# Fact Families (Multiplication and Division): Level 3

## Indicator of progress

Success depends on students knowing that once they can recall a particular multiplication fact, they can use that fact to solve related multiplication and division tasks. The set of related facts is called a fact family.

A sample fact family:

 7 × 5 = 35 5 × 7 = 35 35 ÷ 5 = 7 35 ÷ 7 = 5 35 = 7 × 5

Students will already have met fact families for addition and subtraction. They will not necessarily recognise the links between multiplication and division. Students can feel overwhelmed by how many multiplication and division facts there are to learn, unless they see the links between them.

A good command of basic facts is required for carrying out multiplication and division algorithms.

### Illustration 1: Three different stages

Students pass through three stages:

1. Students recognise the 5 × 7 has the same answer as 7 × 5. If they know the answer to one, then they know the answer to the other.
2. Students can use the known fact to solve missing numbers tasks. For example, if they know 5 × 7 = 35, they can use this fact to solve tasks like 5 × ? = 35 or ? × 7 = 35.
3. Students recognise the relationship between multiplication and division. For example, if they know 5 × 7 = 35, then they know that 35 ÷ 5 = 7 and 35 ÷ 7 = 5.

### Illustration 2

Examples of types of tasks that would be illustrative of using multiplication facts to solve multiplication problems, aligned from the Mathematics Online Interview:

• Question 28 - Sharing teddies on the mat
• Question 29 - Tennis balls task
• Question 30 - Dot array task
• Question 31 - Teddies at the movies

## Teaching strategies

Understanding about fact families builds connections in mathematics and reduces the amount of material that students need to learn. The key teaching strategy is to emphasise that there are related facts that belong together. Once a student knows one fact, they can use this to solve related number sentences with missing numbers. The activities below are illustrated with the fact family of 3 × 4 = 12, but teachers can use fact families from the multiplication tables currently being learned.

Activity 1: Fact families from arrays uses real objects, counters and squared paper as representations of number facts. Students generate fact families themselves.

Activity 2: Recognising different fact families encourages students to group facts into families.

Activity 3: Fact family fortune is a simple game designed to highlight which combinations of three numbers are in the fact family and which ones are not.

Activity 4: Fact family bonanza encourages students to extend the notion of fact family creatively.

### Activity 1: Fact families from arrays

Arrays in common objects
Many objects found at home are arranged in arrays, for example, egg cartons, muffin trays, trays for organising nails and screws, and boxes of chocolates. As well as using real objects, a digital camera can be used to bring pictures into the classroom.

Ask the students to choose an object or picture and write down as many multiplication and division number sentences as they can about their array. For example, consider a muffin tray which is a four by three array. Ask students to write as many number sentences as they can, for example, 3 × 4 = 12 and 12 ÷ 3 = 4. They should also write a sentence or story about these facts (e.g. there are 3 rows of muffins with 4 in each row, so there are 12 muffins on the tray).

Inventive students might write compound sentences, although this is not the primary aim of this activity (e.g. there are 3 rows of 2 chocolate muffins and 3 rows of 2 plain muffins: 3 × 2 + 3 × 2 = 12).

Arrays from materials and squared paper
Demonstrate the fact 4 × 3 = 12 using an array of counters, centicubes, dots or symbols as shown below. Use both array language (3 rows of 4 items / 4 columns of 3 items) and equal groups language (3 groups of 4 items / 4 groups of 3 items) so that students build links between previous ‘equal groups’ understandings and the powerful array model of multiplication.

Note that the array can also be rotated to appear as 4 rows of three, so that 4 × 3 = 12 and 3 × 4 = 12. Rearrangement does not change the relationship between the three numbers.

Cover portions of the pictures or materials and ask questions such as “how many rows of four makes twelve?” (? × 4 = 12) as a missing multiplication question and as a division question (12 ÷ 4 = ?).

Give students some counters and get them to write other number sentences, also explained in words.

Extend to larger numbers by drawing rectangles on squared paper e.g. with a 1mm grid (10 divisions per cm), and then by looking not at the number of small squares but at the area of the rectangle. This extends the models students have for multiplication from equal groups, to arrays, to area of a rectangle. Link to the multiplication tables that the students are currently learning.

### Activity 2: Recognising different fact families

Give students a set of numbers (e.g. 3, 4, 5, 12, 15, 20) and ask them to write as many different multiplication or division number sentences as they can using only numbers from the set (e.g. 3 × 4 = 12 and 12 ÷ 4 = 3). Ask students to group all the number sentences from the same family together. For example, here are eight number sentences from one fact family (there are two more fact families possible with the numbers provided):

3 × 4 = 12

4 × 3 = 12

12 ÷ 4 = 3

12 ÷ 3 = 4

12 = 3 × 4

12 = 4 × 3

3 = 12 ÷ 4

3 = 12 ÷ 4

Note that the last row contains number sentences that some students think are ‘backwards’. They are just as valid as the number sentences in the top row, and it is important that these are included to assist students with the notion that an equals sign can mean ‘balance’ as well as ‘give an answer’. For more information see: The meaning of the equals sign (Level 3) which addresses the need for students recognising that the expressions on either side of an equation have the same value.

Include multiplication facts that the students are currently learning. Stress the usefulness of these links in calculating: “I can’t remember 9 sixes, but I do know 6 nines.”

### Activity 3: Fact family fortune

Here is a list of all the missing number sentences associated with the three by four array. The first column contains the fact family, while the number sentences in the other columns have a missing number. Students need to experience all of these different combinations at some stage.

4 × 3 = 12

4 × 3 =

4 × = 12

× 3 = 12

3 × 4 = 12

3 × 4 =

3 × = 12

× 4 = 12

12 ÷ 4 = 3

12 ÷ 4 =

12 ÷ = 3

÷ 4 = 3

12 ÷ 3 = 4

12 ÷ 3 =

12 ÷ = 4

÷ 3 = 4

12 = 4 × 3

12 = 4 ×

12 = × 3

= 4 × 3

12 = 3 × 4

12 = 3 ×

12 = × 4

= 3 × 4

3 = 12 ÷ 4

3 = 12 ÷

3 = ÷ 4

= 12 ÷ 4

4 = 12 ÷ 3

4 = 12 ÷

4 = ÷ 3

= 12 ÷ 3

Be sure that students understand why some combinations are NOT in the fact family. For example:

 3 ÷ 12 = 4 4 ÷ 12 = 3 4 ÷ 3 = □ □ × 3 = 4

Game: Fact family fortune
Students make themselves two set of three cards, and a resource sheet with two columns: TRUE/ FALSE.

One set (for example, coloured red) contains:

• a card with 12 written on one side
• a card with 3 written on one side and
• a card with 4 written on one side.

The other set of three cards (for example, coloured blue) contains:

• a card with = on one side
• a card with × on one side and
• a card with ÷ on one side.

• Shuffle each set, face down.
• Deal a red card, a blue card, a red card, a blue card and a red card along the table.
• Turn over and read the number sentence, for example 3 ÷ 12 = 4. (If there is no = in the number sentence, deal again).
• Write this ‘fact’ down under either the heading TRUE or the heading FALSE. In this case, it goes under FALSE.
• Students discuss why this is the case.
• Pick up all the cards, shuffle, and play again.
• Place the resulting fact under the appropriate heading e.g. 3 = 12 ÷ 4 goes under the heading TRUE.
• Continue making the list.

Finish the activity with a discussion of which facts are true and which are false.

### Activity 4: Fact family bonanza

There is no need to constrain students’ understanding of fact families to a limited set. The main idea is that many number facts are related, and this makes calculation easier.

This is an open activity where students can exercise creativity and work at their own levels.

Begin with a multiplication statement at any appropriate level (e.g. 5 × 7 = 35, 11 × 8 = 88, 77 × 13 = 1001).

Students make as many facts as they can from this one fact, saying how it is derived from the basic fact. As well as standard ‘fact family’ variations, students may include 11 × 80 = 880, 11 × 16 = twice 88 = 176, 110 x 8 = 880 and division variants, as well as derived facts such as 12 × 8 = 88 + 8 = 96 etc).