hands-on

Grade 3

mathematics

Project Editor Jennifer E. Lawson

Senior Author Dianne Soltess

Mathematics Consultant Meagan Mutchmor

Module Writers Patricia Ashton

Joni Bowman

Betty Johns

Kara Kolson

Suzanne Mole

Winnipeg • Manitoba • Canada

© 2007 Jennifer Lawson

First edition, 2005

Second edition, 2007

Portage & Main Press acknowledges the

financial support of the Government of

Canada through the Book Publishing Industry

Development Program (BPIDP) for our

publishing activities.

All rights reserved. With the exceptions of

student activity sheets and evaluation forms

individually marked for reproduction, no part

of this publication may be reproduced or

transmitted in any form or by any means –

graphic, electronic, or mechanical – without

the prior written permission of the publisher.

Series Editor: Leslie Malkin

Book and Cover Design: Relish Design Ltd.

Illustrations: Jess Dixon

Senior Author: Dianne Soltess

Mathematics Consultant: Meagan Mutchmor

Hands-On Mathematics: Grade 3Revised Manitoba Edition

ISBN 978-1-55379-126-3

ISBN Part 1: 978-1-55379-135-5

ISBN Part 2: 978-1-55379-136-2

Printed and bound in Canada by Prolific Group

100-318 McDermot Avenue

Winnipeg, Manitoba, Canada R3A 0A2

E-mail: books@portageandmainpress.com

Tel: 204-987-3500

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Fax: 1-866-734-8477

PART 1Introduction to Hands-On Mathematics 1

Program Introduction 1

Program Principles 1The Big Ideas of Mathematics 1Hands-On Mathematics Learning

Outcomes 5Program Implementation 8Classroom Environment 9Timelines 9Classroom Management 9Planning Guidelines 10

Assessment 13

The Hands-On Mathematics Assessment Plan 13

Mental Math 26Introduction to Mental Math – Grade 3 26Grade 3 Mental Math Strategies 28

Websites 117

Module 1: Patterns and Relations 121Books for Children 122Introduction 124 1 Ongoing Patterns and Relations Activities 125 2 Repeating Linear Patterns 133 3 Increasing and Decreasing Patterns 144 4 Number-Sequence Patterns 152 5 Patterns in Addition and Subtraction 168 6 Patterns in Multiplication 174 7 Patterns in Structures 185 8 Addition Equations Involving Symbols 192 9 Subtraction Equations Involving Symbols 197Problem-Solving Black Line Master: Patterns and Relations 201References for Teachers 203

Module 2: Statistics and Probability 205

Books for Children 206Introduction 207 1 Literature Connections 208 2 TV Time 213 3 Classroom Carnival 222 4 Looking at Names 230 5 The Typical Grade Three Student 234 6 Weather or Not… 240 7 Symbolic Graphs 250 8 A Year in Time 255 9 Line Plots 259Problem-Solving Black Line Master: Statistics and Probability 263References for Teachers 266

PART 2Module 3: Shape and Space 267

Books for Children (Measurement) 268Books for Children (3-D Objects and 2-D Shapes) 269 Introduction 270 1 Measuring the Passage of Time 271 2 Measuring the Passage of Time in Standard Units 276 3 Days in a Month, Months of the Year 285 4 Introducing Standard Measurement: Metre and Centimetre 296 5 Measuring Objects by Height, Length, and Width 305 6 Comparing and Ordering Objects by Mass 309 7 Investigating Mass 313 8 Perimeter 318 9 More about Perimeter 321 10 Exploring Two-Dimensional Shapes 327 11 Regular and Irregular Polygons 345 12 Identifying the Faces of Geometric Solids (Three-Dimensional Objects) 353

Contents

13 Identifying, Comparing, and Contrasting Geometric Solids 35814 Investigating Nets 36215 Skeletons and Pictorial Representations of Three-Dimensional Solids 372Problem-Solving Black Line Master: Shape and Space 378References for Teachers 382

Module 4: Number Concepts 383

Books for Children 384Introduction 385 1 Personal Numbers 386 2 Rote Counting to 1 000 391 3 Number Paths and Number Charts 399 4 Skip Counting 425 5 Skip Counting by 25s 434 6 How Much is 1 000? 437 7 Walk to 1 000 445 8 One Hundred Pennies 449 9 $1 000 Budget 455 10 Doubling 458 11 Numeral and Number-Word Chains 46312 Fractions 47213 More Fractions: Halves 47514 More Fractions: Fourths 47915 More Fractions: Thirds 48116 Still More Fractions 48217 Comparing Fractions 489 Problem-Solving Black Line Master: Number Concepts 494References for Teachers 498

Module 5: Number Operations 499

Books for Children 500Introduction 502 1 Addition and Subtraction Facts to 18 503 2 More Addition and Subtraction Facts 507 3 One- and Two-Digit Addition and Subtraction 515 4 Addition and Subtraction with a Calculator 521 5 Addition of Hundreds 523 6 Compatible Numbers 529 7 Addition with Three-Digit Numbers 537 8 More Addition with Three-Digit Numbers 540 9 Solving Multiple-Step Problems 546 10 Subtraction with Three-Digit Numbers 547 11 More Subtraction with Three-Digit Numbers 549 12 Choosing a Method for Solving Problems 556 13 Exploring Multiplication – Part One 559 14 Exploring Multiplication – Part Two 56115 Exploring Multiplication – Part Three 563 16 Exploring Multiplication – Part Four 573 17 More Multiplication 575 18 Commutative Property of Multiplication 582 19 Multiplying by 1 and by 0 587 20 Multiplication Facts 59121 Exploring Division – Part One 59422 Exploring Division – Part Two 59523 Exploring Division – Part Three 59624 The Inverse Relationship between Multiplication and Division 602Problem-Solving Black Line Master: Number Operations 609References for Teachers 616

Program Introduction

Hands-On Mathematics focuses on developing

students’ knowledge, skills, and attitudes

through active inquiry, problem solving, and

decision making. Throughout all activities,

students are encouraged to explore, investigate,

and ask questions in order to heighten their own

curiosity about and understanding of the world

of mathematics.

Program Principles

1. Effective mathematics programs involve

students actively building new knowledge

from experience and prior knowledge.

2. The development of students’ understanding

of concepts, flexibility in thinking, reasoning,

and problem-solving skills/strategies form

the foundation of the mathematics program.

3. From a young age, children are interested

in mathematical ideas. This interest must be

maintained, fostered, and enhanced through

active learning.

4. Mathematics activities must be meaningful,

worthwhile, and relate to real-life

experiences.

5. The teacher’s role in mathematics education

is to actively engage students in tasks

and experiences designed to deepen and

connect their knowledge. Children learn

best by doing, rather than by just listening.

The teacher, therefore, should focus on

creating opportunites for students to

interact, in order to propose mathematical

ideas and conjectures, to evaluate their own

thinking and that of others, and to develop

mathematical reasoning skills.

6. Mathematics should be taught in correlation

with other school subjects. Themes and

topics of study should integrate ideas and

skills whenever possible.

7. The mathematics program should

encompass, and draw on, a range of

educational resources, including literature

and technology, as well as people and

places in the local community.

8. Assessment of student learning in

mathematics should be designed to focus

on performance and understanding, and

should be conducted through meaningful

and varied assessment techniques carried

on throughout the modules of study.

The Big Ideas of Mathematics

In order to achieve the goals of mathematics

education and to support lifelong learning in

mathematics, students must be provided with

opportunites to encounter and practice critical

mathematical processes. These processes are

as follows:

Communication

Students need to be given opportunites to

communicate their mathematical ideas through

the use of oral language, reading and writing,

diagrams, charts, tables, and graphs.

For example:

Introduction to Hands-On Mathematics

Introduction 1

▲

Introduction 13

▲

The Hands-On Mathematics Assessment Plan

Hands-On Mathematics provides a variety

of assessment tools that enable you to build a

comprehensive and authentic daily assessment

plan for your students.

Embedded Assessment

Assess students as they work by using the

questions provided with each activity. These

questions promote higher-level thinking skills,

active inquiry, problem solving, and decision

making. Anecdotal records and observations

are examples of embedded assessment:

■ anecdotal records: Recording observations

during mathematics activities is critical in

having an authentic view of a student’s

progress. The Anecdotal Record sheet,

presented on page 15, provides the teacher

with a format for recording individual or

group observations.

■ individual student observations: During

activities when you wish to focus more on

individual students, you may decide to use

the Individual Student Observations sheet,

found on page 16. This black line master

provides more space for comments and

is especially useful during conferencing,

interviews, or individual student

presentations.

Performance Assessment

Performance assessment is planned, systematic

observation and assessment based on students

actually doing a specific mathematics activity.

■ rubrics: To assess students’ performance

on a specific task, rubrics are used in

Hands-On Mathematics to standardize and

streamline scoring. A sample rubric and a

black line master for teacher use are

included on pages 17 and 18. For any

specific activity, the teacher selects four

criteria that relate directly to the learning

outcomes for the specific activity being

assessed. Students are then given

a checkmark point for each criterion

accomplished to determine a rubric score for

the assessment from a total of four marks.

These rubric scores can then be transferred

to the Rubric Class Record on page 19.

Cooperative Skills

To assess students’ ability to work effectively in

a group, teachers must observe the interaction

within these groups. A Cooperative Skills

Teacher Assessment sheet is included on page

20 for teachers to use while conducting such

observations.

Student Self-Assessment

It is important to encourage students to reflect

on their own learning in mathematics. For this

purpose, teachers will find a Student Self-

Assessment sheet on page 21, as well as a

Cooperative Skills Self-Assessment sheet on

page 22.

In addition, a Math Journal sheet is found on

page 23. Teachers can copy several sheets for

each student, cut them in half, add a cover,

and bind the sheets together. Students can

then create title pages for their own journals.

For variety, you may also have students use

the blank back sides of each page for other

reflections. For example, have students draw

or write about:

■ new math challenges

■ favourite math activities

■ math-related book reports

■ new math terminology

Assessment

14 Hands-On Mathematics • Grade 3

Students will also reflect on their own learning

through writing in their math journals.

For young students, self-assessment is best

done through oral discussion, reflecting on

activities done in class. These conversations

can occur as individual student conferences,

small-group discussions, or as whole-class

discussions.

Portfolios

Select, with student input, work to include in

a mathematics portfolio, or in a mathematics

section of a multi-subject portfolio. This can

include activity sheets, patterning samples,

graphs, charts, as well as other written material.

Use the portfolio to reflect the student’s progress

in mathematics over the course of the school

year. Black line masters are included to organize

the portfolio (Portfolio Table of Contents on

page 24 and Portfolio Entry Record on page 25).

Students can be assisted in completing these

sheets by having an adult scribe for them.

Note: Throughout each module of Hands-On Mathematics, suggestions for assessment are provided for several lessons. It is important to keep in mind that these are merely suggestions. Teachers are encouraged to use the assessment strategies presented in a wide variety of ways and to ensure that they build an effective assessment plan using these assessment ideas as well as their own valuable experience as educators.

What Is Mental Math?

Mental math is a core focus of mathematics,

which helps students develop essential skills

involved in working with numbers and assists

them with daily mathematical thinking and

problem solving.

Mental math

■ is visually and cognitively based. Students

do not use manipulatives or paper and

pencils when doing mental math activities or

solving mental math problems

■ involves mental calculating without the use of

external memory aids

■ is a combination of cognitive strategies that

enhances flexible thinking and number sense

■ improves computational fluency by

developing efficiency, accuracy, and flexibility

At any grade level, students should do mental

math activities daily for approximately five

minutes.

The Hands-On Mathematics program focuses

on three types of mental math: subitizing, mental

calculations, and counting.

Subitizing

Subitizing means instantly recognizing random

number patterns. This is sometimes referred

to as flash math. You can use an overhead

projector to flash number patterns or you can

simply flash cards by hand. Display patterns

for a few seconds, and then ask students to tell

what number they saw and how they saw it.

For example:

You can also use various objects for subitizing,

such as bingo chips or interlocking cubes, by

placing sets of them onto an overhead projector.

Subitizing patterns can be regular patterns or

random configurations. For example, students

should instantly recognize “five” in any of the

following configurations:

Four types of subitizing templates are included

in the Hands-On Mathematics program:

1. Number-cube cards display the six sides of

number cubes (dice) with dots in the traditional

number-cube positions.

2. Dot-pattern cards can have more than six

dots, which are not in the traditional number-

cube positions.

Introduction to Mental Math – Grade 3

26 Hands-On Mathematics • Grade 3

▲

I saw 8, orI saw 5 and 3.

Module 1

Patterns and Relations

124 Hands-On Mathematics • Grade 3

Mathematics is the study of patterns and relations. When students begin to recognize and explore the patterns that are inherent in mathematics, it becomes easier for them to understand the relationships among different mathematical concepts. Students need opportunities to discover and explore both patterns that occur in everyday life as well as those revealed through calculators and computers.

In this module, students learn to recognize, describe, extend, and create patterns using real objects, mathematical materials, and numbers. Students first learn about patterns by identifying similarities and differences as they sort. Students explore various sorting activities at the beginning of the module, learning to identify, describe, and classify objects by their attributes. As they start to understand the relationships between objects, students can begin making predictions about patterns. They then proceed to the recognition of visual patterns, auditory patterns, and patterns involving the sense of touch. From recognition, students progress to pattern extension, translation of patterns to other modes, and finally to the creation of their own patterns.

Students learn to create various forms of patterns in this module including patterns using objects, geometric shapes, pictures, numbers, sounds, “touch” actions (for example, tapping), and physical actions (clapping, jumping, and so on). Students should be exposed to all different forms of patterning and should develop skills in transferring patterns from one form to another.

Teachers should also expose students to a wide variety of activities and play with patterns of all kinds including those from different cultures such as the patterns in ancient number systems like Roman numerals. These should consist of linear patterns, symmetrical patterns, repeating patterns, and increasing/decreasing patterns.

Mathematics Vocabulary

Students must learn to recognize and understand the mathematical vocabulary related to the patterns and relations module. A “mathematics word wall” is a valuable reference for students for displaying new vocabulary. Dedicate a classroom bulletin board to your word wall, and display the letters of the alphabet along the top of the bulletin board. Use index cards to record math vocabulary introduced in each lesson, and place these on the board under the appropriate letter of the alphabet. Encourage students to refer to the math word wall during activities and while doing written tasks.

Throughout this module, teachers should use, and encourage students to use, vocabulary such as: attribute, less, more, fewer, shape, pattern,

element, term, repeat, increasing, decreasing,

compare, extend, vertical, horizontal, row,

column, diagonal, and core.

Introduction

168 Hands-On Mathematics • Grade 3

▲

Background Information for Teachers

Grade three students should already have a good foundation in addition and subtraction facts to 20. The activities in this lesson are designed to build on these skills to develop problem-solving and predicting skills. The lesson begins with a review of the use of patterns to solve addition and subtraction equations. Students observe number patterns in addition and subtraction grids by examining rows, columns, and diagonals, and then describe the pattern rule.

Materials

■ overhead projector■ overhead transparency film■ overhead markers■ addition grid (included. Photocopy onto

overhead transparency.) (1.5.1)■ pencils■ graph paper■ magic number generator (Fold a long piece

of cardboard into three sections. Cut out “in”and “out” slots from the middle section, asin the diagram below. The slot must be largeenough to receive an index card. You willneed one magic number generator for eachsmall group of students.)

■ index cards■ markers■ chart paper

Activity: Part One: Review of Patterns in Addition and Subtraction Strategies

Doubles

On the overhead, begin a chart showing doubles addition facts, such as the one below, and ask students to tell you the answers to fill in:

Remind students that doubles are addition facts where both addends are the same. Ask:

■ What number patterns do you see?

Doubles plus one

Explain to students that doubles facts can be used to learn other facts. On the overhead, record the facts 5 + 5 = 10 and 5 + 6 = ___. Ask:

■ What is the same about these two numbersentences?

■ What is different?■ What is the sum of 5 + 6?■ How do you know? (The second addend is

1 more than the second addend in the fact5 + 5, so the sum should also be 1 more.)

Repeat with other doubles-plus-one facts.

Fact families

Record 6 + 5 = ___ on the overhead. Ask:

■ What is the sum?■ Is the sum different from the sum of 5 + 6

(used in the doubles-plus-one example)?■ Why or why not? (Although the addends

have been reversed, the numbers are thesame, so the sum is the same).

Now, ask students to make a subtraction fact using the same numbers (11 – 5 = 6). Ask students for the related fact. List all of the facts in the fact family for these three numbers.

Patterns in Addition and Subtraction5

In

Out

1 + 1 2 + 2 3 + 3 4 + 4 5 + 5 6 + 6 7 + 7 8 + 8 9 + 9

2 4 6

Module 1 • Patterns and Relations 169

▲

Ask:

■ How many facts are in this fact family? (four)

Repeat with other doubles-plus-one fact families.

Now, record 7 + 7 = 14 on the overhead. Ask:

■ Does this addition fact have a related addition fact? (No, because reversing the addends does not change the fact.)

■ What is the related subtraction fact?■ What do you know about doubles-fact

families? (They have only one addition fact and one subtraction fact.)

Expand this activity to include all doubles facts to 20, as well as doubles facts beyond 20.

Activity: Part Two: Addition Grids

Distribute Activity Sheet A (1.5.2), and have students complete the addition grid by finding the sum of each pair of horizontal and vertical numbers. Then, ask students to record the number patterns they observe.

Activity Sheet ANote: Before having students complete this activity sheet, review definitions for the words horizontal and vertical.

Directions to students:

Complete the addition grid by finding the sum of each pair of horizontal and vertical numbers. Then, record any number patterns you observe (1.5.2).

On the overhead, display the transparency of the addition grid (1.5.1). Ask students to share the patterns they discovered on their own addition grids (1.5.2), guiding them to:

■ describe the patterns they observe in the rows

■ describe the patterns they observe in the columns

■ describe the patterns they observe in the diagonals

Now, distribute pencils, and ask students to draw an outline around any 2 x 2 square on their addition grid. Ask:

■ How is your 2 x 2 square like other 2 x 2 squares? How is it different?

■ Do all 3 x 3 squares have the same pattern? (Have students draw 3 x 3 boxes and share their discoveries.)

Tell students to draw a box around any three numbers in a row or column. Ask:

■ What do you notice about the middle number? Does this pattern work for any five numbers in a row or column?

Have students record in their math journals all the discoveries they made from observing their addition grids.

Now, distribute graph paper to students, and have them make their own subtraction grids, numbering the top row 9 to 18 and numbering the left column 0 to 9. Ask students to complete their subtraction grids and then describe the patterns they see. Compare the addition and subtraction grids.

Activity: Part Three: Magic Number Generator

Show students one of the “magic number generators” (see “Materials” list, page 168). Stand behind it, and explain that you are going to record a rule on an index card. When a number goes into the magic number generator, a different number will come out that fits the rule.

Record an addition rule (for example, “+ 5”) on an index card, but do not show it to students. Ask a volunteer to record a number on another index card and insert the card into the “in” slot. If a student inserts a 3, record an 8 on a blank

5

170 Hands-On Mathematics • Grade 3

▲

card, and send it through the “out” slot. If a student inserts a 9, send out a 14, and so on. On the overhead, record the numbers going in and coming out. Repeat until students guess your rule. Then, review the numbers on the overhead showing numbers in and out to verify the rule.

In 3 9 5

Out 8 14 10

Repeat the activity, this time with a subtraction rule. Once students understand how the magic number generator works, have them work in small groups to create their own number generator rules. Provide each group with a magic number generator, index cards, a piece of chart paper, and a marker. Have students in each group take turns making up the rule while the others guess. Encourage students to keep track of the rules on chart paper.

Note: Have students alternate between making vertical charts and horizontal charts:

Problem Solving

If you cut a piece of string fifty times, how many pieces of string will you have? Make a chart to explain your answer.

Note: A reproducible master for this problem can be found on page 201.

Activity Centres

■ Include four sets of cards numbered 1 through 6 at an activity centre along with scrap paper and pencils, and have pairs of students play Race to 365. Tell students to shuffle the four sets of cards together before they begin and place the deck facedown in the centre of the playing space.

Have each player begin with a score of 300. Tell students to take turns drawing a card and adding its value to their running total on scrap paper. A player may choose to pass on a card drawn and not add its value to his/her total, thereby missing a turn. The first person to reach 365 without going over wins.

Variations:

■ Have students play the game with other sets of numbers (for example, 1 through 9).

■ Change the target number to different numbers, to 1000.

■ Start with a large number, and subtract to a target number.

■ Place calculators at an activity centre. Starting with the number 50, have pairs of students take turns subtracting any one-digit number (except 0) from the difference. The first player to reach 0 wins. As a follow-up, ask students:

■ Can you find a strategy to win? ■ Does it matter who goes first?

5

In Out

In

Out

Module 1 • Patterns and Relations 171

■ How would you change your strategy if the rule changed and the first person to reach 0 lost the game?

■ At an activity centre, place a set of cards numbered 0 to 9, and include an extra 9 card. Also include scrap paper and a pencil. Have pairs of students play Minus 9. Tell the pair to place one of the 9 cards face up in front of them, shuffle the rest of the cards, and put them facedown beside the 9. Ask players to take turns drawing two cards from the deck, placing them above the 9, and subtracting 9 from the two-digit number created. If they subtract correctly, they gain a point. Once students have played for a while, have them describe the patterns they observe. Vary the game by having students add 9 to the two-digit number created rather than subtracting 9.

ExtensionsNote: The following activity has students using the constant feature on a calculator. A common way for activating this feature is presented here, but different calculators prompt the feature in different ways. Be sure to experiment with the calculators students will be using before doing this activity.

■ Distribute calculators, and show students how to use the constant feature for repeated addition or subtraction. Have students enter 500 on their calculators and then press “+, 1, =, =, =…” to count-on by 1s. Tell students to continue pressing “=” until the display reaches 530. Next, have students use the constant feature to skip count by 2s, starting at 500 (enter 500, then press “+, 2, =, =, =...”).

Have students predict how long it will take them to use the constant feature to count from 500 to 600 by 1s, stopping exactly at 600. Ask students to enter “500, +, 1” and then wait for you to say “Go!” to begin

pressing the “=.” Time students, and record the best time.

Now, have students estimate how long it will take them to use the constant feature to count from 500 to 600 by 2s. Time students, and discuss the difference between the amount of time it took to count by 1s and the amount of time it took to count by 2s. Have students predict how long it would take them to skip count from 500 to 600 by 3s, 4s and 5s. Encourage students to consider whether they could reach 600 exactly for each of these skip counting numbers.

Repeat the previous activities with students, this time skip counting backward (“600, –, 1, =, =, =…”). Have students say the numbers aloud as they count back by 1s, 2s, 3s, 4s, and 5s. Other variations include:

■ Have students predict how long it would take them to count from 500 to 1000 by 1s or by 10s.

■ Have students skip count forward or backward beginning at random numbers. For example, have students enter “18, +, 5, =, =, =…” on their calculators. Ask students to record the numbers displayed on their calculators and describe the number patterns they see.

■ To extend the magic number generator activities from Activity: Part Three, have students try The Shodor Education Foundation site’s interactive “Number Cruncher” at http://www.shodor.org/interactivate/activities/numbercruncher/index.html.

5

172 – 1.5.1

+

Addition Grid

0 1 2 3 4 5 6 7 8 9

0 0 1 2 3 4 5 6 7 8 9

1 1 2 3 4 5 6 7 8 9 10

2 2 3 4 5 6 7 8 9 10 11

3 3 4 5 6 7 8 9 10 11 12

4 4 5 6 7 8 9 10 11 12 13

5 5 6 7 8 9 10 11 12 13 14

6 6 7 8 9 10 11 12 13 14 15

7 7 8 9 10 11 12 13 14 15 16

8 8 9 10 11 12 13 14 15 16 17

9 9 10 11 12 13 14 15 16 17 18

Portage & Main Press, 2007, Hands-On Mathematics, Level 3, ISBN: 978-1-55379-126-3

1.5.2 – 173

Completing an Addition Grid

5A

+ 0 1 2 3 4 5 6 7 8 9

0

1

2

3

4

5

6

7

8

9

Patterns Observed: __________________________________________________________

___________________________________________________________________________________

___________________________________________________________________________________

___________________________________________________________________________________

Date: __________________________ Name: ___________________________________________

Portage & Main Press, 2007, Hands-On Mathematics, Level 3, ISBN: 978-1-55379-126-3