# Explain the relationship between addition and multiplication equations

### Number families and relationships | nzmaths

Multiplication is defined here as meaning that you have a certain number of of multiplication, and about the connection between multiplication and addition. It helps greatly to break the equations down into smaller parts to solve the What is the Distributive Property of Addition and Multiplication?. How Converting between Addition and Multiplication Makes Math Easier The program has a lot of equations and their graphs built in, and you can start To generalize, y=bx and x=logb(y) encode the same relationship.

### Multiplication concept, plus Multiplication and Addition

Well, whatever blank is, whatever blank is, if you multiply it by five, if you multiply by five, you should get You should get So one way of thinking about it, you could say "what times five is 50?

Hopefully you see the relationship here. If 50 divided by five is 10, then 10 times five is And you could do it the other way around. What is 50 divided by 10 going to be?

How do I know that? Well, five times 10, five times 10, five times 10, is equal to Is equal to So let's keep thinking about this. If someone walked up to you in the street again, and said blank, blank, divided by, divided by two, blank divided by two is equal to, is equal to nine.

How would you figure out what blank is?

**Properties of Addition and Multiplication**

Something divided by two is equal to nine. Well, one way to think about it, and if we just follow here, if you said 50 divided by five is 10, you could say 10 times five is 50, so right over here we could say well, nine times two, nine times two, must be equal to blank.

Once again, discuss what is happening in these equations. 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.

## How Are Addition and Multiplication Related?

Have students suggest a meaning for inverse, then confirm this with a dictionary. Make paper, pencils and felt pens available to the students. Session 3 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. In particular highlight the commutative property of addition. 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. 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. 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.

- Relating division to multiplication
- Number families and relationships

Introduce the word commutative. 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. Have students share their work.

Discussion should focus on highlighting the relationships between addition and subtraction. Activity 3 Introduce the game Families on Board. Model how it is played. 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.

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. 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. 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.

The Hundreds Board is screened so that numbers 1 — 30 only are visible. 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. Numbers are made visible.

The task is completed. The challenge is to leave just two numbers uncovered by counters.