Topics
9. Algorithms
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Australia
Year 6

9.03 Partitioning and ordering

Lesson
Practice
Lesson

Partition

When we partition numbers, we break them into parts. We can use a number expander to help us, just like this one that breaks 423 up into its parts.

Number expander with 4 hundreds, 2 tens, and 3 units.

We can also use a place value table to do the same thing, the choice is yours.

Examples

Example 1

Write 327 in expanded form by following these steps:

a

First, write down the hundreds part.

Worked Solution
Create a strategy

Write the number in a place value table.

Apply the idea

Here is the 327 in a place value table:

HundredsTensUnits
327

We have 3 hundreds which equals 300.

b

Next, write down the tens part.

Worked Solution
Apply the idea
HundredsTensUnits
327

We have 2 tens which equals 20.

c

Now, write down the units (ones) part.

Worked Solution
Apply the idea
HundredsTensUnits
327

We have 7 units which equals 7.

d

Now, write the number in expanded form by adding these three numbers together.

⬚+⬚+⬚

Worked Solution
Create a strategy

Write the numbers found from parts (a) to (c) in each box.

Apply the idea

300+20+7

Idea summary

Partitioning means breaking a number into parts according to each digit's place value.

Algorithms

This word might be new to you, so in the video you'll see how it means that we follow a set of steps. To start out, we'll use an example of getting dressed, then see how we can use an algorithm to sort some coins into order by value.

Loading video...

Examples

Example 2

We want to place these four coins in order by following the algorithm below:

The Image shows Australian 20-cent, 50-cent, 5-cent, and 10-cent coins.

  1. Find the coin with the lowest value.

  2. Place this coin in the first empty position.

  3. Repeat this process for the remaining coins.

a

Following the algorithm, which coin should go in the first position?

A
Australian 50 cent coin
B
Australian 20 cent coin
C
Australian 5 cent coin
D
Australian 10 cent coin
Worked Solution
Create a strategy

Follow steps 1 and 2 of the algorithm.

Apply the idea

Step 1 tells us to find the coin with the lowest value, which is the 5-cent coin. Step 2 of the algorithm tells us that this the coin that should go in the first position.

So the answer is option C.

b

Which coin should go in the second position?

A
Australian 20 cent coin
B
Australian 10 cent coin
C
Australian 5 cent coin
D
Australian 50 cent coin
Worked Solution
Create a strategy

Follow step 3 of the algorithm.

Apply the idea

Step 3 tells us to repeat steps 1 and 2 for the remaining coins.

The remaining coins are the 20,50, and 10-cent coins:

The image shows an Australian 20 cent coin, a 50 cent coin, a crossed out 5 cent coin, and a 10 cent coin.

Step 1 says to find the coin with the lowest value which is the 10-cent coin, and step 2 says to put this in the first empty position, which will be the second position.

So the answer is option B.

c

Which of these options shows the four coins sorted according to the algorithm?

A
An image showing the Australian 20-cent, 50-cent, 5-cent, and 10-cent coins.
B
An image showing the Australian 5-cent, 10-cent, 20-cent, and 50-cent coins.
C
An image showing the Australian 5-cent, 10-cent, 50-cent, and 20-cent coins.
D
An image showing the Australian 50-cent, 20-cent, 10-cent, and 5-cent coins.
Worked Solution
Create a strategy

Repeat the steps of the algorithm excluding the coins found from parts (a) and (b).

Apply the idea
Image of Australian 5-cent and 10-cent coins.

The remaining coins are the 20 and 50-cent coins.

The lowest among them is the 20-cent coin, so this should go in the third position according to steps 1 and 2.

So the algorithm tells us that the order should be:

An image showing the Australian 5-cent, 10-cent, 20-cent, and 50-cent coins.

The answer is option B.

Example 3

We want to order these three digit numbers from smallest to largest by following the algorithm below:

705, \,281, \,106

  1. Order the numbers from the lowest to highest hundreds digit.

  2. Is there more than one number with the same hundreds digit?

    • If no - STOP

    • If yes - Order them from the lowest to highest tens digit.

  3. Is there more than one number with the same hundreds and tens digits?

    • If no - STOP

    • If yes - Order them from the lowest to highest units digit.

  4. STOP

a

List the hundreds digit of each number from smallest to largest.

Worked Solution
Create a strategy

Write the numbers in a place value table.

Apply the idea

Here are the numbers in a place value table:

HundredsTensUnits
705
281
106

Looking at the hundreds, we can see that the hundreds digits in order from lowest to highest are: 1, \,2, \,7.

b

Will the algorithm STOP at step 2?

Worked Solution
Create a strategy

Use the answer from part (a) to follow step 2 of the algoritm.

Apply the idea

Step 2 tells us to STOP if there is not more than one number with the same hundreds digit.

The hundreds digits from part (a) are all different: 1,2,7. So, the algorithm will STOP at step 2.

c

Using your answers above, put the numbers 705,281, and 106 in order from smallest to largest.

Worked Solution
Create a strategy

Write the numbers according to the list of hundreds from part (a).

Apply the idea

Step 1 tells us that we should write the numbers in order of their hundreds digits from lowest to highest:

106, \,281, \,705

Idea summary

An algorithm is a set of steps that tells us what we need to do, and when. It can be used in lots of ways, but we have seen how to use it to sort numbers into different orders.

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