Cuisenaire mats

Purpose

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.

Specific Learning Outcomes
• Use the mathematical symbols of =, <, >.
Description of Mathematics

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.

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:

To assist the development of part-whole thinking pose problems that can be solved using the Cuisenaire mats. For example, if I had seven lollies in one bag and nine lollies in another bag, how many lollies would I have altogether?

Are there easier numbers we could use to solve this problem?

Encourage the students to see that they are simply changing the way the numbers are partitioned by illustrating with Cuisenaire mats. Try several examples, using larger numbers as the students ideas develop.

Required Resource Materials
• Cuisenaire rods for each group of two.
• Magnetic Cuisenaire rods which stick onto a blackboard are very useful.
Activity

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”.

1. 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.
2. 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 Cuisenire 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. 3. 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. 4. 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 + 14 + 12 + 31 + 1 + 1 + 1 + 1or 5 x 1 = 5= 5= 5= 5= 5

Note 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.

5. 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”.
6. 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.

1. 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 give larger numbers. The diagram below shows the first four rows of a 12-mat. 2. 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.)
3. Students should also record some of the relationships between rows of the mat. For example, 6 + 1 + 5 = 3 + 3 + 3 + 3.
4. 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 any one else find something like this that did NOT work?
5. 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 any one else find something like this that did NOT work?

Rows that show a strong visual pattern may also show interesting number patterns.
6. The activity can be repeated using a mat of a different size.
7. 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.

1. 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.
2. Highlight the important points.  This will include observations about addition (for example, 8 + 1 = 7 + 2 = 6 + 3 = etc., and 8 + 1 = 1 + 8, 7 + 2 = 2 + 7, etc.) multiplication (3 x 4 = 4 x 3 etc.,) and the meaning of equality.
3. 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.

1. Older students can be challenged by changing the value of the white rod from 1 to, say 2, or even 0.1.
2. 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.
3. 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.

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