T

HE STANDAR

D

O

F EXCE LLEN

CE

30YEARS

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Better Results.

Common CoRe State StandaRdS oveRview

Courses 1–3

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2 Saxon Math

In the introduction to the Common Core State Standards, a quote from William Schmidt and

Richard Houang (2002) is referenced, stating that standards are coherent if they are:

Incremental lessons aligned to the Common Core Domains and Clusters are distributed across the year, becoming increasingly more complex as mastery is built.

End of the school

year

M

astery of the Common Core builds throughout the year >>>

3 Ratios and Proportional Relationships

3 The Number System

3 Expressions and Equations

3 Geometry

3 Statistics and Probability (Courses 2–3)

3 Functions (Course 3)

In the Saxon Math © 2012 Common Core Edition, the Common Core Domains build throughout the year to support deep mastery:

dif

fic

ult

y

lessons

Begining of the

school year

Greater coherence through a curriculum that is articulated

over time leads to mastery of the Common Core State

Standards. Saxon Math accomplishes this through

its incremental, distributed pedagogy that builds

upon concepts throughout the year, articulating

them over time. This allows students to gain

deep understanding and long-term mastery of

the Common Core State Standards.

”

“ articulated over time as a sequence of topics and performances that are logical and reflect, where appropriate, the sequential or hierarchical nature of the disciplinary content from which the subject matter derives.

— Common CoRe STaTe STanDaRDS, CoReSTanDaRDS.oRG

mastering the

Common CoRe STaTe STanDaRDS

3

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4 Saxon Math saxonmath.com I 800.289.4490 5

Name:

Copyright © Houghton Mifflin Harcourt 15 SAXON MATH ACTIVITIES FOR COURSE 2

A plane section of a solid object is the two-dimensional intersection of a plane with the solid. To model a plane section of a geometric solid, wrap a rubber band around it so the band looks like a straight line segment when viewed from the side. The shape of the rubber band is the shape of the plane section of the solid. Think of a plane section as the flat shape created by a straight cut through the object. Placing a rubber band around an object is a way to model a cut without actually making one.

Notice that the plane section depends on how the rubber band is placed.

1. Choose one of the geometric solid models. Draw and describe the shape of different plane sections of that figure. Share your results with another student who found plane sections for the same figure.

Although you can find plane sections by making a slice through a solid at any angle, the most common plane sections are a vertical slice through its tallest point and a horizontal slice through its widest point. For each figure in the table, identify the shape of these vertical and horizontal slices.

ActivityMaterials:Set of transparent geometric solid models, rubber bands

Identify plane sections of three-dimensional figures.

How can I visualize a plane section of a three-dimensional shape?

Math Focus Essential Question

Plane Sections of Three-Dimensional Figures

USE WITH LESSON 67

ACTIVITY

8

Solid Vertical Slice Horizontal Slice

2. Cube

3. Cone sitting on its base

4. Cylinder sitting on a base

5. Sphere

1 Plane Sections of a Rectangular Pyramid

Use the rubber band method on a rectangular pyramid to show its plane sections. Label the shape of each plane section.

Example

Rectangle Triangle Trapezoid

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The © 2012 Saxon Math Teacher’s manuals include the Standards Success Companions, which provide easy access to Common Core resources.

activities reinforce key Common Core concepts.

Complete Support for the Common Core State Standards

Common Core State Standards for Mathematics: corestandards.org © 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

LaurelTech/HMH Design Pass First Pages Second Pages Digital Pages

Key: CT: Cumulative Test LXA: Lesson Extension Activity PU: Power UpET: Extension Test PA: Performance Activity WP: Written PracticeInv: Investigation PS: Problem SolvingL: Lesson PT: Performance Task

Correlation of Saxon Math Course 1 to the Common Core State Standards for Mathematics Grade 6

Mathematical Practices – These standards are covered throughout the program; the following are examples.

Make sense of problems and 1. persevere in solving them.

Power Up: PS3, PU18, PU27, PS36, PU37, PU44, PS44, PS49, PU54, PS55, PS62, PU70, PS74, PS83, PU87, PU92, PS105, PU110Lessons: L11, WP11, L13, L15, PA2, L22, WP24, L36, WP38, L50, WP50, L66, L68, WP69, L77, WP78, Inv9, WP94, Inv10, L105, WP110, L111Other: PT7

Reason abstractly and quantitatively.2. Power Up: PS13, PS36, PS44, PS49, PS54, PS70, PS92, PS107Lessons: L3, WP3, L4, WP5, L16, WP16, L18, WP25, WP36, L39, WP43, L59, L77, WP77, WP78, L95, Inv10, L103, L118, WP118

Construct viable arguments and 3. critique the reasoning of others.

Power Up: PS3, PS5, PS7, PS15, PS18, PS26, PS28, PS34, PS43, PS56, PS64, PS72, PS83, PS99, PS103, PS110, PS117Lessons: PA2, L16, WP17, WP22, PA8, L51, WP53, WP59, WP62, L64, PA14, L89, WP91, L93, WP93, L97, L109

Model with mathematics.4. Power Up: PS10, PS17, PS24, PS30, PS34, PS39, PS70, PS78, PS98, PS117Lessons: L1, Inv2, L26, WP28, WP31, PA6, WP43, WP52, PA10, Inv6, WP69, WP77, WP81, L83, WP90, L98, WP110, Inv11, L117

Use appropriate tools strategically.5. Power Up: PS10Lessons: L7, WP7, L10, WP10, WP13, L17, WP17, WP19, WP22, PA4, L27, Inv3, WP31, WP46, L48, WP57, L61, L62, WP71, Inv8, WP81, WP107, WP110

Attend to precision.6. Lessons: L7, L8, WP8, L10, WP10, WP11, WP12, WP13, WP15, L28, Inv3, L31, WP31, L32, WP36, WP45, L60, L69, WP71, L78, L81, L90, L102, Inv11, L113, L120, Inv12

Look for and make use of structure.7. Lessons: L5, WP5, L12, WP15, L21, L24, L25, L32, L34, L43, L44, L46, WP46, WP48, L52, WP52, L65, L67, L72, L84, WP85, WP90, L92, WP93, WP94, L113

Look for and express regularity in 8. repeated reasoning.

Power Up: PS1, PS4, PS11, PS12, PS16, PS80, PS94, PS102, PS109Lessons: L10, WP10, L22, WP22, WP23, L25, L29, WP31, L33, L35, L42, WP43, L46, WP48, L55, L56, L57, L63, L74, L75, L76, L85, L94, L99, L112, L115, L116, L117, WP117, WP118

Common Core State Standards Saxon Math Course 1 Italic references indicate foundational.

Ratios and Proportional Relationships 6.RP

Understand ratio concepts and use ratio reasoning to solve problems.

Understand the concept of a ratio and use ratio 1. language to describe a ratio relationship between two quantities.

Power Up: PS36, PS57, PS87, PS91, PS118Lessons: L23, WP24, WP25, WP30, WP31, WP32, WP35, WP36, WP39, WP44, WP54, WP61, WP90, WP98, WP104, WP118Other: CT5, CT11, CT12, CT17

Understand the concept of a unit rate 2. a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship.1

Power Up: PS78, PS91Lessons: L23, WP54, WP57, WP82, WP84, WP98, WP103, WP104, WP107, WP109, WP118Other: CT5, CT11, CT12, CT17

1Expectations for unit rates in this grade are limited to non-complex fractions.

© HMH Supplemental Publishers Inc.Saxon Math Course 1 13

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Problem Solving

R E A L W O R L D

Name:

SAXON MATH ACTIVITIES FOR COURSE 2 16 Copyright © Houghton Mifflin Harcourt

A hut is built in the shape of a half cylinder. The complete cylinder would have a diameter of 20 feet and a height of 48 feet. What are the shape and dimensions of the floor of the hut?

The floor of the hut can be thought of as a plane section through a diameter of the complete cylinder.

The floor is a rectangle. Its length is the same as the height of the cylinder, or 48 feet. Its width is the same as the diameter of the cylinder, or 20 feet.

Tse

Theas

History

Huts like this were actually used by the US military in WWII. They are called Quonset huts.

Practice

a. Multiple Choice Which of the following shapes cannot be a plane section of a cube?

A circle B triangle C rectangle D square

b. Multiple Choice Which of the following shapes does not have any plane section that is triangular?

A cone B square pyramid C cube D cylinder

Name the shape of each plane section of the rectangular prism.

c. d.

e. f.

Problem Solving R E A L W O R L D

g. When a carrot lying on its side is sliced vertically through its diameter, what shape is the plane section formed? When a carrot is sliced at an acute angle to its diameter, what shape is the plane section formed?

h. Reasoning Every plane section of a certain geometric figure is a circle. What is the figure?

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Real-world problem solving and practice activities provide opportunities for students to apply their learning.

Includes detailed correlations to the Standards for Mathematical Practice and Mathematical Content.

Common Core State Standards Companion

StandardsSuccess

C O M P A N I O N1467013

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6 Saxon Math saxonmath.com I 800.289.4490 7

Problem-Solving Overview 1

PROBLEM-SOLVING OVERVIEW

Focus on Problem Solving

As we study mathematics we learn how to use tools that help us solve

problems. We face mathematical problems in our daily lives, in our work, and

in our efforts to advance our technological society. We can become powerful

problem solvers by improving our ability to use the tools we store in our

minds. In this book we will practice solving problems every day.

This lesson has three parts:

Problem-Solving Process The four steps we follow when solving problems.

Problem-Solving Strategies Some strategies that can help us solve

problems.

Writing and Problem Solving Describing how we solved a problem or

formulating a problem.

four-step problem-

solving process

Solving a problem is like arriving at a destination, so the process of solving

a problem is similar to the process of taking a trip. Suppose we are on the

mainland and want to reach a nearby island.

Mainland

Island

Problem-Solving Process

Step 1:

Understand Know where

you are and where you want

to go.

Step 2:

Plan Plan your route.

Step 3:

Solve Follow the plan.

Step 4:

Check Check that you have

reached the right place.

Taking a Trip

We are on the mainland and want to go

to the island.

We might use the bridge, the boat, or

swim.

Take the journey to the island.

Verify that you have reached your

desired destination.

SXN_M6_SE_Problem_solving.indd 1

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In addition to providing comprehensive coverage of the Common Core State Standards for Mathematical Content, Saxon Math © 2012 Common Core edition infuses instruction with Mathematical Practices.

Common Core mathematical Practices are infused throughout the curriculum

Students Persevere in Problem Solving during each lesson, with a focus on a structured four-step problem-solving process.

Students Use appropriate Tools Strategically to solve real-world problems.

See more examples of Saxon Math’s support for the Standards for Mathematical Practice. Full correlations are

found in the Standards Success Companion.

584 Saxon Math Course 1

Solution

To find the mean we add the three times and divide by 3.37.5336.9036.63 37.02111.06 3) 111.06 Kimberly’s mean time was 37.02 seconds. Practice Set a. Infer Ninety students were assigned to four classrooms as equally as

possible. How many students were in each of the four classrooms? 22, 22, 23, 23 b. Movie tickets cost $9.50. Aluna has $30.00. How many movie tickets can she buy? 3 tickets

c. Infer Twenty-eight children need a ride to the fair. Each van can carry six children. How many vans are needed? 5 vans d. Corinne folded an 8 12 in. by 11 in. piece of paper in half. Then she folded

the paper in half again as shown. After the two folds, what are the dimensions of the rectangle that is formed? How can you check your answer? 5 1

2 in. by 4 14 in.; I can check the answer by folding an 8 1

2 in.

by 11 in. piece of paper and measuring it.

11 in.

?

?

8 in.12

e. Kevin ordered four books at the book fair for summer reading. The books cost $6.95, $7.95, $6.45, and $8.85. Find the average (mean) price of the books. $7.55Written Practice Strengthening Concepts

1.(111)

Eighty students will be assigned to three classrooms. How many students should be in each class so that the numbers are as balanced as possible? (Write the numbers.) 26, 27, 27 2.

(111)Four friends went out to lunch. Their bill was $45. If the friends divide the bill equally, how much will each friend pay? $11.25 3.

(49)Shauna bought a sheet of 39¢ stamps at the post office for $15.60. How

many stamps were in the sheet? 40 stamps 4.(82)

Eight cubes were used to build this2-by-2-by-2 cube. How many cubes are needed to build a cube that has three cubes along each edge? 27 cubes

> See Math Conversations in the sidebar.

584 Saxon Math Course 1

Example 5 Instruction Have students note that the quotient is a decimal number because the dividend is a decimal number. Remind students to always place a decimal point in the quotient

whenever a dividend is a decimal number. Practice Set Problem a Infer Students must infer that only whole numbers can be used to describe each classroom because the quotient represents a number of people.

Problem b Error Alert Encourage students who have difficulty finding the solution to model the problem using coin and bill sets (play money). Problem c Infer Extend the practice exercises by asking pairs or small groups of students to write and solve two problems, one in which the quotient needs to be rounded down (as shown

in example 2 and problem b) and one in which the quotient needs to be rounded up (as shown in example 3 and problem c). Ask students to present the problems they wrote to the class and compare answers after the problems are solved.

New Concepts (Continued)

Math Conversations Discussion opportunities are provided below. Problem 2 Analyze Extend the Problem “Suppose the friends decide to leave a 20% tip? What amount of money represents 20% of the bill? Explain how you can use mental math to find the answer.” Sample: Finding 10% of a number is the same as shifting the decimal point in that number one place to the left. So 10% of $45 is $4.50, and double that amount is $9. “What is each friend’s share of the bill if they decide to leave a 20% tip?” $13.50

(continued)

Written Practice

SXN_M6_TM_L111.indd 584

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math Conversations encourage students to Construct Viable arguments and Critique the Reasoning of others.

78 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Course 3

Name Performance Task 23A

Rock climbers use ropes of various lengths to climb natural rock surfaces, such as cliffs, which are commonly called faces or walls. To prepare for a safe climbing trip it helps to divide responsibilities among the climbers. Your task for an upcoming climb is to collect and organize all of the rope for the group.

1. The figure illustrates how a traditional rope-assisted climb can be seen as a right triangle. The hypotenuse of the triangle is equivalent to the minimum rope length between the top of the climb and the belayer, whose job it is to anchor the climber with a rope from the ground. Find Pythagorean triples to determine the minimum amount of rope needed for each climb and then complete the table below. (Use your calculator.) a

cb

belayer

Distance of belayer from base of climb

(a)

Maximum height

of climb ( b)

Minimum rope length from belayer

to top of climb (c)

16 ft 63 ft

6 m 8 m

7 m 24 m

33 ft 56 ft

2. A climber reaches the top of a rock face at 16 meters. You only have 20 meters of rope to anchor the climber. Complete the Pythagorean triple to determine the greatest distance you can stand from the face of the cliff.

62588_InstMstr_037-080.indd 78 3/31/06 5:51:29 PM

Students make Use of Structure to solve problems with hands-on activities.

Performance Tasks reinforce Standards for Mathematical Practice and the Common Core testing format.

Saxon Math™ is a trademark of HMH Supplemental Publishers Inc.© Houghton Mifflin Harcourt Publishing Company. All rights reserved. Printed in the U.S.A. 10/11 MS31649 Z-1500687

800.289.4490 • hmheducation.com

Contact us to learn more!saxonmath.com I 800.289.4490

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