Activities to support Multiplication and division
Where are they now?
Students:
- model equal-sized groups
- count visible groups of items using skip counting to find the total
- calculate the total number of items in an array pattern, using rhythmic counting and skip counting
- know the forward number word sequences when counting in composite units
- know the backward number word sequences when counting in composite units
Where to next?
Students:
- calculate the total number of items when some or all of the groups are concealed
- recall, or easily derive, some multiplication and division facts
- use perceptual markers to represent each groups
- 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:
MA1-1WM describes mathematical situations and methods using everyday and some mathematical language, actions, materials, diagrams and symbols
MA1-2WM uses objects, diagrams and technology to explore mathematical problems
MA1-3WM supports conclusions by explaining or demonstrating how answers were obtained
MA1-6NA uses a range of mental strategies and concrete materials for multiplication and division
How?
How many arrays?
Provide students with a large pile of counters and explain that we are going to investigate how many different arrays we can make that have a total of 24 (for example). Model how to make rows of 1, discussing how the array could be described and how we could work out the total number of counters. Continue investigating arrays for 24, taking photos and documenting the ways students could count to check that there are 24 counters in total. Students could also be supported in using a table to record their investigations, for example:
Teaching Points
Teachers should use this activity to discuss the various ways we can count, supporting students to see how counting in composite units can be a more efficient strategy when compared to counting by ones.
Teachers should also use this to launch investigations about how many arrays can be formed for given numbers and how we explain situations where equal groups cannot be formed. Other aspects such as the relationship between multiplication and division could be explored as well as repeated addition.
Array match
Provide students with a set of matching cards (BLM - Array Match). Students shuffle the cards and then lay them out in 6x4 array. Students take turns to flip over 2 cards, looking for a match. If a match is found, students take the cards and replenish the array from the central pile until no more cards are available. Teachers may choose whether to use a single set of cards or use both sets of cards and leave remaining cards in a central pile for students to replace as they match pairs.
Variations
- Initially, teachers could choose two of the representations for students to match rather than using all 4 corresponding cards
- Students play with all 4 matching cards and score a bonus point for every set of matching cards they are able to collect
- Students score a bonus point if they are able to work out the product for each of the pair they collect
- This could also be played as a version of snap
Arrays in a row
Provide students with one game board (BLM - Array Race), two different coloured counters and three 6-sided numeral dice. Students play in pairs and determine how a game is won, either by completing a vertical, horizontal or diagonal row. Students roll three dice and choose two of the dice to multiply. They then place a counter on that array and must say the total. E.g. 2 threes is 6, or, 2 rows of 3 are 6, for example.
Variations
- Use word cards (e.g. 5 sixes) and numeral cards (e.g. 30) instead of dice
- The student who covers the largest amount of squares wins. As such, students can cover more than one square with their number sentence. For example, if they roll a three and a four, they can cover both the 3 by 4 array and the 4 by 3 array. This aids in seeing the composite nature of multiplication. When playing this way, choosing numbers that make square numbers is not preferable as you can only cover one array
- Use dot arrays (BLM - Array Race Alternative) to support students
Racing back
Provide students with a counter each, two special dice (one red, for example, marked 2, 2, 3, 4, 5, 10. This tells students how many there are in each group. The other, blue for example, marked 1, 1, 2, 2, 3, 3 tells the students how many groups they have), a 120 chart (BLM - 120 Chart), two mini whiteboards and markers. Students start with their counter at 120 and race to be the first to reach 1.
One student starts by rolling the two dice and reading them out aloud to his or her partner. For example, “I have 3 tens”. The student then works out the answer, explain it to their partner who records their strategy. Students should be encouraged to count in multiples to work out the total or record using repeated addition on an empty number line.
The student then moves his or her counter the corresponding number of spaces, using the structure of the 120 chart to move in the most efficient way possible.
Teaching point
Saying and writing multiplication as “3 tens” or “2 fives” encourages multiplicative thinking. Students may still be using repeated addition to solve problems, however, our end goal is to move from additive thinking (10 + 10 + 10) to multiplicative thinking (3 tens).
Fact family jigsaw
Show students a blank fact family jigsaw (BLM - Array Jigsaw) and a 5x10 array. Discuss the different things you can “see”, drawing out both multiplication and division facts, flipping the jigsaw to explore all facts. For example,
Provide pairs of students with their own jigsaw and array patterns to complete. As a class, discuss the various arrays pairs of students examined, exploring the relationship between multiplication and division.
Share it out
Provide students with a collection of counters, a standard 1-6 sided dice to represent the number of groups and a special dice with the numerals 10 – 16 to represent the number of items. Students take turns to roll the dice and use the counters to work out the answer. The first student explains their thinking to a partner who records their work. Any “left-over” counters can be collected by the student. The students swap roles. The student with the largest number of counters at the end is the winner.
Variation
- Change the value of the dice
Hundred chart multiples
Using a hundred chart, ask the students to determine the multiples of a nominated number. This can be achieved through saying the forward number word sequence or using a stressed count for the nominated multiple. If using an interactive Whiteboard, colour in the numerals as the students call them out. Once the sequence of multiples has been identified, have the class repeat the multiples, counting forwards and backwards.
Teaching point
Provide ample opportunities for repeating patterns from the hundred chart, such as counting by fives or tens.
Buzz game
Organise the students in a circle. Have the students begin counting by ones, each taking turns to call out the next number in the sequence. Each time students arrive at a number which is a multiple of five, for example, they call out “buzz” instead of the number. A student who makes an error in counting sits down.
Number line jumps
Provide each student with a mini whiteboard, marker and two dice, one showing the numbers 1, 2, 3, 4, 5 and 6 and another dice showing dot patterns for 2, 5 and 10. Have students roll the dice , and using their whiteboards, work out the answer to their multiplication problem. Encourage students to use repeated addition and counting in multiples to work out a solution, recording their thinking on an empty number line.
Have the students use the calculator to check their counting and then practise counting backwards. On most calculators this can be determined by entering a numeral, pressing the + or - sign twice and then repeatedly pressing the equals sign.
Why?
The strategies students use to solve multiplication and division problems will be dependent upon their knowledge of counting sequences of multiples.
Rolling groups
Provide the students with one numeral dice showing the numbers 1, 2, 3, 4, 5 and 6 and another dice showing dot patterns for 2, 5 and 10. The students will also need a supply of counters. Instruct the students to roll the two dice and construct arrays as indicated by the roll of the dice. The numeral dice indicates the number of rows and the dot dice indicates the number in each row. Students cover the array they’ve made and try to work out the total using rhythmic or skip counting. Have students record their thinking before revealing their array, counting visible items to check.
Variation
- Replace the dice with a pack of numeral cards and a pack of dot cards. Arrange the students into pairs. Instruct one student to choose two cards as instructions for a partner
Teddy tummies
Provide pairs of students with a base board of three teddies (BLM - Teddy Tummies), 30 counters and a recording sheet. Ask the students to place any number of counters onto each of the teddy tummies, concealed from their partner and ensuring each group is equal. Students tell their partner: “I have 4 (for example) counters on each teddy”. The other student has to try to work out the total number of counters used by visualising and skip counting.
The second student tells their partner how many counters they think have been used. The student records their thinking, ensuring they write down how their partner counted. Both students then use counting to check, recording the actual amount. Students swap roles.
Variation
- Increase the number of teddies students use
- Support students in investigating how many ways 30 can be shared into 3 equal groups
- Have students record their work by drawing arrays
Trucking teddies
Complete this activity as for “Teddy tummies” using trucks for a base board (BLM - Trucking Teddies). If available, use plastic teddies instead of counters. For this activity the students will require 40 teddies.
Variation
- Support students in investigating how many ways 40 can be shared into 4 equal groups
Why?
These activities encourage students to visualise groups and use skip counting to work out the product of unseen groups.