# Number families and relationships

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Purpose

The purpose of this unit of three lessons is to develop an understanding of how the operations of addition and subtraction behave and how they relate to one another.

Achievement Objectives
NA2-6: Communicate and interpret simple additive strategies, using words, diagrams (pictures), and symbols.
Specific Learning Outcomes
• Recognise three numbers that are related through the operations of addition and subtraction.
• Recognise that there are two related addition and two subtraction equations in a ‘family of facts’.
• Write and read sets of related addition and subtraction equations.
• Explain, in their own words, the inverse relationship between addition and subtraction.
• Recognise that addition is commutative but that subtraction is not.
• Solve number problems that involve application of the additive inverse.
Description of Mathematics

Algebra is the area of mathematics that uses letters and symbols to represent numbers, points and other objects, as well as the relationships between them. We use these symbols,=, ≠, <, >, and later ≤ and ≥, to express the relationships between amounts themselves, such as 14 = 14, or 14 > 11, and between expressions of amounts that include a number operation, such as 11 + 3 = 14 or 11 + 3 = 16 – 2.

These relationships between the quantities are evident and clearly stated within an equation or expression. This highlights the purpose of an equation or expression, which is to express relationships.

The relationships between the number operations, and the way in which they behave, can be less obvious. Often these operations are part of arithmetic only, as computation is carried out and facts are memorised. However, recognising and understanding the behaviours of, and relationships between, the operations is foundational to success in algebra and arithmetic.

Early on, the relationship between addition and subtraction is explored by connecting the members of a ‘family of facts.’ These ‘fact families’ can be used to facilitate learning of basic facts and to develop a deeper understanding of the relationship between addition and subtraction.

Students often encounter conceptual difficulties in understanding the nature of the operational relationships that exist between three numbers. It is not unusual for students, when asked to write related facts, to write, for example, 3 + 4 = 7, 4 + 3 = 7, 10 – 3 = 7, 10 – 7 = 3. In these statements, equations are correct, but the relationship between 3, 4 and 7 is not understood.

As students encounter more difficult problems, and are required to develop a range of approaches for their solution, having an understanding of the inverse relationship that exists between operations is critical. To ‘just know and understand’ that subtraction ‘undoes’ addition, and that addition ‘undoes’ subtraction becomes very important. For example, it is important to know how and why problems such as 61 – 19 = ☐ can be solved by addition, saying 19 + ☐ = 61, or that the value of n can be found in 12n + 4 = 28, by subtracting 4 from 12n + 4 and from 28. By having a sound understanding of the inverse relationship between addition and subtraction (for example) students are better placed to solve equations by formal means, rather than by simply guessing or following a memorised procedure.

It is important to consider this larger purpose, as we explore ‘families of addition and subtraction facts’. Within this, an equal emphasis must be placed on both operations.

The activities suggested in this series of three lessons can form the basis of independent practice tasks.

Links to the Number Framework
Advanced counting (Stage 4)
Early Additive (Stage 5)

Opportunities for Adaptation and Differentiation

This unit can be differentiated by varying the scaffolding provided or altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. For example:

• using materials (such as the tens frame and coloured counters) to establish, explore, and show relationships between three numbers, rather than progressing to working with numbers only without the support of materials
• removing the expectation that students will record equations and focus on exploring the relationships between the three numbers.
• providing opportunities for students to work in flexible groups, in which they can share their thinking and justify their answers.

Situating the families of facts in familiar additive contexts will appeal to students’ interests and experiences and encourage engagement. Examples may include:

• Native birds in a tree, with birds arriving and flying away. 3 birds are in the tree and 2 more arrive, how many birds now? 3 birds fly away, how many are left in the tree?
• People travelling in a van. How many in the front seast? How many are in the back seat? How many all together? How many left if the people in the front/back seat get out of the van?
• Collecting kai moana. How many pipi in the bucket? How many kina? How many altogether? If we take the pipi out, how many are left in the bucket?
Required Resource Materials
Activity

#### Session 1

SLOs:

• Recognise three numbers that are related through the operations of addition and subtraction.
• Recognise that there are two addition and two subtraction members of a ‘family of facts’.

Activity 1

1. Make available to the students, digit cards, cards with addition, subtraction and equals symbols, tens frames and counters.
Ask students to think of three numbers that they like, between and including 1 and 10. Accept all groups of three numbers, and record them on a class chart.
For example:
1, 2, 3
3, 4, 5
2, 4, 6
4, 7, 10
1, 5, 9
3, 5, 8
Ensure, without pointing this out to the students, that there are some sets that include ‘family of fact’ numbers, For example. eg. 1, 2 ,3, or 3, 5, 8 or 4, 6 10.
Have each pair of students select a set of three numbers for their investigation.
Students take at least four digit cards for each of their chosen numbers, symbol cards, an empty tens frame and counters of two colours.
Pose, “What equations can you make with your numbers? Make a display or record digitally.” (The students with an ‘unrelated’ set of numbers, for example 3, 4, 5 will quickly discover that no equations can be made using all three digits at the same time.)
2. When some have completed the task, have all the students move to look at the sets of equations that have been made using all three digits. For example: 1 + 2 = 3, 2 + 1 = 3, 3 – 1 = 2, 3 – 2 = 1. Tell them to be prepared to explain what they notice.
Discuss their observations, eliciting observations such as, “there are four equations”, “there are two subtraction equations and two addition equations”, “the addition equations are just the other way around (commutative property).”
3. Record one set of four related equations and write Family of Facts on the class chart (this could be digital). Ask whether this is a good name and why. Record student ideas, highlighting the fact that these numbers are related through the operations of addition and subtraction. Families are related.

Activity 2

1. Return to the list created in Activity 1, Step 1. Identify the sets of numbers that have been found to be related by addition and subtraction. Record four equations for some of these. For example:
1, 2, 3
3, 4, 5
2, 4, 6 (2 + 4 = 6, 4 + 2 = 6, 6 – 4 = 2, 6 – 2 = 4)
4, 7, 10
1, 5, 9
3, 5, 8 (3 + 5 = 8, 5 + 3 = 8, 8 – 5 = 3, 8 – 3 = 5)
Ask, “Can you see what we could do to the other groups of numbers to make each of them into a family?” (Change one of the numbers) Accept suggestions and explore ideas.
Take one of these groups, for example 3, 4, 5. Have a student model with counters on a tens frame, 3 + 4. This will show that the third number is 7.

Conclude that 5 can be changed to 7. Explore the other options of changing one of the numbers: 4 to 2, or 3 to 1.
Explore one more example, for example 4, 7, 10. Discuss that instead of 10 this number should be 11. Alternatively, model seven

and highlight the fact that the 4 could be changed to 3. Ask, is there a third thing we could do? (Change 7 to 6).
2. Continue to make available to the students, digit cards, cards with addition, subtraction and equals symbols, tens frames and counters, paper and pencils.
Have students choose to work in pairs or on their own, to explore the other sets of ‘unrelated numbers’ on the list. They should make a change and write the four equations (+-) which result.
3. Have students (pair) share their recordings. Discuss what is the same or different about them (this depends on which number they change). They should draw a box around sets of four equations that they have written that are the same as a partner has recorded.

Activity 3

Conclude this session by summarising on the class chart, the features of a family of related facts: three number members of the family, and four equations, two of addition and two of subtraction.

#### Session 2

SLOs:

• Recognise that there are two addition and two subtraction members of a ‘family of facts’.
• Write and read sets of related addition and subtraction equations.
• Explain, in their own words, the inverse relationship between addition and subtraction.
• Recognise and understand the additive inverse, a (+) - a = 0. (It is not necessary for students to know the name for this.)

Activity 1

1. Begin by reading together the concluding notes from Session 1.
2. Distribute sets of Family Shuffle cards (Copymaster 1) to students.
(Purpose: To recognise related addition and subtraction equations)
Explain how to play.
Each student has one set of 16 shuffled cards. These are dealt out, face up, in a four-by-four array. Two cards (any) are removed from the array and set aside, creating two empty spaces in the array. Individual cards can be slid across or up and down within the array space, but not lifted, till the array shows one complete ‘family’ in each line or a column. The two cards that were set aside are replaced to complete the array.
To increase the challenge of the task, remove one card only, and/or place each ‘family’ in the same order. (eg. two addition equations then two subtraction equations). Students can swap sets and explore other ‘families’.
Students could write their own sets to create alternative puzzles.

Activity 2

1. Write a ‘family’ of teen number equations on the class chart. For example:
14 + 5 = 19, 5 + 14 = 19, 19 – 14 = 5, 19 – 5 = 14.
Ask students what they notice about what is happening with the numbers.
Record observations such as, “you can add the numbers both ways without changing the result”, “when you subtract you write the biggest number first”, “when you take one of the numbers away you get the other number.”
2. Pose the question, “How are addition and subtraction related to each other?” Record student’s ideas.
3. Read to the class (or have written on the class chart) this scenario.
“Sam helped his Gran with lots of jobs. He earned \$5. He helped Grandpa too and he gave Sam \$1. Unfortunately Sam lost the \$1 on his way home.”
Record, or ask a student to write, the equations that express the scenario on the class chart. (5 + 1 – 1 = 5 and 5 + 1 = 6, 6 – 1 = 5).
Discuss what is happening in these equations. Elicit the observation that subtraction is ‘undoing’ addition.
4. Have student pairs discuss, create, and agree upon, a parallel scenario in which subtraction “undoes” addition.
Record several of these in words and in equations on the class chart.
5. Read to the class (or have written on the class chart) this scenario or a similar one relevant for the class.
“Sam had \$6 and lost \$1 on his way home. He did some extra jobs for his Mum and she gave him \$1.”
Record, or ask a student to write, the equations that express the scenario on the class chart. (6 -1 + 1 = 6 and 6 - 1 = 5, 5 + 1 = 6).
Once again, discuss what is happening in these equations. Elicit the observation that addition is ‘undoing’ subtraction.
Explain that we say this relationship between addition and subtraction is known as an inverse relationship. Record this in answer to the question posed in Step 2 above. Have students suggest a meaning for inverse, then confirm this with a dictionary.
6. Explain that each student is to create a small creative A4 poster or slideshow showing what they have learned so far about number ‘families’ and about the relationship between addition and subtraction. Make paper, pencils and felt pens available to the students.
Each student could write their own scenario, including a picture or diagram to show what is happening, writing related equations, and an explanation in words of ‘inverse relationship.’

Activity 3

Conclude by sharing and discussing student work.

#### Session 3

SLOs:

• Recognise that addition is commutative but that subtraction is not.
• Recognise how knowing about number families is helpful for solving problems.
• Solve number problems that involve application of the additive inverse.

Activity 1

1. Begin by sharing the student work from Session 2, Activity 2, Step 6.
Ask “Why is knowing about families of related facts useful?”
List student suggestions. In particular highlight the commutative property of addition. For example: If you know 17 + 5 = 22, you will also know 5 + 17 = 22. Also highlight the related subtraction facts.
2. Distribute the addition and subtraction grids (Copymaster 2) to each student. Use the larger class copies to model how to complete each grid. In particular, show how to complete the subtraction grid, subtracting the numbers down the side from those along the top row, and putting a dot for those that ‘cannot be subtracted’, rather than discussing negative numbers at this point.
Highlight the importance of the students writing their observations about each grid once they are completed. These observations should include number patterns and the fact that the subtraction grid cannot be fully completed.
3. Once completed, have students share what they notice and record their observations.
4. On the class addition grid look for the same sums for both addends.
For example, 2 + 1 = 3 and 1 + 2 = 3, 2 + 3 = 5 and 3 + 2 = 5.

(Discuss the pattern and also notice the pattern of doubles)
Write this statement on the class chart and read it with the students:
We can carry out addition of two numbers in any order and this does not affect the result. Introduce the word commutative.
5. Ask why they can’t complete the subtraction grid in the same way they have the addition grid. Record a student statement that states, in their words, that addition is commutative, subtraction is not.

Activity 2

1. Have students complete the number problems on Copymaster 3. Nana’s party. In giving instructions highlight the importance of the students recording equations and on explaining what is happening with the numbers in the problems.
2. Have students share their work. Discussion should focus on highlighting the relationships between addition and subtraction. Buddy students to support each other if needed.

Activity 3

1. Introduce the game Families on Board. Model how it is played.
(Purpose: to identify three fact family members and to record four related equations.)
The game is played in pairs. The players have one Hundreds Board, 25 counters of one colour each, pencil and paper.
Tens frames showing ten, blank tens frames, and extra counters should be available to the students to model or work out equations if appropriate.

Round one: The Hundreds Board is screened so that numbers 1 – 20 only are visible. Students take turns to place 3 counters on three related numbers. Counters cannot all be placed in the same row.
For example: students cannot cover 3, 4 and 7 because they are all in the same row.
As they place their counters they say and write the four related equations. Students can use tens frames, if needed, to work out or demonstrate the equations and their relationship.
For example:
Player 1

Player 2

Turns continue.
The challenge is to complete the task between them, leaving only two numbers uncovered. If they have more than two uncovered on the first try, they try again with different combinations.

Round two:
The Hundreds Board is screened so that numbers 1 – 30 only are visible. Students take turns to place 3 counters on three related numbers. Counters cannot all be placed in the same row. As they place their counters they say and write the four related equations, using tens frames if needed.
The challenge is to complete the task between them, leaving no more than three numbers uncovered by counters.

Round three:
Numbers 1-50 are made visible. The task is completed. The challenge is to leave just two numbers uncovered by counters.
2. Conclude this lesson and series of lessons, by recording students’ reflections on their learning about addition and subtraction, and the relationship between them. They can share these reflections with a buddy or whānau.
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Level Two