# Number families and relationships

*Keywords:*

AO elaboration and other teaching resources

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.

- 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.

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, by the very nature of an equation or expression, the purpose of which __ is__ to express a relationship.

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, recognizing and understanding the **behaviours of and relationships between the operations**, is foundational to success in *algebra* as well as arithmetic.

One way in which the relationship between the operations of addition and subtraction is often explored early on, and is made more obvious to students, is by connecting the members of a ‘family of facts.’ Whilst these ‘fact families’ have often been used to facilitate learning of basic of facts, they have not always been used well to develop a *deeper* understanding of the relationship between these operations.

Because of this, students often encounter conceptual difficulties in understanding the nature of the operational relationships that exist between the 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 which the 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, if responses to problems are to come from a position of understanding, rather than simply from having been taught a procedure. 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 from12n + 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’. Furthermore, in studying addition and subtraction together, an equal emphasis is placed on both operations. Unfortunately, it is not always the case that these operations are given equal emphasis in classrooms. Subtraction has been the ‘poor *relation*’ that receives less attention. In fact sometimes students are not helped to see that there is any *relationship* at all between addition and subtraction.

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)

**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__

- 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.” (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.) - 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).” - Record one set of four related equations and write
*Family of Facts*on the class chart. Ask whether this is a good name and why. Record student ideas, highlighting that fact that these numbers are**related**through the operations of**addition and subtraction. Families are related.**

__Activity 2__

- 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). - 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. - 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__

- Begin by reading together the concluding notes from Session 1.
- Distribute sets of
cards (Attachment 1) to students.**Family Shuffle**

(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,*, 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.*but not lifted**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__

- 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.” - Pose the question, “How are addition and subtraction related to each other?” Record student’s ideas.
- 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.”

Ask a student to write on the class chart, the equations that express the scenario. (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**. - 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. - 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.”

Ask a student to write on the class chart, the equations that express the scenario. (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 the question posed in Step 2 above. Have students suggest a meaning for inverse, then confirm this with a dictionary. - Explain that each student is to create a small creative A4 poster 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.

Suggest that 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__

- 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. - Distribute the addition and subtraction grids (Attachment 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. - Once completed, have students share what they notice and record their observations.
- 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**. - 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__

- Have students complete the number problems on Attachment 3.
. In giving instructions highlight the importance of the students recording equations and on explaining what is happening with the numbers in the problems.**Nana’s party** - Have students share their work. Discussion should focus on highlighting the relationships between addition and subtraction.

__Activity 3__

- Introduce the game
. Model how it is played.**Families on Board**

(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*Students can use tens frames,**they say and write the four related equations**., to work out or demonstrate the equations and their relationship.*if needed*

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 that 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. - Conclude this lesson and series of lessons, by recording students’ reflections on their learning about addition and subtraction, and the relationship between them.

Dear Parents and Whānau,

In maths this week, the students have been learning about the relationship between addition and subtraction. They have been exploring fact “families” with three numbers, such as 8 + 6 = 14, 6 + 8 = 14, 14 - 6 = 8 and 14 – 8 = 6. They have found out that subtraction “undoes” addition. (Subtraction is known as the inverse operation of addition, and vice versa).

They have played a game called, * Family Shuffle*. Your child will show you how they have played this. You might like to have a turn, then make up your own fact family together and try the game again.

We hope you enjoy the game, and your discussions, as you make up and talk about new families.

Thank you.

Attachment | Size |
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NumberFamiliesAndRelationshipscm1.pdf | 27.1 KB |

NumberFamiliesAndRelationshipscm2.pdf | 573.75 KB |

NumberFamiliesAndRelationshipscm3.pdf | 93.78 KB |