grade 9 applied math workbook
TRANSCRIPT
Lesson 2: Introduction To Integers Name:
6 Represent the following numbers using a tile display.
Use a small shaded square to represent +1 (the red tile)
Use a small unshaded square to represent -1 (the white tile)
a) - 5 b) + 4 c) - 7
d) 0 e) + 1 f) - 2
7 State the opposite of the following integers.
a) - 6 b) - 100 c) + 10 d) 5
8 State the integer that must be combined with the following integers to result in the zero
model.
a) 16 b) - 10 c) + 8 d) -1528
5
TIPS Assignment -
Write an integer expression for each of the following. Do NOT evaluate.
a) The temperature is 14EC, and drops 10EC.
b) You deposit $40 to your bank account,
and then write a check for $15 and
another check for $15.
Lesson 3: Combining Integers 1 Evaluate the following using algebra tiles. Model the expression and show the final answer using tiles and a
number.
a) -3 + 5 b) -2 - 4 c) 4 - 5 d) -3 + 3 2. Evaluate the following WITHOUT a calculator. (Remember: No calculators will be
permitted on the test.)
a) - 2 - 3 b) - 1 - 4 c) - 5 + 2 d) - 3 + 1 e) - 4 + 7 f) - 9 - 6 g) 7 + 9 h) - 12 + 4 i) - 22 - 4 j) - 2 - 6 – 4 k) - 1 - 2 - 3 l) - 3 + 5 - 9 m) 6 - 4 – 6 n) 3 - 5 - 7 o) -3 + 4 + 5 - 2 3. Gasoline is made from crude oil. The price of crude oil often changes from day to day. Suppose that a barrel of
crude oil cost $41.60 at the end of the day on Monday. For the rest of the week, the price changed as shown in this table.
Tuesday Wednesday Thursday Friday - $ 0.09 - $ 0.42 + $ 0.76 - $ 1.14
a) Write an expression to model the value of the stock at the end of the week.
b) Evaluate the expression to model the value of the stock at the end of the week. 4. At the beginning of the month, a family had $640 in their bank account. During the month, they made the
following deposits and withdrawals:
Deposits: $875, $700, $875, $700 Withdrawals: $225, $40, $75, $225, $55, $225, $199, $225
a) Write an expression to model the bank balance at the end of the month.
b) Evaluate the expression to model the bank balance at the end of the month.
Lesson 4: Introduction to Rational Numbers
1 Write the fraction that represents the shaded part of each diagram.
a) b) c) d)
2 Place each number beside the corresponding letter, and reduce if possible.
| | | | | | | | | | | | | | -2 D -1 C A 0 F B 1 E
A B C D E F _____
3 Place each letter on the number line
| | | | | | | | | | | | | | -4 -3 -2 -1 0 1 2
A 2
3 B 1
4
5 C
1
2 D 3
2
3 E 2
1
3 F ‐ 2.65
4 i) Rewrite each rational number in decimal form.
a) 1
4 b) 2
1
4 c) 1
1
3
ii) Rewrite the fractions from smallest to largest.
5 i) State the opposite of each rational number.
ii) Express its opposite in decimal form.
a) 1
4 b) 2
1
7 c) 1
1
2
3 i) Express each rational number in fraction form, then reduce it to lowest terms.
ii) Rewrite the fractions from largest to smallest.
a) 0.6 b)-1.4 c) 0.875 d) 0.75 e)-0.2
4 Express each mixed number as an improper fraction
a) 21
4 b) 2
3
4 c) 1
2
3 d) 2
4
7 e) 1
3
5
Lesson 5: Combining Rational Numbers 1. Simplify. Remember to reduce all answers to lowest terms.
a) 5
6
1
6 b)
3
4
1
4 c) 1
3
3
4
d) 2
3
3
4 e)
3
8
5
4 f)
2
32
5
6
g) 11
2
1
8 h) 35
6 i)
1
21
5
6
j) 11
3
3
4 k) 5
2
3 l) 1
3
1
2
1
4
m) 11
2
2
31
5
6 n) 2
3
4
1
53
2 How many slices are in two pizzas if each pizza is cut into twelve slices.
3 At the start of the week, Mom’s car reads 1
8 of a tank. At the end of the
week, she adds 3
4 of a tank. The drive to the cottage takes
1
3 of a tank of
gas. a) How much gas is left when she arrives at the cottage?
b) Will she have enough gas to make the return trip home? Explain.
Lesson 6: Introduction to Polynomials 1 Complete the sentences below using the following words. binomial, coefficient, constant, exponent, like terms, monomial, term, trinomial, unlike terms, variable
a) In the term, 4x 2, the 4 is the and x is a . The
expression 5a 2 has one . Therefore, this expression is a . b) The expression 2b - 5 has two terms. Therefore, it is a . In this
expression, the second term is a . c) The expression 3y 2 - 2y - 4 is an example of a . d) 3a 2, -5a 2, and 9a 2 all have the same variable and the same .
Therefore, these three terms are . 3a 2, 4b, and 11 are .
2 State the coefficient of each term. a) 10 d 2 b) -3x c) y d) -y 3 State whether each of the following expressions is a monomial, binomial or trinomial or
polynomial. Then, state its degree.
a) 5h b) -9k + 6 c) 77b – 55
d) 2xy 2 + 6y + 8 e) 6x 4y + 5x 3 - 8x 2 + 2x + 9 4 Identify the like terms.
a) 5x 2, 7x, 8xy, 54x, 67, -8y 2, 9x b) 6k, -10, 8k 3, 7k, -9k 2, 12, k 3
Lesson 7: Combining Polynomials 1 Simplify by collecting like terms. State the degree of the final polynomial answer.
a) 3x - 6x b) -2x 2 + 4x 2 c) 3y + 6y d) 4xy - 2xy e)6n + 9 - 3n + 5 f) 4p - p - 6 + 2 g) 2x + 3x + 8y - 7y h) z 3 + 3z 2 - 6 - 3z 3 + 6z - 3 i) 4y 2 - y + 3 + 2y 2 + 2y - 1
k) m 2 - 4m + 1 + 6m 2 - 7m - 10
j) 3x 2 + 7x - 7 - 8x 2 + 5x + 3
l) 5v 2 - v + 6 - 12v 3 - v - 6
Lesson 8: Combining Polynomials Assignment For the triangle to the right, a) Determine the length of each b) Determine the perimeter of side if x = 5 units. the triangle if x = 5 units. (HINT: use your results from a) Side 1 = Side 2 = Side 3 = c) Write an expression for the d) Use the expression from part c perimeter of the triangle. Simplify to determine the perimeter of the expression. The triangle when x = 5 units. e) Compare your results for parts f) Which method was easier? Why (b) and (d). What do you notice?
Lesson 9: Multiplying and Dividing Integers 1 Evaluate the following WITHOUT a calculator. (No calculators will be permitted on the test.) a) 7 x (-4) b) - 5 x (- 8) c) 45 (- 2) d) -12 x (- 3) e) (- 12)(3) f) 11 x 5 g) (- 8) (- 10) h) (- 4) (- 5) i) - 16 ÷ (-8) j) 24 ÷ 3 k) 25 ÷ (-5) l) -24 ÷ (-8) 2 Owing money can be expressed as a negative number. Earning money can be
expressed as a positive number. Write a multiplication expression to model each situation. (For example, suppose you cut three lawns and were paid $10 for each lawn. This can be expressed mathematically as 3 x 10.) Then, evaluate each expression.
a) On three different days, you borrowed $2 from your friend for french fries. b) For three nights a week, you babysat the neighbour’s children. You were paid $10
each time you babysat. 3 One year, the highest temperature in town was 41̊ C and the lowest temperature
was -25 ̊C. What was the range in temperatures for this town? (Hint: Range = highest value - lowest value.)
Lesson 10: Order of Operations 1 Evaluate the following statements in the space shown.
(18 - 6) ÷ 4 2³ + 5 x 6
3(8 - 2) + 6²
6 24 2
3 4 35
10
5 3
3² - 4 x 2 9 4 6
7 1
519 11
18 92
2 The formula h = -5t 2 + 10t + 1 gives the height, in metres, of a
ball above the ground t seconds after it is tossed into the air. Determine the height of the ball after
a) 0s b) 1s c) 2s
Lesson 11: Combining Integers Revisited
1 Evaluate without a calculator. Calculators will NOT be permitted on the test.
a) 3 - (- 9) b) 1 + 3 + (-6)
c) 14 + (- 3) - (- 5) d) 7 - (-3) + 3
2 Follow the order of operations (BEDMAS) to simplify each expression.
a) -3 (2 + 7) - 1 b) (- 4 - 1) x 2 - ( 1 - 3) c) 5 (-2) - 3(- 1) + 14 ÷ 7
d) -2 (2 + 3) - 1 e) 4 ( -3) + 2 ( -2) f) 18 ÷ 2 - (4 x 5)
g) 6 (- 2) + 4 (- 4) - 21 ÷ 7 h) -2( 15 ÷ 3) - 3 ( 27 ÷ 3)
i) (-2)² - 3(-1)³ j) 3 ( -2)³ - ( -3) -3 ( -1 - 1)²
Lesson 12: Multiplying and Dividing Rational Numbers 1 Evaluate without using a calculator.
a) 2
5
3
4 b)
2
3
6
4x c)
5
62
1
5
d)
2
2
31
2
5x e)
5
83 f)
4
9
2
3
g) 4
5
3
10
h) 2
1
4
5
6 i)
4
5
1
10
2 Mrs. May’s children always take Dad’s Oatmeal cookies for lunch. If they
ate 23
4 packages of cookies in five school days,
a) How many packages did they eat each day? Express your answer as a fraction.
b) If there are 20 cookies in each bag, how many cookies did they eat each day?
c) If Mrs. May needs to buy cookies for school lunches for the next two weeks, excluding the two Pizza Days at school, how many packages of cookies should she buy to ensure she has enough cookies. (Please make sure your answer is reasonable.) 3 The waiters at a restaurant have agreed to give one-third of their tips to the
kitchen staff. If a waiter collects $32.75 in tips, how much does he end up keeping?
4 Lisa claims that she can find the answer for
2 1
2
3 in her head. She says
the answer is 32
3. Do you agree? Justify your answer
Lesson 13: Review 1 Evaluate a) - 6 + 13 b) 7 x (-2) c) (- 3) x (- 6) d) ( - 8) x 4 e) 5(- 9) f) (- 42) ÷ (- 6) g) (- 32) ÷ 4 h) (-12) (3) i) - 25 ÷ (-5) j) - 48 ÷ 6 k) 17 - 12 - (-5) l) - 12 x 4 2 One night the temperature fell from - 5 ̊ C to - 15 ̊ C in 3 h. What was the rate of
temperature change per hour? 3 The following outside temperatures were recorded at school at 10:00 am for one
week. Calculate the average outside temperature. Monday + 2 ̊ C, Tuesday + 3 ̊ C Wednesday - 2 ̊ C Thursday - 6 ̊ C Friday - 7 ̊ C
4 The elevation of Mount Everest is 8850 m. The elevation of the Dead Sea is -400 m. What is the difference between the two elevations?
5 Mountain climbers in the Canadian Rockies must prepare for cold temperatures as they approach the top of a mountain. The temperature drops approximately 6 ̊ C for every 1000 m they climb. The height of Mount Logan is 5960 m. The temperature at the bottom of Mount Logan is 14 ̊ C. a) Write an expression to determine the temperature at the top of the mountain. b) Determine the temperature at the top of the mountain. 6 For each of the following polynomials, state their degree and classify them as a monomial, binomial or trinomial. a) 2x – 3 b) 6n c) k 2 + 3k – 7 d) -8y 2 e) 2y 3 - 9y 2 + 6y f) 8q + 5 7 Simplify the following expressions. a) 4a + 2a + 9 + 2 b) 13x - 5x + 7 + 12 c) 2y - 6 + 5y + 1 d) 8n 2 + 3n + 5 - 2n 2 + 4n + 2
8 Evaluate each of the following expressions. Show all of your work.
a) 72
5
b)
3
1
4
3
2 c) 3
1
4
13
2
d)
3
5
6
7 e) 4
2
36
3
4
9 A teaspoon is a common unit of measure for baking. A set of three small measuring
spoons is labelled: 1teaspoon, 1
2 teaspoon,
1
4 teaspoon. Describe two ways to
measure out 3
4 of a teaspoon.
Lesson 1: Solving One-Step Equations with Multiplication and Division
1 Solve the following equations. Use mental math, not a calculator.
a) 2x = 6 b) -3x = 6 c) -2x = - 4
d) 5x = 10 e) 4x = 16 f) 3x = - 15
g) -x = - 5 h) 2x = -8 i) -6x = 12
2 Solve the following equations. Use mental math, not a calculator.
a) b) c)
d) e) f)
A
4
Equations Assignment
Luis knows that the area of a rectangle is 225 cm and the length of the2
rectangle is 45 cm. Help Luis find the width of the rectangle using the
formula, A = Rw.
Lesson 3: Kitchen Relationships
1 For each of the following relationships,
a) Complete the table, label the x axis and y axis on the graph
b) State an equation that relates the variables (include let statements to introduce the variables.)
c) Graph the relation on the grid to the right of each table.
Number of
Tablespoons (b )
Number of
Teaspoons (p)
0
1
2
3
4
Let b represent
Let p represent
Equation: p =
Number of Cups
(c)
Number of Fluid
Ounces (f )
0
1
2
Let c represent
Let f represent Equation: f =
Lesson 3: Kitchen Relationships
Number of Fluid
Ounces (f )
Number of
Tablespoons (b )
0
1
2
3
4
5
Let b represent
Let f represent
Equation: b =
Number of
Litres (L)
N Number of
Millilitres (m)
0
0.25
0.5
0.75
1
Let L represent
Let m represent
Equation: m =
Lesson 4: Kitchen Relationships
1 For the six relations on Pages 3 - 5, complete the table below.
Equation Constant of Variation
2 Use the graphs on Pages 3 - 5 to find more reasonable measurements for
the following converted quantities. Show the interpolation or extrapolation
on the graphs by drawing the vertical and horizontal lines.
a) 21 tsp = tbsp
b) tsp = 2.5 tbsp
c) 10 tbsp = fl oz
d) 32 fl oz = cups
3 State whether you used interpolation or extrapolation in each part of
Question 2. Explain your answer.
a)
b)
c)
d)
Lesson 5: Relationships Beyond the Kitchen Part 1
Independence Day
For the following relationships, fill in the blanks to complete each statement.
1 Distance vs. Time travelled in a car.
Statement: The depends on the
so is the dependent variable and
is the independent variable.
2 The amount of candy you get on Halloween vs. the number of houses you
visit.
Statement: The depends on the
so is the dependent variable and
is the independent variable.
3 The number of detentions a student receives vs. the number of times a
student is late.
Statement: The depends on the
so is the dependent variable and
is the independent variable.
4 The number of hours worked vs. the amount earned.
Statement: The depends on the
so is the dependent variable and
is the independent variable.
Whenever you graph a relationship, the independent variable is plotted on the
axis and the dependent variable is plotted on the axis.
Lesson 5: Kitchen Relationships Part 1
1 The cost of a taxi ride is shown in the table below.
a) Complete the table of values for the total cost in dollars for a taxi ride.
d (distance in km) C (total cost in $)Cost per kilometer
($/km)
0 0
1
2
3 0.90
7
8
9 2.70
b) State the independent variable. c) State the dependent variable.
d) Determine the rate of change
of cost versus the distance
travelled.
e) What does the rate of change
of cost versus the distance
travelled represent?
f) What does this tell you about the relationship between total cost and
distance travelled?
Lesson 5: Kitchen Relationships Part 1
2 A salesman’s commission on his sales is shown in the table below.
a) Complete the table of values for his total commission.
s (total sales in $)C (Total commission
in $)
Commission per dollar
sold
0 0
100
200
300 30
700
800
900 90
b) State the independent variable. c) State the dependent variable.
d) Determine the rate of change
of commission versus total
sales.
e) What does the rate of change
of commission versus total
sales represent?
f) What does this tell you about the relationship between commission and total
sales?
Lesson 6: Relationships Beyond the Kitchen Part 2
1 Using the data from Lesson 5, write an equation including LET statements for
a) The taxi ride b) The salesman’s commission
2 Use the equations created in Question 1, to determine
a) the cost of a 15.5 km taxi ride. b) the number of kilometres
travelled for $ 2.25.
c) the commission earned on $ 250
of sales
d) the sales when a commission of
$150 is paid.
Lesson 6: Relationships Beyond the Kitchen Part 2
A
4
C
5
Name:
Writing Equations Assignment
ABC Construction Company pays $1000 per house to the framer (person who puts
up the frame.)
[3, C] 1 Write an equation relating the pay per house to the total
income for the framer. Remember to introduce the variables
using LET statements.
2 Determine the
[1, C]
[2, A] a) amount of money
earned when 7
houses are framed.
[1, C]
[2, A] b) the number of
houses framed for
$12000.
Lesson 7: Relationships Beyond the Kitchen Part 3 Practicing Your Scales Recall: Range of Data = Highest Data – Lowest Data Scale = Range . Number of Boxes
Round to a nice number 1 Using the data from Lesson 5 and assuming you have a 15 (horizontal) x 20 (vertical) grid,
determine the horizontal and vertical scales for a) The taxi ride Dependent Variable _________________ Axis _________ Boxes ________
Independent Variable _______________ Axis _________ Boxes ________ Distance (km)
0 1 2 3 7 8 9 Range of Data =
Scale =
Cost ($)
0 0.9 2.7 Range of Data =
Scale =
a) The salesman’s commission Dependent Variable _________________ Axis _________ Boxes ________
Independent Variable _______________ Axis _________ Boxes ________ Sales ($)
0 100 200 300 700 800 900 Range of Data =
Scale =
Earnings ($)
0 30 90 Range of Data =
Scale =
Lesson 8: Relationships Beyond the Kitchen Part 4
1 Sketch a graph of the taxi ride on the grid below. Do not forget to label each axis
and place a title on the graph.
Distance (km) Cost ($)
0 0
1 0.3
2 0.6
3 0.9
7 2.1
8 2.4
9 2.70
2 Use interpolation or extrapolation to estimate
a) the cost of a 4 km taxi ride.
b) the cost of a 12.5 km taxi ride.
c) the distance travelled for $3.
d) the distance travelled for 50¢.
3 State whether you used interpolation or extrapolation in each part of Question 2.
a)
b)
c)
d)
4 Verify each answer in Question 2 using the equation you created on Lesson 6.
Lesson 8: Relationships Beyond the Kitchen Part 4
1 Sketch a graph of the salesman’s commission on the grid below. Do not forget to label each
axis and place a title on the graph.
Total Sales ($) Commission ($)
0 0
100 10
200 20
300 30
700 70
800 80
900 90
2 Use interpolation or extrapolation to estimate
a) the commission on sales totalling
$975.
b) the commission on sales totalling
$530.
c) the total sales when the commission is
$64.
d) the total sales when the commission is
$120.
3 State whether you used interpolation or extrapolation in each part of Question 2.
a)
b)
c)
d)
4 Verify each answer in Question 2 using the equation you created on Lesson 6.
T
24
Name:
Lesson 9: Direct Variation TIPS Assignment
1 In Extreme Boards, the first mandatory event is called “Slalom.” In this event,
competitors must travel back and forth on a curved surface.
Number of Tricks, n Style Score, S
0 0
2 4
4 8
6 12
The boarder can earn points for performing trick moves at the top of each swing.
[2] a) Is the Style Score a direct variation of the number of tricks? Explain your answer.
[1] b) Determine the constant of variation for this relation.
[3] c) Write an equation that relates Style Score, S, and the number of tricks, n.
(Remember the LET statements.)
[3] d) When Ally’s turn comes, the top style score is 18. How many tricks must Ally
perform successfully to beat this score?
Name:
Lesson 9: Direct Variation TIPS Assignment
2 Suzanne’s pay for the past three weeks is shown in the table below.
[2] a) Determine Suzanne’s hourly rate of pay for each week.
Hours Worked, h Pay ($), P Hourly Rate
($/h)
20 180
15 135
22 198
[2] b) What does this tell you about the relationship between pay and hours worked?
Explain.
[7] c) Graph pay versus hours worked
[2] d) Use the graph to determine Suzanne’s earnings if she works
i) 10 hours ii) 25 hours
[2] e) How many hours must Suzanne work to earn $250.00?
Name:
Lesson 10: Direct Variations Assignment1 The table shows several different
heights and areas for triangles with a
base of 10 cm.
[2, A] a) Complete the table below.
Height (cm) 0 2 5 10 20
Area (cm²) 0 10 25 50 100
k = DV ÷ IV
[2, C]
[4, A] a) Graph the relationship on the grid below.
[3, C] b) Write an equation that relates the area and the height of the
triangle. (Don’t forget to include LET statements.)
Recall: DV = k (IV )
Name:
Lesson 10: Direct Variations Assignment
[1, C]
[2, A] c) Use interpolation or extrapolation to determine the area of a
triangle with a height of 25 cm. Remember to show your work
on the graph and include a sentence answer.
[1, C]
[2, A] d) Use the equation to determine the area of a triangle with a
height of 25 cm. Remember to include a sentence answer.
[1, C]
[2, A] e) Use interpolation or extrapolation to determine the height of
a triangle with an area of 75 cm . Remember to show your2
work on the graph and include a sentence answer.
[1, C]
[2, A] f) Use the equation to determine the height of a triangle with an
area of 75 cm . Remember to include a sentence answer.2
Lesson 11: Test Part 1 Review
1 Solve the following equations for the unknown variable.
a) 2x = - 8
b) p = 11
c) -6x = 20
d) y = - 9
2 Circle whether each of the following relations is or is not a direct variation. Then, state
the constant of variation.
a) C = 2t
Direct or Not direct
COV
b) P = 50h + 100
Direct or Not direct
COV
c) y = - 2x
Direct or Not direct
COV
d) y = x - 5
Direct or Not direct
COV
3 Samantha babysits her little brother. Her hours of work and pay for one evening are shown
in the table below.
Hours worked 0 1 2 3
Pay ($) 0 5 10 15
a) In this relationship, state the
i) independent variable ii) dependent variable
b) State the constant of variation in the equation. (Remember: k = .)
c) Write an equation to express this relationship. (Remember to include LET
statements.)
d) Use the equation to determine
how many hours she babysits
if she earns $22.50.
e) Is this relationship a direct or not a
direct variation? Explain your answer.
Lesson 11: Test Part 1 Review
3 f) Graph the relation.
g) Look at the graph and explain why this is a direct variation.
h) Use interpolation or extrapolation to determine the
i) her pay for babysitting 7.5 hours.
ii) how many hours she babysits if she earns $12.50.
h) State whether you used interpolation or extrapolation in each part of ( g ).
i) ii)
4 Create a cheat sheet for the test that includes a definition of
• hypothesis • inference • interpolation
• extrapolation • direct variation • constant of variation
• rate of change • independent variable • dependent variable
Lesson 13: Introduction to Ratios
Vegetable Dip (Serves 6)
250 mL sour cream
160 mL mayonnaise
15 mL finely chopped onion
15 mL chopped fresh dill
20 mL chopped fresh parsley
10 mL seasoned salt
Mix all ingredients. Chill for several hours.
Use as a dip for fresh vegetables.
1 Express the following ratios in lowest terms.
a) sour cream to mayonnaise b) onion to sour cream
c) dill to parsley d) dill to salt
e) mayonnaise to salt f) mayonnaise to dill
g) onion to salt h) mayonnaise to parsley
Lesson 14: - Solving Proportions in the Kitchen
Use the Vegetable Dip Recipe on Lesson 13 and proportions to determine the quantity of each ingredient
required for the following numbers of people. Hint: if 24 people are being served.
24 people 15 people
Sour Cream
Mayonnaise
Onion
Dill
Parsley
Salt
Lesson 16: Solving Proportions Out of the Kitchen 1 Based on current cancer rates, 19 out of 50 Canadian women will develop cancer during their lifetime and 43 out of 100 Canadian men will develop cancer during their lifetime. a) Predict how many women out b) Predict how many men out of 30 of 700 are likely to develop are likely to develop cancer cancer. during their lifetime. 2 In 2003, 1.1 million students collected over $5.4 million dollars in pledges for the Terry Fox Run. At this rate, how much money could 1.6 million students raise? 3 A new skateboard park is being constructed in Fletcher’s Meadow. If the ratio of
the height of the ramp to the base is 2:3 a) Determine the base of a ramp b) Determine the height of a ramp
with a height of 15m with a base of 4m.
Lesson 19: Unit Rates
1 Beginning in St. John’s, Newfoundland Terry Fox ran 294 km every week.
a) Express this as a unit rate (per
day).
b) If Terry ran for 143 days, use
the unit rate to determine how
far Terry Fox ran for Cancer
Research.
2 After 143 days, Terry stopped running in Thunder Bay, Ontario. He had
raised $23 million at that time.
a) On average, how much money
did Terry raise per day of his
run?
b) At that time, Canada’s
population was about 24 million
people. On average, how much
had Terry Fox raised from
every Canadian Citizen?
3 About 300,000 people participated in the very first Terry Fox Run which
was held about 1 year after Terry stopped his marathon. If each participant
raised $11.68 each, how much money was raised in total?
Lesson 19: Unit Rates
4 Some doctors suggest taking vitamin E to help prevent cancer. A pharmacist
sells two brands of vitamin E:
Rexall: 175 g for $9.98
or
Life: 260 g for $12.99
Which BRAND is cheaper? Justify your answer.
5 India has been involved with the Terry Fox run for about 10 years. One
India Rupee costs about $0.02 Cdn.
a) Express the ratio of rupee to Canadian dollars in lowest terms.
b) India has raised 400 000 rupees for cancer research. How much is
this worth in Canadian dollars?
Name:
Lesson 20: Conversion Factors Assignment
** Show all work for full or part marks.
You are throwing a party for 30 people. These are the recipes you used.
Vegetable Dip (Serves 6)
250 mL sour cream
160 mL mayonnaise
15 mL finely chopped onion
15 mL chopped fresh dill
20 mL chopped fresh parsley
10 mL seasoned salt
Apple Crisp (Serves 5)
4 cups sliced apples (about 4 medium)
cup packed brown sugar
cup flour
cup oats
cup butter
tsp ground cinnamon
tsp ground nutmeg
1 Determine the conversion factor for each recipe.
Hint: New Measure = k (Old Measure)
[2, A] a) Apple Crisp [2, A] b) Vegetable Dip
[2, C] 2 Can the same conversion factor be used for BOTH recipes?
Explain your answer.
Name:
Lesson 20: Conversion Factors Assignment
[13, A] 3 Use the conversion factor to determine the quantity of each
ingredient required. All measurements must be REASONABLE.
Apple Crisp Vegetable Dip
Apples Sour Cream
Brown Sugar Mayonnaise
Flour Onion
Oats Dill
Butter Parsley
Cinnamon Salt
Nutmeg
Lesson 22: Scale Diagrams
The diagram below represents an overhead view of a typical classroom.
The “scale” of the diagram is 1 square to 50 centimetres.
Recall: Scales may be written using a colon.
1 square : 50 cm.
This means the length of one side of 1 square on the diagram
represents 50 cm inside the classroom.
1 Determine the number of squares required for 1 metre (100 cm).
Lesson 22: Scale Diagrams
2 Count the number of squares along the long wall of the diagram.
squares
Determine the actual length of the classroom in metres.
3 Count the number of squares along the short wall of the diagram.
squares
Determine the actual width of the classroom in metres.
4 Determine the actual perimeter of the classroom in metres.
5 A table will be placed in the classroom. The table is 2 m long and 1.5 m wide.
a) Convert these measurement to centimetres.
Lesson 22: Scale Diagrams
b) Use the scale of 1 square : 50 centimeters to determine the diagram
measures of the table.
c) Draw the table on the diagram accurately using the diagram measures.
6 Four desks will be added to the room. Each desk is 1 m long and 1 m wide.
a) Convert these measurement to centimetres.
b) Use the scale of 1 square : 50 centimeters to determine the diagram
measures of each desk.
c) Draw all four desks on the diagram accurately using the diagram measures.
d) Remember to add a door as was discussed in the reading.
e) Complete your diagram and remember to label the contents of the room and
put your scale on the diagram.
Lesson 24: One Number as a Percent of Another Number
Complete the table below.
Fraction Decimal Percent
60%
1.3
0.74
1.4%
Lesson 24: One Number as a Percent of Another Number
1 Express each of the following as a percent.
a) 230:200 b) 17 to 40 c) 18 out of 30
d) 36.6 to 12.2 e) 1.2 to 80 f) 35 to 20
2 Determine to the nearest tenth of a percent, the percent increase from
a) $50 to $57.50 b) $200 to $400
c) 5 cm to 12 cm d) 18EC to 32EC
3 Determine to the nearest tenth of a percent, the percent decrease from
a) 120 m to 100 m b) $80 to $53.33 c) 18.4 s to 15.9 s
Lesson 25: Fractions, Decimals and Percent 1 Determine the given percent of each number. (Use a constant of variation.) a) 40% of 600 b) 7% of $150 c) 6.5% of $230000 d) 4.5% of 675 e) 8.25% of $89000 f) 8% of $49.75 2 Determine each number. (Use a proportion) a) 20% of what number is 68? b) 40% of what number is 52? c) 45% of what number is 35? d) 7.25% of what number is 1885? 3 A basketball shirt with a regular price of $89.95 is on sale for 25% off.
Both 8% PST and 7% GST apply to the sale price. What is the cost, including taxes on the sale price?
Lesson 27: Test Part 2 Review
1 What is a ratio? (Include some examples.)
2 Reduce each of the following ratios to lowest terms.
a) 27 : 18 b) 150 cm : 5 m
3 What is a proportion? (Include some examples.)
4 Solve each of the following proportions.
a) x : 6 = 20 : 24
b) c)
5 Several Fletcher’s Meadow, Heart Lake and Brathwaite students were surveyed and 15% of
students stated that they ENJOY learning MATH. If 431 students enjoy learning math, how
many students were surveyed.
6 What is a unit rate? (Include some examples.)
Lesson 27: Review
7 State each of the following as a unit rate.
a) Harveys charges $12.85 for 2 burger
combos.
b) Sasha drove 750 km in 9 hours.
8 Two grocery stores have a sale on Freezies
Metro 40 Freezies for $13.99
Fortinos 50 Freezies for $15.99
Which store has the better buy?
9 A recipe that serves 4 people calls for 1 cup of chicken broth and cup of chopped onions.
a) Express the ratio of chicken
broth : onions in lowest terms.
b) Determine the conversion factor if the
recipe must serve 20 people.
c) Use the conversion factor to determine the amount of
i) chicken broth required for the
new recipe.
ii) onions required for the new recipe.
10 Express the following expressions as a percent, correct to the nearest percent.
a)
b) 36 out of 51 c) 250 : 210
Lesson 27: Review
11 Solve the following expressions.
a) 76% of 15 b) 30 % of what number is 52
12 State the formula used to calculate a percent increase/decrease
13 Plasma T. V.s are selling for $3999.99 at Best Buy. Next week they go on sale for
$3250.99. What is the discount as a percent of the regular price?
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Unit 3:
Uh-Oh, It’s NOT DIRECT!
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Lesson 1: Investigation: Uh-Oh, It’s NOT Direct!
A banquet hall charges $100 for the hall and $20 per person for dinner. a) Complete the following table.
Number of People, n Total Cost ($), C Cost per Person
0 100 20
1 120 20
2 140 20
3 160 20
7 7(20) + 100 = 240 20
10 10(20) + 100 = 300 20
15 15(20) + 100 = 400 20
b) How is this table different than the table for a direct variation?
When no people attend the banquet, there is still a charge. In a direct, there would be no charge when no people attend.
c) What is the rate of change of cost with respect to the number of people
attending the dinner? The rate of change of cost with respect to the number of people attending the dinner is $20/person
d) Did we need to complete the table to determine this value? Explain.
No, we did not need to complete the table to determine the value because the rate of change was stated in the original question
e) What does the rate of change of cost represent?
The rate of change of cost with respect to the number of people represents the cost per person.
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Lesson 1: Investigation: Uh-Oh, It’s NOT Direct! f) Sketch the graph of this relationship. g) How does the graph of this relationship differ from the graph of a direct
variation? The graph of a partial variation does not pass through (0,0) h) How is the graph of this relationship the same as the graph of a direct
variation ? They both have a straight line i) Write LET statements to introduce the two variables in this relationship,
then write an equation to express this relationship.
You will need your knowledge of direct variations and an understanding of the differences between direct variations and this relationship to express the relationship using an equation.
Let C represent the total cost of the banquet, in dollars Let n represent the number of people attending the banquet
C = 20n + 100
0
50
100
150
200
250
300
350
400
450
0 5 10 15 20
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Lesson 2: Partial Variations Part 1 Lesson 2: Partial Variations Part 1 1 To rent a car for the weekend it costs $10 plus $0.25/km.
b) Sketch the graph.
c) Write LET statements to introduce the two variables in this relationship, then write an equation to express this relationship.
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Lesson 2: Partial Variations Part 1 2 A race car had a head start of 50 km. It travels at a constant speed of 200
km/h. a) Complete the chart.
b) Sketch the graph.
c) Write LET statements to introduce the two variables in this relationship, then write an equation to express this relationship.
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Lesson 3: Direct and Partial Variations Assignment Decisions, Decisions, Decisions
You are helping a friend decide which cell phone package is best for her to buy. The cost of purchasing the phone is the same in all cases. Crystal Net: Charges $0.30 per minute. Rover: Charges $12 per month, plus $0.20 per minute. Ring Mobility: Charges $15 per month, plus $0.10 per minute.
1 State whether the company’s have a direct or partial variation. Crystal Net Rover Ring Mobility 2 State the rate of change of Cost with respect to time for each
company. Crystal Net Rover Ring Mobility 3 State an equation to represent each relationship including LET
statements for the variables. (Note: Only two LET statements are required because the three companies may use the same variables.) Remember: For DIRECT VARIATION: DV = m (IV)
For PARTIAL VARIATION: DV = m (IV) + b Crystal Net Rover Ring Mobility
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Lesson 4: Partial Variations Part 2 2 A telephone company offers a discount plan for overseas calls.
a) Is this a direct or partial variation? Explain your answer.
b) What is the rate of change of total cost with respect to the number of minutes talked? c) What is the slope of this graph?
d) What does the rate of change of total cost with respect to the number of minutes talked represent?
e) What is the fixed cost of this graph?
f) What is the y - intercept?
g) Write an equation including LET statements to represent this relation.
Cost of Overseas Phone Calls
Cost ($)
Duration (min)
h) Use the graph to estimate
i) The duration of a call that costs $7.00
ii) The duration of a call that costs $15.
iii) The cost of a 40 minute call.
iv) The cost of a 90 minute call.
10 20 30 40 50 60 70 80 90 10
51015202530
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Lesson 5: Graphing From an Equation 1 The Reliable Repair Company charges $25 for a visit plus $60 per hour for labour. a) What is the slope of the line representing this relationship? How does this value relate to
the repair charges? b) What is the y - intercept of the line representing this relationship? How does this value
relate to the repair charges? c) Write an equation including LET statements to relate the two variables. d) Graph the relation. (Don’t forget to
telescope the axis to make life simpler.)
e) Use the graph to determine the cost of a repair that takes 5 hours.
f) Use the equation to determine the
cost of a repair that takes 3 hours. g) Use the graph to determine the
number of hours of labour spent if the bill was for $160.
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Lesson 6: Graphing From an Equation Part 2
1 A soccer league holds an awards banquet at the end of the season. The banquet hall charges $20 per person for the meal plus $500 for the use of the hall.
a) Write an equation to model the cost of the banquet. (Don’t forget LET statements.)
b) Identify the slope and the y - intercept in your formula slope: _________ y - intercept:____________
c) How can you tell the equation describes a partial variation? d) Use the equation to sketch a graph of
the data. e) Use the equation to complete the
table of values for the banquet costs.
Number of People, n
Banquet Cost B ($)
0
20
50
75
100
150
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Lesson 6: Graphing From an Equation Part 2
2 The cost to rent a snowboard for one day is $40. You can rent it for a reduced cost, as shown.
Number of Extra Days
Cost ($)
0 40
1 55
2 70
3 85
a) What is the cost for each extra day?
Explain how you determined this.
b) What is the y-intercept of this variation?
What is the slope of this variation? c) Write an equation to express cost, C,
in terms of additional day, D. (Don’t forget LET statements.)
d) Use the equation to calculate the
total rental cost for a total rental of 8 days.
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Lesson 7: Assignment - The Best Value 2 A law office plans to do some landscaping around their building. They have two estimates: Company A: $240 for a full landscape plan plus $30 per hour to do the work Company B: $60 per hour to do the work (which includes the landscape plan) a) Graph Company A and Company
B on the same axis for upto 9 hours (Use a different colour for each.)
b) For which time value are the costs the same for both Company A and Company B?
c) For which time value is
Company A a better deal than Company
d) State an equation including LET statements to relate the cost and total time of
each company. Company A Company B e) Complete each of the following sentences. I would choose Company A if ... I would choose Company B if ...
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Lesson 10: Slope 1 For each graph, state the rise, run and slope.
a)
Rise =
Run =
Slope =
b)
Rise =
Run =
Slope =
c)
Rise =
Run =
Slope =
d)
Rise =
Run =
Slope=
-4 4
-4
4
x
y
-4 4 8 12
4
4
8
12
x
y
8 16
8
-20 20
-20
20
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Lesson 10: Slope e)
Rise =
Run =
Slope =
f)
Rise =
Run =
Slope = 2 Lines with slope rise to the right.
The greater the positive slope, the steeply the line rises.
The smaller the positive slope, the steeply the line rises. 3 Lines with slope fall to the right.
The greater the negative slope, the steeply the line falls.
The smaller the negative slope, the steeply the line falls. 4 The slope of a line is zero (0).
The slope of a line is undefined.
-4 4
-4
4
x
y
-4 4
-4
4
x
y
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Lesson 13: Graphing Linear Relations and First Differences 1 Fill in the following blanks for the following linear relations. a) y = 2x + 5
slope
y - intercept
Is it a direct or partial variation?
b) y = 3
4x + 3
slope
y - intercept
Is it a direct or partial variation?
c) y = - 4 + x
slope
y - intercept
Is it a direct or partial variation?
d) y = -x
slope
y - intercept
Is it a direct or partial variation?
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Lesson 13: Graphing Linear Relations and First Differences 2 Complete the table of values and determine the first differences for each relation. Then,
graph each relation. Graph (a) on the first grid and (c) on the second grid. Use different colours for each line. (Label each graph.)
a) y = 2x + 5
x y 1st Differences
b) y = 3
4x + 3
x y 1st Differences
c) y = - 4 + x
x y 1st Differences
d) y = -x
x y 1st Differences
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Lesson 14: Assignment The Catering Problem
Maxwell’s Catering Company prepares and serves food for large gatherings like weddings and company parties. They charge a base fee for renting the hall plus a cost per person.
Menu A Chef’s Salad Chicken Kiev
Broiled Potatoes Mixed Vegetables
Ice Cream Coffee or Tea
Menu B French Onion Soup
Chef’s Salad Roast Beef
Baked Potato Steamed Broccoli
Cheese Cake Coffee or Tea
Menu C Shrimp Coctail Chef’s Salad
Steak & Lobster Baked Potato
Glazed Carrots Dessert Crepes Coffee or Tea
The following formulas are used to calculate the ordered pair (n, C ). The number of people served is n. The total cost in dollars is C. Menu A: C = 12n + 200
Menu B: C = 16n + 200 Menu C: C = 20n + 200 1 Complete the table of values for each relation:
Menu A
n C ($)
1ST D
Menu B
n C ($)
1ST D
Menu C
n C ($)
1st D
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Lesson 14: Assignment - The Catering Problem 2 Graph the three relations on the same axis. Graph each Menu with a different
colour. 3 Identify the slope and the C-intercept of the Menu C line. How do these values
relate to the total cost?
How it relates
Slope:
C -intercept:
4 Compare the first differences for the three Menus. How are the first differences
related to the a) graph? b) equation?
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Lesson 14: Assignment - The Catering Problem 5 Compare the graphs of each of the three Menus. How are the graphs the a) same? b) different? 6 For Menu B, what does the ordered pair (120, 2120) mean?
7 Brad and Jennifer, oops I mean Angelina have invited 70 people to their Wedding
anniversary. How much will it cost for
Menu A? Menu B? Menu C? 8 Lui and Sami are planning their wedding. They have $3500 to spend on dinner. They
would like to have Menu C. What is the greatest number of guests they can have at dinner?
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Lesson 15: Non-Linear Relationships Part 1 Remember: To calculate first differences, choose x - values that increase or decrease by
1. 1 Consider the relation, y = 2x + 1 a) Does the data represent a linear or
non-linear relationship? b) If it is linear, is it a direct or partial
variation. c) State the slope of the relation. d) State the y -intercept of the
relation. e)
Sketch the graph of the relation.
f) Determine the first differences.
x y
First Differences
g) What is significant about the value of the first differences?
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2 Consider the relation y = -3x
a) Does the data represent a linear or non-linear relationship?
b) If it is linear, is it a direct or partial variation.
c) State the slope of the relation. d) State the y -intercept of the
relation. Lesson 15: Non-Linear Relationships Part 1 e) Sketch the graph of the relation. f) Determine the first differences.
x y
g) What is significant about the value of the first differences?
First Differences
3 Consider the relation, y = 2x 2
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a) Does the data represent a linear or
non-linear relationship? b) If it is linear, is it a direct or partial
variation. c) Sketch the graph of the relation.
Lesson 15: Non-Linear Relationships Part 1 4 Consider the relation, y = x 2 + 2 a) Does the data represent a linear or
non-linear relationship? b) If it is linear, is it a direct or partial
variation. c) Sketch the graph of the relation. d) Determine the first differences.
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x y
First Differences
e) What is significant about the value of the
first differences? 5 Consider the relation, y = x 2 - 3x a) Does the data represent a linear or
non-linear relationship? b) If it is linear, is it a direct or partial
variation.
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Lesson 15: Non-Linear Relationships Part 1 c) Sketch the graph of the relation. d) Determine the first differences.
x y
First Differences
e) What is significant about the value of the first differences? 6 How are the equations of the linear and non-linear relations different? 7 How are the first differences of linear and non-linear relations different?
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Lesson 16: Graphing Non-Linear Relations and First Differences 1 Determine which of the following relations are linear or non-linear. a)
b)
c)
d)
e)
f) y = x 2 - 2 g) y = -5x h) y = x 3 - 1 i) y = 3x + 4
-2 2 4 6
-2
2
4
6
x
y
-2 2 4 6
-2
2
4
6
x
y
-2 2 4 62
2
4
6
8
x
y
4 8 12 16 20 24
4
8
12
16
20
x
y
2 4 6 8 10 12 14 1
2468101214
x
y
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Lesson 18: Solving One-Step Equations
[2] 1 a) To solve x + 4 = 9, from each side of the equation.
b) To solve m - 6 = 2, to each side of the equation. [6] 2 Solve the following equations. Show all work.
a) y + 3 = 0 b) x + 4 = 9 c) k - 5 = 0
d) n - 2 = -3 e) m - 6 = 2 f) p + 4 = -8
g) 3 + x = 8 h) y - 5 = -1 i) x + 3 = - 1 [2] 3 Use a LEFT SIDE / RIGHT SIDE check to determine if x = 3 is a solution to
x + 4 = 7.
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Lesson 19: Solving Multi-Step Equations 1 Solve each of the following equations. a) 3x - 5 = 13 b) 2j + 9 = 11 c) m - 3 = 19 d) -v = 6 e) -8t = 24 f) 9z - 4 = 14
g) q
53 h)
c
4= - 7
i) -5w = - 35
m) -2h + 1 = -5 n) 3 - p = 11 2 Jacques is enclosing a rectangular swimming pool deck with 144 m of fencing. What is the width of the rectangle if the length is 40 m?
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Lesson 22: Unit Review 1 State whether each of the following relations is a direct or partial variation. a) y = - 3x b) A banquet hall
charges $50 for the hall and $30 per person.
c) C = 1.5t + 5
d) Pay-As-You-Go Cell
Phone Company charges $0.25 per minute talked.
e)
2 State the slope and “ y” -intercept of the following. Then, sketch each graph. a) y = - 3x
slope =
y - intercept =
-4 4
-4
4
x
y
-4 4
-4
4
x
y
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Lesson 22: Unit Review 3 A banquet hall charges $50 for the hall and $30 per person. a) Write an equation, including LET
statements for the relation. b) Graph the relation.
c) Use the equation to determine the
total cost of the banquet for 150 people.
d)Use the graph to determine the total cost of the banquet for 6 people. (Don’t forget to show your work on the graph and the final answer here.)
4 Pay-As-You-Go Cell Phone Company charges $0.25 per minute talked plus $5 for the month. a) Write an equation, including LET
statements for the relation. b) Graph the relation.
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Lesson 22: Unit Review c) Use the equation to determine the
total cell phone bill when a person talks for 300 minutes.
d) Use the graph to determine the total cell phone bill when a person talks for 50 minutes.
5 For each of the following relations, state the rise, run, slope, y -intercept, and equation. a) rise = run = slope = y - intercept = equation :
b)
rise = run = slope = y - intercept = equation :
-4 4
-4
4
x
y
-4 4
5
x
y
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Lesson 22: Unit Review 7 Complete the table of values, then graph each of the following relations. a) y = x + 4
x y (x, y)
-2
0
3
b) y = x ² + 1
x y (x, y)
-2
-1
0
1
2
-6 6
-6
6
x
y
-6 6
-6
6
x
y
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Lesson 22: Unit Review 8 Determine the first differences for the following relations. Then, state whether the
relations are linear or non-linear. a)
x y
-1 3
1 8
3 13
5 18
7 23
Conclusion:
1ST Differences
b)
x y
-7 0
-5 2
-3 5
-1 7
1 10
Conclusion:
1ST Differences
c) y = 3x + 1
x y = 3x + 1
-1
1
3
5
Conclusion:
1ST Differences
d) y = 2x ² - 1
x y = 2x ² - 1
-2
-1
0
1
2
Conclusion:
1ST Differences
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9 Using the scatterplot to the right: a) Identify the independent variable. b) Which point represents a body length of 20 cm and a tail length of 30 cm? c) Which point represents a body length of 60 cm and a tail length of 60 cm? d) What do the coordinates of point G represent?
Animal Body Length vs Tail Length
Tail Length (cm) G F C D A E B
Body Length (cm)
15 Answer True, T or False, F. A partial variation is a straight line through (0, 0). The graph of a scatterplot always has a positive or negative relationship.
10 20 30 40 50 60 70 8
10
20
30
40
50
60
70
Page 1 of 18
Grade 9 Applied Math
Unit 4 Geometry
Page 2 of 18
Lesson 1: Complimentary Angles, Supplementary Angles and The Opposite Angle Theorem
The sum of the angles in any right angle is always ________________. Therefore,
<A + <B = .
AND Therefore,
< A + < B + < C = .
Regardless of the number of angles, the sum of the angles in a right angle is always . These angles are referred to as ____ . Example 1: Determine the value of the unknown angles in each of the following
right angles. a)
b)
A B
A B
C
30.7°
y17.4°
x55.5°
MFM 1P0 Unit 4: Geometry Student Notes
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Lesson 1: Complimentary Angles, Supplementary Angles and The Opposite Angle Theorem
The sum of the angles in any straight line is always .
Therefore,
< A + < B = .
Therefore,
< A + < B + < C = .
Regardless of the number of angles, the sum of the angles in a straight line is always . These angles are referred to as . Example 2: Determine the value of the unknown angles in each of the following
straight angles. (Note: Diagrams are not to scale.) a)
b)
A B
A B
C
m 64.6°18.6°
10.5° x
25.1°
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Lesson 1: Complimentary Angles, Supplementary Angles and The Opposite Angle Theorem
c)
When two or more lines intersect, the angles that are opposite one another are always .
Therefore, < A = < < B = <
y y 3y4x
5x
A
D C
B
MFM 1P0 Unit 4: Geometry Student Notes
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Lesson 1: Complimentary Angles, Supplementary Angles and The Opposite Angle Theorem
AND
Therefore, < A = < < B = < < C = <
Regardless of the number of intersecting lines, opposite angles are always . Example 3: Determine the value of the unknown angles in each of the following
opposite angles. a)
A
F
E D
C
B
45°
D C
B
MFM 1P0 Unit 4: Geometry Student Notes
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Lesson 1: Complimentary Angles, Supplementary Angles and The Opposite Angle Theorem
b)
x+1
F
E Dx+4
x+1
MFM 1P0 Unit 4: Geometry Student Notes
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Lesson 1: Complimentary Angles, Supplementary Angles and The Opposite Angle Theorem
1 In each diagram, calculate the measure of the unknown angle. a)
b) c)
d) e) f)
39° a 115° b c
x + 3
30°
40°
ya52°
130°
bb
MFM 1P0 Unit 4: Geometry Student Notes
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2 Calculate the measure of the unknown angles. You must state and solve two equations to find the four unknown angles.
x + 3
x + 3
y + 40
y
MFM 1P0 Unit 4: Geometry Student Notes
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Lesson 2: Interior and Exterior Angles of Triangles and Quadrilaterals The Sum of the Angles in a Triangle • The sum of the interior angles in a triangle is . • The measure of an exterior angle is equal to the sum of
__________________________________________. • The sum of the exterior angles in a triangle is .
Therefore, < A + < B + < C = . AND
< D = . < E = .
< F = . AND < D + < E + < F = .
The Sum of the Angles in a Quadrilateral • The sum of the interior angles of a quadrilateral is . • The sum of the exterior angles of a quadrilateral is .
Therefore, < A + < B + < C + < D = . AND < E + < F + < G + < H = .
Note: To form exterior angles, each side of the polygon is extended only once.
A
BE
C FD
A
H G
FE
D C
B
MFM 1P0 Unit 4: Geometry Student Notes
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Lesson 2: Interior and Exterior Angles of Triangles and Quadrilaterals b)
c)
38o x80o
p n
mg
42o
85o
95ok
97o j i
MFM 1P0 Unit 4: Geometry Student Notes
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Lesson 2: Interior and Exterior Angles of Triangles and Quadrilaterals 1 In each diagram, calculate the measure of the unknown angle. a)
b)
c)
d)
e)
f)
e
138° 22°a
7 5E b
70°
25°
x y
148°84°
72°c
df
78°126°
93°
75°
125°
130°
ef
g
MFM 1P0 Unit 4: Geometry Student Notes
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Lesson 03: Parallel Line Theorems Alternate Angles
Corresponding Angles
Example 1: Solve for the unknown angles. a)
b)
a
b
a
b
ab
50o
c
d ef g
c
d b50o50oe a
f g
a
b
Co-interior Angles
MFM 1P0 Unit 4: Geometry Student Notes
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Lesson 03: Parallel Line Theorems 1 In the diagram to the right, • and are equal because they are opposite
angles (X pattern) • and are equal because they are
corresponding angles (F pattern) • and are equal because they are alternate
angles (Z pattern) • and add to 180 ° because they are co-
interior angles ( C pattern)2 Determine the measure of each angle marked with a letter in these
diagrams. List the angle relationship for each calculation. a) a = Reason:
b = Reason:
b) c = Reason:
d = Reason: e = Reason:
y
w
z x
ba60 °
c40 °
70 °
ed
MFM 1P0 Unit 4: Geometry Student Notes
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Lesson 03: Parallel Line Theorems
c) f = Reason: g = Reason: h = Reason:
3 If two lines appear to be parallel but are not marked with parallel arrows,
can you apply any of the parallel line theorems? Explain your answer. 4 A parallelogram is formed by two sets of intersecting parallel lines. a) Sketch a parallelogram. Don’t forget to label the parallel sides. b) Label each interior angle, a, b, c and d. c) State all relationships that exist between the four interior angles?
f
hg
75°
MFM 1P0 Unit 4: Geometry Student Notes
Page 16 of 18
Lesson 04: Algebra in Geometry Recall: 1 The sum of the angles in any right angle is always . 2 The sum of the angles in any straight line is always . 3 When two or more lines intersect, the angles that are opposite one another
are always . 4 • The sum of the interior angles in a triangle is .
• The measure of an exterior angle of a triangle is equal to the of the measures of the two interior and opposite angles.
• The sum of the exterior angles in a triangle is . 5 • The sum of the interior angles of a quadrilateral is .
• The sum of the exterior angles of a quadrilateral is . 6 Alternate Angles are .
Corresponding Angles are .
Co-interior Angles add to be .
a
b
a
b
a
b
MFM 1P0 Unit 4: Geometry Student Notes
Page 17 of 18
Lesson 04: Algebra in Geometry Example 1: Solve for the unknown angles. a)
b)
x - 7
2x + 12x + 1
2x + 5 x - 5
2x + 52x + 5
MFM 1P0 Unit 4: Geometry Student Notes
Page 18 of 18
Lesson 04: Algebra in Geometry 1 Rewrite the following expressions using multiplication. Then expand each
expression. The first one is done for you. a) x + 6 + x + 6 + x + 6
= 3 (x + 6) = 3x + 18 b) k + 4 + k + 4 + k + 4 + k + 4 + k + 4 + k + 4 + k + 4 + k + 4 c) 2x - 3 + 2x - 3 + 2x - 3 + 2x - 3 d) - 4m - 1 - 4m - 1 - 4m - 1 - 4m - 1 - 4m - 1 - 4m - 1 2 Determine the measures of each of the following angles using an algebraic
“shortcut.” -x + 5
-2x-x + 5
-x + 5
1 5
Lesson 1: More Exponents Exponential Form: 25 = 2 x 2 x 2 x 2 x 2 5 times
1. Write each of the following as powers a) 5 x 5 =
b) 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12 =
c) s x s x s x s x s =
d) 6a x 6a x 6a x 6a x 6a x 6a x 6a =
e) 2 x u x u x u x u x u x u =
f) 7 x 7 x 7 x 7 = 4 4 4 4
Exponent Rules: i) Power of 1: 51 = 5 ii) Multiplication Law: 54 x 53 = 54+3 = 57 iii) Division Law: 56 52 = 56-2 = 54 iv) Zero Law: 50 = 1 v) Power Law: (53) 2= 53*2 = 56 vi) Negative Law: 5-2 = ( ) 2
5 is the exponent 2 is the Base 25
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2. Using the Exponent Rules, solve the following questions. a) 74 x 73 x 72 b) 44 x 49 42
= 7 4 + 3 + 2
= 79 c) 27 22 x 21 d) ((-2)4) 2
e) 45 x ((4)2) 3 413 f) (-19)0
g) 40 x 70 – 90 h) 6-4
3. Using the Exponent Rules, evaluate the following:
a) k2 x k4 b) k3 ( k2 – 1) c) p (p2 p4)
Lesson 2: Surface Area and Volume
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Surface Area of any object is the sum of all the areas of all the faces of the object. Example: Take this cube of side length 3cm. There are 6 faces to a cube (4 sides, 1 top, 1 bottom). The area of 1 face is s x s = 3 x 3 = 9cm2 S.A. of the cube = 6 * 9 = 54 cm2
Solve for the surface areas of the following objects:
a) l = 3 cm w = 7cm h = 4 cm (Notice here that the top and bottom rectangles are the same the 2 different side rectangles are the same) S.A. of a rectangular prism = Area of all 6 faces = 2(area of 1st side) + 2(area of 2nd side) + 2(area of 3rd side) = = Volume is the amount of space that a figure occupies. Volume of any object = Area of the base x height V = B x H
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Example: The same cube of side length 3. This means the length, width and height of the cube is 3 cm. Volume = Area of the base x height = (3 x 3 ) x 3 = 27 cm3
Solve for the volumes of the following objects: a) The rectangular prism has length = 4cm width = 3cm and height = 7cm Volume = _________________ x __________ = = = b) Triangle base = 12 ft Triangle height = 20 ft Height of the prism = 25 ft
Volume of a Triangular Prism = Area of the Triangle x Height of prism =
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