Grade 7 Mathematics Unit 1 Patterns and · PDF fileUnit 1: Patterns and Relations Grade 7 Math Curriculum Guide 31 Unit 1 Overview Introduction Students will develop their ability

Grade 7 Mathematics Curriculum Outcomes 29

Outcomes with Achievement Indicators

Unit 1

Grade 7 Mathematics

Unit 1

Patterns and Relations

Estimated Time: 21 Hours

[C] Communication [PS] Problem Solving

[CN] Connections [R] Reasoning

[ME] Mental Mathematics [T] Technology

and Estimation [V] Visualization



Unit 1

Unit 1: Patterns and Relations

Grade 7 Math Curriculum Guide 31

Unit 1 Overview

Introduction

Students will develop their ability to explore a variety of situations involving patterns and change.

Students will explore three different ways to represent change - words, tables, and graphs. They will then

move into symbolic representation using simple expressions & equations. The big ideas in this unit are:

• Words and symbols can be used to describe patterns.

• A variable is a symbol that acts as a place holder. It can represent a number or a set of numbers.

• Algebraic expressions are used to describe patterns. Any term in a pattern can be found by

replacing the variable with the term number the student is trying to find.

• The relationship between the term in a pattern and its term number is called a relation.

• Relations can be represented either symbolically, or as tables, or graphically. They can be used

to model situations and solve problems related to those situations.

• A statement of equality between two expressions is called an equation.

Context The students have used number patterns to explore the rules of divisibility. They will extend this

technique and explore divisibility for this entire list; 2, 3, 4, 5, 6, 8, 9.

The students will be introduced to the concept of variables and how to create expressions from

uncomplicated situations and basic statements. They will learn that a variable is a place holder and, as

such, can be replaced. A value can then be determined for the expression. Patterns will be represented by relations and these relations will be used to make predictions and/or solve

problems. Connections will be made between representing relations in three ways; symbolically,

graphically, and in tabular form. Students will determine which method is best for each relation as it

pertains to making predictions or solving problems.

Students will extend this knowledge of expressions and create equations. They will learn that the solution

to an equation is the number that can be used to replace the variable and make the equation a true

statement. To help find solutions to equations students will learn to model and solve equations using

algebra tiles.

Why are these concepts important?

Being able to identify and extend patterns is critical to:

• Problem solving

• Algebraic reasoning

Using variables, employing algebraic reasoning and solving equations are all concepts that can successfully be developed in conjunction with each other. These concepts will be extremely useful in the

student’s future study of math, science, social studies, etc. The ability to reason logically and model

situations to solve problems is a skill that can be utilized in all walks of life and a great many trades and

professions.

“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent

than theirs, it is because they are made with ideas.” Godfrey Harold Hardy (1877 - 1947)

Strand: Number



Unit 1

General Outcome: Develop Number Sense

Specific Outcome

It is expected that students will:

7N1 Determine and explain

why a number is divisible by

2, 3, 4, 5, 6, 8, 9 or 10, and

why a number cannot be

divided by 0. [C, R]

Achievement Indicators:

Elaborations: Suggested Learning and Teaching Strategies

In earlier grades, students have built their understanding of

sequences and number patterns. These patterns are used to

develop the rules of division of larger numbers. It is assumed

that students can:

• recognize number patterns in tables

• extend a table of values using a pattern

• describe the relationships among terms in a table

Exploration of the divisibility rules serves as an excellent

opportunity to extend number sense. Instruction should be

organized so that the students can arrive at the divisibility rules

themselves. For example, see the Explore, p.6 of the student

text. Knowledge of divisibility rules will provide a valuable

tool for mental arithmetic and general development of

operation sense.

Students may have been exposed to the divisibility rules for 2,

5, and 10 since they are simple number patterns. A list of

divisibility rules can be found in the Student Text on p.12.

An alternate rule for divisibility by 8 is that a number is

divisible by 8 if the number is divisible by 4 and the resulting

quotient is even. For 92, think 92 ÷ 4 = 23, since 23 is not

even, 92 is not divisible by 8.

It is important for students to notice that a number divisible by

8, for example, is also divisible by 4 and by 2. In general, if a

number is divisible by another number then it is also divisible

by the factors of that number. The converse of this is not

always true. For example, 24 is divisible by 3 and 6. But it is

not divisible by 18 (3 x 6) since 3 and 6 have 3 as a common

factor. This should be seen as a problem solving opportunity

for students.

It is also important to learn how to test for divisibility on a

calculator. That is, students should realize that the test for

divisibility on a calculator involves dividing to see if the

quotient is a whole number. For example, to find if 276 is

divisible by 8, have students use a calculator to calculate

276 ÷ 8. Since the calculator shows 34.5, it tells them that 276

is not divisible by 8.

7N1.1 Determine if a

given number is divisible

by 2, 3, 4, 5, 6, 8, 9 or 10,

and explain why.

Strand: Number



Unit 1


Suggested Assessment Strategies

Paper and Pencil

1. Once you know that a particular number is divisible by 3 and 5,

how does this help you in determining whether it is divisible by

15? Explain, using an example.

2. Have students create 5 numbers that are divisible by both 2 and

10. Are all these numbers also divisible by 20? Explain.

3. A number is divisible by both 3 and 6. Must it be divisible by

18? Explain.”

4. Ask students to complete the number by filling in each blank

with a digit. Ask them to explain, using divisibility rules, how

they know their answers are correct.

A. 26_ is divisible by 10

B. 154_ is divisible by 2

C. _6_ is divisible by 6

D. 26_ is divisible by 3

E. 1_ 2 is divisible by 9

F. 15_ is divisible by 4

5. There will be 138 people at a party. Can the host fill tables of 5?

Tables of 6? Etc. Support your answer by using divisibility

rules.

Informal Observation

1. Play “Divisibility Challenge” (Master 1.6 in Teacher Resource)

with the class once all the divisibility rules have been

discovered. Note: When extending the game to the 10-sided die

rolling, if a student rolls a 7 then they will have to re-roll since

we have not covered the divisibility rule for 7.

2. Try the divisibility rule matching game which can be found at

the Oswego City School District (http://www.oswego.org/):

http://www.oswego.org/ocsd-

web/match/dragflip.asp?filename=slanedivrules

Resources/Notes

Math Makes Sense 7

Lesson 1.1

Lesson 1.2

Unit 1: Patterns and

Relations

TR: ProGuide, pp. 4–7 &

8–11

Master 1.10, 1.12, 1.22

Master 1.13, 1.23

CD-ROM Unit 1 Masters

ST: pp. 6–9 & 10–13

Practice and HW Book

pp. 4–5 & 6–8

Strand: Number



Unit 1


Specific Outcome




2, 3, 4, 5, 6, 8, 9 or 10, and



(Cont’d)

Achievement Indicators:


It is important to show students how to use a Venn Diagram

which has two loops (see p. 8 in the Student Text) before one

with three loops.

Div.

by 2

Div.

by 3

Div. by 2

Div. by 4

Div. by 2 Div. by 3

Div. by 5

Div. by 2

Div. by 4

Div. by 8

Carroll Diagrams should only be used to compare numbers

using two divisors (see p. 12 in the Student Text).

Students could review the concept of Venn and Carroll

Diagrams by categorizing student in class who are wearing 1)

short sleeve shirt or long sleeve shirt and 2) blue jeans or

other. Once they understand how the diagrams are used,

categorizing numbers based on divisibility can be introduced.

See material in Teacher Resource and Student Text (p.6 - 13).

7N1.3 Determine the

factors of a given number,

using the divisibility rules.

7N1.2 Sort a given set of

numbers based upon their

divisibility, using

organizers such as Venn

and Carroll diagrams.

Strand: Number



Unit 1



Paper and Pencil

1. Create a Carroll diagram or a Venn diagram to sort the

following numbers based on the divisibility rules for 3 and 5:

6, 8, 10, 15, 18, 25, 26, 36, 40, 45, 120.

Extension: Which numbers are also divisible by 15?

2. Create a Carroll or Venn diagram to sort the numbers according

to divisibility by 6 and 9.

30 79 162 3996 23517 31974

3. A. Have students choose a number that is divisible by both 6

and 9. What is the smallest number, other than 1, by which

the chosen number is divisible?

B. Repeat for other chosen numbers.

C. Have students conjecture as to what is the smallest divisor

for any number divisible by both 6 and 9.

Presentation/Portfolio

The principal of Great School is trying to determine the number of

classes of Grade seven students she can have in her school. Use the

divisibility rules to determine the possible number of classes if

there are 240 Grade seven students.

Journal/Portfolio

1. Which statements are true? For those statements that are false,

provide an example to verify your answer.

A. All numbers divisible by 6 are divisible by 3.

B. Some, but not all, numbers divisible by 6 are divisible by

3.

C. No numbers divisible by 6 are divisible by 3.

D. All numbers divisible by 3 are divisible by 6.

E. Some, but not all, numbers divisible by 3 are divisible by

6.

F. No numbers divisible by 3 are divisible by 6.

2. Each of Eli’s four friends has a code number. Keile’s number is

divisible by 3, 5, and 8. Max’s number is divisible by 2 and 3.

Jennifer’s number is divisible by 4 and 5, but not 3. Ben’s

number is divisible by 3 and 5, but not 8. Eli receives a

message signed with the code number 5385 from one of his four

friends. Who sent the message?

Resources/Notes

Math Makes Sense 7

Lesson 1.1

Lesson 1.2

(continued)

Math Makes Sense 7

Lesson 1.1

Lesson 1.2

(continued)

Strand: Number



Unit 1


Specific Outcome




2, 3, 4, 5, 6, 8, 9 or 10, and



(Cont’d)

Achievement Indicators


To avoid an arbitrary rule for not being able to divide by 0, use

the repeated subtraction meaning of division. For example

with 20 ÷ 5 we can say 20 – 5 – 5 – 5 – 5 = 0. You can

subtract 5 four times from 20 until you get to 0, so 20 ÷ 5 = 4.

So, for 6 ÷ 0, ask how many times can you subtract 0 from 6

before you get to 0? There is no answer; you will never get to

0 (6 – 0 – 0 – 0 = 6). So, 6 ÷ 0 is undefined.

Alternately:

6 ÷ 3 = 2. This means that if you had six counters, they could

be separated into two groups of three.

6 ÷ 0 = ? Into how many groups of zero can six counters be

separated? It is not possible to separate the six counters into

groups of 0. Thus, division by 0 is not possible.

7N1.4 Explain, using an

example, why numbers

cannot be divided by 0.

Strand: Number



Unit 1



Presentation/Portfolio

1. Explain why it is not possible to calculate 12 ÷ 0.

2. A. Complete the table:

Div. Stmt. Related Mult. Stmt.

6 ÷ 2 = 3 3 × 2 = 6

10 ÷ 5 = 2 2 × 5 =

14 ÷ 2 = 2 × 7 = 14

15 ÷ = 5 3 × 5 =

÷ 8 = 3 3 × 8 =

12 ÷ 0 = 0 × = 12

B. Explain how the table shows division by 0 is not possible.

Resources/Notes

Math Makes Sense 7

Lesson 1.2

(continued)

Strand: Patterns and Relations (Patterns)

.



Unit 1

General Outcome: Represent algebraic expressions in

multiple ways.

Specific Outcome


7PR4 Explain the difference

between an expression and

an equation. [C, CN]



Students do not need to distinguish between expressions and

equations at this time because it will be addressed in Unit 6.

Note that students will need to identify constant term,

numerical coefficient, and variables in both an expression and

an equation. In Lesson 1.3, the focus should be on identifying

these in expressions only.

Students will need to be able to define variable. Additionally,

they should be able to tell what the variable represents in a

given context.

In the expression 1

62

+k the constant term is 6 (since it is

unchanged by a variable), the numerical coefficient is 1

2 (and

is multiplied by the variable), and the variable is k.

Notes:

• When a numerical coefficient is not written in front of a

variable, it is automatically considered to be 1, e.g. the

numerical coefficient in x + 5 is 1.

• The above expression could also have been written as

62

k+ . To help students see that the numerical coefficient

is 1

2, the expression could be rewritten as

16

2

k+ .

The concept of an algebraic expression should be introduced

using real-life or concrete examples.

(This elaboration is continued on the next two page spread…)

7PR4.1 Identify and

provide an example of a

constant term, numerical

coefficient and variable in

an expression and an

equation.

7PR4.2 Explain what a

variable is and how it is

used in a given

expression.


.



Unit 1


multiple ways.


Paper and Pencil

1. Write an algebraic expression that has a variable h, numerical

coefficient 4 and constant term 11.

2 A grocer has 40 loaves of brown bread on hand orders w white

loaves of bread. She sells her white loaves for $2.00 each.

Describe in words what each of the following expressions

represents.

A. w + 40 (Answer: Total number of loaves of bread

she has)

B. 2w (Answer: amount of money she takes in for

selling all the loaves of white bread).

Portfolio

1. Ask students to create a classroom chart with the following

headings:

Algebraic

Expression

Expression

in Words Variable

Numerical

Coefficient Constant

3b + 1

One more

than 3 times

a number

b 3 1

y + 6

Have students continue to add examples to the chart each day.

Resources/Notes

Math Makes Sense 7

Lesson 1.3


Relations

TR: ProGuide, pp. 14–17

Master 1.14, 1.24


ST: pp. 16–19


pp. 9–11


.



Unit 1


multiple ways.

Specific Outcome








using real-life or concrete examples:

Example A

John earns $4.00 per hour for babysitting his sister’s child. The

table shows how we could calculate his earnings for different

numbers of hours:

Hours 4 × Hours

0 4 × 0

1 4 × 1

1.5 4 × 1.5

2 4 × 2

3.25 4 × 3.25

The word Hours, or H, is referred to as a variable because it

represents the amount that is changing (or varying) in the

situation.

4 × H is an expression that summarizes how we could calculate

John’s earnings.

In algebra, we often abbreviate multiplication statements

involving variables by omitting the multiplication symbol, e.g.

we write 4 × H as 4H. (Note: Ensure students do not confuse

this with their prior experience with placeholders, e.g.

• 4� + 8 When the � is replaced by a 2, the value of the

expression is 50. In this context, the � is a

placeholder that represents a digit.

• 4H + 8 When the H is replaced by 2, we mean 4 × 2 + 8

and the value of the expression is 16.


7PR4.1 Identify and





equation.

(continued)



used in a given

expression.

(continued)

We can abbreviate by

using H for Hours and

then write 4 × H.


.



Unit 1


multiple ways.


Resources/Notes

Math Makes Sense 7

Lesson 1.3

(continued)


.



Unit 1


multiple ways.

Specific Outcome








using real-life or concrete examples:

Example B

For a certain cell phone plan, a customer pays an initial fee of

$10.00 plus an additional fee of $0.05 per text message. One

month, John sent 15 messages on his phone, Daisy sent 3

messages, Judy sent 200. The table shows how to calculate

each person’s costs:

Messages 0.05 × Messages + 10

0 0.05 × 0 + 10

1 0.05 × 1 + 10

1.5 0.05 × 1.5 + 10

2 0.05 × 2 + 10

3.25 0.05 × 3.25 + 10

Variables such as n are used to represent an unknown quantity

or a quantity that can change. Students should also understand,

however, that while in many situations variables can take on

many values (e.g., 4p s= for any value of s), in other

situations they represent a single value (e.g., 3 9x + = ). The

work with equations will be addressed in unit 6.

Example for Discussion

Mary babysits for her neighbour and earns $8.00 per hour.

Students should be able to write an expression to represent her

earnings: 8h. They should also be able to explain what the

variable h represents in this situation: the number of hours she

worked.

7PR4.1 Identify and





equation.

(continued)



used in a given

expression.

(continued)

We can abbreviate

by using M for

Messages and then

write O.5 × M + 10,

or 0.05M + 10.


.



Unit 1


multiple ways.


Resources/Notes

Math Makes Sense 7

Lesson 1.3

(continued)




Unit 1

General Outcome: Use patterns to describe the world and to

solve patterns.

Specific Outcome





(Cont’d)



In order to understand the difference between an expression

and an equation, students should become familiar with writing

expressions before they write equations. (Note: The

relationship of an expression to an equation is similar to the

relationship of a phrase to a sentence. In an English sentence,

we need verbs to describe the relationship between phrases. In

mathematics, an equation is a complete sentence describing the

relationship, by means of an equal sign, between two

expressions.) This outcome will be further solidified once

students learn more about equations in Unit 6 of Math Makes

Sense 7. In Lesson 6.1, students will be expected to distinguish

between algebraic expressions and equations. For now, the

focus of this outcome is to introduce the writing of algebraic

expressions.

An algebraic expression is a mathematical expression that

contains a variable or a combination of operations (+,– ,÷,×)

involving numbers and variables.

For example y, 5y, and 5y + 4 are algebraic expressions.

See relevant examples in Student Text p. 17.

This concept lends itself well to being reviewed orally at the

beginning of subsequent classes. For example, discuss writing

algebraic expressions for the following:

• Chris worked n hours yesterday and 8 more hours

today. Write an expression for his total number of

hours. (Answer: n + 8)

• Loretta earned $5.00 per hour for n hours. Write an

expression for her earnings. (Answer: 5n)

• Dennis has $20 in his pocket. He works for x hours

and earns $6.00 per hour. Write an expression for the

total amount of money he will have.

(Answer: 6x + 20).

7PR4.3 Represent a given

oral or written pattern

using an algebraic

expression.




Unit 1


solve patterns.


Interview

Give an algebraic expression for each phrase:

A. Bob’s salary increased by $25

B. The cost of $2 per hour plus $5 for renting skates

C. Hamburgers at $3 per person

D. 15 more marbles than triple Jane’s marbles

E. 5 more than a number

F. 8 less than a number

G. 3 more than twice a number

H. Double a number

Informal Assessment

Mix up Match up for Algebra: Create a set of algebraic expressions

on index cards. Write the matching word forms on another set of

cards. Randomly distribute the cards among the class and have

students find their partner.

Resources/Notes

Math Makes Sense 7

Lesson 1.3

(continued)




Unit 1


multiple ways

Specific Outcome


7PR5 Evaluate an

expression, given the value

of the variable(s). [CN, R]



To evaluate an algebraic expression, substitute a number for

the variable and carry out the computation. Order of

Operations will likely need to be reviewed prior to a discussion

of evaluating expressions. It will likely be helpful to start by

providing expressions based on real-life contexts and

substituting values for the variables, such as the situations used

on 7PR4.

Calculations with expressions involving division should also

be discussed, e.g.

82

−m

Note that we could also write this as 8 – m ÷ 2, but this

form is not often used in algebra.

Refer to Teacher Resource (p. 14 – 17) and Student Text (p. 16

– 19).

7PR5.1 Substitute a value

for an unknown in a given

expression, and evaluate

the expression.




Unit 1


multiple ways


Paper and Pencil

1. Evaluate each of the following for the given value of the

variable.

A. 3 5p + , for 1p =

B. 32

m− , for 6m =

2. Salma gets $7 per hour to baby-sit. She gets a bonus if she has

to baby-sit past 10 p.m. The expression 7 3h + represents what

Salma was paid last night. She baby-sat from 5:30 p.m. to

10:30 p.m.

A. What is the variable in this expression? Explain what it

represents.

B. What does the constant term “3” represent in the

expression?

C. How much did she earn last night?


Play ‘Substitution Toss’. See Teacher Resource Master 1.7a and

1.7b.

Resources/Notes

Math Makes Sense 7

Lesson 1.3

(continued)

Note: This outcome is

also developed in:

Math Makes Sense 7

Lesson 1.4




Unit 1


solve patterns.

Specific Outcome


7PR1 Demonstrate an

understanding of oral and

written patterns and their

equivalent linear relations.

[C, CN, R]



Consider the pattern of circles:

1 2 3 4

The relationship between the diagram number and the number

of dots in each diagram can be summarized in a table:

Diagram

Number (d) 1 2 3 4 5

Number of

Circles (c) 3 6 9 12 15

The goal is to have students find a relationship between one

quantity (variable) and another, in this case between the

Diagram Number and the Number of Circles. They can

begin by describing the relationship in words and then writing

in symbolic form. For example,

• In words:

“The number of circles in a diagram is equal to

three times the diagram number”.

• Symbolically:

c = 3d

Notes:

• In some resources, relations may be written in the form of

a mapping. For example, we might write the above

relation as

d → 3d

which may be interpreted as “The diagram number, d, is

related to the number of circles, 3d”.

The student text tends to write only the output expression

when representing relations symbolically, but it is

desirable to have students write relations using either an

equation.


7PR1.1 Formulate a linear

relation to represent the

relationship in a given oral

or written pattern.




Unit 1


solve patterns.


Paper and Pencil

1. Look at the diagrams below and answer the following

questions.

A. Draw the next figure.

B. Complete the table of values.

# of squares

(S)

# of dots

(d)

1

2

3

4

5

6

C. Describe the relationship between the number of squares

and the number of dots in words.

D. Write the relationship using the variables S and d.

E. Explain how you can use your relation to determine the

number of dots that would be used for 25 squares.

2.i) Each table represents a relationship between a diagram

number and a dot pattern.

ii) Let n represent the diagram number and let d represent the

number of dots in a diagram. Write a relation between n and

d.

A.

B.

Term Number 1 2 3 4 5

Term 6 7 8 9 10

Term Number 1 2 3 4 5

Term 5 8 11 14 17

Resources/Notes

Math Makes Sense 7

Lesson 1.4


Relations


Master 1.15, 1.25


ST: pp. 20–24


pp. 12–13




Unit 1


solve patterns.

Specific Outcome


7PR1 Demonstrate an




[C, CN, R]

(Cont’d)



• Students may tend to focus on the increasing pattern in the

number of circles, i.e. that the number of circles in one

diagram is three more than the number of circles in the

previous diagram. However, this does not express the

relationship between the two variables. Additionally, it is

not very useful if we want to find the number of circles in,

say, the 50th diagram. The relation, c = 3d, allows us to

find the number of circles in any diagram, e.g. for the 50th

diagram, there would be c = 3(50) circles.

• It is important to start students’ investigation of linear

relationships using concrete models followed by oral and

written descriptions. For the circle diagrams above, one

option is to provide counters to students and allow them to

construct the diagrams.

Students should investigate other patterns by examining

modifications of simpler diagrams. For example:

1 2 3 4

By adding one circle to each diagram, we get a new table:

Diagram

Number (d) 1 2 3 4 5

Number of

Circles (c) 4 7 10 13 16

The linear relationship is now described as follows:

• In words:

“The number of circles in a diagram is equal to

three times the diagram number plus one”.

• Symbolically:

c = 3d + 1

7PR1.1 Formulate a linear

relation to represent the

relationship in a given oral

or written pattern.

(continued)




Unit 1


solve patterns.


Paper and Pencil

The diagrams below show a series of triangular supports for a

bridge.

A. Continue the pattern above up to the seventh diagram.

B. Complete the chart to show pattern growth.

C. Describe in writing how the pattern grows.

D. Write an algebraic expression to show the term (t) and for

the term number (n).

Diagram number: (x) 1 2 3 4 5 6 7

Number of line segments

(y)

Resources/Notes

Math Makes Sense 7

Lesson 1.4

(continued)




Unit 1


solve patterns.

Specific Outcome


7PR1 Demonstrate an




[C, CN, R]

(Cont’d)



Students should also be able to provide contexts or real-life

situations for given relations, e.g.

• D = 2n

• D = 3N + 2

• y = 5x + 1

Example:

Suggest a real-life situation that could be represented by the

relation D = 3N + 2.

.

Students should show how the relation fits the situation. This

relationship could represent someone getting $5 after one hour

of work, $8 after two hours, and so on. Since the amount of

money earned increases by $3 per hour but the person earns $5

after only one hour, there must be a $2 bonus for taking the

job.

There are many other patterns students could investigate that

may be conveniently expressed using variables. For example, a

kitchen floor is being covered with black and white tiles. The

basic design is shown:

Several of these basic designs were fitted together to make a

pattern:


N 1 2 3 4 5

D 5 8 11 14 17

7PR1.2 Provide a context

for a given linear relation

that represents a pattern.

7PR1.3 Represent a

pattern in the

environment, using a

linear relation.




Unit 1


solve patterns.


Pencil and Paper

1. Consider how many people can be seated at “n” tables in the

following situation:

…n tables

4 people 6 people 8 people

at 1 table at 2 tables at 3 tables

A. Make a table showing the number of tables and number of

people in the first five seating arrangements.

B. Describe the pattern in words.

C. Explain what the variable n represents.

D. Use the variable to write a relation for the number of

people that can be seated at n tables.

E. How many people could be seated at 7 tables?

2. A taxi charges a base fare of $4, plus $1 per kilometre traveled.

A. Make a table showing the total cost of the taxi ride for the

first 5 kilometres.

B. Describe the pattern in words.

C. Write a relation for the cost of the taxi ride for d

kilometres.

D. How much would a 10 kilometre taxi ride cost?

3. Suggest a real-life situation that could be represented by each

relation:

A. d = 2n

B. m = 3p + 4

C. y = 5x - 1

Resources/Notes

Math Makes Sense 7

Lesson 1.4


Relations


Master 1.15, 1.25


ST: pp. 20–24


pp. 12–13

Math Makes Sense 7

Lesson 1.4

(continued)


also developed in Lesson

1.7 of Math Makes Sense

7.




Unit 1


solve patterns.

Specific Outcome


7PR1 Demonstrate an




[C, CN, R]

(Cont’d)



Students should be able to make a table of values showing the

number of black and white tiles in the first 5 designs and

describe the pattern.

Black Tiles

(b) 2 4 6 8 10

White Tiles

(w) 10 20 30 40 50

The number of white tiles, w, is five times the number of black

tiles, b, which can be expressed as w = 5b. NOTE: This

describes the relationship between two quantities within the

diagrams and not between the diagram number and number of

tiles.

7PR1.3 Represent a

pattern in the

environment, using a

linear relation.

(continued)




Unit 1


solve patterns.


Resources/Notes

Math Makes Sense 7

Lesson 1.4

(continued)


also developed in Lesson

1.7 of Math Makes Sense

7.




Unit 1


solve patterns.

Specific Outcome


7PR2 Create a table of

values from a linear

relation, graph the table of

values, and analyze the

graph to draw conclusions

and solve problems.

[C, CN, PS, R, V]



Students have had experience graphing in the first quadrant of

the Cartesian Plane. Teachers should be careful not to use

relations that will result in negative output values at this point.

Students have already learned to evaluate an expression by

replacing a variable with a number. The student resource refers

to Input/Output machines. The input number is substituted into

an expression to get the output.

This relationship can also be written as 12 +→ pp .

The previous achievement indicator can be addressed here also

by giving students an input/output machine and asking them to

describe the relation represented in the completed table. (See

student text p. 26-28.)

2 1p + input output

Input

p

Output

2 1p +

1 3

2 5

3 7

4 9

5 11

7PR2.1 Create a table of

values for a given linear

relation by substituting

values for the variable.




Unit 1


solve patterns.


Paper and Pencil

1. For each table, find the output.

A.

B.

C. In the first table the numerical coefficient is 2 and the

constant is 3. In the second table the numerical coefficient

is 3 and the constant is 2. Explain how these differences

affect the output.

{Sample Answer: The first output value for each

table is 5, but the output values in the second table

are growing more quickly because the input values

are being multiplied by 3 instead of 2}

2. Write the relation for the input/output machine.

Resources/Notes

Math Makes Sense 7

Lesson 1.5


Relations


Master 1.11, 1.16, 1.26


ST: pp. 25–28


pp. 14–16

This is a good time to do

the ‘Human Graph”

activity. See Teacher

Resource Master 1.8a

and 1.8b.

? input output

Input

n

Output

?

2

4

6

7

13

19

Input

n

Output

2n + 3

1

2

3

4

Input

n

Output

3n + 2

1

2

3

4




Unit 1


solve patterns.

Specific Outcome







and solve problems.

[C, CN, PS, R, V]

(Cont’d)



This is an extension in that students are now required to graph

the data from the table of values on the coordinate grid.

To plot a point, an input value and its related output value

should be considered as directions to move from the origin

(0,0) to another specific point on the coordinate plane. For

example, if the input value is 4 and the output value is 6, we

move 4 units to the right from the origin, then continue by

moving 6 units upward.

When students are provided with a context for a given linear

relation it does not always make sense to draw a line

connecting the points of a graph.

For example:

Number of

Triangles 1 2 3 4

Number of

Line

Segments

3 5 7 9

Number of Triangles-10 -8 -6 -4 -2 2 4 6 8 10

Number of Line Segments

-10

-8

-6

-4

-2

2

4

6

8

10

Relationship Between Number of Triangles and Number of Line Segments



values, using a linear

relation, and graph the

table of values (limited to

discrete elements).




Unit 1


solve patterns.


Paper and Pencil

1. The diagram below shows a series of triangular supports for

a bridge.

…

A. Continue the pattern above for up to five triangles.

B. Complete the chart to show the pattern.

C. Describe the relationship in words.

D. Predict the number of line segments for 10 triangles and

20 triangles.

E. Write a relation to show how the number of line segments

in related to the number of triangles.

F. Draw a graph to show the relation. Does it make sense to

join the points? Discuss the shape of the graph.

Interview/Journal

Refer to the table in Pencil and Paper question #1. A student was

asked to explain the relationship between the number of triangles

and the number of line segments. The student described the pattern

as follows: “It goes up by 2.” Ask students if they agree or

disagree, and have them explain their reasons.

Resources/Notes

Math Makes Sense 7

Lesson 1.6


Relations


Master 1.17, 1.27

PM 22


ST: pp. 30–34


pp. 17–19




Unit 1


solve patterns.

Specific Outcome







and solve problems.

[C, CN, PS, R, V]

(Cont’d)



We cannot plot points in between the given points since we are

not dealing with parts of triangles and an input value such as

1.25 does not make sense, therefore we should not connect the

points of the graph with a solid line. In cases where the input

values are discrete (such as this case), we do not connect the

points.

Discrete cases also arise when the input numbers are restricted

to natural numbers, whole numbers and – later in unit 8 –

integer values. For example, for the table

Input 0 1 2 3 4

Output 4 4.5 5 5.5 6

If we restrict the input values to only the whole numbers, then

we cannot connect the points of the graph.

Students should be reminded to label the axes, give the graph a

title and use appropriate scales.


values, using a linear

relation, and graph the

table of values (limited to

discrete elements).

(continued)




Unit 1


solve patterns.


Resources/Notes

Math Makes Sense 7

Lesson 1.6

(continued)




Unit 1


solve patterns.

Specific Outcome







and solve problems. [C, CN,

PS, R, V] (Cont’d)



Having already created a table of values and a graph of a given

relation the focus should be on interpreting the graph. Work

can also be done on interpolating (finding a point between two

known points) and extrapolating (finding a point that lies

beyond the known data). This would be done mainly by

“eyeballing” or estimating from the graph. Note that teachers

can use the terms interpolate and extrapolate in their classroom

discussions; however the terminology is not the focus of the

indicator so students are not expected to be responsible for the

terms.

Consider the following graph:

1 2 3 4 5 6 7 8 9 10

25

50

75

100

125

150

175

200

Car's Distance from Home

Dis

tan

ce (

km

)

Time (hours)

Students should describe the general pattern of the graph (the

graph goes upward to the right because the distance increases

as the number of hours travelled increases), and use the graph

to answer questions such as: How far from home was the car at

2 hours? At 4.5 hours? At 6 hours? At 0 hours?

7PR2.4 Describe, using

everyday language in

spoken or written form,

the relationship shown on

a graph to solve problems.

7PR2.3 Sketch the graph

from a table of values

created for a given linear

relation, and describe the

patterns found in the

graph to draw

conclusions; e.g., graph

the relationship between n

and 2n + 3.




Unit 1


solve patterns.


Paper and Pencil

The graph shows the number of swimmers allowed in a pool in

relation to the number of lifeguards on duty.

A. How many swimmers would be allowed for 12

lifeguards?

B. How many lifeguards would be needed for 50 swimmers?

C. Describe the pattern in words.

D. Write a relation for the number of swimmers for n

lifeguards.

E. If there are 12 lifeguards on duty, how many swimmers

are allowed in the pool?

F. Suppose you wanted to find out how many swimmers

would be allowed for 31 lifeguards. Which would be the

easier way to find the answer, extending the graph or

using the relation? Explain.

Resources/Notes

Math Makes Sense 7

Lesson 1.6

(continued)

2 4 6 8 10 12 Number of Lifeguards

Nu

mb

er o

f S

wim

mers

Lifeguards needed for Swimmers

20

40

60

80




Unit 1


solve patterns.

Specific Outcome







and solve problems.

[C, CN, PS, R, V]

(Cont’d)



Present to the class a collection of graphs and relations (in

word and algebraic expression forms) that would describe

them. Students should be able to match the description to the

most appropriate graph and explain the reasoning for their

choice.

For an example see Student Text p.34 #6 and p. 45 #9.

Example:

Faith says that Graph A shows 8 2y x= − , and Graph B shows

8y x= − . Is she correct? Explain how you know.

She is correct. Answers will vary when students explain how

they know.

0 1 2 3 4

Graph A 8

6

4

2

0 1 2 3 4

Graph B 8

6

4

2

7PR2.5 Match a given set

of linear relations to a

given set of graphs.

7PR2.6 Match a given set

of graphs to a given set of

linear relations.




Unit 1


solve patterns.


Journal/Portfolio/Presentation

1. Which relations can be matched with the graph? Explain.

Input Number1 2 3 4 5 6 7 8 9 10

Outp

ut N

um

ber

1

2

3

4

5

6

7

8

9

10

A. y = 2x - 1

B. y = 2x + 1

C. y = 2x

D. y = 1x + 2

E. The output number is equal to double the input number

increased by 1.

F. The output number is equal to double the input number

decreased by 1.

2. A teacher has $60 from left from her class fundraiser. She is

going to buy each student an ice cream cone. One ice cream

costs $2. Which graph best shows the amount of money she has

left after buying various numbers of ice cream cones? Explain.

A. B.

Number of Ice Cream Cones1 2 3 4 5

Am

ou

nt o

f M

on

ey L

eft (

$)

8

16

24

32

40

48

56

64

Number of Ice Cream Cones1 2 3 4 5

Am

ou

nt o

f M

on

ey L

eft (

$)

8

16

24

32

40

48

56

64

3. Which relation matches Graph A in #2? Explain.

A. y = 60 – 5x B. y = 60 – 2x

Resources/Notes

Math Makes Sense 7

Lesson 1.6

(continued)

Unit Review:

Page 45, #9




Unit 1


multiple ways.

Specific Outcome

7PR4. Explain the difference


an equation.

[C, CN]

(Cont’d)



Note: Other relevant material and elaboration can be found at

the beginning of Unit 6.

An algebraic equation is a mathematical statement that two

expressions are equal. For example, 3 6x = and 4 1 11m − = .

It is important for students to understand that equations like

x + 6 = 10 is the same as 10 = x + 6. Each side of the equation

has the same value. The phrase “is equal to” has been added to

equate the two expressions.

One way to develop the concept of equations is to build on the

work with relations. The key here is that we are looking for the

input values when we know the output values. For example,

what is the missing input value in the table:

N 1 3 4 ?

2N + 1 3 7 9 201

We could perform a guess and check routine:

• 2(10) + 1 = 21. An input of 10 is not large enough.

• 2(50) + 1 = 101. An input of 50 is not large enough.

• 2(100) + 1 = 201. Therefore, 100 is the correct input value.

This approach is a valuable one for students to explore.

However, students should also develop facility with writing an

equation to be solved. For the above problem, they could begin

by writing

2(?) + 1 = 201.

Eventually, students should start using the input variable to

write the equation, e.g. 2N + 1 = 201.




using an equation.




Unit 1


multiple ways.


Interview

Give an algebraic equation for each sentence

A. Six less than double a number is 12.

B. A number divided by 4 is 6.

C. Seven decreased by a number is 23.

D. The product of 5 and a number is 35.

E. Angela’s age five years from now is 16.

F. Roger’s height decreased by eleven centimetres is 135 cm.

Portfolio

Provide a table such as the following: Expression in

Words

Unknown

quantity

Choose a

Variable

Algebraic

Expression Equation

Twice Jenny’s

age increased

by 3 is 15.

Jenny’s age j 2j + 3

One more than

3 times a

number is 10.

A number n

Six less than Mark’s age is

10.

4x – 1 = 19

Sample responses

Expression in

Words

Unknown

quantity

Choose a

Variable

Algebraic

Expression Equation

Twice Jenny’s

age increased

by 3 is 15.

Jenny’s

age j 2j + 3 2j + 3 = 15

One more than

3 times a

number is 10. A number n 3n + 1 3n + 1 = 10

Six less than

Mark’s age is 10.

Mark’s

age m m – 6 m – 6 = 10

Four times the

number of

marbles in a

bag decreased

by 1 is 19.

Number

of marbles x 4x – 1 4x – 1 = 19

Resources/Notes

Math Makes Sense 7

Lesson 1.7


Relations


Master 1.18, 1.28


ST: pp. 35–37


pp. 20–21




Unit 1


multiple ways.

Specific Outcome

7PR4. Explain the difference


an equation.

[C, CN]

(Cont’d)



Students should learn to translate verbal statements into

equations, e.g.

“Three more than a number is 8”

• Choose a variable: Let n represent the number.

• Write an expression involving the variable: “Three more

than a number” can be written as n + 3

• Write the equation: n + 3 = 8

Caution: It is unwise to focus on using solely a key word

approach to writing equations. Students should be encouraged

to read verbal statements for meaning. The following examples

will illustrate the pitfalls of sole reliance on key words:

A) Kim has 10 chocolate bars. Pat has 5 times as many

chocolate bars as Kim. How many chocolate bars does

Kim have?

B) Kim has 10 chocolate bars. Kim has 5 times as many

chocolate bars as Pat. How many chocolate bars does

Pat have?

In example A), the key word times implies multiplication

should be used, which is correct for this example. In example

B), on the surface the same key word times implies

multiplication but the correct operation is division. Similarly,

many students will write 5 – x for the statement “5 less than a

number”, but they need to make sense of the statement,

perhaps using several numerical examples, e.g. “5 less than 8”

would be written as 8 – 5.

Other approaches to developing the concept of an equation are

covered in both the Teacher Resource p. 33-35 and the Student

Text p. 35- 37.



using an equation.

(continued)




Unit 1


multiple ways.


Resources/Notes

Math Makes Sense 7

Lesson 1.7

(continued)


.



Unit 1


multiple ways.

Specific Outcome





(Cont’d)



Students were previously expected to be able to identify the

constant term, numerical coefficient, and variable in an

algebraic expression (see Achievement Indicator 7PR4.1).

Students should be able to identify these in an algebraic

equation as well. For example, in the equation 5 7+ =x , the

constant terms are 5 and 7, the numerical coefficient is 1 and

the variable is x.

7PR4.5 Identify and




an equation.


.



Unit 1


multiple ways.


Portfolio

Ask students to create a classroom chart with the following

headings:

Equation Variable Numerical

Coefficient Constant

3b + 1= 7 b 3 1 & 7

Have students continue to add examples to the chart each day.

Resources/Notes

Math Makes Sense 7

Lesson 1.8


Relations


Master 1.19, 1.29

PM 30


ST: pp. 38–42


pp. 22–24




Unit 1

General Outcome: Represent algebraic expressions in multiple

ways.

Specific Outcome


7PR7 Model and solve,

concretely, pictorially and

symbolically, problems that

can be represented by linear

equations of the form:

• ax + b = c

• ax = b

• ,x

b aa

= ≠ 0

where a, b and c are whole

numbers.

[CN, PS, R, V]



At this time, the focus will be on solving equations concretely (e.g.

algebra tiles) and pictorially (e.g. a sketch of algebra tiles or a two-

pan balance), using whole numbers only. Solving linear equations

symbolically (algebraically) will be covered in Unit 6.

Students are expected to use concrete models when solving problems

and equations. They can then draw pictures of their models in order to

move from the concrete stage to the pictorial. It is also important for

students to verify the solutions to equations using their models.

As mentioned in the teacher resource, the yellow algebra tiles used in

the student text represent positive values and the red tiles will represent

negative. Other sets of tiles may vary in their color schemes. Class

agreement should be reached as to which color will represent positive

and which will represent negative. Some teachers may choose to create

a class set that matches the text using ‘fun foam’ or other similar

materials. For the purpose of this section, we will be working with

positive tiles only.

A full description of how to use algebra tiles to solve equations of the

form ax + b = c and ax = b is given in the Teacher Resource p. 36-41

and the Student Text p. 38-41.

(This achievement indicator is continued on the next page…)

7PR7.1 Model a given

problem with a linear

equation; and solve the

equation, using concrete

models, e.g., counters,

integer tiles.




Unit 1

General Outcome: Represent algebraic expressions in multiple

ways.


Paper and Pencil

1. The algebra tile diagram represents an equation.

A. What are the two expressions that make up the equation?

B. What is the equation?

C. Solve the equation. Draw pictures to represent the steps

you took to solve the equation.

2. Use tiles to solve each equation. Draw pictures to represent the

steps you took to solve each equation.

A. 7 + x = 15

B. 4x = 16

C. 23

x=

3. Three more than twice a number is 9.

A. Write an equation that can be solved to find the number.

B. Use tiles to solve the equation.

C. Verify the solution.

Extension

Write an equation such that its solution is your age. Describe a

problem that could be represented by your equation.


Play ‘Equation by Chance.’ See Teacher Resource Master 1.9.

Resources/Notes

Math Makes Sense 7

Lesson 1.8

(continued)




Unit 1


multiple ways.

Specific Outcome


7PR7 Model and solve,

concretely, pictorially and

symbolically, problems that

can be represented by linear

equations of the form:

• ax + b = c

• ax = b

• ,x

b aa

= ≠ 0

where a, b and c are whole

numbers.

[CN, PS, R, V]

(Cont’d)



Students will also have to model solutions for equations of the

form , 0x

b aa

= ≠ . In equations such as 54

x= or

15

4x = ,

students will need to be reminded that the 4

x means that the

whole consists of 4 equal parts and that we are referring to just

1 of those parts since we have just 1x.

54

x= would be modelled with algebra tiles as follows:

where

Three more fourths are needed to complete the whole:

Which means

7PR7.1 Model a given

problem with a linear

equation; and solve the

equation, using concrete

models, e.g., counters,

integer tiles.

(continued)

Represents x

Represents 1

Since four fourths

of x equals 1x, we

now have 1x

equals 20 and the

equation is solved.




Unit 1


multiple ways.


Resources/Notes

Math Makes Sense 7

Lesson 1.8

(continued)




Unit 1

Documents

Grade 7 Mathematics Unit 1 Patterns and · PDF fileUnit 1: Patterns and Relations Grade 7 Math Curriculum Guide 31 Unit 1 Overview Introduction Students will develop their ability