Connected Mathematics® Project 3 - My Pearson Training · PDF filewith Connected...

Copyright © 2014 Pearson, Inc. or its affiliates. All rights reserved.1

This guide explores ways for students to monitor their own learning with Connected Mathematics® Project 3 (CMP™ 3).

It examines a variety of self-assessment resources and explores the benefits of helping students develop tools to monitor their own learning.

Finally, this guide investigates the role that math notebooks play in students’ learning.

Help your students become independent, confident, and reflective learners by using CMP™ 3’s resources with a focus on self-assessment.

Begin each Unit by discussing the following features with your students. All are located in the student edition or online ACTIVe-book.

• Looking Ahead• Mathematical Highlights• Mathematical Practices and Habits of Mind

Think about some of the ways you use numbers. You use them to count and to measure. You use numbers to make comparisons and to describe where places are located. Numbers help you communicate and find information. You use numbers to use technology and to make purchases. Numbers can help you think about situations such as those on the previous page.

Whole numbers have interesting properties and structures. Some numbers can be divided by many numbers. Other numbers can be divided by only a few numbers. Some pairs of numbers have lots of factors in common. Some pairs of numbers share only one factor. Some numbers can be written as products or sums of products.

The Investigations in Prime Time will help you use ideas about the structure of numbers. You will explain some curious patterns and solve problems. You should think about some interesting questions, including the three on the previous page.

3Looking Ahead

CMP14_SE06_U01_FM_LA.indd 3 04/01/13 12:36 AM

LookingAhead

Sometimes people in a stadium are asked to hold up cards creating a display. What rectangular arrangements are possible for such displays in a formation of 100 people?

Insects called cicadas spend most of their lives underground. Many come above ground only every 13 years or 17 years. Why are there no 12-year, 14-year, or 16-year cicadas?

Why does your birthday fall on a different day of the week from one year to the next? Why is the same pattern also true for New Year’s Day and the Fourth of July?

2 Prime Time

CMP14_SE06_U01_FM_LA.indd 2 04/01/13 12:36 AM

Supporting Self-Monitoring

Introduction

Beginning a Unit

Common Core State Standards

Mathematical Practices and Habits of Mind

5Common Core State Standards

In the Connected Mathematics curriculum you will develop an understanding of important mathematical ideas by solving problems and reflecting on the mathematics involved. Every day, you will use “habits of mind” to make sense of problems and apply what you learn to new situations. Some of these habits are described by the Common Core State Standards for Mathematical Practices (MP).

MP1 Make sense of problems and persevere in solving them.

When using mathematics to solve a problem, it helps to think carefully about

• data and other facts you are given and what additional information you need to solve the problem;

• strategies you have used to solve similar problems and whether you could solve a related simpler problem first;

• how you could express the problem with equations, diagrams, or graphs;

• whether your answer makes sense.

MP2 Reason abstractly and quantitatively.

When you are asked to solve a problem, it often helps to

• focus first on the key mathematical ideas;

• check that your answer makes sense in the problem setting;

• use what you know about the problem setting to guide your mathematical reasoning.

MP3 Construct viable arguments and critique the reasoning of others.

When you are asked to explain why a conjecture is correct, you can

• show some examples that fit the claim and explain why they fit;

• show how a new result follows logically from known facts and principles.

When you believe a mathematical claim is incorrect, you can

• show one or more counterexamples—cases that don’t fit the claim;

• find steps in the argument that do not follow logically from prior claims.

CMP14_SE06_U01_FM_MH.indd 5 03/01/13 11:37 PM


Mathematical Highlights

Together, these sections provide students with a preview of the ideas that they will investigate in the Unit.

Review Using the Unit Opener and Using the Mathematical Highlights for ideas on how to utilize these pages to introduce each Unit.

Remind your students to check back in with the Mathematical Highlights page periodically as they work.

Since the goals are written in student-friendly language, students and parents can use them to monitor progress throughout a Unit.

Plan to revisit these goals frequently with your classes. Model the thought process that you would like your students to go through as they review the goals.

Common Core State Standards

Mathematical Practices and Habits of Mind

In the Connected Mathematics curriculum you will develop an understanding of important mathematical ideas by solving problems and reflecting on the mathematics involved. Every day, you will use “habits of mind” to make sense of problems and apply what you learn to new situations. Some of these habits are described by the Common Core State Standards for Mathematical Practices (MP).

MP1 Make sense of problems and persevere in solving them.When using mathematics to solve a problem, it helps to think carefully about

• data and other facts you are given and what additional information you need to solve the problem;

• strategies you have used to solve similar problems and whether you could solve a related simpler problem first;

• how you could express the problem with equations, diagrams, or graphs;

• whether your answer makes sense.

MP2 Reason abstractly and quantitatively.When you are asked to solve a problem, it often helps to

• focus first on the key mathematical ideas;

• check that your answer makes sense in the problem setting;

• use what you know about the problem setting to guide your mathematical reasoning.

MP3 Construct viable arguments and critique the reasoning of others.When you are asked to explain why a conjecture is correct, you can

• show some examples that fit the claim and explain why they fit;

• show how a new result follows logically from known facts and principles.

When you believe a mathematical claim is incorrect, you can

• show one or more counterexamples—cases that don’t fit the claim;

• find steps in the argument that do not follow logically from prior claims.

5Common Core State Standards

CMP14_SE06_U01_FM_MH.indd 5 08/01/13 5:49 AM

Mathematical Highlights

In Prime Time, you will explore important properties of whole numbers. Many of these properties are related to multiplication and division. The Investigations will help you understand relationships among

factors, multiples, divisors, and products. You will also learn how the Distributive Property relates multiplication and addition.

The Investigations in this Unit will help you understand the following ideas.

• Classify numbers as prime or composite

• Recognize which situations call for common factors, common multiples, the least common multiple, or the greatest common factor

• Develop strategies for finding factors and multiples, least common multiples, and greatest common factors

• Recognize and use the fact that every whole number can be written in exactly one way as a product of prime numbers

• Use exponential notation to write repeated factors

• Relate the prime factorization of two numbers to the least common multiple and greatest common factor of two numbers

• Recognize that the Distributive Property relates the multiplicative and additive structures of whole numbers

• Use the properties of operations of numbers, including the Distributive Property and the Order of Operations convention, to write equivalent numerical expressions

• Use factors and multiples to solve problems, and explain some numerical facts of everyday life

When you encounter a new problem, it is a good idea to ask yourself questions. In this Unit, you might ask questions such as:

Will breaking a number into factors help me solve the problem?

What common factors and common multiples do the numbers have?

What do the factors and multiples of the numbers tell me about the situation?

When might it be useful to write a number in factored form or as a sum?

ASK Y

OU

RSEL

F

Factors and Multiples

4 Prime Time

CMP14_SE06_U01_FM_MH.indd 4 03/01/13 11:37 PM


Ask Yourself

Mathematical Practices

Launch

Draw students’ attention to the questions at the bottom of the Mathematical Highlights page. They are labeled, “Ask Yourself.”

Have students use these questions as an advanced organizer. The questions give students concrete examples and help them develop their metacognition. That is, these questions show students how they can practice thinking about their thinking.

Consider posting these questions in your classroom. Add questions that you create as you plan for the Unit. Encourage your students to share their own questions as well. Remind your students to return to these questions as they encounter new Problems.

Find the Mathematical Practices and Habits of Mind feature at the beginning of each Unit. Use this content to develop additional questions that further students’ metacognition.

Show students how the general information on these pages complements the unit-specific information in Looking Ahead and Mathematical Highlights. This section helps students think about how they use the mathematical practices when they solve Problems.

Pose the Focus Question during the Launch phase to guide students’ investigation throughout the lesson.

Ask students to refer to the Focus Question as they monitor their thinking about a Problem.

Remind students to check in with their thought processes as they explore the content in the Problem. Refer them to the questions that you developed during the Unit opening.


Explore

Summarize

As students work in class, incorporate questions that help them monitor their thinking.

Consider questions like the following:

• Why did you use that strategy? • What was difficult about this Problem? How did you address that

difficulty? • How does this compare to strategies that you used in previous

Problems?• How does your work help you make sense of the Focus Question?

Find suggested questions for the Explore phase in the teacher resources.

Use the Summarize phase of the lesson to help your students monitor their progress.

Guide their reflections with the suggested questions from the teacher resources.

Be sure to address the Focus Question in the whole-class discussion. Consider asking students to respond to the Focus Question in writing after the discussion. Have students reflect on the goals of the lesson as well as the processes that they used in their work. Provide individualized feedback to students on their self-reflections.


End of an Investigation

Make sure to include class time for the Mathematical Reflection found at the end of each Investigation.

Reference the pacing guide for suggestions on how much time to devote to each Mathematical Reflection.

Encourage your students to think deeply about their learning by answering the reflection questions.

Give students time to discuss their answers to the Mathematical Reflections together. Also, have them record their individual responses in their math notebooks.

Prompt them to reflect on how their understanding has developed over the course of the Investigation. Refer to the fictional students in the Mathematical Reflections to model self-reflection for your students.

45Investigation 4 Linking Multiplication and Addition

Mathematical PracticesAs you worked on each Problem in this Investigation, you used prior knowledge to make sense of the Problem and applied mathematical practices used by mathematicians. Think back over your work, the ways you thought about the Problems, and how you used mathematical practices in solving the Problems.

Describe one other instance of the Mathematical Practices that you and your classmates used in this Investigation to solve a Problem.

“The Order of Operations convention helps me to make sense of an arithmetic statement. It also helped me make exact calculations when needed in Problem 4.3.”

MP2. Reason abstractly and quantitatively

Class Discussion


Looking Back at a Unit

Revisit strategies that students found helpful in solving Problems. Point out that while their classmates may have approached a Problem differently, they have all made progress as a result of the work that they have done.

Find even more opportunities for self-assessment at the end of each Unit.

Assign the Looking Back section to give your students a way to summarize their learning. The questions in the Looking Back section address content knowledge and connections from the entire Unit.

Choose to use these questions as a review before the Unit test. Suggest that students use the questions to identify any concepts that they need more practice with.

2. A university marching band consists of 60 members. The band director wants to arrange the band into a rectangular array for the halftime activities.

a. In how many ways can she arrange the band? Make a sketch of each arrangement.

b. How many rectangular arrangements are possible if the band adds one member and becomes a 61-member band?

3. The prime factorization of Tamika’s special number is 2 * 2 * 3 * 11 and the prime factorization of Cyrah’s special number is 3 * 3 * 5 * 5.

a. What is the least common multiple of the two special numbers?

b. What is the greatest common factor of the two special numbers?

c. List all the factors of Tamika’s number.

d. Is Tamika’s number even or odd? Is Cyrah’s number even or odd?

e. Is Tamika’s number a square number? Is Cyrah’s number a square number?

4. Shani gave a clue for her secret number.

Clue 1: My number is a factor of 90.

a. Can you determine what Shani’s secret number is?

b. What is the smallest Shani’s number can be? What is the largest Shani’s number can be?

c. Brandon says the secret number must also be a factor of 180. Is he correct?

d. Shani gave a second clue for her secret number.

Clue 2: My number is prime.

Now can you determine what the secret number is?

e. Shani gave a third clue for her secret number.

Clue 3: Twenty-one is a multiple of my secret number.

Now can you determine what the secret number is?

For Exercises 5–7, insert parentheses and addition, subtraction, or multiplication signs on the left side of the equality sign to make a true statement.

5. 3 2 1 = 9

6. 6 4 3 2 = 26

7. 5 7 2 7 = 21

93Looking Back

CMP14_SE06_U01_EM LB.indd 93 03/01/13 11:27 PM

Looking Back

While working on the Problems in this Unit, you investigated some important properties of whole numbers. Finding factors and multiples of numbers and identifying prime numbers helps in answering questions about clocks and calendars, puzzles and games, and rectangular patterns of tiles. Factoring also focuses attention on the properties of even and odd numbers, square numbers, greatest common factors, and least common multiples.

Use Your Understanding: Number PatternsTest your understanding of multiples, factors, and prime numbers by solving the following problems.

1. The Red Top Taxi Company wants to keep its cars in good operating condition. It has a schedule for regular maintenance checks on each car. Oil is to be changed once every 6 weeks. Brakes are to be inspected and repaired every 10 weeks.

a. After a new cab is put in service, is there ever a week when that cab is scheduled for both an oil change and a brake inspection? If so, what is the first such time?

b. Suppose the oil change time is extended to 8 weeks and the brake inspection to 12 weeks. Is there ever a week when the cab is due for both an oil change and a brake inspection? If so, when will such a incident first occur?

92 Prime Time

CMP14_SE06_U01_EM LB.indd 92 04/01/13 2:12 AM


Have students complete the Self Assessment feature as a take-home assignment at the end of each Unit.

At the beginning of the year, model how to respond to the prompts, since self-assessment may be unfamiliar to some students.

Demonstrate how to use work from the Unit to illustrate mathematical concepts.

Provide feedback to your students, not only about their mathematical understanding but also about the evidence of their reflections.

Establish routines for math notebooks that emphasize reflection. In CMP™ 3, a math notebook should be a record of a student’s evolving understanding.

Decide on how your students should organize their notebooks, and use them consistently. Each Unit contains a Notebook Checklist that students can use to organize their notebooks.

Develop a system to ensure that your students complete their notebooks. For example, collect notebooks once a week or after an Investigation. Check your students’ work, and provide feedback to help deepen their analysis of their own reasoning.

Self Assessment

Math Notebooks


Reflection

Portfolios

Review

Make it clear to students that their notebooks should contain not only their work but also their notes about their thoughts and reasoning.

Help students see their notebooks in this manner by reminding them to reflect on their thinking as they work.

Point out opportunities for reflection when your students answer Applications, Connections, Extensions (ACE) exercises, in addition to formal assignments like Mathematical Reflection and Looking Back.

Model for your students how to use Problems that they struggle with as opportunities to ask for help. Explain that monitoring your own thinking does not just occur at the end of an Investigation or Unit. It occurs continuously throughout one’s learning.

Have your students build in-class portfolios as a final record of their learning at the end of a Unit. The portfolio may include the following items:

• Unit Record Sheet (reflecting all assessed items from the Unit)• Two samples of best work selected by the student• All Mathematical Reflections• Quizzes and the Unit Test and Project• Student Self Assessments• Pre- and Post-Assessment Reflections

Give your students control over creating this record of their learning. Encourage them to take ownership of their own thinking. That way, they can develop the independence, confidence, and self-awareness that will help their mathematics understanding continue to grow.

This guide explored ways for students to monitor their own learning with CMP™ 3.It also examined a variety of self-assessment resources and explores the benefits of helping students develop tools to monitor their own learning.

Finally, this guide investigated the role that math notebooks play in students’ learning.

Applications | Connections | Extensions

Applications 1. At some international airports, trains carry passengers between the

separate terminal buildings. Suppose that one such train system moves along a track like the one below.

a. A train leaves the main terminal going east at 10 meters per second. Where will it be in 10 seconds? When will it reach the east terminal?

b. A train passes the main terminal going east at 10 meters per second. Where was that train 15 seconds ago? When was it at the west terminal?

c. A train leaves the main terminal going west at 10 meters per second. Where will it be in 20 seconds? When will it reach the west terminal?

d. A train passes the main terminal going west at 10 meters per second. When was it at the east terminal? Where was it 20 seconds ago?

66 Accentuate the Negative

CMP14_SE07_U02_ACE03.indd Page 66 21/02/13 5:21 AM sw-099 ~/Desktop/g7_u2

Connected Mathematics® Project 3 - My Pearson Training · PDF filewith Connected...

Documents

Transcript of Connected Mathematics® Project 3 - My Pearson Training · PDF filewith Connected...