Math 5
Unit 1
Lesson 4
Mental Math
Playing Solitaire
A popular game on a computer or with cards is called Solitaire. You make
patterns with the cards that are based on number and colour. Here you
can see that the pattern is descending.
There are all kinds of number patterns. They can increase or
decrease. You can use these patterns to come up with easy
ways to do math computations.
Reflection
How do you multiply numbers in your head?
Math 5
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Math 5
Unit 1
Lesson 4: Mental Math
Objectives for this Lesson
In this lesson you will explore the following concept:
• A
pply mental mathematics strategies to number properties
and multiplication
Sometimes you just need to get the answer fast. Want to learn some
tricks that will help you multiply numbers fast? These tricks will also help
you maintain your basic facts.
Skip Counting
This method is great for counting many items quickly or working
multiplication facts.
You should remember how to count by two. Counting by 2 is a great
shortcut to count things.
Cameron and Zach each have a stack of pennies to count. They both
have the same amount, but they need to know how many. If Zach counts
his pennies one at a time (1, 2, 3 …) and Cameron counts his two at a
time (2, 4, 6 …) who would finish first?
Cameron’s results:
2
1-34
4
6
Zach’s results:
8
10
12 ...and so on...
1 2 3 4 5 6 ...and so on...
Math 5
Unit 1
Lesson 4: Mental Math
Skip counting may be done by any grouping. You may remember some
of the more common ones:
Skip Counting
... by two
2, 4, 6, 8, 10, 12,...
... by five
5, 10, 15, 20, 25,...
... by ten
10, 20, 30, 40, 50,...
For working with larger multiplication problems you may want to be
prepared to count by larger numbers:
Skip Counting
... by seven
7, 14, 21, 28,...
... by eight
8, 16, 26, 32,...
... by twelve
12, 24, 36, 48,...
Now let’s use skip counting to complete some patterns.
Now It’s Your Turn
Complete the following on your own paper:
a. Use skip counting by seven to complete:
49
91
112
b. Use skip counting by eight to complete:
96
136
168
c. Use skip counting by fifteen to complete:
45
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Math 5
Unit 1
Lesson 4: Mental Math
Solutions:
a.
b.
c.
49
56
63
70
77
84
91
98
105
112
96
104
112
120
128
136
144
152
160
168
45
60
75
90
105
120
135
More Skip Counting
How can skip counting help you with math problems?
Let’s say you’re stuck with a problem in multiplication like:
8x4
You’ve forgotten the answer and you need to do this fast. Remember
back in first grade when you would use your fingers to count to 10? Now
you can use them to count by other numbers. If you can skip count by
four, then you can solve this problem by skip counting by four, 8 times, to
get the answer:
4, 8, 12, 16, 20,
24, 28, 32
32!
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Math 5
Unit 1
Lesson 4: Mental Math
Track the number of 4s on your fingers (in this case up to 8). When you
get to eight you can stop counting by four and you have your answer!
By skip counting with the larger number you can complete the problem
even faster:
8 x 4Skip count by 8, four times: 8, 16, 24, 32
The answer is 32.
Example 1
Use the mental math strategy of skip counting to solve.
15 x 4
Skip count by 15, four times. The fourth number you get will be
the answer:
15, 30, 45, 60
Another way of doing this is to skip count by 4, fifteen times.
The fifteenth number you get will be the answer.
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60
Most of your multiplication facts can be done in this way. Now try the
following using this skip counting method.
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Math 5
Unit 1
Lesson 4: Mental Math
Let’s Practice
• In your Workbook go to Unit 1, Lesson 4 and complete 1 to 13.
Doubles
Another trick that will help you to be fast at multiplication is using
doubles. The doubles facts for multiplication are very simple to learn.
2 x 2, 3 x 3, 4 x 4, 5 x 5, 6 x 6, 7 x 7, 8 x 8, 9 x 9.
2 x 2 = 4
3 x 3 = 9
4 x 4 = 16
5 x 5 = 25
6 x 6 = 36
7 x 7 = 49
8 x 8 = 64
9 x 9 = 81
Once you have them committed to memory you can use them to find
other multiplication facts.
If you know that: 7 x 7 = 49 and you want to find 7 x 8 = ?
Remember: 7 x 7 = 7 + 7 + 7 + 7 + 7 + 7 + 7
That means that: 7 x 8 = 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7
So, just add 7 to 49 to get 56.
That is one way you can use doubles to help you find the answer to
multiplication facts.
Example 2
Use the mental math strategy of doubles to solve.
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8x9
Math 5
Unit 1
Lesson 4: Mental Math
You know that 8 x 8 = 64 since it is a double.
To get 8 x 9 you need to count up by 8 more.
65, 66, 67, 68, 69, 70, 71, 72
8 x 9 = 72
Example 3
Use the mental math strategy of doubles to solve.
7x6
You know that 6 x 6 = 36 since it is a double.
To get 7 x 6 you need to count up by 6 more.
37, 38, 39, 40, 41, 42
7 x 6 = 42
Let’s Explore
Exploration 1: Patterns of Multiplication
Materials: Unit 1, Lesson 4, Exploration 1 page from your Workbook, 8 pencil crayons
Use the Hundreds Charts found on the Exploration page in your
Workbook. Use a different chart for each set of facts:
1. Shade the answers to the 2s facts from 2 x 1 to 2 x 9 using one colour.
2. Extend the pattern you see to all numbers on the chart.
3. Shade the answers to the 3s using another colour.
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Math 5
Unit 1
Lesson 4: Mental Math
4. Extend the pattern you see to all numbers on the chart.
5. Shade the answers to the 4s using another colour.
6. Extend the pattern you see to all numbers on the chart.
7. Shade the answers to the 5s using another colour.
8. Extend the pattern you see to all numbers on the chart.
9. Shade the answers to the 6s using another colour.
10.Extend the pattern you see to all numbers on the chart.
11.Shade the answers to the 7s using another colour.
12.Extend the pattern you see to all numbers on the chart.
13.Shade the answers to the 8s using another colour.
14.Extend the pattern you see to all numbers on the chart.
15.Shade the answers to the 9s using another colour.
16.Extend the pattern you see to all numbers on the chart.
Halving and Doubling
The halving and doubling strategy can be used when multiplying.
This method involves taking half of one number and doubling the
other to make a multiplication problem that you remember. This is a
great strategy when multiplying by 5 but it can also be used with
other numbers.
The process for halving and doubling: Sample Problem: 14 x 5
1. Double one number:
5 doubled is 10
2. Take half of the second number:
half of 14 is 7
3. Multiply:
10 x 7 = 70
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Therefore, 14 x 5 = 70
Math 5
Unit 1
Lesson 4: Mental Math
If you know that 3 x 10 = 30, then you can halve 6 and double 5 to find
that 6 x 5 = 30.
Example 4
Use the mental math strategy of halving and doubling to solve.
6x5=?
6
x5
Half of 6
5 doubled
3
x 10
30
6 x 5 = 30
Let’s Practice
• In your Workbook go to Unit 1, Lesson 4 and complete 14 to 28.
Annexing
To solve a problem like 70 x 5, you can use a method called annexing.
Since 70 is a multiple of 10, you can ignore the 0 in the number and
break the problem into two parts:
1: Drop the zero from the number and multiply.
7 x 5 = 35
2: Add the zero back to the end of your answer.
350
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Math 5
Unit 1
Lesson 4: Mental Math
Example 5
Use the mental math strategy of annexing to solve.
800 x 6
1. Drop the zeros from the 800 and multiply:
8 x 6 = 48
2. Now add the zeros back to the end:
4800
800 x 6 = 4 800
Using Distributive Property
When problems have two digit numbers you may feel like you are starting
something new. As long as you know your multiplication facts to 10 you
are just repeating the use of these facts.
You learned to break a number into expanded notation using place value.
You can use that understanding with multiplication.
Example 6
Use the mental math strategy of distributive property to solve.
426 x 4
1. Write 426 in expanded notation: 400 + 20 + 6
2. Multiply 4 by each number in the sum (400 x 4) + (20 x 4) + (6 x 4)
Use annexing: 4 x 4 = 16 add two zeros: 1600
4 x 2 = 8 add one zero: 80
3. Add the results of each: 1600 + 80 + 24 = 1 704
1-42
426 x 4 = 1 704
Math 5
Unit 1
Lesson 4: Mental Math
Example 7
Use the mental math strategy of distributive property to solve.
653 x 8
1. Write 653 in expanded notation: 600 + 50 + 3
2. Multiply 8 by each number in the sum: (600 x 8) + (50 x 8) + (3 x 8)
Annexing: 8 x 6 = 48 add two zeros; 4 800
8 x 5 = 40 add one zero; 400
3. Add the results of each: 4 800 + 400 + 24 = 5 224
653 x 8 = 5 224
Patterns with Nine
If the patterns with nine have given you trouble in memorizing your facts
you are going to love this trick.
Think of the groups of nine:
1 group of 9 is 9 1 x 9 = 9
2 groups of 9 is 18 2 x 9 = 18
3 groups of 9 is 27 3 x 9 = 27
4 groups of 9 is 36 4 x 9 = 36 … and so on.
Now, look at the pattern in the results. The tens digit of the answer is
one less than the number multiplied by 9 and the ones digit plus the tens
digit equals 9. Let’s try it on the next one:
5 x 9 = 45
4+5=9
Tens digit: 4
Ones digit: 5
Sum = 9
You can use these to quickly help you work harder problems like a 3-digit
number times a 1-digit number.
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Math 5
Unit 1
Lesson 4: Mental Math
Example 8
Use the mental math strategy of distributive property and patterns with
nine to solve.
592 x 9
Use distributive property:
1. Write 592 in expanded notation; 500 + 90 + 2
2. M
ultiply each by 9; (500 x 9) + (90 x 9) + (2 x 9)
Don’t forget to use your trick!
9 x 5 = 45, 9 x 9 = 81 and 9 x 2 = 18
3. A
dd 4 500 + 810 + 18 = 5 328
592 x 9 = 5 328
Now It’s Your Turn
Use the mental math strategies of annexing, distributive property, and
patterns with nine to solve.
a. 38 x 5
b. 70 x 4
c. 21 x 5
d. 870 x 2
Solutions
a. 190
b. 280
c. 125
d. 1 740
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Math 5
Unit 1
Lesson 4: Mental Math
Let’s Practice
o online to watch the Notepad Tutor: Mental Math – Distributive
G
Property.
• In your Workbook go to Unit 1, Lesson 4 and complete 29 to 39.
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Math 5
Unit 1
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Lesson 4: Mental Math