W1 – Lesson 3: Fractions

Developed by Alberta Distance Learning Centre ....................................................................................................... 1

Preview/Review Concepts W1 - Lesson 3 MathematicsGrade7

W1 – Lesson 3: Fractions

Warm-up:

• Icanusepatternblockstomodelfractions.

Fraction Tiles

How many?

1. are in ? _______ 2. are in ? _______

3. are in ? _______ 4. are in ? _______

5. are in ? _______

Based on the relationships above,

1. If = 1. = _______

2. If = 1. = _______

3. If = 1. = _______

4. If = 1. = _______

2

3

2

6

3

1/6

1/3

1/2

2/3

Preview/Review Concepts W1 - Lesson 3MathematicsGrade7

....................................................................................................... Developed by Alberta Distance Learning Centre2

Now, try some really fun shapes.

1. If + = 1, what is ? _______

2. If + = 1, what is + ? _______

3. If + = 1, what is + ? _______

4. If + = 1, what is ? _______

5. If - = 1, what is + ? _______

1/9

2/8 = 1/4

4

3/7

1



Review:

• Iunderstandfractionbasics.

Practice

Example 1: What fraction of the pie has been removed?

One of eight pieces has been removed; therefore, 18

has beenremoved.

Example 2: What fraction of the eggs is left in the carton?

Ten of twelve eggs are still in the carton; therefore 1012

or 56

is left in the carton.

3 numerator or dividend vinculum

7 denominator or divisor

properfraction: a fraction in which the numerator is lessthan the denominator

Example: 23

mixednumber: the sum of a whole number and a proper fraction

Example: 647

improperfraction: a fraction in which the numerator is greaterthan the denominator

Example: 72

vinculum: the line between the numerator and the denominator, showing that the two numbers are connected



Equivalent Fractions

Example: Write three equivalent fractions for 23

.

46

, 69

, and 812

are all equivalent fractions of 23

.

Multiplying or dividing the numerator and denominator by the same number can generate an unlimited number of equivalent fractions.

Practice

Fill in the blanks to complete the equivalent fractions.

a. 23 15= b.

17 21=

c. 4 18= d. 5 10

12=

e. 7

11 33= f.

618 9

=

g. 8

20 5= h.

33 199

=

i. 2 4

10= j. 3 21

28=

Fractions that represent the same amount are called equivalent fractions.

2 4 2 6 2 83 6 3 9 3 12

X 2

X 2

X 3

X 3

X 4

X 4

10

21

3

3

2 24

23

5 4



Reducing Fractions to Simplest Form

Example: Write the fractions in lowest terms.

a. 9 and 12 are both divisible by 3.

The simplest form of 912

is 34

.

b. 712

Because 7 and 12 do not have any numbers by which they are both divisible, the fraction is already in its simplest form.

Practice

Express each fraction in lowest terms.

a. 2 4= b. 4

12=

c. 6 9= d. 5

15=

e. 7 21

= f. 10 18

=

g. 8 24

= h. 50 100

=

i. 312

= j. 5 25

=

A fraction in simplest form is the fraction that cannot be divided to �nd an equivalent fraction.

9 312 4

3÷

3÷

A fraction in simplestform is also known as a reducedfraction or a fraction expressed in lowestterms.

Answers should always be given in lowestterms.

12

13

23

13

13

59

13

12

15

14



Objective:

• Icanaddandsubtractfractionswiththesamedenominators..

Practice

Solve. Express your answers in lowest terms..

a. + =1 13 3

b. + =1 57 7

c. + =2 49 9

d. 5 3

10 10+ =

e. 1 36 6+ = f. 11 5

20 20+ = g. 3 2

8 8+ = h. 10 8

12 12− =

i. 17 821 21

− = j. 13 315 15

− = k. 9 510 10

− = l. 7 29 9− =

Subtracting fractions with the same denominator.

Keep the denominator the same; subtract the numerators.

Example: 7 312 12

−

7 3 47 3 4 Therefore, 12 12 12

− −

Adding fractions with the same denominator

Keep the denominator the same; add the numerators.

Example: 1 24 4

1 2 31 2 3 Therefore, 4 4 4

23

67

6 29 3=

8 410 5

=

4 26 3=

16 420 5

=58

2 112 6

=

9 321 7

=10 215 3

= 4 210 5

=59



Objective:

• Icanaddandsubtractfractionswiththedifferentdenominators..

Practice

Solve. Express your answers in lowest terms..

a. 2 13 7+ = b.

1 36 4+ =

c. 1 42 9+ = d.

6 310 8

+ =

e. 5 5

20 8+ = f.

3 221 3

+ =

g. 10 112 4

− = h. 17 521 7

− =

i. 9 3

10 20− = j.

7 19 3− =

2 810 40

7 358 40

Example 1:

You need a common denominator. 8 and 10 are both factors of 40.

Change:

Change: Subtract:

7 28 10−

35 8 2740 40 40

−

1 34 12

Example 2:

Common denominator: 12

Change:

Add:

OrCommon denominator: 4

Change:

Add:

1 34 12

3 3 6 112 12 12 2

1 1 2 14 4 4 2

3 112 4

14 3 1721 21 21

+ = 2 9 1112 12 12

+ =

9 8 1718 18 18

+ =

10 25 35 740 40 40 8

+ = =3 14 1721 21 21

+ =

10 3 712 12 12

− =17 15 221 21 21

− =

18 3 15 320 20 20 4

− = =7 3 49 9 9− =

24 15 39+ =40 40 40



Objective:

• Icanmodeltheadditionandsubtractionoffractions.

Grid Model

Note: Use the columns and rows of a grid to represent different denominators.

Example 1: 1 13 4+ =

Example 2: 3 15 3− =

Practice

Draw the grid that represents best the solution. For this activity you do not need to simplify the fraction.

a. + =1 13 3

b. + =1 57 7

c. + =2 49 9

Total squares = 12Number of shaded squares = 7

Answer = 7

12

Total squares = 14Number of shaded squares left = 4

Answer = 4

15

1 of 3 rows = 14

colour in 4 squares 1 of 4 columns = 13

colour in 3 squares

3 of 5 rows = 35

colour in 9 squares1 of 3 columns = 1

3erase 5 squares

2435

1318

2024

2 2=4 6

2 2 =7 5

1 2=2 6



d. 1 24 3+ = e. 1 3

5 4+ = f. 1 3

3 10+ =

g. 3 14 3− = h. 4 2

5 3− = i. 6 2

7 6− =

j. 8 49 6− = k. 7 1

8 2− = l. 2 1

3 4− =

1920

1930

215

2242

512

512

1112

1254

616

1 24 3

1 35 4

133

10

1 4 5 3 12 12

2 10 2 3 15 15

2 14 22 6 42 42

4 36 12 6 54 54

1 8 6 2 16 16 1 3 5

4 12 15



Summary and Practice:

• Usingwhatyoulearnedanswerthefollowingquestions.

1. On the chart below, summarize the steps needed to add or subtract fractions withthe same denominator and fractions with different denominators.

Same Denominators Different Denominators

Steps to add fractions

Steps to subtract fractions

2. Express the fractions in lowest terms.

a. 618

b. 954

c. 1648

d. 1421

3. Using the grid below, write an addition equation represented by the grid.

Answersmayvary

Example:

1. Keepthedenominatorthesame.

2. Addthenumerators,thenreduceifpossible.

Answersmayvary

Example:

1. Findacommondenominator.

2. Changethefractionstoanequivalentfractionwiththenewdenominator.

3. Addthenewnumerators,thenreduceifpossible

Answersmayvary

Example:

1. Keepthedenominatorthesame.

2. subtractthenumerators,thenreduceifpossible.

Answersmayvary

Example:

1. Findacommondenominator.

2. Changethefractionstoanequivalentfractionwiththenewdenominator.

3. Subtractthenewnumerators,thenreduceifpossible

29

=16

=13

=27

=

3 1 145 3 15+ =



4. Model using a grid.

a. 3 16 3+ b. 9 3

10 4−

5. 13

of entries in a pet show are cats. 12

of the entries are dogs.

What fraction of the animals is not a cat or a dog?

6. George spent 28

of his spare time playing video games, 26

practising basketball, 14

practising guitar, and the remainder watching TV. What fraction represents the

time he spent watching TV?

6 340 20

= =

1 1 2 3 53 2 6 6 66 5 16 6 61 of the animals is not a cat or a dog.6

+ = + =

− =

2 2 1 6 8 6 20 58 6 4 24 24 24 24 66 5 16 6 6

1George spent of his time watching TV.6

+ + = + + = =

− =

1518

=



W1 – Lesson 4: Fractions, Decimals, and Percents

Objective:

• Icanchangeanimproperfractiontoamixednumberandback.

Mixed number: sum of a whole number and a proper fraction.

Example: 647

Writing as a mixed number:

Number of whole pizzas: 2

Fraction of remaining pizza: 34

The mixed number is written as: 324

Improper fraction: when the numerator is greater than the denominator.

Example: 72

Writing as a improper fraction:

Total remaining slices: 11

Number of slices per pizza: 4

The improper fraction then is: 114



Converting mixed numbers to improper fractions:

Practice

Express the mixed number as an improper fraction.

a.223= b.

335=

c. 217= d. 16

2=

e.429= f.

136=

Converting improper fractions to mixed numbers:

Practice

Express the improper fraction as a mixed number.

a. 112= b.

125

=

c. 237

= d. 143

=

e. 149

= f. 156

=

Example:

1. Multiply the whole number by the denominator. (3 × 4)

2. Add step 1 to thenumerator (12 + 1)

3. Write step 3 as afraction over theoriginaldenominator

134

134

Example:

1. Divide the numerator by the denominator (13 ÷ 3)

2. Write step 1 asthe whole numberand the remainderas a fraction over the original denominator (4 )1

3

133

83

185

97

132

229

196

152

225

237

243

519

3 12 26 2=



Objective:

• Icanaddandsubtractmixednumbers.

Practice:

Solve. Express your answer in lowest terms.

a. 1 532 7+ = b.

4 11 56 4+ = c.

2 15 26 2+ =

d. 1 36 22 7− = e. 3 13

4 3− = f. 4 22 1

5 3− =

Example 1:

1. Find a common denominator for the fractions.2. Subtract the whole numbers.3. Subtract fractions as usual.

7 and 2 are both factors of 14.

Change:

Change:

Subtract: 12 7 54 2 214 14 14

−

6 14 27 2−

6 12 7 14

1 72 14

Example 2:

Common denominator: 18

Change:

Change:

Add:

4 119 2

4 89 181 92 18

8 9 171 118 18 18

7 10 17 33 3 414 14 14 14

+ = =8 3 111 5 6

12 12 12+ =

2 3 55 2 76 6 6+ =

7 6 16 2 414 14 14

− =9 4 53 3

12 12 12− =

12 10 22 1 115 15 15

− =



Objective:

• Icanconvertbetweenfractionsanddecimalnumbers.

Converting fractions and terminating decimals:

The shaded area of this grid, as a fraction, is: 36100

The shaded area, written as a decimal, is: 0.36

But consider a fraction like 25

: What would it look like on a grid?

Use your calculator to calculate 2 ÷ 5. What do you notice?

Repeating decimal: a decimal number in which a block of one or more digits repeats in a pattern.

Terminating decimal: a number that is complete after a certain number of digits with no repeats.

Fractions with

denominators of 10,

100, or 1000 are easy

to convert into

decimals by using

place values.

Examples:

210 = 0.2 (tenths place)

7100

= 0.07 (hundredths)

41000

= 0.004 (thousandths)

1. Write 2/5 as a fraction with a denominator of 100:100

2. Shade in the same number of squares as thenumerator.

3. Use the place value to determine the decimal: ______

40

0.4

Theanswersarethesame.

Turn any fraction into a decimal by dividing the numerator by the denominator.



Example:

25 10.25100 430.3

10

= =

=

Practice

Convert the fractions to decimals, and the decimals to fractions.

a. 15= b. 6

8= c. 3

20=

d. 625

= e. 1440

= f. 924

=

g. 0.7 = h. 0.22 = i. 0.12 =

j. 0.4 = k. 0.33 = l. 0.54 =

Converting fractions and repeating decimals:

Example 1:

5 0.555 555 555 555... 0.592 0.2857142857142857... 0.2857147

= ⇒

= ⇒

Bar notation: a method of writing a repeating decimal using a bar above the digits to represent a repeat.

Turn any fraction into a decimal by dividing the numerator by the denominator.

Convert a terminating decimal to a fraction by using the place value as the denominator and reduce if necessary.

0.2 0.75 0.15

0.24 0.35 0.375

22 11100 50

=

33100

54 27100 50

=

12 3100 25

=7

10

=4 2

10 5



Example 2:

1 2 3 40.1, 0.2, 0.3, 0.49 9 9 9Predict5 6 7_____, _____, _____9 9 9

= = = =

= = =

Example 3: 0 .27

Think: We know 0.3 is 310

, so the fraction must be a smaller fraction (smaller numerator

or larger denominator) so it may be 210

or3

11. Because 2

10 is non repeating it

must be 311

Check: 3 ÷ 11 = 0.27272727…

Practice


a. 17= b. 2

9= c. 5

11=

d. 6

33= e.

299

= f. 724

=

g. 0.7 = h. 0.72 = i. 0.003 =

j. 0.63 = k. 0.28514 = l. 0.16 =

Convert a repeating decimal to a fraction by looking for repeat patterns.

0.5 0.6 0.7

0.142857 0.2 0.45

0.45

79

711

0.02 0.2916

811

31000

16

27



Objective:

• Icanexpressapercentasadecimalnumber.

Per cent: means “out of 100”

Practice

Write the percent as a decimal.

a. 38% =

b. 29% =

c. 2% =

d. 67% =

e. 8% =

Examples:

78% = 78 ÷ 100 = 0.78

24% = 24 ÷ 100 = 0.24

Convert a percent into a decimal number by dividing by 100.

Examples:

0.777 = 0.777 × 100 = 78%

0.542 = 0.542 × 100 = 54%

Convert a decimal to a percent by multiplying by 100. Remember to round the answer if necessary.

Practice

Write the decimal as a percent.

a. 0.281 =

b. 0.458 =

c. 0.894 =

d. 0.676 =

e. 0.073 =

0.38

0.29

0.02

0.67

0.08

28%

46%

89%

68%

7%



Objective:

• Icanexpressapercentasafraction

Practice

Write the percent as a fraction.

a. 88% =

b. 27% =

c. 12% =

d. 6% =

e. 48% =

Practice

Write the fraction as a percent.

a. 2562

=

b. 3453

=

c. 1920

=

d. =785

e. 57=

Convert a percent into a fraction by putting the percent as the numerator over a denominator of 100. Simplify.

75% = 75

100= 3

4

35% = 35100

= 720

Example:

Convert a fraction to a percent by �rst converting the fraction to a decimal and multiply by 100. Remember to round the answer.

Example: 2836

28 ÷ 36 = 0.777 × 100 = 78%

88 22100 25

=

2710012 3

100 25=

=6 3

100 5048 12

100 25=

40%

64%

95%

71%

8%



Converting mixed numbers to improper fractions:

1. How many squares are shaded: _____

a. Write as a fraction: _____

b. Write as a decimal: _____

c. Write as a percent: _____

2. How many squares are white: _____

a. Write as a fraction: _____

b. Write as a decimal: _____

c. Write as a percent: _____

3. Add the percent of shaded to the percent of white. What do you notice?

59%

0.59

59

41%

0.41

41

Thatittotals100%.

59100

41100



Objective:

• Icanusepercentstosolveproblems.

Practice

a. 25% of 44 =

b. 3% of 70 =

c. 29% of 11 =

d. 48% of 63 =

e. 21% of 89 =

Calculating a number from a percent:

Create two proportional fractions to find the missing number.

Example:

25% of ____ = 10 25% = 25 1 1 10100 4 4 ?

= = the only answer that will fit

in the blank and remain

proportional is 40!

Practice


a. 12% of _____ = 54 b. 20% of _____ = 45

c. 30% of _____ = 27 d. 4% of _____ = 24

Proportion: two equivalent ratios or

fractions ( 14

is proportional to 520

).

To calculate a percent of a number, multiply the number by the decimal equivalent of the percent.

Example: 40% of 600.40 6024

×

450 225

90 600

11

2.1

3.19

30.24

18.69



Problem solving with percents:

Example 1:

45% of a class with 20 students is boys. How many girls are there?

Method 1: 45% of 20 = number of boys

0.45 × 20 = 9 boys

20 students – 9 boys = 11 girls

Method 2: 45% are boys, which means 100% - 45% are girls

= 55% girls

0.55 × 20 = 11 girls

Example 2:

How much would a 15% tip be on a meal costing $48.50?

15% of 48.50 = 0.15 × 48.50 = $7.28

A 15% tip would be $7.28.

Example 3:

35% of a sport shop’s income is from hockey equipment. Last year the shop sold

$4970 in hockey equipment. What was the shop’s total income?

35% of total income = hockey income

35 4970 7 4970 4970 7 710 20 710 $14 200100 ? 20 ?

= = ÷ = × =

The shop made $14 200 last year.



Practice

Solve the word problem.

a. Steven got 1420

on a science test and 3650

on a math test. On which test did he score a

higher mark?

b. 18 adults and 30 children go on a field trip. What percent of the group are children?

c. A pair of jeans advertised for $34.97 is 35% off. What is the sale price of the jeans?

d. Calculate the total price of a CD for $9.65, plus 5% tax.

Science: 14 ÷ 20 = 0.7 = 70%

Math: 36 ÷ 50 = 0.72 = 72%

Hescoredhigheronhismathtest.

18 + 30 = 48 people total

30 ÷ 48 = 0.625 = 63%

63%ofthegrouparechildren.

34.97 × .35 = 12.24

34.97 – 12.24 = 22.73.

Thejeanswere$22.73

9.65 × 0.05 = 0.48

9.65 + 0.48 = 10.13

TheCDcosts$10.13.




• Usingwhatyoulearned,answerthefollowingquestions.

1. Complete the chart.

Mixed number

Improper fraction

Model

126

263

2. Solve.

a. 2 11 45 2+ =

b. 1 33 22 8+ =

c. − =4 25 15 3

d. − =5 19 26 4

3. Convert the fractions to decimals. Decide if the fractions are >, <, or =.

a. 9 8 11 14

b. 3 10 13 17

4 5 91 4 510 10 10

+ =

4 3 73 2 58 8 8+ =

12 10 25 1 415 15 15

− =

10 3 79 2 712 12 12

− =

> <0.82 > 0.57 0.23 < 0.59

136

318

118

283



4. Complete the table.

Fraction Decimal Percent

a. 1125

b. 0.23

c. 63%

d. 48%

e. 0.06

f. 39

5. Laura’s curling team won 9 of 17 games. What percent of the games did her teamwin?

6. A bag of jelly beans has 32% cherry flavoured, 45% lemon, and the rest are grapeflavoured. What percent of the jelly beans are grape flavoured?

9 ÷ 17 = 0.529 × 100 = 53%

Shewon53%ofhergames.

100%-32%-45%=23%

23%ofthejellybeansaregrapeflavoured.

0.44

0.63

0.48

0.33 33%

23%

6%

44%

23100

63100

1225

350



7. Bob mixes a punch with 3 parts ginger ale and 5 parts fruit juice. What percentageof the punch is ginger ale?

8. Jessica and Nelly both earn commission (a percent of any sales they make) at acomputer store. Last month Jessica earned $825 on sales of $5500 and Nelly earned$800 on $5100 sales. Who has the better commission percent?

9. A drum set usually costs $548.00. A set is on sale at Drummers Dream for 40%off. Drum Discounters offers the same set at 20% off, plus a further 20% off thediscounted price. Which store would have the lowest price?

3 + 5 = 8 parts total

3 ÷ 8 = 0.375 × 100 = 37.5%

Thepunchis37.5%gingerale.

Jessica825 ÷ 5500 = 0.15 × 100 = 15%

Nelly800 ÷ 5100 = 0.157 × 100 = 16%

Nellyhasthebettercommission.

Drummers Dream Drum Discounters

548 × 0.40 = 219.2 548 × 0.20 = 109.6548 – 219.2 = 328.8 548 – 109.6 = 438.4

438.4 × 0.20 = 87.68438.4 – 87.68 = 350.72

$328.80 $350.72

DrummersDreamhasthelowestprice.



W3 – Lesson 4: Plots and Plotting on a Cartesian Plane

Objective:

• Icanidentifypartsofacartesianplane.

Cartesian plane

Also called a coordinategrid, the plane is formed when two

perpendicular number lines intersect at 0. The horizontal line is the x-axis,

and the vertical line is known as the y-axis. The point where the two lines

meet is called the origin. Each section of the grid is called a

quadrant (1, 2, 3, and 4).

Quadrant 1

Quadrant 4

Quadrant 2

Quadrant 3

x-axis

y-axis

origin



Objective:

• Icanplotpointsonacartesianplane.

Ordered Pairs

A point on the plane is located by an ordered pair (x, y).

The first integer (x-coordinate) tells how far to go on the x axis. The second integer (y-coordinate) tells how far to go on the y-axis.

Example: P(-5, 7) means a point labeled P that is 5 units left, and 7 units up.

-2 -4 -6 6 4 2

2

4

6

-6

-4

-2

x

y

P(-5, 7)



Practice:

1. a. Plot and label the following points on the grid below.

A(4, 5) B(-3, -1) C(-2, 4) D(0, 7) E(6, 0) F(3, -6)

b. What is the xcoordinate for point C?

c. What are thecoordinates for theorigin?

d. Which quadrant ispoint B in?

e. What is the ycoordinate for point D?

2. a. What are the coordinates of each point?

i. Q ii. R iii. S

b. Plot each point on the grid.

i. W(0, -15)

ii. Z(30, 0)

-2 -4 -6 6 4 2

2

4

6

-6

-4

-2

x

y

y

x -20 20 40 -40

20

40

-20

-40

Q

R

S

-2

(0, 0)

Quadrant 3

7

-2 -4 -6 6 4 2

2

4

6

-6

-4

-2

x

y

A

B

C

D

E

F

(-30, -10) (0, -30) (40, 0)

y

x -20 20 40 -40

20

40

-20

-40

Q

R

S

W

Z



Objective:

• Icancreateshapesanddesignsusingpointsonacartesianplane.

Example: Points J, K, and L are three points of a parallelogram, what coordinates should point M be to complete the figure?

M(30, 20) would complete the parallelogram.

Practice:

1. a. Draw a closed five sided figure on the grid below.

b. List the coordinates of the vertices for the figure you drew.

y

x -20 20 40 -40

20

40

-20

-40

J

K L

-2 -4 -6 6 4 2

2

4

6

-6

-4

-2

x

y

Answers will vary.

Answers will vary.



2. Plot and connect the following series of points.

a. Shape A: (2, 1), (2, 9), (9, 9)

b. Shape B: (-5, 1), (-1, 5), (-5, 9), (-9, 5)

c. Shape C: (6, -1), (9, -4), (4, -9), (1, -6)

d. Shape D: (-3, -2), (-5, -4), (-7, -2), (-9, -4), (-5, -8), (-1, -4)

e. What quadrant is shape D in?

f. Is there a way to tell which quadrant shape A would be in before plotting thepoints?

-2 -4 -6 6 4 2

2

4

6

-6

-4

-2

x

y

-2 -4 -6 6 4 2

2

4

6

-6

-4

-2

x

y

Quadrant 3

Since all the ordered pairs are positive it will be in Quadrant 1.

A

B

DC



3. Points A, B and C are the vertices of a square, what are the coordinates for point Dthat would complete the square? Plot the coordinate and draw the shape.

4. a. Draw a Cartesian plane using appropriate units for the axes so the following points can be plotted.

b. Plot the following points.

(-10, -10), (-5, 0), (20, -15), (30, 5), (5, 15), (10, 25), (-20, 20)

c. Connect the points in order. Which quadrant is the shape pointing to?

y

x -20 20 40 -40

20

40

-20

-40

B

A

C

y

x -20 20 40 -40

20

40

-20

-40

B

A

C

D

Quadrant 2

y

x -20 20 40 -40

20

40

-20

-40

The coordinates are (20, -40).



1. Fill in the blanks.

a. An ordered pair is in Quadrant 1 when the x coordinate is a ____________integer, and the y coordinate is a ____________ integer.

b. An ordered pair is in Quadrant 2 when the x coordinate is a ____________integer, and the y coordinate is a ____________ integer.

c. An ordered pair is in Quadrant 3 when the x coordinate is a ____________integer, and the y coordinate is a ____________ integer.

d. An ordered pair is in Quadrant 4 when the x coordinate is a ____________integer, and the y coordinate is a ____________ integer.

e. When the x coordinate is equal to 0 the point is in no quadrant because it is onthe ____________.

f. When the y coordinate is equal to 0 the point is in no quadrant because it is onthe ____________.

2. a. Plot the following points.

A(0, 30) B(10, 0)

C(-35, 0) D(-15, -15)

E(-20,-45) F(0, -30)

G(20, -45) H(15, -15)

I(35, 0) J(10, 0)

K(0, 30)

b. Connect the points in order from A to J. Which quadrants have a point of thefigure in them?

y

x -20 20 40 -40

20

40

-20

-40

positive

positive

positivepositive

negative

negative

negativenegative

y axis

x axis

y

x -20 20 40 -40

20

40

-20

-40

Quadrants 3 and 4.



3. Patrick was playing a game of Battleship with his brother. If his brothers’ shipswere arranged like the image below, what are the three coordinates Patrick needs towin the game? (You are looking for the coordinates of the unmanned ships.)

4. a. Draw a Cartesian plane using an appropriate unit for the axes.

b. Plot the following points.

(30, 50), (30, 130), (10, 130), (10, 90), (-10, 130), (-30, 90), (-30, 130), (-50, 130),(-50, 50), (-30, 50), (-10, 90), (10, 50)

(4, 10)(3, 7) and(6, 3)

y

x -30 30 60 -60

30

60

-30

-60



5. Plot and connect the points of each series.

6

4

2

- 2- 4- 6642

y

x

- 2

- 4

- 6

a. (-3, -2)+(-1, -2) b. (1, -9)+(1, -5)+(3, -9)+(3, -5)

c. (10, 7)+(10, 5) d. (2, 5)+(0, 5)+(0, 9)+(2, 9)

e. (7, 7)+(8, 5) f. (0, 7)+(1, 7)

g. (-5, -9)+(-5, -5)+(-3, -5) h. (0, -2)+(2, -2)+(2, 0)+(0, 0)+(0, 2)+(2, 2)

i. (-5, -7)+(-4, -7) j. (4, 9)+(4, 5)

k. (-10, 9) + (-12, 8) + (-12, 6) + (-10, 5) +(-10, 7)+(-11, 7)

l. (-6, 5)+(-6, 9)+(-4, 9)+(-4, 5)+(-6, 5)

m. (-3, 5)+(-3, 9)+(-2, 6)+(-1, 9)+(-1, 5)

n. (-9, 7)+(-8, 7) o. (9, 9)+(10, 7)+(11, 9)

p. (3, 9)+(5, 9) q. (-3, 2)+(-1, 2)

r. (6, 5)+(6, 9)+(8, 9)+(8, 7)+(6, 7) s. (-2, -5)+(-2, -9)+(0, -9)+(0, -5)

t. (-2, -2)+(-2, 2) u. (-7, 5)+(-9, 5)+(-9, 9)+(-7, 9)

6

4

2

- 2- 4- 6642

y

x

- 2

- 4

- 6



W3 – Lesson 5: Transformations

Objective:

• Icanmovepointsonacartesianplane.

Translations

Example: A triangle ABC shown below has been moved 6 units to the right and 3 unitsup. If the coordinate of angle B is (-7, -4), what are the coordinates of angle B’.

Answer: Since the slide was (+6, +3), the coordinate for angle B’ is ((-7+6), (-4+3)) or (-1, -1).

Also called a slide, occurs when a figure is moved in one direction in a

straight line from its original position. The final figure is identical to the

original figure.

A

B C

A’

B’ C’

Right 6

Up 3



Practice:

1. Translate the figure 2 units to the right and 1 unit down (2, -1).

2. Translate the figure (4, 0).


y

x

y

x

Note: When translating a figure x refers to the horizontal movement. + = right, – = left y refers to the vertical movement + = up, – = down

y

x



4. Translate the figure (4, -6).

5. Translate the figure (-7, -4).

6. Translate the figure (-3, 3).

y

x

y

x

y

x



Objective:

• Icanrotateafigureonacartesianplane.

Rotations

Example 1: A polygon ABCDE, shown below, is rotated 90° counterclockwise around the origin. If point A was originally (4, 6), what is point A’?

Answer: From the point of rotation, the distance to point A is 6 up and 4 across. Rotating the arrows counterclockwise 90° (or +90°,) but still starting from the point of rotation, has the arrows pointing at (-6, -4).

Therefore, point A’ is (-6, -4)

A rotation turns a shape around a point, called point of rotation. The

rotation can be any angle, and can be clockwise (negative), or

counterclockwise (positive).

A

B

C

A’

B’

C’

D

E

D’ E’

Point of rotation

hint: Trace the image onto tracing

paper, put your pencil onto the

point of rotation, and turn the

tracing paper to see the position of

the final image.



Example 2: A triangle ABC is rotated 180° clockwise around point Q. Draw and label triangle A’B’C’?

Practice:

1. Rotate the figure 90° clockwise about point A.

A’

B’

C’

Q

A

B

C

remember:A circle has 360°.

90° = a quarter turn

180° = a half turn

270° = three quarters turn

y

x

A



2. Rotate the figure 180° about point A.

3. Plot the points W(2, 0), X (5, 0), Y (1, 3). Rotate the figure 180° about the origin.

4. Plot A(1, -2), B(0, 2), C(3, 2), D(3, -3). Rotate the figure 90° about the origin.

y

x

y

x A

y

x

W’

B’

D’

W

B

D

X’

A’

X

C

Y’

C’

Y

A



5. What is the degree of rotation and point of rotation of triangle ABC?

6. What is the degree of rotation and point of rotation of triangle ABC?

y

x

A

A’

B’

C’

B

C

y

x

A

A’

B’

C’

B C

90° about the origin

-90° about the origin



Objective:

• Icanreflectafigureonacartesianplane.

Reflections

Example 1: A polygon ABCDE, shown below, is reflected across the x-axis. If point A was originally (0, 6), what is point A’?

Answer: From the line of reflection, the perpendicular distance to point A is 6. Which means point A’ would be 6 units in the opposite direction from the line of reflection.

Therefore, point A’ is (0, -6)

A reflection or flip, creates a mirror image of the shape. The mirror line

would be a line of symmetry for the shape and its image.

Line of reflection

A

B

C

A’

B’

C’

D

E

D’

E’

hint: Trace the image onto tracing

paper including the mirror line. Flip

the paper over and realign the

mirror line to see the position of the

final image.



Example 2: Draw triangle A’B’C’ when triangle ABC is reflected across line QR.

Practice:

1. Reflect the figure across the y-axis.

Q A

B

C

R

C’

B’ A’

y

x



2. Reflect the figure across the x-axis.

3. Reflect the figure across the given line.


y

x

y

x

y

x



5. Draw the line of reflection.


y

x

A

A’

B’

C’

B C

y

x

A A’

B’

C’

B C




• Usingwhatyou'velearned,answerthefollowingquestions.


a. A word describing a transformation where a figure stays the same except for itslocation is _________________.

b. If point A(5, 0) was translated (2, -1) the new coordinates for A’ would be________.

c. The point that a rotation occurs around is called the ____________________. Itdoes not always have to be the origin.

d. A negative rotation means the image is turned in a __________________ direction.

e. Rotating an image -90° has the same result as rotating it _______°.

f. The line used to create a mirror image is called a ____________________.

g. If point G(-4, 2) was reflected across the y-axis, the coordinates of point G’ wouldbe ______________.

h. If point H(3, -5) was reflected across the x-axis, the coordinates of point H’ wouldbe ______________.


y

x

translation

line of reflection

point of rotation

counterclockwise

A’(7, -1)

G’(4, 2)

H’(3, 5)

270



3. Translate the figure (-3, -2).

4. Rotate the figure -90° about point A.

5. Rotate the figure 180° about point A.

y

x

y

x

A

y

x A





8. Danny created this image of arrows by reflecting itover the y-axis.

a. Would it be possible to make the sametransformation using only a translation? Whyor why not?

b. If Danny wanted to create this image without using a reflection, whattransformation(s) could he use instead?

y

x

y

x

A

A’

B’ C’

B

C

D

D’

No, a translation keeps the imageidentical and will not point the arrow inthe opposite direction.

Because the image is symmetrical, Danny can rotatethe image 180° then translate it into the appropriateposition.



5. Plot and join the coordinates of CDE. C(3, 2), D(-2, 2), E(3, -2).

a. Translate CDE 2 units left and 3 units up to form C’D’E’.

b. Rotate C’D’E’ -90° about the origin forming C”D”E”

6. Plot and join the points to the shape. P(1, 0), Q(4, 0), R(3, 2), S(1, 2).

a. Translate the image 4 units left formingP’Q’R’S’

b. Reflect the image across the x-axis formingP”Q”R”S”. Shade in this figure.

c. Plot figure PQRS again on the second Cartesianplane.

d. Repeat steps a. and b. but start with the reflection,then the translation. (Do step b then step a)

e. Compare the final images. Are they the same? Explain.

y

x

y

x

y

x

Yes the final image is the same since thetransformations were the same. The differencewould be the in between image.

D

P

P

R

R

C

Q

Q

D'

E

S

S

C' D"

P"

P"

R"

R"

P'

P'

R'

R'

Q'

Q'

S'

S'

E'

Q"

Q"

C"

S"

S"

E"


Preview/Review Concepts W2 - Lesson 4 Mathematics Grade 7

W2 – Lesson 4: Statistics

Objective:

• Icanunderstandtermsusedinstatistics.

Collecting Data

Example: Malcolm wanted to know which subject was the most popular at School. He conducted a personal survey asking all the grade 7 students which subject was their favorite. Using tally marks, his data looked like this:

Survey ResultsSubject Tally

MathScienceLanguage ArtsSocial Studies

He used the tally chart to make a frequency table. The table looked like this:

Survey ResultsSubject Tally Frequency

Math 14Science 12Language Arts 8Social Studies 9

Personal Survey: collecting

data in person. Data can also

be collected by mail, telephone,

newspapers, books, or magazines.

Frequency Table: a table

that uses tallies to count the

frequency (number) of data.

Preview/Review Concepts W2 - Lesson 4Mathematics Grade 7


Using the database he just created, Malcolm could now create an organized table:

Organized Survey ResultsMath 14Science 12Language Arts 9Social Studies 8

Using the organized results, Malcolm could answer the following questions.

1. Which subject was the most popular? Math

2. Which subject is the least popular? Language Arts

3. What was the sample size (number ofpeople surveyed)?

14 + 12 + 9 + 8 = 43 people

Database: an organized

collection of data or information.

Sample: a selection of the

total information.



The data are arranged in alphabetical order

Peanut butter cookies sold the most

Snickerdoodles sold the least

Four boxes of shortbread cookies sold

Three boxes of raisin cookies sold

Practice:

Use the pictograph to answer the following questions.

Cookie SurveyChocolate chip

Gingerbread

Peanut Butter

Raisin

Shortbread

Snickerdoodle

Vanilla

Legend: Each represents one box of cookies sold

1. How are the data in the pictograph arranged?

2. Which cookie sold the most?

3. Which cookie sold the least?

4. What was the frequency of shortbread cookies sold?

5. What was the frequency of raisin cookies sold?



10.8

12.4

12.5

16.8

Mean: the arithmetic average.

Objective:

• IcanIcanfindthemeanofasetofdata.

Finding the mean

Example:

Basketball ResultsGame Number 1 2 3 4 5Points scored 24 51 34 22 43

Find the total points over all five games:

24 + 51 + 34 + 22 + 43 = 174

174 points in five games.

Find the mean by dividing the total points by the number of games.

174 ÷ 5 = 34.8 (but since you can’t score 0.8 points in basketball round to 35)

The mean (average) points were 35 per game.

Practice:Find the mean of the following sets of data. Round your answer to one decimal point.

a. 24, 5, 14, 9, 32

b. 11, 5, 8, 6, 20, 15

c. 2, 9, 22, 12, 26, 5, 11

d. 18, 12, 9, 11

How many games:

scored close to 35 points?Only one out of five games.

scored above 35 points?Two out of five games

scored below 35 points?Two out of five games



1.3, 1.5, 1.8, ____, 2.1, 2.2, 2.3(1.8 + 2.1) ÷ 2 = 1.95 or 2

11, 23, 34, ___, 44, 56, 62(34 + 44) ÷ 2 = 39

123, 132, 138, 205, 234138

149, 156, 218, ___, 219, 223, 321(218 + 219) ÷ 2 = 218.5

Objective:

• Icanfindthemedianofasetofdata.

Finding the median

Example:

Some prices are listed below, find the median.

$215, $211, $246, $213, $287

Arrange in ascending order:

$211, $213, $215, $246, $287

Since there is an odd number of numbers, the middle number is the median: $215

Example 2:

Find the median of the numbers.

49, 62, 57, 44, 51, 67, 46, 54


44, 46, 49, 51, _____, 54, 57, 62, 67

Since there is an even number of numbers, the median will be exactly between the two middle numbers:

(51 + 54) ÷ 2 = 52.5

Practice:

Find the median of the following sets of data.

a. 1.3, 2.1, 1.8, 2.3, 2.2, 1.5 b. 149, 218, 321, 156, 223, 219

c. 23, 56, 34, 11, 44, 62 d. $123, $234, $132, $138, $205

Median: the middle value in a

distribution.



3, 15 ,15, 15, 17, 17, 27, 40, 44, 44Mode: 15

11, 11, 12, 12, 13, 14, 15Mode: 11, 12

22, 23, 24, 32, 32, 33, 34Mode: 32

68, 79, 84, 84, 85, 90, 91, 100Mode: 84

Objective:

• Icanfindthemodeofasetofdata.

Finding the mode

Example:

Steven counted the number of people in the cars that stopped in front of the school. Find the mode.

2 3 1 2 4 4 1 3 4 52 2 2 3 3 2 5 2 1 4

Arrange the numbers in ascending order:

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5

Count the frequency of each number:

1’s – 3 2’s – 7 3’s – 4 4’s – 4 5’s – 2

The highest frequency is the mode – 2 people in the car

Example 2:

Find the mode of the following numbers.

1, 2, 7, 2, 4, 5, 1, 7


1, 1, 2, 2, 4, 5, 7, 7

Since there are three modes, we call this a trimodal distribution.

→ 1, 2, 7 are the three modes.

Practice:Find the mode of the following sets of data.

a. 3, 44, 17 , 17, 44, 15, 15, 15, 27, 40 b. 91, 79, 84, 85, 84, 90, 68, 100

c. 23, 32, 34, 22, 32, 24, 33 d. 11, 12, 15, 13, 12, 11, 14

Mode: the most frequently

occurring value in a set of data.

A data set with no repeated

numbers has no mode.

A data set with two modes is

called a bimodal distribution.



Range = 252 – 210 = 42

Range = 40 – 8 = 32 Range = 8 – 2 = 6

Range = 2.3 – 1.1 = 1.2

Objective:

• Icandeterminetherangeofasetofdata.

The rangeExample:

Find the range of 150, 250, 825, 400, 18, and 500

Arrange the numbers in ascending order:

20, 150, 250, 400, 500, 825

Calculate range:

Range = largest value – smallest value = 825 – 20 = 805

The range is 805

Example 2:

A seven day forecast looks like this

Sun Mon Tues Wed Thurs Fri Sat17°C 15°C 20°C 18°C 17°C 18°C 16°C

Find the range of the temperatures

15, 16, 17, 17, 18, 18, 20

Range = largest value – smallest value = 20 – 15 = 5

The temperatures had a range of 5°C during the week.

Practice:Find the range of the following sets of data.

a. 243, 221, 210, 252, 231, 224 b. 1.1, 1.7, 2.3, 1.6, 1.2, 1.4

c. 11, 40, 21, 32, 8 d. 2, 5, 7, 2, 6, 8, 4, 3

range: the difference between

the highest and lowest values.



No, since the accident was a one-time problem, the 45 minute wait should be considered an outlier which makes the wait time less than 5 minutes.

Objective:

• Icananalyzeasetofdatatoidentifyanyoutliers.

Outliers

Example:

A bus company will add another bus if the wait time at a busy bus stop is longer than 5 minutes.

The following numbers represents the time in minutes that people wait at the bus stop.

1, 3, 4, 3, 5, 2, 5, 45, 1, 3, 4, 4, 2, 5, 2, 4, 1, 3, 4, 2

Using this information the mean, median and mode is:

Mean:

(1 + 3 + 4 + 3 + 5 + 2 + 5 + 45 + 1 + 3 + 4 + 4 + 2 + 5 + 2 + 4 + 1 + 3 + 4 + 2) ÷ 20 = 103 ÷ 20 = 5.15 minutes

Median:

1, 1, 1, 2, 2, 2, 2, 3, 3, 3, ____, 3, 4, 4, 4, 4, 4, 5, 5, 5, 45 Median: 3

Mode is 4

The 45 minute wait was because there was an accident on the road. Therefore 45 minutes is an outlier.

If the outlier is removed, the new mean, median and mode is:

Mean:

(1 + 3 + 4 + 3 + 5 + 2 + 5 + 1 + 3 + 4 + 4 + 2 + 5 + 2 + 4 + 1 + 3 + 4 + 2) ÷ 19 = 3.05

Median:

1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5 Median: 3

Mode is 4

Should the bus company add another bus? Explain.

Outliers: a value in a data set

which is far removed in value

from the other data.



67.85 = 67.9

71.2

Unfortunately all student in the class must be included as partof the class average.

In this case, the student is no longer part of the class so hermark can be excluded.

Answers may vary. Example: Yes, because Greg has to work for his money. Or, no, it is not uncommon to do chores to get an allowance.

5, 5, 5, 5, 6, 6, 7, 8, 25

Mean: $8Median: $6Mode: $5

Practice:

1. A class average is the mean of all the marks of each student in the class.

Dave’s class has the following scores.

67%, 78%, 59%, 66%, 72%, 5%, 65%, 69%, 68%, 75%,82%, 77%, 73%, 78%, 67%, 78%, 65%, 66%, 62%, 85%

a. Calculate the mean including the outlier.

b. Calculate the mean without the outlier.

c. The school rewards classes with an average above 70%, should the outlier beincluded when reporting this class average?

d. What if the 0% was the result of a student who had moved away, but her namewas still on the class list?

2. Some friends compared the amount of allowance they get. Their allowance eachweek was:

$5, $7, $5, $25, $6, $5, $8, $6, $5

a. What is the mean, median and mode of the allowances?

b. If Greg receives $25 in allowance partly as payment for helping with farm choresevery day, should his allowance be counted as an outlier? Why or why not?



Since most players do not earn $5000 a game, the best average is the median, at $1250, which best represents the rookies and the five-year players.

Objective:

• Icangivereal-worldexampleswherethemean,medianormodeisthemostappropriatemeasuretousewhenreportingfindings.

Deciding which to use, mean/ median/ mode

Example: A sports team has a different contract for all their players. Rookies get $1000 per game, players over 5 years get $1500 a game, and veterans over 10 years get $5000 a game.

The team roster currently has: 6 rookies, 5 five years, and 1 veteran.

Or

1000, 1000, 1000, 1000, 1000, 1000, 1500, 1500, 1500, 1500, 1500, 5000

In a TV interview, the team manager is asked “abouthowmuchdoeseachplayermake?”

If the manager reports the mean:

(1000 + 1000 + 1000 + 1000 + 1000 + 1000 + 1500 + 1500 + 1500 + 1500 + 1500 + 5000) ÷ 12 = $1541

If the manager reports the median:

1000, 1000, 1000, 1000, 1000, 1000, _____, 1500, 1500, 1500, 1500, 1500, 5000= (1000 + 1500) ÷ 2 = $1250

If the manager reports the mode:

1000, 1000, 1000, 1000, 1000, 1000, 1500, 1500, 1500, 1500, 1500, 5000= $1000

Which amount would be the most accurate representation of the players’ earnings? Why?



Mean = 68%

Median: 74%

Mode: 65%

The median is the best indicator overall of how much Scottunderstands math as a subject.

No, Robin should have used the mean as the average to share the money since it is the value of the bills that is important, not the frequency of each.

Each brother should have gotten $28.80 (rather than $6) whichmeans Robin kept $22.80 from each brother.

Practice:

1. Scott’s math test marks for the year are 65%, 83%, 90%, 84%, 65%, and 21%

a. Calculate the mean

b. Find the median

c. Find the mode

d. If Scott’s parents asked his teacher how Scott was doing which average is thebest representation of what Scott has learned? Why?

2. Robin has some money she has to share with her 4 brothers. The bills she was givenhas are:

1, 1, 1, 1, 5, 5, 5, 5, 20, 100

a. She decides that she will use the modes to distribute the bills, and gives herbrothers each a dollar bill and a five. Is this a fair method to share the money?Why or why not?

b. How much did Robin keep from each of her brothers?



Mode, to tell you which size sold the most numbers.

Mean, to see if most of the bags are the same weight.

Mean, to balance out the expensive days with the cheaper days.

Mode, since more people would have smaller salaries.

85.5

$5.52

84

17 – 2 = 15



1. Susanne scored 75, 83, 92, 79, 96, and 88 points on her spelling tests. What is heraverage score?

2. If different boxes of cookies cost $4.10, $4.32, $4.79, $5.52, $7.18, $7.92 and $9.58,what is the median price?

3. Find the mode of 68, 77, 79, 84, 85, 84, 90, 91, 93, 100

4. Find the range of the values: 2, 5, 6, 6, 8, 10, 12, 17

5. Which measure (mean, median or mode) would you use to describe each of thefollowing cases? Explain.

a. Size of pants sold in a store, if you wanted to send in an order for more pants.

b. Mass of bags of candy, if you were quality control.

c. Cost of lunch over a one month period, if you were planning next month’s budget.

d. Salary of people in a company with only a few managers and lots of entry levelemployees, if you were trying to prove wages are too low.



Answers will vary. Example: low test score because of not studying

Answers will vary. Example: wrong number measured because theruler was upside down.

I would replace the macaroni and cheese because it is currently the least popular menu item.

2012891417135

Fries 2017141312985

Chicken fingersChicken wrapHotdogPizzaTacosHamburgerMacaroni and cheese98

6. Describe a situation when an outlier should be included when calculating anaverage.

7. Describe a situation when an outlier should not be included.

8. Betty was conducting a survey about the school’s cafeteria menu. She asked peoplein the lunch line what food they liked the most. Her results were:

Survey ResultsFood Item Tally Frequency

FriesPizzaHamburgerTacosChicken wrapChicken fingersHotdogMacaroni and cheese

a. Complete the frequency table.

b. Create an organized table listing foods frommost popular to least popular.

c. What was the sample size of this survey?

If you were to replace an item on the menu, which item would you replace.

Organized Survey Results



W2 – Lesson 1: Table of Values and Graphing Linear Equations

Objective:

• Icanusealinearrelationtorepresentapattern.

Finding A Pattern

Figure 1 Figure 2 Figure 3

Question:

What if someone asked you to draw the 9th figure in this series?

Look for the relationship or pattern between the figure numbers and each figure.

→ The only thing that changes arethe number of stars

→ The number of stars is the same as the figure number

→ The moons stay the same in each figure.

Pattern rule:

Figure x = x stars + 3 moons

Answer:

Figure 9: 9 stars + 3 moons

How many stars would Figure 15 have? _______

relation:when two numbers or things

are related in some way.

Example: Canada → Dollar

Japan → Yen

Variable:the part of the pattern that

varies or changes; usually represented

by a letter..

15



Practice


1. Write a pattern rule for the figures.

2. How many dark tiles would there be in Figure 27?

3. How many total tiles will there be in Figure 13?

Figure x = 4x + 3

27 × 4 = 108

4(13) + 3 = 55 tiles



Objective:

• Icancreateatableofvaluesforagivenlinearrelation.


Shape Number(n)

Number of Hexagons(x)

1 3

2 4

3 5

4

5

6

10

24

Creating a pattern rule from a chart

→ The relationship is: shape number + 2

→ The pattern rule: Number of Hexagons x = n + 2

→ Using this pattern rule: Number of Hexagons in Shape number 4 = 4 + 2 = 6

Complete the remainder of the chart using the pattern rule.

6

7

8

12

26



Practice:

1.


a. What is a possible pattern rule?

b. Complete the table of values using the rule.

2. Use the pattern rule given at the top of the table to complete the tables.

Pattern Number (n)

Number of circles (x)

1 22 43 64 85122537

n = p - 1

PatternNumber(p )

Numberofcircles(n=p-1)

1 023451050

n = 2p

PatternNumber(p )

Numberofcircles(n=2p)

1 223451050

n = 2p - 1

PatternNum-ber(p )

Numberofcircles(n=2p-1)

1 023451050

10

1 4 3

2 6 5

3 8 7

4 10 9

9 20 19

49 100 99

24

50

74

Number of circles in Figure n = 2n



Objective:

• Icanusealinearrelationtocreateatableofvaluesandthengraphit


→ The pattern rule: Number of boxes in Fig. x = 2x + 3

→ A table of values for the figures above:

By extrapolating (extending) the line, we can use the graph to predict how many squares would be in other figures. Predict how many squares would be in Fig. 12.

Figure Number

Number of squares

1 52 73 94 115 136 157 178 19

02468

1210

14161820222426

0 1 2 3 4 5 6 7 8 9 10

Nu

mb

er o

f S

qu

ares

Figure Number

27 squares



Practice:

a. Patrick is arranging tables and chairs for a banquet.

What is the pattern rule describing the number of chairs, c, needed for each number of tables, t?

b. Complete the table, and graph the results

c. If Patrick has 10 tables lined up, how many chairs wouldhe need?

Number of Tables (t)

Number of Chairs (c)

12345678

c = 3t + 4

7

00

2

1 2

4

3 4

6

5 6

8

7 8

10

9 10

1214161820222426283032

10

13

16

19

25

28

22

34

Nu

mb

er o

f C

hai

rs

Number of Tables



Objective:

• Icandescribethepatternsfoundinthegraphtodrawconclusions.

Graph each of the following table of values in different colours on the same grid. Make a key to identify each line produced.

a. b.

c. d.

y = n

n y

12345

y = n + 3

n y

12345

y = 2n

n y

12345

y = n - 1

n y

12345

1 4

02

2 5

14

3 6

26

4 7

38

5 8

410



a. What is the differencebetween graphs a. and b?

b. What is the differencebetween graphs a. and c?

c. What is the differencebetween graphs a. and d?

Same graph 3 spaces higher

Graph c is 2 times steeper.

Same graph 1 space lower

0123456789

1011

y=n

y=n-1

y=2n

y=n+3

0 1 2 3 4 5 6 7 8 9 10





1. Hilary started with $50 in her piggy bank and saved $5 each week from herallowance.

a. What is the pattern relation that would describe the amount of money (t) in thebank for nweeks?

b. Complete a table of values for this relation, and graph the relation.

Week (n) Money (t)12345

c. After 20 weeks, how much money would Hilary have in the bank?

Total money = 5n + 50

5560657075

5(20) + 50 = $150

0 0 1 2 3 4 5 6 7 8

10

9 10

2030405060708090

100110

Tot

al C

ost

Week



2. Consider the following graph.

a. What is the relation represented by this graph?

b. The data for input value 3 is missing. What should bethe output value for input value 3?

c. What would be the output value for an input value of20?

3. A pizza with cheese costs $8.50. Each additional topping is $0.50.

a. Write a relation for the cost of a pizza with n number of extra toppings.

b. What is the cost of a pizza with 8 extra toppings?

c. On the first Thursday of the month the pizzas go on sale for $5.00. What wouldthe new relation look like?

d. What is the cost of the 8 topping pizza on the first Thursday of February?

10

1211

13

00 1 2

123

3 4 5 6 7

456789

Out

put V

alue

Input Value

Output value = 2x - 1

Output value = 5

Output value = 39

Cost = 0.5n + 8.50

Cost = 0.5(8) + 8.50 = $12.50

Cost = 0.5n + 5

Cost = 0.5(8) + 5 = $9



4. Admission to Wacky World is $6. Each ride is an additional $2.

a. What is the pattern rule showing how total cost is related to the number ofrides?

b. Complete a table of values for this relation, and graph the relation.

Rides (r) Total Cost (c)

12345

c. Circle the point on the graph that would show the total cost of going on 8 rides.

d. If Charles goes into the park with $35, what is the maximum number of rides hecan go on?

Cost (c) = 2r + 6

810121416

00 1 2 3 4 5 6 7 8

2

9 10

468

10121416182022

Tot

al C

ost

Rides

14 rides and $1.00 leftover.35 - 6 = 2929 = 14.52



W2 – Lesson 3: Algebra and Verifying Linear Equations

Objective:

• Icansolveandverifythesolutiontoalinearequation.

Solving equations using algebra

Example: 2x + 9 = 13

Undo addition: subtract 9 frombothsidesoftheequalsign.

2x+9-9=13-9

2x=4

Undo multiplication: divide bothsides by 2.

2x 4=2 2

x=2

Example 2: x -4 = 23

Undo subtraction: add 4 to bothsides.

x -4+ 4 = 2+43x = 63Undo division: multiply bothsides by 3.

x ×3 = 6×33x=18

remember: whatever you

do to one side, you do to

the other side too.

To solve an equation, isolate the variable by “undoing” the operations surrounding the variable

Do Un-do+ -- +× ÷÷ ×



4x – 2 + 2 = 14 + 2 3n + 5 – 5 = 11 – 5

d = 4w = 5

4x = 16 3n = 6

w + 45 – 45 = 50 – 45 2d – 6 + 6 = 2 + 62d = 8

x = 4 n = 2

Practice:

Solve each equation, showing each algebra step.

a. 4x – 2 = 14 b. 3n + 5 = 11

c. w + 45 = 50 d. 2d – 6 = 2

Verifying a solution

Example: 2x + 9 = 13 Our solution was x = 2.

Verify: Left side: 2(2) + 9 Right side: 134 + 9 13

Since the left side of the equal sign ends up being the same as the right side, we know the solution is correct.

Example 2: x/3- 4=2Our solution was x = 18.

Verify: Left side: (18)/3- 4 Right side: 26 – 42

Verify a solution by substituting the answer back into the original equation.



Left side: Left side:

2(9) – 7

(18) - 53

18 – 7 6 – 5 11 1

Right side: 11 Right side: 1

Correct Correct

Left side: Left side:(6) + 32 2(7) + 7

3 + 3 14 + 7 6 21


Incorrect Incorrect

Left side: Left side:(20) + 64 6(2)- 11

5+6 12 – 11 11 1


Correct Incorrect

Practice:

Verify if the given solution is correct. Show all steps. Note: Some will be incorrect

a. 2x – 7 = 11, x = 9 b. 5 1, 183y y− = =

c. 3 8, 62n n+ = = d. 2d+7=17, d=7

e. 6 11, 204g g+ = = f. 6d-11=13,d=2



Unit tile: represented by a single

tile. Each one tile represents +1.

Variable tile: represented by a

long rectangular tile. Each tile

represents one unit of a variable.

Objective:

• Icanverifythesolutiontoalinearequationusingmanipulatives.

Verifying equations with algebra tiles

Solve: x + 4 = 9

Left side: Right side

Remove4unittilesfromeachsidetoisolatethex-tile

Eachx-tileequals5unittiles(x=5)

Verify: x + 4 = 9, x = 5

Left side

Right side

(9 tiles)

→Since the left side has the same number of

tiles as the right side, we know the solution

is correct.



Correct

Practice:

Verify each given solution using Algebra tiles.

Verify: x – 2 = 3, x = 5

Left side

Right side

Verify: 2x + 1 = 3, x = 1

Left side

Right side

Verify: 3x – 3 = 6, x = 3

Left side

Right side



Objective:

• Icanuseequationstosolvewordproblems.

Writing and solving equations from word problems

Example:At the bakery loaves of bread cost $2.00 per loaf. How many loaves can be bought with $10.00?

a. Write the equation representing the solution.

2b = 10

b. Solve the equation, verify the equation.

2b = 10

2 102 2b=

b = 5

Left side: 2b → 2(5) → 10

Right side: 10

LS = RS, therefore the solution is correct



13 + h = 21

2n - 5 = 7

13 -13 + h = 21 - 13h = 8

Verify:

Left side:13 + (8)21

Right side: 21

2n - 5 + 5 = 7 + 52n=12n=6

The mystery number is 6

Practice:

1. In Greg’s class, 13 people play hockey. The class has 21 students.

a. Write an equation to find the number of students who don’t play hockey.

b. Solve the equation, verify the solution.

1. Jake is thinking of a number. He multiplies it by two and then subtracts five. Theresult is seven.

a. Write an equation to represent the solution.

b. Solve the equation to find the mystery number.



subtract

add

4

6

multiply 2

divide 3

5x-7=185x-7+7=18+75x=25x=5

4n+8=204n+8-8=20-84n=12n=3

Verify:Left side:5(5)-725-718

Verify:Left side:4(3)+812+820

Right side: 18

Right side: 20


• Icanuseequationstosolvewordproblems.


→ A variable, x, has 4 added to it, we need to _____________ ______ from both sides to isolate x.

→ A variable, n, has 6 subtracted from it; we need to _____________ ______ to both sides to isolate n.

→ A variable, g, is multiplied by 3, to isolate g we need to _____________ by ______ on both sides.

→ A variable, w, is divided by 2; to isolate w we need to _____________ by ______ on both sides.

2. Solve and verify, showing all steps.

a. 5x – 7 = 18

b. 4n + 8 = 20



Left side

Right side

2y + 6 – 6 = 16 – 62y = 10y = 5

h + 3 - 3 = 6 - 32h = 32h ×2 = 3×22h = 6

w + 4 – 4 = 11 – 4w = 7

Left side

Right side

Left side

Right side

3. Solve. Verify your solution using Algebra tiles.

a. w + 4 = 11

b. 2y + 6 = 16

c. 3 62h+ =



f – 3 = 7

f – 3 + 3 = 7 + 3f = 10

Verify:Left side:(10) – 37

Right side: 7

No, Peter is incorrect because p needs to stand for the number of spaces between the posts not including the gate space.

3 (8) + 524 + 529

He actually needs 29 pieces.

4. Sarah invited some friends over to play video games. Three friends left after onegame, but seven friends stayed to play more. How many friends did Sarah haveover?

a. Write an equation

b. Solve the equation, verify the solution.

5. Peter wants to build a fence around his garden. Heneeds 3 cross pieces between each fence post, exceptthe gate which needs 5 pieces. Before going to thelumber store, Peter determines he will need 3p + 5pieces of wood, where p is the number of posts.

a. If Peter’s garden has 9 fence posts, he decides he’ll need 32 pieces of wood. IsPeter correct? Why or why not?

b. How many pieces of wood does Peter actually need?



c = 4S + 6L

44 = 4S + 6(2)44 = 4S + 1244 – 12 = 4S +12 – 1232 = 4S8 = S

38 = 4S + 6(3)38 = 4S + 1838 – 18 = 4S +18 – 1820 = 4S5 = S

She can have 8 small tables

She can have 5 small tables

She can have 4 large tables

48 = 4(6) + 6L48 = 24 + 6L48 – 24 = 24 – 24 + 6L 24 = 6L4 = L

6. Beth is setting up a restaurant with large and small tables. She decides that eachsmall table will have 4 chairs, and each large table will have 6 chairs.

a. Write an expression for the number of chairs (c) using s for small tables, and l forlarge tables.

b. How many tables of the other size can Beth set up if she has:

i. 2 large tables and 44 chairs?

ii. 3 large tables and 38 chairs?

iii. 6 small tables and 48 chairs?



8m +15 = 718m + 15 – 15 = 71 – 15 8m = 56m = 7 It will take 7 months to get 71 songs.

7. Warren has 15 songs downloaded to his MP3 player. He downloads 8 new songseach month. After how many months will Warren have 71 songs?

W1 – Lesson 3: Fractions

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