Download Gr.7 Math Study Guide PD - St. John Paul II Collegiate

Strand: Numbers
General Outcomes: Develop number sense.
Specific Outcomes:
1. Determine and explain why a number is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, and why a number
cannot be divided by 0.
 Determine if a given number is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, and explain why.
Notes:
Divisibility: When one term (“dividend”) is divided by another (“divisor”), there is no remainder. (i.e. It goes in
evenly)
Example:
(Write both ways and label dividend, divisor, quotient, and remainder) 15 ÷ 3 = 5 R 0  We can say that 15 is
divisible by 3.
15 ÷ 2 = 7 R 1  We can say that 15 is not divisible by 2.
RULES
A number is divisible by 2 if the number is even (or last digit is even).
Example: 238 904 is divisible by 2.
Example: 179 is not divisible by 2.
A number is divisible by 3 if the sum of the digits is divisible by 3.
Example: 441  4+4+1 = 9, which is divisible by 3. Therefore, 441 is divisible by 3.
Example: 167  1+6+7 = 14, which is not divisible by 3. Therefore, 167 is not divisible by 3.
A number is divisible by 4 if the last two digits are divisible by 4.
Example: 65 224  24 is divisible by 4. Therefore, 65 224 is divisible by 4.
Example: 903  3 is not divisible by 4. Therefore, 903 is not divisible by 4.
A number is divisible by 5 if the last digit is a 5 or 0.
Example: 735 is divisible by 5.
Example: 480 is divisible by 5.
Example: 112 is not divisible by 5.
A number is divisible by 6 if it is divisible by 2 AND 3.
Example: 372 (even so div by 2, 3+7+2 = 12 so div by 3) is divisible by 6.
Example: 406 (is div by 2, but not 3) is not divisible by 6.
Example: 87 (is div by 3, but not 2) is not divisible by 6.
A number is divisible by 8 if the last 3 digits are divisible by 8.
Example: 15 800  800 is divisible by 8. Therefore, 15 800 is divisible by 8.
Example: 23 015  15 is not divisible by 8. Therefore, 23 015 is not divisible by 8.
A number is divisible by 9 if the sum of the digits is divisible by 9.
Example: 32 931  3+2+9+3+1 = 18 which is divisible by 9. Therefore, 32 931 is divisible by 9.
Example: 912  9+1+2 = 12 which is not divisible by 9. Therefore, 912 is not divisible by 9.
A number is divisible by 10 if the last digit is 0.
Example: 230 is divisible by 10.
Example: 123 456 789 is not divisible by 10.
Questions:
1. 2__5 - Fill in the blank with any number(s) between 0 and 9 to make the number divisible by 3.
Solution = 2 +__ + 5 = 7
2+2+5=9÷3=3
2 + 5 + 5 = 12 ÷ 3 = 4
2 + 8 + 5 = 15 ÷ 3 = 5
2. If you use 1035 tires to build cars, will any wheels be left over? Explain.
Solution = 1035 (35 ÷ 4 = 8.75). Therefore, the answer is NO. The last two digits of the
number is not divisible by four. There would be three tires left over.
Manipulatives:


Blocks for dividing into even groups
100s chart to see patterns in division

Sort a given set of numbers based upon their divisibility, using organizers such as Venn and
Carroll diagrams
Notes:
Venn Diagram – a diagram using circles or other shapes, to show the relationship between sets.
Carroll Diagram – a sorting diagram.
Questions:
3, 5, 6, 9, 10, 12, 15, 20, 21, 24, 25, 27, 30
1.
Which of the above numbers are divisible by 3, 5, and 15? Sort the numbers into a Venn
diagram.
Solution =
2. a) Sort the following numbers into the given Carroll Diagram:
6, 81, 75, 13, 63, 12, 101, 36, 54, 51, 28, 99
Solution =
Divisible by 3
Divisible by 9
81, 63, 36, 54, 99
Not Divisible by
9
6, 75, 12, 51,
Not Divisible by 3
13, 101, 28,
b) Are there any boxes left empty? If so why.
Solution = Yes, the “Not Divisible by 3 / Divisible by 9” box is empty because a number
divisible by 9 in automatically divisible by 3.

Determine the factors of a given number, using the divisibility rules.
Questions:
1. Find the factors of 36 using divisibility rules.
Solution = 36 x 1 because all number are divisible by 1
18 x 2 because 36 is an even number
12 x 3 because the sum of the digits ( 3 + 6 = 9) is divisible by 3
9 x 4 because the last two digits are divisible by 4
6 x 6 because 36 is divisible 2 and 3

Explain, using an example, why numbers cannot be divided by 0
Questions:
1.
If you have 12 chocolate bars and you are the sole person on a deserted island, can you
share them equally with anyone?
Solution = No, because there is no one to share with.
Demonstrate an understanding of the addition, subtraction, multiplication and division of
decimals to solve problems (for more than 1-digit divisors or 2-digit multipliers, the use of
technology is expected).
Notes
Adding and Subtracting Decimals
1. Line up decimals
2. Drag decimal straight down
3. Add/subtract as usual
Multiplying Decimals
1. Line up the numbers on the right - do not align the
decimal points.
2. Count the number of “jumps” or decimals places in each
of your numbers.
3. Starting on the right, multiply each digit in the top
number by each digit in the bottom number, just as with
whole numbers.
4. Add the products.
5. Add the total number of “jumps” or decimals places to
your answer
Dividing Decimals
1. Divisor must be a WHOLE number.
2. If the divisor is not a whole number, move decimal point to right to make it
a whole number and move decimal point in dividend the same number of
places.
3. Put decimal point directly above decimal point in the dividend.
4. Divide as usual. Keep dividing until the answer terminates or repeats.
5. Check your answer. Multiply quotient by divisor. Does it equal the
dividend?
BEDMAS
1.
2.
3.
4.
Do operations in brackets first.
Evaluate exponents.
Do division and multiplication in the order they appear – from left to right.
Do addition and subtraction in the order they appear – from left to right.
B – Brackets ( )
E – Exponents 3³
D – Division ÷
M- Multiplication x
A – Addition +
S- Subtraction –

Solve a given problem involving the addition of two or more decimal numbers.
Question
Six Boy Scouts have a piece of rope. They want to see exactly how long the rope would be altogether.
Their lengths of rope are 1.59 m, 1.51 m, 1.60 m, 1.41 m, 1.54 m and 2.09 m. How long is their rope?
Solution = 1.59 + 1.51 + 1.60 + 1.41 + 1.54 + 2.09 = 9.74 m

Solve a given problem involving the subtraction of decimal numbers.
Solution = 1.00 – 0.608 = 0.392

Solve a given problem involving the multiplication of decimal numbers.
Question
Which 10 × 10 grid shows the product of 0.4 × 0.7?
Solution = B because yellow represents 0.4, blue represents 0.7, green represents the
product 0.28.

Solve a given problem involving the multiplication or division of decimal numbers with 2-digit
multipliers or 1-digit divisor (whole numbers or decimals) without the use of technology.
Question
A single bus fare costs $1.75. A monthly pass costs $37.75. Tom estimates that he will ride the
bus 25 times this month. Should he buy a monthly pass?
Solution = 25 x 1.75 = $36.25

It is better to buy individual passes.
Solve a given problem involving the multiplication or division of decimal numbers with more than
2-digit multipliers or 1-digit divisor (whole numbers or decimals) with the use of technology.
Question
A nickel is 0.185 cm thick. How thick is a roll of nickels worth $2.00? Use your calculator.
Solution = 2.00 ÷ 0.05 = 40 nickels × 0.185 = 7.4 cm

Place the decimal in a sum or difference, using front-end estimation; e.g., for 4.5 + 0.73
+256.458, think 4 + 256, so the sum is greater than 260.
Question
Mr. Sattler is moving a statue from his house to his parents. The truck he is moving it in can
only hold 650 kg. If Mr. Sattler weighs 92.6 kg, his dad weighs 89.8 kg and his mom weighs 53.7
kg and the piano weighs 402.5 kg, will the truck hold the Mr. Sattler and his parents, as well as,
the piano?
Solution = 93 + 90 + 54 + 400 = 635 therefore everyone and the piano will fit.

Place the decimal in a product, using front-end estimation; e.g., $12.33 x 2.4, think $12 x 2, so
the product is greater than $24.
Question
Garrett wants to cover a pool that measures 12.3 m long and 9.7 m wide. Estimate how big a
cover does he need to cover the pool?
Solution = 12 x 10 = 120 m²

Place the decimal in a quotient, using front-end estimation; e.g., for 51.50 m ÷ 2.1, think 50 m ÷
2, so the quotient is approximately 25m.
Question
Ming earned $150.35 last week. If she worked 15.4 h, estimate how much she earns for each
hour.
Solution = 150 ÷ 15 = $10/hour

Check the reasonableness of solutions, using estimation.
Question
Find the actual and the estimated answer. Make sure your two answers are reasonable.
48.8 – 22.5
Solution =

48.8  49
- 22.5
-23
26.3
26
Solve a given problem that involves operations on decimals (limited to thousandths), taking into
consideration the order of operations.
Question
Paula would like to place a mat around a photo in a picture frame. The picture frame is 1.6 m
long and 1.35 m wide. The photo is 0.75 m long and 0.3 m wide.
Select the expression that will calculate the area of the mat.
a. 0.75 0.3 + 1.6 1.35
c. 1.6 + 1.35 0.75 + 0.3
b. 1.6 1.35 0.75 0.3
d. 1.6 1.35 + 0.75 0.3
Solution = The answer is B. The area of the frame (1.6 x 1.35) – the area of the picture
(.75 x 0.3).
Solve problems involving percents from 1% to 100%.
Notes
Percent to Fraction
Fraction to Percent
30% = 30 = 3
100 10
 remove percent sign
 put number over 100
 reduce fraction to lowest terms
7 = 14 = 14%
50 100
 change fraction to equivalent fraction with
100 as the denominator
 when it is out of 100, that is your percent
 add percent sign
Decimal to Percent
Percent to Decimal
78% = 78 = 0.78
100
 remove percent sign
 put number over 100
 move decimal in numerator 2 spaces to
left

0.03 = 3%


move decimal 2 spaces to right
add percent sign
Express a given percent as a decimal or fraction.
Question
What percent of a dollar does each decimal represent?
a) $0.01
b)$ 0.72
Solution = a) 1 %

b) 72%
c) $1.00
c) 100%
Solve a given problem that involves finding a percent.
Question
Tracy and Tristan both earn a percent of any sales they make at a furniture store. Last month,
Tracy earned $750 on sales of $5000, and Tristan earned $825 on sales of $6875. Who earns
the greater percent?
Solution = Tracey – 750 ÷ 5000 x 100 = 15%
Tracey ears a greater percent.
Tristan – 825 ÷ 6875 x 100 = 12%

Determine the answer to a given percent problem where the answer requires rounding, and
explain why an approximate answer is needed
Question
Colin is on a first date. The restaurant bill is $40.00 and he wants to leave a 15% tip. He only
has a $50 bill. Will he be able to leave a 15% tip?
Solution = Think about it like this . . . 10% of $40.00 is $4.00 and half of that is another
$2.00 (4 +2 = 6) therefore the cost of the meal and 15% tip will be $46.00. Colin can
breathe easy.
Demonstrate an understanding of the relationship between positive terminating decimals and
positive fractions and between positive repeating decimals and positive fractions.
Notes
Writing Fractions as Decimals
- If the denominator IS out of 100, simply take the numerator and move the decimal two places
to the left.
Ex. 45_ = 0.45
100
- If the denominator is 10, move the decimal one place to the left.
 The number of zeros in the power of ten (10, 100, 1000, etc.) determines the number of
places to move the decimal in the numerator from right to left.
Ex. 76 = 7.6
76_ = 0.76
76_ = 0.076
10
100
1000
- If the denominator is not a power of 10, you need to determine if the denominator can be
expressed as an equivalent fraction with a denominator which is a power of ten.
If yes – create the equivalent fraction and follow the steps from above.
Ex. 3_ = 6_ = 0.6
OR 3_ = 60 = 0.60
5
10
5
100
If no – you need to divide the numerator by the denominator.
Ex. 7_ = 0. 7 7 7 7 7 7 . . .
9
** If the denominator is a 9, the decimal will be the numerator repeated. **
Types of Decimals
Terminating Decimal – when you divide the numerator by the denominator, the
remainder is always zero . . . the number ends!
Ex. 3_ 8√ 3 = 0.375 OR 3_ = 375 = 0.375
8
8
1000
** A decimal that is able to be expressed as an equivalent fraction with a power of ten
denominator will always be terminating. **
Repeating Decimal – when the remainder is not zero and the decimal repeats itself.
Ex. 5_ = 0.454545
11
- When there is a repeating decimal, a bar is placed above the first digit or set being repeated
and the rest are dropped OR three periods are put behind.
Ex. 0.8333 – 0.83
0.8333 – 0.83 . . .
Predict the decimal representation of a given fraction, using patterns; e.g., 1/11 = 0.9090 repeating,
2/11 = 0.181818 repeating, 3/11 = ?
Question
Given the following decimal representations of fractions:
1
2
3
4
= 0.090909… , 11 = 0.181818… , 11 = 0. 272727… , 11 = 0.363636… ,
11
5
9
a) predict the decimals for 11 and 11
b) predict the fraction which will have 0.636363… as a decimal
Solution:
5
9
a) Using the pattern above, 11 = 0.454545….. and 11 = 0.818181…
7
b) 0.636363… = 11

Match a given set of fractions to their decimal representations.
Ex: Fill in the chart below
Decimal
Solution
Fraction
1
5
Decimal
0.20
0.50
Fraction
1
5
1
2
1
4
8
25
1
9
3
4
45
99
54
100
0.50
0.25
0.25
8
25
1
9
0.32
0.11…
0.75
0.75
0.45…
0.45…
54
100

0.54
Sort a given set of fractions as repeating or terminating decimals.
Question
Order the following fractions from smallest to largest
1/5
1/2
4/5
15/20 7/9
1/10
2/9
31/99 6/50
Solution:
First express each fraction as a decimal. Students should find an equivalent fraction with
a denominator of 10 or 100 in order to convert to a decimal.
0.20
0.50
0.80
0.75
0.77… 0.10
0.22… 0.31… 0.12
Then order the decimals from smallest to largest.
0.10

0.12
0.20
0.22… 0.31… 0.50
0.75
0.77… 0.80
Express a given fraction as a terminating or repeating decimal.
Question
1/5
4/10
5/9
32/99 4/25
Solution
Again, try to express each fraction as an equivalent fraction over 100. Repeating
fractions (over 9, 99, 999, etc) are the numerator repeated.
0.20

0.40
0.5…
0.32… 0.16
Express a given repeating decimal as a fraction.
Question
Express as a fraction:
0.46…
0.535…
0.8…
0.327…
Solution:
Move the decimal point the required number of spaces to the right, to make the
decimal a whole number.
This whole number becomes the numerator
Denominator: However many spaces the decimal point was moved to make a whole
number is how many nines are in the denominator
46/99

535/999
8/9
327/999
Express a given terminating decimal as a fraction.
Question
Express each decimal as a fraction:
0.25
Solution:
0.1
0.8
0.72
0.131
0.453
Move the decimal point the required number of spaces to the right, to make the
decimal a whole number.
This whole number becomes the numerator
Denominator: However many spaces the decimal point was moved indicates what
power of 10 we place in the denominator (ex: 0.25 decimal was moved two places, so
102 = 100 in the denominator)
25/100

1/10
8/10
72/100
131/1000
453/1000
Provide an example where the decimal representation of a fraction is an approximation of its
exact value.
Demonstrate an understanding of adding and subtracting positive fractions and mixed
numbers, with like and unlike denominators, concretely, pictorially and symbolically (limited
to positive sums and differences).

Model addition and subtraction of a given positive fraction or given mixed number, using
concrete representations, and record symbolically.

Determine the sum of two given positive fractions of mixed numbers with like denominators.
Question
𝟏
𝟐
𝟒
𝟒
𝟐 +𝟑
=?
Solution:
Add fractions (add numerators, denominator stays the same). 1/4 + 2/4 = 3/4
Add whole numbers. 2 + 3 = 5
The solution is the whole number and the fraction.
5
3
4
OR:
Convert the mixed numbers to improper fractions, then add numerators.
9
4

14
+
=
4
23
4
Determine the difference of two given positive fractions or mixed numbers with like
denominators.
Question
1
2
4
4
5 -3
=?
Convert mixed numbers to improper fractions, then subtract numerators.
21
4

-
14
4
=
7
4
Determine a common denominator for a given set of positive fractions or mixed numbers.
Question
Express each fraction over a common denominator
3 4
,
4 5
Solution:
First, multiply denominators. This will create a common denominator.
20
20
Next, cross multiply. 3x5 = 15, and 4x4 = 16
15 16
,
20 20

Determine the sum of two given positive fractions or mixed numbers with unlike denominators.
Question
3
4
+
4
5
Solution:
Express each fraction over a common denominator
First, multiply denominators. This will create a common denominator.
20
20
Next, cross multiply. 3x5 = 15, and 4x4 = 16
15 16
,
20 20
Finally, add numerators. Denominator stays the same.
15
20
+
16
20
=
31
20
Simplify a given positive fraction or mixed number by identifying the common factor between
the numerator and denominator.
Question
Put the fraction
4
10
into simplest form (lowest terms).
Solution:

Simplify the solution to a given problem involving the sum or difference of two positive fractions
or mixed numbers.
Question
3
1
In Jennifer’s class, 4 of the students were born in Alberta and 7 were born in
Saskatchewan. How many more students were born in Alberta than in Saskatchewan?
Solution:
Subtract the two fractions.
The least common multiple of 4 and 7 is 28.
Use the equivalent fractions with 28 as the denominator.
Subtract 4 parts from 21 pats and the parts are all 28ths. There are 17 parts left. In
17
Jennifer’s class, 28 more students were born in Alberta than in Saskatchewan.
Solve a given problem involving the addition or subtraction of positive fractions or mixed
numbers, and determine if the solution is reasonable.
Question
1
3
4
Caleb mixed 12 cans of yellow paint with 24 cans of blue paint. He used 35 of these cans
to paint a room. How much paint is left?
Solution
Demonstrate an understanding of addition and subtraction of integers, concretely, pictorially
and symbolically.

Explain using concrete materials such as integer tiles and diagrams, that the sum of opposite
integers is zero.
Question
Pictorially show the following: (+4) + (-4) = 0
Solution
Red = Positive
Yellow = Negative
Every red disk will cancel out one yellow disk.
Illustrate, using a number line, the results of adding or subtracting negative and positive
integers; e.g., a move in one direction followed by an equivalent move in the opposite
direction results in no net change in position.
Question
Using a number line solve the following equation (+5) + (-5)
Solution

Add two integers, using concrete materials or pictorial representations, and record the process
symbolically.
Question
Using integer discs solve the following equation : (+6)+ (-4) = ??
Solution
4 red disks will cancel out 4 yellow disks
We are left with 2 red (positive) disks.
Therefore our answer is (+2)

Solve a given problem involving the addition and subtraction of integers.
Rule
Example
+(+)
3+(+2) = 3 + 2 =
5
-(-)
6-(-3) = 6 + 3 =
9
+(-)
7+(-2) = 7 - 2 =
5
-(+)
8-(+2) = 8 - 2 =
6
Two like signs become a
positive sign
Two unlike signs become a
negative sign
Example: What is 5+(-2) ?
From above: +(-) becomes a negative sign.
5+(-2) = 5 - 2 = 3
Answer: 5+(-2) = 3
Compare and order positive fractions, positive decimals (to thousandths) and whole numbers by using
: benchmarks, place value, equivalent fractions and/or decimals.

Order the numbers of a given set that include positive fractions, positive decimals and/or whole
numbers in ascending or descending order; verify the results, using a variety of strategies.
Question
Place the following values in order from smallest to largest.
0.6, ½, 1.2, ¾, 0.01
Solution
Use one whole as your bench mark to help order the values. Convert each into a decimal
fraction to make it easier to compare.
or
0.01, ½, 0.06, ¾, 1.2

Identify a number that would be between two given numbers in an ordered sequence or on a
number line.
Identify 4 different numbers that fall between (-3) and (+1)


Answers will vary
Examples: -2, -1, 0, 0.5

Position fraction with like and unlike denominators from a given set on a number line, and
explain strategies used to determine order.
Question
Arrange the following fractions on a number line and explain the strategy used.
1
3
, ,
2 4
5
,
8
1
,
4
3
8
Strategy: Convert all of the fractions to decimals and compare. Then order on a number line.
𝟏
𝟐
𝟑
= 0.5

𝟒
𝟓
= 0.75
𝟖
= 0.625
𝟏
𝟒
= 0.25
𝟑
=0.375
𝟖
Order the numbers of a given set by placing them on a number line that contains benchmarks,
such as 0 and 1 or 0 and 5.
Question
Place the following fractions on a number line. Remember to use your bench marks.
¾, ¼, 3⁄8, ½, 5⁄8
Solution
Position a given set of positive fractions, including mixed numbers and improper fractions, on
a number line; and explain strategies used to determine position.
Question
Arrange the following fractions on a number line and explain the strategy used.
1
2
3
, 4,
7
5
4
4
,
2
, 18
Strategy: Convert all of the fractions to decimals and compare. Then order on a number line.
1
= 0.5
2
3
= 0.75
4
7
= 1.75
4
5
= 1.25
4
4
1 8 = 1.5
Match a given set of fractions to their decimal representations.
Question
Write the fraction as a terminating decimal.
Solution = create an equivalent fraction with 100 as your denominator. Therefore, 20 x
5 = 100, what you do to the bottom you must do to the top 17 x 5 = 85. The answer is
0.85.
Sort a given set of fractions as repeating or terminating decimals.
Question
Indicate whether the following fractions are repeating or terminating.
2
3
4
22
a) 5
b) 9
c) 11 d) 7
Solution = a and d are both terminating decimals while b and c are repeating.
Express a given fraction as a terminating or repeating decimal.
Question
Write the fraction as a repeating decimal.
Solution = 0.45 . . .
Express a given repeating decimal as a fraction.
Question
Create a fraction that is equivalent to the repeating decimal
.
Solution = Take the numbers being repeated (324) and make it the numerator. The
324
denominator will be 999 because there are three digits in the numerator. 999
Express a given terminating decimal as a fraction.
Question
Write a fraction that is equivalent to the decimal number 0.45
Solution = Remove the decimal so that you have a whole number (45) this is your
numerator. Your denominator will be a power of ten. The number of zeros in your
denominator will be equivalent to the number of digits after the decimal in the original
45
question. 100
Provide an example where the decimal representation of a fraction is an approximation of its exact
value.
Question
If you have a pizza cut into four pieces and you are sharing equally with a friend, represent your
share using a decimal.
Solution = Half of four is two which is equivalent to 0.5
Demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with
like and unlike denominators, concretely, pictorially and symbolically (limited to positive sums and
differences).
Notes
Changing an Improper Fraction to a Mixed Number
Improper fraction – when the numerator is larger than the denominator.
Mixed number – a whole number and a fraction.
Adding and Subtracting Fractions with the Same Denominators
1) Denominator stays the same.
2) Add or subtract the numerators (depending on the question)
Adding and Subtracting Fractions with the Different Denominator
1) Create a common denominator.
-
the easiest way to do this is multiply the denominators together
2) What you do to the bottom, you do to the top.
3) Add or subtract the numerators and place your answer over the new denominator (common
denominator).
Adding and Subtracting Mixed Numbers
Adding 1) focus on the fractions – if the denominators are the same follow the rules.
- if the denominators are different follow the rules.
2) If your answer is an improper fraction turn it into a mixed number.
3) After adding the fractions, add the whole numbers
Subtracting – 1) Turn into an improper fraction.
2) Follow the rules for adding and subtracting fractions.
3) Remember to turn your answer back into a mixed number.
Model addition and subtraction of a given positive fraction or given mixed number, using concrete
representations, and record symbolically.
Question
Using fraction strips, solve the following equation
Solution:
Strand: Patterns and Relations (Patterns)
General Outcomes: Use patterns to describe the world and to solve problems.
Specific Outcomes:
1. Demonstrate an understanding of oral and written patterns and their equivalent linear
relations.

Formulate a linear relation to represent the relationship in a given oral or written pattern.
Question
a) Amy works in a snack bar in a community centre. She earns $9 an hour. One weekend, she
made $45. How many hours did she work that weekend?
Answer: 9h = 45
h = 45 / 9
h=5
b) Identify the linear relation represented by the graph.
Answer: Y= 2x

Provide a context for a given linear relation that represents a pattern.
Question
9h = 45
Answer: $9 earned per hour, for a unknown number of hours, results in a total of $45 earned.

Represent a pattern in the environment, using a linear relation.
Question
The average rainfall per month is 15mm. A total of 135mm falls. How much time has passed?
Answer: 15m = 115
M = 115 / 15
M = 10
2. Create a table of values from a linear relation, graph the table of values, and analyze the
graph to draw conclusions and solve problems.

Create a table of values for a given linear relation by substituting values for the variable.
Question
Create a table of values for the linear relation y = x + 6. (one point)
Solution Note - Answers may vary

x
1
2
3
4
y
7
8
9
10
Create a table of values, using a linear relation, and graph the table of values (limited to discrete
elements).
Question
Complete the table of values and graph the points in this table of values.
Y = 10d
D
y
1
2
3
4
Solution
Complete the table of values and graph the points in this table of values.
Y = 10d
D
1
2
3
4
y
10
20
30
40

Sketch the graph from a table of values created for a given linear relation, and describe the
patterns found in the graph to draw conclusion; e.g. graph the relationship between n and 2n + 3
Question
Graph the pattern. In words describe the relationship between the figure number and
the number of squares.
Figure number
1
2
3
4
5
Number of squares
4
5
6
7
8
Solution:
The number of squares in each figure is twice the figure number plus three.
s

Describe, using everyday language in spoken or written form, the relationship shown on a graph
to solve problems.
Question
In words, describe the relationship shown on the graph.
Solution: The relation value is the variable value plus three. (y = x + 3)

Match a set of linear relations to a set of graphs.
Question
Identify the linear relation represented by the graph.

a. Y = x
c. y = x + 3
b. Y = 2x
d. y = x – 4
Solution: Create a table of values. By viewing the table of values you can see that if you
take the variable (x) value and multiply it by 2, you will get the relation (y) value.
X
Y
1
2
2
4
3
6
4
8
5
10
6
12
7
14
8
16
Strand: Patterns and Relations (Variables and Equations)
General Outcomes: Represent algebraic expressions in multiple ways.
Specific Outcomes:
3. Demonstrate an understanding of preservation of equality by: modeling preservation of
equality, concretely, pictorially and symbolically; applying preservation of equality to solve
equations.

Model the preservation of equality for each of the four operations, using concrete materials or
pictorials representations; explain the process orally; and record the process symbolically.
Question
Show the balance of an equation for each of the four operation using words and pictures.
Solution #1
a) Add up the left side (12).
b) In order for there to be preservation of equality (both sides being equal), the right side must
equal 12 as well.
c) Multiply the two numbers on the right side (12)
d) compare the results from both sides (both sides equal 12, therefore there is balance).
Solution #2
a) find the answers to both sides of the balance scale (left = 10 and right = 10)
b) we have demonstrated balance (preservation of equality) because both sides are equal.
Write equivalent forms of a given equation by applying the preservation of equality, and verify, using
concrete materials; e.g., 3b = 12 is the same as 3b + 5 = 12 + 5 or 2r =7 is the same as 3(2r)=3(7).
Question
Write equivalent forms of a given equation, solve and verify using pictures.
Solution:

Solve a given problem by applying preservation of equality.
Question
Solve 2n + 5 = 19 by applying preservation of equality.
Solution:
Substitute the value 7 in for n to verify balance.
2 (7) + 5 = 19
14 + 5 = 19
19 = 19
4. Explain the difference between an expression and an equation.

Identify and provide an example of a constant term, numerical coefficient and variable in an
expression and an equation.
Question
What is a constant term? Give an example of a constant term in an expression and an
equation.
Solution:
Constant Term : A quantity that does not change; for example, in the expression 2 x
n + 5, 5 is a constant term. In the equation 2 x n + 5 = 13, 5 is the constant term.
Question: What is a numerical coefficient? Give an example of a numerical
coefficient in an expression and an equation.
Solution:
Numerical Coefficient: The multiplier of a variable; for example, in the expression 2
x n + 5, 2 is the numerical coefficient of n. In the equation 2 x n + 5 = 13, the 2 is the
numerical coefficient.
Question
What is a variable? Give an example of a variable in an expression and an equation.
Solution:
Variable: Variables are (usually) letters or other symbols that represent unknown
numbers or values. In the expression 2 x n + 5, 2 the variable is n. In the equation
2 x n + 5 = 13, the variable is n.

Explain what a variable is and how it is used in a given expression.
Question
What is a variable and how is it used in a given expression?
Solution: The following are examples of algebraic expressions and equations containing
variables.
2x + 5 = 10, the variable here is x
7y + 10 = 24, the variable here is y
a2 + b2, the variables here are a and b
The variable in the above equations is the unknown factor.
Solved Example on Variable
Identify the variables in the equation, y = 3x + 2.
Choices:
A. 2 and y
B. x and y
C. x and 3
D. 2 and 3
Correct Answer: B
Solution:
Step 1: Variables are letters or other symbols that represent unknown numbers
or values.
Step 2: Therefore, x and y are the variables in the given equation
y = 3x + 2.

Provide an example of an expression and an equation, and explain how they are similar and
different.
Question
How are expressions and equations similar and different? Provide a specific
example.
Solution:
Expression: may contain one or more operations but does not have an equal sign
Examples: 3n, b + 4, 2p - 7
Equation: may contain one or more operations and does have an equal sign
Examples: 3n = 6, b + 4 = 13, 2p – 7 = 37
Example: 3n (expression) and 3n = 6 (equation)


Similarities: both have a numerical coefficient and a variable.
Differences: the difference between the two is that an equation has an equal
sign. In this example, the equation has an equal sign and the answer to the
equation ( =6)
5. Evaluate an expression given the value of the variable(s).

Substitute a value for an unknown in a given expression, and evaluate the expression.
Question
Evaluate the expression 3y + 2y when 5 = y.
1. Replace each letter with its assigned value.
The first step: 3(5) + 2(5)
(Use Parentheses!)
2. Perform the operations, to find the value of the expression.
The second step:
3(5) + 2(5) = 15 + 10 = 25
Model and solve, concretely, pictorially and symbolically, problems that can be represented
by one-step linear equations of the form x + a = b, where a and b are integers.

Represent a given problem with a linear equation; and solve the equation, using
concrete models, e.g., counters, integer tiles.
Question
Solve the following linear equation using a concrete model.
x 3 9)
Solution
Three less than a number is -9. Students should be able to write an equation for the number
sentence, and then solve using algebra tiles. (In the diagram below, represents a negative,
represents a positive.)
To model this equation, students need to recall that subtracting 3 is equivalent to adding
negative 3
To isolate
positive
make zero
tiles to the
equality.
from both sides.
the variable tile, add 3
tiles to the left side to
pairs. Add 3 positive
right side to preserve
Remove the zero pairs
Students can verify the solution by replacing x, the variable tile, with 6 negative tiles. They can
also verify by replacing x with -6 in the equation.

Draw a visual representation of the steps required to solve a given linear equation.
Question
Using pictures, solve the following linear equation: 2x + 1 = 5
Solution:

Solve a given problem, using a linear equation.
Question
Solve the following equation: 2x + 1 = 5
Solution:
2x + 1 = 5
2x = 5 – 1
2x = 4
X–4÷2
X=2

Substitute a possible solution for the variable in a given linear equation into the
original linear equation to verify the equality.
Question
Determine the solution to the equation 4 + 5a = 19. Verify your answer via
substitution.
Solution
4 + 5a = 19
Check: 4 + 5a = 19
4 – 4 + 5a = 19 – 4
4 + 5 (3) = 19
5a = 15
4 + 15 = 19
5a ÷ 5 = 15 ÷ 5
19 = 19
a=3
Strand: Shape and Space (Measurement)
General Outcomes: Use direct and indirect measurement to solve problems.
Specific Outcomes:
1. Demonstrate an understanding of circles by:
a. Describing the relationship among radius, diameter and circumference
b. Relating circumference to pi
c. Determining the sum of the central angles
d. Constructing circles with a given radius or diameter
e. Solving problems involving the radii, diameters and circumferences of circles.

Illustrate and explain that the diameter is twice the radius in a given circle.

Illustrate and explain that the circumference is approximately three times the diameter in a
given circle.
Question
C=πxd
C = 15
Therefore, C ÷ π = d
15 ÷ 3.14 = 4.7
See image below.

Explain that, for all circles, pi is the ratio of the circumference to the diameter (c/d) and its value
is approximately 3.14.
C = 3.14 x d
OR
C = πd

Explain, using an illustration, that the sum of the central angles of a circle is 360°.
Students should be familiar with the concept of a right angle (90o). As illustrated above, the
sum of the central angles is the sum of four right angles (360o)

Draw a circle with a given radius or diameter, with and without a compass.
Ex: Use a compass to draw a circle of radius 4 cm.
Step 1: Use a ruler to set the distance from the point of the compass to the pencil's lead at
4 cm.
Step 2: Place the point of the compass at the centre of the circle.
Step 3: Draw the circle by turning the compass through 360º.
Solve a given contextual problem involving circles.
Question
A n n e i s r i d i n g a h o r se w h i c h i s t i e d to a p o le w i t h a 3 .5 m p i ec e o f ro p e a n d h e r
f r i e n d L a u r a i s r i d i n g a d o n k ey w h i c h i s 2 m f ro m t h e s am e ce nt e r p o i nt . C a l c u l a te
t h e d i s t a n ce t r av e l led b y e a c h w h e n t h e y h a v e ro t at e d 1 0 t i m es a r o u n d t h e c e nt r e .
Solution
Annie – C = 𝜋 𝑥 𝑑
Laura – C = 𝜋 𝑥 𝑑
3.14 x 3.5
3.14 x 2
= 10.99m x 10 = 109.9 m
=6.28 m x 10 = 62.8m
Ex: A pizza has a circumference of 38 inches. A square pizza box has a length of 14 inches.
Will the pizza fit into the pizza box? Draw a diagram to explain.
2. Develop and apply a formula for determining the area of: triangles, parallelograms, and
circles.

Illustrate and explain how the area of a rectangle can be used to determine the area of a
triangle.
Question
Triangle:
14cm
A=
Rectangle:
A=bxh
8cm
Rectangle Area = b x h
= 14 x 8
=112cm
Triangle Area =
= 14 x 8
2
= 56cm

Generalize a rule to create a formula for determining the area of triangles.
The area of a rectangle is equal to (length) x (width), or (base) x (height). Any rectangle
can be divided into two equal triangles. Therefore, the resulting area of one of the
triangles is half the area of the rectangle, or ½ (base) x (height).
Even if the triangle doesn’t immediately look like half a rectangle, the formula still
applies. Note how the triangle below (bottom left) can be “cloned” and manipulated to
form a rectangle.

Illustrate and explain how the area of a rectangle can be used to determine the area of a
parallelogram.
Area of the rectangle:
(B+A) x (H)
Area of a triangle:
½ (A)(H).
Two triangles, so total area = 2 x ½(A)(H) = (A) (H)
Therefore, area of the parallelogram = (B + A)(H) – (A)(H)
= (BH)+(AH) – (AH)
= (BH)
which is base x height

Generalize a rule to create a formula for determining the area of parallelograms.
Refer to the above diagram and explanation, as it demonstrates how to generalize the
formulas for rectangles and triangles to create a formula for the area of a parallelogram
Illustrate and explain how to estimate the area of a circle without the use of a formula.
Question
Using graph paper illustrate and explain how you would estimate the area of a
circle. You cannot use a formula!!!
Solution
-
Using a compass, draw a circle on grid paper.
Count the squares inside the circle and estimates the area.
Draws a square outside the circle and calculate the area of the square.
Draw a square inside the circle and calculates the area of the square.
Estimate the area of the circle by relating it to areas of the outer and inner squares.
(Hint: average the areas of the two squares.)

Apply a formula for determining the area of a given circle.
Question
4cm
A= 3.14 x 4²
A = 3.14 x 16
A = 50.24 cm

Solve a given problem involving the area of triangles, parallelograms and/or circles.
Question
Calculate the area of the shaded ring. Explain.
𝐴 = 𝜋𝑟 2
A = 3.14 x 1.5²
A = 7.065
𝐴 = 𝜋𝑟 2
A = 3.14 x 3²
A = 28.26
A = 28 26 – 7.065
A = 21.195 m
Strand: Shape and Space (3-D Objects and 2-D Shapes)
General Outcomes: Describe the characteristics of 3-D objects and 2-D shapes,
and analyze the relationships among them.
Specific Outcomes:
3. Perform geometric construction, including:
a. Perpendicular line segments: lines that intersect at right angles (90°).
b. Parallel line segments: Describes lines in the same plane that never cross or intersect.
c. Perpendicular bisector: A line which cuts a line segment into two equal parts at
90°.
d. Angle bisectors. A line that divided an angle into two equal parts.

Describe examples of parallel line segments, perpendicular line segments, perpendicular
bisectors and angle bisectors in the environment.
Perpendicular line segments: T - intersection
Parallel line segments: railroad tracks
Perpendicular bisectors: 4 way intersection
Angle bisectors: tying a swing on to a tree branch

Identify line segments on a given diagram that are parallel or perpendicular.
Question
Which pair of lines in the diagram are parallel?
Solution: Line segments DE and CB are parallel

Draw line segment perpendicular to another line segment, and explain why they are
perpendicular.
Question
A
Line segment AB is perpendicular to line segment CD because
they intersect each other at 90°
C

B
D
Draw a line segment parallel to another line segment, and explain why they are parallel.
A
B
C
D
Line segment AB is parallel to line segment CD because they are on the same plane and never
cross or intersect.

Draw the bisector of a given angle, using more than one method, and verify that the resulting
angles are equal.
Use of a compass
After doing this
Start with angle PQR that we will bisect.
1. Place the compass point on the angle's
vertex Q.
Your work should look like this
After doing this
2. Adjust the compass to a medium wide
setting. The exact width is not important.
3. Without changing the compass width, draw
an arc across each leg of the angle.
4. The compass width can be changed here if
desired. Recommended: leave it the same.
5. Place the compass on the point where one
arc crosses a leg and draw an arc in the interior
of the angle.
Your work should look like this
After doing this
6. Without changing the compass setting repeat
for the other leg so that the two arcs cross.
7. Using a straightedge or ruler, draw a line
from the vertex to the point where the arcs
cross
Done. This is the bisector of the angle ∠PQR.
Your work should look like this
Protractor method
After doing this
Your work should look like this
Start with angle PQR that we will bisect.
Steps: a) measure the angle with your protractor
b) divide the angle in half
c) find the half measure on your protractor
d) draw a line from the vertex straight out. This is the angle bisector.

Draw the perpendicular bisector of a line segment, using more than one method, and verify the
construction.
Protractor Method
After doing this
Start with a line segment
PQ.
Your work should look like this
After doing this
1. Place the compass on
one end of the line segment.
2. Set the compass width to
a approximately two thirds
the line length. The actual
width does not matter.
3. Without changing the
compass width, draw an arc
on each side of the line.
Your work should look like this
After doing this
4. Again without changing
the compass width, place
the compass point on the
other end of the line. Draw
an arc on each side of the
line so that the arcs cross
the first two.
5. Using a straightedge,
draw a line between the
points where the arcs
intersect.
Your work should look like this
After doing this
Your work should look like this
6. Done. This line is
perpendicular to the first
line and bisects it (cuts it at
the exact midpoint of the
line).
Strand: Shape and Space (Transformations)
General Outcomes: Describe and analyze position and motion of objects and
shapes.
Specific Outcomes:
4. Identify and plot points in the four quadrants of a Cartesian plane, using integral ordered
pairs.

Label the axes of a four quadrant Cartesian plane, and identify the origin.

Identify the location of a given point in any quadrant of a Cartesian plane, using an integral
ordered pair.
Question
(-2 , -3)

Plot the point corresponding to a given integral ordered pair on a Cartesian plane with units of
1,2,5 or 10 on its axes.
Plot the point (1,3)

Draw shapes and designs in a Cartesian plane, using given integral ordered pairs.
A (-3 , 5) B (-2, 3) C(-2, 2)

D(-4, 1)
E(-4,3)
Create shapes and designs, and identify the points used to produce the shapes and designs, in
any quadrant of a Cartesian plane.
A (2, -3)
B(5, -4)
C(2, -4)
5. Perform and describe transformations (translations, rotations or reflections) of a 2-D shape in
all four quadrants of a Cartesian plane (limited to integral number vertices).
(It is intended that the original shape and its image have vertices with integral coordinates.)
The three main Transformations are:

Rotation
Turn!
Reflection
Flip!
Translation
Slide!
Identify the coordinates of the vertices of a given 2-D shape on a Cartesian plane.
New Image Coordinates
S’ (-4, -1) K’(-5, -2)
J’(-3, -2)

Describe the horizontal and vertical movement required to move from a given point to another
point on a Cartesian plane.
(2, 3) Horizontal movement is 2 (right) and the vertical movement is 3
(up)

Describe the positional change of the vertices of a given 2-D shape to the corresponding vertices
of its image as a result of transformation, or successive transformation, on a Cartesian plane.
A (1,6) A’(3,2)
B (4,6) B’(6,2)
C (3,3) C’(0,5)
The original image was translated (2, -3)

Determine the distance between points along horizontal and vertical lines in a Cartesian plane.
The distance between point A and B


up
Starting point A - to get to B… (5,1)
This means move 5 units to the right and 1 unit

Perform a transformation or consecutive transformations on a given 2-D shape, and identify
coordinates of the vertices of the image.
Original
Translation
New Image
(6,0)

F (-5, 5)
F’(1,5)
C(-2, 5)
C’(4,5)
P(-3, 0)
P’(3,0)
Describe the image resulting from the transformation of a given 2-D shape on a Cartesian plane
by identifying the coordinates of the vertices of the image.
Transformation of the original image (-4,2)
The original image slid 4 units to the left and 2 units up
B(3,-4)
L(0,0)
L’(-4,2)
E(-1,-2)
E’(-5,0)
B’(-1,-2)
Strand: Statistics and Probability (Data Analysis)
General Outcomes: Collect, display and analyze data to solve problems.
Specific Outcomes:
1. Demonstrate an understanding of central tendency and range by:
a. Determining the measures of central tendency (mean, median, mode) and range
b. Determining the most appropriate measures of central tendency

Determine mean, median and mode for a given set of data, and explain why these values may be
the same or different.
Notes
The mean is a number that can represent the centre of a set of numbers.
 Use linking cubes: Show each number with cubes and then rearrange the cubes
to make rows of equal length.
Ex.
What is the mean of the data set? 3 4 4 9
The mean of the data set is 5.
 Use symbols: Add the numbers in the data set together and then divide by the
numbers in the set.
Ex.
What is the mean of the data set? 3 4 4 9
3 + 4 + 4 + 9 = 20
20 ÷ 4 = 5
The mean of the data set is 5.
The mode is the number that occurs most often in a set of data.
 Use symbols: Find the number in the data set that occurs most often. There can
be no mode or several modes.
Ex. What is the mode of the data set? 5 6 8 11 8 9 8 5
The number that occurs most often is 8. The mode of the data set is 8.
The median of a set of data is the middle number when the data are arranged in order.
 Use symbols: Arrange the numbers in the data set from least to greatest and
select the middle number. If there is an even number, find the two middle
numbers and then divide by 2.
Ex. What is the median of the set of data? 39 44 43 40 39 44 37
37 39 39 40 43 44 44
40 is the middle number, so it is the median.
Ex. What is the median of the set of data? 39 44 43 40 39 44 37 42
37 39 39 40 42 43 44 44
40 and 42 are the middle numbers, so divide by 2.
(40+42)÷2=41
41 is the median of the set of data.
The mean and mode are sometimes called average.

Determine the range for a given set of data.
The range of a data set tells how spread out the data are. It is the difference between the
greatest and the least numbers in the set.
 Use symbols: Find the difference between the largest number and the smallest
number in the data set.
Ex. What is the median of the set of data? 39 44 43 40 39 44 37
The largest number is 44 and the smallest number is 37.
44-37=7
The range of the data set is 7.
Ex.
Daily High Temperature Change
Date
Temperature
Feb. 1
-24°C
Feb. 2
-24°C
Feb. 3
-22°C
Feb. 4
-17°C
Feb. 5
-8°C
Feb. 6
-23°C
Feb. 7
-24°C
Calculate the central tendencies and range for
the daily high temperatures using the table.
Find the mean:
(-24°C)+(-24°C)+(-22°C)+(-17°C)+(-8°C)+(-23°C)+(-24°C)=142°C
Find the mode:
-24°C occurs three times
Find the median:
-24°C -24°C -22°C -17°C -8°C -23°C -24°C
Find the range:
The smallest number is -24°C and largest number is -8°C.
(-8°C)-(-24°C)=(-8°C)+(+24°C)=16°C

Provide a context in which the mean, median or mode is the most appropriate measure of
central tendency to use when reporting findings.
Question
Match the following cases with the appropriate measure of central tendency
a. John is keeping track of all the shirt
sizes in his store he has 2 smalls, 9
mediums and 4 larges.
b. Susan’s basketball coach keeps track of
the number of baskets she makes in
each game (5, 7, 3, 6, 8, 15)
c. Mike is trying to determine how many
weeks it will take to save for his bike.
Over the last 4 weeks he has made
Mean
Range
Median
$85, $76, $91, $88.
d. Bob would like to find the age
difference of his teachers in his school.
Mode
Answer
a. Mode – John wants to stock the most popular size which is the number that occurs most
often.
b. Median – Coach wants the average number of baskets, median is best when there is
outliers.
c. Mean – Mike wants to find the average amount of money he makes, mean is the best
measure.
d. Range – Bob wants to know the difference of the teachers ages, so find the range of the
ages.

Solve a given problem involving the measures of central tendency.
The mean, median, and mode are measures of central tendency.
Ex. The average age of 3 brothers is 16. The oldest brothers are 15 and 24.
What is the age of the youngest brother?
𝑥+15+24
3
3×
Add together the three ages, using a variable for the unknown age.
= 16
𝑥+15+24
3
= 16 × 3
Divide the added ages by three to find the age of the third brother.
Use opposite operations to both sides to remove the dividing by
three.
𝑥 + 15 + 24 = 48
Add the ages of the two known brothers.
𝑥 + 39 = 48
𝑥 + 39 − 39 = 48 − 39
Use opposite operations to both sides to determine the age of the brother.
𝑥=9
The youngest brother is 9 years old.
2. Determine the effect on the mean, median and mode when an outlier is included in a data set.

Analyze a given set of data to identify any outliers.
Notes
A number in a data set that is very different form the other numbers is an outlier.
Ex. The number of baskets that George scored in 15 games during his high school
basketball season were: 12, 10, 18, 12, 13, 20, 28, 19, 12, 9, 3, 18, 17, 9, 13.
28 is an outlier because it is much larger than the rest of the data in the set.
3 is an outlier because it is much smaller than the rest of the data in the set.

Explain the effect of outliers on the measures of central tendency for a given data set.
Question
Ex. The number of baskets that George scored in 15 games during his high school
basketball season were: 12, 10, 18, 12, 13, 20, 28, 19, 12, 9, 3, 18, 17, 9, 13.
Identify the outliers in the above set:
How did they affect the mean? median? mode?
The outliers are 3 and 28.
Without Outliers
With Outliers
Mean
(12+10+18+12+13+20+28+19+12+9+3+18+17+9+13)÷15=14.2
Mean
(12+10+18+12+13+20+19+12+9+18+17+9+13)÷13=14
Median
3,9, 9, 10, 12, 12, 12, 13, 13, 17, 18, 18, 19, 20, 28
(13+13) ÷2=13
Median
3,9, 9, 10, 12, 12, 12, 13, 13, 17, 18, 18, 19, 20, 28
(13+13) ÷2=13
Mode
3,9, 9, 10, 12, 12, 12, 13, 13, 17, 18, 18, 19, 20, 28
12 is the mode
Mode
9, 9, 10, 12, 12, 12, 13, 13, 17, 18, 18, 19, 20
12 is the mode
The outliers affected the mean, but they did not affect the median or the mode.

Identify outliers in a given set of data, and justify whether or not they are to be included in
reporting the measures of central tendency.
Question
Ex. These data show the daily temperatures, in degrees Celsius, for two weeks in the
summer in Nelson, B.C.:
23, 25, 22, 25, 28, 24, 25, 24, 25, 25, 52, 24, 20, 22
a) Find the mean, median, mode, and range for these data.
b) Identify the outlier. Why is the outlier so much greater than the other temperatures?
c) Calculate the mean, median, mode, and range for these data without the outlier.
d) When reporting the average daily temperature, should the outlier be included?
Answer
a)
Mean (23+25+22+25+28+24+25+24+25+25+52+24+20+22)÷14=26°C
Median 20, 22, 22, 23, 24, 24, 24, 25, 25, 25, 25, 25, 25, 52 (24+25) ÷2=24.5°C
Mode The mode is 25°C.
Range 52°C - 20°C = 30°C
b) The outlier is 52°C. It is probably a recording error, because 52°C is a very high
temperature.
c)
Mean (23+25+22+25+28+24+25+24+25+25+24+20+22)÷13=24°C
Median 20, 22, 22, 23, 24, 24, 24, 25, 25, 25, 25, 25, 25
24°C
Mode The mode is 25°C.
Range 28C - 20°C = 28°C
d) The 52°C would not be included when reporting the data set because it is probably
an error.

Provide examples of situations in which outliers would and would not be used in reporting the
measures of central tendency.
Notes
An outlier is much greater than or much less than most of the numbers in the data set.
Outliers sometimes occur as a result of error in measurement or recording. In these cases,
outliers should be ignored.
Sometimes an outlier is an important piece of information that should not be ignored. For
example, if one student does much better or much worse than the rest of the class on a
test.
Outliers may not always be obvious. Identifying outliers is then a matter of choice.
Ex. These data show the daily temperatures, in degrees Celsius, for two weeks in the
summer in Nelson, B.C.:
23, 25, 22, 25, 28, 24, 25, 24, 25, 25, 52, 24, 20, 22
When reporting the average daily temperature, should the outlier be included?
Answer
The 52°C would not be included when reporting the data set because it is probably an
error.
Ex. Here are the marks out of 100 on an English test for students in a Grade 7 class:
21, 23, 24, 24, 27, 29, 29, 29, 32, 37, 37, 38, 39, 40, 50, 50, 51, 54, 56, 57, 58, 59, 61, 71,
80, 99
When reporting the average test mark, should the outlier be included?
Answer
The outliers should be used when reporting the average test mark. To understand how
the class is performing, all test marks should be included.
3. Construct, label and interpret circle graphs to solve problems.

Identify common attributes of circle graphs, such as:
o Title, label and legend
 Graphs need to have a clear title and labels to illustrate the data represented.
o The sum of the central angles is 360°
 To find the size of a central angle, multiply the percent (as a decimal) by 360°
o The data is reported as a percent of the total, and the sum of the percents is equal to
100%.
 Circle graphs can show data sets that equal 100% only.
Question
a. Give a title to the above graph
b. What percentage of books borrowed
were science?
c. How many degrees does the History
pie take up?
Answer
a. Types of Books Borrowed
b. 11+15+18+9+11+21=85
100-85=15%
18
c. 18%=100 × 360° = 0.18 × 360° = 64.8°

Create and label a circle graph, with and without technology, to display a given set of data.
Question
In a survey of 50 people: 30 people like chocolate, 15 people like strawberry, 5 people
like vanilla.
Make a table and draw these results on a graph.
Favourite Flavour
# People
Chocolate
30
Strawbery
15
Vanilla
5
Total
50
% People
Number of Degrees of a full circle
( ̊)
Answer
Chocolate
Strawberry
Vanilla
30
50
× 100 = 60%
15
50
60
100
× 360° = 216°
30
× 100 = 30%
5
100
5
× 100 = 10%
50
100
× 360° = 108°
× 360° = 36°
Favourite Flavour
# People
% People
Chocolate
30
60%
Number of Degrees of a full circle
( ̊)
216 ̊
Strawberry
15
30%
108 ̊
Vanilla
5
10%
36 ̊
Total
50
100%
360 ̊
Vanilla
36
Strawberry
108
Chocolate
216

Find and compare circle graphs in a variety of print and electronic media, such as newspapers,
magazines and the Internet.
Question
The Morning Mix bag is 100g, but the Super Snack Mix bag is 150g. Which bag has the
largest mass of raisins?
Super Snack Mix:
13% x 150g = 0.13 × 150g = 19.5g
Morning Snack Mix: 17% x 100g = 0.17 × 100g = 17.0g
Super Snack Mix has more raisins.

Translate percentages displayed in a circle graph into quantities to solve a given problem.
Question
This graph shows the world’s gold production for a particular year. In this year, the
world’s gold production was approximately 2300 t. About how much gold would have
been produced in each country?
a) Canada
b) South Africa
Answer
Canada
South Africa

7% x 2300t = 0.07 x 2300 t = 161 t
35% x 2300t = 0.35 x 2300 t = 805 t
Interpret a given circle graph to answer questions.
Question
Use the following circle graph to answer these questions.
a) Which group of viewers would represent 25% of the demographics?
b) What groups make up about 1/3 of the viewers?
Answer
a) 20 to 29 years or 40 years and over and 30 to 39
years
b) 40 years and over, 30 to 39 years, and 0 to 12
years
Strand: Statistics and Probability (Chance and Uncertainty)
General Outcomes: Use experimental or theoretical probabilities to represent
and solve problems involving uncertainty.
Specific Outcomes:
4. Express probabilities as ratios, fractions and percents.

Determine the probability of a given outcome occurring for a given probability experiment, and
express it as a ratio, fraction and percent.
 The probability of an event occurring is:
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑂𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝐹𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑡𝑜 𝑡ℎ𝑎𝑡 𝐸𝑣𝑒𝑛𝑡
𝑇𝑜𝑡𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑂𝑢𝑡𝑐𝑜𝑚𝑒𝑠
Question
In a bag of pencils there are 4 red, 5 green and 2 blue.
a. Write the probability of picking a red pencil in 3 different ways
𝑃(𝑟𝑒𝑑) =
4
11
𝑃(𝑟𝑒𝑑) =
4
̅̅̅
= 0. ̅36
11
4
̅̅̅̅ × 100 = 36. 36
̅̅̅̅%
𝑃(𝑟𝑒𝑑) = 11 = 0. 36
b. What is the ratio of green to non-green pencils
Green : Non-Green = 5 : 6

Provide an example of an event with a probability of 0 or 0% (impossible) and an example of an
event with a probability of 1 or 100% (certain).
Question
A bag contains 1 green, 6 red, 8 black and 10 yellow balls. What is the probability of
choosing a white ball? What is the probability of choosing a ball?
Answer
The probability of choosing a white ball is 0, so there is a 0% chance of choosing a white
ball.
This is an impossible event.
The probability of choosing a ball is 1, so there is a 100% chance of choosing a ball.
This is a certain event.
5. Identify the sample space (where the combined sample space has 36 or fewer elements) for a
probability experiment involving two independent events.

Provide an example of two independent events, such as:
o Spinning a four section spinner and an eight-sided die
o Tossing a coin and rolling a twelve-sided die
o Tossing two coins
o Rolling two dice
and explain why they are independent.
Question
Ex. What is the probability of tossing two coins and having them both show heads?
Using a Tree Diagram:
Coin 1 Coin 2 Outcomes
Heads Heads/Heads
Heads
Tails
Heads/Tails
Heads Tails/Heads
Tails
Tails
Tails/Tails
The probability of both coins showing heads is ¼.
Two events are independent events when one event does not affect the other event.

Identify the sample space (all possible outcomes) for each of two independent events, using a
tree diagram, table or other graphic organizer.
Question
One box on a shelf contains 1 pencil, 1 pen. A second box contains 2 balls and 2 tacks.
Lee randomly chooses an item from each box, how many possible outcomes (sample space)
could occur?
Answer
Shelf 1
Pencil
Pen
Shelf 2
Ball
Ball
Tack
Tack
Ball
Ball
Tack
Outcomes
Pencil/Ball
Pencil/Ball
Pencil/Tack
Pencil/Tack
Pen/Ball
Pen/Ball
Pen/Tack
Tack
Pen/Tack
There are 8 possible
outcomes.
The probability of both coins
showing heads is ¼.
6. Conduct a probability experiment to compare the theoretical probability (determined using a
tree diagram, table or other graphic organizer) and experimental probability of two
independent events.

Determine the theoretical probability of a given outcome involving two independent events.
Question
Find the probability of rolling a even number on a regular 6 sided dice and pulling a black card
from regular deck of 52 cards.
Answer
The two events are independent because one event does not affect the outcome of
another.
Draw a tree diagram to determine the sample space.
Die Cards
Red
1
Black
Red
2
Black
Red
3
Black
Red
4
Black
Red
5
Black
Red
6
Black

Outcomes
1/Red
1/Black
2/Red
2/Black
3/Red
3/Black
4/Red
4/Black
5/Red
5/Black
6/Red
6/Black
There are 12 total outcomes.
There are 6 outcomes that include even numbers.
Of those 6 outcomes, 3 of them include red.
So, rolling an even number and pulling a black card
3
happens 12 times.
Conduct a probability experiment for an outcome involving two independent events, with and
without technology, to compare the experimental probability with the theoretical probability.
Question
Ex. Students were spinning a 3 color spinner. The following data was recorded. Draw a
diagram of the spinner and label appropriately:
Answer:
Green
Red
Since the color red was the most prominent color spun,
it is the largest sector on the spinner.
Blue
Both green and blue will be similar in size because they
have a similar number of outcomes.

Solve a given probability problem involving two independent events.
Question
Ex. Create a tree diagram to determine the probability of flipping heads AND rolling an
even number?
Answer
Colour
Total
Number
of Spins
Red
Green
Blue
20
10
10
40
Event 1:
Flip a coin
START
Event 2:
Roll the die
Possible
Outcomes
The probability of
flipping a ‘heads’ and
rolling ‘an even
1
number’ is 4.