AdditionList words that indicate addition.


Ans

wer

Sum Total AddPlus Increased byMore ThanAltogether

Slide 106 / 172

SubtractionList words that indicate subtraction.


SubtractionList words that indicate subtraction.


Ans

wer

Minus Difference Take away Less than SubtractDecreased byLessFewer thanSubtracted from

Slide 107 / 172

MultiplicationList words that indicate multiplication.


MultiplicationList words that indicate multiplication.


Ans

wer

Product TimesOfTwice...Double...Multiplied by

Slide 108 / 172

DivisionList words that indicate division.


DivisionList words that indicate division.


Ans

wer

Divided by Quotient of Half...FractionsDivisible byDivisibility

Slide 109 / 172

Be aware of the difference between "less" and "less than".

For example:

"Eight less three" and "three less than eight" are equivalent expressions, so what is the difference in wording?

Eight less three: 8 - 3Three less than eight: 8 - 3

When you see "less than", take the second number minus the first number.

Less and Less Than


Be aware of the difference between "less" and "less than".

For example:

"Eight less three" and "three less than eight" are equivalent expressions, so what is the difference in wording?

Eight less three: 8 - 3Three less than eight: 8 - 3

When you see "less than", take the second number minus the first number.

Less and Less Than


Mat

h Pr

actic

eMP6: Attend to precision.Make sure that students know the order of the numbers when "less than" (and "more than" w/ addition) appear.

Additional Q's to help:What number is _ less than _? How did you get your answer? (MP2)- Note: Fill in the blanks with any numbers.

Slide 110 / 172

As a rule of thumb, if you see the words "than" or "from" it means you have to reverse the order

of the two numbers or variables when you write the expression.

Reverse the Order

Examples: · 8 less than b means b - 8· 3 more than x means x + 3· x less than 2 means 2 - x

Slide 111 / 172

The many ways to represent multiplication.

How do you represent "three times a"?

(3)(a) 3(a) 3 a 3a

The preferred representation is 3a.

When a variable is being multiplied by a number, the number (coefficient) is always written in front of the variable.

The following are not allowed:

3xa ... The multiplication sign looks like another variable

a3 ... The number is always written in front of the variable

Multiplication

Slide 112 / 172

How do you represent "b divided by 12"?

b ÷ 12

b ∕ 12

b12

Representation of Division

Slide 113 / 172

Sort the words by operation.

Quotient Product

Sum TotalRatio

Difference

Less Than

More Fraction

Multiply

Per


Sort the words by operation.

Quotient Product

Sum TotalRatio

Difference

Less Than

More Fraction

Multiply

Per


Ans

wer

The example on this slide addresses MP6.

Slide 114 / 172

Three times j

Eight divided by j

j less than 7

5 more than j

4 less than j

1 2 3 4 5 6 7 8 90 + - . ÷

Translate the Words into Algebraic Expressions Using the Red Characters

j


Three times j

Eight divided by j

j less than 7

5 more than j

4 less than j

1 2 3 4 5 6 7 8 90 + - . ÷

Translate the Words into Algebraic Expressions Using the Red Characters

j


Ans

wer

* Note numbers and symbols are infinitely cloned.

Slide 115 / 172

The sum of twenty-three and m

Write the Expression


The sum of twenty-three and m



Ans

wer

23 + m

Slide 116 / 172

The product of four and k



The product of four and k



Ans

wer

4k

Slide 117 / 172

Twenty-four less than d



Twenty-four less than d



Ans

wer

d - 24

Slide 118 / 172

**Remember, sometimes you need to use parentheses for a quantity.**

Four times the difference of eight and j



**Remember, sometimes you need to use parentheses for a quantity.**

Four times the difference of eight and j



Ans

wer

4(8-j)

Slide 119 / 172

The product of seven and w, divided by 12



The product of seven and w, divided by 12



Ans

wer

7w12

Slide 120 / 172

The square of the sum of six and p



The square of the sum of six and p



Ans

wer

(6+p)2

Slide 121 / 172

63 The sum of 100 and h

A 100 h

B 100 + h

C 100 - h

D 100 + h 200


63 The sum of 100 and h

A 100 h

B 100 + h

C 100 - h

D 100 + h 200


Ans

wer

B

Slide 122 / 172

64 The quotient of 200 and the quantity of p times 7

A 200 7p

B 200 - (7p)

C 200 ÷ 7p

D 7p 200


64 The quotient of 200 and the quantity of p times 7

A 200 7p

B 200 - (7p)

C 200 ÷ 7p

D 7p 200


Ans

wer

A

Slide 123 / 172

65 Thirty five multiplied by the quantity r less 45

A 35r - 45

B 35(45) - r

C 35(45 - r)

D 35(r - 45)


65 Thirty five multiplied by the quantity r less 45

A 35r - 45

B 35(45) - r

C 35(45 - r)

D 35(r - 45)


Ans

wer

D

Slide 124 / 172

66 a less than 27

A 27 - a

B a 27

C a - 27

D 27 + a


66 a less than 27

A 27 - a

B a 27

C a - 27

D 27 + a


Ans

wer

A

Slide 125 / 172

67 Which expression represents "6 more than x"?

A x - 6

B 6 ∙ x

C x + 6

D 6 - x




67 Which expression represents "6 more than x"?

A x - 6

B 6 ∙ x

C x + 6

D 6 - x



Ans

wer

C


Slide 126 / 172

68 Which expressions represent "the sum of 3 and n"? Select all that apply.

A 3n

B n + 3

C 3 + n

D n + n + n

E n3

From PARCC EOY sample test #6



68 Which expressions represent "the sum of 3 and n"? Select all that apply.

A 3n

B n + 3

C 3 + n

D n + n + n

E n3

From PARCC EOY sample test #6


Ans

wer

B & C


Slide 127 / 172

Now, we know how to translate a mathematical sentence in words to a mathematical expression in symbols.

Next, we need to practice translating from English sentences to mathematical sentences.

Then, we can translate from English sentences to mathematical expressions.

Translating English Sentences to Mathematical Sentences

Slide 128 / 172

Write a mathematical sentence, in words, for the below. Then translate that into a mathematical expression.

The total amount of money my friends have, if each of my seven friends has x dollars.

Translating From English Sentences

7 multiplied by x

7x

click for mathematical sentence

click for mathematical expression

Slide 129 / 172



12 added to x

x + 12


click for mathematical expression

My age if I am x years older than my 12 year old brother

Slide 130 / 172



The total of 15 minus 5 divided by 2

(15-5)/2click for mathematical expression


How many apples each person gets if starting with 15 apples, 5 are eaten and the rest are divided equally by 2 friends.

Slide 131 / 172



d divided by s

d/sclick for mathematical expression


My speed if I travel d meters in s seconds

Slide 132 / 172



r multiplied by 28

28rclick for mathematical expression


How much money I make if I earn r dollars per hour and work for 28 hours

Slide 133 / 172



6 less than two times h

2h - 6click for mathematical expression


My height if I am 6 inches less than twice the height of my sister, who is h inches tall

Slide 134 / 172

69 The total number of jellybeans if Mary had 5 jellybeans for each of 4 friends.

A 5 + 4 B 5 - 4

C (5)(4)

D 5 ÷ 4


69 The total number of jellybeans if Mary had 5 jellybeans for each of 4 friends.

A 5 + 4 B 5 - 4

C (5)(4)

D 5 ÷ 4


Ans

wer

C

Slide 135 / 172

70 If n + 4 represents an odd integer, the next largerodd integer is represented by

A n + 2B n + 3C n + 5D n + 6

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.


70 If n + 4 represents an odd integer, the next largerodd integer is represented by

A n + 2B n + 3C n + 5D n + 6



Ans

wer

D

Slide 136 / 172

71 Jenny earns $15 an hour waitressing plus $150 in tips on a Friday night. What expression represents her total earnings?

A 150 - 15h

B h 150

C 15h + 150

D 15 + h


71 Jenny earns $15 an hour waitressing plus $150 in tips on a Friday night. What expression represents her total earnings?

A 150 - 15h

B h 150

C 15h + 150

D 15 + h[This object is a pull tab]

Ans

wer

C

Slide 137 / 172

72 Bob's age if he is 2 years less than double the age of his brother who is z years old?

A 2z + 2

B z 2

C 2z - 2

D z - 2

Ans

wer

Slide 138 / 172

When choosing a variable, there are some letters that are often avoided:

l, i, t, o, O, s, S

Why might these letters be avoided?

It is best to avoid using letters that might be confused for numbers or operations.

In the case above (1, +, 0, 5)

Variables

click to reveal

Slide 139 / 172

73 Bob has x dollars. Mary has 4 more dollars than Bob. Write an expression for Mary's money.

A 4xB x - 4C x + 4D 4x + 4


73 Bob has x dollars. Mary has 4 more dollars than Bob. Write an expression for Mary's money.

A 4xB x - 4C x + 4D 4x + 4


Ans

wer

C

Slide 140 / 172

74 The width of the rectangle is five inches less than its length. The length is x inches. Write an expression for the width.

A 5 - xB x - 5C 5xD x + 5


74 The width of the rectangle is five inches less than its length. The length is x inches. Write an expression for the width.

A 5 - xB x - 5C 5xD x + 5


Ans

wer

B

Slide 141 / 172

75 Frank is 6 inches taller than his younger brother, Pete. Pete's height is P. Write an expression for Frank's height.

A 6PB P + 6C P - 6D 6


75 Frank is 6 inches taller than his younger brother, Pete. Pete's height is P. Write an expression for Frank's height.

A 6PB P + 6C P - 6D 6


Ans

wer

B

Slide 142 / 172

76 A dog weighs three pounds more than twice the weight of a cat, whose weight is c pounds.

Write an expression for the dog's weight.

A 2c + 3B 3c + 2C 2c + 3cD 3c


76 A dog weighs three pounds more than twice the weight of a cat, whose weight is c pounds.

Write an expression for the dog's weight.

A 2c + 3B 3c + 2C 2c + 3cD 3c


Ans

wer

A

Slide 143 / 172

77 Write an expression for Mark's test grade, given that he scored 5 less than Sam who earned a score of x.

A 5 - xB x - 5C 5xD 5


77 Write an expression for Mark's test grade, given that he scored 5 less than Sam who earned a score of x.

A 5 - xB x - 5C 5xD 5


Ans

wer

B

Slide 144 / 172

78 Tim ate four more cookies than Alice. Bob ate twice as many cookies as Tim. If x represents the number of cookies Alice ate, which expression represents the number of cookies Bob ate?

A 2 + (x + 4)

B 2x + 4C 2(x + 4)D 4(x + 2)



78 Tim ate four more cookies than Alice. Bob ate twice as many cookies as Tim. If x represents the number of cookies Alice ate, which expression represents the number of cookies Bob ate?

A 2 + (x + 4)

B 2x + 4C 2(x + 4)D 4(x + 2)



Ans

wer

C

Slide 145 / 172

79 Marshall took $36.75 to the state fair. Each ticket into the fair costs x dollars. Marshall bought 3 tickets. Write and expression that represents the amount of money in dollars, that Marshall had after he bought the tickets.

Students type their answers here

From PARCC PBA sample test non-calculator #5



79 Marshall took $36.75 to the state fair. Each ticket into the fair costs x dollars. Marshall bought 3 tickets. Write and expression that represents the amount of money in dollars, that Marshall had after he bought the tickets.

Students type their answers here

From PARCC PBA sample test non-calculator #5


Ans

wer

$36.75 - 3x


Slide 146 / 172



page3svg





Mat

h Pr

actic

e This lesson addresses MP1

Additional Q's to address MP standards:What is the problem asking? (MP1)How could you start the problem? (MP1)

page3svg

Slide 147 / 172


When evaluating algebraic expressions, the process is fairly straight forward.

1. Write the expression.

2. Substitute in the value of the variable (in parentheses).

3. Simplify/Evaluate the expression.

Slide 148 / 172

Evaluate (4n + 6)2 for n = 1

Write:

Substitute:

Simplify:

(4n + 6)2

(4(1) + 6)2

(4 + 6)2

(10)2

100

Slide 149 / 172

Evaluate 4(n + 6)2 for n = 2

Write:

Substitute:

Simplify:

4(n + 6)2

4((2) + 6)2

4(8)2

4(64)

256

Slide 150 / 172

Evaluate (4n + 6)2 for n = 2

Write:

Substitute:

Simplify:

(4n + 6)2

(4(2) + 6)2

(8 + 6)2

(14)2

196

Slide 151 / 172

108

114

130128118

116

106

Let x = 8, then use the magic looking glass to reveal the correct value of the expression

12x + 23

104

Slide 152 / 172

118128

130

114

20800

72

4x + 2x3

24

Let x = 2, then use the magic looking glass to reveal the correct value of the expression

Slide 153 / 172

80 Evaluate 3h + 2 for h = 3


80 Evaluate 3h + 2 for h = 3


Ans

wer 3(3) + 2

11

Slide 154 / 172

81 Evaluate 2(x + 2)2 for x = 8


81 Evaluate 2(x + 2)2 for x = 8


Ans

wer 2(-10 + 2)2

2(-8)2

128

Slide 155 / 172

82 Evaluate 2x2 for x = 3


82 Evaluate 2x2 for x = 3


Ans

wer 2(3)2

2(9)

18

Slide 156 / 172

83 Evaluate 4p - 3 for p = 20


83 Evaluate 4p - 3 for p = 20


Ans

wer 4(20) - 3

80 - 3

77

Slide 157 / 172

84 Evaluate 3x + 17 when x = 13


84 Evaluate 3x + 17 when x = 13


Ans

wer 3(13) + 17

39 + 17

56

Slide 158 / 172

85 Evaluate 3a for a = 12 9


85 Evaluate 3a for a = 12 9


Ans

wer

3(12) 9

369

4

Slide 159 / 172

86 Evaluate 5x2 - 4 when x = 3.




86 Evaluate 5x2 - 4 when x = 3.



Ans

wer

5(3)2 - 45(9) - 445 -4

41


Slide 160 / 172

87 Evaluate 4a + for a = 8, c = 2 ca


87 Evaluate 4a + for a = 8, c = 2 ca


Ans

wer

4(8) +

32 + (4)

36

82

Slide 161 / 172

88 Evaluate 3x + 2y for x = 5 and y = 12


88 Evaluate 3x + 2y for x = 5 and y = 12


Ans

wer 3(5) + 2( )

15 + 1

16

12

Slide 162 / 172

89 Evaluate 8x + y - 10 for x = and y = 50

14


89 Evaluate 8x + y - 10 for x = and y = 50

14


Ans

wer 8( ) + 50 - 10

2 + 50 - 10

42

14

Slide 163 / 172

90 What is the value of a2 + 3b ÷ c - 2d, when a = 3, b = 8, c = 2, and d = 5?

From PARCC EOY sample test calculator #11



90 What is the value of a2 + 3b ÷ c - 2d, when a = 3, b = 8, c = 2, and d = 5?

From PARCC EOY sample test calculator #11


Ans

wer

(3)2 + 3(8) ÷ (2) - 2(5)

9 + 3(8) ÷ (2) - 2(5)

9 + 12 - 10

11


Slide 164 / 172

Glossary &

Standards


page3svg


Glossary &

Standards



Teac

her N

otes Vocabulary Words are bolded

in the presentation. The text box the word is in is then linked to the page at the end of the presentation with the word defined on it.

page3svg

Slide 165 / 172

Back to

Instruction

Coefficient The number multiplied by the variable and is located in front of the variable.

4x + 2 These are not coefficients. These are constants!

Tricky!1x + 7

- 1x2 +18

When not present, the coefficient is assumed to be 1.

7 3 5

page14svg

Slide 166 / 172

Back to

Instruction

Constant A fixed number whose value does not change. It is either positive or negative.

4x + 2 7x 3y3z

These are not constants. These are coefficients!

Tricky!

7

4

69

1108

0.45

1/2π

page352svg

Slide 167 / 172

Back to

Instruction

The Distributive PropertyA property that allows you to multiply all the terms on the inside of a set of parenthesis by a term on the outside of the parenthesis.

a(b + c) = ab + ac

a(b + c) = ab + ac

a(b - c) = ab - ac

3(x + 4) = 48 (3)(x) + (3)(4) = 48

3x + 12 = 48 3x = 36 x = 12

2(3+4)=(2x3)+(2x4)

23 4

page101svg

Slide 168 / 172

Back to

Instruction

ExpressionAn expression contains: number,

variables, and at least one operation.

4x + 2

7x = 21

11 = 3y + 2

11 - 1 = 3z + 1

Remember!

7x "7 times x"

"7 divided by x"

7x

page24svg

Slide 169 / 172

Back to

Instruction

Like Terms Terms in an expression that have the same

variable raised to the same power.

3x

5x15.7x

x 1/2x

-2.3x

27x3

-2x3

x3

1/4x3

-5x3

2.7x3

5x3

5x

5x25

5x4

NOT LIKE

TERMS!

page121svg

Slide 170 / 172

Back to

Instruction

Order of Operations

Please Excuse My Dear Aunt Sally

The rules of which calculation comes first in an expression.

Parentheses, Exponents, Multiplication or Division, Addition or Subtraction

page2svg

Slide 171 / 172

Back to

Instruction

VariableAny letter or symbol that represents a

changeable or unknown value.

4x + 2 l, i, t, o, O, s, S

x y zu v

any letter towards end of alphabet!

page13svg

Slide 172 / 172

Throughout this unit, the Standards for Mathematical Practice are used.

MP1: Making sense of problems & persevere in solving them.MP2: Reason abstractly & quantitatively.MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics.MP5: Use appropriate tools strategically.MP6: Attend to precision.MP7: Look for & make use of structure.MP8: Look for & express regularity in repeated reasoning.

Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used.

If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.


Throughout this unit, the Standards for Mathematical Practice are used.

MP1: Making sense of problems & persevere in solving them.MP2: Reason abstractly & quantitatively.MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics.MP5: Use appropriate tools strategically.MP6: Attend to precision.MP7: Look for & make use of structure.MP8: Look for & express regularity in repeated reasoning.

Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used.

If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.


Mat

h Pr

actic

e

Documents

Slide 1 / 172content.njctl.org/courses/math/6th-grade-math/... · Slide 5 / 172 Expressions Algebra extends the tools of arithmetic, which were developed to work with numbers, so