high school math solution reviewer’s guide · high school math solution reviewer’s guide ......

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High School Math Solution Reviewer’s GuideWe are honored to provide you with Carnegie Learning’s High School Math Solution.

Other providers think of blended learning as a simple mix of print resources for instruction and software for practice. Carnegie Learning delivers a deeper level of blended learning, where students are given opportunities to learn together (using printed resources) and learn individually (using MATHia, previously known as Cognitive Tutor, software or printed Skills Practice materials).

We invite you to follow this Step-by-Step Reviewer’s Guide to learn more about us and our program as you embark upon a new math journey. Let’s begin your review.

STEP 1

Check out your box.

This box contains the essentials:

¨ This Reviewer’s Guide!

¨ Teacher Implementation Guides, Volume 1 & Volume 2

¨ (2) Student Consumable Textbooks with MATHia access

¨ (1) Student Skills Practice Book (if students do not have regular access to technology)

¨ Access to Carnegie Learning’s Virtual Sample Kit

STEP 2

Find out more about our learning philosophy and spend time with a teacher and students through www.carnegielearning.com/loudoun

Watch the following three videos:

Video 1: “Carnegie Learning Overview”

Video 2: “The Carnegie Learning Way: A Story About Math Transformation.”

Video 3: “Transforming the Math Classroom”

http://www.carnegielearning.com/loudoun

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STEP 3

Grab the Teacher Implementation Guide. Read the front matter and examine teacher notes.

FRONT MATTER• FM 12–FM 19 will take you through the powerful research behind the Carnegie Learning High School Math

Solution and talk about the Three Big Ideas in which students will engage in during instruction.

• FM 20–FM 21 will show you the progression of content through a module overview of the course. To see the Virginia SOL correlations, go to www.carnegielearning.com/loudoun.

• FM 22–FM 23 discusses how our resource helps you align your teaching to student learning. Each lesson was designed to engage and cultivate student knowledge and understanding of mathematics through peer interaction and discourse.

• FM 42–FM 50 illustrates the type of problems students will encounter while working through lessons in the resource. The problems include Worked Examples, Thumbs Up/Thumbs Down, and Who’s Correct.

TEACHER NOTESThe Teacher’s Implementation Guide includes suggestions for grouping students, targeted questions to guide learning, and a lesson map to help facilitation. The guide also includes common student misconceptions and ELL support.

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88 Chapter 2 Graphs, Equations, and Inequalities

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guiding Questions for Discuss Phase, Question 1• Is the airliner ascending or descending in the problem situation?

•What is the height of the airliner before the descent?

• Is the airliner descending at a constant rate?

•What is the constant rate at which the airliner descends?

Problem 1A scenario is given with a negative slope that is similar to the scenario in the previous lesson that had a positive slope. Students will identify the independent and dependent quantities including their units of measure, and the rate of change. They then complete a table of values and calculate several inputs and outputs. The values in the table are used to write an expression to represent the problem situation, and the expression is used to write a function. Several expressions are then given and students identify the input value, the output value, the x-intercept, the y-intercept and the rate of change in terms of their units of measure, contextual meaning, and mathematical meaning. Students will sketch a simple graph of the situation labeling the axes with the independent and dependent quantities and their units of measure. Students calculate several outputs both algebraically and graphically and compare the results. Exact answers are distinguished from approximate answers with respect to the multiple representations.

grouping•Ask a student to read the

scenario. Discuss and and complete Question 1 as a class.

•Have students complete Questions 2 through 5 with a partner. Then share the responses as a class.

PROBLEM 1 As We Make Our Final Descent

At 36,000 feet, the crew aboard the 747 airplane begins making preparations to land. The plane descends at a rate of 1500 feet per minute until it lands.

1. Compare this problem situation to the problem situation in Lesson 2.1, The Plane! How are the situations the same? How are they different?

The independent quantity and dependent quantity are the same with time being the independent quantity and height being the dependent quantity in both problem situations.

The rate of change is different for each situation. In this situation, the plane is now descending as opposed to taking off.

2. Complete the table to represent this problem situation.

Independent Quantity Dependent Quantity

Quantity Time Height

Units minutes feet

0 36,000

2 33,000

4 30,000

6 27,000

12 18,000

20 6000

Expression t 21500t 1 36,000

3. Write a function, g(t), to represent this problem situation.

g(t) 5 21500t 1 36,000

Think about the pattern you

used to calculate each dependent quantity

value.

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grouping•Ask a student to read the

information aloud. Discuss as a class.

•Have students complete Questions 6 through 8 with a partner. Then share the responses as a class.

guiding Questions for Share Phase, Questions 6•Using the table of values,

between what two times is the airliner 14,000 feet above the ground?

•Do the two lines on the graph intersect at a point closer to 12 minutes or closer to 20 minutes?

•Do the two lines on the graph intersect at a point closer to 12 minutes or closer to 16 minutes?

•Does the intersection of the two lines on the graph look like it is halfway between 12 minutes and 16 minutes?

•What equation was used to solve for t?

•How many steps were used to solve the equation?

•What was the first step in the solution of the equation?

•Which method gave you an exact answer, using the table, using the graph, or solving the equation?

You have just represented the As We Make Our Final Descent scenario in different ways:

• numerically, by completing a table,

• algebraically, by writing a function, and

• graphically, by plotting points.

Let’s consider how to use each of these representations to answer questions about the problem situation.

6. Determine how long will it take the plane to descend to 14,000 feet.

a. Use the table to determine how long it will take the plane to descend to 14,000 feet.

The plane will descend to 14,000 feet between 12 and 20 minutes. The plane will descend to 14,000 feet in approximately 14 minutes.

b. Graph and label y 5 14,000 on the coordinate plane. Then determine the intersection point. Explain what the intersection point means in terms of this problem situation.

See graph.

The graph of y 5 14,000 crosses the graph of g(t) 5 21500t 1 36,000 about halfway between t 5 12 and t 5 16. I estimate that it will take the plane about 14 minutes to reach 14,000 feet because that is halfway between 12 and 16.

c. Substitute 14,000 for g(t) and solve the equation for t. Interpret your solution in terms of this problem situation.

g(t) 5 21500t 1 36,000

14,000 5 21500t 1 36,000

14,000 2 36,000 5 21500t 1 36,000 2 36,000

222,000

________ 21500

5 21500t _______ 21500

14.667 5 t

(14.667, 14,000)

At about 14.667 minutes, the plane is at 14,000 feet.

d. Compare and contrast your solutions using the table, graph, and the function. What do you notice? Explain your reasoning.

Using the table, I can only estimate the time. When plotting the points on my graph, I was able to approximate the intersection of the graphs of both functions. When I used the function, I was able to determine an exact time when the plane would descend to 14,000 feet.

http://www.carnegielearning.com/loudoun

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STEP 4

Grab one of the Student Consumable Texts and log into MATHia through our Virtual Sample Kit. Work the problems. Have fun!

Carnegie Learning’s High School Math Solution uniquely integrates our textbook resources with MATHia (previously known as Cognitive Tutor), our intelligent math learning platform.

Students will receive instruction and work through interactive lessons within the textbook. Each lesson in the series delivers three big ideas through our Engage, Develop, and Demonstrate instructional model. It is through this model that students will be motivated, develop their conceptual understanding of mathematics, and demonstrate their learning.

• Engage: We activate student thinking by tapping into prior knowledge and real-world experiences. Our lessons provide an introduction that generates curiosity and plants seeds for deeper learning.

• Develop: We build a deep understanding of mathematics — through a variety of activities, real-world problems, sorting activities, worked examples, and peer analysis — in an environment where collaboration, conversations and questioning are routine practices.

• Demonstrate: We provide opportunities for students to reflect on and evaluate what was learned. Ongoing formative assessment underlies the entire experience, driving real-time adjustments, next steps, insights, and measurements.

And MATHia punctuates student learning by seamlessly aligning with textbook topics and providing problems for further exploration. This component of the program mimics a human tutor and carefully assesses how individual students approach problem solving. MATHia adjusts and provides the necessary help each student needs in order to achieve on-grade level skill mastery.

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Problem 1 as We Make Our Final Descent

At 36,000 feet, the crew aboard the 747 airplane begins making preparations to land . The plane descends at a rate of 1500 feet per minute until it lands .

1. Compare this problem situation to the problem situation in Lesson 2 .1, The Plane! How are the situations the same? How are they different?

2. Complete the table to represent this problem situation .

Independent Quantity Dependent Quantity

Quantity

Units

0

2

4

6

18,000

6000

Expression t

3. Write a function, g(t), to represent this problem situation .

Think about the pattern you

used to calculate each dependent quantity

value.

Evaluating Linear Functions

© 2017 Carnegie LearningProblem: gof001 Client Version: 2.4.27 Server Version: 2.4.27

Go to ProblemUnit Overview Step-by-Step Sample Problem

Carnegie Learning

System Help Glossary Maria Hernandez

https://www.carnegielearning.com/…

Step-by-Step Example

Given:

h(z) = –8.1z – 6

Answer each question using the worksheet.

1. If h(z) = –8.1(7) – 6, what is z?

2. Evaluate h(z) for z = 3.

3. Evaluate h(z) for z = 8.

h(z)z

Question 1

Question 2

Question 3

–8.1(7) – 6 –62.7

3

8

Substituted h(z) Result

Workspace Introduction

Get started with Evaluating Linear Functions by studying the sample problem, then do the Step-by-Step example.

Begin Step-by-Step ExampleSkip and go to problem

Example:

To evaluate a function in one variable means to determine the value of the function by replacing the variable with a given value, and simplifying the result. For instance, suppose that you want to evaluate the function f(x) for the value x = 1. You could represent this as f(1). Substitute the value of 1 for x into the function and simplify.

a. If f(x) = 4x – 7, what is f(5)?

b. Evaluate h(x) = –0.5x + 3 for x = –2.

a. From the expression f(5), you know that x = 5. So, substitue 5 for x in the

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Through MATHia, you will have access to reports that predict the trajectory of student learning, performance on standardized assessments, the amount of time spent on the platform, as well as content mastery. MATHia’s leadership reports provide administrators and instructional leaders with interactive data so they can view student performance across the district, buildings, and even at the class level.

Note: If your students have limited access to technology, please see the Student Skills Practice Book.

To experience MATHia from a student’s perspective, go to our Virtual Sample Kit, where all of Carnegie Learning’s print resources and MATHia demos are found: www.carnegielearning.com/vsk-login

For your username and password, please refer to your printed Reviewer’s Guide or contact:[email protected]

You can explore the problems in MATHia by selecting High School and clicking on Try MATHia Software.

Leadership Report

Carnegie Learning


REPORTSHello, Jane Doe

©2017 Carnegie Learning Inc

EXPORTPRINTAVERAGE USAGE PER STUDENTMinutes spent in MATHia per student per week

1 2 Next3 4 i

Acre Middle SchoolByte Middle SchoolCarat Middle SchoolCelsius Middle SchoolFahrenheit Middle SchoolHertz Middle SchoolJoule Middle SchoolKelvin Middle SchoolOunce Middle SchoolPascal Middle SchoolTonne Middle SchoolTorr Middle School

Average (12 Schools)08/21/1

6

09/18/1

6

10/16/1

6

11/13/1

6

12/11/1

6

01/08/1

7

02/26/1

7

03/05/1

7

300

350

250

200

150Min

utes

Weeks

100

50

0

On Track

Average Usage 134

01/15/17

Kelvin Middle School

12 Schools selected 87 Classes selected 2 Class Categories selected 2 Class Profiles selected Period: 08/21/16 to 03/05/17

AVERAGE USAGE

90

On Track 90 minutesApproaching On Track: 60-89 minutes

minutesper week 23%

AVERAGE SYLLABUS COMPLETION

ACTIVE STUDENTS

44% 440 out of 1000

On Track 90%Approaching On Track: 70-89%

59%

APLSE PERFORMANCEOn Track 70-100%Approaching On Track 50%-69%

https://www.carnegielearning.com/vsk-login/overview

Carnegie Learning

Carnegie Learning


REPORTSHello, Jane Doe

©2017 Carnegie Learning Inc

APLSE Report Standards ReportSession Report Student Detail Report

Student Detail Report: All Students Export Print

Student Name % SyllabusComplete

Average PerformanceScore (out of 100)

Current ModuleTime on Task(in minutes)

ModulesCompleted

UnitsCompleted

Adams, Janet 76 100 One-Step Equations … 637 4 10Baker, Lindsey 44 88 Numeric and Algebraic … 90 3 6

Bartlett, Michael 53 77 Numeric and Algebraic … 707 3 8

Franklin, Sam 45 78 Numeric and Algebraic … 422 3 6

Harrison, Sara 49 61 Numeric and Algebraic … 566 3 6Jacobs, Beth 35 92 Ratios, Rates, and Percents … 551 2 5

Ming, Jo 32 90 Ratios, Rates, and Percents … 300 2 5Peterson, Phillip 0 0 Not Started 0 0

Ruiz, Marco 36 87 Ratios, Rates, and Percents … 101 2 5

7Trazzoli, Kim 49 100 Numeric and Algebraic … 96 3

0

Tresport, Jacob 4 100 One-Step Equations and … 8 1 1Williams,Shania 5 100 Fraction and Decimal … 27 0

Zeller, Trevor 1 99 Pre-Launch Protocol 1 0

0

0

WorkspacesCompleted

56

37

39

33

36

26

24

27

36

0

3

4

1

Student Usage Totals

http://www.carnegielearning.com/vsk-login

mailto:MHaney%40carnegielearning.com?subject=Loudoun%20Username/Password%20Request

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Once on the MATHia Software page, click on the button that says Launch High School Student Experience. This will take you into the student launch page with a changeable student avatar.

If you’d like to experience various problems, click into the various workspaces located under the Launch the High School Experience button.

You can then explore sample High School course modules from Algebra I, Geometry, and Algebra II. These modules will take you into unit overviews and problems for you to try.

https://www.carnegielearning.com/vsk-login…

Carnegie Learning

GlossarySystem Help John Doe

© 2017 Carnegie LearningClient Version: 2.4.31 Server Version: 2.4.31

Carnegie Learning

Getting Started

1 Unit1 Workspace

1 of 1 Workspaces Complete

Review

1 Relating Quantities and Reasoning with Equations4 Units17 Workspaces


Let’s Go!

Linear and Exponential Relationships7 Units24 Workspaces


3 Descriptive Statistics

2 Units7 Workspaces


4

2

TOTAL 00:00

31

THIS SESSION 00:00

31

Time:Problems:

Workspaces:

1%Syllabus Completion


https://www.carnegielearning.com/vsk-login…

Carnegie Learning

Evaluating Linear Functions

© 2017 Carnegie LearningProblem: gof001 Client Version: 2.4.27 Server Version: 2.4.27

I’m DoneUnit Overview Step-by-Step Sample Problem Hints Solve It

Carnegie Learning

System Help Glossary Maria Hernandez


Given:

r(x) = –4x – 3

Answer each question using the worksheet.

1. Evaluate r(x) for x = 3.

2. Evaluate r(x) for x = –9.

3. What is the result of evaluating r(–10)?

r(x)x

Question 1

Question 2

Question 3 r(–10)

3

–9

Substituted r(x) Result

Progress

3

SkillsEnter value of variable.

Enter function with value of variable.

Evaluate function.

Progress to Mastery

GlossarySystem HelpHome John Doe

Carnegie Learning

Identifying Quantities

Go to WorkspaceCurrent Workspace 4 of 6: Evaluating Linear FunctionsBack to Module

HSMS: Function Overview > Unit Overview

50%

WorkspacesFunction Overview

WHY THIS MATTERS:

WHAT YOU’LL LEARN:

Progress

1


• Recognize appropriate graphs for real-world situations.• Compare a variety of function families.• Identify the intercepts of a function.• Identify minimum or maximum of a function.• Identify domain and range of a function.• Use function notation to represent a specific output value.• Using function notation, evaluate a function for a specific input.

WHAT THIS CONNECTS TO:You have learned that functions are relationships between quantities where exactly one input maps to each output. You will use this knowledge to explore different families of functions that share characteristics.

Key Termsindependent quantitydependent quantityslopelinear functionexponential functionquadratic functionunit rate of change

Introduction to Function Families2Understanding Linear Functions3Evaluating Linear Functions4

Exploring Graphs of Linear Functions5Identifying Key Characteristics of Graphs of Functions6

Let’s Go!

Review

Review

Review


Function Overview

© 2017 Carnegie LearningClient Version: 2.0.448 Server Version: 2.0.448

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Chapter 2 Assessments 815

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Pre-Test

Name Date

1. An elevator in a high-rise building moves upward at a constant rate. The table shows the height of the elevator above the ground floor after various times.

Time Height

Units Seconds Feet

0 0

1 12

2 24

3

4.5

5

Expression t

a. What are the dependent and independent quantities in this problem situation? Explain your reasoning.

b. Determine the unit rate of change for the problem situation.

c. Complete the table. Write an expression that represents the height for an arbitrary time t seconds in the last row.

d. Use function notation to determine the height of the elevator at a time of 14 seconds.

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2 1. Jasmin is practicing her juggling act. She tosses one ball straight up in the air to a maximum height

of 16 feet. She knows she can then toss the second ball any time the first ball is at a height greater than 10 feet. Which graph correctly represents this problem situation using an oval to represent the time when she can toss the second ball?

a.

0

16

18

12

8

4

14

10

6

2

4 8 12 16 182 6 10 14x

y

Time (seconds)

Hei

ght

(feet

)

b.

0

16

18

12

8

4

14

10

6

2

4 8 12 16 182 6 10 14x

y

Time (seconds)

Hei

ght

(feet

)

c.

0

16

18

12

8

4

14

10

6

2

4 8 12 16 182 6 10 14x

y

Time (seconds)

Hei

ght

(feet

)

d.

0

16

18

12

8

4

14

10

6

2

4 8 12 16 182 6 10 14x

y

Time (seconds)

Hei

ght

(feet

)

Standardized Test Practice

Name Date

STEP 5

And while you’re in our Virtual Sample Kit, you can look at all of the print resources and assessments, too.

Again, your review team has immediate access to all print resources (student consumable textbook, Skills Practice Book, and Student Assignments) online through our Virtual Sample Kit. The Student Assignments book allows you to extend lesson activities to students who need more of a challenge. The Skills Practice books builds students computational fluency and is used in the event technology is not available to students on a regular basis. Assessments include Pre- and Post-Tests, Mid-Chapter Tests, End-of-Chapter Tests, and Standardized Test Practice.

As you conduct your review, please contact us with any questions. We’re here to help!

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2 1. Elena works a part-time job after school to earn money for a summer vacation. She is paid a constant

rate for each hour she works. The table shows the amounts of money that Elena earned for various amounts of time that she worked.

Time Worked Amount Earned

Units Hours Dollars

2.5 22.50

3 27.00

3.5 31.50

4.5

5

6

Expression t

a. What are the dependent and independent quantities in this problem situation? Explain your reasoning.

b. Determine the unit rate of change for the problem situation.

c. Complete the table. Write an expression that represents the amount of money Elena earns for an arbitrary time worked of t hours.

d. Use function notation to determine the amount of money that Elena earns for working 7.5 hours.

Post-Test

Name Date

high school math solution reviewer’s guide · high school math solution reviewer’s guide ......

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