western canadian teacher guide - sd67 (okanagan...

Teacher Guide

Western

Western Canadian

Unit 1: Number Patterns

1UNIT

“Logical patterns exist and area regular occurrence inmathematics. They can berecognized, extended, orgeneralized. The same patterncan be found in physical andgeometric situations as well asin numbers.”

John A. Van de Walle

Mathematics Background

What Are the Big Ideas?

• Patterns exist throughout mathematics.

• Describing a number pattern gets students to think about how tomake and state a generalization.

• A pattern can be identified, extended, and created using words,numbers, and symbols.

• An equation shows that two mathematical expressions are equal.

How Will the Concepts Develop?

Students investigate many forms of numerical and geometric patterns.They describe patterns mathematically using words, pictures, andsymbols.

Students analyse patterns to determine how they change or grow. Theylook for relationships in patterns. Students make general statementsabout these relationships, then predict what will happen if a pattern isextended. They apply this strategy to problem solving.

Students are introduced to the concept of an equation as a statement thattwo expressions are equal. Students balance equations involvingaddition and subtraction by determining missing terms.

Why Are These Concepts Important?

Recognizing patterns and making generalizations are skills thattouch all strands of mathematics and provide the foundationfor algebraic reasoning.

Finding missing terms in an equation prepares students for moreadvanced algebra that they will encounter later, in both mathematicsand science.

FOCUS STRANDPatterns and Relations

SUPPORTING STRANDSNumber Concepts and NumberOperations

Number Patterns

ii Unit 1: Number Patterns

Unit 1: Number Patterns iii

Lesson 1:Patterns in ChartsLesson 2:Exploring Number PatternsLesson 3:Number Patterns with aCalculator

Curriculum Overview

General Outcomes• Students investigate, establish, and

communicate rules for, andpredictions from, numerical andnon-numerical patterns, includingthose found in the community.

Specific Outcomes• Students identify and explain

mathematical relationships andpatterns. (PR1)

• Students make and justifypredictions, using numerical andnon-numerical patterns. (PR2)

General Outcomes• Students investigate, establish, and

communicate rules for, andpredictions from, numerical andnon-numerical patterns, includingthose found in the community.

• Students demonstrate a numbersense for whole numbers 0 to 10 000.

• Students apply arithmeticoperations on whole numbers, andillustrate their use in creating andsolving problems.

LaunchCalendar Patterns

Cluster 1: Understanding Number Patterns

Cluster 2: Understanding Equations

Specific Outcomes• Students identify and explain

mathematical relationships andpatterns. (PR1)

• Students represent and describenumbers to 10 000 in a variety ofways. (N7)

• Students use manipulatives,diagrams, and symbols in aproblem-solving context, todemonstrate and describe theprocess of addition and subtractionof numbers up to 10 000. (N12)

Show What You Know

Unit ProblemCalendar Patterns

Lesson 4: Equations Involving AdditionLesson 5:Equations Involving SubtractionLesson 6:Strategies Toolkit

iv Unit 1: Number Patterns

Curriculum across the Grades

Material for This Unit

Gather old calendars for the Unit Problem.

Grade 3

Students sort concretelyand pictorially, using twoor more attributes.

Students use objects andconcrete models toexplain the rule for apattern, such as thosefound on addition andmultiplication charts.

Students make predictionsbased on addition andmultiplication patterns.

Students represent anddescribe numbers to 1000in a variety of ways.

Students usemanipulatives, diagrams,and symbols, in aproblem-solving context,to demonstrate theprocesses of addition andsubtraction to 1000, withand without regrouping.

Grade 4

Students identify andexplain mathematicalrelationships andpatterns, using:grids/tables/objects;Venn/Carroll/treediagrams; graphs; objectsor models; technology.

Students make and justifypredictions, usingnumerical and non-numerical patterns.

Students represent anddescribe numbers to 10 000 in a variety ofways.

Students usemanipulatives, diagrams,and symbols in aproblem-solving context,to demonstrate anddescribe the process ofaddition and subtractionof numbers up to 10 000.

Grade 5

Students develop charts torecord and revealpatterns.

Students describe how apattern grows, usingeveryday language inspoken and written form.

Students construct andexpand patterns in twoand three dimensions,concretely and pictorially.

Students generate andextend number patternsfrom a problem-solvingcontext.

Students predict andjustify pattern extensions.

Students read and writenumerals to 100 000.

Students recognize,model, and describemultiples, factors,composites, and primes.

Students add and subtractdecimals to hundredths,concretely, pictorially,and symbolically.

Number Search For Extra Support (Appropriate for use after Lesson 1)Materials: hundred chart (PM 13), Number Search(Master 1.7)

The work students do: Students work individuallywith a hundred chart and use patterns to add 10, 20,or 30 to a number.

Students choose a number less than 50 and mark iton a hundred chart. They add 10 to the number andnote where the sum is on the chart. Students start withthe same number, then add 20 and note where thesum is on the chart. Students start with the samenumber again, add 30, then predict where the sumwill be on the chart.

Take It Further: Students choose a number lessthan 50 and describe how they can use patterns in ahundred chart to add 9, 18, or 27.

Logical/MathematicalIndividual Activity

Additional Activities

Twenty-One For Extra Practice (Appropriate for use after Lesson 5)Materials: Snap Cubes, Twenty-One (Master 1.9)

The work students do: Students work in pairs. Theobject of the game is to make the other player removethe last Snap Cube.

Students connect 21 Snap Cubes to form a chain.Players take turns. They remove 1, 2, or 3 Snap Cubesfrom the chain each time. The player who removes thelast cube loses the game.

Students play the game several times and discuss any patterning strategies they used.

Take It Further: Challenge students to play Twenty-One by removing 2, 3, or 4 cubes each time.

Kinesthetic/SocialLogical/MathematicalPartner Activity

Patterns to the NinesFor Extra Practice (Appropriate for use after Lesson 3)Materials: calculators, Patterns to the Nines(Master 1.8)

The work students do: Students work individuallyto determine patterns in the given calculations. Theydescribe the pattern rule for each set of calculationsand use it to predict the missing products.

Students check their predictions using a calculator.

Take It Further: Challenge students to discover theirown patterns with nines. For example, 4 x 9 = 36, 4 x 99 = 396, 4 x 999 = 3996, and so on.

Logical/MathematicalIndividual Activity

Make It Work Extension (Appropriate for use after Lesson 6)Materials: Pattern Blocks, Make It Work (Master 1.10)

The work students do: Students work in pairs.Statements are given with different figures representingdifferent numbers. The same figure always representsthe same number. For example, if one hexagon has thevalue 10, then all hexagons have the value 10.

Students determine a value for each figure to makeeach statement an equation.

Take It Further: Challenge students to use PatternBlocks to create equations with missing terms for apartner to complete.

Kinesthetic/Social Partner Activity

Unit 1: Number Patterns v

3 x 9 = 273 x 99 = 2973 x 999 = 29973 x 9999 = ____

99 x 12 = 118899 x 23 = 227799 x 34 = 336699 x 45 = ____

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

vi Unit 1: Number Patterns

Planning for Unit 1

Planning for Instruction

Lesson Time Materials Program Support

Suggested Unit time: 1–2 weeks

Unit 1: Number Patterns vii

Purpose Tools and Process Recording and Reporting

Planning for Assessment

2 Unit 1 • Launch • Student page 4

Calendar Patterns

LESSON ORGANIZER

Curriculum Focus: Activate prior learning about patterns.Vocabulary: pattern, row, column, diagonal

10–15 min

L A U N C H

ASSUMED PRIOR KNOWLEDGE

Students recognize that patterning results from repetition.Students can extend a pattern from a pattern rule.Students can describe a pattern.

Teaching notes for Cross Strand Investigation It’s All in the Can!are in the Additional Assessment Support module.

✓✓✓

ACTIVATE PRIOR LEARNING

Invite students to examine the calendar pages inthe Student Book.

Ask questions, such as:• What does each calendar page show?

(One month)• Why are there 7 columns?

(One for each day of the week)

Discuss the first question in the Student Bookand record students’ answers on the board.(Sample answers: As you go down a column, thenumber increases by 7. Both January and October starton Monday. Go down one square then across onesquare to the right, and the number increases by 8.)

Ask:• How could I use one of these patterns to add

9 + 7? (Start at 9, go down one square to 16.)• How could I use a pattern to subtract 7 from

12? (Start at 12, go up one square to 5.)

Discuss the second question in the Student Book.Elicit from students that some patterns stay thesame when the first day of the month is onMonday instead of Sunday.

Ask: • If July 7 falls on a Thursday, on what day of

the week will July 14 fall? (Thursday)• If January 1 falls on a Friday, on what day of

the week will January 9 fall? (Saturday)

Tell students that, in this unit, they willinvestigate number patterns. They will usenumber patterns to solve problems, and toexplore equations. At the end of the unit,students will look for more patternsin calendars.

LITERATURE CONNECTIONS FOR THE UNIT

The King’s Chessboard by David Birch. New York: ScottForesman (Pearson K–12), 1993. ISBN 0140548807 The king insists on rewarding a wise man for his services. Thewise man asks for a payment of rice for each square of theking’s chessboard, the amount to be doubled each day. Soon theroyal granaries are almost empty. Will the proud king concedethat he has been outwitted?

DIAGNOSTIC ASSESSMENT

What to Look For

✔ Students recognizethat patterning resultsfrom repetition.

✔ Students can extenda pattern from apattern rule.

✔ Students candescribe a pattern.

What to Do

Extra Support:

Students who have difficulty finding a pattern may benefit from being directed towhere a pattern might be found. For example, ask, “Can you find a pattern in thiscolumn of this chart?”Work on the skill during Lesson 1.

Students who have difficulty extending a number pattern in a chart may benefitfrom writing out the numbers found in one row or column.Work on this skill during Lesson 1.

Students who cannot describe a pattern may benefit from modelling the patternwith concrete materials. For example, they could use counters in groups of 7 tomodel moving down a column in a calendar.Work on this skill during Lessons 1 and 2.

Unit 1 • Launch • Student page 5 3

Some students may benefit from using the virtualmanipulatives on the e-Tools CD-ROM.

The e-Tools appropriate for this unit include Place-Value Blocks.These can be used in place of, or to support the use of,Base Ten Blocks.

REACHING ALL LEARNERS

4 Unit 1 • Lesson 1 • Student page 6

Patterns in Charts

Key Math Learnings1. Patterns in a chart can be described by using numbers and

by using their positions in the chart.2. Patterns in a chart can be extended by following a number

pattern rule or a position pattern rule.3. Charts are tools for identifying and extending patterns.

LESSON ORGANIZER

Curriculum Focus: Investigate patterns in charts. (PR1, PR2)Teacher Materials� hundred chart transparency (PM 13)� overhead projector markers � 1-cm grid transparency (PM 20)Student Materials Optional� hundred charts (PM 13) � 10 + 10 addition chart (PM 14)� pencil crayons � Step-by-Step 1 (Master 1.11)� 1-cm grid paper (PM 20) � Extra Practice 1 (Master 1.18)Vocabulary: position pattern, number pattern, pattern ruleAssessment: Master 1.2 Ongoing Observations:Number Patterns

40–50 min

L E S S O N 1

BEFORE Get S tar ted

Invite students to examine the hundred chart inthe Student Book. Ask:• What is the number pattern in the coloured

squares? (3, 6, 9, 12, 15, 18, …)• How would you describe the number pattern

without saying all the numbers?(Start at 3. Count on by 3.)

• How else could you describe this pattern?(The diagonal that starts with 3 and goes down tothe left is shaded. After that, every 3rd diagonal thatgoes down to the left is shaded.)

Present Explore. Use a hundred charttransparency to model the activity. Colour thenumbers 1, 5, 9, 13, 17, 21, 25, 29, 33, and 37(the first 10 numbers in a pattern). Askstudents to extend the pattern by five morenumbers. Shade these numbers on the chart asthey answer.

TEACHING TIP

Save PaperMake a transparency from any black

line master by photocopying ontotransparency film. Students can use

non-permanent markers on thetransparency. When finished, erase themarkings and re-use the transparency.

Alternatively, laminate a copy of amaster and use it with non-permanent

markers. Or, tape a blank transparencyto a copy of the master.

If you photocopy onto a transparency,tape it to a sheet of white paper for

easy reading and safe storage.

132, 142, 143, 213260, 600, 602, 620287, 792, 879, 927

Alternative ExploreMaterials: 10 + 10 addition chart (PM 14)Ask students to shade the even numbers on the addition chart.Have them describe how to create this pattern using numbers.(Start at 2. Count on by 2.) Then, students should describe thepatterns in the positions of the shaded squares.

Early Finishers Have students look for patterns on a 9 + 9 addition chart (PM 14: 10 + 10 addition chart, with the 10th row and 10thcolumn deleted). They should describe the patterns using numberpattern rules and position pattern rules.

Common Misconceptions ➤Students create a pattern that can be extended in more than

one way.How to Help: Ensure students write enough numbers to identify,predict, and extend the pattern.

ESL StrategiesESL students may respond non-verbally, or with one or twowords. Students with speech emergence may use short phrasesor sentences.

Unit 1 • Lesson 1 • Student page 7 5


Ask:• How would you describe how to make this

pattern? (Start at 1. Count on by 4.)

DURING Exp lore

Ongoing Assessment: Observe and Listen

Ask questions, such as:• How many numbers do you need to shade

before trading your patterns? (10)• How could you describe how you made your

pattern?(I started at 1 and shaded every fifth square.)

• How else can you describe it?(Start at 1. Count on by 5.)

• What strategy did you use to extend thepattern? (I continued to shade every fifth square.)

AFTER Connec t

Invite volunteers to describe their partner’spattern to the class. Have them describe it intwo different ways.

Ask:• What do we mean when we say the numbers

follow a pattern rule? (We can describe thepattern by saying the number it starts with, andwhat number to count on by each time.)

• What do we mean when we say the positionof the numbers follows a pattern rule?(We can describe the pattern using the positions ofthe numbers in the hundred chart.)

Use the pattern rules in Connect. Elicit fromstudents that these patterns in a hundred chartcan be described using number or positionpattern rules.

Numbers Every DayEncourage students to use place value to order the numbers. Forexample, in the first set of numbers, 213 has the greatesthundreds digit, and is the greatest number. Ensure studentsunderstand that they are to order from least to greatest.


Discuss how this information helps us check apattern. For example, we can check a numberpattern by comparing it with the positionpattern in the chart. If the position pattern isinaccurate, the number pattern probably is too.

As students extend patterns by counting on,they are practising addition facts andmultiplication facts.

Prac t i ce

The description of the position pattern isinformal; students may describe a positionpattern in different ways. If students havedifficulty describing the rule for the positionpattern, accept their informaldescription instead.

Question 1 requires a hundred chart and pencilcrayons. Question 4 requires 1-cm grid paper.Have hundred charts available for Question 4and Reflect.

Assessment Focus: Question 4

Students create patterns on the 5-wide hundredchart. They understand that number andposition patterns still exist when thedimensions of the hundred chart change.

Students who need extra support to completeAssessment Focus questions may benefit fromthe Step-by-Step masters (Masters 1.11 to 1.15).

Sample Answers 1. Student answer should include a hundred chart with all

multiples of 3 and 4 shaded. a) 12, 24, 36, 48, 60, 72, 84, 96 are shaded in both colours.

They make a diagonal position pattern.b) Start at 12. Count on by 12. Or, start at 12, go down 1

square, and across 2 squares to the right.2. b) Start at 2 and count on by 3. The answer is the date on

which a number in this pattern falls on a Wednesday. 3. a) The 3rd and 8th columns are shaded.

b) 103, 108, 113, 118, 123,...c) Position pattern: start at 103, and go down 1 square each

time. Start at 108, and go down 1 square each time. Number pattern: start at 103, and count on by 5.

4. a) In any column, the numbers increase by 5. In any row, thenumbers increase by 1. In any diagonal going down to theright, the numbers increase by 6. In any diagonal goingdown to the left, the numbers increase by 4. The numbersin the 5th column follow the pattern: start at 5, multiply by1, 2, 3, 4, …, 20.

b) Some patterns in a 10-wide hundred chart are differentfrom the patterns in a 5-wide hundred chart. For example,in any column of a 10-wide hundred chart, the numbersincrease by 10. In any diagonal going down to the right,the numbers increase by 11. Some patterns in a 10-widehundred chart are the same as in a 5-wide hundred chart.For example, in any row, the numbers increase by 1.

17

5. The patterns have the same numbers. A pattern rule for thefirst pattern is: start at 4 and multiply by a number thatincreases by 1 each time. A pattern rule for the secondpattern is: start at 4 and count on by 4. This is the same asmultiplying 4 by a number that increases by 1 each time.

6. a) The coloured squares lie along a diagonal going down tothe right.

b) The pattern is all even numbers beginning with 20 andending with 30.

c) Start at 20 and count on by 2 until you reach 30.

REFLECT: A number pattern on a hundred chart also has aposition pattern. If the position pattern changes, then there isa mistake in the number pattern. For example, there is amistake in this number pattern: 3, 7, 11, 15, 19, 23, 26, 31,35, 39. I can shade the numbers on a hundred chart and lookfor a mistake in the position pattern:

26 does not fit the pattern, so it must be a mistake.


ASSESSMENT FOR LEARNING

What to Look For

Understanding concepts ✔ Students understand that numbers in a

chart may follow number pattern rulesand position pattern rules.

Applying procedures✔ Students can identify, extend, and

create a pattern in a chart using anumber pattern rule and a positionpattern rule.

Communicating✔ Students use the correct terminology

when describing patterns.

What to Do

Extra Support: Students can do the Additional Activity,Number Search (Master 1.7).Students can use Step-by-Step 1 (Master 1.11) to completequestion 4.

Extra Practice: Have students create a position pattern byshading in squares on a hundred chart (PM 13). Challenge students to describe the number pattern rule that fitsthe position pattern. Students can complete Extra Practice 1 (Master 1.18).

Extension: Have students create a 5 by 5 multiplication chart ongrid paper. Discuss with students the patterns on this multiplicationchart. Ask them to describe number patterns and position patternsin the chart.

Recording and ReportingMaster 1.2 Ongoing Observations:Number Patterns

Making ConnectionsMath Link: Discuss the pattern in the minutes going down eachcolumn of the train schedule: 30, 38, 44, 49, 54, 00, 07. Askwhat this pattern means. (The trains take the same time to makethe same trip, no matter when they begin the trip.)

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40


Exploring NumberPatterns

Key Math Learnings1. Number patterns may be repeating patterns, growing

patterns, or shrinking patterns.2. The core of a repeating pattern is the smallest part

that repeats.

LESSON ORGANIZER

Curriculum Focus: Identify and extend number patterns. (PR1, PR2)Student Materials Optional� 2-column charts (PM 17) � Snap Cubes

� Step-by-Step 2 (Master 1.12)� Extra Practice 1 (Master 1.18)

Vocabulary: repeating pattern, term, core, growing pattern,shrinking patternAssessment: Master 1.2 Ongoing Observations:Number Patterns

40–50 min

L E S S O N 2

Numbers Every DayStudents could round each number to the nearest 10 beforeadding. Or, they could round only the number being added tothe nearest 10 before adding.


Review some of the number patterns fromLesson 1. Write these patterns on the board: 3, 6, 9, 12, 15, 18, …103, 108, 113, 118, 123, ...

Ask:• How are these patterns the same?

(The numbers get bigger. I add the same number allthe way through each pattern.)

• How are these patterns different? (I add 3 inthe first pattern. I add 5 in the second pattern.)

Present Explore. Students should record eachpattern before they extend it.

DURING Exp lore


Ask questions, such as:• Which numbers repeat in the first pattern?

(1, 1, 2, 2)

• How did you extend the pattern?(I repeated the numbers 1, 1, 2, 2.)

• What is the pattern rule?(Write 1, 1, 2, 2. Repeat these numbers.)

• What is the pattern rule for the secondpattern? (All the odd numbers, starting with 1)

• What other pattern rule could you write forthe second pattern? (Start at 1. Add 2 each time.)

• How did you find the pattern rule for thethird pattern? (I subtracted each number from theone before it to see how the numbers were changing.)

• What is the pattern rule? (Start at 1. Add 1.Increase the number you add by 1 each time.)

• What is the pattern rule for the fourthpattern? (Start at 91. Subtract 4 each time.)

Watch for students who do not see the patterns.Encourage them to add or subtract consecutivenumbers, using a calculator if they need to.Ensure that students extend their patterns totest their pattern rules.

1, 1, 2, 213, 15, 17, 1922, 29, 37, 4671, 67, 63, 59


Alternative ExplorePost two repeating patterns, two growing patterns, and twoshrinking patterns on the board. Have students compare andcontrast the patterns. For example, 1, 2, 5, 1, 2, 5, … and 1, 2, 5, 1, 1, 2, 5, 1, …2, 4, 6, 8, 10, … and 3, 7, 11, 15, 19, …30, 29, 27, 24, 20, 15, … and 30, 26, 22, 18, 14, …Have students copy the patterns, and then write the next 4 termsin each pattern (or as far as they can with the shrinkingpatterns).

Early Finishers Have students create repeating patterns with cores of differentlengths. Students can then trade patterns and try to identify thecores of their partner’s patterns.

Common Misconceptions ➤Students confuse a repeating pattern with a growing or

shrinking pattern.How to Help: Have students look for a core first. If they cannotfind a core, then the pattern is not a repeating pattern. It mustbe a growing or shrinking pattern.

ESL Strategies ESL students can create and tell a story to a classmate based ona number pattern. For example, a story for the pattern 2, 4, 8,16, 32 could be, I folded a sheet of paper in half five times andcounted the parts each time.


about 80.about 80.about 80.about 190.

AFTER Connec t

Invite students to share some number patternsthey created. Have them record the patterns onthe board. Discuss a repeating pattern, agrowing pattern, and a shrinking pattern.

Ask:• What do we mean when we say that a

pattern repeats?(The same numbers appear in the same order.)

• What do we mean when we say that apattern grows?(The numbers increase every time.)

• Can a pattern repeat and grow?(No, because in a growing pattern, each newnumber is larger than the number before. This isnot true with a repeating pattern.)

Use the patterns in Connect to introduce thewords term and core. Illustrate the three typesof patterns and pattern rules.

Point out the method of using arrows labelledwith the operation and a number to recordrelationships in a pattern.

Have students extend each pattern for severalmore terms.

Students may notice that a shrinking patterncannot go on forever. Eventually, when wesubtract, we reach 0, or a number close to 0from which we cannot subtract a greaternumber.

Sample Answers 2. The patterns are alike because each has 5 terms, and you add

the first number to get the next term each time. The patternsare different because you add different numbers.

3. a) Start at 2. Add 2 each time. b) Repeat the core 8, 7, 6.c) Start at 85. Subtract 9 each time. d) Start at 6. Add 7 each time.

4. 5, 6, 8, 11, 15, …; Start at 5. Add 1, and then increase thenumber added by 1 each time.5, 7, 9, 11, 13, …; Start at 5. Add 2 each time.5, 10, 20, 40, 80, …; Start at 5, multiply by 2 (or add thenumber to itself) each time.

5. a) This pattern increases in a predictable way, so it is agrowing pattern. The next terms are 8, 8, 8, 8, 8, 8, 8, 8.

b) 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, …; Start at 3,increase the number by 1, and increase the number oftimes the number is repeated by 1.

7. a)

Start at 1. Increase the numberadded by 1 each time.b) 29 cmc) 92 cm; I added the lengths in

the table.


Studying patterns helps students generalizeand develop algebraic reasoning.

For example, in the pattern 2, 4, 6, 8, 10, the1st term is 2, or 2 � 1. The 2nd term is 4, or 2 � 2. The 3rd term is 6, or2 � 3, and so on. To get the value of any term in this pattern,multiply the number of the term by 2.

In later years, students will learn to write sucha generalization using symbolic representation.

Prac t i ce

Students could use strips of coloured paper tomodel the pattern in question 7. Students canuse a 2-column chart for questions 7 and 8.


Students understand that a growing patternincreases by a predictable amount. Theyunderstand that a pattern must have a sufficientnumber of terms to be able to identify it and towrite its pattern rule. Some students mayinclude different types of growing patterns. Forexample, patterns that use addition of the samenumber, addition of numbers that increase eachtime, and repeated multiplication

3, 10, 17, 24, 31, 38

10, 12, 14

Start at 5. Add 5 each time.

Start at 50. Add 50 each time.

8, 7, 6 - core: 8,7,649, 40, 31 41, 48, 55

59

1520

928

39

core: 2,9,1

8. a) b) 120 cmc) 735 cm; the lengths are

multiples of 5.

9. Sandor was framing pictures. Each picture was bigger thanthe last. The first frame used 50 cm of wood. The second used80 cm. The third used 110 cm. This pattern continues. Howmuch wood did the 6th frame use? How much wood didSandor need to buy to make 6 frames?(Answer: 6th frame: 200 cm; total wood: 750 cm)

REFLECT: A repeating pattern has a core. This is the fewestnumbers that repeat. For example, 3, 4, 2, 3, 4, 2, 3, 4, 2,… . The core is 3, 4, 2. A growing pattern increases. You canmake a growing pattern by adding or multiplying repeatedly.For example, 3, 7, 11, 15, 19, … The pattern rule is: start at3 and add 4 each time. A shrinking pattern decreases. Youcan make a shrinking pattern by starting with a large number,then taking away a number repeatedly. For example, 90, 85,80, 75, 70, … The pattern rule is: start at 90 and subtract 5each time.



What to Look For

Understanding concepts✔ Students understand that a number

pattern may repeat, grow, or shrink.

✔ Students understand that a repeatingnumber pattern can be described bya core.

✔ Students understand that a growingor shrinking number pattern can bedescribed by a pattern rule.

Applying procedures✔ Students can identify, extend, and

create patterns using pattern rules orcores.

What to Do

Extra Support: Have students model the pattern 1, 5, 9, 13, …with Snap Cubes of different colours. They then investigate therelationships between consecutive terms by comparing the colours.Students can use Step-by-Step 2 (Master 1.12) to complete question 4.

Extra Practice: List 5 number patterns, labelled A, B, C, D, andE. List, in a different order, the cores or pattern rules for thesenumber patterns. Have students match each pattern with its coreor pattern rule. Students can complete Extra Practice 1 (Master 1.18).

Extension: Have students create square patterns with Snap Cubes.Students determine which numbers of cubes can be arranged intosolid squares, such as, . They record these numbers, then writea pattern rule for them. (1, 4, 9, 16, 25, … Start at 1, add 3, thenincrease the number added by 2 each time.)


Making ConnectionsLiteracy: Discuss the growing pattern in the story, The King’sChessboard.


Number Patterns witha Calculator

Key Math Learnings1. Calculators can be used to explore number patterns.2. Calculators allow us to create and extend patterns with

large numbers.

LESSON ORGANIZER

Curriculum Focus: Use a calculator to explorenumber patterns. (PR1)Teacher Materials� overhead calculator (optional)Student Materials Optional� 4-function calculators � Step-by-Step 3 (Master 1.13)

� Extra Practice 2 (Master 1.19)Vocabulary: keystrokesAssessment: Master 1.2 Ongoing Observations:Number Patterns

40–50 min

L E S S O N 3


Review how to use a calculator with your class.To add repeatedly with a TI-10:

The display shows the two numbers beingadded in the top left. The sum is in the bottomright. In the bottom left is a count of how manytimes the addition has been done.

For calculators other than a TI-108 or TI-10,consult the instruction manual to determinehow to perform repeated additions andrepeated multiplications before startingthis lesson.

Demonstrate the introductory activity on anoverhead calculator. Alternatively, workthrough the activity as a class.

Present Explore. Ensure students understandthat they should make a number pattern usingrepeated addition, and another number patternusing repeated multiplication.

Remind students to record their patterns.

DURING Exp lore


Ask questions, such as:• Why is it important to record the numbers?

(The calculator only shows one number at a time,and I need to look at all the numbers at once tosee patterns.)

• How does a calculator help you explorenumber patterns? (It is faster, and I can workwith larger numbers easily.)

Numbers Every DayReview the concept of odd and even numbers. Remind studentsof the meanings of the terms product and sum. Suggest they useguess and check, trying various numbers, then using the resultsto refine their guesses.


Alternative ExploreMaterials: calculatorsGive students the start number 99. Have them create a pattern by adding this number repeatedly. They should record thepattern. Have them describe any patterns they find in the ones,tens, and hundreds digits.

Early Finishers Students work in pairs and challenge each other to guess patternrules. One student enters a start number on the calculator. Hepresses + or � , enters a second number, then presses = once.He tells his partner the start number, then gives her the calculator.She presses = repeatedly until she finds the pattern rule.

Use the instructions on pages 12 and 13 to modify this activity fora TI-10 calculator.

Common Misconceptions ➤Students cannot identify patterns in the ones digits or

the tens digits.How to Help: Remind students to extend the pattern far enoughto correctly identify the cores of any repeating patterns. Patternsin the tens digits may have long cores. Point out that this is onereason we use calculators to help identify these patterns.


Making ConnectionsMath Link: Ask students to describe a melodic ostinato theyhave heard. For example, students will likely be familiar withadvertising jingles that contain repeating melodies.

Watch for students who key in repeatedmultiplication incorrectly. To multiply repeatedly on a TI-10:

The display shows the two numbers beingmultiplied, the product, and the number oftimes the multiplication has been done.

Ensure students record every number andcheck for mistakes.

AFTER Connec t

Invite students to record some patterns on theboard. Have them identify any patterns in theones digits or tens digits.

Have students create the patterns in Connectwith calculators.

Elicit from students that calculators allow us toextend growing patterns quickly and to seerepeating patterns within the digits.

Prac t i ce

All questions require a calculator.


Students record the extension of the pattern.They recognize that they are adding twodifferent 2-digit numbers which have the samedigits in reverse order.

They extend the pattern to 19 + 91. Studentsmay describe the rule in different ways. Therule works until the numbers “run out.”

3, 6, 12, 24, 48, 96, 192, 384, 768

101, 92, 83, 74, 65, 56, 47, 383 � 19 = 57For example, 109 + 1 + 1 = 111, or 55 + 53 + 3 = 111

Sample Answers1. After the first term 3, the ones digits follow a repeating pattern

with core 6, 2, 4, 8. There is no apparent pattern in the tens digits.

2. After the first term 101, the tens digits form the shrinkingpattern: start at 9, subtract 1 each time. The ones digits followthe growing pattern: start at 1, add 1 each time.

5. The first numbers increase by 1. The second numbers increaseby 10. The sums increase by 11. The pattern rule for the sums is: start at 33, add 11 until youreach 110. The rule does not work beyond 19 + 91 = 110. Or, the tens and ones digits in the sums increase by 1. Therule does not work beyond 18 + 81 = 99.

6. The answer is either 198 or 1089. For example,526: 625 – 526 = 99; 99 + 99 = 198317: 713 – 317 = 396; 396 + 693 = 1089

7. 256 � 4 is the same as 256 + 256 + 256 + 256.Priya can press 256 + , then to get 1024.

REFLECT: A calculator allows me to create number patternsquickly. I can then explore patterns in the ones digits and tensdigits. For example, the pattern made by starting at 99, thenadding 99 each time, is 99, 198, 297, 396, 495, 594, 693,792, 891, 990, 1089… There is a repeating pattern in theones digits, with the core 9, 8, 7, 6, 5, 4, 3, 2, 1, 0.



What to Look For

Understanding concepts ✔ Students understand that calculators

allow them to explore numberpatterns quickly.

✔ Students understand that numbers ingrowing patterns and shrinkingpatterns can contain repeatingpatterns in their digits.

Applying procedures✔ Students can use the correct

keystrokes to extend and createnumber patterns on a calculator.

What to Do

Extra Support: Have students record and extend patterns in a place-value chart. Students can use Step-by-Step 3 (Master 1.13) to completequestion 5.

Extra Practice: Students can explore the Additional Activity,Patterns to the Nines (Master 1.8). Students can complete Extra Practice 2 (Master 1.19).

Extension: Write the core 2, 5, 7, 0. Have students use theircalculators to find a growing pattern that contains this core in thetens digits. (For example, 25, 50, 75, 100, 125, 150, 175,200, …)


, 143, 154, 165, 274, 170, 66

, 1304, 1392, 1480

= 33 15 + 51 = 66 18 + 81 = 99= 44 16 + 61 = 77 19 + 91 = 110= 55 17 + 71 = 88 (Some students may stop

at 18 + 81 = 99)

808


L E S S O N 4

Equations InvolvingAddition

LESSON ORGANIZER

Curriculum Focus: Explore patterns in equationsinvolving addition. (PR1) (N7, N12)Teacher Materials� 10 + 10 addition chart transparency (PM 14)Student Materials Optional� 30 counters � blank addition charts (PM 16)� 2 squares of paper � 10 + 10 addition charts (PM 14)� calculators � decks of cards


Vocabulary: addition fact, sum, equationAssessment: Master 1.2 Ongoing Observations:Number Patterns

40–50 min


Use a 10 + 10 addition chart transparency onthe overhead projector to identify addition factsfor the number 6. Show how to model theseaddition facts using counters. Alternatively,draw circles on the board to represent counters,as in the Student Book.

Present Explore. Suggest that as one studentmodels the addition fact, the other records it.

DURING Exp lore


Ask questions, such as:• How did you organize your addition facts?

(We made a list starting with the smallest numberon the left side of the + sign, and increasing by 1each time; 1 + 12 = 13, 2 + 11 = 13, and so on.)

• How do you know when you have all theaddition facts? (The number of addition facts is equal to the sum. When we write all the additionfacts for 11, we have 11 facts.)

AFTER Connec t

Invite two students to model their additionfacts with counters on the overhead projector oron the board. Record these facts in a systematicway. Ask:

• How can you describe the patterns in theaddition facts? (As the number on one side of the+ sign increases by 1, the number on the other sidedecreases by 1, but the sum stays the same.)

Introduce the word equation. Elicit fromstudents that an equation uses an equal sign toshow that two amounts are equal.

Introduce the addition facts where one of theadded numbers is 0.

Key Math Learnings1. An equation is a statement that two things are equal. 2. Number patterns can be used to find a missing term

in an equation.

Curr i cu lum Focus

This lesson introduces the term equation to describe a numbersentence. The concept level of the lesson matches your Grade 4 curriculum requirements.

Use Connect to illustrate different ways to writean equation.

Ensure students understand that an equationcan be written more than one way. Forexample, 2 + 4 = 6 can be written as: 4 + 2 = 6;6 = 4 + 2, and 6 = 2 + 4.

Stress that knowing addition facts makes iteasier to find missing numbers in an equation.

This work with equating addition statementsis an introduction to the algebra students willlearn in later grades. An equation such as

+ 3 = 7 becomes x + 3 = 7, and studentssolve for the specific unknown, or variable, x.

Prac t i ce

Have counters available. Question 6 requires acalculator.


Students record all possible addition facts forthe sum of the two numbers given. Onestrategy is to be systematic; start at 0, record thecorrect addend to make the statement anequation, then move to 1; record the nextaddend that makes the statement an equation;and so on. When finding all ways to make theequations, some students may consider factssuch as 3 + 6 = 9 and 6 + 3 = 9 to be different.Some students may not include 0 asone number.


Early Finishers Have students make an 11 + 11 to 20 + 20 addition chart (PM 16). They use the addition facts in the chart to writeequations with missing terms, then trade them with a partner,who completes the equations.

Common Misconceptions ➤Students cannot find a missing term in an equation.How to Help: Suggest students try several different strategiesfor adding numbers. They could count on, or use doubles ornear doubles where relevant. For example, in 5 + __ = 9, theycould use near doubles to say that 5 + 5 = 10, so 5 + 4 = 9.

➤Students incorrectly identify sums as being equal.How to Help: Have students use counters to find the sum ofeach pair of numbers, then compare the two sums.


Making ConnectionsScience: Finding a missing term in an equation is like addingmasses to one pan of balance scales until they balance. Forexample, place 50 g in one pan. Add masses to the other panuntil the scales balance. Students can explore creating equationsby balancing masses; for example, 10 g + 10 g = 20 g.

about 20.about 30.about 25.about 10.

Numbers Every DayStudents could round each number to the nearest 10, thensubtract. Or, students could round only the numberthey subtract.



What to Look For

Understanding concepts ✔ Students understand that an equation

is a statement that two things areequal.

Applying procedures✔ Students can make equations using

addition facts.

What to Do

Extra Support: Have students use a 10 + 10 addition chart(PM 14) to find a missing term in an equation. Students can useStep-by-Step 4 (Master 1.14) to complete question 5.

Extra Practice: Students play in pairs. They place a deck ofcards, with face cards removed, face-down. An ace is 1. Player1 draws 2 cards. He arranges them with a + and an = sign toshow an equation with a missing term. Player 2 draws a thirdcard. If Player 2 can complete the equation with this card, shescores a point. If not, Player 1 draws a card and tries to completethe equation. The winner draws 2 new cards and makes a newequation. They repeat 5 times. Students can complete Extra Practice 2 (Master 1.19).

Extension: Students play a game similar to that in ExtraPractice, but use more cards and more + squares.


Sample Answers1. 4 + 16 = 20

5 + 15 = 20 6 + 14 = 20 19 + 1 = 20I wrote the statements in order, adding 1 more to the firstnumber each time until I reached 19.

5. a) 0 + 9, 1 + 8, 2 + 7, 3 + 6b) 0 + 15, 1 + 14, 2 + 13, 3 + 12, 4 + 11, 5 + 10, 6 + 9c) 0 + 13, 1 + 12, 2 + 11, 4 + 9, 5 + 8, 6 + 7d) 0 + 18, 1 + 17, 3 + 15, 4 + 14, 5 + 13, 6 + 12, 7 + 11,

8 + 10, 9 + 9I started with 0 as the first missing number, then used 1, 2,3, and so on, leaving out the statement that was the same asthe one I was given.

6. I found the sum on one side, then used guess and check tofind the missing number on the other side. I could alsosubtract the number on the other side from the sum.

7. 1 + 1 + 8; 1 + 2 + 7; 1 + 3 + 6; 1 + 4 + 5; 2 + 2 + 6; 2 + 3 + 5; 2 + 4 + 4; 3 + 3 + 4I found 8 ways to do it.

REFLECT: Knowing the pattern allows you to predict otheraddition facts. Once you know the addition facts, you can usethem to fill in missing pieces of equations. In the equation 5 + = 9 + 7, I make a pattern by taking 1 away from 9 and adding 1 to 7: 9 + 7, 8 + 8, 7 + 9, 6 + 10,5 + 11. When I reach 5 + 11, I know 11 is the missingnumber.

.

.

.==

40

5

10

99

84

78772

Yes NoYesNo


Equations InvolvingSubtraction

Key Math LearningNumber patterns can be used to find missing terms inequations involving subtraction.

LESSON ORGANIZER

Curriculum Focus: Explore patterns in equationsinvolving subtraction. (PR1) (N7, N12)Student Materials Optional� counters � 2 squares of paper


Vocabulary: subtraction fact, differenceAssessment: Master 1.2 Ongoing Observations: NumberPatterns

40–50 min

L E S S O N 5


Review the idea that an addition fact has tworelated subtraction facts. Explain that equalsubtraction facts can be written as equations inthe same way as equal addition facts.

Present Explore.

DURING Exp lore


Ask questions, such as:• How did you organize your subtraction facts?

(We recorded them with the smallest numbersubtracted first, and then increased it by 1 each time.)

Students should realize that an equationinvolving subtraction can be written in morethan one way. For example, 12 – 7 = 5 can also bewritten as 12 – 5 = 7, 7 = 12 – 5, and 5 = 12 – 7.

AFTER Connec t

Ask:• Why does the difference increase as the

number you subtracted decreases?(If you subtract less, more is left over.)

Review the idea that an equation is a statementthat two things are equal.

Prac t i ce

Have counters available.


Students should recognize that many numberswill work in these equations. Some studentswill choose numbers close to those given onone side of the equation, and will use a pattern.Others may find examples using much largernumbers.

Numbers Every DayStudents use their knowledge of growing patterns to find apattern rule. Ensure students can create the pattern with theircalculator if it uses different keystrokes.



What to Look For

Understanding concepts ✔ Students understand that addition

facts and subtraction facts are related.

Applying procedures✔ Students find missing terms in

equations involving subtraction.

What to Do

Extra Support: Have students use an 10 + 10 addition chart(PM 14) to find answers to subtraction facts. Practice with factfamilies should help strengthen students’ ability to workwith equations. Students can use Step-by-Step 5 (Master 1.15) to complete question 5.

Extra Practice: Students can play the Additional Activity game,Twenty-One (Master 1.9). Students can complete Extra Practice 3 (Master 1.20).

Extension: Challenge students to find 3 different equations thatuse both addition and subtraction on one side and the samenumber on the other side. For example, 12 = 15 + 5 – 8; 12 = 10 – 1 + 3; 100 – 98 + 10 = 12


Sample Answers1. 14 – 4 = 10

14 – 5 = 914 – 6 = 8 14 – 14 = 0The pattern is complete when I have subtracted 1 more eachtime, and reached 0 as the difference.

5. a) 18 – 6 = 19 – 718 – 6 = 20 – 818 – 6 = 152 – 140

b) 20 – 3 = 21 – 420 – 3 = 22 – 520 – 3 = 117 – 100

I used a pattern. In part a, I started with a number 1 morethan 18, and subtracted a number 1 more than 6.

REFLECT: I use related facts to find the missing number in anequation. For example, I write – 10 = 6. I use a relatedaddition fact. I know that 10 + 6 = 16, so 16 – 10 = 6, andthe missing number is 16.

Alternative ExploreMaterials: 24 counters, 2 squares of paper with one labelled –,and the other labelled =.Students create subtraction facts using 20 to 24 counters. Theychoose a number between 8 and 12. They use the counters andpaper squares to model and record all the subtraction facts for thatnumber. Students then organize the facts as they look for patterns.They write how finding subtraction facts with counters is differentfrom finding addition facts.


.

.

.9 8 19

410

16 10

3, 53, 103, 153, 203Start at 3, add 50 each time.

No YesNoYes


Strategies Toolkit

Key Math Learnings1. Interpret a problem involving equations.2. Solve a problem involving equations using guess and check.

LESSON ORGANIZER

Curriculum Focus: Interpret a problem and select anappropriate strategy. (N7, N12)Student Materials Optional

� countersAssessment: Masters PM 1 Inquiry Process Check List,PM 3 Self-Assessment: Problem Solving

40–50 min

L E S S O N 6


Present Explore. Review the key steps inproblem solving: understand the problem, planhow to solve the problem, solve it, then lookback to check the solution.

DURING Exp lore


Ask questions, such as:• Which strategy will you use to solve the

problem? (Try different numbers, then changethem if they do not work.)

• Does your solution work for all threeequations? (No, we had to try different numbers.)

Watch for students who do not understand thata figure retains its value for all three equations.For example, if a circle is worth 3 in the 1stequation, it is also worth 3 in the 2nd and 3rdequations.

Have students share their solutions with theclass. They should explain any strategies used.

AFTER Connec t

Work through the problem in Connect with theclass. Record each problem solving step on theboard and refer to it as you proceed.

Tell students that for the “plan” stage, they canpick one of several strategies to solve theproblem.

Ask:• Why do you think guess and check is a good

strategy? (Once you try a few numbers, you canuse the results to help pick new numbers.)

• Why is it important to “look back”? (To checkif your numbers work for all the equations)

Prac t i ce

Encourage students to refer to the Strategies listto assist in selecting an appropriate strategy.

= 1= 6= 4

= 5= 2= 3

♥

■▲●

▲●

♥ ♥♥ ♥

♥


Common Misconceptions ➤Students have difficulty “guessing” a number when using a

guess and check strategy.How to Help: Suggest students work with two equations first.They should look for things that the equations have in common.For example, the first equation in Explore has one more trianglethan the second. The first equation is worth “one triangle” morethan the second. So, one triangle is the difference between 17and 11.


Sample Answers3. Two cubes balance 1 pyramid, so a pyramid has twice the

mass of a cube. Three pyramids balance 1 sphere, so asphere has 3 times the mass of a pyramid. The cube has theleast mass. The sphere has the greatest mass.

REFLECT: I used guess and check in question 2. The 2nd equation has 2 As and 2 Bs equal 14. The 3rd equation has A and 2 Bs equal 11. This equation has1 less than the 2nd equation, so A = 14 – 11 = 3.In the 2nd equation, 14 = 3 + 3 + B + B, so B = 4. In the 1st equation, 12 = 3 + 4 + C, so C = 5.


What to Look For

Problem solving✔ Students can follow the “understand,

plan, solve, and look back” steps tosolve a problem.

✔ Students can use a guess and checkstrategy to find numbers that satisfyan equation.

What to Do

Extra Support: Students may benefit from modelling eachequation with counters. Have them manipulate the counters untilthey find an arrangement that works for each equation. Forexample, to model the first equation in Explore, use 17 counters.Arrange them in 4 groups, with 2 of the groups identical, andthe other 2 groups different.

Extra Practice: For extra practice with problem solving,students can do the Additional Activity, Make It Work(Master 1.10).

Extension: Challenge students to use two different problem-solving strategies to solve each problem.

Recording and ReportingPM 1 Inquiry Process Check ListPM 3 Self-Assessment: Problem Solving

= 10= 6= 2

A = 3B = 4C = 5

22 Unit 1 • Games • Student page 24

Number the Blocks

Take It Further: Students can create a new game by changing the value of eachPattern Block. For example, the yellow hexagon could have theleast value, and the green triangle the greatest value. Or, theperson with the fewest points wins.

LESSON ORGANIZER

Student Materials Optional� Pattern Blocks (PM 25) � calculators

20 min

G A M E


Invite a student to read the game rules.

Ensure students understand they can only scorepoints for blocks that touch entire sides. Theycannot score points for blocks that share onlyvertices or parts of sides.

Suggest students use a calculator to add thepoints at the end of the game.

DURING Game

As students play, ask questions such as:• Does the order in which you place the blocks

change the points you score?(Yes. Connect two separate blocks by placing a thirdblock, then you get points for all three blocks.)

• Can you find different ways to obtain a scoreof 12? (Yes. For example, touch 2 yellow hexagons,or 3 blue rhombuses, or 3 red trapezoids, and so on.)

• What is the fewest number of points you canget in one turn?(6, by placing one green triangle to touch anothergreen triangle)

• What arrangement would give the maximumnumber of points? (I place the centre block. I get42 points: see diagram below.)

AFTER

After students have played the game a fewtimes, invite volunteers to share their scoreswith the class. Have students who achievedhigh scores explain their strategies.

SHOW WHAT YOU KNOW

Sample Answers1. Pattern rule for numbers shaded in both colours: start at the

start number, count on by 10, or start at the start number andshade that column. Student answer should include a hundredchart shaded to show the pattern he or she described.

2. a) Start at 2. Count on by 4.b) Start at 37. Count back by 3.c) Repeat the core: 18, 19, 20.

3.

7th layer has 49 cubes. The totalnumber of cubes is 140.

4. a) Start at 5. Double the number each time.b) Start at 200. Subtract 12 each time.

5. a) 0 and 11; 1 and 10; 2 and 9; 3 and 8; 4 and 7b) 0 and 10; 1 and 9; 3 and 7; 4 and 6; 5 and 5

6. a) 5 b) 20

Unit 1 • Show What You Know • Student page 25 23


What to Look For

Reasoning; Applying concepts ✔ Questions 5 and 6: Student understands that an equation represents two things that are equal.

Accuracy of procedures✔ Question 1: Student can identify and create number patterns in a chart.

✔ Question 2: Student can correctly identify a pattern rule and extend a pattern.

✔ Question 4: Student can use a calculator to identify pattern rules and extend patterns.

✔ Questions 5 and 6: Student can find missing terms in an equation.

Problem solving✔ Question 3: Student can organize information about a pattern and use it to solve a problem.

Recording and ReportingMaster 1.1 Unit Rubric: Number PatternsMaster 1.4 Unit Summary: Number Patterns

LESSON ORGANIZER

Student Materials� Question 1: hundred charts (PM 13)� Question 4: 4-function calculatorsAssessment: Masters 1.1 Unit Rubric: Number Patterns,1.4 Unit Summary: Number Patterns

80, 160, 320140, 128, 116

Remind students of the calendars in theUnit Launch.

Present the Unit Problem. Invite volunteers toread the instructions for each part of theproblem and clarify any points.

Have one student read aloud the Check List toensure all students understand what their workshould include. Discuss possible ways inwhich students might present their work.

Make blank calendar pages available. Studentscan fill in the dates to match their real calendarpages. They may record their patterns on thesepages, or use them to test pattern rules.

Suggest that students use calculators to helpthem add large numbers.

Calendar Patterns

Sample ResponsePart 1

In each square the sums are always the same. For example, 9 + 17 = 10 + 169 is 1 less than 10. 17 is 1 more than 16.This pattern is true for all 2 by 2 squares.Pattern rule: The sums of the diagonals in a 2 by 2 square on a calendar are the same.

The sums of diagonals in a 3 by 3 square are the same. For example, 10 + 26 = 12 + 24.12 is 2 more than 10. 24 is 2 less than 26.The sums of diagonals in a 4 by 4 square are the same. For example, 1 + 25 = 4 + 224 is 3 more than 1. 22 is 3 less than 25.

LESSON ORGANIZER

Student Grouping: 4Student Materials� Calendar page (Master 1.6)Assessment: Masters 1.3 Performance Assessment Rubric:Calendar Patterns, 1.4 Unit Summary: Number Patterns

40–50 min

U N I T P R O B L E M

24 Unit 1 • Unit Problem • Student page 26

Part 2

The differences of the numbers in the diagonally oppositecorners of a 2 by 2 square are always 8 or 6. The differences of the numbers in the diagonally oppositecorners of a 3 by 3 square are always 16 or 12.

Reflect on the UnitI learned that number patterns can grow, shrink, or repeat. Youcan write a rule that tells how to create a number pattern. Onceyou know the rule, you can extend the pattern.

I can use pattern rules to help match the day of the week to adate. I can look on a calendar to see on which day the 7th falls.If I go down one square on a calendar, the number goes up by7, so the 14th, the 21st, and the 28th would be the same day ofthe week as the 7th.

Unit 1 • Unit Problem • Student page 27 25


What to Look For

Understanding concepts ✔ Students describe patterns in the

numbers and the positions of thenumbers in a calendar.

Applying procedures✔ Students identify and describe patterns

accurately, using number pattern rulesand position pattern rules.

Communicating✔ Students describe patterns clearly and

use mathematical terminology.

What to Do

Extra Support: Make the problem accessible.

Some students may have difficulty identifying patterns other thanthose outlined in Parts 1 and 2. Point out how other patternsmight be generated by adding rows or columns.

Provide students with a list of numbers that corresponds to apattern in the calendar. For example, list the sums of the rows ina 3 by 3 square. Have students use this list to identify the patternin the calendar.

Recording and ReportingMaster 1.3 Performance Assessment Rubric: Calendar PatternsMaster 1.4 Unit Summary: Number Patterns

Copyright © 2004 Pearson Education Canada Inc. 26

Evaluating Student Learning: Preparing to Report: Unit 1 Number Patterns This unit provides an opportunity to report on the Patterns and Relations strand. Master 1.4 Unit Summary: Number Patterns provides a comprehensive format for recording and summarizing evidence collected.

Here is an example of a completed summary chart for this Unit: Key: 1 = Not Yet Adequate 2 = Adequate 3 = Proficient 4 = Excellent

Strand: Patterns and Relations

Reasoning; Applying concepts

Accuracy of procedures

Problem solving

Communication Overall

Ongoing Observations 2 2 2 2 2 Strategies Toolkit 1 1 Work samples or portfolios; conferences

2 2 1 2 2

Show What You Know 2 2 2 2 2 Unit Test 2 3 2 not assessed 2 Unit Problem Calendar Patterns

2 3 2 2 2

Achievement Level for reporting 2(C)

Recording How to Report Ongoing Observations

Use Master 1.2 Ongoing Observations: Number Patterns to determine the most consistent level achieved in each category. Enter it in the chart. Choose to summarize by achievement category, or simply to enter an overall level. Observations from late in the unit should be most heavily weighted.

Strategies Toolkit (problem solving)

Use PM 1: Inquiry Process Check List with the Strategies Toolkit (Lesson 6). Transfer results to the summary form. Teachers may choose to enter a level in the Problem solving column and/or Communication.

Portfolios or collections of work samples; conferences, or interviews

Use Master 1.1 Unit Rubric: Number Patterns to guide evaluation of collections of work and information gathered in conferences. Teachers may choose to focus particular attention on the Assessment Focus questions. Work from late in the unit should be most heavily weighted.

Show What You Know Master 1.1 Unit Rubric: Number Patterns may be helpful in determining levels of achievement. #5 and 6 provide evidence of Reasoning; Applying concepts; #1, 2, 4, 5, and 6 provide evidence of Accuracy of procedures; #3 provides evidence of Problem solving; all provide evidence of Communication.

Unit Test Master 1.1 Unit Rubric: Number Patterns may be helpful in determining levels of achievement. Part A provides evidence of Accuracy of procedures; Part B provides evidence of Reasoning; Applying concepts; Part C provides evidence of Problem solving; all parts provide evidence of Communication.

Unit performance task Use Master 1.3 Performance Assessment Rubric: Calendar Patterns. The Unit Problem offers a snapshot of students’ achievement. In particular, it shows their ability to synthesize and apply what they have learned.

Student Self-Assessment Note students’ perceptions of their own progress. This may take the form of an oral or written comment, or a self-rating.

Comments Analyse the pattern of achievement to identify strengths and needs. In some cases, specific actions may be planned to support the learner.

Learning Skills

PM 4: Learning Skills Check List Use to record and report throughout a reporting period, rather than for each unit and/or strand.

Ongoing Records

PM 10:Summary Class Records: Strands PM 11: Summary Class Records: Achievement Categories PM 12: Summary Record: Individual Use to record and report evaluations of student achievement over several clusters, a reporting period, or a school year. These can also be used in place of the Unit Summary.


Name Date

Unit Rubric: Number Patterns Not Yet

Adequate Adequate Proficient Excellent


• shows understanding by demonstrating, explaining, and applying: – relationships in

patterns – equivalence in

simple numerical equations

may be unable to demonstrate, explain, or apply: – relationships in



partially able to demonstrate, explain, and use: – relationships in



appropriately demonstrates, explains, and uses: – relationships in



in various contexts, appropriately demonstrates, explains, and uses: – relationships in


simple and more complex numerical equations


• accurately identifies and applies pattern rules; extends number patterns

• finds the missing terms in an equation

• represents numbers to 10 000

often makes major errors in: – identifying pattern

rules – applying pattern

rules – extending number

patterns – finding missing

terms in an equation

– representing numbers to 10 000

makes frequent minor errors in: – identifying pattern






makes few errors in: – identifying pattern






makes no errors in: – identifying pattern






Problem-solving strategies

• uses appropriate strategies to solve and create problems that involve number patterns; includes use of calculator

may be unable to use appropriate strategies to solve and create problems that involve patterns

with limited help, uses some appropriate strategies to solve and create problems that involve patterns; partially successful

uses appropriate strategies to solve and create problems that involve patterns successfully

uses appropriate, often innovative, strategies to solve and create problems that involve patterns successfully

Communication • explains reasoning

and procedures clearly

• uses appropriate

language to describe number patterns (e.g., pattern rule, core, growing pattern, repeating pattern)

unable to explain reasoning and procedures clearly uses few mathematical terms appropriately

partially explains reasoning and procedures clearly uses some mathematical terms appropriately

explains reasoning and procedures clearly uses appropriate mathematical terms

explains reasoning and procedures clearly, precisely, and confidently uses a range of appropriate mathematical terms with precision

Master 1.1


Name Date

Ongoing Observations: Number Patterns The behaviours described under each heading are examples; they are not intended to be an exhaustive list of all that might be observed. More detailed descriptions are provided in each lesson under Assessment for Learning.

STUDENT ACHIEVEMENT: Number Patterns Student Reasoning;

Applying concepts Accuracy of procedures

Problem solving Communication

Explains and applies concepts related to: – identifying and

extending patterns

– equivalence

Accurately identifies and extends a number pattern Finds missing


Uses appropriate strategies to solve and create number pattern problems

Uses appropriate language to describe number patterns Explains reasoning

and procedures

*Use locally or provincially approved levels, symbols, or numeric ratings.

Master 1.2


Name Date

Performance Assessment Rubric: Calendar Patterns

Not Yet

Adequate Adequate Proficient Excellent

Reasoning; Applying concepts • shows

understanding and ability to apply concepts by describing and explaining pattern rules

shows little understanding; may be unable to describe or explain pattern rules

gives a partially appropriate description and explanation of pattern rules; may be vague or incomplete

gives an appropriate and complete description and explanation of pattern rules

gives clear, appropriate, and detailed descriptions and explanations of pattern rules

Accuracy of procedures • identifies and

describes patterns accurately

makes major errors in identifying and describing patterns

makes frequent minor errors in identifying and describing patterns

makes few errors in identifying and describing patterns

makes no errors in identifying and describing patterns

Problem-solving strategies • uses appropriate

strategies to identify and investigate number patterns on a calendar

uses very limited strategies for investigating number patterns on a calendar; may rely only on those described in Steps 1 and 2

uses some appropriate strategies for investigating number patterns on a calendar including one that is not described (Step 3)

uses appropriate and effective strategies for investigating number patterns on a calendar, including at least two that are not described (Step 3)

uses innovative and effective strategies for investigating number patterns on a calendar, including at least two that are not described (Step 3)

Communication • uses appropriate

mathematical terminology (e.g., pattern rule, core, growing pattern, repeating pattern)

• shows thinking clearly

uses few appropriate mathematical terms

unable to show thinking clearly

uses some appropriate mathematical terms

shows thinking with some clarity

uses appropriate mathematical terms

shows thinking clearly

uses a range of appropriate mathematical terms clearly and precisely

shows thinking clearly, precisely, and confidently

Master 1.3


Name Date

Unit Summary: Number Patterns Review assessment records to determine the most consistent achievement levels for the assessments conducted. Some cells may be blank. Overall achievement levels may be recorded in each row, rather than identifying levels for each achievement category. Most Consistent Level of Achievement*

Strand: Patterns and Relations



Problem solving

Communication OVERALL

Ongoing Observations

Strategies Toolkit (Lesson 6)

Work samples or portfolios; conferences

Show What You Know

Unit Test

Unit Problem: Calendar Patterns

Achievement Level for reporting

*Use locally or provincially approved levels, symbols, or numeric ratings. Self-Assessment:

Comments: (Strengths, Needs, Next Steps)

Master 1.4


Name Date

To Parents and Adults at Home … Your child’s class is starting a mathematics unit on number patterns. From traffic to petals on a flower—patterns are how we make sense of the world around us. Patterns occur regularly in mathematics. As children learn to analyse patterns, they develop powerful reasoning skills that will help them make sense of mathematics and science. In this unit, your child will:

• Use charts to display patterns. • Identify the rule for a number pattern. • Extend number patterns. • Create number patterns. • Use patterns to solve problems. • Investigate equations.

Patterns occur in many different forms. Encourage your child to look for patterns around the home, and talk about them. You may find regular repeating patterns—maybe you mark the calendar to remind you of soccer practice every week. Other patterns may be growing or shrinking—the number of cookies remaining in the jar, if it’s “take one at a time.” Here’s a game you can play with your child that creates a growing pattern of words. Growing List Word Game Think of words to describe a cat or other animal. Each player repeats the words said by previous players in the correct order, and adds a new word at the end of the list. The first player starts by saying, for example, “My cat is an adorable cat.” The next player must repeat this but add a new descriptive word. For example, “My cat is an adorable, black cat.” A player is out of the game when he or she cannot repeat the list or fails to provide a new word.

Master 1.5


Name Date

Calendar Page

SATU

RD

AY

FRID

AY

THU

RSD

AY

WED

NES

DA

Y

TUES

DA

Y

MO

ND

AY

SUN

DA

Y

Master 1.6


Name Date

Additional Activity 1: Number Search

Work on your own. Use a hundred chart.

Choose a number less than 50.

Circle the number on your hundred chart. Add 10 to your number. Where did you land on the chart? Start at the same number.

Add 20. Where did you land on the chart?

Start at the same number.

Add 30. Predict where you will land on the chart.

Describe how you can use patterns in a hundred chart to add 10, 20, or 30 to a number. How could you add 40 or 50 to a number? Take It Further: Describe how you can use patterns in the hundred chart to add 9, 18, or 27 to a number.

Master 1.7


Name Date

Additional Activity 2: Patterns to the Nines

Work on your own.

You will need a calculator.

Look at the 3 products in List A.

What patterns do you see?

Use the patterns to predict the next 3 products.

Check your predictions with a calculator.

How can you extend the pattern?

Use a calculator to check.

Repeat the activity for List B.

List A

3 x 9 = 27

3 x 99 = 297

3 x 999 = 2997

3 x 9999 = _____

3 x 99 999 = _____

3 x 999 999 = _____

List B

99 x 12 = 1188

99 x 23 = 2277

99 x 34 = 3366

99 x 45 = ____

99 x 56 = ____

99 x 67 = ____

Take It Further: Use a calculator to find other patterns with nines.

Master 1.8


Name Date

Additional Activity 3: Twenty-One

Work with a partner.

Use 21 Snap Cubes.

The goal is to make the other player remove the last cube.

How to play:

• Join 21 Snap Cubes to form a chain.

• Take turns to remove 1, 2, or 3 Snap Cubes from the chain.

• The player who removes the last cube loses the game.

Play the game several times.

Discuss any patterning strategies you used to win.

Take It Further: Play Twenty-One by removing 2, 3, or 4 cubes each time.

Master 1.9


Name Date

Additional Activity 4: Make It Work

Work with a partner.

Use Pattern Blocks.

Find the value for each block that makes each statement an equation.

The same blocks represent the same number.

Each different block represents a different number.

+ = 20

+ = 15

Take It Further: Create your own number puzzle with Pattern Blocks. Trade puzzles with your partner. Solve your partner’s puzzle.

Master 1.10


Name Date

Step-by-Step 1 Lesson 1, Question 4 Step 1 Fill in the missing numbers on this 5-wide hundred chart. Step 2 Count on by 2s. Shade these numbers with one colour. Step 3 Count on by 5s. Shade these numbers with a second colour. Step 4 Start at 8 and count on by 10s. Shade these numbers with a third colour. Step 5 Use this 10-wide hundred chart. Repeat the number patterns from Steps 2, 3, and 4. Step 6 How are the patterns in the two charts the same? ________________________________

________________________________

________________________________

Step 7 How are the patterns different?

________________________________

________________________________

________________________________

1 2 3 5

6 7 8 9

11 14 15

17 19

21 24

26 27 30

33 35

36 38 39

42 45

48

51 55

56 57 59

62 63

67 69

71 74 75

76 78

82

86 90

96 98 100

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Master 1.11


Name Date

Step-by-Step 2 Lesson 2, Question 4 Step 1 Here is a number pattern: 5, 8, 11, 14, 17

Is this a growing pattern?

________________________________________________________________

How do you know?

________________________________________________________________

What is the pattern rule?

________________________________________________________________

Step 2 Continue this pattern:

5, 15, 25, 35, ___, ___, ___

Is this a growing pattern?

________________________________________________________________


________________________________________________________________

Step 3 Use 5 as a start number. Write your own growing pattern.

5, ___, ___, ___, ___, ___


________________________________________________________________

Master 1.12


Name Date

Step-by-Step 3 Lesson 3, Question 5 Step 1 12 + 21

How are 12 and 21 different?

How are they the same?

___________________________________

Step 2 Continue the pattern. Fill in the blanks. Find each sum.

Step 3 Write a rule for the pattern in the sums.

_______________________________________________________

_______________________________________________________

Step 4 The next statement is 19 + 91. Find the sum. ___________________

Step 5 What would the next statement be? Find the sum. _______________

Step 6 Does the pattern rule in Step 3 always work? ___________________ Explain your answer. _______________________________________________________

_______________________________________________________

12 + 21 = ___ 13 + 31 = ___ 14 + 41 = ___

___ + ___ = ___ ___ + ___ = ___ ___ + ___ = ___ ___ + ___ = ___

Master 1.13


Name Date

Step-by-Step 4 Lesson 4, Question 5 Step 1 Find the sum: 5 + 4 = ___

Step 2 Write two different numbers that have the same sum as in Step 1.

___ + ___ = ___

Step 3 What other pairs of numbers have the same sum as in Step 1?

5 + 4 = ___ + ___ 5 + 4 = ___ + ___

5 + 4 = ___ + ___ 5 + 4 = ___ + ___

Step 4 How do you know when you have found all the pairs of numbers with

the same sum?

_______________________________________________________

_______________________________________________________

_______________________________________________________

Step 5 Repeat Steps 1 to 4 with each pair of numbers.

b) 8 + 7 = ___ c) 10 + 3 = ___ d) 16 + 2 = ___

Master 1.14


Name Date

Step-by-Step 5 Lesson 5, Question 5 Step 1 Find the difference: 18 – 6 = ___

Step 2 Write two different numbers that have the same answer as in Step 1.

___ – ___ = ___

Step 3 What other pairs of numbers have the same answer as in Step 1?

18 – 6 = ___ – ___ 18 – 6 = ___ – ___ 18 – 6 = ___ – ___

How did you find the numbers?

__________________________________________________

__________________________________________________

__________________________________________________

Step 4 Repeat Steps 1 to 3 for 20 – 3 = ___ – ___ .

Master 1.15


Name Date

Unit Test: Unit 1 Number Patterns Part A 1. Use 1-cm grid paper. Make a 6-wide number chart from 1 to 48. This chart will have 8 rows. a) What patterns do you see in the rows? Columns? Diagonals? b) Start at 4. Shade every 5th number.

Describe the position pattern.

c) Write a rule for the number pattern you shaded.

Use the rule to find the next 6 numbers in the pattern.

Master 1.16


Name Date

Unit Test continued 2. a) Find the pattern rule for this pattern.

Write the next 3 terms.

67, 61, 55, 49, ___, ___, ___

b) Look at the pattern in part a. Add 1 to each number. Write the first 5 terms in this pattern. Write the new pattern rule.

c) Look at the pattern in part a.

Use the same start number. Subtract 1 from the number you take away. Write the first 5 terms in this pattern. Write the new pattern rule.

Part B 3. Find all the pairs of numbers that make this statement an equation.

11 + 3 = ___ + ___ Use a pattern rule so you know when you have found all possible ways.

Master 1.16b


Name Date

Unit Test continued 4. Each figure represents a different number.

Find the number that each figure represents.

+ + = 11

8 = +

Explain how you solved the problem. Part C 5. Write as many different patterns as you can that begin with 1, 2, 3. Tell about each pattern you write.

Master 1.16c


Name Date

Sample Answers Unit Test – Master 1.16 Part A 1. a) As you go across a row, each number

increases by 1. As you go down a column, each number

increases by 6. In a diagonal going down and to the right,

each number increases by 7. In a diagonal going down and to the left,

each number increases by 5. b) Start at 4. Go down 1 square, then

1 square left. Then start at 24 and go down 1 square, then 1 square left.

c) Start at 4. Count on by 5. 49, 54, 59, 64, 69, 74 2. a) 43, 37, 31

Rule: Start at 67. Subtract 6 each time. b) 68, 62, 56, 50, 44 Rule: Start at 68. Subtract 6 each time. c) 67, 62, 57, 52, 47 Rule: Start at 67. Subtract 5 each time.

Part B 3. I can do this 7 ways. 11 + 3 = 14, so I begin with 0 + 14. I added 1 to the first number and subtracted 1

from the second number until I reached 6 + 8. 0 + 14; 1 + 13; 2 + 12; 3 + 11; 4 + 10; 5 + 9;

6 + 8 If I continue, I would get the same pairs of

numbers, but in reverse order. 4. Square = 3, triangle = 5

The 1st equation is 2 squares and 1 triangle make 11. The second equation is 1 square and 1 triangle make 8. So, the extra square in the 1st equation is 11 – 8 = 3. So, in the 2nd equation, if the square is 3, then the triangle is 5. I tried 3 and 5 in the first equation and they worked.

Part C 5. 1, 2, 3, 4, 5, 6, … This is a growing pattern that

increases by 1 each time. 1, 2, 3, 1, 2, 3, 1, 2, 3, … This is a repeating

pattern with a core: 1, 2, 3. 1, 2, 3, 3, 2, 1, 1, 2, 3, 3 ,2, 1, … This is a

repeating pattern with a core: 1, 2, 3, 3, 2, 1. 1, 2, 3, 5, 7, 10, 13, 17, 21, … This is a growing

pattern where you start at 1, add 1 two times, then add 2 two times, then add 3 two times, and so on.

Master 1.17


Extra Practice Masters 1.18–1.21 Go to the CD-ROM to access editable versions of these Extra Practice Masters.

Program Authors

Peggy Morrow

Ralph Connelly

Bryn Keyes

Jason Johnston

Steve Thomas

Jeananne Thomas

Angela D’Alessandro

Maggie Martin Connell

Don Jones

Michael Davis

Linden Gray

Sharon Jeroski

Trevor Brown

Linda Edwards

Susan Gordon

Copyright © 2004 Pearson Education Canada Inc., Toronto, Ontario

All Rights Reserved. This publication is protected by copyright,and permission should be obtained from the publisher prior toany prohibited reproduction, storage in a retrieval system, ortransmission in any form or by any means, electronic, mechanical,photocopying, recording, or likewise. For information regardingpermission, write to the Permissions Department.

This book contains recycled product and is acid free.

Printed and bound in Canada

2 3 4 5 6 – TC – 09 08 07 06 05 04

western canadian teacher guide - sd67 (okanagan...

Documents