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Teacher GuideUnit 1: Number Patterns

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1UNIT

Number Patterns

ii Unit 1: Number Patterns

Mathematics Background

What Are the Big Ideas?

• Patterns can be generalized using words and symbols.

• The same pattern can be represented in different ways.

• Patterns exist in numbers and in geometry.

• Patterns can be used to represent situations and to solve problems.

• An equation is a statement that two expressions have the same value.

How Will the Concepts Develop?

Students will build on previous experience with number patternsinvolving one operation to gain experience in patterns with twooperations. They will use recursive patterns that create a new term based on the value of a previous term.

Number patterns are used to find ways to recognize the divisibility of a given number by a given factor.

Students use the number sense reviewed and developed to solveequations by guess and check or inverse operations.

Why Are These Concepts Important?

Students need to recognize and develop strong patterning skills, whetherthese are with numbers or figures. Learning to recognize a pattern and tobe able to describe it recursively or to relate the term value (output) tothe term number (input) will strengthen a student’s ability to deal withalgebraic concepts. The patterns are a concrete way for students todevelop an understanding of a variable and its use in equations andalgebraic expressions.

A strong number sense skill is significant for more advanced work withsolving equations and simplifying algebraic expressions. Recognizing the divisibility of a number is one way of developing this skill.

“Pattern searching is at theheart of many activities,especially in the algebraicreasoning strand. Patterns innumber and in operations playa huge role in helping studentslearn about and master basicfacts and continue to be a majorfactor into the middle and high school years.”– John A. Van de Walle

FOCUS STRANDPatterning and Algebra

SUPPORTING STRANDNumber Sense and Numeration

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Curriculum Overview

Unit 1: Number Patterns iii

Overall Expectations• Describe and represent

relationships in growing andshrinking patterns. (6m55)

• Use variables in simple algebraicexpressions and equations todescribe relationships. (6m56)

Specific Expectations• Make tables of values, for growing

patterns given pattern rules, inwords, then list the ordered pairsand plot the points in the firstquadrant. (6m58)

• Describe pattern rules that generatepatterns by adding or subtracting aconstant, or multiplying or dividingby a constant, to get the next term,then distinguish such pattern rulesfrom pattern rules that describe thegeneral term by referring to theterm number. (6m60)

• Determine a term, given its termnumber, by extending growing and shrinking patterns that aregenerated by adding or subtractinga constant, or multiplying ordividing by a constant, to get the next term. (6m61)

Launch:Crack the Code!

Show What You Know

Unit Problem:Crack the Code!

Lesson 4:Solving Equations

Lesson 5: Exploring Integers

Lesson 6:Strategies Toolkit

Cluster 1 — Developing Number Patterns

Cluster 2 — Applying Number Patterns

Overall Expectations• Describe and represent

relationships in growing andshrinking patterns. (6m55)

• Use variables in simple algebraicexpressions and equations todescribe relationships. (6m56)

Specific Expectations• Demonstrate an understanding of

different ways in which variablesare used. (6m63)

• Solve problems that use two orthree symbols or letters as variablesto represent different unknownquantities. (6m65)

• Determine the solution to a simpleequation with one variable, throughinvestigation using a variety oftools and strategies. (6m66)

Lesson 1: Input/Output Machines

Lesson 2: Number Patterns

Lesson 3:Patterns in Division

The codes refer to the 2005 Revised Curriculum.

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Curriculum across the Grades

iv Unit 1: Number Patterns

Materials for This Unit

Student will need calculators to extend some patterns. You may wish toprovide students with manipulatives to build models associated with thenumber patterns. Lesson 4 Explore requires numbered tetrahedrons andtwo different coloured counters. A large thermometer will help introduceinteger concepts in Lesson 5.

Grade 5

Students create, identify,and extend patterns innumber and geometry.They represent a numberpattern presented in atable of values bybuilding a model of the pattern.

Students make a table of values for a recursivepattern.

Students exploregrowing and shrinkingpatterns and makepredictions.

Students use a variety of tools and strategies to solve simple equations involvingaddition, subtraction,multiplication, ordivision and one- or two-digit numbers.

Grade 6

Students describerecursive pattern rules.They distinguishrecursive pattern rulesfrom pattern rules thatdescribe the generalterm by referring to the term number.

Students extendrecursive patterns todetermine a term, givenits term number.

Students solve problemsthat use two or threesymbols or letters asvariables to representdifferent unknownquantities.

Students use a variety oftools and strategies tosolve simple equationswith one variable.

Grade 7

Students use a variety oftools to represent lineargrowing patternsdifferent ways.

Students use concretematerials to investigatelinear growing patternsand to make predictions.

Students use algebraicexpressions with oneoperation to representthe general term of alinear growing pattern.They evaluate algebraicexpressions and relatethis to using the generalterm to determine thevalue of a term.

Students comparerecursive pattern ruleswith pattern rules thatuse the term number to describe the general term.

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Additional Activities

Unit 1: Number Patterns v

Code BreakerFor Extension (Appropriate for use after Lesson 1)Materials: Code Breaker (Master 1.9)

The work students do: Students work in pairs.They are given part of a codebook. Studentsdetermine the pattern used to change the letter valuesto new ones. Then students decode the message.

Take It Further: Challenge students to make uptheir own codebook based on some operations onthe values of the letters of the alphabet.

Logical/Mathematical

Partner Activity

Starting PointFor Extra Practice (Appropriate for use after Lesson 3)Materials: Starting Point (Master 1.11)

The work students do: Students work alone. Thestudent writes a pattern rule with two operations. Forexample: multiply by 2, then add 1. The student thenchooses 3 different starting numbers and applies thepattern rule to each number to write the first 10 termsfor each pattern. Then the student compares theterms in each pattern and describes any similarities.Students extend their patterns to check their ideas.

Take It Further: Students trade patterns with aclassmate and identify the pattern rules.


Individual Activity

It’s Getting Smaller!For Extra Practice (Appropriate for use after Lesson 2)Materials: It’s Getting Smaller! (Master 1.10),calculators

The work students do: Students work alone. The student begins with 1 000 000 and makes up a pattern rule for a shrinking pattern. The studentwrites the first 7 terms of the pattern. Then thestudent trades patterns with a classmate. Each writesthe pattern rule for the classmate’s pattern and thenwrites the next 3 terms.

Take It Further: Have students explore the totalnumber of terms in their pattern. All the terms mustbe 0 or greater.


Individual Activity

And the Number Is . . .For Extra Practice (Appropriate for use after Lesson 4)Materials: And the Number Is … (Master 1.12)

The work students do: Students work in pairs.They take turns writing a simple equation with oneunknown number and one operation. Students havetheir partners solve the equations.

Take It Further: Challenge students to write andsolve equations with more than one missing numberand/or more than one operation.


Partner Activity

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Planning for Unit 1

vi The right to reproduce or modify this page is restricted to purchasing schools.This page may have been modified from its original. Copyright © 2006 Pearson Education Canada Inc.

Planning for Instruction

Lesson Time Materials Program Support

Suggested Unit time: 2 weeks

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Unit 1: Number Patterns vii

Purpose Tools and Process Recording and Reporting

Planning for Assessment

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L A U N C H

Crack the Code!

2 Unit 1 • Launch • Student page 4

LESSON ORGANIZER

Curriculum Focus: Activate prior learning about number patterns.Vocabulary: number patterns, codes

ASSUMED PRIOR KNOWLEDGE

Students can extend a pattern from a pattern rule.Students can recognize and describe different patterns.

✓✓

10–15 min

ACTIVATE PRIOR LEARNING

Have students read the Unit Launch. Invite anystudents who have used Morse code to sharesome of their experiences. Ask:

• What is a code? (A code is a series of symbols, characters, or signalsused to communicate.)

Discuss the first question in the Student Book.You may wish to record students’ answers onthe board or on chart paper. (One reason to code is to communicate with machines.Another reason to code is to keep the messages secret.Braille and sign language are special codes that allowblind people to read and deaf people to communicate.)

Discuss the second question in the Student Book.(The least number, 0, is represented with 5 dashes andno dots. As the numbers increase by 1, the number ofdashes decreases by 1. The dashes are replaced withleading dots, until the code represents 5, which is

5 dots. As the numbers continue to increase, thenumber of dots decreases by 1. Now the dots arereplaced with leading dashes.)

Discuss the third question in the Student Book.(I think the number 503 would be written as 5 dots, 5 dashes, then 3 dots and 2 dashes: ••••• ----- •••--.)

Tell students that, in this unit, they will learnmore about recognizing and describing numberpatterns. They will use patterns to explore avariety of problems.

They will use patterns to create a binary code to produce a secret message.

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Unit 1 • Launch • Student page 5 3

LITERATURE CONNECTIONS FOR THE UNIT

Codes, Ciphers and Secret Writing by Martin Gardner.Dover Publications, 1984.ISBN 0486247619This book is about different kinds of coding and ciphers and their related instruments used throughout history.

DIAGNOSTIC ASSESSMENT

What to Look For

✔ Students can extenda pattern from apattern rule.

✔ Students canrecognize anddescribe differentpatterns.

What to Do

Extra Support:Use simple patterns, such as 1, 3, 5, 7, … or 2, 4, 6, 8, …, and have students describe the rule. Have students use the rule to extend each patternby writing the next 3 terms.

Students who have difficulty describing number patterns should be encouragedto examine how adjacent terms change. For example, they can find the differencesbetween consecutive terms or the quotients of consecutive terms and look forpatterns in the differences or quotients.


L E S S O N 1

Input/Output Machines

4 Unit 1 • Lesson 1 • Student page 6

LESSON ORGANIZER

Curriculum Focus: Explore Input/Output machines with twooperations. (6m58, 6m60, 6m61)Student Materials Optional

� Step-by-Step 1 (Master 1.13)� Extra Practice 1 (Master 1.20)

Vocabulary: Input/Output machine; pattern ruleAssessment: Master 1.2 Ongoing Observations: Number Patterns

Key Math Learnings1. A pattern rule can be illustrated with an Input/Output machine.2. The input and output can be recorded in a table. This table

can be used to illustrate two types of pattern rules.

40–50 min

B E F O R E G e t S t a r t e d

Discuss the Input/Output machine shown atthe top of page 6 in the Student Book. Make a table on the board. Have students find theoutput for input numbers from 1 to 5. Ask:

• How can you find the output when the input is 9? (I can find the output two ways. First, I can multiply9 by 7 to get 63. Second, when the input numbersare consecutive numbers beginning with 1, the inputnumbers are the term numbers. I can continue thepattern in the output numbers to find the value ofthe ninth term.)

Present Explore. Ensure students understandthat the output from the first part of themachine is the input for the second part.

D U R I N G E x p l o r e

Ongoing Assessment: Observe and Listen

Ask questions, such as:

• Which operations and numbers did you choose? (I chose add 2, then multiply by 4.)

• How do you find the output when the input is 3? (I add 2 to 3 to get 5, then multiply 5 by 4 to get 20.)

• What is a pattern rule for the output numbers? (Start at 12. Multiply by 4 each time.)

• What other pattern rule can you find? (The output numbers are multiples of 4, starting at 12.)

63


Unit 1 • Lesson 1 • Student page 7 5

In the second example, to discover the numbersand operations, identify these patterns:

• The input numbers increase by 1.• The output numbers increase by 3.

This shows that the input numbers aremultiplied by 3. The second part of the rule isdiscovered by comparing the results of triplingthe input numbers with the output numbers. Ask:

• Could the output be 24? How do you know? (No; I counted on by 3s in the output column,beginning at 13: 16, 19, 22, 25, …. Since I did notget 24, the output in this pattern cannot be 24.)

• What other reason is there that the outputcould not be 24? (Since we multiply by 3 and subtract 2, each output is 1more than a multiple of 3. But 24 is a multiple of 3,so it cannot be an output number in this pattern.)

REACHING ALL LEARNERS

Early FinishersHave students use their Input/Output machine from Explore.They predict what happens to the output numbers if theyreverse the order of operations and numbers in theInput/Output machine. Students find the outputs to checktheir prediction.

Common Misconceptions➤ Students do not know how to identify a pattern rule for

a table of values.How to Help: Have students practise with patterns that use multiplication, then addition or subtraction. Studentscalculate the differences of consecutive terms in the outputcolumn. These differences should be constant. This constant is the number each input is multiplied by. Students can thenmultiply each input number by this constant and compare theproduct with the output. This difference is the number that isadded or subtracted in the second part of the machine.

A F T E R C o n n e c t

Invite students to record their input/output tableson the board. Have other students identify thepattern rules and extend the patterns.

Present the examples in Connect. After you reviewthe first example, ask:

• Could the output be 15? How do you know? (No; the input is multiplied by 2, which results in an even number, then we add 6, and the result isstill an even number, so the output is always even.)

• Which input number has an output of 20?How do you know? (7; I continued the table. The input numbers were 5, 6, 7, and 8 and the corresponding outputs were16, 18, 20, and 22. So, an input of 7 has an outputof 20.)


Making ConnectionsA coded message is used to keep information from everyoneexcept those to whom it is sent. Rather than just replacing eachletter of the message with a number, the encoder will use one ortwo operations on these numbers to make the code harder todetermine. Students could investigate these types of codes andhow they relate to the security of countries and businesses.

Sample Answers1. a)

b)

2. a) The Input/Output machines use the same numbers andoperations but they perform them in a different order.

b) The outputs from the machine in part b are 5 greater than the outputs from the machine in part a.

c) There is only one output number for each input number.When you multiply a given number by a certain numberthere can only be one answer. The same is true for addition.Therefore, you can only get one output number for eachinput number.


P r a c t i c eBefore students begin Practice, point outquestions 3 and 4. They have input numbersdifferent from consecutive whole numbersbeginning with 1.

Assessment Focus: Question 5

Students should recognize that, since the outputnumbers increase by 4 each time, this suggeststhat � 4 is one operation and number. Studentscould use guess and check with logicalreasoning to find that the second operation and number is � 1. Alternatively, studentsmight identify the output numbers as 1 morethan a multiple of 4.

Students who need extra support to complete theAssessment Focus questions may benefit fromusing the Step-by-Step masters (Masters 1.13 to 1.17).

818283848

Input

Output

1

7

2

13

3

19

4

25

5

31

Input

Output

1

12

2

18

3

24

4

30

5

36


4. The last two output numbers are incorrect because they donot fit the pattern rule. 42 � 6 � 5 � 7 � 5 � 1254 � 6 � 5 � 9 � 5 � 14

5. a) � 4, � 1b)

6. a) � 5, � 3b)

REFLECT: To find the numbers and operations used in anInput/Output machine, I check that the input numbers are in order. I subtract each output number from the numberabove it to see if the differences are equal. If they are, thenthis number is what I multiply each input by. I multiply aninput number by this factor and decide if any number needsto be added or subtracted to get the output number. I checkthese numbers and operations with other inputs and outputs.


ASSESSMENT FOR LEARNING

Recording and ReportingMaster 1.2 OngoingObservations: Number Patterns

What to Look For

Knowledge and Understanding✔ Students can use an Input/Output

machine and given input numbers to generate output numbers.

Thinking✔ Students can determine the contents

of an Input/Output machine, given a table of values.

Application✔ Students can identify erroneous data in

a table of input and output numbers.

What to Do

Extra Support: Have students use a calculator to find thedifferences in consecutive output numbers.Students can use Step-by-Step 1 (Master 1.13) to completequestion 4.

Extra Practice: Students can complete Extra Practice 1(Master 1.20).

Extension: Students can complete the Additional ActivityCode Breaker (Master 1.9).

Numbers Every DayStudents should test their strategies on a calculator.225 � 14 � (224 � 1) � 14 or (226 � 1) � 1410.5 � 3.2 � 10.4 � 0.1 � 3.2 or 10.6 � 0.1 � 3.2

Input

Output

7

29

9

37

11

45

Input

Output

70

17

90

21

110

25


L E S S O N 2

Number Patterns


see how each term related to the one before it. Imultiplied each term by 2, and got 2, 10, 26, 58,122, …. I noticed that if I then added 3 to eachnumber, I got the pattern. So, the pattern rule is:Start at 1. Multiply by 2, then add 3 each time.)

• How did you get the rule for the secondpattern?(I tried the same strategy as the first pattern. Inoticed that each term, after the first, is 1 more than a multiple of 3: 3, 9 � 1, 30 � 1, 93 � 1,282 � 1, …. I multiply each term by 3, then add 1to get the next term. So, the pattern rule is: Start at 3.Multiply by 3, then add 1 each time.)

• What is the rule for the third pattern? (Start at 300. Subtract 2 each time.)

• What type of pattern is this? How do you know?(It is a shrinking pattern; the terms get smaller.)

LESSON ORGANIZER

Curriculum Focus: Identify, extend, and create number patterns. (6m60, 6m61)Student Materials Optional� calculators � Step-by-Step 2 (Master 1.14)

� Extra Practice 1 (Master 1.20)Vocabulary: recursive patternAssessment: Master 1.2 Ongoing Observations: Number Patterns

Key Math LearningA number pattern may be described recursively. That is, thepattern rule is applied successively to each term, rather than to input numbers.

40–50 min

290, 288, 286850, 2551, 7653

125, 253, 509


Discuss the number patterns preceding Explore.Students should identify the first pattern as a shrinking pattern. The second pattern is analternating pattern. The third pattern is analternating growing pattern. Have studentsextend each pattern for several terms.

Present Explore. Encourage students to try to findmore than one way to describe each pattern.




• How did you find the pattern rule for the first pattern? (I subtracted terms and got 4, 8, 16, 32, so I knewthe rule was not multiplying each input number by the same number. I used guess and check to

Numbers Every DayFind “friendly” numbers for easier subtraction. Adjust the onesand tens digits so they are the same, then readjust the answer.


Making ConnectionsMath Link: Finding the distance a thunderstorm is away is justone example of using a pattern rule for approximations. Patterningrules can help estimate the amount of tax payable on an item orconvert between metric and imperial units.


P r a c t i c eIn question 4b, some students may consider thepattern is not recursive because it is the squarenumbers. The pattern is recursive since eachterm is created by adding the next odd numberto the preceding term.


Students’ answers may vary from the simplestpattern: “Start at 4. Add 3 each time;” to morecomplicated patterns, such as: “Start at 4.Multiply by 8, then subtract 25 each time.”Students might try to “outdo” each other byusing very large numbers.


Early FinishersHave students find different rules for the patterns in Practice.For example, the pattern rule in question 3c can be writtendifferent ways.

Common Misconceptions➤ Students have difficulty identifying recursive patterns

when the preceding term is multiplied by a constantbefore another constant is added.

How to Help: Have students generate simple patterns first.Begin at 1, multiply each term by 2, then add 1 to generate1, 3, 7, 15, 31, …. Then have students begin at 1, multiplyeach term by 2, then add 2 to generate 1, 4, 10, 22, ….Continue with similar patterns, multiplying, then adding orsubtracting small numbers each time.

ESL StrategiesStudents may have difficulty writing pattern rules. Allow themto describe the rules orally or visually to you or to anotherstudent during Practice exercises. Help students relate thecorrect word to the operation they have said or drawn. Havestudents highlight this in a notebook or journal that studentscan use during assessments.


Review students’ patterns and the strategiestheir classmates used to identify the patterns.Use the patterns in the text and the onesstudents created to review recursive patterns.Each term after the first term is created byapplying the same rule to the preceding term.The first pattern preceding Explore and thepatterns in Explore are recursive.

Show students another way to describe the firstnumber pattern.1 � 1 � 5 � 4; 6 � 2 � 5 � 4; 11 � 3 � 5 � 4;16 � 4 � 5 � 4; 21 � 5 � 5 � 4To get any term, multiply the term number by 5,then subtract 4. For example, the 20th term is20 � 5 � 4 � 96.


Sample Answers1. a) 10, 15, 25, 45, 85,165

b) 10, 48, 238, 1188, 5938, 29 688c) 10, 10, 10, 10, 10, 10d) 10, 40, 190, 940, 4690, 23 440

2. All patterns have the same first term. The patterns in parts a and b and in parts c and d have the same operations, but the constants involved are reversed.

3. a) Multiply by 3, then subtract 2 each time; 244, 730; 19 684b) Start at 250. Subtract 20 each time; 150, 130; 70c) Start at 3. Multiply by 2, then subtract 2 each time;

34, 66; 514d) Start at 2. Multiply by 2, then add 1 each time;

95, 191; 15355. 4, 7, 13, 25, 49 Multiply by 2, then subtract 1 each time.

4, 7, 16, 43, 124 Multiply by 3, then subtract 5 each time.4, 7, 10, 13, 16 Add 3 each time.4, 7, 28, 175, 1204 Subtract 3, then multiply by 7 each time.

6. a) 54; Start at 5. Multiply by 2, then add 2 each time; 894b) 243; Start at 300. Multiply by 3, then subtract 2 each

time; 167

REFLECT: To find the pattern rule for the pattern from question 4a,I subtracted consecutive terms: 9 � 4 � 5; 19 � 9 � 10; ….The difference doubles each time, so I think I need to multiplyby 2. I checked: 4 � 2 � 8; 9 � 2 � 18; …. Each time theproduct is 1 less than it should be. So, I need to add 1 eachtime. The pattern rule is: Start at 4. Multiply by 2, then add 1 each time.


Yes

Yes

YesYes

� 508

� 108� 88

� 55


Recording and ReportingMaster 1.2 Ongoing Observations:Number Patterns

What to Look For

Knowledge and Understanding✔ Students understand how a recursive

pattern is formed.

Thinking✔ Students can identify the pattern

rule for a given recursive pattern.

Application✔ Students can use a pattern rule to

write a recursive pattern.

Communication✔ Students can explain how they

identify recursive pattern rules.

What to Do

Extra Support: Have students generate recursive patternsfirst by adding a small number each time. Then proceed tomultiplying by 2, and adding 1 each time. Gradually increasethe size of the numbers used to generate the patterns, andinclude subtraction with multiplication.Students can use Step-by-Step 2 (Master 1.14) to completequestion 5.

Extra Practice: Students do the Additional Activity It’s GettingSmaller! (Master 1.10).Students can complete Extra Practice 1 (Master 1.20).

Extension: Have students generate shrinking recursivepatterns. Ask them to predict when the pattern “breaks down,”then check their predictions.


L E S S O N 3

Patterns in Division


LESSON ORGANIZER

Lesson Focus: Use patterns to explore divisibility rules.Student Materials Optional� calculators � Step-by-Step 3 (Master 1.15)

� Extra Practice 2 (Master 1.21)Vocabulary: multiple, divisibility rulesAssessment: Master 1.2 Ongoing Observations: Number Patterns

Key Math LearningPatterns in numbers with a particular factor can be summarizedas divisibility rules.

40–50 min

64, 8200, 4164, 72, 9974They are all even.


The ability to quickly recognize divisibilitywith numbers other than 2 and 5 provides a foundation for future success.

Write the numbers at the top of page 13 on the board. Allow students to take turns circlingthe numbers that are divisible by 2. State adivisibility rule for 2.

Present Explore. Ensure students understand the problem.




• How do you know when a number is divisibleby 3? (When I divide a number by 3, there is no remainder.)

• What do you notice about the sums of the digits? (All the sums are multiples of 3.)

• How can you check if a number is divisibleby 3, without dividing? (I add the digits of the number. If the sum of thedigits is divisible by 3, then the number is divisibleby 3.)

• What do you notice about the multiples of 4? (In all the numbers, the tens and ones digits togetherare divisible by 4.)

Curr i cu lum FocusIn this lesson, students use patterns to explore divisibilityrules. While this material is not a curriculum outcome,this lesson is recommended as a prerequisite foridentifying composite numbers and prime numbers.


Sample Answers2. It has 10 as a factor.3. 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126,

135, 144, 153, 162, 171, 180; For any number divisible by9, the sum of the digits is divisible by 9.

5. 732, 5028, 102 330; I chose each number by making sure the ones digit was even and the sum of the digits was divisible by 3.

6. a) If a number is divisible by 4, then it is also divisible by 2.Any multiple of 8 is also a multiple of 4.


After discussing the divisibility rule for 6, ask: • If a number is divisible by 3 and by 4, which

other number is it divisible by? (12)• What is a divisibility rule for 12?

(If the sum of the digits is a multiple of 3, and if the last 2 digits are divisible by 4, then the number is divisible by 12.)

P r a c t i c eStudents should use the divisibility rules toanswer the questions. A calculator can be usedto check the answers.


Students should realize that, since 4 is a factorof 8, any number that is divisible by 8 is alsodivisible by 4. So, the loop for 8 is contained inthe loop for 4. Also, since 2 is a factor of 4, theloop for 4 is contained in the loop for 2.


Early FinishersHave students write or find divisibility rules for 15, 16, 18,and 20.

Common Misconceptions➤ Students have difficulty remembering the divisibility rule

for each number.How to Help: Allow students to use a calculator to findwhether a number is a multiple.

• How can you check if a number is divisibleby 4? (If the number has 2 digits, I have to divide. If thenumber has more than 2 digits, I divide the numberformed by the tens and ones digits by 4. If there is no remainder, I know the number is divisible by 4.)

Listen for the language of mathematics asstudents discuss the patterns: “dividingexactly,” “repeating decimals,” and other termsthat indicate an understanding of divisibilityand patterns.

Watch for students who try to use the samedivisibility rule for 4 as for 3. The sum of thedigits is not helpful for divisibility by 4.


Review the divisibility rules for 5, 6, and 8.Divisibility rules for 7 and 9 are covered in Practice.


b, c)

7. Test 777, which is divisible by 7: 77 � 2 � 7 � 77 � 14 � 63;63 � 7 � 9, so the rule works. Test 121, which is not divisibleby 7: 12 � 2 � 1 � 12 � 2 � 10; 10 is not divisible by 7,so the rule works. The rule appears to work.

REFLECT: I chose 2520. It ends in 0, which is an even number,so it is divisible by 2, by 5, and by 10. Add the digits. Thesum of the digits is divisible by 9, so the number is divisibleby 3 and by 9. Since the number is even, it is also divisibleby 6. The last 3 digits form a number that is divisible by 8,so the number is divisible by 4 and by 8. Using the rule fromquestion 7, 252 � 2 � 0 � 252; 25 � 2 � 2 � 21; since 21is divisible by 7, the number is divisible by 7.


Numbers Every DayEncourage students to explain the strategies they used to findeach product.

Divisible by 2

Divisible by 4

Divisible by 81046 3662

322 7894

541900 460

28382

10081784

2241088

� 7500

� 4200� 273

� 6000

3, 43

43, 4

33

66

96, 9

9Neither



What to Look For

Knowledge and Understanding✔ Students understand the divisibility

rules for 2 to 10, excluding 7.

Thinking✔ Students can solve problems related

to divisibility rules.

Communication✔ Students can explain how they

identify whether a given number is a multiple of a certain number.

Application✔ Students can apply the divisibility rules

to identify multiples.

What to Do

Extra Support: Students can use Step-by-Step 3 (Master 1.15) to complete question 6.

Extra Practice: Students can do the Additional ActivityStarting Point (Master 1.11).Students can complete Extra Practice 2 (Master 1.21).

Extension: Challenge students to find a divisibility rule for 11.Have students try to find as many different divisibility rules for11 as they can.(Add alternate digits, for example, the 1st, 3rd, 5th, and so on,then the 2nd, 4th, 6th, and so on. Subtract the sums. If the resultis 0 or 11, then the number is divisible by 11.)


L E S S O N 4

Solving Equations


LESSON ORGANIZER

Curriculum Focus: Find the value of a missing number in an equation. (6m63, 6m65, 6m66)Student Materials Optional� tetrahedrons � Step-by-Step 4 (Master 1.16)� 2 colours of counters � Extra Practice 2 (Master 1.21)� What’s My Number

Game Board (Master 1.6)Vocabulary: equation, inverse operationAssessment: Master 1.2 Ongoing Observations: Number Patterns

Key Math Learnings1. A simple equation can be solved using a variety of

strategies, including guess and check.2. The solution to an equation is the number that makes the

two sides of the equation equal.

40–50 min

NoYes

YesNo

YesYes


Write the statements on page 16 of the StudentBook on the board. Invite volunteers to circlethe statements that are equations.

Present Explore. Read the game instructions as aclass. Use the following example to illustratewhat students are expected to do: Find themissing number to make this equation true: � � 8 � 13Do not solve the equation but have studentsthink of ways they might find the missingnumber.

After you copy Master 1.6, cut the paper toseparate the game boards. Give each pair ofstudents one game board.



As students play the game, listen and watch fordifferent strategies that students use to find themissing number. Ask questions, such as:

• How do you find the missing number whenthe operation is addition? (I subtract the number added to the missing numberfrom the larger number on the other side of theequal sign.)

• How do you find the missing number whenthe operation is subtraction? (I use guess and check with logical reasoning, until I reach the answer.)

• How do you find the missing number whenthe operation is multiplication? (I divide the number multiplied by the missingnumber into the larger number on the other side of the equal sign.)



one way to solve an equation, but there is only one correct answer. When students useguess and check, encourage them to estimatebefore guessing.

P r a c t i c eIn questions 2 and 5, make sure studentsunderstand that the square and the trianglerepresent different numbers.

Encourage students to draw diagrams forquestions 4 and 6 as part of their solutions. In questions 3 and 5, students need to completethe operations where both numbers are knownbefore finding the missing number(s).


Students need to simplify each equation first.Because there is more than one number missing,there will be more than one possible answer.


Early FinishersHave students solve the equations on the game board theydid not use in Explore.

Common Misconceptions➤ Students have difficulty making reasonable guesses for the

missing number(s) when they use guess and check to solvean equation.

How to Help: Have students try changing the numbers in an equation to “friendly” numbers or rounding the numberson each side of the equation to make an easier problem tosolve first.

• How do you find the missing number whenthe operation is division? (I multiply the number divided into the missingnumber with the number on the other side of theequal sign.)


Have volunteers take turns to show how theysolved an equation on the game board. Ask:

• How did you solve the equations? (I used guess and check until I got the correct answer.I learned from each guess and check to change myguess higher or lower. Sometimes I used the inverseoperation. For example, if the equation involvedmultiplying, I used division to solve it.)

Review the equations from Connect. Askstudents which other strategies they could use to solve these equations. It is important for students to realize there may be more than


Sample Answers2. The pairs of whole numbers that have the product 36 are:

1 � 36, 2 � 18, 3 � 12, 4 � 9, or the reverse of each ofthese pairs. I cannot use 6 � 6 since the different figuresrepresent different numbers.

5. a) Any pair of numbers with a difference of 3b) 1 � 7, 2 � 6, 3 � 5, 6 � 2, 7 � 1, 8 � 0, 0 � 8c) 1 � 48, 2 � 24, 3 � 16, 4 � 12, 6 � 8, 8 � 6, 12 � 4,

16 � 3, 24 � 2, 48 � 1d) Any pair of numbers with a difference of 2

REFLECT: If the missing number is in an addition equation, I subtract. For example, 27 � � � 56; � � 56 � 27, so � � 29If the missing number is subtracted in an equation, I subtract.For example, 27 � � � 5; � � 27 � 5, so � � 22If the missing number has a number subtracted from it, I add.For example, � � 27 � 36; � � 36 � 27, so � � 63If the missing number is multiplied, I divide.For example, 11 � � � 297; � � 297 � 11, so � � 27If the missing number is the divisor, I divide.For example, 420 � � � 12; � � 420 � 12, so � � 35If the missing number is the dividend, I multiply.For example, � � 12 � 35; � � 12 � 35, so � � 420


Numbers Every DayTo add mentally, change the question into the sum of “friendly” numbers.



What to Look For

Knowledge and Understanding✔ Students understand more than one

method for solving an equation.

Thinking✔ Students can use an equation to solve

a problem.

Application✔ Students can solve simple equations

with one variable.

What to Do

Extra Support: Allow students to use a calculator with guessand check if their numeracy skills are weak. Students can useStep-by-Step 4 (Master 1.16) to complete question 5.

Extra Practice: Students can do the Additional Activity Andthe Number Is … (Master 1.12).Students can complete Extra Practice 2 (Master 1.21).

Extension: Have students create a pair of equations that eachhave two different missing numbers, but both equations containthe same missing numbers. For example: � � � � 24 and � � � � 11Students trade equations with a classmate and solve theirclassmate’s equations.

7 � 13 � � � 30

� 3314

� 9326

10 m

4159

9126

11 m

16 � � � 176

444

4253

14255


W O R L D O F W O R K

Fraud Investigator

Unit 1 • World of Work • Student page 19 17

Many different careers involve collectinginformation and analysing it for patterns ortrends. Discuss other careers that involve usingpatterns on the job.

Credit card companies keep track of the typesand places of spending by card holders. Anychanges in these spending habits may signalpossible theft of a credit card.

Police look for patterns in how certain types ofcrimes are committed to help identify criminals.

Statisticians look for patterns in the data they collect to help predict future events. The statistician could be a biologist, anenvironmentalist, a geologist, or an economist.


L E S S O N 5

Exploring Integers


LESSON ORGANIZER

Lesson Focus: Use integers to describe quantities with sizeand direction.Student Materials Optional� calculators � Step-by-Step 5 (Master 1.17)

� Extra Practice 3 (Master 1.22)Vocabulary: integersAssessment: Master 1.2 Ongoing Observations: Number Patterns

Key Math Learnings1. Integers are numbers that can be created from

subtraction patterns.2. Integers appear in many areas of the real world.

optional

Curr i cu lum FocusIn this lesson, students use integers to describe quantities with size and direction. This material is not required by the curriculum until grade 7.


Discuss the introductory temperature examples.

Present Explore. Encourage students to use theirpatterning skills to complete the activity in Explore.



Watch for students who want to subtract 1 eachtime in the temperature column. Since a 150-maltitude increase results in a 1°C temperaturedecrease, then for double the change in altitudewe should double the change in temperature.


• What is the pattern rule for the altitudes? The temperatures? (Start at 1000. Add 300 eachtime. Start at 6. Subtract 2 each time.)

• How do you find each temperature? (I subtract 2 from 6 to get 4; then I subtract 2 from 4to get 2, and so on.)


Have a volunteer draw the completed chart onthe board.

Altitude (m) Temperature (°C)

6

4

2

0

�2

�4

1000

1300

1600

1900

2200

2500


Numbers Every DayInvite students to share the strategies they used to find the numbers.


Ask:

• How did you know what the temperature at2200 m was? (I continued the pattern. I kept subtracting 2. I thoughtabout a thermometer; 2°C below 0°C is �2°C.)

Review the thermometer and number line inConnect. Discuss other places students have usednegative numbers.

P r a c t i c eStudents may write a � sign in front of apositive number, but it is not necessary.


Students may create an altitude/temperaturechart as in Explore to estimate the temperature at the top of the mountain. Students coulddetermine the number of 150-m changes from566 m to 2244 m and subtract this number from23°C to represent the drop of 1°C every 150 m.They could apply the same strategy to thestarting temperature of 9°C.


Early FinishersHave students draw number lines, then show each integer inquestion 2 on a number line.

Common Misconceptions➤ Students have difficulty understanding why a scenario that

involves a debt or spending money can be represented bya negative integer.

How to Help: Have students think about opposites. You mayhave $10 or you may owe $10. You may earn $25 or youmay spend $25. For each situation, have students draw anarrow on a number line and label it with a positive or anegative integer.

ESL StrategiesStudents who are unfamiliar with the language in Exploremay need some help understanding words such as “altitude”and “sea level.” Be sure to pair these students with studentswho can help them understand the language.

1 � 2 � 3� 1 � 2 � 3� 6


Sample Answers2. a) I earned $125 mowing lawns.

b) I spent $22 of the money I earned.c) It is 900 m below sea level.d) I flew at an altitude of 42 000 m above sea level.e) I walked 4 steps forward.

3. a) The temperature is about 12°C at the top of the mountain.I know the temperature decreases 2°C for each increase of 300 m. I made a chart as I did in Explore.

b) If the temperature is 9°C at the bottom, then the temperatureat the top is below 0°C because a drop of 12°C will go below0°C. This can be illustrated using a chart similar to that inpart a. The temperature at the top of the mountain will beabout �2°C.

REFLECT: A height above sea level is represented by a positiveinteger. For example, a plane flies at 10 000 m. A depth belowsea level is represented by a negative integer. For example, asubmarine travels at �50 m.


18°C24°C

15°C27°C

�150

11 000�400

Temperature (°C)

23

21

19

17

15

13

11

Altitude (m)

566

866

1166

1466

1766

2066

2366



What to Look For

Knowledge and Understanding✔ Students understand an integer

represents a quantity with both size and direction.

Application✔ Students can use patterns to

list integers.✔ Students can represent integers

on a number line.

What to Do

Extra Support: Have students use a real or modelthermometer to see the transition from positive numbers to negative numbers in Explore.Students use Step-by-Step 5 (Master 1.17) to complete question 3.

Extra Practice: Students can work with a partner to writescenarios similar to those in question 1. One student writes ascenario, then the other student writes an integer to illustrate it.Students can complete Extra Practice 3 (Master 1.22).

Extension: Have students make up a problem similar to thatin Explore or question 3. They could use an atlas to find theheight of a mountain. Students trade problems with aclassmate, then solve each other’s problem.


G A M E

Equation Baseball

Unit 1 • Game • Student page 23 21

LESSON ORGANIZER

Student Materials� number cube, labelled 1 to 6� Equation Baseball Game Cards (Master 1.7)� Equation Baseball Game Boards (Master 1.8)� game pieces

optional


Organize students into groups of 4.

Invite volunteers to read through the gamedirections. To ensure students understand how to play the game, have volunteers model a round of the game with you.

Distribute game boards and sets of game cards.Ensure students understand that the circlednumber tells how many bases you get to moveif you solve the expression correctly.

D U R I N G G a m e

As students play the game, circulate and notewhich expressions give the most difficulty. Besure to discuss strategies for solving theseexpressions after the game has concluded.

Allow students to check the solution with acalculator if any disputes arise.

A F T E R

Invite students to discuss their experienceplaying the game. Have them suggest revisionsor variations.


L E S S O N 6

Strategies Toolkit


LESSON ORGANIZER

Lesson Focus: Get unstuck. (6m1, 6m7)

Key Math LearningA variety of strategies can be used to get unstuck when solvinga problem.

40–50 min

� � 3� � 5� � 6


Present the problem in Explore. Have studentswork in pairs to solve the problem. Theyshould record the strategies they used to solve the problem.




• How did you solve the problem? (I used guess and check. Since 2 � 6 did not work, I tried other pairs of numbers that sum to 8.)

• What other strategy could you use to solvethe problem? (In the first equation, one square plus one triangleequal 8. In the second equation, two triangles equalone circle. I could use this information to write thethird equation 8 � two triangles equal 14.)

• What number does each figure represent? (� � 3; � � 5; � � 6)


Sample Answers1. The difference between each term and the previous term is

the sum of the previous 2 differences.

REFLECT: I had trouble solving Practice question 2. I had troublethinking about the problem with symbols, so I thought itthrough in words. For example, to figure out what number �represented, I thought of pairs of numbers with a differenceof 4. Then I noticed in the second equation that the numberrepresented by � is double the number represented by �.



Early FinishersHave students try to find another method for solving the equations.

Common Misconceptions➤ Students have difficulty seeing the relationships among

the equations.How to Help: Explain to students that the same equation maybe expressed in a different way. A square plus a triangle � 8is the same as a triangle plus a square � 8. Switching thesymbols in addition and multiplication statements may helpstudents see possible relationships among equations.

� � 4� � 8� � 5

41, 67, 108


Discuss the strategies presented in Connect. Askstudents to share any experiences they havehad with difficult problems and the strategiesthey used to get unstuck.

P r a c t i c eEncourage students to use the problem-solvingstrategies to solve each problem.


SHOW WHAT YOU KNOW

24 Unit 1 • Show What You Know • Student page 26

Encourage students to show work in anorganized way so that their solutions showhow they solved the problems.

Sample Answers1.

4. a) 35, 40; start at 5, add 9, decrease the number youadd by 1 each time; 49

b) 370, 345; start at 470, subtract 25 each time; 270c) 33, 65; start at 3, add 2, double the number you add

each time; 513d) 41, 122; start at 1, add 1, triple the number you add

each time; 3281

5. a) Recursive; multiply by 10 then add 1.b) Recursive; add 10.c) Not recursived) Recursive; multiply by 2 then add 9.

6. a) 2, 9, 16, 23, 30, 37, . . .b) Start at 2. Add 7 each time.

7. Divisible by 5: a, b, e, f, since they end in a 0 or a 5.Divisible by 8: a, c, e, since the last digit is even and thelast three digits are divisible by 8.

8. a) The loops should overlap because an even numberdivisible by 9 will also be divisible by 6.

b)

9. b) 6 is divisible by 3.

LESSON ORGANIZER

AssessmentMaster 1.1 Unit Rubric: Number PatternsMaster 1.4 Unit Summary: Number Patterns

40–50 min

7, 21, 63, 189, 5677, 11, 15, 19, 23

7, 19, 55, 163, 4877, 15, 31, 63, 127

– 4� 3

Input Output

2

7

12

17

22

27

32

37

1

2

3

4

5

6

7

8

Divisible by 6 Divisible by 9

330 639

2295

10 217 187

858

5598

12 006


Unit 1 • Show What You Know • Student page 27 25


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

What to Look For

Knowledge and Understanding✔ Questions 1 and 3: Students create and extend patterns and write pattern rules.

✔ Question 9: Students understand the divisibility rules.

Thinking✔ Questions 6, 8, and 12: Students apply patterning strategies to problem-solving situations.

Application✔ Questions 2 and 4: Students recognize relationships and use them to summarize and

generalize patterns.

✔ Questions 10 and 11: Students solve simple equations with one variable.

12

817

6

12 cm

1613

�100140

1296

Yes; 9No

1224


U N I T P R O B L E M

Crack the Code!

26 Unit 1 • Unit Problem • Student page 28

Sample ResponsePart 1

Place-value chart for base 2

Part 2

10011� 1(16) � 0(8) � 0(4) � 1(2) � 1(1)� 16 � 2 � 1 = 19 S

1101 � 1(8) � 1(4) � 0(2) � 1(1) � 8 � 4 � 1 � 13 M1001 � 1(8) � 0(4) � 0(2) � 1(1) � 8 � 1 � 9 I1100 � 1(8) � 1(4) � 0(2) � 0(1) � 8 � 4 � 12 L101� 1(4) � 0(2) � 1(1) � 4 � 1 � 5 EThe message is SMILE.

Part 3

A sample secret message to encode: PATTERNP is the 16th letter of the alphabet. In binary code: 1(16) � 0(8) � 0(4) � 0(2) � 0(1) � 10000A is the 1st letter of the alphabet. In binary code: 1T is the 20th letter of the alphabet. In binary code: 1(16) � 0(8) � 1(4) � 0(2) � 0(1) � 10100E is the 5th letter of the alphabet. From the chart, 5 is 101in base 2.R is the 18th letter of the alphabet. In binary code: 1(16) � 0(8) � 0(4) � 1(2) � 0(1) � 10010N is the 14th letter of the alphabet. In binary code: 1(8) � 1(4) � 1(2) � 0(1) � 1110The message would be:10000, 1, 10100, 10100, 101, 10010, 1110

LESSON ORGANIZER

Student Grouping: 2Assessment:Master 1.3 Performance Assessment Rubric: Crack the Code!Master 1.4 Unit Summary: Number Patterns

40–50 min

Base 10Number

Base 2Number16s 8s 4s 2s 1s

123456789

10

11110100

110011001

1010101010

11011

100101110111

100010011010

11


Reflect on the UnitI have used patterns from Input/Output machines. In thesepatterns, the output is related to the input. These patterns couldbe described using the Input/Output machine as the pattern rule,or the input and the output could each be described by a recursivepattern rule. I used patterns to find divisibility rules for the numbersfrom 2 to 9. I also used patterns to solve simple equations with onevariable and more complex equations with 2 or more variables.

Unit 1 • Unit Problem • Student page 29 27


Recording and ReportingMaster 1.3 Performance Assessment Rubric: Crack the Code!Master 1.4 Unit Summary: Number Patterns

What to Look For

Knowledge and Understanding✔ Students understand how to extend

the pattern in the base 2 chart.

Thinking✔ Students can create and solve a

message using the binary code.

Application✔ Students can use the pattern of place

values to decode the message.

What to Do

Extra Support: Make the problem accessible.Some students may have difficulty working with the base 2number system. They may benefit from scaffolding the steps.Ask questions, such as:

• What digits can be used in a base 2 number?• What patterns do you see in the base 2 numbers?

Some students may need to continue the base 2 chart beyondthe number 10 and write the appropriate letter beside eachnumber. Other students may only need a list of the alphabetand the base 10 number associated with each letter for easier decoding.


The right to reproduce or modify this page is restricted to purchasing schools. This page may have been modified from its original. Copyright © 2006 Pearson Education Canada Inc.

28

Evaluating Student Learning: Preparing to Report:Unit 1 Number Patterns This unit provides an opportunity to report on the Patterning and Algebra 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: Most Consistent Level of Achievement*

Strand: Patterning and Algebra

Knowledge and Understanding

Thinking Communication Application Overall

Ongoing observations 3 2 4 3 3Work samples or portfolios; conferences

3 3 3 3 3

Show What You know 3 3 3 3 3Unit Test 3 2 4 3 3Unit Problem: Crack the Code!

3 3 3 3 3

Achievement Level for reporting 3

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 mostly heavily weighted.

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 later in the unit may be more heavily weighted.

Show What you Know Teachers may choose to assign some or all of these questions. Master 1.1 Unit Rubric: Number Patterns may be helpful in determining levels of achievement.

Unit Test Master 1.1 Unit Rubric: Number Patterns may be helpful in determining levels of achievement.

Unit performance task Use Master 1.3 Performance Assessment Rubric: Crack the Code!. 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’ perception of their own progress. This may take the form of an oral or written comment, or 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 RecordsPM 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 clusters, a reporting period, or a school year. These can also be used in place of the Unit Summary.

Name Date


29

Unit Rubric: Number Patterns

Categories/Criteria Level 1 Level 2 Level 3 Level 4 Knowledge and Understanding • shows understanding of

concepts by demonstrating and explaining: – mathematical

relationships among and between patterns

– patterning rules (and choice of a pattern rule)

– multiples and factors; prime and composite numbers (to 100)

demonstrates limited understanding in demonstrations and explanations of: – mathematical


– patterning rules – multiples and

factors; prime and composite numbers (to 100)

demonstrates some understanding in demonstrations and explanations of: – mathematical




demonstratesconsiderableunderstanding in demonstrations and explanations of: – mathematical




demonstrates thorough understanding in demonstrations and explanations of: – mathematical




Thinking• plans and effectively

carries out patterning strategies to pose and solve problems

uses patterning strategies to pose and solve problems with limited effectiveness

uses patterning strategies to pose and solve problems with some effectiveness

uses patterning strategies to poseand solve problems with considerable effectiveness

uses patterning strategies to poseand solve problems with a high degree of effectiveness

Communication • explains reasoning and

procedures clearly, using appropriate terminology and symbols

limited effectiveness; unable to explain reasoning and procedures clearly; rarely uses appropriate terms and symbols

some effectiveness; explains reasoning and procedures with some clarity and use of appropriate terms and symbols

considerableeffectiveness; explains reasoning and procedures clearly, using appropriate terms and symbols

a high degree of effectiveness; explains reasoning and procedures clearly and precisely, using the most appropriate terms and symbols

Application applies patterning skills with limited effectiveness; makes major errors or omissions in: – identifying,

extending, and creating patterns (2 variables)

– analysing pattern rules

– finding the value of a missing termor factor

applies patterning skills with some effectiveness;somewhat accurate with several minor errors or omissions in: – identifying,




applies patterning skills with considerable effectiveness;generally accurate with few minor errors or omissions in: – identifying,




applies patterning skills with a high degree of effectiveness; accurate and precise with very few or no errors in: – identifying,




• applies number patterning skills and concepts appropriately to: – identify, extend, and

create patterns (2 variables)

– analyse pattern rules – find the value of a

missing term or factor and make connections to real-world applications of integers

makes limited connections to real-world applications of integers

makes somewhat effective connections to real-world applications of integers

makes considerably effective connections to real-world applications of integers

makes highly effective connections to real-world applications of integers

Master 1.1

Name Date


30

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 PatternsStudent Knowledge and

Understanding Thinking Communication Application

• Demonstrates and explains pattern rules, relationships, multiples, and factors; prime and composite numbers

• Uses patterning strategies to poseand solve problems

• Explains reasoning and procedures clearly, using appropriate terms

• Accurately uses patterning skills to: – identify, extend, and

create patterns – find the value of a

missing term/factor– multiply and divide

Level 1 – very limited; Level 2 – somewhat or limited; Level 3 – satisfactory; Level 4 – thorough

Master 1.2

Name Date


31

Performance Assessment Rubric: Crack the Code!

Categories/Criteria Level 1 Level 2 Level 3 Level 4 Knowledge and Understanding • shows understanding

of number patterns in explanations of procedures and results

demonstrates limited understanding in explanations of procedures and results

demonstrates some understanding in explanations of procedures and results

demonstratesconsiderableunderstanding in explanations of procedures and results

demonstrates thorough understanding in explanations of procedures and results

Thinking• plans and effectively

carries out patterning strategies to decode and create messages using binary code

uses patterning strategies to decode and create messages using binary code with limited effectiveness

uses patterning strategies to decode and create messages using binary code with some effectiveness

uses patterning strategies to decode and create messages using binary code with considerableeffectiveness

uses patterning strategies to decode and create messages using binary code with a high degree of effectiveness

Communication • expresses and organizes

procedures, results, and reasoning clearly

limited effectiveness; unable to express and organize procedures, results, and reasoning clearly

some effectiveness; expresses and organizes procedures, results, and reasoning with some clarity

considerableeffectiveness;expresses and organizes procedures, results, and reasoning clearly

a high degree of effectiveness;expresses and organizes procedures, results, and reasoning clearly

Application • applies number patterning

skills and concepts appropriately to complete a place-value chart for base 2

limited effectiveness; makes major errors or omissions in completing a place-value chart for base 2

some effectiveness; somewhat accurate in completing a place-value chart for base 2, with several minor errors or omissions

considerableeffectiveness;generally accurate in completing a place-value chart for base 2, with few minor errors or omissions

high degree of effectiveness; accurate and precise in completing a place-value chart for base 2, with very few or no errors

Master 1.3

Name Date


32

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:Patterning and Algebra

Knowledge and Understanding

Thinking Communication Application Overall

Ongoing Observations

Work samples or portfolios; conferences

Show What You Know

Unit Test

Unit Problem:Crack the Code!

Achievement Level for Reporting

*Use Ontario Achievement Levels 1, 2, 3, 4.

Self-Assessment:

Comments: (Strengths, Needs, Next Steps)

Master 1.4

Name Date


33

To Parents and Adults at Home …

Your child's class is starting a mathematics unit on number patterns. Understanding and producing patterns are important skills that help develop your child's number sense and algebraic reasoning skills.

Patterns occur in nature, art, and many everyday activities. Patterns can be described using numbers, words, models, and formulas.

In this unit, your child will: • Write a pattern rule for a number pattern. • Identify, extend, and create patterns. • Use patterns to pose and solve problems. • Use patterns to explore divisibility rules. • Find the value of a missing term or factor. • Use patterns to explore integers.

Patterns occur in many forms. Help your child see the patterns that occur in his or her everyday life — in pictures, numbers, or sequences of performing steps.

Master 1.5

Name Date


34

What’s My Number? Game Boards

1 2 3 4

1 8 × = 112 216 = 48 + – 89 = 24 9 × = 279

2 + 76 = 145 245 = × 7 – 23 = 98 318 = 279 +

3 514 + = 629 136 = 230 – 648 = 9 × ÷ 7 = 81

4 – 108 = 47 96 + = 147 5 × = 240 229 = 38 +

1 2 3 4

1 717 = 402 + 223 = 87 + ÷ 9 = 83 9 × = 297

2 + 314 = 541 6 × = 306 – 32 = 81 318 = 59 +

3 496 = 8 × 173 = 221 – 118 + = 294 – 91 = 33

4 – 213 = 46 74 + = 147 5 × = 255 480 = × 12

Master 1.6

Name Date


35

Equation Baseball Game Cards

÷ 7 = 3 8 × = 56 4802 + = 7803 6 × 8 = 24 +

– 13 = 22 7 × = 84 17 + 17 = 20 + × 9 = 30 – 3

19 + = 51 67 + 43 = 20 × 21 = 255 + 345 = × 9

22 – = 3 × 9 = 108 ÷ 8 = 70 26 + 24 = 500 ÷

57 + = 78 15 × = 150 16 × 3 = 8 × 639 ÷ = 71

– 26 = 43 83 – 44 = 700 ÷ = 140 28 + 53 = × 9

9 × = 54 1 + 21 + = 67 7 × 9 = × 3 28 ÷ = 12 ÷ 3

180 ÷ 10 = 444 ÷ 4 = 15 × 2 = ÷ 2 450 – 120 = – 90

+ 43 = 79 77 + 90 = 328 ÷ 8 = × = 60

÷ 10 = 35 10 × 12 = – 225 = 500 ÷ = 5

23 + = 40 15 × 99 = 7 × 4 × 25 = × = 108

46 – = 16 14 × 7 = 4 × 50 = 2 × ÷ = 7

49 ÷ = 7 13 + 60 + 17 = 50 × 50 = 22 – 22 = 361 ×

Master 1.7

Name Date


36

Equation Baseball Game Board

Score Cards

Names Scores

Master 1.8

Name Date


37

Additional Activity 1: Code Breaker

Here is part of a codebook.

Actual A B C D E New Letter O T Y D I

Work with a partner.

How is the code created?

Explain why D remained D.

Complete the code table for the rest of the alphabet.

Decode the following message: AII EGK ROFIV

Take It Further: Create your own codebook. Use it to create a coded message.Trade messages with another pair of students. Try to crack your classmates’ code.

Master 1.9

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Additional Activity 2: It’s Getting Smaller!

Work on your own.

Begin with 1 000 000.

Make up a pattern rule for a shrinking pattern.

Write the first 7 terms of your pattern.

Trade patterns with a classmate.

Write the pattern rule for your classmate’s pattern and write the next 3 terms.

Take It Further:What is the total number of terms in your pattern? All the terms must be 0 or greater.

Master 1.10

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Additional Activity 3: Starting Point

Work on your own.

Write a pattern rule with two operations. For example: Multiply by 2, then add 1.

Choose 3 different starting numbers. Apply the pattern rule to each number. Write the first 10 terms for each pattern.

Compare the terms in each pattern. Describe any similarities among the patterns. How often do numbers divisible by 2 appear?

Look for numbers divisible by 3, 5, 9, or other numbers. Describe any patterns where these numbers appear.

Extend your patterns to check your ideas.

Take It Further:Trade patterns with a classmate. Identify your classmate’s pattern rules.

Master 1.11

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Additional Activity 4: And the Number Is …

Work with a partner.

Take turns to write a simple equation with one missing number and one operation.

Have your partner solve the equation.

Take It Further: Write and solve equations with more than one missing number and/or more than one operation.

Master 1.12

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Step-by-Step 1

Lesson 1, Question 5

The table shows the input and output from a machine with two operations.

Step 1 What is the pattern rule for the output numbers?

______________________________________

______________________________________

Step 2 List the first six multiples of 4.

____________________________________________________________

Compare the multiples of 4 with the

output numbers. What do you notice?

____________________________________________________________

Step 3 What do you do to each input number

to get each output number?

____________________________________________________________

Step 4 Use your rule from Step 3.

Find the output for each input number:

Master 1.13

Input Output 1 5 2 9 3 13 4 17 5 21 6 25

Input Output 7 28 9 10

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Step-by-Step 2


The first two terms of a recursive pattern are 4 and 7.

Step 1 How can you get 7 from 4? Find the missing number.

4 + ____ = 7

Use this number and operation to write the next 3 terms:

4, 7, ____, ____, ____

Write the pattern rule for this pattern.

____________________________________________________________

Step 2 How can you get 7 from 4 using multiplication followed by subtraction?

Find the missing numbers: 4 × ____ – ____ = 7

Use these numbers and operations to write the next 3 terms:

4, 7, ____, ____, ____

Write the pattern rule for this pattern.

____________________________________________________________

Step 3 Can you get 7 from 4 using addition followed by multiplication?

If so, find the missing numbers: (4 + ____) × ____

If possible, write the next 3 terms and write the pattern rule for this pattern.

4, 7, ____, ____, ____

____________________________________________________________

Step 4 Try different numbers and operations.

Write any other recursive patterns you find that begin with 4, 7, ….

____________________________________________________________

Master 1.14

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Step-by-Step 3


Step 1 Which of these numbers is divisible by 2? By 4? By 8?

1046 322 460 1784 28

54 1088 224 382 3662

If a number is divisible by 4, it is also divisible by _____.

If a number is divisible by 8, it is also divisible by _____ and by _____.

Step 2 Draw a Venn diagram with 3 loops.

Label the loops “Divisible by 2,” “Divisible by 4,” and “Divisible by 8.”

How did you draw the loops? Why did you draw them that way?

____________________________________________________________

____________________________________________________________

Step 3 Sort the numbers from Step 1 into the Venn diagram in Step 2.

How did you know where to place each number?

____________________________________________________________

Step 4 Write 3 different 4-digit numbers: _____, _____, _____

Place each number in the Venn diagram in Step 2.

Master 1.15

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Step-by-Step 4


Step 1 To solve the equation – = 7 – 4:

Find the difference: 7 – 4 = _____

Find another pair of numbers whose difference

is equal to the difference above: 7 – 4 = ____ – ____

Step 2 To solve the equation 5 + 3 = + :

Find the sum: 5 + 3 = _____

Find another pair of numbers whose sum

is equal to the sum above: 5 + 3 = ____ + ____

Step 3 To solve the equation 16 × 3 = × :

Find the product: 16 × 3 = _____

Find another pair of numbers whose product

is equal to the product above: 16 × 3 = ____ × ____

Step 4 To solve the equation 8 – 6 = – :

Find the difference: 8 – 6 = _____

Find another pair of numbers whose difference

is equal to the difference above: 8 – 6 = ____ – ____

Step 5 For each equation in Steps 1 to 4, explain how you chose the numbers.

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

Master 1.16

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Step-by-Step 5


For every 150 m above sea level, the temperature decreases about 1°C. The Brienzer-Rothorn Railway takes passengers from an altitude of 566 m to an altitude of 2244 m on the Rothorn mountain.

Step 1 Suppose the temperature at the bottom of the mountain is 23°C.

Complete the table. Predict the temperature at the top of the mountain.

__________________________

__________________________

__________________________

__________________________

Step 2 Now suppose the temperature when you get on the train at the bottom of

the mountain is 9°C. Will the temperature at the top of the mountain be

above or below 0°C? Explain how you know.

___________________________________________________________

___________________________________________________________

Master 1.17

Altitude (m) Temperature (°C)

566 23 716866

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Unit Test: Unit 1 Number Patterns

Part A 1. Complete the table for this Input/Output machine.

Input Output 23456

2. Write the first 5 terms for this pattern rule: Start at 2. Multiply by 3, thensubtract 1 each time: _____, _____, _____, _____, _____

3. How can you tell if 1245 is divisible by 3 and by 5?

________________________________________________________________

________________________________________________________________

4. Use an integer to represent each situation.

a) Regean spent $14 on her new shirt. _________

b) Katherine won $50 in the art contest. _________

Part B 5. Draw an Input/Output machine that would give

the numbers in the table.

Master 1.18

Input Output

1 1 2 5 3 9 4 13 5 17

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Unit Test continued

6. Write the next two terms in this pattern: 4, 5, 7, 11, 19, _____, _____ Then write the pattern rule.

________________________________________________________________

7. Which of these numbers is divisible by 4? By 9? How do you know? 140 612 150 7590 1089

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

8. Find each missing number.

a) 14 + 17 = _____ + 8 b) 32 × 4 = _____ × 16

c) 24 ÷ 6 = 28 ÷ _____ d) 12 – _____ = 21 – 14

9. Which of these patterns are recursive? Explain how you know.

a) 2, 5, 8, 11, 14, … b) 1, 6, 4, 9, 7, 12, 10, …

c) 1, 3, 9, 27, … d) 4, 12, 6, 18, 9, 27, …

________________________________________________________________

________________________________________________________________

Part C 10. The first two terms of a recursive pattern are 3 and 7. What might the pattern be?

Give two different answers. Write the pattern rule for each pattern.

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________

Master 1.18b

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Unit Test continued

11. Draw a Venn diagram with 2 loops labelled “Divisible by 10” and “Divisible by 6.” How did you know how to draw the loops? Sort these numbers. 2325 570 3135 186 6750 882 5110 10 830

________________________________________________________________

________________________________________________________________

12. The perimeter of an isosceles triangle is 19 cm. The length of the unequal side is 5 cm. What is the length of each of the other two sides?

________________________________________________________________

________________________________________________________________

Master 1.18c

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Unit Test Sample Answers

Unit Test – Master 1.18

Part A 1.

2. 2, 5, 14, 41, 122

3. 1245 is divisible by 3 because the sum of the digits is divisible by 3. It is divisible by 5 because the ones digit is 5.

4. a) –14 b) +50

Part B 5.

6. 35, 67; Start at 4. Multiply by 2, then subtract 3 each time.

7. Divisible by 4: 140, 612; The number formed by the last 2 digits is divisible by 4. Divisible by 9: 612, 1089; The sum of the digits is divisible by 9.

8. a) 23 b) 8 c) 7 d) 5

9. a and c are recursive because the next term is produced by a set of operations on the previous term.

Part C 10. 3, 7, 11, 15, 19, 23, …; Start at 3.

Add 4 each time. 3, 7, 15, 31, 63, 127, …; Start at 3.Multiply by 2, then add 1 each time.

11.

I overlapped the loops because numbers such as 30 and its multiples are evenly divisible by 6 and by 10.

12. 7 cm

Input Output

2 0

3 3

4 6

5 9

6 12

Master 1.19


50

Extra Practice Masters 1.20–1.23

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

Program Authors

Peggy Morrow

Ralph Connelly

Jason Johnston

Bryn Keyes

Don Jones

Michael Davis

Steve Thomas

Jeananne Thomas

Nora Alexander

Linda Edwards

Ray Appel

Cynthia Pratt Nicolson

Carole Saundry

Ken Harper

Jennifer Paziuk

Maggie Martin Connell

Sharon Jeroski

Trevor Brown

Copyright © 2006 Pearson Education Canada Inc.

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.Pages identified as line masters may be copied for classroom use.

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