Maths teacher, mentor teacher, ed consultant fascinated with discovery learning and inquiry, problem solving, assessment, visualization & making a difference.
Richmond, BC
Joined July 2009
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When I had extended the squares question on a checkerboard, I asked, "How many rectangles of any size is on an m x n grid?" The students started with simple cases, organized counting and then one had a breakthrough by organizing the data on a grid. A 6x4 has 210 rectangles Pt.1
And once a class makes this discovery, how many verifications will they need to do to believe their hypothesis?
Attached is my first thought for younger students (middle schoolers who know what a power of 2 is)...
2025^2 and in 2026, we'd need to add 2x2025+1 to this last number to create the next consecutive square
For those with 99, If you are going to solve it by sequential patterns, you need to monitor the first number (odd, odd, odd, odd, even). This then hints at some missing sequential patterns like below. Better to generalize on a row: e.g. shown in 2 ways in the table below:
Visuals are great. I use the same image for division too. Is it 6 ÷ into 2 groups of 3 or 6 ÷ into 2 groups of 3. Visuals can teach a deeper conceptual understanding. Is doubling and halving understood? If I double the number of groups, the groups will be half the size etc.
Discovered (uncovered) by a Gr. 11 student. Find a power of 10 that your divisor goes into with a small remainder. Ex: 10000 ÷ 17 =588 r 4 so break of 4-digit bundles
I like Dotty 6's for buddy classes pairing early primary with intermediates. Littles have a great chance of beating the bigs while learning lots of friendly facts to 6. For older classes, I like 10000 (Farkle) as it involves applied probability and risk/reward strategies
I like to do something similar. I have them reverse engineer how some students were counting from their mathematical expression. Here is a partial list. The shapes are repeated so when they talk to others, they might see/hear an alternate solution. Ss learn formal writing too.
My take is with Dave Hewitt's Grid Algebra: What happens when you move left and right, up and down in different rows and columns of a multiplication table? What journey can you go on to move efficiently from one 'cell' to another cell? Can you reverse the operations?
