In this unit students use Cuisenaire rods (or other equivalent material that fits together precisely) to make “number mats” that illustrate a variety of numerical patterns and can be visually appealing. They formally record the number relationships in the mat.

- Use addition and multiplication to find number combinations that 'make' a given result.

It is important for students to know the meaning of the equality sign. Realising that “=” indicates that the two expressions on either side of it are equal is a key step on the road to algebra. These two expressions have the same status - one does not have to be the “answer” of the other.

This very important idea is fundamental to a sound platform on which to proceed to algebra proper at a later level. It is introduced here by way of a geometric technique that links numbers via Cuisenaire rods.

This unit could be repeated at a higher level by assigning a value other than 1 to the white rod.

#### Links to Numeracy

This unit provides an opportunity to develop students’ number knowledge in the area of Grouping and Place Value. It also provides a way develop early part-whole thinking in the addition and subtraction domain as it allows students to clearly see the different ways a number can be partitioned.

To develop students knowledge of groupings within 5 and ten focus their attention while they are working with the 5-mat and the 10-mat.

*Can you find 2 numbers that join together to make 10 on the 10-mat?*

*Can you find two different numbers?*

*How many different combinations can you find?*

List the combinations as they are identified. Encourage students to see the relationships between the two addends: as one increases, the other decreases. This may also be illustrated using the Cuisenaire mats:

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:

- Simplify the task by working with smaller numbers, such as 5-mats, and make the task more complex by working with larger number mats such as 12-mats.
- Some students may be able to move from using the structured physical representation of Cuisenaire rods to drawing diagrams to support their thinking.

The context in this unit can be adapted to recognise cultural diversity and student interests to encourage engagement. For example, in place of a Cuisenaire mat students could work with images of a waka ama with a defined number of seats, with coloured shirts to represent number patterns. For example in a 5-waka, there can be 3 red and 2 blue shirts (3 + 2), 4 red and 1 blue shirt (4 + 1), or 3 red, 1 blue and 1 green shirt (3 + 1 + 1).

- Cuisenaire rods for each pair of students.
- If they are available Magnetic Cuisenaire rods which stick onto a blackboard are very useful.

#### Getting Started

Here the concept of a 5-mat is introduced. It is constructed from combinations of Cuisenaire rods that all have the same length as the yellow rod. The 5-mat is a device to help students explore equality of combinations of numbers. It also helps them to see that “=” means “is equal to”.

- Give each pair of students a set of Cuisenaire rods and allow time for free play if they are new to the students. During the free play, encourage building activities that lead to comparison of the length of the rods and activities that fit them together tightly.
- Conduct a class discussion about the lengths of the rods. Begin by making a staircase of the rods in increasing length. Then by covering the rods with the unit (white rods), establish the lengths as 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 times the length of the white rod.

(When Cuisenaire rods are exactly made in units of 1 cm, some students may also be able to check the length by measuring.)

Draw a clearly labelled diagram on the blackboard, for reference.

- Introduce the idea of a 5-mat. First: students take a 5-rod and put together one other combination of rods that makes 5. For example, different students might make 5 as 1 + 3 + 1 (white, green, white), 4 + 1 (pink, white), 2 + 3 (red + green), or as 5 whites. Then these different combinations can be put together as shown in the diagram to make a 5-mat. Of course there are many different 5-mats, but they are all rectangles with the yellow rod (5) as one side.

- Number combinations in the 5-mat. Students suggest the number combinations demonstrated by the 5-mat on the board. The mat above has:
1 + 3 + 1

4 + 1

2 + 3

1 + 1 + 1 + 1 + 1

or 5 x 1= 5

= 5

= 5

= 5

= 5Note that it is important to keep the numbers in the order that they appear on the mat: 2 + 3 and 3 + 2 would be different rows of the 5-mat. Note also that there are two relationships shown by the row of 5 whites; one is an addition and one a multiplication.

- Linking relationships together.

Explain that the 5-mat also shows other relationships that are true. For example, 1 + 3 + 1 = 4 + 1 and 5 x 1 = 2 + 3 etc. This use of equality may seem strange to students because it does not give an answer on the right hand side. Explain that “=” means “is equal to”. -
Students write down relationships from the 5-mat and read out their favourites to the class.

#### Exploring

Here the concept of equal is explored further using mats of different sizes.

- Students pick a mat to make of a given size. Controlled choice of the size of the mat can allow for individual differences. The more able students should be challenged by being given larger numbers. The diagram below shows the first four rows of a 12-mat.

- Ask the students to record the relationships shown on their mat. The mat in the diagram illustrates many relationships. For instance, 3 + 3 + 3 + 3 = 12 and 4 x 3 = 12. (Note that 4 x 3 is interpreted as 4 groups of 3 here and not 3 groups of 4.)
- Students should also record some of the relationships between rows of the mat. For example, 6 + 1 + 5 = 3 + 3 + 3 + 3.
- Initiate a class discussion on interesting examples: for example, 5 + 7 = 7 + 5.

*Did anyone else find something like this?*

Is 4 + 8 = 8 + 4? Why?

*Did anyone else find something like this that did NOT work?* - Other interesting examples that are worth discussing are things like 4 x 3 = 3 x 4.

*Did anyone else find something like this?*

Is 2 x 6 = 6 x 2? Why?

Did anyone else find something like this that did NOT work?

Rows that show a strong visual pattern may also show interesting number patterns. - The activity can be repeated using a mat of a different size.
- Turn the situation around.

*Make me a mat that shows that 4 + 7 = 2 + 9.*

What other equalities can a mat like this show?

Make me a mat that shows that 2 x 5 = 3 + 7.

What other equalities can a mat like this show?

Let the students pursue this aspect of the problem in pairs.

#### Reflecting

This section brings together all that the students have discovered so far.

- The students can make a poster of their work on a large piece of paper individually or in a pair. Selected students can report their most interesting findings to the class.
- Highlight the important points. This will include observations about addition (for example, 8 + 1 = 7 + 2 = 6 + 3, 8 + 1 = 1 + 8, and 7 + 2 = 2 + 7) multiplication (3 x 4 = 4 x 3) and the meaning of equality.
- Is it true that 4 + any number = that same number + 4? Why? Why not?

Is it true that 2 x any number = that same number x 2? Why? Why not?

**Possible extensions**

The ideas above can be extended for more able students and older students. So this unit could be used with students at Levels 3 or even 4.

- Older students can be challenged by changing the value of the white rod from 1 to, say 2, or even 0.1.
- Carefully removing a rod from a number mat leads to a natural setting for equation solving. For example, removing the dark green rod from the mat above, leads to equations such as

? + 1 + 5 = 12; and

? + 1 + 5 = 3 x 4. - A variety of other questions can be asked within the context of the number mats and checked visually. For example, I am making a 16-mat: Can I make a row just out of the light green rods (3-rod)?

Answering this could lead to a statement such as 5 x 3 + 1 = 16.

*Use a mat to check whether*2 x 5 + 4 = 6 + 1 + 7*or*5 + 3 x 4 = 7 + 9.

Dear parents and whānau,

This week in maths we have been using coloured plastic Cuisenaire rods to make " Cuisenaire mats" which help to show that the equal sign “=” says that the numbers on either side of it are the same. Here is an example of a 12 mat. (The white rod = 1, the red = 2, the green = 3, the pink = 4 and so on.) Each line of colours = 12.

Looking at the green and pink rows, this shows 3 + 3 + 3 + 3 = 4 + 4 + 4. This can be read as 3 + 3 + 3 + 3 **is the same as** 4 + 4 + 4. Talk with your child about what other equations can be written using the 12 mat.