We are wrapping up a short unit on multiplication. It began with an unexpected debate. We had been doing a lesson on how to add, subtract and multiply with multiples of ten when the commutative property of addition came up. Students knew that they could "flip flop" the numbers in an addition problem and, after reflection, they agreed that they could not do the same for subtraction. Could you flip multiplication? About a third of the class wasn't sure. "Three groups of two is different than two groups of three." The pro-commutative group tried to convince us but struggled until we looked at multiplication as an array, specifically, a rectangular candy bar. Was the candy bar with three rows of two squares the same as the candy bar with two rows of three squares. Everyone agreed that it was.

Sensing a rich vein, I became the owner of a candy company who wanted students to design the best box to hold 24 cube chocolates. This gave students the opportunity to experiment with arrays and multiples. Many chose to build their boxes out of graph paper - a great spacial exercise.

When everyone had a chance to present their ideas to the shareholders. I explained that we now were facing a new challenge - we needed to find the most flexible number of candies so that we could change our box often. Consumers like things to be "new," afterall, so we wanted to know the number of chocolates that would give us the most boxing flexibility. The table the students created allowed us to explore primes and square numbers as well as see which numbers had the most factors.

Our multiplication minds warmed up, I introduced

multiplication wrestling (which you may recognize from your algebra days as the F.O.I.L. method). This is a multiplcation algorithm that emphasizes the "groups of" nature of multiplication. Students can see how quantities can be broken done into smaller quantities that are easier to multiply and then reassembled to find the final product. We used base ten blocks to create a concrete representation of what we were doing with the numbers on the board. I was excited to see several students setting up their own problems with the blocks when we had open work later in the day.

Today we explored the

Sieve of Eratosthenes, an ancient tool for finding prime numbers. By crossing out the multiples of a prime, you begin to winnow down the remaining integers, slowly revealing only prime numbers. (The link above is a slick interactive web version.) Don't worry, we only went to 100 or 200, not the two million plus digit number that is the largest prime number discovered yet.

I hope this overview gives you an idea of how we are attempting to weave more concrete, computational math into authentic, engaging experiences. By following students' ideas and needs, we help them construct the understanding that "math makes sense."