## CPM Educational Program

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- Core Connections Integrated I, 2013
- Core Connections Algebra 1, 2013
- Core Connections Geometry, 2013
- Core Connections Algebra 2, 2013
- Core Connections Integrated I, 2014
- Core Connections Integrated II, 2015
- Core Connections: Course 1
- Core Connections: Course 2
- Core Connections: Course 3
- Core Connections Integrated III, 2015

## 3.1 Functions and Function Notation

- ⓑ yes (Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)

w = f ( d ) w = f ( d )

g ( 5 ) = 1 g ( 5 ) = 1

m = 8 m = 8

y = f ( x ) = x 3 2 y = f ( x ) = x 3 2

g ( 1 ) = 8 g ( 1 ) = 8

x = 0 x = 0 or x = 2 x = 2

- ⓐ yes, because each bank account has a single balance at any given time;
- ⓑ no, because several bank account numbers may have the same balance;
- ⓒ no, because the same output may correspond to more than one input.
- ⓐ Yes, letter grade is a function of percent grade;
- ⓑ No, it is not one-to-one. There are 100 different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade.

No, because it does not pass the horizontal line test.

## 3.2 Domain and Range

{ − 5 , 0 , 5 , 10 , 15 } { − 5 , 0 , 5 , 10 , 15 }

( − ∞ , ∞ ) ( − ∞ , ∞ )

( − ∞ , 1 2 ) ∪ ( 1 2 , ∞ ) ( − ∞ , 1 2 ) ∪ ( 1 2 , ∞ )

[ − 5 2 , ∞ ) [ − 5 2 , ∞ )

- ⓐ values that are less than or equal to –2, or values that are greater than or equal to –1 and less than 3
- ⓑ { x | x ≤ − 2 or − 1 ≤ x < 3 } { x | x ≤ − 2 or − 1 ≤ x < 3 }
- ⓒ ( − ∞ , − 2 ] ∪ [ − 1 , 3 ) ( − ∞ , − 2 ] ∪ [ − 1 , 3 )

domain =[1950,2002] range = [47,000,000,89,000,000]

domain: ( − ∞ , 2 ] ; ( − ∞ , 2 ] ; range: ( − ∞ , 0 ] ( − ∞ , 0 ]

## 3.3 Rates of Change and Behavior of Graphs

$ 2.84 − $ 2.31 5 years = $ 0.53 5 years = $ 0.106 $ 2.84 − $ 2.31 5 years = $ 0.53 5 years = $ 0.106 per year.

a + 7 a + 7

The local maximum appears to occur at ( − 1 , 28 ) , ( − 1 , 28 ) , and the local minimum occurs at ( 5 , − 80 ) . ( 5 , − 80 ) . The function is increasing on ( − ∞ , − 1 ) ∪ ( 5 , ∞ ) ( − ∞ , − 1 ) ∪ ( 5 , ∞ ) and decreasing on ( − 1 , 5 ) . ( − 1 , 5 ) .

## 3.4 Composition of Functions

( f g ) ( x ) = f ( x ) g ( x ) = ( x − 1 ) ( x 2 − 1 ) = x 3 − x 2 − x + 1 ( f − g ) ( x ) = f ( x ) − g ( x ) = ( x − 1 ) − ( x 2 − 1 ) = x − x 2 ( f g ) ( x ) = f ( x ) g ( x ) = ( x − 1 ) ( x 2 − 1 ) = x 3 − x 2 − x + 1 ( f − g ) ( x ) = f ( x ) − g ( x ) = ( x − 1 ) − ( x 2 − 1 ) = x − x 2

No, the functions are not the same.

A gravitational force is still a force, so a ( G ( r ) ) a ( G ( r ) ) makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but G ( a ( F ) ) G ( a ( F ) ) does not make sense.

f ( g ( 1 ) ) = f ( 3 ) = 3 f ( g ( 1 ) ) = f ( 3 ) = 3 and g ( f ( 4 ) ) = g ( 1 ) = 3 g ( f ( 4 ) ) = g ( 1 ) = 3

g ( f ( 2 ) ) = g ( 5 ) = 3 g ( f ( 2 ) ) = g ( 5 ) = 3

[ − 4 , 0 ) ∪ ( 0 , ∞ ) [ − 4 , 0 ) ∪ ( 0 , ∞ )

Possible answer:

g ( x ) = 4 + x 2 h ( x ) = 4 3 − x f = h ∘ g g ( x ) = 4 + x 2 h ( x ) = 4 3 − x f = h ∘ g

## 3.5 Transformation of Functions

The graphs of f ( x ) f ( x ) and g ( x ) g ( x ) are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.

g ( x ) = 1 x - 1 + 1 g ( x ) = 1 x - 1 + 1

g ( x ) = − f ( x ) g ( x ) = − f ( x )

-2 | 0 | 2 | 4 | |

h ( x ) = f ( − x ) h ( x ) = f ( − x )

-2 | 0 | 2 | 4 | |

15 | 10 | 5 | unknown |

Notice: g ( x ) = f ( − x ) g ( x ) = f ( − x ) looks the same as f ( x ) f ( x ) .

2 | 4 | 6 | 8 | |

9 | 12 | 15 | 0 |

g ( x ) = 3 x - 2 g ( x ) = 3 x - 2

g ( x ) = f ( 1 3 x ) g ( x ) = f ( 1 3 x ) so using the square root function we get g ( x ) = 1 3 x g ( x ) = 1 3 x

## 3.6 Absolute Value Functions

using the variable p p for passing, | p − 80 | ≤ 20 | p − 80 | ≤ 20

f ( x ) = − | x + 2 | + 3 f ( x ) = − | x + 2 | + 3

x = − 1 x = − 1 or x = 2 x = 2

## 3.7 Inverse Functions

h ( 2 ) = 6 h ( 2 ) = 6

The domain of function f − 1 f − 1 is ( − ∞ , − 2 ) ( − ∞ , − 2 ) and the range of function f − 1 f − 1 is ( 1 , ∞ ) . ( 1 , ∞ ) .

- ⓐ f ( 60 ) = 50. f ( 60 ) = 50. In 60 minutes, 50 miles are traveled.
- ⓑ f − 1 ( 60 ) = 70. f − 1 ( 60 ) = 70. To travel 60 miles, it will take 70 minutes.

x = 3 y + 5 x = 3 y + 5

f − 1 ( x ) = ( 2 − x ) 2 ; domain of f : [ 0 , ∞ ) ; domain of f − 1 : ( − ∞ , 2 ] f − 1 ( x ) = ( 2 − x ) 2 ; domain of f : [ 0 , ∞ ) ; domain of f − 1 : ( − ∞ , 2 ]

## 3.1 Section Exercises

A relation is a set of ordered pairs. A function is a special kind of relation in which no two ordered pairs have the same first coordinate.

When a vertical line intersects the graph of a relation more than once, that indicates that for that input there is more than one output. At any particular input value, there can be only one output if the relation is to be a function.

When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input.

not a function

f ( − 3 ) = − 11 ; f ( − 3 ) = − 11 ; f ( 2 ) = − 1 ; f ( 2 ) = − 1 ; f ( − a ) = − 2 a − 5 ; f ( − a ) = − 2 a − 5 ; − f ( a ) = − 2 a + 5 ; − f ( a ) = − 2 a + 5 ; f ( a + h ) = 2 a + 2 h − 5 f ( a + h ) = 2 a + 2 h − 5

f ( − 3 ) = 5 + 5 ; f ( − 3 ) = 5 + 5 ; f ( 2 ) = 5 ; f ( 2 ) = 5 ; f ( − a ) = 2 + a + 5 ; f ( − a ) = 2 + a + 5 ; − f ( a ) = − 2 − a − 5 ; − f ( a ) = − 2 − a − 5 ; f ( a + h ) = 2 − a − h + 5 f ( a + h ) = 2 − a − h + 5

f ( − 3 ) = 2 ; f ( − 3 ) = 2 ; f ( 2 ) = 1 − 3 = − 2 ; f ( 2 ) = 1 − 3 = − 2 ; f ( − a ) = | − a − 1 | − | − a + 1 | ; f ( − a ) = | − a − 1 | − | − a + 1 | ; − f ( a ) = − | a − 1 | + | a + 1 | ; − f ( a ) = − | a − 1 | + | a + 1 | ; f ( a + h ) = | a + h − 1 | − | a + h + 1 | f ( a + h ) = | a + h − 1 | − | a + h + 1 |

g ( x ) − g ( a ) x − a = x + a + 2 , x ≠ a g ( x ) − g ( a ) x − a = x + a + 2 , x ≠ a

a. f ( − 2 ) = 14 ; f ( − 2 ) = 14 ; b. x = 3 x = 3

a. f ( 5 ) = 10 ; f ( 5 ) = 10 ; b. x = − 1 x = − 1 or x = 4 x = 4

- ⓐ f ( t ) = 6 − 2 3 t ; f ( t ) = 6 − 2 3 t ;
- ⓑ f ( − 3 ) = 8 ; f ( − 3 ) = 8 ;
- ⓒ t = 6 t = 6
- ⓐ f ( 0 ) = 1 ; f ( 0 ) = 1 ;
- ⓑ f ( x ) = − 3 , x = − 2 f ( x ) = − 3 , x = − 2 or x = 2 x = 2

not a function so it is also not a one-to-one function

one-to- one function

function, but not one-to-one

f ( x ) = 1 , x = 2 f ( x ) = 1 , x = 2

f ( − 2 ) = 14 ; f ( − 1 ) = 11 ; f ( 0 ) = 8 ; f ( 1 ) = 5 ; f ( 2 ) = 2 f ( − 2 ) = 14 ; f ( − 1 ) = 11 ; f ( 0 ) = 8 ; f ( 1 ) = 5 ; f ( 2 ) = 2

f ( − 2 ) = 4 ; f ( − 1 ) = 4.414 ; f ( 0 ) = 4.732 ; f ( 1 ) = 5 ; f ( 2 ) = 5.236 f ( − 2 ) = 4 ; f ( − 1 ) = 4.414 ; f ( 0 ) = 4.732 ; f ( 1 ) = 5 ; f ( 2 ) = 5.236

f ( − 2 ) = 1 9 ; f ( − 1 ) = 1 3 ; f ( 0 ) = 1 ; f ( 1 ) = 3 ; f ( 2 ) = 9 f ( − 2 ) = 1 9 ; f ( − 1 ) = 1 3 ; f ( 0 ) = 1 ; f ( 1 ) = 3 ; f ( 2 ) = 9

[ 0 , 100 ] [ 0 , 100 ]

[ − 0.001 , 0 .001 ] [ − 0.001 , 0 .001 ]

[ − 1 , 000 , 000 , 1,000,000 ] [ − 1 , 000 , 000 , 1,000,000 ]

[ 0 , 10 ] [ 0 , 10 ]

[ −0.1 , 0.1 ] [ −0.1 , 0.1 ]

[ − 100 , 100 ] [ − 100 , 100 ]

- ⓐ g ( 5000 ) = 50 ; g ( 5000 ) = 50 ;
- ⓑ The number of cubic yards of dirt required for a garden of 100 square feet is 1.
- ⓐ The height of a rocket above ground after 1 second is 200 ft.
- ⓑ The height of a rocket above ground after 2 seconds is 350 ft.

## 3.2 Section Exercises

The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.

There is no restriction on x x for f ( x ) = x 3 f ( x ) = x 3 because you can take the cube root of any real number. So the domain is all real numbers, ( − ∞ , ∞ ) . ( − ∞ , ∞ ) . When dealing with the set of real numbers, you cannot take the square root of negative numbers. So x x -values are restricted for f ( x ) = x f ( x ) = x to nonnegative numbers and the domain is [ 0 , ∞ ) . [ 0 , ∞ ) .

Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the x x -axis and y y -axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate − ∞ − ∞ or ∞ . ∞ . Combine the graphs to find the graph of the piecewise function.

( − ∞ , 3 ] ( − ∞ , 3 ]

( − ∞ , − 1 2 ) ∪ ( − 1 2 , ∞ ) ( − ∞ , − 1 2 ) ∪ ( − 1 2 , ∞ )

( − ∞ , − 11 ) ∪ ( − 11 , 2 ) ∪ ( 2 , ∞ ) ( − ∞ , − 11 ) ∪ ( − 11 , 2 ) ∪ ( 2 , ∞ )

( − ∞ , − 3 ) ∪ ( − 3 , 5 ) ∪ ( 5 , ∞ ) ( − ∞ , − 3 ) ∪ ( − 3 , 5 ) ∪ ( 5 , ∞ )

( − ∞ , 5 ) ( − ∞ , 5 )

[ 6 , ∞ ) [ 6 , ∞ )

( − ∞ , − 9 ) ∪ ( − 9 , 9 ) ∪ ( 9 , ∞ ) ( − ∞ , − 9 ) ∪ ( − 9 , 9 ) ∪ ( 9 , ∞ )

domain: ( 2 , 8 ] , ( 2 , 8 ] , range [ 6 , 8 ) [ 6 , 8 )

domain: [ − 4 , 4], [ − 4 , 4], range: [ 0 , 2] [ 0 , 2]

domain: [ − 5 , 3 ) , [ − 5 , 3 ) , range: [ 0 , 2 ] [ 0 , 2 ]

domain: ( − ∞ , 1 ] , ( − ∞ , 1 ] , range: [ 0 , ∞ ) [ 0 , ∞ )

domain: [ − 6 , − 1 6 ] ∪ [ 1 6 , 6 ] ; [ − 6 , − 1 6 ] ∪ [ 1 6 , 6 ] ; range: [ − 6 , − 1 6 ] ∪ [ 1 6 , 6 ] [ − 6 , − 1 6 ] ∪ [ 1 6 , 6 ]

domain: [ − 3 , ∞ ) ; [ − 3 , ∞ ) ; range: [ 0 , ∞ ) [ 0 , ∞ )

domain: ( − ∞ , ∞ ) ( − ∞ , ∞ )

f ( − 3 ) = 1 ; f ( − 2 ) = 0 ; f ( − 1 ) = 0 ; f ( 0 ) = 0 f ( − 3 ) = 1 ; f ( − 2 ) = 0 ; f ( − 1 ) = 0 ; f ( 0 ) = 0

f ( − 1 ) = − 4 ; f ( 0 ) = 6 ; f ( 2 ) = 20 ; f ( 4 ) = 34 f ( − 1 ) = − 4 ; f ( 0 ) = 6 ; f ( 2 ) = 20 ; f ( 4 ) = 34

f ( − 1 ) = − 5 ; f ( 0 ) = 3 ; f ( 2 ) = 3 ; f ( 4 ) = 16 f ( − 1 ) = − 5 ; f ( 0 ) = 3 ; f ( 2 ) = 3 ; f ( 4 ) = 16

domain: ( − ∞ , 1 ) ∪ ( 1 , ∞ ) ( − ∞ , 1 ) ∪ ( 1 , ∞ )

window: [ − 0.5 , − 0.1 ] ; [ − 0.5 , − 0.1 ] ; range: [ 4 , 100 ] [ 4 , 100 ]

window: [ 0.1 , 0.5 ] ; [ 0.1 , 0.5 ] ; range: [ 4 , 100 ] [ 4 , 100 ]

[ 0 , 8 ] [ 0 , 8 ]

Many answers. One function is f ( x ) = 1 x − 2 . f ( x ) = 1 x − 2 .

- ⓐ The fixed cost is $500.
- ⓑ The cost of making 25 items is $750.
- ⓒ The domain is [0, 100] and the range is [500, 1500].

## 3.3 Section Exercises

Yes, the average rate of change of all linear functions is constant.

The absolute maximum and minimum relate to the entire graph, whereas the local extrema relate only to a specific region around an open interval.

4 ( b + 1 ) 4 ( b + 1 )

4 x + 2 h 4 x + 2 h

− 1 13 ( 13 + h ) − 1 13 ( 13 + h )

3 h 2 + 9 h + 9 3 h 2 + 9 h + 9

4 x + 2 h − 3 4 x + 2 h − 3

increasing on ( − ∞ , − 2.5 ) ∪ ( 1 , ∞ ) , ( − ∞ , − 2.5 ) ∪ ( 1 , ∞ ) , decreasing on ( − 2.5 , 1 ) ( − 2.5 , 1 )

increasing on ( − ∞ , 1 ) ∪ ( 3 , 4 ) , ( − ∞ , 1 ) ∪ ( 3 , 4 ) , decreasing on ( 1 , 3 ) ∪ ( 4 , ∞ ) ( 1 , 3 ) ∪ ( 4 , ∞ )

local maximum: ( − 3 , 60 ) , ( − 3 , 60 ) , local minimum: ( 3 , − 60 ) ( 3 , − 60 )

absolute maximum at approximately ( 7 , 150 ) , ( 7 , 150 ) , absolute minimum at approximately ( −7.5 , −220 ) ( −7.5 , −220 )

Local minimum at ( 3 , − 22 ) , ( 3 , − 22 ) , decreasing on ( − ∞ , 3 ) , ( − ∞ , 3 ) , increasing on ( 3 , ∞ ) ( 3 , ∞ )

Local minimum at ( − 2 , − 2 ) , ( − 2 , − 2 ) , decreasing on ( − 3 , − 2 ) , ( − 3 , − 2 ) , increasing on ( − 2 , ∞ ) ( − 2 , ∞ )

Local maximum at ( − 0.5 , 6 ) , ( − 0.5 , 6 ) , local minima at ( − 3.25 , − 47 ) ( − 3.25 , − 47 ) and ( 2.1 , − 32 ) , ( 2.1 , − 32 ) , decreasing on ( − ∞ , − 3.25 ) ( − ∞ , − 3.25 ) and ( − 0.5 , 2.1 ) , ( − 0.5 , 2.1 ) , increasing on ( − 3.25 , − 0.5 ) ( − 3.25 , − 0.5 ) and ( 2.1 , ∞ ) ( 2.1 , ∞ )

b = 5 b = 5

2.7 gallons per minute

approximately –0.6 milligrams per day

## 3.4 Section Exercises

Find the numbers that make the function in the denominator g g equal to zero, and check for any other domain restrictions on f f and g , g , such as an even-indexed root or zeros in the denominator.

Yes. Sample answer: Let f ( x ) = x + 1 and g ( x ) = x − 1. f ( x ) = x + 1 and g ( x ) = x − 1. Then f ( g ( x ) ) = f ( x − 1 ) = ( x − 1 ) + 1 = x f ( g ( x ) ) = f ( x − 1 ) = ( x − 1 ) + 1 = x and g ( f ( x ) ) = g ( x + 1 ) = ( x + 1 ) − 1 = x . g ( f ( x ) ) = g ( x + 1 ) = ( x + 1 ) − 1 = x . So f ∘ g = g ∘ f . f ∘ g = g ∘ f .

( f + g ) ( x ) = 2 x + 6 , ( f + g ) ( x ) = 2 x + 6 , domain: ( − ∞ , ∞ ) ( − ∞ , ∞ )

( f − g ) ( x ) = 2 x 2 + 2 x − 6 , ( f − g ) ( x ) = 2 x 2 + 2 x − 6 , domain: ( − ∞ , ∞ ) ( − ∞ , ∞ )

( f g ) ( x ) = − x 4 − 2 x 3 + 6 x 2 + 12 x , ( f g ) ( x ) = − x 4 − 2 x 3 + 6 x 2 + 12 x , domain: ( − ∞ , ∞ ) ( − ∞ , ∞ )

( f g ) ( x ) = x 2 + 2 x 6 − x 2 , ( f g ) ( x ) = x 2 + 2 x 6 − x 2 , domain: ( − ∞ , − 6 ) ∪ ( − 6 , 6 ) ∪ ( 6 , ∞ ) ( − ∞ , − 6 ) ∪ ( − 6 , 6 ) ∪ ( 6 , ∞ )

( f + g ) ( x ) = 4 x 3 + 8 x 2 + 1 2 x , ( f + g ) ( x ) = 4 x 3 + 8 x 2 + 1 2 x , domain: ( − ∞ , 0 ) ∪ ( 0 , ∞ ) ( − ∞ , 0 ) ∪ ( 0 , ∞ )

( f − g ) ( x ) = 4 x 3 + 8 x 2 − 1 2 x , ( f − g ) ( x ) = 4 x 3 + 8 x 2 − 1 2 x , domain: ( − ∞ , 0 ) ∪ ( 0 , ∞ ) ( − ∞ , 0 ) ∪ ( 0 , ∞ )

( f g ) ( x ) = x + 2 , ( f g ) ( x ) = x + 2 , domain: ( − ∞ , 0 ) ∪ ( 0 , ∞ ) ( − ∞ , 0 ) ∪ ( 0 , ∞ )

( f g ) ( x ) = 4 x 3 + 8 x 2 , ( f g ) ( x ) = 4 x 3 + 8 x 2 , domain: ( − ∞ , 0 ) ∪ ( 0 , ∞ ) ( − ∞ , 0 ) ∪ ( 0 , ∞ )

( f + g ) ( x ) = 3 x 2 + x − 5 , ( f + g ) ( x ) = 3 x 2 + x − 5 , domain: [ 5 , ∞ ) [ 5 , ∞ )

( f − g ) ( x ) = 3 x 2 − x − 5 , ( f − g ) ( x ) = 3 x 2 − x − 5 , domain: [ 5 , ∞ ) [ 5 , ∞ )

( f g ) ( x ) = 3 x 2 x − 5 , ( f g ) ( x ) = 3 x 2 x − 5 , domain: [ 5 , ∞ ) [ 5 , ∞ )

( f g ) ( x ) = 3 x 2 x − 5 , ( f g ) ( x ) = 3 x 2 x − 5 , domain: ( 5 , ∞ ) ( 5 , ∞ )

- ⓑ f ( g ( x ) ) = 2 ( 3 x − 5 ) 2 + 1 f ( g ( x ) ) = 2 ( 3 x − 5 ) 2 + 1
- ⓒ f ( g ( x ) ) = 6 x 2 − 2 f ( g ( x ) ) = 6 x 2 − 2
- ⓓ ( g ∘ g ) ( x ) = 3 ( 3 x − 5 ) − 5 = 9 x − 20 ( g ∘ g ) ( x ) = 3 ( 3 x − 5 ) − 5 = 9 x − 20
- ⓔ ( f ∘ f ) ( − 2 ) = 163 ( f ∘ f ) ( − 2 ) = 163

f ( g ( x ) ) = x 2 + 3 + 2 , g ( f ( x ) ) = x + 4 x + 7 f ( g ( x ) ) = x 2 + 3 + 2 , g ( f ( x ) ) = x + 4 x + 7

f ( g ( x ) ) = x + 1 x 3 3 = x + 1 3 x , g ( f ( x ) ) = x 3 + 1 x f ( g ( x ) ) = x + 1 x 3 3 = x + 1 3 x , g ( f ( x ) ) = x 3 + 1 x

( f ∘ g ) ( x ) = 1 2 x + 4 − 4 = x 2 , ( g ∘ f ) ( x ) = 2 x − 4 ( f ∘ g ) ( x ) = 1 2 x + 4 − 4 = x 2 , ( g ∘ f ) ( x ) = 2 x − 4

f ( g ( h ( x ) ) ) = ( 1 x + 3 ) 2 + 1 f ( g ( h ( x ) ) ) = ( 1 x + 3 ) 2 + 1

- ⓐ ( g ∘ f ) ( x ) = − 3 2 − 4 x ( g ∘ f ) ( x ) = − 3 2 − 4 x
- ⓑ ( − ∞ , 1 2 ) ( − ∞ , 1 2 )
- ⓐ ( 0 , 2 ) ∪ ( 2 , ∞ ) ; ( 0 , 2 ) ∪ ( 2 , ∞ ) ;
- ⓑ ( − ∞ , − 2 ) ∪ ( 2 , ∞ ) ; ( − ∞ , − 2 ) ∪ ( 2 , ∞ ) ;
- ⓒ ( 0 , ∞ ) ( 0 , ∞ )

( 1 , ∞ ) ( 1 , ∞ )

sample: f ( x ) = x 3 g ( x ) = x − 5 f ( x ) = x 3 g ( x ) = x − 5

sample: f ( x ) = 4 x g ( x ) = ( x + 2 ) 2 f ( x ) = 4 x g ( x ) = ( x + 2 ) 2

sample: f ( x ) = x 3 g ( x ) = 1 2 x − 3 f ( x ) = x 3 g ( x ) = 1 2 x − 3

sample: f ( x ) = x 4 g ( x ) = 3 x − 2 x + 5 f ( x ) = x 4 g ( x ) = 3 x − 2 x + 5

sample: f ( x ) = x g ( x ) = 2 x + 6 f ( x ) = x g ( x ) = 2 x + 6

sample: f ( x ) = x 3 g ( x ) = ( x − 1 ) f ( x ) = x 3 g ( x ) = ( x − 1 )

sample: f ( x ) = x 3 g ( x ) = 1 x − 2 f ( x ) = x 3 g ( x ) = 1 x − 2

sample: f ( x ) = x g ( x ) = 2 x − 1 3 x + 4 f ( x ) = x g ( x ) = 2 x − 1 3 x + 4

f ( g ( 0 ) ) = 27 , g ( f ( 0 ) ) = − 94 f ( g ( 0 ) ) = 27 , g ( f ( 0 ) ) = − 94

f ( g ( 0 ) ) = 1 5 , g ( f ( 0 ) ) = 5 f ( g ( 0 ) ) = 1 5 , g ( f ( 0 ) ) = 5

18 x 2 + 60 x + 51 18 x 2 + 60 x + 51

g ∘ g ( x ) = 9 x + 20 g ∘ g ( x ) = 9 x + 20

( f ∘ g ) ( 6 ) = 6 ( f ∘ g ) ( 6 ) = 6 ; ( g ∘ f ) ( 6 ) = 6 ( g ∘ f ) ( 6 ) = 6

( f ∘ g ) ( 11 ) = 11 , ( g ∘ f ) ( 11 ) = 11 ( f ∘ g ) ( 11 ) = 11 , ( g ∘ f ) ( 11 ) = 11

A ( t ) = π ( 25 t + 2 ) 2 A ( t ) = π ( 25 t + 2 ) 2 and A ( 2 ) = π ( 25 4 ) 2 = 2500 π A ( 2 ) = π ( 25 4 ) 2 = 2500 π square inches

A ( 5 ) = π ( 2 ( 5 ) + 1 ) 2 = 121 π A ( 5 ) = π ( 2 ( 5 ) + 1 ) 2 = 121 π square units

- ⓐ N ( T ( t ) ) = 23 ( 5 t + 1.5 ) 2 − 56 ( 5 t + 1.5 ) + 1 N ( T ( t ) ) = 23 ( 5 t + 1.5 ) 2 − 56 ( 5 t + 1.5 ) + 1
- ⓑ 3.38 hours

## 3.5 Section Exercises

A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.

A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output.

For a function f , f , substitute ( − x ) ( − x ) for ( x ) ( x ) in f ( x ) . f ( x ) . Simplify. If the resulting function is the same as the original function, f ( − x ) = f ( x ) , f ( − x ) = f ( x ) , then the function is even. If the resulting function is the opposite of the original function, f ( − x ) = − f ( x ) , f ( − x ) = − f ( x ) , then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

g ( x ) = | x - 1 | − 3 g ( x ) = | x - 1 | − 3

g ( x ) = 1 ( x + 4 ) 2 + 2 g ( x ) = 1 ( x + 4 ) 2 + 2

The graph of f ( x + 43 ) f ( x + 43 ) is a horizontal shift to the left 43 units of the graph of f . f .

The graph of f ( x - 4 ) f ( x - 4 ) is a horizontal shift to the right 4 units of the graph of f . f .

The graph of f ( x ) + 8 f ( x ) + 8 is a vertical shift up 8 units of the graph of f . f .

The graph of f ( x ) − 7 f ( x ) − 7 is a vertical shift down 7 units of the graph of f . f .

The graph of f ( x + 4 ) − 1 f ( x + 4 ) − 1 is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of f . f .

decreasing on ( − ∞ , − 3 ) ( − ∞ , − 3 ) and increasing on ( − 3 , ∞ ) ( − 3 , ∞ )

decreasing on ( 0 , ∞ ) ( 0 , ∞ )

g ( x ) = f ( x - 1 ) , h ( x ) = f ( x ) + 1 g ( x ) = f ( x - 1 ) , h ( x ) = f ( x ) + 1

f ( x ) = | x - 3 | − 2 f ( x ) = | x - 3 | − 2

f ( x ) = x + 3 − 1 f ( x ) = x + 3 − 1

f ( x ) = ( x - 2 ) 2 f ( x ) = ( x - 2 ) 2

f ( x ) = | x + 3 | − 2 f ( x ) = | x + 3 | − 2

f ( x ) = − x f ( x ) = − x

f ( x ) = − ( x + 1 ) 2 + 2 f ( x ) = − ( x + 1 ) 2 + 2

f ( x ) = − x + 1 f ( x ) = − x + 1

The graph of g g is a vertical reflection (across the x x -axis) of the graph of f . f .

The graph of g g is a vertical stretch by a factor of 4 of the graph of f . f .

The graph of g g is a horizontal compression by a factor of 1 5 1 5 of the graph of f . f .

The graph of g g is a horizontal stretch by a factor of 3 of the graph of f . f .

The graph of g g is a horizontal reflection across the y y -axis and a vertical stretch by a factor of 3 of the graph of f . f .

g ( x ) = | − 4 x | g ( x ) = | − 4 x |

g ( x ) = 1 3 ( x + 2 ) 2 − 3 g ( x ) = 1 3 ( x + 2 ) 2 − 3

g ( x ) = 1 2 ( x - 5 ) 2 + 1 g ( x ) = 1 2 ( x - 5 ) 2 + 1

The graph of the function f ( x ) = x 2 f ( x ) = x 2 is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.

The graph of f ( x ) = | x | f ( x ) = | x | is stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up.

The graph of the function f ( x ) = x 3 f ( x ) = x 3 is compressed vertically by a factor of 1 2 . 1 2 .

The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.

The graph of f ( x ) = x f ( x ) = x is shifted right 4 units and then reflected across the vertical line x = 4. x = 4.

## 3.6 Section Exercises

Isolate the absolute value term so that the equation is of the form | A | = B . | A | = B . Form one equation by setting the expression inside the absolute value symbol, A , A , equal to the expression on the other side of the equation, B . B . Form a second equation by setting A A equal to the opposite of the expression on the other side of the equation, − B . − B . Solve each equation for the variable.

The graph of the absolute value function does not cross the x x -axis, so the graph is either completely above or completely below the x x -axis.

The distance from x to 8 can be represented using the absolute value statement: ∣ x − 8 ∣ = 4.

∣ x − 10 ∣ ≥ 15

There are no x-intercepts.

(−4, 0) and (2, 0)

( 0 , − 4 ) , ( 4 , 0 ) , ( − 2 , 0 ) ( 0 , − 4 ) , ( 4 , 0 ) , ( − 2 , 0 )

( 0 , 7 ) , ( 25 , 0 ) , ( − 7 , 0 ) ( 0 , 7 ) , ( 25 , 0 ) , ( − 7 , 0 )

range: [ – 400 , 100 ] [ – 400 , 100 ]

There is no solution for a a that will keep the function from having a y y -intercept. The absolute value function always crosses the y y -intercept when x = 0. x = 0.

| p − 0.08 | ≤ 0.015 | p − 0.08 | ≤ 0.015

| x − 5.0 | ≤ 0.01 | x − 5.0 | ≤ 0.01

## 3.7 Section Exercises

Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that y y -values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no y y -values repeat and the function is one-to-one.

Yes. For example, f ( x ) = 1 x f ( x ) = 1 x is its own inverse.

Given a function y = f ( x ) , y = f ( x ) , solve for x x in terms of y . y . Interchange the x x and y . y . Solve the new equation for y . y . The expression for y y is the inverse, y = f − 1 ( x ) . y = f − 1 ( x ) .

f − 1 ( x ) = x − 3 f − 1 ( x ) = x − 3

f − 1 ( x ) = 2 − x f − 1 ( x ) = 2 − x

f − 1 ( x ) = − 2 x x − 1 f − 1 ( x ) = − 2 x x − 1

domain of f ( x ) : [ − 7 , ∞ ) ; f − 1 ( x ) = x − 7 f ( x ) : [ − 7 , ∞ ) ; f − 1 ( x ) = x − 7

domain of f ( x ) : [ 0 , ∞ ) ; f − 1 ( x ) = x + 5 f ( x ) : [ 0 , ∞ ) ; f − 1 ( x ) = x + 5

a. f ( g ( x ) ) = x f ( g ( x ) ) = x and g ( f ( x ) ) = x . g ( f ( x ) ) = x . b. This tells us that f f and g g are inverse functions

f ( g ( x ) ) = x , g ( f ( x ) ) = x f ( g ( x ) ) = x , g ( f ( x ) ) = x

not one-to-one

[ 2 , 10 ] [ 2 , 10 ]

1 | 4 | 7 | 12 | 16 | |

3 | 6 | 9 | 13 | 14 |

f − 1 ( x ) = ( 1 + x ) 1 / 3 f − 1 ( x ) = ( 1 + x ) 1 / 3

f − 1 ( x ) = 5 9 ( x − 32 ) . f − 1 ( x ) = 5 9 ( x − 32 ) . Given the Fahrenheit temperature, x , x , this formula allows you to calculate the Celsius temperature.

t ( d ) = d 50 , t ( d ) = d 50 , t ( 180 ) = 180 50 . t ( 180 ) = 180 50 . The time for the car to travel 180 miles is 3.6 hours.

## Review Exercises

f ( − 3 ) = − 27 ; f ( − 3 ) = − 27 ; f ( 2 ) = − 2 ; f ( 2 ) = − 2 ; f ( − a ) = − 2 a 2 − 3 a ; f ( − a ) = − 2 a 2 − 3 a ; − f ( a ) = 2 a 2 − 3 a ; − f ( a ) = 2 a 2 − 3 a ; f ( a + h ) = − 2 a 2 + 3 a − 4 a h + 3 h − 2 h 2 f ( a + h ) = − 2 a 2 + 3 a − 4 a h + 3 h − 2 h 2

x = − 1.8 x = − 1.8 or or x = 1.8 or x = 1.8

− 64 + 80 a − 16 a 2 − 1 + a = − 16 a + 64 − 64 + 80 a − 16 a 2 − 1 + a = − 16 a + 64

( − ∞ , − 2 ) ∪ ( − 2 , 6 ) ∪ ( 6 , ∞ ) ( − ∞ , − 2 ) ∪ ( − 2 , 6 ) ∪ ( 6 , ∞ )

increasing ( 2 , ∞ ) ; ( 2 , ∞ ) ; decreasing ( − ∞ , 2 ) ( − ∞ , 2 )

increasing ( − 3 , 1 ) ; ( − 3 , 1 ) ; constant ( − ∞ , − 3 ) ∪ ( 1 , ∞ ) ( − ∞ , − 3 ) ∪ ( 1 , ∞ )

local minimum ( − 2 , − 3 ) ; ( − 2 , − 3 ) ; local maximum ( 1 , 3 ) ( 1 , 3 )

( − 1.8 , 10 ) ( − 1.8 , 10 )

( f ∘ g ) ( x ) = 17 − 18 x ; ( g ∘ f ) ( x ) = − 7 − 18 x ( f ∘ g ) ( x ) = 17 − 18 x ; ( g ∘ f ) ( x ) = − 7 − 18 x

( f ∘ g ) ( x ) = 1 x + 2 ; ( g ∘ f ) ( x ) = 1 x + 2 ( f ∘ g ) ( x ) = 1 x + 2 ; ( g ∘ f ) ( x ) = 1 x + 2

( f ∘ g ) ( x ) = 1 + x 1 + 4 x , x ≠ 0 , x ≠ − 1 4 ( f ∘ g ) ( x ) = 1 + x 1 + 4 x , x ≠ 0 , x ≠ − 1 4

( f ∘ g ) ( x ) = 1 x , x > 0 ( f ∘ g ) ( x ) = 1 x , x > 0

sample: g ( x ) = 2 x − 1 3 x + 4 ; f ( x ) = x g ( x ) = 2 x − 1 3 x + 4 ; f ( x ) = x

f ( x ) = | x − 3 | f ( x ) = | x − 3 |

f ( x ) = 1 2 | x + 2 | + 1 f ( x ) = 1 2 | x + 2 | + 1

f ( x ) = − 3 | x − 3 | + 3 f ( x ) = − 3 | x − 3 | + 3

f − 1 ( x ) = x - 9 10 f − 1 ( x ) = x - 9 10

f − 1 ( x ) = x - 1 f − 1 ( x ) = x - 1

The function is one-to-one.

## Practice Test

The relation is a function.

The graph is a parabola and the graph fails the horizontal line test.

2 a 2 − a 2 a 2 − a

− 2 ( a + b ) + 1 − 2 ( a + b ) + 1

f − 1 ( x ) = x + 5 3 f − 1 ( x ) = x + 5 3

( − ∞ , − 1.1 ) and ( 1.1 , ∞ ) ( − ∞ , − 1.1 ) and ( 1.1 , ∞ )

( 1.1 , − 0.9 ) ( 1.1 , − 0.9 )

f ( 2 ) = 2 f ( 2 ) = 2

f ( x ) = { | x | if x ≤ 2 3 if x > 2 f ( x ) = { | x | if x ≤ 2 3 if x > 2

x = 2 x = 2

f − 1 ( x ) = − x − 11 2 f − 1 ( x ) = − x − 11 2

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## Eureka Math Grade 3 Module 5 Answer Key | Engage NY Math 3rd Grade Module 5 Answer Key

Eureka Math Grade 3 Module 5 Answer Key extends and deepens students’ knowledge of understanding fractions as equal partitions of a whole. Knowledge becomes more formal if you work with area models and the number line from our EngageNY Eureka Math 3rd Grade Module 5 Solutions Key. You can download Eureka Math Third Grade Module 5 Solutions for free of cost via quick links available. Take your preparation to the next level by practicing from the Eureka Math Grade 3 Answer Key Module 5 Fractions as Numbers on the Number Line Answer Key on a daily basis.

## EngageNY Math Grade 3 Module 5 Answer Key | Eureka Math 3rd Grade Module 5 Answer Key

Eureka Math Grade 3 Module 5 Solutions Key includes the topics Partitioning a Whole into Equal Parts, Unit Fractions and their Relation to the Whole, Comparing Unit Fractions and Specifying the Whole, Fractions on the Number Line, Equivalent Fractions, etc. Access the Topicwise Eureka Math Grade 3 Module 5 Answer Key via quick links available below. You just have to tap on the respective topic at your convenience and prepare the concepts within quite easily.

Eureka Math Grade 3 Module 5 Fractions as Numbers on the Number Line

Eureka Math Grade 3 Module 5 Topic A Partitioning a Whole into Equal Parts

- Eureka Math Grade 3 Module 5 Lesson 1 Answer Key
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Eureka Math 3rd Grade Module 5 Topic B Unit Fractions and Their Relation to the Whole

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Engage NY Math 3rd Grade Module 5 Topic C Comparing Unit Fractions and Specifying the Whole

- Eureka Math Grade 3 Module 5 Lesson 10 Answer Key
- Eureka Math Grade 3 Module 5 Lesson 11 Answer Key
- Eureka Math Grade 3 Module 5 Lesson 12 Answer Key
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Eureka Math Grade 3 Module 5 Mid Module Assessment Answer Key

EngageNY Math Grade 3 Module 5 Topic D Fractions on the Number Line

- Eureka Math Grade 3 Module 5 Lesson 14 Answer Key
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3rd Grade Eureka Math Module 5 Topic E Equivalent Fractions

- Eureka Math Grade 3 Module 5 Lesson 20 Answer Key
- Eureka Math Grade 3 Module 5 Lesson 21 Answer Key
- Eureka Math Grade 3 Module 5 Lesson 22 Answer Key
- Eureka Math Grade 3 Module 5 Lesson 23 Answer Key
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- Eureka Math Grade 3 Module 5 Lesson 27 Answer Key

Engage NY Grade 3 Module 5 Topic F Topic F Comparison, Order, and Size of Fractions

- Eureka Math Grade 3 Module 5 Lesson 28 Answer Key
- Eureka Math Grade 3 Module 5 Lesson 29 Answer Key
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Eureka Math Grade 3 Module 5 End of Module Assessment Answer Key

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## Home > INT2 > Chapter 9 > Lesson 9.3.1

Lesson 9.1.1, lesson 9.1.2, lesson 9.1.3, lesson 9.1.4, lesson 9.2.1, lesson 9.2.2, lesson 9.3.1, lesson 9.3.2, lesson 9.3.3, lesson 9.3.4, lesson 9.4.1.

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## Go Math Grade 1 Chapter 3 Answer Key Pdf Addition Strategies

Go Math Grade 1 Chapter 3 Answer Key Pdf: Students of First Grade can keep a track of their preparation level using the Go Math Grade 1 Answer Key Chapter 3 Addition Strategies. By Solving Problems from the Go Math Grade 1 Ch 3 Answers you can enhance your fundamentals both at Chapter Level and Lesson Level. The HMH Go Math Grade 1 Answer Key Chapter 3 Addition Strategies covers questions belonging to Lessons, Assessments, Homework & Practice, Review Tests, etc. You can download the First Grade Go Math Chapter 3 Answer Key and access the preparation material no matter whichever corner you are preparing.

## Addition Strategies Go Math Grade 1 Chapter 3 Answer Key Pdf

To become proficient in the Concepts of Go Math Grade 1 Chapter 3 Addition Strategies all you need to do is be thorough with the concepts and solve the various questions from it. Access the Topicwise HMH Go Math Grade 1 Ch 3 Addition Strategies Solution Key by simply clicking on the direct links available. Enhance your conceptual knowledge taking the help of these quick resources and become a pro in the subject.

Addition Strategies

- Addition Strategies Show What You Know – Page 128
- Addition Strategies Vocabulary Builder – Page 129
- Addition Strategies Game: Ducky Sums – Page 130
- Addition Strategies Vocabulary Game – Page(130A-130B)

Lesson: 1 Algebra • Add in Any Order

- Lesson 3.1 Algebra • Add in Any Order – Page(131-136)
- Algebra • Add in Any Order Homework & Practice 3.1 – Page(135-136)

Lesson: 2 Count On

- Lesson 3.2 Count On – Page(137-142)
- Count On Homework & Practice 3.2 – Page(141-142)

Lesson: 3 Add Doubles

- Lesson 3.3 Add Doubles – Page(143-148)
- Add Doubles Homework & Practice 3.3 – Page(147-148)

Lesson: 4 Use Doubles to Add

- Lesson 3.4 Use Doubles to Add -Page(149-154)
- Use Doubles to Add Homework & Practice 3.4 – Page(153-154)

Lesson: 5 Doubles Plus 1 and Doubles Minus 1

- Lesson 3.5 Doubles Plus 1 and Doubles Minus 1 – Page(155-160)
- Doubles Plus 1 and Doubles Minus 1 Homework & Practice 3.5 – Page(159-160)

Lesson: 6 Practice the Strategies

- Lesson 3.6 Practice the Strategies – Page(161-166)
- Practice the Strategies Homework & Practice 3.6 – Page(165-166)

Mid-Chapter Checkpoint

- Addition Strategies Mid-Chapter Checkpoint – Page 164

Lesson: 7 Add 10 and More

- Lesson 3.7 Add 10 and More – Page(167-172)
- Add 10 and More Homework & Practice 3.7 – Page(171-172)

Lesson: 8 Make a 10 to Add

- Lesson 3.8 Make a 10 to Add – Page(173-178)
- Make a 10 to Add Homework & Practice 3.8 – Page(177-178)

Lesson: 9 Use Make a 10 to Add

- Lesson 3.9 Use Make a 10 to Add – Page(179-184)
- Use Make a 10 to Add Homework & Practice 3.9 – Page(183-184)

Lesson: 10 Algebra • Add 3 Numbers

- Lesson 3.10 Algebra • Add 3 Numbers – Page(185-190)
- Algebra • Add 3 Numbers Homework & Practice 3.10 – Page(189-190)

Lesson: 11 Algebra • Add 3 Numbers

- Lesson 3.11 Algebra • Add 3 Numbers – Page(191-196)
- Algebra • Add 3 Numbers Homework & practice 3.11 – Page(195-196)

Lesson: 12 Problem Solving • Use Addition Strategies

- Lesson 3.12 Problem Solving • Use Addition Strategies – Page(197-202)
- Problem Solving • Use Addition Strategies Homework & Practice 3.12 – Page(201-202)
- Addition Strategies Chapter 3 Review/Test – Page(203-206)

Curious George

## Addition Strategies Show What You Know

Model Addition

2 + 3 = 5

Therefore, there are 5 in total

Use Symbols to Add Use the picture. Write the addition sentence.

Answer: 2 + 4 = 6

Explanation:

The number of red bugs = 2

The number of green bugs = 4

Total : 2 + 4 = 6

Therefore, the total number of bug pictures are 6.

The number of purple butterflies = 3

The number of orange butterflies = 1

Total butterflies : 3 + 1 = 4

Therefore, there are 4 butterflies

Add in Any Order

Total : 1 + 4 = 5

Total : 4 + 1 = 5

## Addition Strategies Vocabulary Builder

Visualize It Write the addends and the sum for the addition sentence

Understand Vocabulary Use a review word to complete the sentence.

Question 1. 4 and 3 in 4 + 3 = 7 are ___

In 4 + 3 = 7 , 4 and 3 are addends.

Question 2. 4 + 3 = 7 is an ___

4 + 3 = 7 is an Addition sentence.

Question 3. 4 cubes and 3 cubes are put together to __ the groups.

4 cubes and 3 cubes are put together to 7 the groups.

## Addition Strategies Game: Ducky Sums

## Addition Strategies Vocabulary Game

Going Places with GOMATH! words

Concentration

Materials 2 sets of word cards How to Play Play with a partner.

- Mix the cards. Put the cards in rows blank side up.
- If the words match, keep the pair of cards.
- If the words do not match, turn the cards blank side up again.
- The other player takes a turn.
- Find all the pairs. The player with more pairs wins.

The Write Way

## Lesson 3.1 Algebra • Add in Any Order

Essential Question What happens if you change the order of the addends when you add?

Listen and Draw

Answer: Addition sentence : 7 + 8 = 15

= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Answer: Addition sentence ( changing the order ) : 8 + 7 = 15

= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Math Talk MATHEMATICAL PRACTICES

Describe how knowing the fact 7 + 8 helps you find 8 + 7.

Model and Draw

Answer: 5 + 6 = 11 Fact : Even after changing the order of addends , the sum is same .

Share and Show MATH BOARD

Add. Change the order of the addends. Add again.

Answer: 8 + 9 = 17 Addition sentence after changing the order of addends = 9 + 8 = 17

Answer: 6 + 7 = 1 3 Addition sentence after changing the order of addends = 7 + 6 = 13

Answer: 7 + 5 = 12 Addition sentence after changing the order of addends = 5 + 7 = 12

Answer: 2 + 8 = 10 Addition sentence after changing the order of addends = 8 + 2 = 10

Answer: 9 + 2 = 11 Addition sentence after changing the order of addends = 2 + 9 = 11

Answer: 8 + 4 = 12 Addition sentence after changing the order of addends = 4 + 8 = 12

On Your Own

MATHEMATICAL PRACTICE Attend to Precision Add. Change the order of the addends. Add again.

Answer: 9 + 6 = 15 Addition sentence after changing the order of addends = 6 + 9 = 15

Answer: 0 + 6 = 6 Addition sentence after changing the order of addends = 6 + 0 = 6

Answer: 8 + 3 = 11 Addition sentence after changing the order of addends = 3 + 8 = 11

Answer: 5 + 9 = 14 Addition sentence after changing the order of addends = 9 + 5 = 14

Answer: 4 +5 = 9 Addition sentence after changing the order of addends = 5 + 4 = 9

Answer: 8 + 5 = 13 Addition sentence after changing the order of addends = 5 + 8 = 13

Question 13. THINK SMARTER Nina uses the number sentence 3 + 7 = 10 to tell about her toy trucks. What other number sentence could Nina write to tell about her trucks using the same addends? __ = __ + __

Answer: Given, Addition sentence : 3 + 7 = 10 Other Addition sentence : 10 = 7 + 3

Answer: Given , Addition sentence = 4 + 7 = 11 Other addition fact : 7 + 5 = 11

Problem Solving • Applications WRITE Math

Write two additional sentences you can use to solve the problem. Write the answer.

Question 15. Roy sees 4 big fish and 9 small fish. How many fish does Roy see? __ fish __ + __ = __ __ + __ = __

Explanation: Given, Number of big fishes = 4 Number of small fishes = 9 Total number of fishes = 4 + 9 = 13 Therefore, Roy sees 13 fishes Addition sentences: 4 + 9 = 1 3 9 + 3 = 13

Question 16. THINK SMARTER Justin has 6 toys. He gets 8 more toys. How many toys does he have now? __ toys __ + __ = __ __ + __ = __

Explanation: Given, Justin has 6 toys. Number of toys he gets more = 8 Total : 6 + 8 = 14 Therefore, the total number of toys = 14 Addition sentences: 6 + 8 = 14 8 + 6 = 14

Answer: 5 + 5 = 10

Explanation : Given, Anna has 2 groups of pennies and also total number of pennies = 10 The addition sentence can be written with same addends as 5 + 5 = 10

Question 18. THINK SMARTER Write the addends in a different order. 3 + 4 = 7 __ + __ = 7

Answer: 4 + 3 = 7

TAKE HOME ACTIVITY • Ask your child to explain what happens to the sum when you change the order of the addends.

## Algebra • Add in Any Order Homework & Practice 3.1

Answer: 7 + 3 =10 3 + 7 = 10

Answer: 4 + 7 = 11 7 + 4 = 11

Answer: 9 + 8 = 17 8 + 9 = 17

Problem Solving

Write two additional sentences you can use to solve the problem.

Question 4. Camila has 5 shells. Then she finds 4 more shells. How many shells does she have now? __ + __ = __ __ + __ = __

Explanation : Given that, The number of shells Camila have = 5 The number of shells she finds more = 4 Total: 5+4= 9 Therefore, the number of shells she have now = 9 Addition sentences : 5 + 4 = 9 4 + 5 = 9

13 can be shown as the sum of 6 and 7 6 + 7 = 13

Lesson Check

Question 1. What is another way to write 7 + 6 = 13? 6 + 7 = __

Answer: 6 + 7 = 13

Grade 1 Go Math Chapter 3 Lesson 1 Answer Key Question 2. What is another way to write 6 + 8 = 14? 8 + 6 = __

Answer: 8 + 6 = 14

Spiral Review

Question 3. What is the sum? Write the number.

Question 4. How many nests are there? Write the number. 2 nests and 1 more nest __ nests

Explanation : 2 + 1 = 3 Therefore, the total number of nests = 3

## Lesson 3.2 Count On

Essential Question How do you count on 1, 2, or 3?

Start at 9. How can you count on to add? Add 1.

Add 1. 9 + 1 =10 Add 2. 9 + 2 = 11 Add 3. 9 + 3 = 12

Math Talk MATHEMATICAL PRACTICES Analyze How is counting on 2 like adding 2?

Answer: Greater addend : 6 6 + 2 = 8

Answer: Greater addend : 6 6 + 3 = 9

Answer: Greater addend = 6 6 + 1 = 7

Answer: Greater addend : 7 7 + 1 = 8

Answer: Greater addend = 7 7 + 2 = 9

Answer: Greater addend : 7 7 + 3 = 10

MATHEMATICAL PRACTICE Circle the greater addend. Count on to find the sum.

Answer: Greater addend : 9 1 + 9 = 10

Answer: Greater addend : 8 3 + 8 = 11

Answer: Greater addend : 8 1 + 8 = 9

Answer: Greater addend : 6 1 + 6 = 7

Answer : Greater addend : 9 9 + 3 =12

Answer: Greater addend : 7 7 + 2 = 9

Answer: Greater addend : 6 2 + 6 = 8

Answer: Greater addend : 5 + 3 = 8

Answer: Greater addend : 7 7 + 1 = 8

Answer: Greater addend = 7 3 + 7 =10

Answer: Greater addend : 9 9 + 2 = 11

Answer: Greater addend : 4 3 + 4 = 7

Question 19. GO DEEPER Adam has 6 hats. Molly has 3 hats. They stack all their hats. Then Blake puts 2 more hats on the stack. How many hats are on the stack? __ + __ = ___ hats __ + __ = ___ hats

Answer: 6 + 3 = 9 9 + 2 = 11

Question 20. MATHEMATICAL PRACTICE Explain Terry added 3 and 7. He got a sum of 9. His answer is not correct. Describe how Terry can find the correct sum. _______________ _ _ _ _ _ _ _ _ _ _ _ _______________ _______________ _ _ _ _ _ _ _ _ _ _ _ _______________

Answer: No, he is wrong. The sum of 3 and 7 is 10 Addition sentence : 3 + 7 = 10 Starts at 7 and counts by 3, means 8, 9, 10 So, the answer is 0.

Draw to solve. Write the addition sentence.

Explanation : Given that, The number of oranges Cindy and Joe pick = 8 The number of more oranges they pick = 3 Total : 8 + 3 = 11 oranges Therefore, the number of total oranges = 11

________________________________________________________________________________________________________

1 2 3 4 5 6 7 8 9 10 11

Which three numbers can you use to complete the problem?

Explanation : Jennifer has 7 stamps Number of more stamps = 3 Addition sentence : 7 + 3 = 10

TAKE HOME ACTIVITY • Have your child tell you how to count on to find the sum for 6 + 3.

## Count On Homework & Practice 3.2

Circle the greater addend. Count on to find the sum.

Answer: Greater addend : 8 8 + 2 = 10

Answer: Greater addend : 7 1 + 7 = 8

Answer: Greater addend : 9 3 + 9 = 12

Answer: Greater addend : 5 5 + 3 = 8

Go Math 1 Circle The Greater Addend Count on to Find the Sum Question 5. Jon eats 6 crackers. Then he eats 3 more crackers. How many crackers does he eat? __ + __ = __ crackers

Explanation : Number of crackers Jon eat = 6 Number of crackers he ate more = 3 Total : 6 + 3 = 9 Crackers Therefore, Jon ate 9 crackers .

Answer: 9 + 3 = 12

Question 1. Count on to solve 5 + 2. Write the sum __

Answer: 5 + 2 = 7

Question 2. Count on to solve 1 + 9. Write the sum. __

Answer: 1 + 9 = 10

Question 3. What does the model show? __ + __ = ___

Answer: 3 + 3 = 6

Answer: 4 + 2 = 6

## Lesson 3.3 Add Doubles

Essential Question What are doubles facts?

__ + __ = __

Answer: 2 + 2 = 4

Math Talk MATHEMATICAL PRACTICES Use Tools Describe how your model shows a doubles fact.

Answer: Answer:

1 + 1 = 2 2 + 2 = 4

_____________________________

Answer: 4 + 4 = 8

___________________________

_______________________________________________

Answer: 6 + 6 = 12

____________________________________________________________________________

Answer: 7 + 7 = 14

______________________________________________________________________________

Answer: 8 + 8 = 16

________________________________________________________________________

________________________________________________________________

Answer: 10 + 10 = 20

Write a doubles fact to solve.

Answer : 16

Explanation : The number of apples meg kept in a basket = 8 The number of apples Paul kept in a basket = 8 Total : 8 + 8 = 16 apples Therefore, total number of apples in the basket = 16

Answer : 9 + 9 = 18

Explanation : Given, Total number of people in the party = 18 Also given , The number of boys is the same as the number of girls. So, 9 + 9 = 18.

Answer: The number sentence with doubles fact = 4 + 4 = 8

TAKE HOME ACTIVITY • Have your child choose a number from 1 to 10 and use that number in a doubles fact. Repeat with other numbers.

## Add Doubles Homework & Practice 3.3

Question 4.

Go Math Grade 1 Pdf Lesson 3.3 Eat It or Toss It Answer Key Question 5. There are 16 crayons in the box. Some are green and some are red. The number of green crayons is the same as the number of red crayons. __ = __ + __

Answer: 16 = 8 + 8

Given , The total number of crayons in the box = 16 Also give, the number of green crayons is the same as the number of red crayons. Now, By using doubles fact we can write number sentence as: 8 + 8 = 16

Question 1. Write a doubles fact with the sum of 18. __ + __ = 18

Answer: A doubles fact with the sum of 18 = 9 + 9 = 18

Question 2. Write a doubles fact with the sum of 12. __ + __ = 12

Answer: A doubles fact with the sum of 12 = 6 + 6 = 12

Explanation: The sum of 3 and 2 is 5 Number sentence : 3 + 2 = 5

Question 4. Draw circles to show the numbers. Write the sum. __ + __ = __

## Lesson 3.4 Use Doubles to Add

Essential Question How can you use doubles to help you add?

There are __ fish.

Look for Structure How does knowing 3 + 3 help you solve the problem?

Answer: 6 + 7 = 13 By using doubles fact , 13 can be shown as 6 + 6 + 1 = 13

Answer : 9 + 8 = 17 8 + 8 + 1 = 17 So, 9 + 8 = 17

Answer: 5 + 5 + 1 = 11 So, 5 + 6 = 11

Answer: 7 + 8 = 15 By using doubles fact , 7 + 7 + 1 = 15

Answer : 5 + 4 = 9 By using doubles fact , 4 + 4 + 1 = 9

Answer : 17

Explanation : Given , the total number of leaves = 17 Also given , Mandy has the same number of red and yellow leaves And also find another yellow flower By using doubles plus one effect , we can show addition sentence as : 8 + 8 + 1 = 17 Therefore, Number of red flowers = 8 Number of yellow flowers = 9.

GO DEEPER Explain Would you use count on or doubles to solve? Why?

Answer: 3 + 3 + 1 = 7 Doubles plus one fact.

Answer : 9 and count on 3

Problem Solving • Applications

Explanation: The number of red cubes = 7 The number of yellow cubes = 8 Total : 7 + 8 = 15 Number sentence with double fact = 7 + 7+ 1 = 15 So, 7 + 8 = 15

TAKE HOME ACTIVITY • Ask your child to show you how to use what he or she knows about doubles to help solve 6 + 5.

## Use Doubles to Add Homework & Practice 3.4

Answer: 5 + 6 = 11 The number sentence with doubles fact = 5 + 5 + 1 = 11

Answer: 9 + 8 = 17 The number sentence with doubles fact = 8 + 8 + 1 = 17

Use doubles to help you add.

Question 3. 8 + 7 = __ Answer: Doubles plus one fact.

Explanation : 8 + 7 = 15 The number sentence with doubles fact : 7 + 7 + 1 = 15

Question 4. 6 + 5 = __

Answer: Doubles plus one fact.

Explanation : 6 + 5 = 11 The number sentence with doubles plus one fact : 5 + 5 + 1 = 11

Question 5. 7 + 6 = __

Answer: Doubles plus one fact .

Explanation : 7 + 6 = 13 The number sentence with doubles plus one fact : 6 + 6 + 1 = 13

Solve. Draw or write to explain.

Go Math 1st Grade Lesson 1 Homework 3.4 Question 6. Bo has 6 toys. Mia has 7 toys. How many toys do they have? __ toys

Explanation: The number of toys Bo have = 6 The number of toys Mia have = 7 Total : 6 +7 = 13 Therefore, the total number of toys they both have = 13

Question 1. Use doubles to find the sum of 7 + 8. Write the number sentence. __ + __ + __ = __ Answer: The sum of 7 + 8 = 15 The number sentence with doubles plus one fact : 7 + 7 + 1 = 15

Answer: 8 – 6 = 2

Answer: 7 – 2 = 5 Therefore, there are 5 fewer black kittens are there than gray kittens.

## Lesson 3.5 Doubles Plus 1 and Doubles Minus 1

Essential Question How can you use what you know about doubles to find other sums?

How can you use the doubles fact, 4 + 4, to solve each problem? Draw to show how. Complete the addition sentence.

- 4 + 4 + 1 = 9 – Doubles plus one fact
- 4 + 4 – 1 = 7 – Doubles minus one fact.

Explain what happens to the doubles fact when you increase one addend by one or decrease one addend by one.

Answer: 2 + 2 = 4 2+ 2+ 1 = 5 doubles plus one 2 + 2 – 1 = 3 – doubles minus one.

Answer : 3 + 3 = 6 3 + 3 + 1 = 7 – Doubles plus one 3 + 3 – 1 = 5 – Doubles minus one

Answer : 4 + 4 = 8 4 + 4 + 1 = 9 – Doubles plus one 4 + 4 – 1 = 7 – Doubles minus one

MATHEMATICAL PRACTICE Make Connections Add. Write the doubles fact you used to solve the problem.

Explanation : 8 + 9 = 17 The number sentence with doubles plus one fact : 8 + 8 + 1 = 17

Answer: Doubles plus one fact . Explanation : 2 + 3 = 5 The number sentence with doubles plus one fact : 2 + 2 + 1 = 5

Explanation : 7 + 6 = 13 The number sentence with doubles plus one fact : 6 + 6 + 1 = 15

Explanation : 6 + 5 = 11 The number sentence with doubles plus one fact : 5 + 5 + 1 = 15

Explanation : 3 + 4 = 7 The number sentence with doubles plus one fact : 3 + 3 + 1 = 7

Explanation : 4 + 5 = 9 The number sentence with doubles plus one fact : 4 + 4 + 1 = 15

Answer : 13

Explanation : The number of toy ducks Brianna

THINK SMARTER Add. Write the doubles plus one fact. Write the doubles minus one fact.

Answer : 6 + 6 = 12 6 + 6 + 1 = 1 3 6 + 6 – 1 = 11

Answer : 9 + 9 = 18 – No 5 + 5 = 10 -Yes 4 + 4 = 8 – Yes

TAKE HOME ACTIVITY • Have your child explain how to use a doubles fact to solve the doubles plus one fact 4 + 5 and the doubles minus one fact 4 + 3.

## Doubles Plus 1 and Doubles Minus 1 Homework & Practice 3.5

Add. Write the doubles fact you used to solve the problem.

Explanation : 8 + 7 = 15 The number sentence with doubles plus one fact : 7 + 7 + 1 = 15

Explanation : 6 + 7 = 13 The number sentence with doubles plus one fact : 6 + 6 + 1 = 13

Explanation : 4 + 3 = 7 The number sentence with doubles plus one fact : 3 + 3 + 1 = 7

Explanation : 2 + 1 = 3 The number sentence with doubles plus one fact : 1 + 1 + 1 = 3

Explanation : 3 + 2 = 5 The number sentence with doubles plus one fact : 2 + 2 + 1 = 5

Question 7. Andy writes an additional fact. One addend is 9. The sum is 17. What is the other addend? Write the additional fact. __ + ___ = 17

Answer : 9 + 8 = 17 Therefore, other addends = 8

Answer : 4 + 5 = 9 By using doubles plus one fact , 4 + 4 + 1 = 9

Answer: The number sentence with double plus one : 2 + 2 + 1 = 5

Question 2. Which doubles fact helps you solve 8 + 7 = 15? Write the number sentence. __ + __ + __

Answer: 8 + 7 = 15 The number sentence with doubles plus one fact : 7 + 7 + 1 = 15

Answer : 7 + 2 = 9 Therefore, total number of dogs = 9

Answer : 2 + 1 = 3

## Lesson 3.6 Practice the Strategies

Essential Question What strategies can you use to solve addition fact problems?

Math Talk MATHEMATICAL PRACTICES Look for Structure Why is the sum the same when you use different strategies?

Answer : 4 + 1 =5 5+ 1 = 6 6+ 1= 7 7 + 1 = 8

5+ 2= 7 6 + 2 = 8 7 + 2 = 9 8 + 2 = 10

Answer : 6 + 3 = 9 7 + 3 = 10 8 + 3 = 11 9 + 3 = 1 2

Answer : 7 + 7 = 14 8 + 8 = 16 9 + 9 =18 10 + 10 = 20

Answer : Doubles plus one : 5 + 5 + 1 = 11 6 + 6 + 1 = 13 Doubles minus one : 7 + 7 – 1 = 15 8 + 8 -1 = 17

Question 6. 9 + 9 = __

Question 7. 7 + 1 = __

Question 8. 5 + 3 = __

Question 9. 2 + 9 = __

Answer : 2 + 9 = 11

Question 10. 7 + 3 = __

Answer : 7 + 3 = 10

Question 11. 7 + 7 = __

Answer : 7 + 7 = 14

Question 12. 6 + 5 = __

Answer : 6 + 5 = 11

Question 13. 2 + 8 = __

Answer : 8 + 2 = 10

Question 14. 8 + 8 = __

Answer : 8 + 8 = 16

Question 15. 8 +9 = __

Answer : 8 + 9 = 17

Question 16. 9 + 3 = __

Answer : 9 + 3 = 12

Question 17. 7 + 8 = __

Answer : 7 + 8 = 15

THINK SMARTER Make a counting on the problem. Write the missing numbers.

Answer : The birds on a tree = 7 The number of birds flew there = 3 Total : 7 + 3 = 10

TAKE HOME ACTIVITY • Have your child point out a doubles fact, a doubles plus one fact, a doubles minus one fact, and a fact he or she solved by counting on. Have him or her describe how each strategy works.

## Practice the Strategies Homework & Practice 3.6

Answer : 8 + 1 = 9

Answer : 1 + 7 = 8

Answer : 8 + 3 = 11

Answer : 5 + 5 = 10

Answer : 8 + 7 = 15

Answer : 6 + 3 = 9

Answer : 6 + 6 = 12

Make a counting on problem. Write the missing numbers.

Question 10. __ apples are in a bag. __ more apples are put in the bag. How many apples are in the bag now? __ apples

Answer : 5 apples are in a bag. 2 more apples are put in the bag. Total number of apples = 5 + 2 = 7

Question 1. Which strategy would you use to find 2 + 8? Explain how you decided. _________________________ _________________________

Answer : Count on. 8 + 2 = 10

Question 2. What is the sum of 9 + 9? Write the number. __

Answer: 18 The sum of 9 + 9= 18

Question 3. What is the sum of 5 + 2 or 2 + 5? Why is the sum the same? _________________________ _________________________

Answer: The sum of 5 + 2 or 2 + 5 = 7 The sum is same because

Question 4. How many flowers are there? Write the number. 3 flowers and 3 more flowers __ flowers

Answer: 6 3 + 3 = 6

## Addition Strategies Mid-Chapter Checkpoint

Concepts and Skills

Answer : 4 + 8 = 12

Answer : 9 + 7 = 16

Answer: Greater addend : 7 3 + 7 = 10

Answer : Greater addend : 6 6 + 3 = 9

Answer : Greater addend : 7 7 + 1 = 8

Answer : Greater addend : 8 2 + 8 = 10

Question 9. 7 + 8 = __

Answer : 7 + 8 = 15 Doubles fact : 7 + 7 + 1 = 15

Question 10. 6 + 7 = __

Answer : 6 +7 = 13 Doubles fact : 6 + 6 + 1 = 13

Question 11. 9 + 8 = __

Answer : 9 + 8 = 17 Doubles fact : 8 + 8 +1 = 17

Answer : Count on fact to show sum of 8 = 7 +1 = 8 Doubles fact plus one = 4+ 4+ 1 = 8

## Lesson 3.7 Add 10 and More

Essential Question How can you use a ten frame to add 10 and some more?

Answer: 10+ 5 = 15

Math Talk MATHEMATICAL PRACTICES Reasoning Explain how your model shows 10 + 5.

Answer: 10 + 3 = 13

Answer : 10+ 1 = 11

Answer : 10 + 2 = 12

Answer : 10+ 4 = 14

Answer : 10+ 7 = 17

Answer: 10 + 8 = 18

Answer: 10 + 2 = 12

Answer: 10 + 6 = 16

Answer: 10+9 = 19

Answer: 10+ 1 = 11

Answer : 4 + 10 = 14

Answer : 5 + 10 = 15

Answer : 10 + 3 = 13

0 + 10 = 10

Answer: 10+ 4 = 14

Answer: 1 7

Explanation : The number of crayons Marina have = 10 The number of crayons she gets more = 7 Addition sentence : 1 0 + 7 = 17 Therefore, total crayons Marina have = 17

TAKE HOME ACTIVITY • Have your child choose a number between 1 and 10 and then find the sum of 10 and that number. Repeat using other numbers.

## Add 10 and More Homework & Practice 3.7

Draw red ○ to show 10. Draw ○ yellow to show the other addend. Write the sum.

Answer : 10 + 7 = 17

Answer : 10+ 5 = 15

Draw red and yellow ○ to solve. Write the addition sentence.

Question 3. Linda has 10 toy cars. She gets 6 more cars. How many toy cars does she have now? __ + __ = __ toy cars

Explanation: The number of toys Linda have = 10 The number of toys she gets more = 6 Total : 10 + 6 = 16 Therefore, the total number of toys she have now = 16

Answer: The addition sentence : 10 + 8 = 18

2. 5 + 5 = 10

3. 8 + 2 = 10

Explanation : The number of large turtles : 3 The number of small turtles = 1 Number sentence : 3 + 1 = 4 Therefore, total number of turtles = 4

## Lesson 3.8 Make a 10 to Add

Essential Question How do you use the make a ten strategy to add?

Answer: 9 + 6 = 15 9 + 1 = 10 10 + 5 = 15

Math Talk MATHEMATICAL PRACTICES Use Tools Why do you start by putting 9 counters in the ten frame?

Answer : 9 + 5 = 14 9 + 1 = 10 So, 10 + 4 = 14

Answer : 4 + 7 = 11 7 + 3 = 10 10 + 1 = 11

Answer : 9 + 8 = 17 9 + 1 = 10 10+ 7 = 17

MATHEMATICAL PRACTICE Use a Concrete Model

Answer : 8 + 2 = 10 10 + 3 = 13 So, 5 + 8 = 13

Draw to make a ten. Write the missing number.

Answer : 16 times

Question 7. 10 + 8 has the same sum as 9 + __.

Answer: 10+ 8 = 18 9 + 9 = 18

Question 8. 10 + 7 has the same sum as 8 + __.

Answer: 10 + 7 = 17 8 + 9 = 17

Question 9. 10 + 5 has the same sum as 6 + __.

Answer : 10 +5 = 15 6 + 9 = 15

Question 10. GO DEEPER Write the numbers 6, 8, or 10 to complete the sentence __ + __ has the same sum as __ + 8

Answer: 10+ 6 = 18 has the same sum of 8 + 8.

Answer: 7 + 3 = 10 10 + 1 = 11 So, 7 + 4 = 11

TAKE HOME ACTIVITY • Cut off 2 cups from an egg carton or draw a 5-by-2 grid on a sheet of paper to create a ten frame. Have your child use small objects to show how to make a ten to solve 8 + 3, 7 + 6, and 9 + 9.

## Make a 10 to Add Homework & Practice 3.8

Use red and yellow ○ and a ten frame. Show both addends. Draw to make a ten. Then write the new fact. Add.

Answer: 5 + 5 = 10 10 + 2 = 12 So, 5 + 7 = 12

Answer: 9 + 1 = 10 10 + 4 = 14 So, 9 + 5 = 14

Solve. Question 3. 10 + 6 has the same sum as 7 + __.

Answer: 9 10 + 6 = 16 7 + 9 = 16

Answer: 5 + 7 = 12 5 + 5 = 10 10 + 2 = 12

Answer: Number sentence : 10 + 5 = 15

Answer: Number sentence : 10 + 4 = 14

Question 3. What is the sum of 4 + 6? Write the sum. __

Answer: 4 + 6 = 8 The sum of 4 + 6 = 8

Explanation : The number of big flowers = 2 The number of small flowers = 4 Number sentence : 2 + 4 = 6 Therefore, Total number of flowers = 6

## Lesson 3.9 Use Make a 10 to Add

Essential Question How can you make a ten to help you add?

Math Talk MATHEMATICAL PRACTICES Represent Describe how the drawings show how to make a ten to solve 6 + 7.

Write to show how you make a ten. Then add.

Answer: 8 + 2 + 2 = 12 10 + 2 = 12 So, 8 + 4 = 12

Answer: 7 + 3 = 10 10 + 2 = 12 So, 5 + 7 = 12

THINK SMARTER Write to show how you make a ten. Then add.

Answer: 8 + 2 = 10 10 + 5 = 15 So, 7 + 8 = 15

Answer: 9 + 1 = 10 1 0 + 7 = 17 So, 9 + 8 = 17 .

MATHEMATICAL PRACTICE Use Models

THINK SMARTER Use the model. Write to show how you make a ten. Then add.

Answer: 8 + 2 + 4 10 + 4 = 14 So, 8 + 6 = 14 Therefore, There are 14 balls of clay.

Use the clues to solve. Draw lines to match.

Question 7. GO DEEPER Look at Exercise 6. Han eats one apple. Now he has the same number of apples as Luis and Mike. How many red and green apples could he have? __ red apples and __ green apples

Answer : 7 red apples and 7 green apples

- 8 + 2 + 2 = Yes
- 5 + 4 + 3 = No
- 6 + 7 + 3 = Yes

## Use Make a 10 to Add Homework & Practice 3.9

Answer : 9 + 1 + 6 9 + 1 = 10 So, 9 + 6 = 15

Answer : 8 + 2 + 3 8 + 2= 10 So, 5 + 8 = 13

Answer: Ann – 10 green grapes and 6 red grapes. Gia – 7 green grapes and 9 red grapes.

Answer : 8 + 2 = 10 10 + 2 = 14

Answer: 5 – 5 = 0

Answer : 8 – 2 = 6

## Lesson 3.10 Algebra • Add 3 Numbers

Essential Question How can you add three addends?

Math Talk MATHEMATICAL PRACTICES Apply Which two addends did you add first? Explain.

Answer: 5 + 2 + 3 = 10

Addition sentences :

- 7 + 3 = 10
- 5 + 5 = 10

Answer: 3 + 4 + 6 = 13

- 3 + 10 = 13
- 7 + 6 = 13

MATHEMATICAL PRACTICE

Answer: 7 + 3 + 1 = 11 Addition sentences :

- 7 + 4 = 11
- 10 + 1 = 11

Answer: 3 + 6 + 3 = 12

- 6 + 6 = 12
- 3 + 9 = 12

GO DEEPER Solve both ways.

Answer: 2 + 3 + 7 = 12

- 2 + 10 = 12
- 5 + 7 = 1 2

Answer: Addition sentence : 2 + 2 + 3 = 7

Answer: 6 = 4 + 2 6 = 5 + 1

Answer: The addition sentences shows the sum 12 are : 1. 2 + 2 + 8 = 12 2. 6 + 0 + 6 = 12 The addition sentences shows the sum 13 are :

- 3 + 5 + 5 = 13
- 4 + 4 +5 = 13

TAKE HOME ACTIVITY • Have your child draw to show two ways to add the numbers 2, 4, and 6.

## Algebra • Add 3 Numbers Homework & Practice 3.10

Explanation: The two addition sentences to find the sum are

- 5 + 6 = 11
- 9 + 2 = 11

Question 2. Choose three numbers from 1 to 6. Write the numbers in an addition sentence. Show two ways to find the sum.

Answer: 3 + 5 + 2 = 10

Question 1. What is the sum of 3 + 4 + 2? Write the sum. __

Explanation: The sum of 3 + 4 + 2 = 9 Therefore, sum = 9

Question 2. What is the sum of 5 + 1 + 4? Write the sum. __

Explanation: The sum of 5 + 1 + 4 = 10 Therefore, sum = 10

Explanation: The sum of 3 and 7: 3 + 7 =10

Answer: Number sentence : 4 + 2 = 6 Therefore, the number of cows in the barn = 6

## Lesson 3.11 Algebra • Add 3 Numbers

Essential Question How can you group numbers to add three addends?

Apply Describe the two ways you grouped the numbers to add.

Choose a strategy. Circle two addends to add first. Write the sum. Then find the total sum. Then use a different strategy and add again.

Answer: Two addends : 6 , 4 Sum : 6 + 4 = 10 Total sum : 6 + 4 + 2 = 12

Answer: Two addends : 3 , 4 Sum : 3 + 4 = 7 Total sum : 3 + 4 + 4 = 11

Answer: Two addends : 2 , 5 Sum : 2 + 5 = 7 Total sum : 2 + 5 + 0 = 7

Answer: Two addends : 5 , 5 Sum : 5 + 5 = 10 Total sum : 5 + 4 + 5 = 14

MATHEMATICAL PRACTICE Use Repeated Reasoning Choose a strategy. Circle two addends to add first. Write the sum.

Answer: Two addends : 8 , 2 Sum : 8 + 2 + 2 =12

Answer : Two addends : 6, 0 Sum : 6 + 0 + 8 = 14

Answer : Two addends : 3 , 4 Sum : 3 + 4 + 6 = 13

Answer: Two addends : 2 , 3 Sum : 2 + 3 + 7 = 12

Answer: Two addends : 7 , 7 Sum : 7 + 7 + 2 = 16

Answer: Two addends : 1 , 1 Sum : 1 + 9 + 1 = 11

Answer: Two addends : 4 , 4 Sum : 5 + 4 + 4 = 13

Answer : Two addends : 5 , 5 Sum: 5 + 5+ 5 = 15

Question 13. THINK SMARTER Susan has 7 shells. Kai has 3 shells. Zach has 5 shells. How many shells do they have? __ + __ + __ + __ = __ shells

Explanation : The number of shells Susan has = 7 The number of Shells Kai have = 3 The number of shells Zach have = 5 Total : 7 + 3 + 5 = 15 Therefore, the number of shells they have = 7+3+5 = 15

THINK SMARTER Write the missing addends. Add.

Answer: The possible addend be 0,8 ; 1,7; 2,6; 3,5; 4,4 ; 5,3 ; 6,2 ; 7,1 ; 8,0 The sum of the number sentence is 11

Answer: Possible addends are : 0,7; 1,6; 2,5; 3,4; 4,3; 5,2; 2,6; 1,7; 7,0. The sum of number sentence = 10

Draw a picture. Write the number sentence.

Explanation: The number of cats Maria have = 3 The number of cats Jim have = 2 The number of cats Cheryl have = 5 Total : 3 + 2 + 5= 10 Therefore, the total number of cats Maria have = 10

Explanation: The number of small turtles = 5 The number of medium turtles = 0 The number of big turtles = 4 Total : 5 + 0 + 4 = 9 Therefore, the number of turtles tony have = 9

Answer: The sum of blue and red fish is 7 Total number of fishes in the tank = 13 Number of gold fishes = 6 Let, the number of blue fishes = 3 the number of red fishes = 4 Total red and blue fishes = 3 + 4 = 7 Number sentence : 3 + 4 + 6 = 14

Answer: 2 + 3 + 4 = 9

TAKE HOME ACTIVITY • Have your child look at Exercise 18. Have your child tell you how he or she decided which numbers to use. Have him or her tell you two new numbers that would work.

## Algebra • Add 3 Numbers Homework & practice 3.11

Choose a strategy. Circle two addends to add first. Write the sum.

Answer: Two addends : 7, 3 Sum : 7 + 3 + 3 = 13

Two addends : 2 , 2 Sum : 2 + 2+ 6 = 10

Answer: Two addends : 6 , 6 Sum : 6 + 6 +3 = 15

Answer: Two addends : 0 , 8 Total : 2 + 0 + 8 = 10

Question 5. Don has 4 black dogs. Tim has 3 small dogs. Sue has 3 big dogs. How many dogs do they have? __ + __ + __ = __ dogs

Explanation : The number of dogs Don have = 4 The number of dogs Tim have = 3 The number of dogs Sue have = 3 Total : 4 + 3 +3 = 10 Therefore, the number of dogs they have = 10

Answer : 6 + 4 + 4 = 14

Question 1. What is the sum of 4 + 4 + 2? __

Answer: 4 + 4 + 2 = 10

Answer: 7 + 3 = 10 Total sum : 7 + 3 + 2 = 12

Question 3. Write a doubles plus one fact for the sum of 7. __ + __ = __

Answer: A doubles plus one fact for the sum of 7 : 3 + 4 = 7 3 + 3 + 1 = 7

Answer: Number sentence : 8 + 3 = 11

## Lesson 3.12 Problem Solving • Use Addition Strategies

Essential Question How do you solve addition word problems by drawing a picture?

Megan put 8 fish in the tank. Tess put in 2 more fish. Then Bob put in 3 more fish. How many fish are in the tank now?

Unlock the Problem

Explanation : Given, Megan put in 8 fish. Tess put in 2 fish. Bob put in 3 fish. Total : 8 + 2 + 3 = 13 Therefore, total number of fishes = 13

HOME CONNECTION • Your child will continue to use this chart throughout the year to help him or her unlock the problem. In this lesson, your child used the strategy draw a picture to solve problems.

Try Another Problem

Draw a picture to solve.

Answer : 15

Explanation : The number of green toy cars = 9 The number of yellow toy cars = 1 The number of blue toy cars = 5 Total : 9 + 1 + 5 = 15 Therefore, the total number of toy cars he have now = 15

Math Talk MATHEMATICAL PRACTICES Reasoning Explain how using make a ten helps you solve the problem.

Write an Equation Draw a picture to solve.

Explanation : The number of marbles Ken kept in a jar = 5 The number of Lou kept in a jar = 0 The number of marbles Mae kept in jar = 5 Total : 5 + 0 + 5 = 10 marbles Therefore, there are 10 marbles in the jar.

Answer : 11

Explanation : The number of kites Ava have = 3 The number of kites Lexi have = 3 The number of kites Fred have = 5 Total : 3 + 3 + 5 = 11 Kites Therefore, the number of kites they have = 11

Explanation : The number of books Al got from library = 8 The number of books Ryan got from library = 7 The number of books Dee got from library = 1 Total : 8 + 7 + 1 = 16 books Therefore, the number of books they have = 16

Explanation : The number of letters Pete sends = 4 The number of letters he sends more = 3 Then also the number of letters he sends more = 2 Total : 4 + 3 + 2 = 9 letters Therefore, the number of letters Pete sends = 9

On Your Own WRITE Math

Solve. Draw or write to show your work.

Explanation : Total number of basketball cards = 15 The number of cards he gives away = 8 Now, the number basketball he have now = 15 – 8 = 7 Therefore, the number of cars he have now = 7

Explanation : Total number of pencils = 14 The number of pencils Haley have = 6 The number of pencils Mac have = 4 Total : 6 + 4 = 10 The number of pencils Sid have = 14 – 10 = 4 Therefore, Sid have 4 pencils.

Question 8. GO DEEPER 12 marbles are in a bag. Shelly takes 3 marbles. Dan puts in 4. How many marbles are in the bag now? __ marbles

Explanation : The total number of marbles in a bag = 12 The number of marbles Shelly takes out = 1 2 – 3 = 9 The number of marbles Dan puts in = 4 9 + 4 = 13 Therefore, total number of marbles in the bag = 13

Question 9. THINK SMARTER Eric has 4 pencils. Sandy gives Eric 3 pencils. Tracy gives Eric 5 more pencils. How many pencils does Eric have in all? Eric has ☐ pencils in all.

Explanation : The number of pencils Eric have = 4 The umber of pencils Sandy gives Eric = 3 The number of pencils Tracy gives Eric = 5 Total : 4 + 3 + 5 = 12 Therefore, the total number pencils Eric have = 12.

TAKE HOME ACTIVITY • Ask your child to look at Exercise 8 and tell how he or she found the answer.

## Problem Solving • Use Addition Strategies Homework & Practice 3.12

Explanation : The number of crayons Franco have = 5 The number of crayons he gets more = 8 The number of crayons he gets more = 2 Total number of crayons = 5 + 8 + 2 = 15 Therefore, the number of crayons he have now = 15

Answer : 14

Explanation: The number of blocks Jackson have = 6 The number of blocks Jackson gets more = 5 + 3 = 8 Total : 6 + 8 = 14 Therefore, Jackson have 14 blocks.

Answer : 12

Explanation : The number of gifts Ava have = 7 The number of gifts Ava gets more = 2 + 3 = 5 Total : 7 + 5 = 12 Therefore, Ava have 12 gifts.

Explanation : The number of large rocks Jeb have = 4 The number of medium rocks Jeb have = 4 The number of small rocks Jeb have = 7 Total : 4 + 4 + 7 = 15 Therefore, Jeb have 15 rocks.

Answer: The total number of stones = 14 Number sentence: 3 + 4 + 7 = 14

Explanation : The number of gray stones = 3 The number of black stones = 4 The number of white stones = 7 Total : 3 + 4 + 7 = 14

Answer: The Total number of stickers = 17 Number sentence: 3 + 6 + 8 = 17

Explanation: The number of red stickers = 3 The number of pink stickers = 6 The number of green stickers = 8 Total: 3 + 6 + 8 = 17

Question 3. What is the sum of 2 + 4 or 4 + 2? Write the number. ____

Explanation: The sum of of 2+4 or 4+2 = 6

Answer: The total number of pens = 9 Number sentence : 6 + 3 =9

Explanation: The number of black pens = 6 The number of blue pens = 3 Total : 6 + 3 = 9

## Addition Strategies Chapter 3 Review/Test

Question 1. Write the addends in a different order. 5 + 4 = 9 __ + __ = __

Answer: The addition sentence with the addends in a different order = 4 + 5 = 9

Question 2. Count on from 4. Write the number that shows 1 more. __

Answer: The addition sentence with doubles fact : 5 + 5 = 10 Therefore, The sum of cubes = 10

Answer: 3 + 4 = 3 + 3 +1 So, 3 + 4 = 7

- 7 + 7 = 14
- 8 + 8 = 14

Answer: Addition sentence on 2 fact to show a sum of 10 8 + 2 = 10 Addition sentence with doubles fact to show a sum of 10. 5 + 5 = 10

Explanation: Given, The model shows 8 + 5 = 13. Addition sentence with 10 fact that has same sum = 10 + 3 = 13

Answer: 1. 7 + 3 + 2 – Yes 2. 7 + 5 +5 – Yes 3. 5 + 4 + 7 – No

Explanation : The number of red cubes = 2 The number of blue cubes = 3 The number of yellow cubes = 3 Now, The addition sentence : 2 + 3 + 4 = 9 Total sum of cubes = 9

Question 11. Write two ways to group and add 4 + 2 + 5. __ + __ = __ __ + __ = __

Answer: 4 + 2 + 5 = 11 The two ways are: 6 + 5 = 11 8 + 3 = 11

Answer: 4 + 2 + 4 = 10 Beth sees = 10 birds

Final Words:

Learn the basics like addition, subtraction from the primary level itself. Refer to Go Math Grade 1 Answer Key Chapter 3 Addition Strategies to score good marks in the exams. Download the free pdf of HMH Go Math 1st Grade Answer Key Chapter 3 from this page.

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Eureka Math Grade 3 Module 1 Lesson 9 Homework Answer Key. Question 1. Dan organizes his stickers into 3 rows of four. Irene adds 2 more rows of stickers. Complete the equations to describe the total number of stickers in the array. ... Given 9 × 3 we wrote 9 × 3 as (10 - 1) × 3 = (10 × 3) - (1 × 3) = 30 - 3 = 27 or 9 x 2 = 27 ...

Eureka Math Grade 3 Module 3 Lesson 9 Exit Ticket Answer Key. Simplify to find the answer to 18 × 3. Show your work, and explain your strategy. Answer: 18 x 3 = 54. Explanation: In the above-given question, given that, 18 x 3 = 54. 3 x 18 = 54. Eureka Math Grade 3 Module 3 Lesson 9 Worksheet Answer Key. Question 1. Use the array to complete ...

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Lesson 3 Homework 3• 1 Lesson 3: Interpret the meaning of factors—the size of the group or the number of groups. Name Date Solve Problems 1-4 using the pictures provided for each problem. 1. There are 5 pineapples in each group. How many pineapples are there in 5 groups? a.

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9.1 Systems of Linear Equations: Two Variables; 9.2 Systems of Linear Equations: Three Variables; 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables; 9.4 Partial Fractions; 9.5 Matrices and Matrix Operations; 9.6 Solving Systems with Gaussian Elimination; 9.7 Solving Systems with Inverses; 9.8 Solving Systems with Cramer's Rule

@AE##$, @EH##$ 9. plane ABC, plane EFG, plane ADH, plane BCG 10. plane EFG 11. alternate interior 12. corresponding 13. alternate interior 14. alternate exterior 15. consecutive interior 16. alternate exterior

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Our resource for Algebra 1: Homework Practice Workbook includes answers to chapter exercises, as well as detailed information to walk you through the process step by step. With Expert Solutions for thousands of practice problems, you can take the guesswork out of studying and move forward with confidence. Find step-by-step solutions and answers ...

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form of a quadratic function with a = -4.9, b = 0, and c = 400. d) This is not quadratic. The function contains 3 factors with the variable (w). This makes the function cubic, not quadratic. . 2. a) b) Vertex (-1, 4) 1, 9) Axis of symmetry x = -1 x-intercepts -3 and 1 4 and 2 y-intercept -3 8 Direction upward downward Max/min min y ...

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CC3 Selected Answers - Free download as PDF File (.pdf), Text File (.txt) or read online for free. This document contains selected answers for lessons in a math course. It provides short answers to math problems across several lessons on topics like operations with integers, fractions, expressions, equations, ratios and proportions. The answers range from single values to brief explanations ...

Introduction to Systems of Equations and Inequalities; 7.1 Systems of Linear Equations: Two Variables; 7.2 Systems of Linear Equations: Three Variables; 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables; 7.4 Partial Fractions; 7.5 Matrices and Matrix Operations; 7.6 Solving Systems with Gaussian Elimination; 7.7 Solving Systems with Inverses; 7.8 Solving Systems with Cramer's Rule

Find step-by-step solutions and answers to College Algebra - 9780321729682, as well as thousands of textbooks so you can move forward with confidence. ... Section 1.9: Distance and Midpoint Formulas; Circles. Page 231: Review Exercises. Page 235: Chapter 1 Test. Exercise 1. Exercise 2. Exercise 3. Exercise 4. Exercise 5. Exercise 6. Exercise 7 ...

Eureka Math Grade 3 Module 5 Answer Key extends and deepens students' knowledge of understanding fractions as equal partitions of a whole. Knowledge becomes more formal if you work with area models and the number ... Eureka Math Grade 3 Module 5 Lesson 9 Answer Key; Engage NY Math 3rd Grade Module 5 Topic C Comparing Unit Fractions and ...

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Draw a picture of the birds. Answer: 4 + 2 + 4 = 10. Beth sees = 10 birds. Final Words: Learn the basics like addition, subtraction from the primary level itself. Refer to Go Math Grade 1 Answer Key Chapter 3 Addition Strategies to score good marks in the exams. Download the free pdf of HMH Go Math 1st Grade Answer Key Chapter 3 from this page.

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