One of the larger goals of math in the intermediate grades is to learn about proportional reasoning (multiplicative thinking) as applied to fractions, percentages, proportions, ratios and patterning. This provides a solid foundation from which higher levels of mathematics can be explored, including geometry, linear and quadratic functions, and data management.
In the primary and junior grades, among other mathematical constructs, students work with decomposing larger numbers into smaller quantities, primarily as addends, and eventually working towards factors as used in multiplication.
Cuisenaire rods are a useful tool in math, allowing students to easily compose and decompose small quantities while showing the proportional relationships. For example,
Each block can represent an even proportion of units. For example, if we call the white rod "1" and the red rod "2", then it follows that light green is "3" and so on up to 10.
We can combine rods to create larger numbers, such as 12 by using red and orange, 20 by using 2 oranges etc.
We can also assign a larger value than "1" as our base (white) unit. If we assign it as "5", then we can count by 5's up to 50 without the need to add additional rods.
Using these relationships, intermediate students can use these rods to explore the concepts of proportion, ratios, fractions, and the distributive property.
Factoring and the Distributive Property With Cuisenaire Rods
Using the principles listed above, students can use the rods to factor a larger number. For example,
in the following photo, 12 is factored in a variety of ways:
In the dark green row, we see 2 rods of 6 or 2 x 6. Likewise, in the red row, we see the reversed version where there are 6 rods of 2, or 6 x 2. Through the use of many examples, students can determine through exploration the commutative property of multiplication, and also the relationship between multiplication and division.
Factor trees can also be built in this way, by further reducing the larger factors. Eventually, each component of each row can be factored down to the prime number components.
If we decided to use a base unit other than one, we could compare similar numbers based on proportional relationships. For example, if we chose "3" as our base for the above example, the top row would be worth 36. The second row would be 2 rods of 18. The third row would be 3 rods of 12, and so on. We could also build the 36 by using more orange and red rods and keeping the base unit as 1.
In the following picture, "9" has been divided into various rows in a somewhat different way:
Similarly, the 4th row also has two different colours.
In the 5th row, however, we see a pattern emerge. There are 3 instances of red-white, or 2, 1 in this row. How might we express this mathematically?
This is an example of the distributive principle of multiplication. How else might we use cuisenaire rods to show this?
So far we've been looking at rows of rods to compare relative values. How might we move into an extra dimension where we can compare both rows and columns?
Eventually, algebra tiles will help with these visualizations, however, the challenge of combining grids to visualize proportions allows for concrete abstraction that will eventually lead into linear and quadratic applications.