Square Area -- Pixel Art

Years before children learn about the concept of multiplication, they are actually using this concept every day as they observe rectangular grids of 2x2, 2x3, 3x3, 3x4, 5x6, 10x10 and so forth. They recognize these numbers as 4, 6, 9, 12, 30, and 100, respectively. What they really mean is that number of square units--a two dimensional concept.

We have spent many lessons teaching them to be comfortable with counting by multiples of numbers. Counting by ones is taught to them from early on although is a shame that they are taught to start with the integer ONE. Alternatively, they could have learned to start with ZERO, or even better, NEGATIVE 10, THEN NEGATIVE 9 and so on. In fact, at the end of every Mathletes class, i have them choose a negative number such as -100 and we count by 10s to zero. Even in my kidnergarten class this week, we counted by 12s from -144 (incidentally, the number 144 is called a "gross").

Counting by twos is the logical next choice but the children seem to be more comfortable with counting by 10s and then 5s. Then 2s come next. Threes are more challenging; sevens are deadly; but elevens are welcome given the palindromic nature of the first nine multiples.

This week's lesson focuses on the children choosing strategies to find square area. Some of the younger children are tempted to count by their favorite easy multiple: one. The objective here is to find rectangles that are large and easy to manage. After the students started with 2x3 and 2x2, they took risks with 5x4 and finally with 10x.... on the larger pictures. From time to time, I would catch a student counting by ones and after I showed them that they could have done a 10x8 rectangle giving them the same 80 square units, they were convinced that adding rows or multiplication was much easier.

I attach here both pixel art pdfs (one is easier than the other). I also attach a sample solution for a Turtle. In this example, I created 5 rectangles with square area in the double digits and then consolidated the remaining single digit rectangles into sums of 10 with a remainder. There are thousands of permutations of how the rectangles could be configured. The goal is to find the optimal strategy that is manageable by their individual skill level. If they can count by any multiple, I challenged them simply create the fewest rectangles; if they can only count by twos and fives, then that was their objective.

Kindergarteners and first graders should only be focused on creating rectangles and indicating the square area of individual rectangles. Second through fifth graders should be focused on the first objective and also adding all of the smaller square area sections to obtain a sum. During class, I told them that if they come within 20 of the actual square area, this it is an accomplishment. Some of them are so precise that they were able to find the actual square area. Of course, some K/1st graders are ready to add numbers vertically and should certainly try.

Now for the part everyone enjoys, I gave the kids blank graph paper so they could create their own pixel art and find the square area. I also encouraged them to color in the pixel artwork I made for them.