Stage 2 multiplication and division
- use a variety of mental strategies to solve multiplication and division problems
- find multiples of a number by skip counting or making arrays
- recognise division and multiplication as opposite operations
Activities to support the strategies
Students need to be explicitly taught numerous strategies before they can use multiplication as an operation and recognise division as its opposite operation, for example, skip counting, repeated addition and forming arrays to represent multiplication.
Students need to be aware that there are a number of strategies that they can use to solve multiplication and division questions. More competent students are able to use multiplication to solve a problem and division to double check that same question.
Some activities for developing mental computation in multiplication and division are detailed below.
There is related strategy in the Talking about Patterns and Algebra resource.
Activity 1 - developing mental computation
Students look for different number patterns on a hundreds chart when counting by 3, 4, 5, 6 an so on.
- Use counters to show the patterns. If counting by 4s, for example, students should count 1, 2, 3 and cover the number using a counter, repeating this for a few rows until they can see a pattern and can continue it for the whole chart. Ask students to describe the patterns that have been made, for example, multiples of 3 make a diagonal pattern, multiples of 5 make a vertical pattern.
- Pairs of students are given a number from 2 to 10 and asked to find the pattern for their number on the hundreds chart.
- Students can explore and share their results on the Notebook file patterns of multiples. Click two pages to the screen in Notebook so you can see the questions and the chart at the same time.
Multiplication table square
This activity provides students with a holistic view of multiples (times tables) and assists students in seeing the reciprocal nature of multiples. For example, if I know that 3 x 7 is 21 then I also know that 7 x 3 is 21.
If students are only exposed to ‘times tables’ charts they will see each fact as distinct and a separate piece of knowledge to learn.
When we present multiples in a two-way table, students begin to see the commutative property of multiplication. Give students time to highlight the facts they know, they will soon see there are only a few pockets of facts they need to focus on. This information can be added to over time.
Whisper or stress counting
The teacher leads the class in counting by whispering the numbers not in the sequence and emphasising those that are part of the number pattern.
Skip or rhythmic counting
Students should be given opportunities to hear and say number sequences with lots of body movements to assist, such as claps, finger clicks and slaps and so on.
Multo – using flash cards
- Prepare 100 flash cards with the multiplication facts 1 x 1 through to 10 x 10.
- Students are given a 4 x 4 grid in which they must write 16 different numbers. The winner is the first student to get four numbers in a row, column or diagonal. On completion of a row of 4 the winning student calls out “Multo”!
- As each flash card is shown, students cross off that product from their game boards. The teacher may decide to have the students read the card aloud and say the answer before they check it off. This is a good way to reinforce prior learning.
- At the completion of a game the teacher runs through the flash cards already shown and students again say the question and provide the answer. This is a check that the winner does indeed have a “correct” grid.
- After the game has been played several times students soon discover that this activity differs from “bingo-style” games in that players can increase their chances of winning in several ways. Students work out for themselves, or with a little help from group discussion, that some numbers are “better” than others. Twenty-four is a “good” number because there are four cards which give that product (6 x 4, 4 x 6, 8 x 3, 3 x 8) whereas only one card (5 x 5) will give the answer 25.
- If students investigate this further they may discover that there are 9 “best” numbers having four chances of being drawn (6, 8, 10, 12, 18, 20, 24, 30, 40). Four numbers have three chances: 4, 9, 16, Students may decide to use the results of this investigation when choosing the numbers to place in the grid. Some students place the “best” numbers on the eight squares which occupy diagonals because they say that these squares have three chances of winning, while the other 8 squares have only two chances.
Students spread themselves around the room. Call out a multiplication, such as 4 x 8, and asks a student for a response. If the student correctly answers, they may take one step towards another student and attempt to tag them out of the game. This game can be used for finding the factors of a number or for division facts.
Activity 2 – using array flash cards
Create flash cards of various arrays to represent multiplication facts. In the same way that we use dice patterns as flash cards, show students different arrays very quickly.
- Then ask how many dots did you see?
Allow students time to give various answers, as they answer students need to say how many rows and how many dots in each row.
- Show the card again and discuss the answer.
Students need a visual representation of the fact to create a mental image of what that number looks like. Students may start out by trying to count all the dots individually, as students practice, they will begin to look across and down to work out how many dots per row and how many rows. This will assist students in forming concepts of area. These array patterns can also be used to show commutative properties by turning the cards sideways. Students in Stage 2 need to be able to tell the difference between three rows of six and six rows of three.
You can also extend the use of these cards to show fractions of collections and to explain division and to cut and paste the arrays in different orientations to show factors
Say to the students - under this cover there are 6 cards, and under each card there are 3 dots. How many dots are there altogether?
Students discuss the following strategies that could be used to solve the problem.
- Linking multiplication facts with the division facts - make students aware that their knowledge of times tables is closely related to division, for example, 6 groups of 3 = 18 and 18 divided by 6 = 3.
- Skip Counting – to solve the above problem students could count by threes across the cards.
- Working Backwards – the students are given the final answer and they need to find out how the problem was solved using multiplication or division
Use arrays to solve division as well as multiplication problems, for example, What is 28 divided by 4?
Students use 28 counters to make an array which has 4 counters in each column. Count the number of columns to find the answer: 28 divided by 4 = 7.
Use this 4 x 7 array to solve other multiplication problems.
Cover all but 2 columns.
- How many counters are showing?
- How many counters are covered?
Skip count to find the answer.
Provide all students with a 10 x 10 dot array and two strips of cards. Ask them to make a variety of arrays and work out the answer. Relate these arrays to multiplication facts.
Using the same arrays, ask the students to find factors of a given number of dots.
Students can generate division facts from their arrays as well.
Activity 3 – solving word problems
The teacher demonstrates how to solve a word problem using the Newman’s questions shown below as a guide.
- Students collaboratively work with a partner to solve other problem using these questions as a guide.
Activity 4 – division with remainders
Students in pairs can play a game of Remainders Count. Two students need three 1 to 6 dot dice.
- The first player throws all three dice. The player uses two of the dice to form a two-digit number that can be divided by the remaining dice, for example, the three dice could show a 2, a 6, and a 3. The player could make a 23 and divide it by 6 or a 26 and divide it by 3.
- The first player determines the answer and records the remainder as their score, for example, if the player made a 23 and divided it by 6, the remainder would be 5. However, if they make a 26 and divide it by 3, their remainder would only be 2. The next player has their turn. The aim is to achieve the largest possible remainder. The play continues until someone reaches 20.