Name: ____________________________________________
Study Guide for Math Enrichment 5th- 8th
Model Multiplication
There are several ways to solve multiplication problems. One way is to consider multiplication as repeated
addition.
Example
What is the product of 5 × 4?
Solution
Multiplying 5 by 4 is the same as adding together 5 groups of 4.
5×4
= 20
4 + 4 + 4 + 4 + 4 = 20
Models can be used to solve multiplication problems. Models help by showing the number of groups and
number of objects in each group.
Example:
The two grids below each have 32 squares.
4 × 8 = 32
8 × 4 = 32
The first grid shows four groups of eight squares.
The first grid represents the number sentence, 4 × 8 = 32.
The factors are 4 and 8. The product is 32.
The second grid shows eight groups of four squares.
The second grid represents the number sentence, 8 × 4 = 32.
The factors are 8 and 4. The product is 32.
Converting word problems into expressions can be helpful for problem solving. Locate the information that is
needed, as well as pick the operations that will be used to solve.
*There are several ways to solve division problems.
One way is to consider division as repeated subtraction.
Example 1: What division problem is the same as 49 - 7 - 7 - 7 - 7 - 7 - 7 - 7 ?
Solution: Seven is subtracted from 49 seven times. So, the division problem is 49 ÷ 7 = 7.
UNDERSTAND THE PROBLEM
1. Read the problem carefully. Do I understand what the problem is asking?
2. Picture the problem. Can I draw or make a model of it?
SOLVE THE PROBLEM
3. Think about what you are being asked. Is my answer going to be large or small?
4. Decide what operation to use. Do I add, subtract, multiply, or divide?
REVIEW YOUR ANSWER
5. Re-read the problem and your answer. Does my answer make sense with the problem?
6. Check your work. Did I calculate (add, subtract, multiply, or divide) correctly?
Example 1:
Keisha has 4 boxes in her closet.There are 9 sweatshirts in each box. How many sweatshirts does Keisha have
in the boxes?
Solution:
Multiply the number of boxes, 4, by the number of sweatshirts in each box, 9.
4 × 9 = 36 sweatshirts
To solve a number sentence, get the symbol by itself on one side of the equal sign. Remember to perform the
same operation on both sides to keep the equation balanced.
Example 1:
Which number for ? makes the sentence true?
? × 4 = 12
Solution:
Remember, division is the opposite of multiplication. So, to solve this problem, divide both sides by 4.
?×4
= 12
? × 4 ÷ 4 = 12 ÷ 4
?
=3
Finally, substitute ? = 3 into the original number sentence to check the answer.
The commutative property of multiplication states that numbers can be multiplied in any order without affecting the
results.
Examples:
2 × 3=6
3 × 2=6
5 × 9=45
9 × 5=45
The associative property of multiplication states that numbers to be multiplied can
be grouped in any way without affecting the results.
Examples:
6
(3 × 2) × 1
=× =6
1
Distributive property states that multiplying a sum (or difference) by a
number is the same as multiplying each number of the sum (or
difference) by the number and then adding (or subtracting) the products.
Examples:
3 × (2 + 1) = (3 × 2) + (3 × 1) = 6 + 3 = 9
5 × (7 - 4) = (5 × 7) - (5 × 4) = 35 - 20 = 15
Multiplication and division are opposite operations.
6 × 4 = 24 and 24 ÷ 6 = 4
Example
8 × 6 = 48
Using the multiplication equation above, solve the division problem below.
48 ÷ 8 = ???
Solution
Since multiplication and division are opposite operations, 48 ÷ 8 = 6. Multiplication and division fact families
are multiplication and division facts which are related to each other.
Example Fact Families
3 × 5 = 15
5 × 3 = 15
15 ÷ 3 = 5
15 ÷ 5 = 3
9 × 7 = 63
7 × 9 = 63
63 ÷ 9 = 7
63 ÷ 7 = 9
8 × 10 = 80
10 × 8 = 80
80 ÷ 8 = 10
80 ÷ 10 = 8
Multiplication can be used to solve many kinds of problems.
For example, it is helpful when solving problems where there are groups of objects.
Example:Patrick has several shapes on his desk.
Other than just counting, how could Patrick determine the number of shapes on his desk?
Solution:
Start by separating the shapes by type.
There are seven circles, seven squares, seven diamonds, and seven pentagons.
Patrick has 4 types of shapes and there are 7 of each shape.
The total number of shapes could be obtained by addition 7 + 7 + 7 + 7 = 28, or by multiplication 4 × 7 = 28.
DIVISION is the inverse (opposite) operation of multiplication, so that if a × b = c then a = c ÷ b.
Ways to write c divided by b.
. In the equation a = c ÷ b, a is the quotient, c is the dividend,
and b is the divisor quotient = dividend ÷ divisor or
Converting word problems into number sentences can be helpful for problem solving.
Locate the information that is needed, as well as pick the operations that will be used to solve.
Example 1:
Jeremy went to the Die-Cast Car collectors convention and bought five cars for five dollars each. He then
bought one car for six dollars. Use the number line above to find which equation below can be used to find how
much Jeremy spent on the cars.
A. 5 + $5 × $6 = t
B. 5 × $5 × $6 = t
C. 5 × $5 + $6 = t
D. 5 + $5 + $6 = t
Solution: Start by finding how much Jeremy spent on the five cars he bought for five dollars. Skip count by 5s
five times.
This expression would be 5 × $5. Next, add the $6 spent on one car.
Put all of the information together into one equation. Use a t to represent the total amount Jeremy spent.
B. 5 × $5 + $6 = t