Mathematics Developmental Continuum P-10

Early Division Ideas: 2.25

Supporting materials

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Indicator of Progress

Success depends on students being able to identify that the operation of division applies in both partition and quotition situations.

Prior to achieving this indicator, students will recognise division applies to partition (sharing) situations, but may not recognise quotition situations.

Being able to describe a real world situation with the correct mathematical operation(s) is an essential skill for using mathematics. To read about partition and quotition division see:

More About Construction of Number Sentences.

 

Illustration 1: Recognising partition division situations

Partition is the first division situation recognised by students. Students achieving Level 2 can write the number sentence 20 ÷ 4 = 5 to describe partition (sharing) division situations, such as:

  • Matthew, Nathan, Oliver and Pat shared 20 apples, so there were 5 apples for each boy.
  • Twenty books were put on 4 shelves, with an equal number on each shelf. There were 5 books on each shelf.
  • Twenty people were divided into 4 teams. How many people were in each team? (Answer: There were 5 people on each team.)
  • Twenty people sat on four rows of chairs, with the same number in each row, so there were 5 people per row.

Note: As the figure above shows, in a partition division, the total number of items is given, along with the number of equal groups. The answer is the number of items per group. So in the situations above the answers are 5 apples per boy, 5 books per shelf, 5 people per team and 5 people per row.

 

Illustration 2: Recognising quotition division situations

Recognition of quotition division situations follows recognition of partition. A student who understands that division applies to quotition situations can write the number sentence 20 ÷ 4 = 5 to describe situations, such as the following:

  • A fruit shop sold apples in packs of 4. So 20 apples would fill 5 packs.
  • A horse has 4 feet. How many horses can be shod with 20 horseshoes? (Answer: 5 horses)
  • It costs $4 to enter a cat in the cat show. Some people paid $20 in entrance fees. How many cats did they enter? (Answer: They entered 5 cats.)
  • I want to make some 4 m ribbons from a length of 20 m. I can make 5 ribbons.

These quotition situations are often summarised as “How many 4s in 20?”

Note: As the figure above shows, in a quotition division, the total number of items is given, along with the number of items per group. The answer is the number of groups. So in the situations above the answers are 5 packs (i.e. groups of apples), 5 horses (i.e. groups of 4 feet), 5 cats (i.e. groups of $4) and 5 ribbons (i.e. groups of 4 m).

 

Illustration 3: Illustrating division situations with diagrams

An important step towards understanding division is for students to be able to link the real situations to abstract diagrams that illustrate the mathematical structure. For example, students should be able to see that all the situations in Illustration 1 above (which are all partition) can be illustrated by the figure on the left below. Similarly, all the situations in Illustration 2 above (which are all quotition) can be illustrated by the figure on the right below.

 

Diagram illustrating 20 ÷ 4 = 5 as partition Diagram illustrating 20 ÷ 4 = 5 as quotition

20 counters grouped into four groups

20 counters grouped into five groups
There are 20 items and 4 groups, so there are 5 items per group. There are 20 items, and they are grouped into 4s, so there are 5 groups.

 

Illustration 4

Examples of the types of the tasks that would be illustrative of division concepts, aligned from the Mathematics Online Interview:

  • Question 28 - Sharing teddies on mats
  • Question 31 - Teddies at the movies

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Teaching Strategies

Before this stage, division is introduced in partition (sharing) situations. Sharing into equal groups is a familiar situation, and it can be easily undertaken with common materials (e.g. counters etc.). The result of the action can be illustrated with simple diagrams and linked to multiplication facts.

Students working towards Level 3 extend their understanding of division to quotition. They will similarly model it with diagrams, learn to recognise that quotition problems can be expressed with division, and solve it (at this level) with materials or by knowledge of multiplication facts.

Activity 1: Modelling quotition introduces quotition ideas through a story problem, solved with concrete materials, with the solution process recorded diagrammatically and then linked to the known multiplication facts.
Activity 2: A first look at remainders capitalises on students’ understanding of the real situations of the story problems to extend division beyond multiplication facts.
Activity 3: Number line and repeated subtraction provides an alternative pictorial model for quotition division, which is easier to generalise than discrete concrete materials when moving beyond whole numbers.

NOTE TO TEACHERS:

Each of the four basic operations of arithmetic applies to real world situations that differ in context (whether the story is about lollies or pets, for example) and which also differ in logical structure. In preparing to use mathematics in their everyday life, students must encounter many situations with different logical structures. The main logical structures are described here:

More About Construction of Number Sentences.

When students learn about division and practise doing division, sets of examples should include a range of story problems, so that students work with both partition and quotition structures.

Teachers need to be aware of the different types of division, so they can present students with both structures. Students should be able to solve both partition and quotition problems by division, but they do not need to know the terms partition and quotition.

Examples of suitable story problems are given in the Illustrations above as well as here:

More About Construction of Number Sentences.

In each case, students should write the division number sentence as part of the solution.

Beyond Level 3, students will also use division in situations that are neither partition nor quotition. For example, a common situation involves multiplicative comparison situations such as “I eat three times as much as she does” or “I earn 20% more than she does”.

 

Activity 1: Modelling quotition

Step 1: Select a word problem solved by quotition division, such as one of the examples below.

  • A fruit shop sold apples in packs of 4. How many packs can be filled with 20 apples? (Answer: 5 packets)
  • A horse has 4 feet. How many horses can be shod with 20 horseshoes? (Answer: 5 horses )
  • It costs $4 to enter a cat in the cat show. Some people paid $20 for their cats. How many cats did they enter? (Answer: They entered 5 cats)
  • I want to make 4 m ribbons from a length of 20 m. How many ribbons can I make? (Answer: 5)

Step 2: Demonstrate how to solve the problem (e.g. 20 apples in packs of 4) with concrete materials and how to record the process with a diagram.

  • Select 20 counters to represent apples. Take out one group of 4 representing a pack, then take out another group of 4 and continue until there are no counters left. (Note for teachers: quotition has this name because there is a ‘quota” of 4 apples per pack. Also note the ‘repeated subtraction’ action here.)
  • Count the number of groups of counters (5).
  • Interpret in the context of the problem: 20 apples can fill 5 packs.
  • Draw a simple diagram. Ask students to identify the 5 packs of apples.
  • Annotate the record with the number sentence and a verbal description.

There are 4 apples in each pack. 20 apples can fill 5 packs. 20 ÷ 4 = 5

Step 3: Ask students what they notice about the result and how it links to other things that they know. Help students make the link with the known multiplication fact 5 × 4 = 20. Ask students to point out the 5 groups of 4 apples, which makes 20 apples altogether, thereby explaining their results. Write the multiplication fact on the written record.

Step 4: Choose a range of quotition problems with different contexts and different numbers, and ask students to solve them using concrete materials and diagrams, and to make the link to the known multiplication facts as above.

Step 5: As students consolidate this knowledge, they can omit the concrete materials and the drawings, but they should still link their answers explicitly to the known multiplication fact.

Step 6: Later, include a mix of partition and quotition problems, using concrete materials where required, drawing diagrams and linking to the multiplication fact.

 

Activity 2: A first look at remainders

The story problems used in Activity 1 easily lead to division with remainders. Students can see this in an informal way at this level, and just record the number left over.

For example, change the problem:

A fruit shop sold apples in packs of 4. How many packs can be filled with 20 apples?

to:

A fruit shop sold apples in packs of 4. How many packs can be filled with 21 apples?

Following the same procedure of concrete materials and diagrams, students will see that there are 5 full packs of apples and one apple is left over. Record this as:

21 ÷ 4 = 5 rem 1

Later the answer can be recorded as:

21 ÷ 4 = 5 1/4

This means that there are 5 1/4 packs of apples (which makes sense because the 1 apple left over is a quarter of a full pack). Many students think that the remainder means a quarter of an apple, rather than a quarter of a pack of apples.

 

Activity 3: Number line and repeated subtraction

Students can illustrate quotition division on a number line, following the same general steps as in Activity 1. The links to repeated subtraction can also be seen this way. The number line can be abstract (numbers only as here) or in a context (e.g. 20 means 20 metres or 20 dollars etc).

Examples:

How many 4s in 20? Start at 20 on the number line then move down in steps of 4, until you reach 0. There are 5 steps so there are 5 fours in 20.

Number line and repeated subtraction
How many 3s in 18? Start at 18 on the number line then move down in steps of 3, until you reach 0.  There are 6 steps so there are 6 threes in 18.

Number line and repeated subtraction
How many 3s in 20? Start at 20 on the number line then move down in steps of 3, until you reach 2.  It is not possible to go any further in steps of 3. There are 6 threes in 20, and 2 left over.

Number line and repeated subtraction

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Further Resources

The following resource contains sections that may be useful when designing learning experiences:

Digilearn object *

Dividing it up: grouping tool – students use a dividing tool to make equal shares of stationery such as pens, pencils or crayons. Students complete a sentence describing a number operation. For example, pack 24 crayons into packets of 5. Students predict how many packets are needed and identify how many items are left over.
(https://www.eduweb.vic.gov.au/dlr/_layouts/dlr/Details.aspx?ID=4264)

Dividing it up: hardware – students use a dividing tool to make equal shares of hardware items such as nails, bolts or screws. For example, pack 32 bolts into packets of 3. Students predict how many packets can be filled and how many items will be left over. Students check their prediction. Students complete a sentence describing the number operations, including the fraction of a packet remaining.
(https://www.eduweb.vic.gov.au/dlr/_layouts/dlr/Details.aspx?ID=4263)

* Note that Digilearn is a secure site; SMART::tests login required.