This document provides examples and explanations of factors, multiples, common factors and multiples, prime numbers, and using area models and the distributive property to multiply polynomials. It includes examples of adding, subtracting, and multiplying polynomials. Check your understanding questions are provided to have students practice these skills.

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Name: ___________________________________ Date: __________________________
Master 3.1a Activate Prior Learning:
Factors and Multiples
The multiples of a number are determined by multiplying the number by 1, 2, 3, 4, and so on,
or by skip counting.
For example, the multiples of 12 are: 12, 24, 36, 48, ...
Multiples that are the same for 2 numbers are common multiples.
• To determine the first 3 common multiples of 4 and 6:
The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …
The multiples of 6 are: 6, 12, 18, 24, 30, 36, …
12, 24, and 36 appear in both lists.
So, 12, 24, and 36 are the first 3 common multiples of 4 and 6.
A factor is a number that divides exactly into another number.
For example, 1, 2, 3, 4, 6, and 12 are the factors of 12.
Each number divides into 12 with no remainder.
Factors that are the same for 2 numbers are common factors.
A prime number is a number with exactly 2 factors, 1 and itself.
• To determine the factors of 40:
40 ÷ 1 = 40 1 and 40 are factors.
40 ÷ 2 = 20 2 and 20 are factors.
40 ÷ 4 = 10 4 and 10 are factors.
40 ÷ 5 = 8 5 and 8 are factors.
The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, and 40.
Some of the factors are prime numbers.
The factors of 40 that are prime numbers are 2 and 5.
Check Your Understanding
1. List the first 6 multiples of each number.
a) 5 b) 11 c) 18 d) 25
2. Determine the first 3 common multiples of each pair of numbers.
a) 2 and 5 b) 3 and 9 c) 7 and 3 d) 8 and 10
3. Determine the factors of each number. List the factors that are prime numbers.
a) 15 b) 20 c) 24
d) 45 e) 60 f) 100
4. Determine the common factors of each pair of numbers.
a) 16 and 24 b) 15 and 45 c) 18 and 42 d) 20 and 30
Master 3.1a 37
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2. Name: ___________________________________ Date: __________________________
Master 3.1b Activate Prior Learning: Using an Area
Model to Multiply
We can use an area model to represent the multiplication of whole numbers.
For example, 3 × 4 can be represented as
3 groups of 4.
So, 3 × 4 = 12
We can use Base Ten Blocks to multiply.
• To multiply: 3 × 14
Arrange 3 rows of 1 ten and 4 ones.
Multiply the tens. 3 × 10 = 30 Multiply the ones: 3 × 4 = 12
Add: 30 + 12 = 42
So, 3 × 14 = 42
We can use a rectangle model to multiply.
• To multiply: 21 × 12
Write each factor in expanded form: (20 + 1) × (10 + 2)
Draw a rectangle with side lengths 20 + 1 and 10 + 2.
Divide the rectangle into 4 smaller rectangles.
Determine the area of each smaller rectangle.
The product is the sum of the areas.
So, 21 × 12 = (20 × 10) + (20 × 2) + (1 × 10) + (1 × 2)
= 200 + 40 + 10 + 2
= 252
This model illustrates the distributive property.
Check Your Understanding
1. Use an area model to determine each product.
a) 5 × 13 b) 7 × 18 c) 14 × 15 d) 21 × 17 e) 25 × 22
38 Master 3.1b
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3. Name: ___________________________________ Date: ________________________
Master 3.1c Activate Prior Learning:
Polynomials
To add polynomials, group like terms, then combine the terms by adding their coefficients.
• To add: (8x + 7) + (3x2 + 2x – 3)
(8x + 7) + (3x2 + 2x – 3) Remove the brackets.
= 8x + 7 + 3x2 + 2x – 3 Collect like terms.
= 3x2 + 8x + 2x + 7 – 3 Combine like terms.
= 3x2 + 10x + 4
To subtract polynomials, use the properties of integers.
Subtracting an integer is the same as adding the opposite integer.
So, to subtract a term, add the opposite term.
• To subtract: (3n2 + 7n) – (2n2 – 4n)
(3n2 + 7n) – (2n2 – 4n) Subtract each term.
= 3n2 + 7n – (2n2) – (–4n) Add the opposite term.
= 3n2 + 7n – 2n2 + 4n Collect like terms.
= 3n2 – 2n2 + 7n + 4n Combine like terms.
= n2 + 11n
To multiply a polynomial by a monomial, use the distributive property to multiply each
term in the polynomial by the monomial.
• To multiply: 4c(–3c + 4)
4c(–3c + 4) = 4c(–3c) + 4c(4) Multiply.
= –12c2 + 16c
• To multiply: –3(–4y2 + y – 7)
–3(–4y2 + y – 7) = (–3)(–4y2) + (–3)(1y) + (–3)(–7)
= 12y2 – 3y + 21
Check Your Understanding
1. Add or subtract.
a) (6x + 3) + (2x + 5) b) (2x2 + 6x – 5) + (–4x2 – 3x + 7)
c) (5a – 8) – (2a + 3) d) (3a2 – 2a + 6) – (–2a2 + 7a – 9)
2 2
e) (–7 + 3d – 2d) + (8 – 4d + 3d)
f) (5e – 9 + 2e2) – (2e2 – 9 + 5e)
g) (10v – 5v2 – 2) + (3v – 7v2 – 1)
h) (m – 3m2 – 5) – (3m2 + 5 – m)
2. Multiply.
a) 2w(4w + 5) b) 5(3w2 – 4w – 9)
c) 3s(–3 – 5s) d) –4(9u2 – 2u + 5)
e) –5(3v – v2 – 3) f) –2z(7 – 3z)
Master 3.1c 39
The right to reproduce or modify this page is restricted to purchasing schools.
This page may have been modified from its original. Copyright © 2010 Pearson Canada Inc.