Conjectures in Grade 9

Today I was working with a grade 9 class as they are starting up their number sense activities. We talked about multiplication as repeated addition and as the area model. We introduced the idea of commutative operations, and looked at how we can use doubling and halving to help us understand how to use our multiplying by 10 skills to be able to multiply by 5

Students tried to fill in as many multiplications they could do in 5 minutes, then we debriefed some of the patterns we saw.

Students were curious about the patterns along the diagonal. They saw that the blue diagonal went odd, even, odd even, odd, even. The adjacent diagonals are all even, then the next adjacent are alternating even and odd again.

They made some conjectures about what’s going on.

next we tried to see if it’s true. We used some big numbers to check if it’s still valid for numbers off our grid.

students chose 112×1011. We used area model to multiply the numbers.

we see that oddxeven=even yet again!

we also had a conjecture that oddxodd=odd.

Finally we noticed that when we draw all the shaded numbers like an area model the rectangles are squares.

After a brief BEDMAS review we ended with a round of max/min dice game, where we start with a framework, then madlibs style we fill in the numbers from a 10 sided dice. We were aiming for the biggest number we could get.