lesson 9 homework 3.3 answer key

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Eureka Math Grade 3 Module 3 Lesson 9 Answer Key

Teachers and students can find this Eureka Book Answer Key for Grade 3 more helpful in raising students’ scores and supporting teachers to educate the students. Learning activities are the best option to educate elementary school kids and make them understand the basic mathematical concepts like addition, subtraction, multiplication, division, etc. Grade 3 elementary school students can find these fun-learning exercises for all math concepts through the Eureka Book.

Engage NY Eureka Math 3rd Grade Module 3 Lesson 9 Answer Key

Students of Grade 3 can get a strong foundation by referring to the Eureka Math Book. It was developed by highly professional mathematics educators and the solutions prepared by them are in a concise manner for easy grasping. To achieve high scores in Grade 3, students need to solve all questions and exercises included in Eureka Grade 3 Book.

Eureka Math Grade 3 Module 3 Lesson 9 Application Problems Answer Key

Solve the following pairs of problems. Circle the pairs where both problems have the same answer.

Question 1. a. 7 + (6 + 4) b. (7 + 6) + 4

Answer: a. 17. b. 17.

Question 2. a. (3 × 2) × 4 b. 3 × (2 × 4)

Answer: a. 24. b. 24.

Question 3. a. (2 × 1) × 5 b. 2 × (1 × 5) Answer: a. 10. b. 10.

Question 4. a. (4 × 2) × 2 b. 4 × (2 × 2)

Answer: a. 16. b. 16.

Question 5. a. (3 + 2) × 5 b. 3 + (2 × 5)

Answer: a. 25. b. 13.

Explanation: In the above-given question, given that, a. 5 x (3 + 2) 3 + 2 = 5. 5 x 5 = 25. b. (2 x 5) + 3. 2 x 5 = 10. 10 + 3 = 13.

Question 6. a. (8 ÷ 2) × 2 b. 8 ÷ (2 × 2)

Answer: a. 8. b. 2.

Explanation: In the above-given question, given that, a. 2 x (8 / 2) 8 / 2 = 4. 2 x 4 = 8. b. (2 x 2) / 8. 2 x 2 = 4. 8 / 4 = 2.

Question 7. a. (9 – 5) + 3 b. 9 – (5 + 3)

Answer: a. 7. b. 1.

Explanation: In the above-given question, given that, a. 3 + (9 – 5) 9 – 5 = 4. 4 + 3 = 7. b. (5 + 3) – 9. 5 + 3 = 8. 9 – 8 = 1.

Question 8. a. (8 × 5) – 4 b. 8 × (5 – 4)

Answer: a. 36. b. 8.

Explanation: In the above-given question, given that, a. 4 – (8 x 5) 8 x 5 = 40. 40 – 4 = 36. b. (5 – 4) x 8. 5 – 4 = 1. 1 x 8 = 8.

Eureka Math Grade 3 Module 3 Lesson 9 Problem Set Answer Key

Answer: The number of rows = 3. the number of columns = 12.

Explanation: In the above-given question, given that, The number of rows = 3. the number of columns = 12. 3 x 12 = 36.

Explanation: In the above-given question, given that, The number of rows = 3. the number of columns = 12. (3 x 3) x 4. 9 x 4 = 36.

Answer: The number of rows = 3. the number of columns = 14.

Explanation: In the above-given question, given that, The number of rows = 3. the number of columns = 14. 3 x 14 = 42.

Explanation: In the above-given question, given that, The number of rows = 3. the number of columns = 14. (2 x 3) x 7. 6 x 7 = 42.

Answer: 3 x 16 = 3 x (2 x 8) (3 x 2) x 8. 6 x 8 = 48.

Explanation: In the above-given question, given that, the equation 3 x 16. 3 x ( 2 x 8 ) ( 3 x 2 ) x 8. 6 x 8 = 48.

Answer: 2 x 14 = 2 x (2 x 7) (2 x 2) x 7. 4 x 7 = 28.

Answer: 3 x 12 = 3 x (3 x 4) (3 x 3) x 4. 9 x 4 = 36.

Answer: 3 x 14 = 3 x (2 x 7) (3 x 2) x 7. 6 x 7 = 42.

Answer: 15 x 3 = 5 x 3 x 3 (5 x 3) x 3. 15 x 3 = 45.

Answer: 2 x 15 = 2 x (5 x 3) (5 x 3) x 2. 15 x 2 = 30.

Question 3. Charlotte finds the answer to 16 × 2 by thinking about 8 × 4. Explain her strategy.

Answer: 16 x 2 = 32. 8 x 4 = 32.

Explanation: In the above-given question, given that, 16 x 2 = 8 x 4. 16 x 2 = 32. 8 x 4 = 32.

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

Answer: The number of rows = 3. the number of columns = 16.

Explanation: In the above-given question, given that, The number of rows = 3. the number of columns = 16. 3 x 16 = 48.

Explanation: In the above-given question, given that, The number of rows = 3. the number of columns = 16. (3 x 2) x 8. 6 x 8 = 48.

Answer: The number of rows = 4. the number of columns = 18.

Explanation: In the above-given question, given that, The number of rows = 4. the number of columns = 18. 4 x 18 = 72.

Explanation: In the above-given question, given that, The number of rows = 4. the number of columns = 18. (4 x 2) x 9. 8 x 9 = 72.

Answer: 14 x 3 = 2 x 7 x 3 (3 x 2) x 7. 6 x 7 = 42.

Answer: 12 x 3 = 4 x 3 x 3 (3 x 3) x 4. 9 x 4 = 36.

Answer: 20 x 2 = 40. 30 x 2 = 60. 35 x 2 = 70. 40 x 2 = 80.

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How to Use a Math Medic Answer Key

Written by Luke Wilcox published 3 years ago

Answer key might be the wrong term here. Sure, the Math Medic answer keys do provide the correct answers to the questions for a lesson, but they have been carefully designed to do much more than this. They are meant to be the official guide to teaching the lesson, providing specific instructions for what to do and say to make a successful learning experience for your students.

Before we look at the details of the answer key, let's make sure we understand the instructional model first.

Experience First, Formalize Later (EFFL)

A typical Math Medic lesson always has the same four parts: Activity, Debrief Activity, QuickNotes, and Check Your Understanding. Here are the cliff notes:

Activity: Students are in groups of 2 - 4 working collaboratively through the questions in the Activity. The teacher is checking in with groups and using questions, prompts, and cues to get students to refine their communication and understanding. As groups finish the activity, the teacher asks students to go to the whiteboard to write up their answers to the questions.

Debrief Activity: In the whole group setting, the teacher leads a discussion about the student responses to the questions in the activity, often asking students to explain their thinking and reasoning about their answers. The teacher then formalizes the learning by highlighting key concepts and introducing new vocabulary, notation, and formulas.

QuickNotes: The teacher uses direct instruction to summarize the learning from the activity in the QuickNotes box - making direct connections to the learning targets for the lesson.

Check Your Understanding: Students are then asked to apply their learning from the lesson to a new context in the Check Your Understanding (CYU) problem. This can be done individually or in small groups. The CYU is very flexible in it's use, as it can be used as an exit ticket, a homework problem, or a quick review the next day.

How Do I See EFFL in the Answer Key?

You will see EFFL in the answer key like this:

Anything written in blue is something we expect our students to produce. This might not be quite what we expect by the end of the lesson, but provides us with a starting point when we move to formalization.

Anything written in red is an idea added by the teacher - the formalization of the learning that happened during the Activity. Students are expected to add these "notes" to their Activity using a red pen or marker.

What Do Students Write Down For Notes?

By the end of the lesson, students will have written down everything you see on the Math Medic Answer Keys. The most important transition is when students finish the Activity and we move to Debrief Activity. "Students, now is the time for you to put down your pencils and get out your your red Paper Mate flair pens" We give each student a Paper Mate flair pen at the beginning of the school year and tell them they must cherish and protect it with their life. They all think we should be sponsored by Paper Mate (anyone have any leads on this?)

The lessons you see on Math Medic are all of the notes we use with our students. We do not have some secret collection of guided notes.

Do Students Have Access to Answer Keys?

Yes! Any student can create a free Math Medic account to get access to the answer keys. We often send students to the website when they are absent from a lesson or when we don't quite finish the lesson in class. We are comfortable with students having access to these answer keys because we do not think Math Medic lessons should be used as a summative assessment or be used for a grade (unless it's for completion). Our lessons are meant to be the first steps in the formative process of learning new concepts.

Math Medic Help

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 )

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

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

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 ]

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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Go Math! 3 Student Edition, Grade: 3 Publisher: Houghton Mifflin Harcourt

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Home > INT2 > Chapter 9 > Lesson 9.3.3

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 K Answer Key Chapter 3 Represent, Count, and Write Numbers 6 to 9

Go Math Grade K Chapter 3 Answer Key complements your preparation and helps you attain better grades in exams easily. Elementary School Kids will find the HMH Go Math Grade K Answer Key Chapter 3 extremely helpful as all of them are given with a clear-cut explanation. Download the Topicwise Go Math Kindergarten Answers for free of cost and ace up your preparation. Look no further and practice using the HMH Grade K Go Math Solution Key on a regular basis.

Go Math Grade K Chapter 3 Answer Key Represent, Count, and Write Numbers 6 to 9

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Represent, Count, and Write Numbers 6 to 9

• Represent, Count, and Write Numbers 6 to 9 Show What You Know – Page 116
• Represent, Count, and Write Numbers 6 to 9 Vocabulary Builder – Page 117
• Represent, Count, and Write Numbers 6 to 9 Game: Number Line Up – Page 118
• Represent, Count, and Write Numbers 6 to 9 Vocabulary Game – Page(118 A-118 B) 

Lesson: 1 Model and Count 6

• Lesson 3.1 Model and Count 6 – Page(119-124)
• Model and Count 6 Homework & Practice 3.1 – Page(123-124)

Lesson: 2 Count and Write to 6

• Lesson 3.2 Count and Write to 6 – Page(125-130)
• Count and Write to 6 Homework & Practice 3.2 – Page(129-130)

Lesson: 3 Model and Count 7

• Lesson 3.3 Model and Count 7 – Page(131-136)
• Model and Count 7 Homework & Practice 3.3- Page(135-136)

Lesson: 4 Count and Write to 7

• Lesson 3.4 Count and Write to 7 – Page(137-142)
• Count and Write to 7 Homework & Practice 3.4 – Page(141-142)

Mid-Chapter Checkpoint

• Represent, Count, and Write Numbers 6 to 9 Mid-Chapter Checkpoint – Page 140

Lesson: 5 Model and Count 8

• Lesson 3.5 Model and Count 8 – Page(143-148)
• Model and Count 8 Homework & Practice 3.5 – Page(147-148)

Lesson: 6 Count and Write to 8

• Lesson 3.6 Count and Write to 8 – Page(149-154)
• Count and Write to 8 Homework & Practice 3.6 – Page(153-154)

Lesson: 7 Model and Count 9

• Lesson 3.7 Model and Count 9 – Page(155-160)
• Model and Count 9 Homework & Practice 3.7 – Page(159-160)

Lesson: 8 Count and Write to 9

• Lesson 3.8 Count and Write to 9 – Page(161-166)
• Count and Write to 9 Homework & Practice 3.8 – Page(165-166)

Lesson: 9 Problem Solving • Numbers to 9

• Lesson 3.9 Problem Solving • Numbers to 9 – Page(167-172)
• Problem Solving • Numbers to 9 Homework & Practice 3.9 – Page(171-172)
• Represent, Count, and Write Numbers 6 to 9 Chapter 3 Review/Test – Page(173-176)

Curious George

Rides are popular at fairs. • What can you tell me about this ride?

Represent, Count, and Write Numbers 6 to 9 Show What You Know

Explore Numbers to 5

Compare Numbers to 5

Write Numbers to 5

This page checks understanding of important skills needed for success in Chapter 3.

DIRECTIONS 1. Circle the dot cards that show 3. 2. Circle the dot cards that show 5. 3. Write the number of cubes in each set. Circle the greater number. 4. Write the numbers 1 to 5 in order.

Represent, Count, and Write Numbers 6 to 9 Vocabulary Builder

DIRECTIONS Point to sets of objects as you count. Circle two sets that have the same number of objects. Tell what you know about sets that have more objects or fewer objects than other sets on this page.

Represent, Count, and Write Numbers 6 to 9 Game: Number Line Up

Number Line Up

DIRECTIONS Play with a partner. Place numeral cards as shown on the board. Shuffle the remaining cards and place them face down in a stack. Players take turns picking one card from the stack. They place the card to the right to form a number sequence without skipping any numbers. The number sequence can be forward from 0 or backward from 5. If a player picks a card that is not next in either number sequence, the card is returned to the bottom of the stack. The first player to complete a number sequence wins the game. MATERIALS 2 sets of numeral cards 0–5

Represent, Count, and Write Numbers 6 to 9 Vocabulary Game

Going Places with GOMATH! Words

DIRECTIONS Players take turns. A player chooses a secret word from the Word Box and then sets the timer. The player draws pictures to give hints about the secret word. If the other player guesses the secret word before time runs out, he or she puts a counter in the chart. The first player who has counters in all his or her boxes wins.

MATERIALS timer, drawing paper, two-color counters for each player

The Write Way

DIRECTIONS Trace the 8. Draw to show what you know about 8. Reflect Be ready to tell about your drawing.

Lesson 3.1 Model and Count 6

Essential Question How can you show and count 6 objects?

Listen and Draw

DIRECTIONS Place a counter on each ticket in the set as you count them. Move the counters to the ten frame. Draw the counters.

Share and Show

DIRECTIONS 1. Place a counter on each car in the set as you count them. Move the counters to the parking lot. Draw the counters. Say the number as you trace it.

Explanation : Number of Yellow counters are 4 Number of Red counters are 4 So, to make a sum of 6 we add two counters. That is represented in 4 different ways. 1 red + 5 yellow, 2 red + 4 yellow, 3 red + 3 yellow, 5 red + 1 yellow all these expressions on adding gives the sum 6.

DIRECTIONS 2. Trace the number 6. Use two-color counters to model the different ways to make 6. Write to show some pairs of numbers that make 6.

DIRECTIONS 3. Six people each bought a bucket of popcorn. Count the buckets of popcorn in each set. Circle all the sets that show six buckets. 4. Draw to show a set of six objects. Tell about your drawing.

HOME ACTIVITY • Ask your child to show a set of five objects. Have him or her show one more object and tell how many objects are in the set.

Model and Count 6 Homework & Practice 3.1

DIRECTIONS 1. Trace the number 6. Use two-color counters to model the different ways to make 6. Color to show the counters below. Write to show some pairs of numbers that make 6.

Lesson Check

Spiral Review

DIRECTIONS 1. Trace the number. How many more counters would you place in the ten frame to model a way to make 6? Draw the counters. 2. Count and tell how many are in each set. Write the numbers. Compare the numbers. Circle the number that is less. 3. Count and tell how many. Write the number.

Lesson 3.2 Count and Write to 6

Essential Question How can you count and write up to 6 with words and numbers?

DIRECTIONS Count and tell how many cubes. Trace the numbers. Count and tell how many hats. Trace the word.

DIRECTIONS 1. Look at the picture. Circle all the sets of six objects. Circle the group of six people.

DIRECTIONS 2. Say the number. Trace the numbers. 3–6. Count and tell how many. Write the number.

Problem Solving • Applications

DIRECTIONS 7. Marta has a number of whistles that is two less than 6. Count the whistles in each set. Circle the set that shows a number of whistles two less than 6. 8. Draw a set of objects that has a number of objects one greater than 5. Tell about your drawing. Write how many objects.

Count and Write to 6 Homework & Practice 3.2

DIRECTIONS 1. Say the number. Trace the numbers. 2–5. Count and tell how many. Write the number.

DIRECTIONS 1. How many school buses are there? Write the number. 2. Count and tell how many are in each set. Write the numbers. Compare the numbers. Circle the number that is greater. 3. How many counters would you place in the five frame to show the number? Draw the counters.

Lesson 3.3 Model and Count 7

Essential Question How can you show and count 7 objects?

DIRECTIONS Model 6 objects. Show one more object. How many are there now? Tell a friend how you know. Draw the objects.

DIRECTIONS 1. Place counters as shown. Count and tell how many counters. Trace the number. 2. How many more than 5 is 7? Write the number. 3. Place counters in the ten frame to model seven. Tell a friend what you know about the number 7.

DIRECTIONS 4. Trace the number 7. Use two-color counters to model the different ways to make 7. Write to show some pairs of numbers that make 7.

Explanation: To show the number 7 there are 7 colored triangles drawn. 7 triangles are drawn to represent  the number 7. Number 7 is represents beside the triangles in the above figure.

DIRECTIONS 5. A carousel needs seven horses. Count the horses in each set. Which sets show seven horses? Circle those sets. 6. Draw to show what you know about the number 7. Tell a friend about your drawing.

HOME ACTIVITY • Ask your child to show a set of six objects. Have him or her show one more object and tell how many objects are in the set.

Model and Count 7 Homework & Practice 3.3

DIRECTIONS 1. Trace the number 7. Use two-color counters to model the different ways to make 7. Color to show the counters below. Write to show some pairs of numbers that make 7.

DIRECTIONS 1. Trace the number. How many more counters would you place in the ten frame to model a way to make 7? Draw the counters. 2. Count and tell how many are in each set. Write the numbers. Compare the numbers. Circle the number that is less. 3. Count and tell how many. Write the number.

Lesson 3.4 Count and Write to 7

Essential Question How can you count and write up to 7 with words and numbers?

DIRECTIONS 1. Look at the picture. Circle all the sets of seven objects.

HOME ACTIVITY • Show your child seven objects. Have him or her point to each object as he or she counts it. Then have him or her write the number on paper to show how many objects.

Count and Write to 7 Homework & Practice 3.4

DIRECTIONS 1. Count and tell how many erasers. Write the number. 2. How many counters would you place in the five frame to show the number? Draw the counters. 3. Count and tell how many cubes. Write the number.

Represent, Count, and Write Numbers 6 to 9 Mid-Chapter Checkpoint

Concepts and Skills

DIRECTIONS 1. Use counters to model the number 7. Draw the counters. Write the number. 2–3. Count and tell how many. Write the number. 4. Circle all the sets of 7 whistles.

Lesson 3.5 Model and Count 8

Essential Question How can you show and count 8 objects?

DIRECTIONS Model 7 objects. Show one more object. How many are there now? Tell a friend how you know. Draw the objects.

DIRECTIONS 1. Place counters as shown. Count and tell how many counters. Trace the number. 2. How many more than 5 is 8? Write the number. 3. Place counters in the ten frame to model eight. Tell a friend what you know about the number 8.

DIRECTIONS 4. Trace the number 8. Use two-color counters to model the different ways to make 8. Write to show some pairs of numbers that make 8.

DIRECTIONS 5. Dave sorted sets of balls by color. Count the balls in each set. Which sets show eight balls? Circle those sets. 6. Draw to show what you know about the number 8. Tell a friend about your drawing.

HOME ACTIVITY • Ask your child to show a set of seven objects. Have him or her show one more object and tell how many.

Model and Count 8 Homework & Practice 3.5

DIRECTIONS 1. Trace the number 8. Use two-color counters to model the different ways to make 8. Color to show the counters below. Write to show some pairs of numbers that make 8.

DIRECTIONS 1. Trace the number. How many more counters would you place in the ten frame to model a way to make 8? Draw the counters. 2. Count and tell how many are in each set. Write the numbers. Compare the numbers. Circle the number that is greater. 3. Count and tell how many. Write the number.

Lesson 3.6 Count and Write to 8

Essential Question How can you count and write up to 8 with words and numbers?

DIRECTIONS Count and tell how many cubes. Trace the numbers. Count and tell how many balls. Trace the word.

DIRECTIONS 1. Look at the picture. Circle all the sets of eight objects.

DIRECTIONS 7. Ed has a number of toy frogs two greater than 6. Count the frogs in each set. Circle the set of frogs that belong to Ed. 8. Robbie won ten prizes at the fair. Marissa won a number of prizes two less than Robbie. Draw to show Marissa’s prizes. Write how many.

HOME ACTIVITY • Show eight objects. Have your child point to each object as he or she counts it. Then have him or her write the number on paper to show how many objects.

Count and Write to 8 Homework & Practice 3.6

DIRECTIONS 1. Count and tell how many bees. Write the number. 2. Count and tell how many are in each set. Write the numbers. Compare the numbers. Circle the number that is greater. 3. Count and tell how many beetles. Write the number.

Lesson 3.7 Model and Count 9

Essential Question How can you show and count 9 objects?

DIRECTIONS Model 8 objects. Show one more object. How many are there now? Tell a friend how you know. Draw the objects.

DIRECTIONS 1. Place counters as shown. Count and tell how many counters. Trace the number. 2. How many more than 5 is 9? Write the number. 3. Place counters in the ten frame to model nine. Tell a friend what you know about the number 9.

DIRECTIONS 4. Trace the number 9. Use two-color counters to model the different ways to make 9. Write to show some pairs of numbers that make 9.

DIRECTIONS 5. Mr. Lopez is making displays using sets of nine flags. Count the flags in each set. Which sets show nine flags? Circle those sets. 6. Draw to show what you know about the number 9. Tell a friend about your drawing.

HOME ACTIVITY • Ask your child to show a set of eight objects. Have him or her show one more object and tell how many.

Model and Count 9 Homework & Practice 3.7

Explanation: The above image represent the model 9. Above image represents the different ways to make 9 using two-color counters. Pair of numbers are written that make 9.

DIRECTIONS 1. Trace the number 9. Use two-color counters to model the different ways to make 9. Color to show the counters below. Write to show some pairs of numbers that make 9.

DIRECTIONS 1. Trace the number 9. How many more counters would you place in the ten frame to model a way to make 9? Draw the counters. 2. Count and tell how many are in each set. Write the numbers. Compare the numbers. Circle the number that is greater. 3. Count and tell how many. Write the number.

Lesson 3.8 Count and Write to 9

Essential Question How can you count and write up to 9 with words and numbers?

DIRECTIONS Count and tell how many cubes. Trace the numbers. Count and tell how many ducks. Trace the word.

DIRECTIONS 1. Look at the picture. Circle all the sets of nine objects.

Explanation: Number of admit card = 9 4 cards each in first and middle rows. and 1 card in the last row. 4 + 4 + 1 = 9.

Explanation: Number of admit card = 9 4 cards each in first and last columns. and 1 card in the middle column. 4 + 1 + 4 = 9

DIRECTIONS 2. Say the number. Trace the numbers. 3–6. Count and tell how many. Write the number

DIRECTIONS 7. Eva wants to find the set that has a number of bears one less than 10. Circle that set. 8. Draw a set that has a number of objects two greater than 7. Write how many.

HOME ACTIVITY • Ask your child to find something in your home that has the number 9 on it, such as a clock or a phone.

Count and Write to 9 Homework & Practice 3.8

DIRECTIONS 1. Count and tell how many squirrels. Write the number. 2. How many birds are in the cage? Write the number. 3. How many counters are there? Write the number.

Lesson 3.9 Problem Solving • Numbers to 9

Essential Question How can you solve problems using the strategy draw a picture?

Unlock the Problem

DIRECTIONS There are seven flags on the red tent. Trace the flags. The blue tent has a number of flags one greater than the red tent. How many flags are on the blue tent? Draw the flags. Tell a friend about your drawing.

Try Another Problem

DIRECTIONS 1. Bianca buys five hats. Leigh buys a number of hats two greater than 5. Draw the hats. Write the numbers. 2. Donna wins nine tokens. Jackie wins a number of tokens two less than 9. Draw the tokens. Write the numbers.

DIRECTIONS 3. Gary has eight tickets. Four of the tickets are red. The rest are blue. How many are blue? Draw the tickets. Write the number beside each set of tickets. 4. Ann has seven balloons. Molly has a set of balloons less than seven. How many balloons does Molly have? Draw the balloons. Write the number beside each set of balloons.

On Your Own

DIRECTIONS 5. There are six seats on a teacup ride. The number of seats on a train ride is two less than 8. How many seats are on the train ride? Draw the seats. Write the number. 6. Pick two numbers between 0 and 9. Draw to show what you know about those numbers.

HOME ACTIVITY • Have your child say two different numbers from 0–9 and tell what he or she knows about them.

Problem Solving • Numbers to 9 Homework & Practice 3.9

DIRECTIONS 1. Sally has six flowers. Three of the flowers are yellow. The rest are red. How many are red? Draw the flowers. Write the number beside each set of flowers. 2. Tim has seven acorns. Don has a number of acorns that is two less than 7. How many acorns does Don have? Draw the acorns. Write the numbers.

Explanation: In the picture above there are 5 books.

DIRECTIONS 1. Pete has 5 marbles. Jay has a number of marbles that is two more than 5. How many marbles does Jay have? Draw the marbles. Write the numbers. 2. Count and tell how many books. 3. Count and tell how many are in each set. Write the numbers. Compare the numbers. Circle the number that is greater.

Represent, Count, and Write Numbers 6 to 9 Chapter 3 Review/Test

DIRECTIONS 1. Circle all the sets that show 6. 2. Circle all the sets that show 7. 3–4. Count and tell how many. Write the number.

Explanation: Number of spinner given are 8 . The number 8 is represented.

DIRECTIONS 5. Match each set to the number that tells how many. 6–7. Count to tell how many. Write the number. 8. The ten frame shows 5 red counters and some yellow counters. Five and how many more make 9? Choose the number.

DIRECTIONS 9. Jeffrey has 8 marbles. Sarah has a number of marbles that is one greater than 8. Draw the marbles. Write the number for each set of marbles. 10. Choose all the ten frames that have a number of counters greater than 6.

Explanation: Number of turtles in the pond = 2 less than 9

= 9 – 2 = 7. 7 turtles in the pond.

Explanation: A set of objects that is 2 more than 6 = 6 + 2 = 8. 8 objects are drawn to represent the number 8.

DIRECTIONS 11. The number of turtles in a pond is 2 less than 9. Draw counters to show the turtles. Write the number. 12. Draw a set that has a number of objects that is 2 more than 6. Write the number.

Conclusion:

We believe the knowledge shared on Go Math Grade K Answer Key Chapter 3 has helped you be on the right track. If you have any queries do leave us your doubts via the comment box so that our team can guide you. Bookmark our site to avail latest info on Gradewise Go Math Answers in a matter of seconds.

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