At a Glance...

B2.3 Use mental math strategies to multiply whole numbers by 10, 100, and 1000,divide whole numbers by 10, and add and subtract decimal tenths, and explain the strategies used.

Skill: Using Mental Math Strategies to Multiply Whole Numbers by 10, 100 and 1000

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The ability to use mental math to multiply by 10, 100 and 1000 and divide by 10 is based on the 10:1 ratio that exists between place values. In order to represent this ratio, a place-value mat is used.

Multiplying a whole number by 10 can be visualized as a shifting fo the digit(s) to the left by one column on a place-value mat.

For example,

The student can also visualize 3 rods of \(1 \times 10\) to perform the multiplication \(3 \times 10\).

Then the product of 3 × 10 is 30.

Multiplying a whole number by 100 can be visualized as a shifting of the digit(s) to the left by two columns on the place-value mat.

For example,

The student can also visualize 3 flats of \(\ 10\; \times 10\) to perform the multiplication \(\ 3 \times 100\).

Then the product of \(\ 3 \times 100\) is 300.

Multiplying a whole number by 1000 can be visualized as a shifting of the digit(s) to the left by three columns on the place-value mat.

For example,

The student can also visualize 3 large cubes of \(\ 10 \times 10 \times 10\) to perform the multiplication \(\ 3 \times 1\;000\).

Then the product of \(\ 3 \times 1\;000\) is 3000.

Skill: Using Mental Math Strategies to Divide a Whole Number by 10

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The ability to use mental math to divide by 10 is based on the 10:1 ratio that exists between place values. In order to properly represent this ratio, a place-value mat is used.

Dividing a whole number by 10 can be visualized as a shifting of the digit(s) to the right by one column on the place-value mat.

For example,

The student can also visualize 2 rods of \(\ 10 \times 10\) to perform the division \(\ 210 \div 10\). They can decompose 210 into 100 + 100 + 10 and then divide each by 10.

There is a group from \(100 \div 10\;\) to \(\;1 \times 10 = 10\).

There is another group from \(100 \div 10\;\) to \(\;1 \times 10 = 10\).

There is another group from \(10\; \div \;10\;\) à \(\;1 \times 1\; = 1\).

The quotient of the division \(210 \div 10\) is 21.

Students need to learn to master the effect of multiplication and division by 10 and multiples of 10. The explanation of these operations often boils down to the statement, "When you multiply by 10, you add a 0 and when you divide by 10, you take away a 0." This statement is not recommended because it does not take into account the understanding of operations and students are encouraged to apply a mechanical shortcut without being able to make a connection during multiplication or division by numbers such as 10, 100, or 1000.

It is helpful for students to recognize that computations such as \(5\; \times \;10\) ou \(130 \div 10\) can be thought of as \(5 \times 1\) tens; 13 tens ÷ 1 ten. We can then better understand the appearance or disappearance of the digits " 0 " in the computations.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 90-92.

Skill: Using Mental Math Strategies to Add and Subtract Decimal tenths, and Explain the Strategies Used

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To add decimal numbers effectively, students need to understand the place value of the digits that make up each number and factor this into their calculations. Students must also recognize that the decimal point is a marker that identifies the place value of the digits.

When adding decimal numbers, the concept of grouping is used just as when adding whole numbers. For example, just as one can add 3 hundreds to 8 hundreds to form 11 hundreds, one can add 3 tenths to 8 tenths to form 11 tenths. Since the decimal system does not allow two digits to be placed in the same position, students need to understand the concept of regrouping.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 98.

During subtraction, it is important, as it was with addition, to consider the place value of the digits that make up the numbers. The strategies for subtracting decimal numbers are essentially the same as those used for subtracting whole numbers.

The student may conclude that one and one-tenth is the same as eleven-tenths. Using words, it is easier to perform the subtraction of eleven tenths minus eight tenths, which results in 3 tenths.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 101.

Mental Math

Everyday life presents many opportunities to perform computations with decimal numbers. For example, shopping and measuring involve decimal numbers. Estimation skills and mental computational skills are characteristics of number sense and operations sense. A variety of mental computational strategies can be used including rounding, decomposition, and compensation, among others. Here are some examples of their use in mental math situations.

Source: Guide d'enseignement efficace des mathématiques de la 4^e à la 6^e année, p. 91.

Example

Annie wants to know how much wood to buy to build a dog house for her dog. The doghouse measures :

1.2 m in height

1.7 m in length

0.6 m in width

Estimation:

\(\begin{align}1.2 + 1.7 + 0.6 &\approx 1 + 2 + 1 \\ &\approx 4 \end{align}\)

Strategy: Addition Using Compensation

I know that 0.8 + 0.2 = 1 so I mentally move 0.7 and 0.1 to 0.2. I then have 2 ones, 1 one and 5 tenths.

Strategy: Addition Using Decomposition and Compensation

Example

Casimir's dog gave birth to 2 puppies. The vet explains that puppies must weigh at least 24.7 kg when they are six months old to be healthy. Casimir wants to know how much more each puppy should weigh. The first puppy weighs 3.8 kg and the second puppy weighs 4.4 kg.

Estimation:

Puppy 1: \(\begin{align}24.7 - 3.8 &\approx 25 - 4\\ &\approx 21 \end{align}\) Puppy 2: \(\begin{align}24.7 - 4.4 &\approx 25 - 4\\ &\approx 21 \end{align}\)

Strategy: Subtraction Using Decomposition

I mentally break down each term into ones and tenths.

\(\begin{array}{l}24.7 - 3.8 = (23 - 3) + \left( {1.7 - 0.8} \right)\\\quad \quad \quad \quad \;\, = 20 + 0.9\\\quad \quad \quad \quad \;\; = 20.9\;{\rm{kg}}\quad \quad \end{array}\) \(\begin{array}{l}24.7 - 4.4 = (24 - 4) + \left( {0.7 - 0.4} \right)\\\quad \quad \quad \quad \;\, = 20 + 0.3\\\quad \quad \quad \quad \;\; = 20.3\;{\rm{kg}}\quad \quad \quad \end{array}\)

The mass of puppy 1 must increase by 20.9 kg.

The mass of puppy 2 must increase by 20.3 kg.

Strategy: Compensation Using the Number Line

Puppy 1:

I identify 24.7 and 3.8 on the number line and represent the distance between the two numbers with a rectangle. To make subtraction easier, I mentally move my rectangle to the left to 3 instead of 3.8. My rectangle is moved by 0.8 on both sides. Since the size of my rectangle remains unchanged, that is, the gap remains the same, the end of it is now 23.9. My subtraction becomes \(\ 23.9 - 3\).

It is much easier to take 3 ones from 23.9.

\(\ 23.9 - 3 = 20.9\)

The mass of puppy 1 must increase by 20.9 kg.