Stage 3

3.08 Modelling division

Lesson

Ideas

Are you ready?
Division review of half and half again
Division review in other methods

Are you ready?

There are some great strategies for division , that we've used for smaller numbers, up to 3 digits long. How many can you remember?

Examples

Example 1

Find 769\div3 by doing the following:

Find 600\div 3.

Worked Solution

Create a strategy

Write 600 as a multiple of 100.

Apply the idea

\displaystyle 600\div3	\displaystyle =	\displaystyle 100\times6\div3	Rewrite 600 as 100\times6
	\displaystyle =	\displaystyle 100\times2	Divide 6 by 3
	\displaystyle =	\displaystyle 200	Multiply 100 by 2

Find 150\div3.

Worked Solution

Create a strategy

Write 150 as a multiple of 10.

Apply the idea

\displaystyle 150\div3	\displaystyle =	\displaystyle 10\times15\div3	Rewrite 150 as 10\times15
	\displaystyle =	\displaystyle 10\times5	Divide 15 by 3
	\displaystyle =	\displaystyle 50	Multiply 10 by 5

Find 18\div3.

Worked Solution

Create a strategy

Rewrite the division as a multiplication.

Apply the idea

18\div3 can be written as 3 \times ⬚ =18. Since 3\times 6 = 18:

18\div3=6

Using the fact that 769=600+150+18+1, complete the statement with the missing numbers:

3 goes into seven hundred sixty nine ⬚ times with a remainder of ⬚.

Worked Solution

Create a strategy

Divide both sides of the number sentence by 3.

Apply the idea

\displaystyle 769 \div 3	\displaystyle =	\displaystyle 600 \div 3 +150\div 3+18\div 3+1\div 3	Divide all the numbers by 3
	\displaystyle =	\displaystyle 200+50+6 + 1 \div 3	Use the previous results
	\displaystyle =	\displaystyle 256+1\div 3	Add the whole numbers

Since we can't divide 1 by 3, \,\,\, 1 must be the remainder. So:

3 goes into seven hundred sixty nine 256 times with a remainder of 1.

Idea summary

The part of a number that cannot be divided into equal groups is called the remainder.

Division review of half and half again

Dividing by one number can often help us when we need to divide by a different number, as we see here.

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Examples

Example 2

Find the value of 40\div4.

Worked Solution

Create a strategy

We halve both numbers to make it easier to divide the numbers.

Apply the idea

Divide each number by 2.

40\div 2 = 20

4\div 2 = 2

By the half and half again method we can use these results in our division.

\displaystyle 40\div 4	\displaystyle =	\displaystyle 20 \div 2	Use the results from halving
	\displaystyle =	\displaystyle 10	Divide 20 by 2

Idea summary

To make a division easier, we can divide both numbers by 2.

We can also think of dividing by 4 as dividing by 2 twice.

Division review in other methods

Dividing by other digits means we can use things like partitioning or the area model to help us.

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Examples

Example 3

Let's use an area model to find the answer to 133\div 7.

We set up the area model using a rectangle like this:

A rectangle with a total area of 133 and a width of 7.

Now if we don't know straight away what 133\div 7 is, we start with something we do know, like groups of 10.

Find the area used so far if we take out 10 groups of 7.

A rectangle with a total area of 133 and a width of 7 divided into 2 rectangles. Ask your teacher for more information.

Worked Solution

Create a strategy

"Groups of" means multiply.

Apply the idea

10 groups of 7 can also be written as: 10\times 7.

\displaystyle \text{Area of left rectangle}	\displaystyle =	\displaystyle 10\times7	Multiply the sides
	\displaystyle =	\displaystyle 70

How much area is remaining?

Worked Solution

Create a strategy

The area remaining is the area of the left rectangle. Subtract the area found in part (a) from the total area of the rectangle.

Apply the idea

\displaystyle \text{Area of left rectangle}	\displaystyle =	\displaystyle 133-70	Subtract the areas
	\displaystyle =	\displaystyle 63

The remaining area is 63.

A rectangle divided into 2 rectangles with areas of 70 and 63. Ask your teacher for more information.

What is the width of the second rectangle?

Worked Solution

Create a strategy

Use the area of the rectangle found in part (b).

Apply the idea

We know that the area of the second rectangle is 63 and the height is 7. So the width will be the area divided by the height, or 63\div7.

\displaystyle 63\div 7

\displaystyle =

\displaystyle 9

Divide 63 by 7

The width of the second rectangle is 9.

Using the area model above, what is 133\div7?

Worked Solution

Create a strategy

133\div7 is the total width of the rectangle in part (c). So we need to add the widths.

Apply the idea

\displaystyle \text{Total width}	\displaystyle =	\displaystyle 10+9	Add the widths
	\displaystyle =	\displaystyle 19

133\div7=19

Idea summary

Multiplication strategy	Division strategy	Example	Calculation
\text{repeated addition}	\text{repeated subtraction}	32\div 8	\text{subtract } 8 \text{ from } 32 \\\text{until we get to zero}
\text{double-double}	\text{half-half}	32\div 4	\text{half of } 32, \text{ and then} \\\text{half of that}
\text{partitioning} \\\text{by place value}	\text{partitioning} \\\text{by place value}	230\div 5	\text{split } 230 \text{ into } \\200+30,\\ \text{and divide each } \\\text{part by } 5 \text{ separately}
\text{splitting number}	\text{splitting number}	56\div8	\text{split } 56 \text{ into } \\40+16,\\ \text{and divide each } \\\text{part by } 8 \text{ separately}

Outcomes

MA3-6NA

selects and applies appropriate strategies for multiplication and division, and applies the order of operations to calculations involving more than one operation

3.08 Modelling division

Ideas

Are you ready?

Examples

Example 1

Division review of half and half again

Examples

Example 2

Division review in other methods

Examples

Example 3

Outcomes

MA3-6NA

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