Fact Families (Addition and Subtraction): 1.75
Supporting materials
Indicator of Progress
Success depends on students knowing that once they can recall a particular addition fact, they can use that fact to solve a variety of related addition and subtraction tasks. The set of related facts is called a family.
Before this, students will see number facts as unrelated, and hence might feel burdened by how many facts there are to learn.
Illustration 1: Three different stages
Students pass through three stages:
i) Students recognise that 4 + 6 has the same answer as 6 + 4. If they know the answer to one, then they know the answer to the other.
ii) Students can use the known fact to solve missing numbers tasks. For example, if they know 6 + 4 = 10, they can use this fact to solve tasks like 6 + ? = 10 or ? + 6 = 10.
iii) Students recognise the relationship between addition and subtraction, and that if they know 6 + 4 = 10, then they know that 10 − 6 = 4 and 10 − 4 = 6.
Illustration 2: Diagnostic Assessment Task
Give students a set of numbers (eg 3, 4, 7, 6, 10) and ask the students to write as many different number sentences as they can using only numbers from the set, using addition or subtraction. For example, 3 + 7 = 10 and 10 - 6 = 4. Ask them to group all the number sentences from the same family together. For example:
| 4 + 3 = 7 | 3 + 4 = 7 | 7 - 4 = 3 | 7 - 3 = 4 |
| 7 = 4 + 3 | 7 = 3 + 4 | 3 = 7 - 4 | 4 = 7 - 3 |
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 means balance rather than just give an answer
Illustration 3
Examples of the types of tasks that would be illustrative of addition and subtraction concepts, aligned from the Mathematics Online Interview:
- Question 18 - I have 9 teddies and you have 4 teddies. How many teddies do we have altogether?
- Question 19 - Work out the number of biscuits left when the start amount was 8 and 3 were eaten.
- Question 20 - Work out the number of strawberries when the start amount was 12 and 9 were eaten using counting back, counting down or counting up from strategies.
- Question 21 - Basic addition and subtraction – using counting on, counting back, counting down to or counting up from strategies.
Teaching Strategies
Understanding about fact families builds connections in mathematics and reduces the amount of material that students need to learn. The three activities below get students to generate fact families themselves.
Activity 1: Fact families using materials uses concrete representations of number facts to give meaning to the symbolic expressions of the facts in the fact family.
Activity 2: Domino fact families and Activity 3: Dice fact families, both use a random process to give each student (or group of students) their own fact families to explore. Class discussion of the responses of various students is essential to bring closure to the activity.
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.
Activity 1: Fact families using materials
You can demonstrate the fact 4 + 3 = 7 using counters, Unifix, centicubes or dots and symbols as shown below.
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Note that these can also be made or drawn in various spatial arrangements (including as columns). Rearrangement does not change the relationship between the three numbers. By covering portions of the pictures or arrangements of materials, you can ask questions such as 4 + ? = 7.
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Here is a list of all the related facts associated with these pictures. Note that the first column contains the complete number sentence, while the sentences on the right have a number missing. Students do not need to generate all of these for any one number fact, but it is important that they do generate a variety of types of facts.
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4 + 3 = 7
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4 + 3 = □
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4 + □ = 7
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□ + 3 = 7
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3 + 4 = 7
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3 + 4 = □
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3 + □ = 7
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□ + 4 = 7
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7 − 4 = 3
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7 − 4 = □
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7 − □ = 3
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□ − 4 = 3
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7 − 3 = 4
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7 − 3 = □
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7 − □ = 4
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□ − 3 = 4
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7 = 4 + 3
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7 = 4 + □
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7 = □ + 3
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□ = 4 + 3
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7 = 3 + 4
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7 = 3 + □
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7 = □ + 4
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□ = 3 + 4
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3 = 7 − 4
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3 = 7 − □
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3 = □ − 4
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□ = 7 − 4
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4 = 7 − 3
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4 = 7 − □
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4 = □ − 3
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□ = 7 − 3
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Activity 2: Domino fact families
Dominoes can be used to produce fact families. For example, the following domino shows 6 + 4 = 10 and students could then be asked to write all the related number sentences. Note that the domino with one blank side should stimulate class discussion as 4 + 0 = 4 .
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Ask the students to choose a domino and write down as many number sentences as they can about their domino. Ask students to make number sentences with missing numbers for their partner to solve.
Activity 3: Dice fact families
Students roll two dice and write as many facts as they can using the two numbers rolled.
Whilst supervising this activity, the teacher can ask students to explain what each fact means in terms of the dice. For example, if the student threw 2 and 1 on the two dice, they might say:
"2 + 1 = 3" - This means that if there are 2 spots showing on one die and 1 spot showing on the other, there are 3 spots showing altogether.
"3 − 2 = 1" - This means that if there are 3 spots showing altogether, and there are 2 spots showing on one of the die, then there must be 1 spot showing on the other.
