Activities to support Multiplication and division
Where are they now?
Students:
- count by ones to work out the product (the total when combining equal groups of objects) and does not pay attention to the group structure when counting
- need visual or tactile clues to complete arithmetical tasks
Where to next?
Students:
- make equal groups by using one-to-one dealing
- identify equal groups
- see groups as a countable unit, that is, they count in composite units (for example, counting by twos towork out the product)
- use rhythmic and skip counting to count groups of visible objects
- know the forward number word sequences when counting in composite units
- know the backward number word sequences when counting in composite units
Outcomes
The following activities provide opportunities for students to demonstrate progress towards the following outcomes. A student:
MAe-1WM describes mathematical situations using everyday language, actions, materials and informal recordings
MAe-2WM uses objects, actions, technology and/or trial and error to explore mathematical problems
MAe-3WM uses concrete materials and/or pictorial representations to support conclusions
MAe-6NA groups, shares and counts collections of objects, describes using everyday language, and records using informal methods
How?
Finding equal groups
Demonstrate making equal groups of objects from classroom items. Ask the students the following questions:
- How many objects are in each group?
- How many equal groups are there?
- How many objects are there altogether?
- Are the groups equal? How can we prove it?
Discuss the definition of the words “equal” and “groups”. Give students a selection of photo cards that represent objects organised into equal groups and those that are not. Ask the students to categorise the photos into those that show equal groups and those that do not show equal groups. Ask the students to share their thinking by taking a photo of their sorting to present back to the class. Teachers should support the students in proving how they know that groups are equal and how they know that collections of objects are not equal by using counting, one-to-one matching, comparing and contrasting. Teachers should ensure that images include a range of objects of various sizes, colour and shape.
Trains
Construct train carriages from milk cartons or similar materials. Instruct the students to place equal numbers of Lego® people, or similar items, into each of the carriages. Ask questions similar to those outlined in Finding equal groups (above).
Making groups
Give students a large collection of counters, a mini whiteboard and a marker. Using two dice, explain to the students that one dice (the red one, for example) will tell us how many groups we need to make and the blue dice will tell us how many items we need to place in each group. Roll the first dice and have students make the corresponding number of “groups” by drawing large circles on their whiteboards. Roll the second dice and say how many items we
need in each group. Ask the students to place the corresponding number of counters in each group. Model how to explain what is present on the mini whiteboard, for example:
“I have 3 groups and there are 2 counters in each group. That means that all of my groups are equal because they have the same amount. I can prove that by counting. Let me show you.”
Demonstrate that each group has 2 counters by counting and labelling each collection. Demonstrate how to calculate the total number of counters on the mini whiteboard by counting by twos, explaining:
“Since there are two items in each group, I can use a more efficient counting strategy and count by twos to work out the total number of counters.”
Demonstrate how to count by twos and record the total, reinforcing that the last number word we say tells us the total (which in this case, is called the product as we are counting in composite units).
Variation
- Use sharing mats (BLM - Sharing Mats) to form equal groups by sharing items into equal groups and/or combining them into a total amount
Why?
Students need to develop concepts of making and counting equal groups to solve multiplication and division problems.
Twos
Model the process of counting by twos. Present a pile of counters to the students. Have the students roll the dice to determine how many groups of two need to be made. Then drag two counters at a time from the pile, skip counting by twos to determine the total number of counters. Write a statement to record what took place, e.g. I made 5 groups of two. I counted by twos to work out there are 10 counters in total. Follow the same process to model counting by threes, counting backwards by twos, etc.
Mail sort
Pin a row of four envelopes to a board. Ensure the board allows students easy access as they will need to be able to reach the envelopes to complete this activity. Write a numeral, for example three, on the outside of each envelope in the row. Instruct the students to cut out pictures from magazines which they will use to “post” into the envelopes (alternatively, students could post counters, paddle pop sticks, or other items around the classroom). Students “post” the correct number of items into the envelopes according to the numeral written on the outside. Discuss with the students the number of groups and the total number of items posted. Model methods of counting in multiples using rhythmic or stressed counting in order to ascertain to total number of items.
Teaching point
It is important to reinforce student understanding that the last number word we say tells us the total amount even when we are counting in groups of numbers.
Variation
- Use zip lock bags (reinforced along the edges) to make and display equal groups where students can deposit a particular amount into each bag. Students label each group and the total number of items in the collection
Why?
Students need to view a group of items as one countable item to develop multiplication and division concepts.
Canisters
Provide the students with a group of objects such as feathers, counters or popsticks as well as a given number of canisters (or cups). Ask the students to estimate how many items they could put into each canister to make equal groups and record their estimations. Allow the students to check their estimations by using the concrete materials to solve their problem, sharing out the items into each group. Have the students record their thinking using diagrams or by taking photos, labelling how many items they had, how many groups they had, how many items could be shared into each group, etc. For example:
Teachers may need to record for the student while they explain their thinking.
Variations
- Have the students investigate how many equal groups a particular number can be shared into. For example, how many ways can we share 10 so that we make equal groups? What happens when we share ten into 5 groups? What happens when we share ten into 4 groups? What happens when we share ten in 3 groups?
- Echidnas: Make three or four echidnas from clay or plasticine. Provide the students with a collection of toothpicks. Have the students place equal groups of toothpicks into each echidna. Ask the students to determine the total number of toothpicks, using rhythmic counting
- Kookaburras: provide a collection of feathers. Instruct the students to place equal groups of feathers onto cardboard outlines of birds. Ask the students to determine the total number of feathers, using rhythmic counting
Why?
Students need to develop concepts of making and counting equal groups to solve multiplication and division problems. They also need to know how to identify and describe situations when equal groups can be formed and situations when they cannot be formed.
Rhythmic counting
As a class or in small groups, ask students to collect items to form a specified number (for example, 2). Students gather their objects and sit in a circle. Explain that because everyone has the same number of items in their collection, our groups are equal and so instead of counting by ones, we can count in equal groups to work out the total. Lead the students in counting the total number of objects by counting around the circle by twos, emphasising the multiple count. The teacher may like to show this counting on a hundreds chart. Record the numbers students say on a number line, identifying only those that have been said. If required, support students by whispering the unsaid number.
Why?
The development of counting in multiples supports the understanding of the concepts of multiplication and division.
Variations
- Change the number of items in each collection (for example, have students make groups of 3)
- Start counting from a different amount, for example, show the students that you have one item and so you will start counting on from 1, use a hundreds chart to show the pattern
- Start from the total (20, for example), and count backwards as students remove their objects from view
Body percussion
Using body actions, accentuate the multiple count when finding the total number of specified groups. For example, to stress the count for multiples of three, direct the students to tap their heads for the first count, tap their shoulders for the second count and click their fingers for the third. Then repeat the pattern while counting.
Percussion instruments
This activity is similar to Body percussion. Use percussion instruments to stress the beat and count.
Demonstrate a pattern such as: “Soft, soft, loud, soft, soft, loud.”
After the students have practised, model a counting sequence and then have the class join in the counting sequence.
As the students become competent at rhythmic counting, voice the stressed numbers only in the count. Students could, for example, complete the following pattern. For the first and second count they tap their knees. For the third count they tap their knees and call out “three”. They then continue the pattern, voicing only the numbers which are multiples of three.
Have the students form a double circle, with both circles facing each other. Instruct one circle to stand still and chant a number sequence, accenting the numbers which are multiples of a nominated number.
Have the other circle take sideward steps in one direction to the beat of the count. On the accented count the students who are moving clap hands with the partner opposite at that count.