We have been playing with the BERCS cards for multiplication. Cuisenaire Rods provide students such a brilliant image of number they are becoming very fluent with the transfer to equations.
Take a look at these pictures:
Multiplication is not “just” addition. It is an operation in its own right. Multiplication moves in at least two directions. It represents change by a unit or “factor of”. The change moves in one direction as we are first building “facts”. Then it can be seen as moving in two directions, but the unit is different for each direction. So a 3 x 4 moves in two directions. Units of 3 are repeated four times one way and units of 4 are repeated three times in the other direction.
Multiplication is proportional. The growth is a “times as many” growth. Addition moves one at a time, multiplication moves in “equal sets” or units. Students need to stop thinking of ones as only single objects and start referring to sets as one set of. Place value is based on this idea of “unitizing” . The ones become a ten, the tens become a hundred… The unit are all related back to a “one” and then new ones are created.
Times as many refers to the change created by multiplying with a unit. The cuisenaire rods are an ideal tool as they are already designed in unit lengths. Three is not seen as 3 independent ones but rather as a 3 rod. A unit of 3. The lime green is 3 so looking at the picture below you have an image of 3 repeated 4 times or 3 times 4. The purple rod is one more so it must represent four. Building with the purples represents multiplication by 4.
Multiplication is dimensional. It involves growth in 2 directions. So you can see that the purple rod, which represents a unit of 4, is embedded in the 3 x 4. You can see the 4 repeats three times, but moving in the opposite direction. So 3 x 4 = 4 x 3.
Multiplication is commutative you can switch the order of the factors (or dimensions) without affecting the product or area they cover. When you read 4 x 3, you can think of four groups or units of 3 or three groups or units of 4. The result is 12.
Every “fact” can be built from 2 separate but related units. Once built, you can cut it out from (cm) grid paper. Looking at the grid model you can see how the intersection of the 2 factors or units being multiplied results in a product or area that is measured with square units.
Now the labels tell you which way the “units” are oriented.
You can use a fact you know to learn more facts with ease. So I used the 3 x 4 to build 6 x 4. Now I know a second fact.
AND JUST AS IMPORTANT OR MAYBE MORE; I just practiced the DISTRIBUTIVE PROPERTY.
Division is not an operation on its own. Division is a part of multiplication. If you can multiply, you can divide. The curriculum stresses that students should learn multiplication and division in a related way. You can see the division in this fact by thinking backwards. Instead of building 6 x 4 to get to 24, you need to see the 24 and ask yourself how to take it apart using either 6 or 4. 24÷ 4 looks like this: A rectangle that fills 24 square units. One side is built with units of 4. So what times 4 equals 24. In your mind you can see the rods for four. How many fit inside a rectangle of 24?
As soon as I think the first two rods: 2 x 6 = 12, I know that it will be 4 rods to get to 24. And that makes sense because 4 x 6 = 24.