Complete Book 2018 - 2019

Table of Contents Review ----------------------------------------------------------------------------------- 2 – 5 Area & Perimeter ------------------------------------------------------------------- 6 – 24 Fractions (with Number Sense)-------------------------------------------------- 25 – 76 Average/Number Sense Review ------------------------------------------------ 77 – 82 Decimals ------------------------------------------------------------------------------ 83 – 106 Percent --------------------------------------------------------------------------------- 107 – 118 Integers -------------------------------------------------------------------------------- 119 – 139 Lines & Angles --------------------------------------------------------------------- 140 – 158 Triangles ------------------------------------------------------------------------------- 159 – 168 Geometry Review ------------------------------------------------------------------- 169 – 174 Transformations -------------------------------------------------------------------- 175 – 184 Data Management ------------------------------------------------------------------ 185 – 200 End of Year Review ----------------------------------------------------------------- 200 – 212 Rubrics ---------------------------------------------------------------------------------- 213 – 214

1

Converting Measurements

meters

Practice:

 4 cm = ______ mm  7000 m = _____ km  8 m = ______ cm

 60 mm = _____ cm  8 cm = ______ mm  300 cm = _____ m

 1 cm = ______ mm

 7 cm = ______ mm

 2 km = ______ m

 10 m = ______ cm

2

WHICH OPERATION TO USE…. without solving the following problems, write whether you would: add, subtract, multiply or divide…

1. There are 544 students at Hadley, and 255 are girls. How many boys are at Hadley?

2. Sam bought 18 baseball cards, 22 hockey cards and 57 football cards. How many cards does he have in total?

3. Jack has 74 cards in his collection. He gave 38 to his friend because he doesn’t want them anymore. How many cards does he have now?

4. At the store, bubble gum cost 2$ per pack. Sally bought 8 packs. How much did it cost?

5. Sarina borrowed $40 each from 10 different people. How much did she borrow altogether?

6. Tim has $12 in his wallet. He’s going to a movie which costs $9. How much money will he have left?

3

7. Andrew spent 10 minutes eating breakfast 15 minutes eating lunch and 20 minutes eating dinner. How much time did he spend eating so far today? If he does this every day, how much time will he spend eating in a week?

8. Chris bought a pair of shorts on sale for $10. The usual price is $40. How much did he save? How much would he pay if he bought five?

9. Jane works at the store for $11 an hour. She worked 25 hours last week. What was the amount on her pay check? How much would it cost to pay a staff of 7 workers for 8 weeks at the same rate?

10. Melanie babysits for $10 an hour. Each night she babysits she makes $50. How many hours does she babysit in a night? How many weeks would it take to save up for a 400$ bicycle if she babysits twice a week

4

ROUNDING: Step 1: Underline the digit to be rounded Step 2: Write the 2 possible rounding positions Step 3: Follow the rule: ▪ The digit to the right is a 4 or less, the underlined digit stays the same ▪ The digit to the right is a 5 or greater, the underlined digit becomes one greater ▪ All the remaining digits can be replaced by zero

Example: 69.098 round to nearest tenths 69.100

69.000

the digit to the right of 0 is a 9 = 69.1 or 69.100

Round to the nearest 10: 54: ____

79: ____

12: ____

303: _____

486: _____ 160:_____

Round to the nearest 100: 732: ____

569: _____ 284: _____ 455:_____

2408:_____ 4347:_____

Round to the nearest 1000: 1348: _____

5027: _____

1595: _____

6387: _____

3811: ______

Round to the nearest tenths: 3.73: _____

0.77: _____

2.5: _____ 0.04: _____ 9.76: ______ 6.38: ______

Round to the nearest hundredths: 19.989:_______

3.562: _______

0.345: ______

459.525: _______

5

Area and Perimeter

** height is always 900 to the base**

Label base (b or B) and height (h) on each polygon.

6

VOCABULARY Fill in the Blanks from the following list Right Parallelogram Trapezoid Area Perpendicular Rhombus

Quadrilateral Rectangle Complex Polygon Height (h) Triangle Congruent

Polygon Square Root Base (b or B) Square Perimeter Parallel lines

___________: The same size or equal. ___________: If you extend them in either direction they will never connect ___________ or ____________: Two words that mean 900 ___________: Any shape made of connected line segments. ___________: Any four-sided Polygon ___________: A quadrilateral that has 2 pairs of parallel sides ___________: A parallelogram with 4 congruent sides ___________: A parallelogram with 4 right angles ___________: A parallelogram with 4 right angles and 4 congruent sides. ___________: The operation used to find the side lengths of a square. ___________: A three-sided polygon. ___________: The distance around a shape. Units are cm, m, km, etc… ___________: The space covered by a shape. Units are cm2, m2, km2, etc… ___________: Any one side of a polygon. ___________: The distance, at 900, from the base of a polygon, to the “top” of the polygon.

___________: A quadrilateral with 1 pair of parallel sides. ___________: A shape made of more than 1 polygon put together. 7

VOCABULARY Sketch an example of each Congruent Lines

Polygon

Parallel Lines

Quadrilateral (label b and h)

Right (or Perpendicular) Angle

Parallelogram (label b and h)

Rhombus (label b and h)

Rectangle (label b and h)

Square (label b and h)

Complex Polygon (label every base and height)

Trapezoid (label b, B, and h)

Triangle (label b and h)

8

Take Note! Perimeter: add up all the sides (the outside of the polygon)

Practice: Find the perimeter of each  Square with side length of 6 cm

 Rectangle with side lengths of 4 cm and 7 cm

 Parallelogram (with no right angles): 4 cm 4.2 cm

4.2 cm 4 cm

 Trapezoid 3 cm 3.2 cm

3.2 cm 2 cm

 Triangle: sides = 7 cm, 6 cm and 8 cm

9

Take Note!

AREA is different for different polygons… …so ALWAYS WRITE the Area Formula!

Square: Area = base x height A=bxh

** height is always 900 to the base**

Rectangle: A = b x h

Parallelogram: A = b x h Rhombus: A = b x h Triangle: A = b x h 2

OR OR

Trapezoid: A = [(B + b) x h] 2

A = d1 x d2 ÷ 2

(d = diagonal)

A=bxh÷2

(notice both bases are in brackets, so add first, then x h, then ÷ 2)

OR

A = (B + b) x h ÷ 2

Practice: Construct each of the following and find the area and perimeter.  Square with base = 3 cm

 Rectangle with base = 11 cm and height = 4 cm

10

 Triangle with base = 5 cm and height = 3 cm

 Parallelogram (with no right angles) with base = 6 cm and height = 2 cm

 Trapezoid with BASE= 7 cm, base = 4 cm and height = 4 cm

 Rhombus with sides = 3 cm

11

Finding base and height when the area is known: Take NOTE! Square: what multiplied by itself gets the given area? For a square ONLY:

Example: Area of a square is 100 cm 2

bxh 10 x 10

Side length =

Take Note!

Area

Rectangle and Parallelogram: what base and height multiplied together gets the given area? Example: Area of rectangle = 56 cm 2

Take Note!

b x h 8 x 7 Example: Area of parallelogram = 36 cm 2

b x h 9 x 4

There are many possible answers for the base and height for each example

Take Note!

Triangle: since the formula is base x height ÷ 2, when the area is known, it needs to multiplied by 2 first, then look for a base x height that gives the new area. Example: Area of triangle = 12 cm 2 x 2 = 24

b x h 6 x 4

Take Note! Verify: A = b x h ÷ 2= 6 x 4 ÷ 2 = 24 ÷ 2 = 12 cm2

Trapezoid: since the formula is [(BASE + base) x height] ÷ 2, when the area is known, it needs to multiplied by 2 first, then look for a base x height that gives the new area, then split the bases … some for the bottom and some for the top. Example: Area of trapezoid = 14 cm 2 x 2 = 28

b x h 7 x 4

Take Note!

B+ b 4 + 3 (B and b cannot be equal) Verify: A = (B + b) x h ÷ 2 = (4 + 3) x 4 ÷ 2 = 7 x 4 ÷ 2= 28 = 14 cm2

12

Practice: Construct the following polygons  Square with an area of 36 cm2

 Rectangle with an area of 16 cm2

 Triangle with an area of 5 cm2

 Parallelogram with an area of 21 cm2

13

 Trapezoid with an area of 27 cm2

 Parallelogram with an area of 42 cm2 and a base of 6 cm.

 Triangle with an area of 15 cm2 and a height of 5 cm.

 Trapezoid with an area of 18 cm2 and a height of 3 cm.

 Complex polygon with a perimeter of 26 cm.

14

Area and Perimeter Practice:

 Harvey built a patio in the back yard. a) What is the area of the patio? b) What is the total area of the patio and grass? c) How can you find the area of the grass? Show your work.

1440 cm

5.6 m Grass

Patio

66 dm

Grass

5.6 m



1440 cm

a) Find the height of a parallelogram if the area is 35 cm2 and the base is 5 cm.

b) Find the base of a parallelogram if the area is 126 cm2 and the height is 9 cm.

c) Find the area of a parallelogram if the base is 11 cm and the height is 7 cm.

15

 Measure the height of the following polygons:

a)

b)

 Find the area and perimeter of the following polygon:

10 cm 3 cm 6 cm

6 cm

 Measure and find the perimeter of the following polygon:

16

 Mr. Nadeau grows tomatoes in a square-shaped garden and lettuce in a rectangular-shaped garden. The two gardens have the same width. If the perimeter of the rectangle is 11 m and its length is 3 m, what is the perimeter of the square garden?

 The diagram below shows a fence around a yard. With the given measurements can you determine how many metres of fencing are required? How much money will be required if the cost of fencing is $35 per metre.

3.5 m

Area = 15.75 m2

 A can of paint covers an area of 18 m2. How many cans of paint are needed to paint two rectangular walls of a warehouse that are 9 m by 6 m and one ceiling that is 9 m by 8 m?

 The hotel swimming pool measures 20 m by 8 m. Management wants to put a 2.5 m wide deck around the pool. Calculate the area of the deck.

17

AREA & PERIMETER REVIEW Multiple choice. Circle the correct answer. 1. What is the area and perimeter of this rectangle? (2 marks) a) b) c) d)

35mm2 and 24mm2 35mm and 24mm 35mm and 24mm2 35mm2 and 24mm

Height = 5 mm

Base = 7 mm

2. What is the area of this Triangle? (2 marks) a) b) c) d)

72cm2 72cm 24cm 36cm2

height = 6 cm Base = 12 cm

Short Answer. (1/2 mark each) Shape

Formula for area

1. Square 2. Rectangle

3. Parallelogram

4. Rhombus

5. Trapezoid

6. Triangle

18

Question 7 is worth 1 mark 7. Shape

Formula for perimeter

b a

a b

8. Convert the following lengths (1/2 mark each) 7 cm

= __________ mm

5000 m = __________ km 4m

= __________ cm

10 cm

= __________ mm

440 dam = __________ km 688 mm = __________ m 25 dm

= __________ cm

1200 cm = __________ km

Long Answer. Show all your work. 9. Calculate the area of the following trapezoid. Remember units. (4 marks)

base = 15 cm Height = 6 cm Base = 12 cm

Area = ________________ 19

10. Calculate the area of the following complex polygon. Remember units. (6 marks)

20 m 2m

10 m

5m

Area = ______________

11. What is the perimeter of a rectangle if the area is 56 cm2 and other side measures 8 cm? (4 marks)

56 cm2

8 cm

Perimeter = _________________

20

Application Questions. Your work, every little step, is graded according to the RUBRIC at the back of the workbook. (10 marks each)

12.Home Garden Center Danny wants to fertilize a garden that measures 3.8 metres by 5.2 metres. He has a $25 gift card for the Home Garden Center. One bag of fertilizer covers an Area of 3.5m². Each Bag costs $4.40, all taxes included. Will the gift certificate cover the cost of the fertilizer? Show your work

Methods

40 32 24 16

8

0

Is the Gift Certificate enough to

Calculations

40 32 24 16

8

0

cover the cost? (yes or no):_________

Organization 20 16 12

4

0

8

21

13.Security Zones A construction zone requires a fence to be installed for security reasons. The diagram below indicates the dimensions of the zone that needs to be fenced in. 34m

5m

28m

42m

Your supervisor asks you to study the following proposals submitted by two companies. Company 1: One metre of fencing costs $26. Installation cost is $1500 Company 2: Fencing is sold in 6 metres. Each section costs $190. The installation is $1000 Which company offers the best deal and what will be the cost of the fencing?

Methods

40 32 24 16

8

0

The cheapest deal is offered by:

Calculations

40 32 24 16

8

0

___________________________

Organization 20 16 12

4

0

8

22

14.Hawks Renovation Marc and Anthony’s Construction company is expanding, so they’ve decided they need a new sign. The original sign is shown below. The new sign will have a base that is 3 times greater than the original and a height that is 3 times greater than the original.

a. Calculate the area of the new sign. Use a diagram to show your work.

b. Marc insists that they need to put a neon feather border around the sign. How long does this border need to be?

c. If the border costs $3/m and the sign material costs $2/m2, how much more will the new sign cost compared to the original size?

Methods

4

32 24 16

8

0

Calculations

4

32 24 16

8

0

Organization

2

16 12

4

0

8

23

Review Score:

Multiple Choice:

Short Answer:

Long Answer:

Application Questions:

TOTAL

8 8 14 30

60

24

FRACTIONS Parts-and-Whole For each of the following, be accurate by measuring with a ruler  If this rectangle is one whole, find one-fourth

 If this rectangle is one whole, find two-thirds

 If this rectangle is one whole, find five-thirds

 If this rectangle is one whole, find three-eighths

 If this rectangle is one whole, find three-halves

25

 If this rectangle is one-third, what could the whole look like?

 If this square is three-fourths, what could the whole look like?

 What fraction of the big square does the small square represent? (In other words, how many times can the small square fit into the larger one?)

Whole

 What fraction is the large rectangle if the smaller one is the whole?

Whole

26

 If the rectangle for each below is one whole,

a) find one-sixth

b) find two-fifths

c) find seven-thirds

 If the following rectangle represents two-thirds, what could the whole look like?

27

 If the following rectangle is one-sixth, what does the whole look like?

 If the following rectangle is four-thirds, what does the whole look like?

 If the following triangle represents one-half, what does the whole look like?

28

Improper and Mixed Numbers Improper Fractions: 7 , 13 , 3 , 9 4

• •

7

2

5

more than a whole (the numerator is larger than the denominator) can always be written as mixed number ( a whole number and a fraction)

Method  Make wholes 22 What makes a whole with this fraction? 7

7 7

How many 7 can be made out of 22 ? 7

7 = 1 whole 7

7

7 = 1 whole 7

7 = 1 whole ( 21 total so far) and 1 is left. 7 7 7

So, 22 = 3 1 7

7

Method  Divide the numerator by the denominator 3 22 = 7

7 22 - 21 1

=3 1 7

Note: A fraction line is a division line.

Remainder 1 becomes the numerator

The denominator does not change

29

Practice: Choose a method to write each improper fraction as a mixed number.  7 = 3

 25

 17

9

5



45 10

 31

 19



8 2

 14



11

5

4

4 1

Making an improper fraction from a mixed number: Method  3

5 8

The 3 means there are 3 wholes: 8 , 8 , 8 then there’s 5 = 29 8

8

8

8

8

do not change the denominator

30

Method 

6 3

Denominator (bottom) x whole number + Numerator (top), over the same denominator

4

4×6+3 4

27 4

=

Practice: Choose a method to write each mixed number as an improper fraction.

 3 2

 7 5

 4 8

 1 3

2 1

 6 4

3

4

 5 4 4

7

5

2 2 5

11

9

1 5

12

31

Compare the following fractions. Which fraction in each pair is GREATER?

Use size of the parts, closer to 0, 1/2, 1, drawings and/or models. DO NOT USE MULTIPLICATION OR COMMON DENOMINATORS



2 5



4 and 5 9 9



3 8

and 2 9

and 4

10



7 and 7 8 3



3 and 9 4 10



3 and 4 8 7

32

5 and 6 11 11





3 5

and 3 7



9 8

and 4



5 9

and 4

3

8



8 and 8 13 15



11 and 8 8 5



1 6

and 1 7

33

Fractions and Number Lines Find the fraction each letter represents  A

B

C

0

1

A: ______

B: ______

2

C: _______

 B C

A

0 A: ______

1 B: ______

C: _______

 Construct an appropriate number line and place the following points: A: 1

2

B: 3

C: 5

4

4

D: 14 8

 On each number line, place a point representing 1 whole. a) 0

5 9

34

b)

0

5 12

c)

10 18

d)

21 26

e)

1 2

35

Number Sense Factors: Two whole numbers multiplied together are factors of the product For example 1, 2, 3, and 6 are factors of 6 Each number divides into 6 with no remainder

Prime Numbers: A prime number can only be divided by itself therefore it has only 2 factors * 1 is not a prime number because it only has one factor, 1

Here are the first few prime numbers: 2

3

5

41 43 89 97

7 47

11 53

13

17 61

19 67

23 71

29 73

31 79

37 83

101 ….

Composite Numbers: Has more than 2 factors 8 is a composite number because its factors are 1, 2, 4, 8

Example: List the factors of 24 24:

1, 2, 3, 4, 6, 8, 12, 24

Practice: List the factors of each  45

 28

7 36

Greatest Common Factor (GCF): The largest common factor between 2 numbers Example: What is the GCF between 12 and 16

12: 1, 2, 3, 4, 6, 12 16: 1, 2, 4, 8, 16 Common Factors: 1, 2, 4

GCF = 4

Practice: Find the GCF for the following.  18 and 24

 21 and 49

 Beth is obsessed with being organized. She even organizes her treats! She has 24 red skittles, 36 green skittles and 48 yellow skittles. Beth puts them ALL, no leftovers, into piles that each have the same amount of each colour of skittle. (Ex: maybe each pile has 1 red, 3 green and 2 yellow). Give at least 2 examples of what the piles could look like.

 Scott is about to simplify the following fraction in one step using the GCF. How can he do it?

18 63

37

Multiples: Counting by the given number Example: List the multiples of 3 3: 3 6 9 12 15 18 21 24 27 …

Practice: List the multiples of:  12 5 Lowest Common Multiple (LCM) The smallest number that is common between 2 or more numbers (it is the first common number that shows up) Example: Find the LCM between 8 and 12 8:

8, 16, 24, 32, 40 LCM = 24

12: 12, 24

Practice: Find the LCM between…  4 and 5  In order to maintain the ship’s speed, three furnaces were filled at exactly 7 A.M. Afterwards, the first furnace was filled every 4 hours, the second every 3 hours and the third every 2 hours. At what time were these three furnaces next filled at the same time?

 The school is planning a BBQ for about 200 students. They will be serving hamburgers and pop. At Costco’s, the buns come in bags of 18 and the patties in boxes of 24. How many of each must the school buy to make about 200 hamburgers and not have any extra buns or patties leftover?

38

Practice: Read the questions below and place an X in the box you think describes the question type. DO NOT SOLVE.  Shannon, Sam, and Jordan are adding fractions. They know that to simplify fractions they must find an equivalent number that will divide into both the numerator and the denominator. 15 What number did they choose when simplifying 25 ?

GCF type problem

 LCM type problem

 Today Mariel, Natasha, and Jenny are going to the video store to rent one DVD each. Mariel rents a DVD every 12 days, Natasha every 16 days, and Jenny every 18 days. How many times during the next year (365 days) will these three people go to rent a DVD on the same day?

GCF type problem

 LCM type problem

 Ms. Mckinnon often asks her math class to work in groups. • • • •

On Monday, she put her students into groups of 3 and had no students left over. On Wednesday, she had the students work in groups of 2 but had 1 student left over. On Friday, the students worked in groups of 5 but there were 2 students left over. No students were absent on any of these days.

What was the smallest number of students that she could have in her class?

GCF type problem

 LCM type problem

I challenge you to solve this one…☺

39

 Three buses leave the station at 8:00 a.m., heading out on different routes. The buses complete each of their round trips back to the station in the following times: Bus 1: Bus 2: Bus 3:

45 minutes 30 minutes 20 minutes

If they continue their round trips on schedule, at what time will all three buses first meet back at the station?

GCF type problem

 LCM type problem

 You are with your friends at the corner store purchasing candies. You have chosen 105 sour cherries, 75 raspberry sours, and 45 sour keys. How many friends will you need in order to divide your candies so that each person obtains equal numbers of sour cherries, raspberry sours, and sour keys? What are their names?

GCF type problem

 LCM type problem

 Mr. Clare wishes to cover the ceiling of the gym with square tiles. The dimensions of the room are 30m by 18m. What is the largest square tile that will fit perfectly in the ceiling?

GCF type problem

 LCM type problem

 Scott, and Blake are adding fractions. They know that to add fractions they must find an equivalent fraction where the denominators are the same. 3 1 What denominator did they choose when adding 4 + 5 ?

GCF type problem

 LCM type problem

40

Equivalent Fractions: Fractions that mean the same amount of the whole

///////////// ///////////// 2 4

2 shaded out of 4 boxes is the same as 1 shaded out of 2 boxes if the wholes are the SAME SIZE

////////////////////////////// 1 2 REMEMBER: the wholes we are comparing are the same size

Practice Write two (2) equivalent fractions for the following situations. 

 /////// ///////

//////// ////////

 /// ///

/// ///

//// ////

//// ////

//// ////

41

Writing Equivalent Fractions To write equivalent fractions, multiply or divide the numerator and denominator by the same factor: Examples: 7

x2

8 x2

= 14 16

14 ÷ 7 = 2 5 35 ÷ 7

REMEMBER the “Golden Rule”: “What you do to the top, you do to the bottom”

How to tell if fractions are equivalent:  Is the numerator and denominator multiplied or divided by the same factor?  Cross-multiply; if the products (answers) are the same, the fractions are equivalent. Example: 8 and 4 12 6

8 x 6 = 48 12 x 4 = 48

these are EQUIVALENT fractions

42

Practice: Which of the following situations show equivalent fractions? Show how you know (multiply or divide by the same factor, or cross multiply). A. Stephanie ate 2 of her Kit Kat bar; Sam ate 4 of his Kit Kat bar. 5

10

B. Kathy drove 28 km, Ken walked 7 km and Kim ran 14 km. 40

10

20

C. 1 of Tim’s money was loonies and 3 of Jim’s were loonies. 3

6

D. Jack got 24 on his test. Jake got 80 . 30

100

E. There are 16 boys in Ms. Mckinnon’s class. There are 13 girls in Ms. Macleod`s class. 26

32

F. Scott shot 8 baskets, Paul shot 12 and Steve shot 4 . 12

18

6

G. Sue ate 7 of her pizza. Steve ate 13 of his pizza. 8

16

43

H. Stan read 100 pages of his book; Jan read 120 pages and Frank read 80 pages. 105

130

90

I. Ann made 5 serves during the volleyball game; Nathalie made 8 serves. 6

9

J. Dan ate 14 pieces of skittles; Harry ate 7 pieces. 15

30

K. Nancy read 84 pages of her book and Beth read 252 pages. 105

315

L. Roxanne drank 75 ml of her juice. Rick drank 190 ml. 100

200

44

Simplifying Fractions: writing equivalent fractions in lowest terms. Example:

6 3 can be simplified to by dividing both the denominator and numerator by the same factor, 2. 8 4 6 ÷2 = 3 8 ÷2 4

Practice: Express in the simplest form.

 3 _______ 6

 16 _______ 48

 18 _______ 36

 4 _______ 5

 10 _______

 15 _______

 11 _______

 120 _______

40

32

35

180

STOP and Review…  Fill in the blank ___________is on top and it tells us _________________________________________________________

6 13

________________is on bottom and it tells us _________________________________________________

 Fill in the blanks

# parts 2

Fraction 1 2 1 3

Word Half

Quarters 45

 Place the following fractions on the number line below: 2 1 , 5

0

3 3 2 , , 1 5 10 10

1

2

 Place the following fractions in the proper column in the table below.

3 38

5 4

Proper fractions

21 20

3 5

Improper Fractions

2 33

12 6

Mixed Number

 Write a single fraction (improper) for 3 3 . How do you know you are right? 7

46

 Rewrite as a mixed number or as an improper fraction as necessary.

23 → 3

9→ 6

25 → 9 4 5 → 13

1 36 →

3 7 → 12

 Which is greater? Briefly explain why. 4 or 3 5 4

11 or 10 10 11

7 or 3 8 8

22 or 4 50 8

2 or 2 3 5

13 or 7 25 16

 Place in order from least to greatest. 5, 6, 7¸ 3, 11 8 11 8 2 22

 Find 3 fractions equivalent to the following fractions: a) 5 __________________ 6

b) 8 _____________________ 11

 Which fraction is not equivalent to the other fractions? a) 4 , 8 , 9 , 16 , 32 ________ 5

10 15

20

40

b) 18 , 2 , 14 , 6 , 20 ________ 27

3

21 15

30

 Which of the following fractions are already simplified? 16 15 23 24 7 , , , , _______________________________ 24 27 24 32 12

47

Adding and Subtracting Fractions ➢

The denominators have to be the same before we add or subtract the numerators

➢

We add or subtract the numerators only

➢

DO NOT ADD OR SUBTRACT THE DENOMINATORS!

➢

If the denominators are not the same, we must find a common denominator.

➢

Rewrite the fractions with the common denominator.

➢

Simplify if possible and rewrite as a mixed number if needed.

Example : 2 + 4 = 6 = 1 1 5

5

5

Example :

5

7 12

−

1 3

=

𝟕 ×1 𝟏𝟐 ×1

−

𝟏 ×4 𝟑 ×4

=

7 12

−

4 12

×𝟕 ×𝟗 Example : 7 + 6 = 7 ×𝟕 + 6 ×𝟗 = 49 54 = 103 = 1 40 9 7 9 7 63 63 63 63

Practice: Find the sum.  5 + 6=

 2+ 2=

 5 +1 =

 3+5=

 4 + 3=

 1+ 3=

 2+ 3=



11

4

3

11

12

4

5

5

5

10

1 + 2

2 = 3

6

2

6

8

 1+ 3= 6

8

48

=

3 12

Practice: Find the difference.  6 - 5= 11

 4 - 2=

11

5

 5 -1 =

5

6

 3 - 5=

 4 - 3=

 3 - 2=

1  2 - =

4

4

12

5

3

3

6

 1 - 3=

10

2

8

 3 - 1=

2

8

6

Practice: Add or subtract.  9 - 3=

 3 + 2=

 3 - 1=

 5+ 2=

 8- 3=

 2+ 6=

10

8

5

7

7

9

 1+ 1+ 2= 3

4

5

 8 - 3+ 1= 15

10

5

5

4

5

3

6

11

 2 + 5+ 3= 3

6

4

 5+ 1 - 5= 6

3

12

49

Adding Mixed Numbers Method  • Add the whole numbers • Add the fractions; DO NOT FORGET TO HAVE A COMMON DENOMINATOR • Add the whole number to the fraction • Simplify to the lowest terms if needed Example: 4

1 3 +2 = 3 4

Step  4 + 2 = 6 Step 

3 1 3 1 ×4 + = + ×3 ×4 4 ×3 3 4 3

Step  6 + 1

=

4 9 13 1 + = =1 12 12 12 12

1 1 =7 12 12

Method  • Write the mixed numbers as improper fractions • Add the fractions; DO NOT FORGET TO HAVE A COMMON DENOMINATOR • Simplify where possible and rewrite as a mixed number if needed Example: 4

1 3 13 11 52 33 85 1 +2 = 3x4+1 +4x2+3= + = + = =7 3 4 3 4 12 12 12 12 3

4

Practice: Choose a method to find the sum.  32+14=

 41 +65 =

 23+2 7 =

 11 +25 +31 =

3

4

9

10

4

3

6

6

4

50

Subtracting Mixed Numbers Method  Borrowing • • • • • •

Subtract the fractions first; DO NOT FORGET TO HAVE A COMMON DENOMINATOR If the subtraction cannot be performed, borrow 1 from the first whole number Make a whole in fractional form using the common denominator Subtract the whole numbers Subtract the fractions; simplify if possible Cannot take 25 from 24 Add the whole number(s) and the fraction

Example: 7

4 5 - 2 = 5 6

Step  7

Step  6

4 5

×6 ×6

− 25 6

×5 ×5

= 7

24 25 -2 30 30

30 24 25 54 25 + - 2 = 6 -2 = 30 30 30 30 30

Step  6 – 2 = 4 6

30 30

is the same as 7 wholes

Step 

54 25 29 29 = =4 30 30 30 30

Method  • • •

Write the mixed numbers as improper fractions Subtract the fractions; DO NOT FORGET TO HAVE A COMMON DENOMINATOR Simplify if possible and rewrite as a mixed number if needed

Example: 4 1 - 2 3 = 3 x 4 + 1 - 4 x 2 + 3 = 13 - 11 = 52 - 33 = 19 = 1 7 3

4

3

4

3

4

12

12

12

12

Practice: Choose a method and find the difference.  3 1 – 1 11 = 3

18

 61 –45 = 4

6

51

9–57 =

 81 –25=

24 – 2 7 =

 17 – 4 2

9

5

2

10

9

5

Solve the following problems.  Beth ate 1 of one cheese pizza and Scott ate 5 of the same pizza. How much 4

8

pizza was eaten? How much was left?

 Harvey’s gas tank showed 11 full at the beginning of the week. On Friday, the 16 1 gas gauge read full. How much gas did he use in a week? 3

 Anne worked 2 1 hours on Monday, 3 3 hours on Wednesday and 4 1 on 2

4

4

Friday. How many hours did she work in total?

52

 Ms. Mckinnon travels 21 km to work. She stops for coffee 15 3 km into her 4

drive. What’s the distance left from the coffee shop to Ms. Mckinnnon’s work?

 The Nadeau family drove from Ottawa to Cambridge to see relatives. They drove for 3 1 hours, stopped for 3 hour for lunch and continued to Cambridge 3

4

for another 2 1 hours. 2 a) How long were they driving?

b) How long did the total trip take?

 Anne bought 7 meters of rope for a school project. She used 5 7 of it. How 8

much rope was not used?

 Beth planted 2 1 rows of beans, 3 2 rows of peppers and 4 rows carrots. 2

3

a) How many rows of vegetables did she plant?

b) How many more carrots than peppers did she plant?

53

Multiplying Fractions (Do not need common denominators) • • •

‘of’ means multiply

Multiply the numerators together (the two top numbers) Multiply the denominators together (the two bottom numbers) Simplify if possible and rewrite as a mixed number if needed Example:

5 3 15 3 x = = 7 5 35 7

Whole number multiplied by a fraction • •

The whole number can be written as a fraction with a denominator of 1 Follow the multiplication rule Example: 9 x

3 9 3 27 3 = x = =6 4 1 4 4 4

Mixed Number multiplied by a Mixed Number • • •

Write the mixed numbers as improper fractions Follow the multiplication rule Simplify if possible and rewrite as a mixed number if needed Example: 2

2 1 8 5 40 4 1 x1 = x = =3 =3 3 4 3 4 12 12 3

Practice: Find the product.  2 x 3=

 11 x 1 =

 4x 3=

 3 of 8 =

 7x 2=

 3 of 1 =

 3x 5=

 6 of 3 =

3

4

4

7

6

4

10

11

5

5

9

8

10

2

 5x 2= 12

7

54

Practice: Solve.  2 x 1 x 3=

 2 x11 =

 2 of 2 1 =

 32 x 43=

3

5

4

5

3

2

3

12 x15= 9

7

 5 2 of 4 =

8

7

 During the summer Scott work 4 1 hours for 8 weeks. How many hours did he 2

work in total?

 What is 1 of 60? 5

 Harvey takes 1 1 weeks to paint a house. How many weeks will it take to paint 3

15 houses on the block?

 How many minutes are there in 5 2 hours? 3

55

Dividing Fractions (Do not need common denominators) • • • • •

Keep the first fraction the same Change the division to multiplication Write the reciprocal of the second fraction (switch the numerator and the denominator of around) Follow the multiplication rule (multiply the numerators together and the denominators together) Simplify if possible and rewrite as a mixed number if needed

•

Mixed Numbers: write the mixed numbers as improper fractions, then follow the above steps

Example:

4 2 4 3 12 2 1  = x = =1 =1 5 3 5 2 10 10 5

Example: 2

3 1 13 7 13 3 39 4  2 =  = x = =1 5 3 5 3 5 7 35 35

Practice: Find the quotient.  4 3= 9

5

 3 1= 10

15

 1 1= 4

4

 1 3= 2

6 2=

 3 21 =

31 15=

42 35 =

3

3

8

7

2

3

6

56

Practice: Solve. 22 2= 7

44 4=

5

9

1341 21 = 5

3

4

 Sally is getting ready to cut a 20 meter ribbon into smaller pieces of 3 meters 5

each. How many 3 meter pieces of ribbon will she have? 5

 Scott and Vitto have 3 of a pizza to share. How much will each boy get? 4

 How many boards 1 1 meters long can be cut from a board that is 11 1 meters long? 2

2

 You are going to a birthday party and bring 10 litres of ice-cream. You estimate that each guest will eat 1 1 cup (there are 4 cups in one litre). 3

How many guests can be served ice-cream?

12÷11÷3= 3

5

57

EXPONENTS

35 = 3 X 3 X 3 X 3 X 3 = 243 3 = BASE 5

= EXPONENT

The exponent tells you how many times the base is multiplied by itself Special Cases: 71 = 7 3330 = 1 (anything to the power of zero is one) 3squared =32 4cubed = 43

Practice: Simplify.  (6)3 =

 (2)4 =

 102 =

 60 =

 05 =

 42 =

 42 + 3 1 =

 53 – 660 =

 82 ÷ 42 =

 31 x 2 3 =

Practice: Find the missing value.  3? = 27

 4? = 1

 ?2 = 16

 ?2 = 49

 ?4 = 0

 5? = 125

58

Exponent Practice  Write in expanded form and solve. a) 73 _____________

b) 82 __________

c) 2160 _____________

d) 1421 ___________

e) 25 __________

f) 34 ______________

 Solve the following. a) 5 + 24 ____________________ b) 23 – 40 __________________ c) 2 x 72 ____________________ d) 24 ÷ 41 __________________  Solve. a) 6squared ______________

b) 2cubed ________________

 One group of 4 students in a grade seven class did a survey about favourite pizza. Each person called 4 people, and they ask those people to call 4 people each. In turn those people asked to call 4 people each, and those individuals called 4 people each. How many people were called?

 Simplify 40 x 51 + 22 - 340

59

Order of Operations: BEDMAS B = brackets E = exponents D = division M = multiplication A = addition S = subtraction

A

S

Whatever comes first from left to right

D

M

Whatever comes first from left to right

E B Start

Remember: Follow the order of operations inside the brackets

Example: 9 + (7 x 20 + 6 ÷ 3) x 13

9 + (7 x 1 + 6 ÷ 3) x 1 9 +( 9

7 +

+

9

9

+

2

) x 1 x 1

9

18 60

Practice:  3 + [(28  7) + (6 – 3)]

 24  4 x 2 + (3 – 2)

 Anne had to calculate the following: 60 – (7 – 3)2 ÷ 2 This is what she did:

Step 1: 60 – (4)2 ÷ 2 Step 2: 60 –16 ÷ 2 Step 3: 44 ÷ 2 Step 4: 22

Is Anne’s answer correct? Explain.

 (6  23  42) + 14  2

 52  14 − 18  17

 43 − [72 − 90]

3+45

61

 (8 − 5)2 + 3  (7 − 2)

 12  3 − 2 − 1 + 5  9

 (8 – 3) + 20 x 4

 8 x 7 – (32 + 4) x 2

 (8 − 5)2 + 3  (7 + 2)

 (3 + 5)  (24 − 4)  0

 {14 + (36 ÷ 9) x 2 } ÷ 11 + 71

 16 + 31 x 0 + ( 33 – 7)

62

Practice: Translate the following and solve.  Add the square of five to thirty, and then subtract the cube of three.

 Square the sum of five and three; then subtract the product of four and seven.

 From the square of eleven, subtract the quotient of forty-eight divided by six.

 Claudia buys tools that cost $81.00 including taxes. She gives the cashier three $20 bills, two $10 bills and one $5 bill. Write the BEDMAS expression needed to calculate how much change Claudia should receive. Solve it.

 Beth is participating in a horse jumping show. The 7 judges awarded the following scores: Technical skills: 5, 7, 6, 8, 5, 3, 10 Timing Skills: 4, 8, 8, 9, 6, 6, 8 The final score is calculated as follows: 1. The highest and lowest mark of each category is removed. 2. The remaining technical skills are added and the sum is multiplied by 4. 3. The remaining timing skills are added and the sum is multiplied by 6. 4. The two results obtained are added.

What is Beth’s final score?

63

Order of Operations with Fractions using BEDMAS B = brackets E = exponents D = division M = multiplication A = addition S = subtraction

A

S

D

M

Whatever comes first from left to right Whatever comes first from left to right

E

B START

Example:

3 𝟏 𝟐 1 ÷2+( ) − 5 𝟑 10 𝟑 1 1 ÷𝟐+ − 𝟓

𝟑 𝟏𝟎

9

𝟏

1

𝟗

10

+ − 𝟑𝟕 𝟗𝟎

−

𝟏 𝟏𝟎

10

Rough Work Do the exponent first

Do the division next

Get a common denominator for the addition Get a common denominator for the subtraction

28

1

1

1

× 3 =9 3 3 1 3 x = 5 2 10

27 10 37 + = 90 90 90 37 9 28 = 90 90 90

90 14 45

Simplified answer: dividing the numerator and denominator by the factor 2

64

Practice: Solve following the order of operations (work in your notebook if you need more space)  51-31 +3 7 =

 3+3x 1=

 1 x 1+ 2 = 4 2 3

 2 3

5

4

10

8

4

2

÷

1 + 4x 1 2 5 4

2

 3+ 1x 2+11= 5

2

3

2

=

 4÷ 3+ 2÷12= 7

7

3

3

 3÷ 2x11 - 1 = 8

3

3

2

65

Extra Practice (do in your notebook) A.

3 2 + 10 5

3 G. 4

-

B.

3 2 +9 6

3 H. 8

2 + 4

C.

8 3 12 - 10

I.

5 D. 15 - 1 3

9 E. 12 +

3 F. 4

+

3 2 3 4 + 12 - 24

3 J. 4

2 12

2 6

3 K. 8

2 L. 3

2 4

-

2 6

+

6 7

2 - 5

1 1 3 1 M. 6 + 3 x 4 ÷ 2

66

Application Questions See the Rubric at the end of the book for a marking guide.

1. School Garden Amy’s students decided to use a rectangular area in the school yard to make a garden. They are drawing up a plan of this garden. On this plan, they have drawn a central path. The area will be divided up as follows: ➢ The central path takes up one fifth of the total area. ➢ Beans will take up 1 of the total area. 4

➢ Tomato plants will take up

3 10

of the total area.

➢ Flowers will take up the rest of the total area. Use a diagram to show how the garden could be laid out. Explain how you found the fraction taken up by flowers. Write the fraction of the area that will be taken up by flowers.

Methods

40 32 24 16

8

0

Calculations

40 32 24 16

8

0

4

0

Organization 20 16 12

8

67

2. Coca-Cola Expenses Mr. Nelson makes $50,000 a year. Of this income, 1/7 goes towards food and beverage. Of this amount, ¼ goes towards Coca-Cola products. 1/3 of this amount is spent on the diet Coke for Mrs. Nelson. The other 2/3 is spent on the always delicious and refreshing straight up Coke. a) How much is spent on straight up Coke?

b) If 3/13 of his Coke spending is spent on chocolate, how much money is spent on chocolate?

Methods

40 32 24 16

8

0

Calculations

40 32 24 16

8

0

4

0

Organization 20 16 12

8

68

3. Ticket Sales $$$ Four players on the Hawks Basketball Team are selling tickets as a fundraiser. Each player had a different number of tickets to sell.

The table below summarizes the number of tickets sold in relation to the number of tickets each had to sell. Player Abby Lily Aisha Cloie

(Tickets Sold) / (Tickets had to sell) 7 9 7 16 7 8 7 18

1. Which player had the best rate of ticket sales? (use calculations) 2. Use your understanding of fractions to explain how you could arrive at this conclusion without doing any calculations.

69

Methods

40 32 24 16

8

0

Calculations

40 32 24 16

8

0

4

0

Organization 20 16 12

8

70

Fraction Assignment: Choose one of the following to hand in for marking

 The Pizza Contest  Sharing Brownies 71

72

 The Pizza Contest Four students in grade 7 were having a pizza contest. Each said he/she could eat the most pizza. Each student ordered a large from the restaurant. Here is what each ate: Scott: I asked to have my large pizza cut into 12 equal pieces. I ate

1 3

of it first. It tasted great! Then I decided to eat 1

1

6

12

full, so I ate . Finally, I stuffed down

1 4

more. I was getting a bit

more, and couldn’t take another bite.

Anne: I asked to have my large pizza cut into 9 equal pieces. I ate

2 3

of it first. Then I ate

1 9

more. I was getting pretty full so I took one piece and cut it in half.

I ate one of those pieces and couldn’t eat anymore.

Beth: I asked to have my large pizza cut into 18 equal pieces.

ullur I ate

1 3

of it first. It tasted great! Then I decided to eat

2 9

of it. I was getting a bit

full, so I managed to eat one more piece of my pizza.

Harvey: I told the restaurant not to cut my pizza. I did it myself into equal pieces, and first I ate 4 pieces. It wasn’t too bad. Then I ate 3 more pieces and finally, I ate 4 more pieces. There was only 1 piece of my pizza left.

Who was the winner of the pizza contest? Use the next page to explain your thinking and show your work. You can use the circles the circles to help represent the pizzas.

73

Show your explanations, drawings and calculations here:

Scott

Anne

Beth

Harvey

The winner of the pizza contest was: Scott

Anne

Beth

Harvey

74

 Sharing Brownies Part A: How much would each person get? Show how you find your answer.  5 brownies for 2 people

 3 brownies for 4 people

 4 brownies for 6 people  2 brownies for 4 people

 4 brownies for 8 people

 5 brownies for 4 people

Part B: If all the brownie pieces started out as the same size, which group above got the most brownies per person? Which group got the least per person? Did any group get the same amount? Show your work.

75

76

Average (Mean) Calculated by adding up all the numbers of the set and dividing by the number of numbers in the set Example:

Paul’s test results 75%

80%

99%

70%

The average of Paul’s results is: 75 + 80 + 99 + 70 = 324 324 ÷ 4 = 81

Paul’s average is 81%

Practice:  The value of 6 houses in Mr. Jones’ neighbourhood is: $260,053 $2543,101 $273,233

$279,000 $256,679 $495,999

What is the average value of the houses in Mr. Jones’ neighbourhood?

 Anne’s pay for the pass 5 weeks was: $212.00, $195.50, $333.33, $245.45 and $303.03. What was her average pay over those five weeks?

 Determine the missing number or numbers. a) The average of 12, 16, 22 and _____ is 15.

b) Two numbers are missing from 51, 68, 47, 32, 41, ____, ____ to equal an average of 47.

77

Number Sense Review  Which of the following is the difference of 25 and 5? (Circle the letter of choice) A)

125

C)

5

B)

20

D)

30

 Which of the following is the sum of 25 and 5? A)

125

C)

5

B)

20

D)

30

 Which of the following is the quotient of 25 and 5? A)

125

C)

5

B)

20

D)

30

 Which of the following is the product of 25 and 5? A)

125

C)

5

B)

20

D)

30

 Which of the following equations is true?

A)

151 = 1

C)

108 = 105  103

B)

83 = 8  3

D)

14 = 4

 What does 32 + 23 = equal? A)

17

C)

55

B)

15

D)

6

78

 Which of the following statements is true?

A)

32 + 2 0 = 3 2 + 1

C)

100 = 10

B)

18 = 2  33

D)

53 = 5  3

 In which of the following can the  be replaced by an equal sign (=)?

A)

12  13

C)

32  23

B)

28  82

D)

50  51

 Which of the following represents the expression 54? A)

5x4

C)

4x5

B)

5x5x5x5

D)

125

 Find the GCF and LCM of 24 and 18. Show your work in the tables provided below.

Factors of 24 Factors of 18 GCF=

Multiples of 24 Multiples of 17 LCM=

79

 List the first ten prime numbers

 List all the factors of 50

 List the first ten multiples of 8.

 Express one million as a power of ten.

 A construction company has 24 foremen, 56 carpenters, and 88 labourers.

The president of the company wants to form crews of the same number of each position. a) How many crews will there be?

b) What will each crew look like? (How many of each position?)

 What is the lowest number than can be divided by 1, 2, 3, 4, 5, 6, 7, 8, 9,

10, without a remainder?

 Scott has two jobs. He gets paid every 14 days from his busboy job and

every 30 days from his lawn-mowing job. How many days will it take Scott to receive both cheques on the same day?

80

 The Army Cadet Camp can accommodate 150 boys and 120 girls. The

leaders want to form as many teams as possible having an equal number of boys and girls. How many girls will be in each team?

 Claudia buys tools that cost $81.00. She gives three $20 bills,

two $10 bills and one $5 bill. Write the BEDMAS expression needed to calculate how much change Claudia should receive. Solve it.

Practice: Solve.  The Hadley Chess Club is made up of 9 students whose average age is 12. The teacher decides to join the team, and the average age increases to 14. What is the age of the teacher?

 Scott loves video games. His average for his favourite game is 4960 points after playing 18 games. He would like to increase his average to 5000 points. How many points must he get in his next game?

81

Interesting Numbers ASSIGNMENT

Your task is to create a mini poster (8 x10) describing a chosen number o You must choose a number o The number should be displayed somewhere on the poster where it stands out (in the middle, a corner…) o On different locations on the mini poster, describe properties of your number o Examples of properties: factors, multiples, divisibility… o You must create at least 2 mathematical sentences (BEDMAS problems), where the answer is your number. Your sentence must include at least one (can be more) of each of the following: ✓ Brackets ✓ Exponent ✓ Each operation (+, -, x, ÷) o Give some interesting fact(s) about your number. For example I chose 26. My interesting facts were: The number on my favourite hockey player’s jersey (Mats Naslund), the number of letters in the alphabet, and apparently you can solve a rubrics cube in 26 moves or less.

NOTES: • You may choose to do this in another format (Ex: PowerPoint presentation)

• See the RUBRIC at the back of the workbook for the marking guide.

82

Decimals Decimals are a part of a whole (just like fractions)

PLACE VALUE

Thousands 1000

Hundreds 100

Tens 10

Ones 1

1000

100

10

1

Tenths 1 10 0.1

Hundredths 1 100 0.01

Thousandths 1 1000 0.001

Ten thousandths 1 10000 0.0001

What is the decimal point for? To determine the position of the ‘ones’ place … the ‘ones’ position is left of the decimal

83

Practice: Write the following decimals in digits. a) One hundred twenty-two and four tenths ________________________ b) Three and seventy-five thousandths _____________________ c) Zero and eight hundredths ______________________ d) Two thousand four and three hundred two thousandths _____________________ e) Six hundred and twenty-three ten thousandths ____________________

Write the following decimals in words. a) 713.56 _____________________________________________________ b) 0.605 _____________________________________________________ c) 303.003 ____________________________________________________ d) 2.0645 _____________________________________________________ e) 9009.09 ____________________________________________________

What is the position of the 8 in each of the following? 380

_________________________________________

1 855 234 _________________________________________ 0.89

_________________________________________

8.216

_________________________________________

245. 708

_________________________________________

84

Ordering & Comparing: • • •

First look at the whole numbers Look at the tenths position…the largest digit is the larger number Then look at the hundredths position, and so forth

Example: Order from greatest to least: 0.36

0.058

0.375

0.4

No whole #s

0.36

0.058

3 10

0.375

0 10

0.4

3 10

6 100

4 10

7 100

Greatest to Least: 0.4

0.375

0.36

0.058

Practice: Place <, >, or =

23.03

23.30

101. 89

11.89

3.13

3.013

211. 46

211.406

0.454

0.407

0.56

0.74

85

Place in order of least to greatest: 1.33

0.67

0.607

0.76

0.706

1.03

0.70

_____, _____, _____, _____, _____, _____, ______

Write the following decimals in increasing order: 2.34

2.27

2.165

_______ _______

2.086 _______

_______

Graphing Decimals on a Number Line: Remember our number system is based on tens... which means we have to divide the space between two numbers into 10 equal parts for tenths and divide each of these into 10ths for hundredths and so forth.

STEPS 1. Determine your smallest and largest number 2. Start with the number just before the smallest number and end with the number just after the largest number 3. Divide between by the whole numbers with 10 equal parts, these will be tenths. 4. If there are hundredths to graph, make 10 equal parts between each tenth.

REMEMBER: do not just show the numbers to be graphed... show the equally divided parts 86

Example:

Graph 2.1, 2.3, 2.5 and 2.9

2.1 is the smallest, so my number line will start at 2 2.9 is the largest, so it will end with 3 2.0 2.1 2.2 2.3

Example:

2.4 2.5 2.6 2.7 2.8 2.9

3

Graph 2.31, 2.34, 2.35 and 2.39

2.31 is the smallest, so my number line will start at 2.3 2.39 is the largest, so it will end with 2.4

2.30

2.31

2.32

2.33

2.34

2.35

2.36

2.37

2.38

2.39

2.40

Practice:  Put the following decimals on a number line a) 3.33

b) 1.8

3.39

0.65

3.32

1.6

3.35

0.75

c) [CHALLENGE] 0.333

0.9

0.9

0.67

1.5

0.70

0.607

0.055

87

 On each number line, determine the value of each point. A

B

C

0 A: ________

2 B: _________

C: _________

B

A 5.60

A: ________

6.20

B: _________

C: _________

B

A 11.235

A: ________

C

C

11.241

B: _________

C: _________

ROUNDING: Step 1: Underline the digit to be rounded Step 2: Write the 2 possible rounding positions Step 3: Follow the rule: ▪ The digit to the right is a 4 or less, the underlined digit stays the same ▪ The digit to the right is a 5 or greater, the underlined digit becomes one greater ▪ All the remaining digits can be replaced by zero

Example: 69.098 round to nearest tenths 69.100

69.000

the digit to the right of 0 is a 9 = 69.1 or 69.100

88

Practice:  76.099 round to nearest tenths __________  0.5403 round to nearest hundredths __________  35.078 round to nearest ones _________  Scott is a cross-country runner. He has a choice of five courses. He decides to choose the course whose distance, rounded to the nearest tenth of a kilometre, has the digit 8 (his lucky number) in the tenths position. Course Name Through the Woods One Hill after Another Flat Lands Through the Creek The Challenge

Distance (km) 9.576 7.882 10.742 9.777 8.089

Which course did he choose? _____________________________________

STOP and Review:  Order the following least to greatest. 25.009

2.5009

0.2509

_______, ________, _________,

2.509 _________

 Put < , >, or = 4.44

44.4

0.303

78.2

78.22

809.101

0.030

6.05

6.050

809.110 89

 Put the following on a number line 3.7

3.45

3.92

3.05

3.67

 The cashier at the restaurant is working with a calculator to get the total amount of each bill after he calculates the tax. He is confused and doesn’t know what to tell the customers what they owe because there are too many numbers. Please help him decide how much each customer must pay. Customer 1: $65.5266 Customer 2: $102.7108 Customer 3: $33.8542

_________________ _________________ _________________

 What is the position of the 6 in each number?

6502 ________________________________________________________ 6.502 ________________________________________________________ 5.62 _________________________________________________________ 7. 06 _________________________________________________________

 Circle which number is greater between the pairs.

a) 4.15 and 4.16

b) 13.32 and 13.23

d) 0.306 and 0.362

e) 103.99 and 10.399

c) 25.05 and 24.5

90

 What about time? Place in order from longest to shortest time: (distance race)

2:33

3:03

3:45

2: 29

2:59

3:19

________, _______, _______, _______, _______, ________

 Place in order from the slowest to fastest time:

5:003

5:030

5:303

5:330

5:033

4:599

_______, _______, _______, _______, _______, _______

 Round the following numbers to the position indicated.

Number

Nearest Tenths

Nearest Hundredths

Nearest Thousandths

6.7493 2.4688 0.2671 0.9987

 The distance between Beth and her friend’s house is 7.846 km. a) What is the distance between their houses, rounded to the nearest tenth of a kilometre? ___________________________ b) What is the distance between their houses, rounded to the nearest hundredth of a kilometre? ___________________________

c) Round the distance to the nearest ones place. __________________ 91

Adding and Subtracting Decimals: Line up the decimals according to their place value and fill in empty space after the decimal with zeroes. Example: 4.89 + 0.0074

4.8900 + 0.0074 4.8974

Example: 70 – 6.974

70.000 - 6.974 63.026

Practice:  Estimate, and then find the sum. a) 3 + 212.09 + 0.1

b) 1009.2 + 14 + 222.006 + 0.76

c) 17.1 + 0.808 + 2

d) 101 + 1.01 + 10.01 + 1.101

92

 Estimate, and then find the difference. a) 205.6008 – 48.567

b) 707 – 70.7

c) 200 – 4.89

d) 8.4 – 4.066

 Find the missing number. a) 43.12 + ______ = 187.332

b) 43.1 – _____ = 6.431

c) ______ + 17.3 = 38.1

d) ______ – 5.08 = 14.7

 Answer the following questions using the decimals from the box. a) Use 2 decimals whose sum is 24.24

12.34 25.008

b) Use 2 decimals whose difference is 13.108

11.9 0.004

c) Use 3 decimals whose sum is 37.352 93

Multiplying Decimals • • • •

Multiply like whole numbers Count the digits after the decimal for each number Starting from the end of the answer, move left the number of spaces you just counted Place the decimal

Example: 24.33 x 7.9 3 digits after the decimal

2433 x 79 21897 + 170310 192207 = 192.207

Practice: Find the product.  35.88 x 1.3

 8.7 x 0.6

 123.4 x 3

 0.55 x 0.7

 Scott worked 24 hours last week. He makes $9.55 per hour. He got $44.50 in tips. How much money did Scott make last week?

94

Dividing Decimals If there is no decimal outside (the divisor), divide like whole numbers, and when you get to the decimal, put it up, and then continue. 9.398 Example: 56.39 ÷ 6

6

56.390 - 54 23 - 18 59 - 54 50

You are usually asked to go to 3 places after the decimal

Practice: Find the quotient.  58.7 ÷ 7

 94.436 ÷ 28

 0.0294 ÷ 6

 0.63 ÷ 9

 48.5 ÷ 10

 28.16 ÷ 4

95

If there is a decimal outside (the divisor), move it until the number is whole, and move the decimal inside the same number of spaces. If there are less spaces inside, add zeroes. Remember, there are no remainders. 22.5 Example: 7.2 ÷ 0.32

0.32 7.2

= 32 7200 - 64 80 - 64 160 - 160 0

Practice: Find the quotient.  78.5 ÷ 7.23

 55.3 ÷ 0.9

 34.416 ÷ 1.8

 18 ÷ 2.3

 Blake stacked 12 blocks which measured 45.6 centimetres. What is the height of each block?

96

Order of Operations and Decimals: follow the rules of BEDMAS Practice:  4.3 + 5 x 3.03

 7.23 + 1.22

 124.8 – 4.1 x (12.8 – 5.7) ÷ 5

 3.12 – 0.42 + 0.55

97

 (3.2 + 5.01) – (2.3 – 0.58 ÷ 10)

 9.8 + [4.51 x 1.5 – (6.15 – 1.7)]

 9.8 x (4.5 ÷ (1.5 – 0.5) – 1.70)

 34.2 – (25.3 + 0.003)

98

Decimal Practice:  Find the missing number. a) 5 x _____ = 45

b) 12 ÷ _____ = 3.75

c) 14.3 ÷ _____ = 5.2

d) 3.2 x _____ = 17.92

 The service elevator can lift a maximum of 650 kg. A shipment of the following just arrived. Can all the boxes be delivered in one trip on the elevator? Explain your reasoning. 52.4 kg 103.22 kg

89.3 kg

68.3 kg

131.89 kg

205.99 kg

 Anne stops to get gas at 97.4 cents per litre. Her total cost was $35.38. On her way home she notices the price of gas at another station was 1.6 cents cheaper. Anne is upset about the money she could have saved. How much could she have saved?

99

 The table below gives the distance of different trails in a park. Trail

Distance (km) 14.354 7.08 19.46 9.489 3.765

Trail A Trail B Trail C Trail D Trail E

Anne wants to hike about 24 km, but not more than 24 km. Give two possible combinations of trails she could take.

 If a = 54.93,

b = 37.14,

c = 9.54, find the value of:

i) a + b + c

ii) (a – b) + c

iii) a – ( b + c)

iv) a – (b – c)

 Scott’s favourite playlist has 8 songs. The length of each song is given below. What is the total time of Scott’s favourite playlist? Times:

3:55

3:01

2:58

4:23

3:46

2:51

4:04

3:49

100

Extra Practice: NO CALCULATOR  33.353 + 2.08 + 8709.0035 + 11

 45 – 9.085

 231.231 x 1.2

 67.9 ÷ 2.3

 702 – 70.2

101

SHOW YOUR WORK to find the area of the following polygons. The polygons are not drawn to scale. a)

7 cm

h= 8 cm 5.5 cm

2.3 cm

9.1 cm

7 cm

b) 8.4 cm

7 cm

c) 3.8 cm 7.4 cm

2.1 cm

7.1 cm

d) 3.1 cm

4.1 cm

102

GO HADLEY GO!

During the Hadley Track and Field Meet the computer broke just before the competition started. In order to determine the winner of each event and the overall winner of the meet, our class was asked to compare and order the results of the three events (javelin, long jump and the 1000 meter race), which were written on paper.

PART A 1. Use the results to determine the first, second and third place winner of each event. 2. At the Hadley Track and Field Meet, points are awarded for each place (1st through to 8th). A table with the value of each place is given. Give the points to each athlete according to their placing in each event. 3. Calculate the total points of all three events to determine the overall competition winner.

PART B Sketch the javelin and the long jump results to scale (like a number line). Draw a javelin field and a long jump area, and accurately divide and place the distance of each athlete’s result on both areas. THESE MUST BE CLEAR, NEAT AND ACCURATE!

~See the RUBRIC at the back of the workbook for the marking Guide ~

103

FIRST PLACE SECOND PLACE THIRD PLACE FOURTH PLACE FIFTH PLACE SIXTH PLACE SEVENTH PLACE EIGHTH PLACE

500 POINTS 400POINTS 300 POINTS 250 POINTS 200 POINTS 150 POINTS 100 POINTS 50 POINTS

Javelin Results: Name Andrew Melissa Stacey Tim Megan Lucas Shawn Kyla

Distance (m) Place 75.86 74.09 74.89 75.87 73.98 75.98 75.08 74. 40

Long Jump Results: Name Distance (m) 4 Andrew 1 Melissa Stacey Tim Megan Lucas Shawn Kyla

Place

Points

Points

8 11 1 12 2 1 3 3 1 4 7 1 8 5 1 12 7 1 12 23 1 24

104

1000 meter Race: Name Andrew Melissa Stacey Tim Megan Lucas Shawn Kyla

Time (mins) 4:03 5:24 5:59 3.59 5:20 5:07 4:30 5:42

Place

Points

AWARD CEREMONY: Javelin: ___________ _____________ _____________ (First Place)

(Second place)

(Third place)

Long Jump: ___________ _____________ _____________ (First Place)

(Second place)

(Third place)

1000m Race: ___________ _____________ _____________ (First Place)

(Second place)

(Third place)

105

Order of Winners (Overall points) Name

Total Points

106

Percent

Fraction

Decimal

Percent

Fraction to Decimal: Numerator ÷ Denominator (fraction line equals a division line) Example:

5 = 5 ÷ 8 = 0.625 8

Decimal to Percent: Decimal x 100 Example: 0.55= 0.55 x 100 = 55%

Percent to Decimal: Percent ÷ 100 Example: 78% = 78 ÷ 100 = 0.78

Percent to Fraction: Write the percent over 100 and simplify Example: 85% =

85 ÷ 5 17 = 100 ÷ 5 20

Decimal to Fraction: The last digit to the right of the decimal gives us the denominator Example: 0.53 = 3 is in the hundredths place =

53 100

Example: 3.224 = 4 in the thousandths place = 3

224 ÷ 8 28 =3 125 1000 ÷ 8

Practice:  Convert the fractions to decimals. a) 3 = 4

b) 2 = 3

c) 3 2 = 5

 Convert the decimals to percents. a) 0.44 =

b) 1.78 =

c) 2.624 = 107

 Convert the fractions to percent. a) 3 =

b) 16 =

7

20

c) 4 5 = 6

 Convert the percents to fractions. a) 74% =

b) 42.75% =

c) 15 1 % = 2

 Convert the decimals to fractions. a) 0.93 =

b) 2.22 =

c) 0.01 =

 Convert the percents to decimals. a) 2% =

b) 79% =

c) 135% =

 Complete the following table. Fractions

Decimals

Percents

1 4

0.64 87% 1.45 42 3

175% 0.05

108

Percent REMEMBER: The word “OF” in math means multiply Percent ‘of’ means multiply Example: 62% of 500 students are girls. How many are girls?

METHOD 1: Before using percent (%), we write it as a decimal, then multiply: 99% = 99 ÷ 100 = 0.99

0.62 x 500 = 310 girls

------------------------------------------------------------------------------------------------------------

Example: 62% of 500 students are girls. How many are girls?

62 ? = 100 500

62 x 500 ÷ 100 = 310 girls

METHOD 2: Cross-Multiply and Divide. Use the percent out of 100

Practice  Calculate the percent of each number: a) 20% of 215

b) 58% of 150

c) 250% of 32

d) 18.5% of 140

109

 42% of the 400 students at Hadley are in grade 7. How many students are in grade 7?

 Anne, Beth and Colleen are softball players. This season, Anne had 65 base hits out of 160 at-bats. Beth batting average was 0.399, and Colleen hit safely 40% of the time. Who was the best hitter?

 There are 600 students at a school, 57% of the students are girls. How many of the students are boys?

 Beth received 16% of the $55 895.00 family inheritance. How much money did she get?

110

TAXES We pay two taxes: G.S.T. = 5%

Example: Pants = $17.00 GST 5% PST 9.975%

(Goods and Services Tax) GST: 17.00 x 0.05 = 0.85

P.S.T. = 9.975% (Provincial Sales Tax)

PST: 17.00 x 0.09975 = 1.695 rounds to 1.70 G.S.T.

5% = 0.05

P.S.T.

9.975% = 0.09975

We pay a percent ‘of ’ taxes

Total: 17.00 + 0.85 + 1.70 = $19.55 Total cost of the pants: $19.55

DISCOUNT: To deduct or subtract from a cost or price (on sale)

Example:

Rollerblades: $129.99

20% off

Step 3: Taxes: GST (5%) and PST (9.9975%) 103.99 x 0.05 = 5.1995 = $5.20 (GST)

Step 1:

Step 2:

0.20 x 129.99 = 25.998 = $26.00 = discount

103.99 x 0.09975 = $10.37 (PST)

129.99 – 26.00 = $103.99 = sale price Total Cost: 103.99 + 5.20 + 10.37 = $119.56

PROFIT: making money Method 1: 1. Turn the percent to a decimal by dividing by 100 2. Multiply the decimal by the original price 3. Add the calculated amount to the original price to make a new selling price

Example: You bought a 32G IPod for $329.99. You want to sell it and make a 25% profit. How much should you sell it for? 25% = 0.25 0.25 x 329.99 = 82.4975 = 82.50 (profit) 329.99 + 82.50 = $412.49 (selling price)

111

Method 2: 1. Add the percent of profit to 100. 2. Cross-Multiply and Divide

Example: You bought a 32G IPod for $329.99. You want to sell it and make a 25% profit. How much should you sell it for?

125 ? = 100 329 .99 125 x 329.99 ÷ 100 = $412.49 (selling price)

Practice  At what price must a skateboard be sold to make a 20% profit if its cost price is $55?

 Scott bought a watch, which was on sale with 15% off. The regular price was $53.33. How much did he pay for the watch? Do not forget GST and PST.

112

 A bicycle with a retail price of $340 is reduced 20%. What is the discount?

 Harvey bought a $1455 television set, $299.89 speaker and a $79.99 DVD player. What was the total cost including GST and PST?

 Two stores are selling the same camera at a regular price of $859. During a sale, the first store offers a 15% discount on the regular price, while the second store reduces the price by $130. Which store has the better deal?

113

MORE Practice:  Anne, Scott and Beth won $65 000 in a lottery. The jackpot will be divided among them based on their money they put towards the ticket. Thus, Anne should get 35 % of the jackpot and Scott gets 1 of the winnings. How much 4

money should Beth get?

 Anne- Marie loves reading historical books. Yesterday, she read 172 pages of a 480-page book. This morning she read 44 more pages. She thinks she has read more than half of the book. What percentage of pages has she read? Is her thinking correct?

 If the sales tax is 15%, how much money are we paying for every dollar spent?

 On a normal day, Beth spends 25% of her time at school, 34% sleeping and 12% eating. What percent of her time is left to do other things?

 What percent of the following figure is dotted? 

 

 

 

 

 



 



            114

Application Questions See the Rubric at the end of the book for a marking guide.

1. Band Practice In the school band: • 1/3 of the musicians walk to practice • 25% of them are driven by their parents • 15 musicians take the bus. If this accounts for everyone in the band, how many people are there in the school band?

Methods

40 32 24 16

8

0

Calculations

40 32 24 16

8

0

4

0

Organization 20 16 12

8

115

2. Shopping for Hiking Trip Some students from Hadley are shopping for a hiking trip. They each want to buy a tent, a sleeping bag, a gas stove and a rain jacket. They want to buy all their equipment at the same store. They have the flyer from two different stores(below): Brian’s Cabin and The Great Outdoors.

$98

The Great Outdoors Tent $225 $98.47 $210.90

Sale 1/3 off $42.30

$261

$55

15% off all products!!! Stove$185

How much will they pay at each store and which will be the least expensive option?

116

Brian’s Cabin

The Great Outdoors

Methods

40 32 24 16

8

0

Calculations

40 32 24 16

8

0

4

0

Organization 20 16 12

8

117

3. You get a scratch card when you get to the cash at a department store. You get 20% off your entire purchase. Fill in the chart below to complete your bill.

Item 3 Shirts 1 pair of Jeans 2 Movies

Cost per item $ 20.99 $ 48.99 $ 15.99 Subtotal

Total Cost $ $ $ $

20% Discount from scratch card

$

New Sale Price

$

Sales Tax

Your final Bill

GST: 5% PST: 9.975% $

Show all your work

Methods

40 32 24 16

8

0

Calculations

40 32 24 16

8

0

4

0

Organization 20 16 12

8

118

Integers •

They are positive and negative WHOLE numbers

•

The zero is neutral

•

The sign tells the direction of the number: ➢ Positive means to the right of zero on a number line ➢ Negative means to the left of zero on a number line

•

Every positive number has an opposite negative number of the same size. For example: -88 is the opposite of +88 because both are the same distance from zero. This means –88 and +88 has an absolute value of 88

Practice: Write an integer for each.  6 units to the left of 11 on a number line.

 7 units to the right of -2 on a number line.

 The stock market went down 291 points today.

 A loss of $35,535 on an investment.

 20 below zero.

 Deposit $1,556 into a bank account.

 The opposite of 201.

 8 units to the left of -4 on a number line.

Put the integers in order from least to greatest.  8, 5, -10, -3, 9, -6, -4, 11, 2, 7, -7

 6, 4, -11, 17, 18, -14, 7, 21

 -40, 44, -51, 24, 5, -48, -50, 49

 -5, -51, 21, -61, 42, -66, 5, 39, -31, -71, 31, 66

119

B

I

N

G

O

INTEGER BINGO

On the next page are a series of Integers, Phrases and Operations 1. Cut out each of the integers, phrases and operations; 2. Match each phrase or operation to an integer; verify your answers with your teacher. 3. Write 24 of the integers to the above BINGO card; 4. Get some bingo chips & you are ready to play…

120

INTEGER BINGO! ☺ Cut out each of these rectangles, there are 52 integers, phrases and operations in total. After you cut them out, match the integer with the phrase or operation. Once your teacher has checked your matches you will write JUST the integers onto your BINGO card! ☺ -

9

1 6 -

17 3

-

16 5

20 -

2

13 -

1

24

-

6

temperature started at -5 C, it rose 13 5 units to the left of 11 on a number line Add six to negative one Nine plus negative twelve Five more than a positive five Negative nine increased by nine Which is greater? 11 or -14 Two greater than negative one The opposite of negative 108 Negative two plus negative twenty Four left of negative twenty Six to the right of negative three Negative twenty increased by six

10

20 below zero.

-

11

The opposite of 27

8

7 units to the right of -16 on a number line Three greater than negative seven Four less than two

-

7

108 -

3

20

38 0 -

27

-

4

-

14

-

22

Six above seven Seven less than negative ten Three subtract ten Three more than negative four Eight less than negative eight Five less than twenty five The sum of negative two and 40 Two decreased by eight 121

122

Practice:  Place <, > or =. a) -5

-6

b) 11

-11

c) -22

2

 Which integer is the correct answer to the following? a) The greatest integer less than zero ________ b) The integer before -30 ________ c) The opposite of 7 ________ d) One greater than -5 _______ e) Two less than -12 _______

 Answer the following using integers from -6 to 6. a) The integers less than 3 ________________ b) The integers greater than -3 _______________ c) The integers less than -1 _______________ d) The integers greater than -2 ______________

 Cars need good batteries, especially during the cold Canadian winters. Battery A is guaranteed to start at a temperature of -40oC and battery B at a temperature of -52oC. Scott thinks battery A is better in cold weather than battery B, because -40 is greater than -52. Do you agree? Why or why not?

123

Adding Integers RULE #1 •

If the signs are the same, pretend they are not there, add the numbers and put the sign of the numbers in the question with the answer (+) + (+) = + (-) + (-) = -

RULE #2 •

If the signs are different, find the difference (subtract the smaller number from the larger number), and the sign of the answer is whichever there is more of in the question (+8) + (-5) = +3 (-15) + (+6) = -9

•

Integers of the same absolute value cancel each other out to equal zero (+7) + (-7) = 0

Practice: Add.  -7 + 5 =

 21 + -14 =

 -80 + 90 =

 -16 + -2 =

 +2 + +6 =

 -13 + -2 =

 40 + (-5) =

 (-4) + 4 =

 (-9) + (-9) =

Practice: Find the sum.  -10 + -2 + 9 =

 6 + -2 + -11 + 5 =

 -1 + -20 + +6 =

 Last Monday the temperature was -23 oC. On Tuesday it rose 3 degrees and dropped by 8 degrees on Wednesday. What was the temperature on Wednesday?

124

 In the following table, calculate the sum of the integers in each column and each row: -5 -3 8 -4 -6

-7 3 -4 8 -2

6 9 -10 -5 -4

-1 -5 7 -1 3

-3 -4 -1 8 -2

 If a = -13 and b = 7, what is the answer in each of the following: a) a + b = _____________________

b) b + a = _____________________

What do you notice about the addition of integers?

 How is -13 + 7 different from -7 + 13? Explain.

 Complete the following: a) -6 + ___ = -4

c) ___ + 5 = -1

e) -3 + ___ = -8

b) -2 + ___ = 0

d) 7 + (-3) = ___

f) ___ + (-9) = -6

Here’s a challenge: complete the following pyramid, given that each number written in a rectangle is equal to the sum of the two integers in the rectangle right below it. -

3

-

8

-

4

-

2

+

1

125

Subtracting Integers Whenever we subtract integers, we ADD the OPPOSITE Example: 

Example: 

+5

– +3 =

+5

+ -3 = 2

-9

– -7 =

-9

+ +7 = - 2

Example:  7 – +6 = 7 + -6 = 1

Example:  -8 – 2 = -8

+ -2 = -10

Practice:  -7 – 5 =

 21 – -14 =

 -80 – 90 =

 -16 – (-2) =

 +2 – +6 =

 5 – 11 =

 18 – (-7) =

 -15 – (+9) =

 (-3) – (-3) =

 (3 – 9) – (-5 – 4) =

 9 – (4 – 6 ) – -7 =

Practice: Determine the value of the expressions if a = -8 and b = 3. i) iii)

a – b = _______________

ii) b – a = _________________

Compare the difference of a – b to b – a. What do you notice? Will the same happen with other integers? Verify using examples.

126

Find the sum or difference.  (8 – 10 + 13) + (-6 + 4 – 11)

 (- 5 + 17 – 24) – (3 – 9 – 5)

 (2 – 9) – (33 – 24 – 6)

 (- 6 + 2 – 8) + (- 9 – 11 + 13)

 What is the difference between – 44 oC and 7 oC?

 On one particular day in Aylmer, it was 3 oC during the day and -8 oC at night. On the same day in Buckingham, it was 1 oC during the day and -9 oC at night. Which town had the greater range in temperature?

 Determine the value of each of the following expressions if a = -6, b = 5 and c = -11. i.

a+b–c

ii. a – b + c

iii. (a – b) + (c – a)

iv. (a + b) – (c – b)

 Complete the table below.

a 3 -4 -9

b 5 -5 4

a+b

a –b

127

 Complete the more challenging table below.

a 7

b

a–b

a+b -8

-10

-4 -6

5

Multiplying Integers Example: (-7) x (+2)

o

positive X positive = positive

o

negative X negative = positive

(-) x (+) = (-)

o

positive X negative = negative

7 x 2 = 14

o

negative X positive = negative

Therefore,

(-7) x (+2) = (-14)

If there is more than two numbers multiplied together: Example: (-4) X (-3) x 2  -3 ⚫ Means multiply +12

X 2  (-3) +24

 (-3) = - 72 ( )( ) Means multiply

Practice:  -7 x 5 =

 21 x -14 =

 (-80)(90) =

 (-16)(-2) =

 +2  +6 =

 4 x -8 =

 11 -3 =

 (-9)  (-3) =

 2 x -2  -1 x (-4) x 3 = 128

Dividing Integers The rules to follow are the same as multiplying:

Example: (-16) ÷ (+2)

o

positive ÷ positive = positive

(-) ÷ (+) = (-)

o

negative ÷ negative = positive

16 ÷ 2 = 8

o

positive ÷ negative = negative

o

negative ÷ positive = negative

Therefore, (-16) ÷ (+2) = (-8)

If there is more than two numbers divided together Example: (-12) ÷ (-3) ÷ 2 ÷ -1 (+4) ÷ 2 ÷ -1 (+2) ÷ -1 = (- 2)

Practice:  -70 ÷ 5 =

 21 ÷ -3 =

 -80 ÷ 10 =

 -16 ÷ -2 =

 +24 ÷ +6 =

 12 ÷ -4 =

 (-28) ÷ (-4) =

 (27) ÷ -9 =

 -36 ÷ 9 ÷ -2 =

 54 ÷ -6 ÷ 3 =

 Beth, Anne, and Scott guessed the temperature one cold morning. Beth’s guess was 3oC too high. Anne guessed -4oC. Scott’s guess was 2oC lower than Anne’s. Beth’s guess was 1oC lower than Scott’s. What was the temperature?

129

Multiply and Dividing Integer Practice  -3 x 2 x -1 x -5 =

 6 x -4 ÷ -3 ÷ - 2

 The table below shows temperatures for one week in March. What is the mean and range for that week? Temperature oC -4 -3 1 0 -7 -6 -2

Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday

 Find the value of the following expressions if a = -36, b = -4 and c = 9. i.

a÷b=

iii.

(a ÷ c) x b =

ii. iv.

bxc= bxc÷a=

 Determine the value of the following powers. a) (-2)4 =

b) -15 =

c) 42 =

 Solve. a) -32 x 2 =

b) 5 x (-2)3 =

c) -23 ÷ -4 =

d) (-4 – 3)2 =

 Complete the following table.

a 16

b

axb

-2 -11

26

a÷b -8 -4

-52

130

Integers and the Order of Operations (BEDMAS) Follow the rules of BEDMAS (brackets, exponents, [division, multiplication], [addition, subtraction]). If an integer is in the problem, first follow BEDMAS. When it is time to work with the integer, follow the rule for that integer. Example: 21 ÷ (-3) + (-6)2 -6

x

-6

= 36

21 ÷ (-3) + 36

IMPORTANT (-7) + 36 29

- 2

3 =3x3=-9

(-3)2 = -3 x -3 = 9 Practice:  22 – (-22 ÷ -2) x +6 ÷ 4 + -10

 2  -11 + -5 – 8 ÷ -4

 -13 + (-33 – 7) ÷ +17 + -180

 5 – 12 ÷ 2 + (-36) + 37

131

 -28 ÷ -4  -3 – (-9 + -9)

 -52 ÷ (-14 + 9) – 15 + (+5)

 Add brackets in the appropriate places to make the answer correct. a) -3 + 5 x -7 + 3 = -23

b) 4 – 6 x -3 + 9 = 15

c) 6 + -4 x -2 + 16 ÷ 7 = -2

d) -7 + 3 x -3 + 10 – 4 = 2

 Use the numbers in the first column once, write an order of operations to give the answer in the second column of the following table.

Numbers -4,

2, 3 -7, -6, 4 -4, 2, 5 -7, -3, -1

Answer -5 14 -10 24

Order of Operations

 (-6 + 3) x [(7 – 10)2 + (5 – 3 x 23)]

132

Integers & BEDMAS Find the mistakes and correct them by redoing each problem to the side. a) 12 ● -3 + 4 ● -9 -36

36

0

b) -11 + -3 + -12 - 9 -14

+ -12 - 9 -26

–9

-17

c) -15 ● -3 ÷ -5 -45

÷ -5 9

d)

-10

●3+6-4

-10 -10

● 9 -4 ●

5

-50 Integer Review

133

STOP and Review  Write the appropriate symbol (<, > or =) in the circle. A)

+

D)

(-5)

-2

5

B)

-20

-22

C)

41

-42

F) –

(-4)

0

E)

(-3) x 4

22

-5

(-2)3

 Last December, Beth kept a record of the outdoor temperature taken at the same time each day for five days and gave it to her Science teacher Mr. Ross. Here are her results : DAY OF THE WEEK Monday Tuesday Wednesday Thursday Friday

TEMPERATURE (in C) -5 -2 0 3 -1

Explain why Monday has the largest digit, 5, yet is the coldest day.

 Which number is the result of the following chain of operations? −2 + 3  4 − 2 a) -6

b) 8

c) 2

d) -14

 On the number line, which two integers are the same distances (equidistant) from 2? a) -1 and 3 b) -5 and 7 c) -2 and 6 d) -8 and 4 Show the number line. 134

 Gina operates the elevator in a large department store. She starts on the ground floor (0) and takes her first group of shoppers to the 3rd floor. Next she takes 2 shoppers down 4 floors; then she goes back up 5 floors with 5 shoppers and finally takes 1 shopper down 4 floors. Which chain of operations will allow you to find the floor where Gina let off her last shopper? 5th floor 4th floor 3rd floor 2nd floor 1st floor Ground floor 1st basement nd

2

basement

3rd basement

a) c)

3 + (-4) + 5 + (-4) 3 + 4 + 5 + (4)

b) d)

3 + (-2) + (-4) + 5 + 5 + (-1) + (-4) 3 + (-4) + 5 + (-1)

Where does Gina end up?

 A submarine, 52 metres below sea level, descends another 25 metres. A missile is fired 165 metres straight up from the submarine. Which of the following mathematical expressions best describes how many metres above the ocean surface the missile reaches? a) c)

-52 +165 -25 + 165

b) d)

-52 + -25 + 165 165

135

 Classify the following numbers into the appropriate column. ½ , 0, 0.25, -5, 45, 4 , -33, 33, (-5)2, 3.6 Integers Not integers

 The table below shows the maximum temperatures recorded on March 2 at different places in the world. Places Ottawa Hull Florida Japan British Columbia

Maximum Temperature (C) -8 -7 20 13 0

Which of the following lists the places in order from the coldest maximum temperature to the warmest? a)

Ottawa, Hull, Florida, Japan, BC

b)

Florida, Japan, BC, Hull, Ottawa

c)

Hull, Ottawa, BC, Japan, Florida

d)

Ottawa, Hull, BC, Japan, Florida

 With words and numbers give an example of an integer in your daily life. Example: It is twelve degrees below zero. It is -12° C. A) An example of a positive integer (you cannot use temperature)

B) An example of a negative integer (you cannot use temperature) 136

 Draw a number line below (use a ruler) and label the positive and negative side. Show where the numbers 5, -13, 0 and -5 lie.

Solve the following expressions. No calculator. – 10

32 – (13)

42 + 63

-23

12 x 7

8  (-2)

20 x (-20)

(-8) x 6

-40 + 52

(-5)3

15 – 28

64  4

13 + -40

12 – (-8)

37 + (-13)

(-36)  6

(-25)  (-5)

(-13) x (-5)

40  (-2) 2

13 + (-12) – 5

(3 + 2)  (-24  6 − 2)  -6

(-12) x (-6)  (-3)

(-2 + 4) + -3 + (7 + -9)

137

Application Questions! See the Rubric at the end of the book for a marking guide.

1. Passing the Time Chris, Melodi, Jenna and Evan are waiting for their movie to start. They amused themselves by trying to express the number 24 in different ways. Which one of them was correct? Show the work. a)

Chris says : 2 + 8  2 + 4

b)

Melodi says : 18 + (-2) − 8  3

c)

Jenna says : 3  6 + 12  2

d)

Evan says : (12 − 4) + 4  2

Methods

40 32 24 16

8

0

Calculations

40 32 24 16

8

0

4

0

Organization 20 16 12

8

138

2. BEDMAS Challenge! Create a BEDMAS problem with an answer of -3 You must include: ✓ At least one set of brackets ✓ At least one exponent ✓ All four operations (+, -, x, ÷) at least once You must solve the problem to show that it works.

Methods

40 32 24 16

8

0

Calculations

40 32 24 16

8

0

4

0

Organization 20 16 12

8

139

Lines and Angles 1.

Point shows position. •A

2.

Straight line is a continuous set of points going on forever in both directions:

3.

Ray is a line with one endpoint. The other goes on forever.

G 4.

Line segment is a line with two endpoints.

J

5.

K

Vertex is the point where two rays meet to form an angle.

Vertex

6.

V

Congruent means the same size, shape, angles, lengths… 

symbol 



7.

Parallel lines run along each other but never cross. Symbol is ||

8.

Angle is a figure formed by two rays with a common end point.

140

9.

Right angles measure 90°.

10.

Perpendicular lines meet at 90º.

11.

Straight angle is an angle that measures 180°, a line.

180°

12.

Acute angles measure between 0° and 90°.

12. Obtuse angles measure between 90° and 180°.

13. Reflex angles measure between 180º and 360º

141

Practice: Find the measure of each angle, label it and state what type of angle it is. 











142

Practice: Use a protractor to construct the following angles.  550

 3400

 900

 2300

 230

 1550

 1800

 870

143

Geometric Properties 1. Complementary angles are two angles that add to 90°. ABD + DBC = 90

A

D C

B

2. Supplementary angles are two angles that add to 180°. XYW+WYZ = 180

W

180°

Z

X Y

3. Vertically Opposite angles are made when two lines intercept. They are congruent

c a

b

d

4. Perpendicular Bisector cuts lines in half creating two equal segments.

Angles a and b are vertically opposite, which means they are the same measure. Angles c and d are vertically opposite, which means they are the same measure.

means equal measure

144

5. Angle bisectors cut angles in half. The angles are the same measure (congruent).

6. Adjacent angles share a ray. Shared ray

Practice:  Construct line segment AB = 7.5 cm and use a compass to bisect the segment. Verify your work.

145

 Construct angle BGN = 85º. Use a compass to bisect the angle. Verify your work.

 Construct angle HEN = 130º. Use a compass to bisect the angle. Verify your work.

 Construct line segment JR = 8.6 cm and use a compass to bisect the segment. Verify your work.

146

Solving Angles: We use the properties of lines and angles to find unknown angles.

Complementary angles add to 90° ?

?

60°

Therefore,

? = 90° - 60° = 30°

When a line is a BISECTOR it will be made clear. NEVER ASSUME that a line is a bisector when it does not say so!

When an angle is bisected, ÷2 to get the smaller angles on both sides

Both sides of a bisector are equal.

A B

AD bisects angle CAB…

600

BD bisects angle ABC…

? D

C angle CAB = 800

…so DBC=600 too!

…800 ÷ 2 = 400 so CAD=400 and DAB=400

Supplementary angles add to 180°

124°

Therefore,

?

124°

?

? = 180° - 124 = 56°

Vertically opposite angles are congruent (equal). A is opposite 160°; therefore A = 160° and B is opposite 20°; therefore B = 20°

A

20° 160°

B

147

Try this: 120o

120° ? ?

?

?

?

This  is a symbol meaning 90 or a right angle.

?

Practice: Without a protractor, using the properties of lines and angles, find the missing angles. Explain your reasoning. G



? A

?

C

? 1330

?

E

?

Justification

K

?

930 H I

D Name and measure

J



B

Name and measure

Justification

148



R

bisector



V U

S

? ?

130

P

T

T

840

310

? W

X

?

?

Q

? Z

Name and measure

Justification

Y

Name and measure

Justification

P





F

Q

?

? ?

G

A

E

47o

?

?

Name and measure

34o

N

L

?

28o D

B

?

R

?

M

C

Justification

Name and measure

Justification

149

C



B

A

D

O

E

Angle AOB = 41o Angle DOE = ______ because __________________________________________ Angle BOC = ______ because __________________________________________ Angle AOE = ______ because __________________________________________

150

Parallel Lines and Transversals A transversal is a line that cuts through two parallel lines.

Transversal

Properties of Parallel Lines and Transversals: Corresponding angles are equal 1 2 3 4

= 5 = 6 = 7 = 8

Alternate interior angles are equal 3 = 6 4 = 5

Interior angles on the same side of the transversal add up to 180 3 + 5 = 180o 4 + 6 = 180

The above properties for parallel lines and transversals can be used to find missing angles without a protractor. 151

Practice: Find the angles marked “?” without a protractor. Explain your reasoning. N

D R ?



G

H

E



C

?

? ?

P

M

F

560

B

L

A

960

S

K Q

Name and measure

Name and measure

Justification

Justification

Q

M M



L

X

H



260

Y

?

?

G ?

? A

K

1220

E

J

Name and measure

F

B

N

Justification

Name and measure

P

Justification

152

Practice: Find the missing angles.  In the figure on the right, AB ll DC . Angle DCE is 55. Angles PAC and RAB are straight angles.

P A

B

R

E

What is the measure of angle PAB? Give a reason (in words) for each step or calculation.

55  D

C

 Given the figure on the right and the following information:

1 C

Rays BA and BC are perpendicular, angle BCE measures 44 and angle EBC measures 35. a) b)

44 E

Explain why angle 1 measures 44. Explain why angle 2 measures 55.

35 2 A

B

153

154

Lines and Angles Assignment

NAME:________________ Due:__________________

~See the RUBRIC at the back of the workbook for the marking Guide ~ Part A: Lines Construct the following on another sheet of paper. 1. Construct line segment AB = 7 cm. Perform a Perpendicular bisector to the line segment. 2. Construct parallel lines CM and HN 3 cm apart. 3. Construct two congruent lines CD and EF = 6.6 cm.

Part B: Angles Construct the following on another sheet of paper. 4. Construct ∟ABC = 110▫. What is this angle called? 5. Construct ∟EFG = 210▫. What type of angle is this? 6. Construct ∟HIJ = 75▫. What is this angle called? 7. ∟ABC = 90▫. What type of angle is this? 8. Bisect one of the angles above. 9. Construct ∟BGN = 180▫. What type of angle is this? 10. Construct examples of complementary and supplementary angles. Explain what type of angles they are.

Part C: Answer the following on these pages. 11. What is the measure of the following angles? What type of angles are they? How do you know? a)

b) ?

? 155

For Questions 12 – 17, use geometric properties to find the unknown angles. SHOW YOUR WORK IN A JUSTIFICATION TABLE 12. What is the measure of angles CEA and DEB? A

32▫

B

? E

?

C

D

13. What are all the missing angles? H

G I

F

48▫ K J

14. Solve the unknown angle. ?

32°

15. Solve for all the missing angles.

W

120°

Y

120°

V

? Z

?

?

X

156 W

16. Solve the “?” angles. D

? C

P

Q

55° ?

L

B

M

? A

17. Solve the following angles.

75 E

115 C A B

D

F

Answers: a)

A=

b)

B=

Because

Because

c)

d)

C=

D=

Because

Because

e)

f)

E=

Because

F=

Because

157

158

Triangles Types of Triangles  Scalene Triangle: All sides are different

 Isosceles Triangle: 2 sides the same

 Equilateral Triangle: all sides are equal (congruent)

 Acute Triangle: All angles are less than 90

 Obtuse Triangle: One angle is greater than 90

 Right Triangle: one angle = 90

We read this triangle as: ABC or BCA or CAB or ACB or CBA or BAC

A

B

C 159

Practice: Measure each side of the following triangles to determine whether each is equilateral, isosceles or scalene. 











Practice: Measure each angle of the following triangles to determine whether each is a right, an acute or an obtuse triangle.













160

Practice: Classify each triangle according to its sides and angles.













Constructing Triangles  Construct an equilateral triangle with 3.7 cm sides.

 Construct a triangle with sides AB = 6 cm, BC = 8 cm, and ABC = 500.

161

 Construct a

CMN, where CM = 4 cm, CN = 5 cm, and MN = 6 cm.

 Construct triangle CAT where CA = 7.7 cm, C = 320 and A = 570.

 Construct an isosceles triangle where two sides are 4 cm each. What is the measure of the 3rd side?

 Construct an obtuse triangle with one angle measuring 1200 and one side measuring 4.9 cm. Indicate the measures of the other angles and sides.

162

Solving Triangles:  Solve the unknown angle in the following triangle.

The sum of all 3 angles of any triangle = 180

x 49

x=

46

 Solve the unknown angles in the following triangle.

x= y=

x y 122

69

 Solve the unknown angles in the following triangle.

? ?

?=

?

 Solve the unknown angles in the following triangle.

? 110

?= ?

 Solve the unknown angles in the triangle.

1040

570

163

Lines, Angles and Triangles: Finding Missing Angles Practice  Angle ao = 700 WX // YZ EF // GH

ao = _____o bo = _____o co = _____o

G

E W

ao

bo

X

do co Y

Z

F

H

Explanation:



a = ____° b = ____°

c 82

a

c = ____°

48

83

b Explanation:

164



bo

a = ____° b = ____° c = ____° co

63o 97o

Explanation: 43o ao

E

EAB = ____°



AB ll CD

A

AFD = ____°

B

DFG = ____°

50o

D

C F

Explanation:

G

165

166

Assignment: Triangles ~See the RUBRIC at the back of the workbook for the marking Guide ~ 1. Construct and label the following, and give a second name to each: ✓ Equilateral triangle ✓ Isosceles triangle ✓ Scalene triangle ✓ Acute Triangle ✓ Right Triangle ✓ Obtuse Triangle

2. Construct ▲ABC such that: BC = 8.2 cm AC = 6.4 cm < BCA = 72 3. Construct ▲XYZ such that YZ = 8 cm

XZ = 6 cm

XY = 5 cm

4. Construct ▲CMT with one side of 5cm and two angles of 40 and 60

5. You will need a ruler, protractor and a compass. Follow these instructions to construct several triangles with common sides. Sketch the triangles before you draw them. ✓ ✓ ✓ ✓ ✓ ✓ ✓

Construct ▲ABC with BC = 6 cm,
167

168

------- GEOMETRY REVIEW ------ Home Garden Center Danny wants to fertilize a garden that measures 3.8 metres by 5.2 metres. He has a $25 gift certificate for the Home Garden Center. One bag of fertilizer covers an area of 3.5 m2. Each bag costs $4.40, all taxes included. Will the gift certificate cover the cost of the fertilizer? Show your work

 Right On! Construct a right triangle where the length of one side is 4 cm and the length of another side is 6 cm. If you were to compare your triangle with those of several classmates, do you think they would all be identical? Explain.

169

 Floor Plan

The music room in a school has a trapezoid shape. The diagram below shows a part of the floor of the music room. A

B

C

a) Using your protractor, measure angle C to the nearest degree. b) Complete the floor plan of the music room by constructing angle A to measure 135.

 Road Work On a blueprint, Lawrence has to mark in a service road which passes through point A and is parallel to autoroute 74. Following these instructions, draw this parallel line. AU

TO RO U

TE

74

A 170

 Angles In the figure below, the measure of angle FOA is 55.

E B O A

F

What is the measure of angle EOA? State the geometric property that you can use to find this measure. Do not use a protractor.

 Which one? Triangle ABC is right-angled at B.

A

130 B

C

Which of the following statements is true? How do you know? A)

The measure of angle A is 40.

C)

The measure of angle A is 50.

B)

The measure of angle A is 45.

D)

The measure of angle A is 55.

171

Application Questions See the Rubric at the end of the book for a marking guide.

1. Mystery Angle

P

In the figure on the right, m AE = m BE

A

B

R

Angle BEC is 65. Angles PAC and RAB are straight angles.

E

65

What is the measure of angle RAP? D

C

Give a reason (in words) for each step or calculation.

Angle RAP =

_______0

Methods

40 32 24 16

8

0

Calculations

40 32 24 16

8

0

4

0

Organization 20 16 12

8

172

2. Tiling the Patio Handy Home Supplies sells patio blocks in three different shapes, shown below. The dimensions and cost of each block are also given. A customer called the store and asked which block would be the least expensive to use to completely cover a 4 metre by 6 metre rectangular patio. Square block $3.25 each

Rectangular block $2.85 each

20 cm 40 cm 50 cm 40 cm

Triangular block $1.30 each

40 cm

20 cm

Which patio block would you advise the customer to purchase? Be sure to justify your answer clearly.

173

The cheapest option is: ___________________

Methods

40 32 24 16

8

0

Calculations

40 32 24 16

8

0

4

0

Organization 20 16 12

8

174

Transformations: mathematical term for “movement”

 Translation: The slide Ex. Shift to right

→

 Reflection: a flip (like a mirror image). Ex. reflected vertical axis

or horizontal axis I am called the reflection line

 Rotation: moving around Ex. 90 rotation

→

The shape does not change in any way…. these transformations are movements only.

175

Translations A translation arrow (t) tells us the direction and distance the shape moves.

Ex

t

Step 1: Draw an axis that is perpendicular to the translation t

Step 2: Move vertices the same distance as the arrow but perpendicular to the axis to create the image.

Step 3: Connect the dots. B B’ A C A’

C’

The image is always denoted with primes

176

Practice: Translate the following shape A

B

C

t

D

C

t

A

T

177

Reflections A reflection is like a mirror, shows the inverted image of the shape that is an equal distance from the reflection line (mirror’s surface) Step 1: Measure the perpendicular distance from a vertex to the reflection line (pathways have to be 900 to the reflection line). Measure the same distance to locate the vertex of the image (on the other side).

s

The reflection line (s) is like a mirror.

Step 2: Repeat for each vertex and connect the dots. Do not forget the label each vertex

s

Practice: Reflect the following

s A B

C

D

E

178

s A

B

C

Perform the following transformations:

s

A

C

B

t

D

179

Rotations

A rotation revolves (clockwise or counter clockwise) all points of the shape about a centre of rotation.

Step1: Draw a line from the centre of rotation to a vertex

Step 2: Using a protractor, measure the angle and draw a line of equal length to find the vertex of the image.

130

Step 3: Repeat for each vertex and connect the dots.

130

180

Practice:  Rotate the following triangle 70o clockwise from the center of rotation.

A

B

C

⚫

 Rotate the following trapezoid 115o counter clockwise from the center of rotation.

⚫

181

Practice  The following object is rotated about the centre of rotation by how many degrees? Answer: _______________don’t forget the degrees sign!

r

 Draw the image of the figure below as reflected about the reflection line. Show your construction lines.

s

182

 Translate the following t A

C

B

D

183

The Art of Transformations ~See the RUBRIC at the back of the workbook for the marking Guide ~ ❖ You will perform on separate pages, a repeated translation, reflection and rotation to create a symmetrical drawing. ❖ There has to be at least 6 transformations for each type using as much as the page as possible. ❖ How to start: Page 1: draw a figure with vertices and a translation arrow ( t ) ✓ Translate this figure using the movement of the arrow at least 6 times ✓ Hint: start at the corner of your page and do not start with a huge figure Page 2: draw a figure with vertices and a reflection line (s) ✓ Reflect this figure across the reflection line at least 6 times ✓ Hint: Be creative with this one: for example, you can reflect across, then put the reflection line beneath the figures you have done to reflect down the page Page 3: Use the center of the page as the origin. State the degree of rotation you will use. Draw a figure with vertices outside the point of origin. ✓ Rotate the figure around the page ✓ Hint: rotate 360  in total (a full circle with your figure). In order to get at least 6 rotations, the degree of rotation may be less 90

COLOR YOUR DESIGNS: EACH TRANSFORMATION A DIFFERENT COLOR

184

Unit 6: Data Management Collecting data Data is collected in a Frequency Table.

Title

Data Category

Eye Colour Green

Tally |||| ||

Brown

|||

Blue

||||

Purple

||

Eye color Frequency 7 3

||||

Total

10 2 22

Percentage, % 7 22 x 100 = 32% 3 22 = 14% 10 22 = 45% 2 22 = 9% 100%

Data name

Mean: another word for ‘average’ Calculated by adding up all the numbers of the set and dividing by the number of numbers in the set.

Range: the difference between the largest and smallest number in the data

Example: Use the following test scores to find the mean and range: 40, 60, 95, 40, 61, 61, 55, 35, 35, 100, 61, 40, 72, 72, 72, 72, 55, 61, 35, 22, 100, 61, 61, 61, 72, 72, 72, 100, 14, 61

Mean = 1818 ÷ 30 = 60.6 Range = 100 – 14 = 86

185

Practice: 1. Find mean, and range from the data below. Make a few comments about your findings. a) 3, 2, 7, 5, 1, 2, 6, 8, 12, 10, 9

b) 15, -10, 25, -20, 10, 15, 20, -30, -15, -10

c) 88%, 75%, 89%, 99%, 100%, 80%, 66%, 92%, 79%

2. Construct a Frequency Table A recent survey asked some grade 7 students “What is your favourite subject at school?” The results are as follows GH, A, Mu, G, G, E, M, M, G, M, G, G, A, F, GH, A, Mu, E, A, A, Mu, A, G, M (M=math, G=gym, E=English, F=French, GH=Geo or History, A=art, Mu=music)

186

Modes of representation: A picture, mode of representation, can give a better understanding of a set of data. All graphs must have 3 titles: overall title, horizontal and vertical titles Bar graph:

Millions of People

Population by Province / Territory (2016) 16 14 12 10 8 6 4 2 0 AB

BC

MB

NB

NL

NT

NS

NU

ON

PE

QC

SK

YT

Province or Territory

Notice the bars are the same width, and each bar is the same distance apart.

% of the Total Population

Histograms:

Age Distribution of Canada (2016) 30 25 20 15 10 5 0 0-19

20-39

40-59

60-79

80-

Age Range

DIFFERENCE between a histogram and a bar graph: Each bar on a histogram represents a RANGE OF DATA, where each bar on a bar graph represents a specific category.

187

Broken-Line graph:

Shows a CHANGE over time.

Circle graph: (Pie Graph)

188

How to make a Circle graph: • Find the TOTAL of the numbers • Decimal = each number ÷ total number • Percent = decimal x 100 • Degrees of the sector = decimal x 360

Example: Females seen at the mall with different hair color Hair Color Blond Brown Black Red Grey Total

Amount 33 29 17 9 37 125

Decimal

Percent %

Degrees

33 ÷ 125 =0.264 29 ÷ 125=0.232 17 ÷ 125 =0.136 9 ÷ 125 =0.072 37 ÷ 125 =0.296

0.264 x 100 =26.4

0.264 x 360 = 950

0.232 x 100 = 23.2

0.232 x 360 = 83.50

0.136 x 100 = 13.6

0.136 x 360 = 490

0.072 x 100 = 7.2

0.072 x 360 = 25.90

0.296 x 100 = 29.4

0.296 x 360 = 106.60

100%

3600

1

Females seen at the mall with different hair color

189

 Construct a histogram with the following data. Make 3 comparative statements about the data. Average Rainfall by Season in Canada (1981-2010) Season

Rainfall (mm)

Dec-Feb March-May June-Aug Sept-Nov

74.5 192 269 258.3

Statements: ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ 190

 Construct a broken line graph with the following data. Make 3 comparative statements about the data.

Ice Cream Sales from April 24th – 28th

Statements:

Mon Tue Wed Thu Fri Sat Sun $410 $440 $550 $420 $610 $790 $770

Broken Line Graph:

191

 DISTRIBUTION OF GRADE 7 FRENCH MARKS MARKS

TALLY

FREQUENCY

20 - 29 30 - 39 40 - 49 50 - 59 60 - 69 70 - 79 80 - 89 90 - 99

What graph is best to display the above data? __________________________ Create the graph.

192

 Bicycles Sales Month

Number of bicycles sold

March

7

April

12

May

23

June

19

July

8

August

5

What graph is best to display the above data? __________________________ Create the graph.

193



Temperature (0C)

Average Temperature During a Week in May, 2017 20 18 16 14 12 10 8 6 4 2 0 Mon

Tues

Wed

Thurs

Fri

Sat

Sun

Day of the Week

a) Which day was the coldest? ________________________ b) What is the temperature range for the week?

c) What is the mean temperature for the week?

d) What is the difference between Monday’s and Thursday’s temperature?

 The ages of 30 Grade 7 students are listed below. Using this data construct a frequency table. 12, 14, 12, 11, 13, 12, 12, 15, 13, 12, 13, 13, 13, 14, 12, 13, 12, 13, 13, 11, 13, 12, 12, 13, 12, 13, 13, 14, 12, 13.

194

 The graph below shows how the student population is distributed among the various grades of a secondary school. STUDENT POPULATION DISTRIBUTION OF A SECONDARY SCHOOL NUMBER OF STUDENTS

300 280 260 240 220 200 180 160 140 120 Sec 1

Sec 2

Sec 3

Sec 4

Sec 5

GRADE LEVEL

Use this graph to construct a circle graph.

195

 The circle graph below shows the activities in which 350 high school students enrolled. In the diagram, the angle of the Art sector measures 108. Sports activities were chosen by 40 % of the students.

How many students enrolled for computers?

 The table below lists the average monthly temperature recorded at the station in the first 5 months of the year. Average monthly temperatures for first five months of the year Month January February March April May

Temperature in C -10 -8 -2 1 11

Draw a broken-line graph to represent the average monthly temperatures recorded during this period which the students can hang up at the entrance to their booth.

196

 Beth’s conducted a survey about favourite television shows among her classmates. The table below represents the results. Type of show

Comedy

Reality Shows

Action

Number of Students

Adventure Time

5

Rick and Morty

4

Brooklyn Nine-Nine

3

Project Runway

3

America’s Got Talent

5

Queer Eye

10

The Flash

1

Agents of S.H.I.E.L.D.

7

Super Girl

2

Total

Draw a circle graph that accurately represents the favorite television shows Beth’s classmates watch.

197

198

Data Management Project

Our School

Part 1 You will prepare a survey and create a tally sheet You will organize, classify, display and analyze the data collected from your peers. Step 1 Create the survey ➢ Decide what you are going to survey of the Hadley Student population. Ex. What time do Hadley students wake up? ➢ Build a frequency table to tally the data. Don’t forget a title, data category, data name, tally and frequency. This will be handed in. Step 2 Conduct the survey



Have at least twenty (20) people outside of our class participate in your survey. Fill in your frequency table.

Step 3 Change your data to at least 2 different modes of representation (hand done) ❖ Bar Graph ❖ Line Graph ❖ Pictogram ❖ Circle Graph: one must be a circle graph Step 4 Analysis and Conclusion in a paragraph Look at your data; this is called analysis. Some questions to answer might be:  Is there one choice greater/lesser than the others? Greatest or least?  Is there an increasing/decreasing trend?  Does anything standout or surprise you? If so analyse further by telling me why you think this might be true?  What is the mean and range?  Comment on your data in a paragraph. Make conclusions about what your data can tell somebody. The more you write, the better your analysis.  Do you think the results would be the same in other junior high schools? Explain.  Prepare a report.

199

Part 2 Present the results to the rest of the class. ❑

Prepare your data and graphs on bristol board paper

CHECKLIST Your work should include: ✓ ✓ ✓ ✓

The survey question and organized/classified data At least 2 different graphs (one is a circle graph) Justifications to the graphs you chose You analysis and explanations (refer to the question provided)

~See the RUBRIC at the back of the workbook for the marking Guide ~

200

Geometry Review  How would you get angle 3 if you know angles 1 and 2? in the diagram below? Explain yourself clearly using mathematical language and numbers.

2

3

1

 You are about to renovate your room. Professionals will install a new floor, baseboards and moulding. You consulted different sales flyers, and have made the following choices : ✓

Hardwood flooring on sale at a cost of $102.10 per box. Each box will cover 1.85 m2.

✓

Baseboards on sale at $5.98 per 2.4 m and moulding on sale at $8.45 per 2.4 m. Taxes are included in these sale prices.

Here is the floor plan of your room. The walls have a height of 200 cm

3.8 m 8.5 m

2.8 m This is not drawn to scale

4.7 m 7m

With the choices that you have made, will your parents be able to renovate your room within a budget of $4 400 planned for this renovation? Justify your answer

 The winners of a Ottawa Marathon are invited onto two rectangular red carpets to receive their trophies. The area of each carpet is 15 m 2. The carpets are not identical. The perimeter of one carpet is a whole number. The perimeter of the other carpet is a decimal. Both carpets must fit onto a platform that is 8 metres in length and 8 metres wide. Find dimensions for 2 carpets that meet these conditions. Draw a sketch to show how they fit on the platform.

201



F

Given the figure on the right and the following information:

1 C

Rays AF and DE are parallel, and angle BCA measures 45.

45

a) What is the measure of angle 1? Why? b) What is the measure of angle 2? Why? c) What is the measure of angle BCF? Why? d) What is the measure of angle CAB? Why?

2 A

40

E

B

D  Harvey built a patio in the back yard. a) What is the area of the patio? b) What is the area of the grass? c) What is the total area of the patio and grass? Show your work.

740 cm

4.3 m Grass

Patio

6.6 m

Grass

3.6 m

0.144 hm



Given rectangle ABCD with diagonals AC and BD, shown on the right. Line EF passes through side AD of the rectangle.

C

B 35 E

O

A

D

F

Which statement below is TRUE? A)

Line EF is perpendicular to segment BC.

B)

Diagonal BD is the right bisector of diagonal AC.

C)

Angles AOB and COD are vertically opposite and complementary.

D)

Angles BOC and COD are adjacent and supplementary. 202

Percent Review  The grade 7 students at your school are organizing a trip to Quebec City at the end of the school year. You are one of the 25 students chosen to participate. The student council is using the Travel in Comfort Travel Agency who has arranged a fare of $350/person, taxes included. To help you finance this trip, you have all participated in a fundraising campaign of selling oranges. Now the students are ready to see how much money was raised. In total, you and the others have sold 550 cases of oranges at $22 each! Now the supplier, the school and the parent committee (who delivered the cases) need to be paid. This is urgent! Here are the obligations that you must consider: o The supplier must receive 35% of your funds; o You must give $125 to the school for the publicity they gave you; o

You must give

1 of your funds to the parent committee for deliveries. 10

Determine the amount of money that each student will receive from participating in this fundraiser. Did they raise enough money each for the trip? 

FRACTION

DECIMAL

PERCENT

2 5 0.04 86% 1.12

 The Canadian Snow Goose clothing store is having a blow out summer sale. You walk into the store with $150 of birthday money. The following items caught your eye and fit perfectly: ▪ ▪ ▪ ▪

Shirt: Regular Price $49.99; on sale for 30% off Jeans: Regular Price $79.97; on sale for 15% off Shorts: Regular Price $24.00; on sale for half off Canvas shoes: Regular Price $33.33; on sale for 25% off

Do you have enough money to buy all 4 items? Don’t forget the taxes, GST 5%, PST 8.5%  Beth and Anne are analyzing the discounts which are being offered by travel agents. Beth thinks that the discounts are all the same! Is she right? 25 % of $30

0.03 of $25

0.25 of $30

5 of $30 20

9 of $25 30

30 % of $25

Compare each of the above discounts offered and give evidence of your thinking, don’t forget to justify your answer with mathematical arguments.

203

Fractions Review: Solve and Simplify  10 + 4

 5 - 5

7

21

8

 3 ÷ 2 4

 6 x 7

12

11

6÷ 2

5

 3 3 +7 4

3

 16 3 - 11 5 8

8

5

7

9– 2 2 +1 1 +3 5

6

3

4

6

2 3 3 x2 1 ÷4 1 5

2

 3 - 5 - 3 +

3

8

 Square ABCD on the right is divided into several other squares of different sizes.

6

4

1 2

A

B

D

C

What fraction of square ABCD does the shaded part represent?

 You are going to a birthday party and bring 8 litres of ice-cream. You estimate that each guest will eat 1 1 cup (there are 4 cups in one litre). 4

How many guests can be served ice-cream?  Beth’s test results are as follows: History: 9 out of 15 Geography: 14 out of 25 Science: 3 out of 5 Math: 7 out of 12 Which test was her best result? Which was her worst? 204

Number Sense Review  List the factors of each: 55:

36:

11:

 Find the GCF (Greatest Common Factor) for 12 and 20  Michael, Lennox and Gina bought hockey cards. Each package contains the same number of cards. Michael bought 120 cards. Lennox bought 180 cards. Gina bought 96 cards. What is the greatest number of cards that could come in one package?

 Harvey goes to the movies every 2 weeks. His brother Brian goes every 3 weeks, and his sister Lucy goes every 4 weeks. All three were at the movies together on Tuesday. When will all three be at the movies at the same time again?

 Where would you put brackets to get an answer of 7 in the following question: 22 – 3 x 4 – 27 ÷ 2 + 7 = ______  9 + (7 x 30 + 6 ÷ 3) x 15

 33- [(21  7) + (6 x 3)]

 (7 − 3)2 + 3  (7 − 3)

 Solve the following. a) 5 + 24 ________________ b) 23 – 40 ________________

c) 2 x 72 ______________

d) 24 ÷ 41 ________________ e) 7squared ________

f) 5cubed ________

 Beth would like to go to horseback riding camp this summer and her parents have agreed as long as she helps to pay for the camp fees. She has taken a part-time job babysitting after school in order to earn some money and has determined that she needs to earn an mean of 95$ each month for five months. After four months of saving, she is a little worried that her earnings have not been quite adequate. These are her monthly earnings so far : $95, $109, $91, $82. She has one more month to save. What is the least amount of money she can earn in this last month to reach her goal?

 This is the problem: “Square 7, add 3 to this number and divide the sum by 4.” A student has attempted to write the correct numbers and operations for this problem in math class. The student has written: 72 + 3 ÷ 4 but the teacher insists it is incorrect. Explain the student’s error and correct it.

205

Integers Review  11 - -3 =  -5 + 8 =  -12 ÷ -2 =  11 – 3 x 4 - -3

 -7 – -6 =  4 x -5 = -40 ÷ 10 =  8 ÷ -2 - -3

 -12 – 4 =  -3 x - 3 =  7 ÷ (6 ÷ -6) + 1  10 + -15 ÷ 3 + -4

 - 7 + -4 =  7 + -3 =  6x2=  6 ÷ -3 =  (2 x 8) - (-2 x 8)  8 + -9 + -7 x -2

 Nines is a very popular card game designed for three players. Each player begins the game with a score of 9, and the object is to bring the score to 0. Each player must win 4 tricks per hand to maintain his/her score. For each hand, each player’s point total will be adjusted according to the number of tricks he/she has won: 0 tricks → + 4 5 tricks → - 1 1 trick → + 3 6 tricks → - 2 2 tricks → + 2 7 tricks → - 3 3 tricks → + 1 8 tricks → - 4 4 tricks → 0 John says that player C has won. Is he right?

Player A B C

Tricks won per hand 2 4 6 5 6 8 4 5 3 0 4 4 3 4 4 2 5 4 0 6 4 6 6 8 7 5 7 5 5 3 3 5 3 4 4 7 1 2 5

 In golf, par is the number of strokes needed to get the ball into the hole. For example, par 4 means that normally you can get the ball into the hole in 4 strokes. If you take 5 strokes, your score that turn is +1, and if you take 3 strokes, your score is -1. If you take 4 strokes, your score is 0. At the end of the course, the final score shows how many strokes it has taken someone to complete the course. You could be at par, under par or over par. Obviously, the fewer strokes you used to complete the course, the better. To help you understand, your Harvey shows his last score card from his last game. Unfortunately it’s quite old, and missing some information. You want to know if Harvey was right when he told you that his score was -2. Check to see if Harvey calculated correctly, by finding the missing information.

Hole 1 Hole 2 Hole 3 Hole 4 Hole 5 Hole 6 Hole 7 Hole 8 Hole 9 TOTAL

SCORE CARD : Harvey Number of Score Strokes 3 +6 -2 2 9 7 1 +1 0 3 8

 What is the mean of the following temperatures (0C)?:

-7, -6,

Par 5 3 4 4 3 5 5 36 0, 1, -1, 1, -8

206

C2 Mock Exam Part 1 Questions 1- 3 Multiple Choice: 4 marks each 1.

Which expression has the greatest value? A)

12 − (5 + 4 × 32)

C)

(420 + 8)  (11 - 2)

B)

(5 + 3)2  8 × 2

D)

24 − (5 × 6  3) + 4

2. Leo spends all of his free time in front of his computer. There, he spends one sixth of his time downloading music, half of his time playing online games, and the rest of his time is spent chatting. What fraction of his time does Leo spend chatting? A)

1 4

C)

1 3

B)

1 6

D)

1 12

3. Janice bought 2 novels regularly priced at $24 and $18. They were reduced by 25%. How much did Janice save? A)

$6.00

C)

$17.00

B)

$10.50

D)

$31.50

Questions 4 -5 Short Answer: 4 marks each 4. Construct a trapezoid with an area of 8 cm2 and a height of 4 cm.

207

5. Place the following four values on the number line provided. Label your points on the number line using the corresponding letter. A) 40%

0

B)

3 15

1

C) 1.8

D)

24 20

2

Questions 6 – 7 Applications: 10 marks each

6. Joe wrote 5 Math tests this term. His Math test results are shown on the graph below:

Joe wants to calculate his average for the term. • • • •

He remembers that his result on the 1st test was a multiple of 4. His result on the 3rd test was divisible by 3, and was 14% higher than his 4th test. His result on the 4th test was 4% lower than his result on the 2nd test. His result on the 5th test was 94%.

What is Joe’s math test average for the term? 208

7. Cassie operates a cupcake business. She needs to replace her oven and must decide whether she should buy a new one or rent one. She wants to spend the least amount of money she can over the next 6 months. She also needs to calculate how much she will be spending in total over the next 6 months once her decision has been made. Here are her expenses: Monthly fixed expenses: She spends $1500 per month in rent and other fixed expenses. Monthly cost of Ingredients: Cassie needs the following ingredients each month: Her needs for 1 month

Her expenses

80 bags of flour

1 bag of flour costs $9.45

288 eggs

a dozen (12) eggs cost $3.50

52 sticks of butter

4 sticks of butter cost $4.95

25 bags of sugar

1 bag of sugar costs $5.05

Decision: Cost of Buying or renting an oven • Buy the oven for $484.80 • Rent the oven for $79.60 for each month Knowing that Cassie wants to spend the least amount, what is the total amount she will spend during the next 6 months of business? All prices include taxes Expense

Monthly cost

Cost for 6 months

Rent and other Fixed expenses Ingredients Oven Total Cassie will spend $ ______________ during the next 6 months of business. 209

C2 Mock Exam Part 2 Questions 1- 3 Multiple Choice: 4 marks each 1.

Simply the following expression, respecting the order of operations:

2 1 1   + 5  120 %  0.75 2

(

2.

) 17 20

A)

8

1 4

C)

1

B)

7

3 4

D)

8

1 2

The drawing below represents a wooded property that must be fenced in. The perimeter of the fence is 337 m. 0.18 km

2m

0.4 hm

?

230 dm

8 dam

What is the length of the missing side? A)

120 m

C)

0.12 m

B)

12 m

D)

1.2 m

210

3. George is a marine biologist and is currently looking at the migration patterns of whales. To begin his research, he dives 60 metres below the surface level of the water. He then rises 25 metres to get a better look at the whales as they come up for air. He then swims down 45 metres. Finally, he swims up 15 metres to snap a photo of a whale. What is George’s depth when he snaps the photo? A)

55 metres below the surface

C)

95 metres below the surface

B)

65 metres below the surface

D)

25 metres below the surface

Questions 4 -5 Short Answer: 4 marks each 4. Draw the following lines on triangle ABC:

A

a) the bisector of angle C. b) the altitude from the vertex A. C

B

5. Consider the triangles below and answer the four questions. The triangles are not necessarily drawn to scale. R 42

F

45 

G

E

136

P

a) b) c) d)

What type of angle is RPQ? What type of triangle is PQR? What is the measure of angle is EFG? What type of triangle is GEF?

211

212

RUBRICS These are guides that show you how your work is graded.

Application Problem RUBRIC RULE: METHODS is the “Boss” category. This means that the categories cannot score higher than the METHODS score. METHODS 40 Correct methods are used

32

24

Correct Correct Correct method method is method is is shown shown shown for for for most some steps almost steps (3 mistakes) all steps (2 (1 mistakes) mistake) CALCULATIONS 40 32 24 16 One or A major Verify all All two mistake 2 major calculations (should calculations minor (EX: mistakes OR use 2 ways) are correct mistakes order of more than 3 (EX: ops, minor mis-calc, wrong mistakes misformula) copy or or 3 missing minor #) mistakes ORGANIZATION 20 16 12 8 Labels, neatness, clarity.

Correct method is shown for every step. Formulas must be shown.

16

Steps are labelled, work is neat, connections are clear.

Missing one of the three.

(Mini Version) Methods 40 32 24 16

8

0

Calculations

8

0

4

0

40 32 24 16

Organization 20 16 12

8

Missing two of the three.

8

0

Correct method is shown for one step OR an attempt is made

No work shown

8

0

An attempt is made.

No work shown

4

0

Work is Incomplete complete and but disorganized disorganized

No work shown

213

Assignment RUBRIC RULE: “Completeness” is the “Boss” category. This means that the categories cannot score higher than the COMPLETENESS score. COMPLETENESS

METHODS and CALCULATIONS

ORGANIZATION Neatness and Clarity (easy to read).

40

30

20

10

0

Complete

Almost complete

Some work is done

An attempt is made.

No work shown

40

30

20 10 Correct Correct Correct methods Correct methods methods and methods and and and calculations calculations calculations calculations sometimes. rarely almost mostly. always. 20 15 10 5 Complete work that is neat, and easy to read.

(Mini Version) Completeness 40 30 20 10 Methods & Calculations Organization

Complete work missing one of the two.

No work shown

0 No work shown

0

40 30 20 10

0

20 15 10

0

5

Work is Incomplete complete and but not neat disorganized or clear.

0

214