Today's learning walk: division reigns supreme

While out buying ham and cheese croissants at Cafe Fanny for a special Mother's Day breakfast yesterday, my 7 year old daughter asked me how to divide 120 into three equal parts. She quickly understood that 60 was half of 120, so 120/2 = 60, and with a little bit of prompting she understood that 30 was half of 60, so 120/4 = 30, but 120/3 remained a puzzle to her. This morning riding our bikes in to school, I tried to break it down a different way; first, we determined that 120 is made of up 12 groups of 10. Then, we agreed that 12/2 = 6, and 12/4 = 3, after which she was able to reach the conclusion that 12/3 = 4. At which point we counted by 10s (aka multiplied) to reach the conclusion that 120/3 = 40.

While doing a 45 minute walk around the 2nd-5th classes today, I saw numerous ways in which students were engaged in mathematical thinking building their division skills.

One group of second grade students was working on static subtraction, using either a bead frame or working mentally with just a paper and pencil. Subtraction is an important concept on its own, but it is also foundational to division, which can be conceived of as repeated subtraction (the inverse of how multiplication can be seen as skip counting, which is repeated addition). The other group of second grade students was working with the division board, a Montessori material that literally makes division visual (in this image, 12 beans have been divided into 4 equal groups of 3 per group).

Third grade students were also working in two groups. One was doing

"pencil & paper" work on a group of problems that included addition, subtraction, multiplication and division of numbers with four digits. The other was learning to write the remainder of division problems such as 32/10 in more sophisticated mathematical expressions, from "3 r 2" (where "r" = "remainder) to 3 & 2/10, to 3.2, using materials including the Stamp Game, Fraction Tiles, and Test Tube Division. BTW, the Stamp Game is an incredible material that can be used to teach all four operations (in this image, 2432 x 4 = 9724), and some important extensions of those ideas - you can check it out here.

In 5th grade, students were working on graphing. In the story problem, one child had grown at a steady rate from age 2-10, while the other had grown more rapidly from age 2-4, and then less rapidly from age 4-10. They had been given the choice of either completing a table showing the two rates of growth and then plotting those lines, or drawing lines to represent the growth and then determining the amount of growth in interval to fill in the table. In a fascinating bit of insight into 5th grade mentality, every single one of the 21 students had chosen to draw the lines of change first - which left them to work out some complicated division problems involving decimals.