In grade 3 we have been working hard for the last two months on multiplication and thinking a lot about multiplication facts. The common core standards say that by the end of grade 3, students need to know the product of all one digit numbers from memory.
Here is where this standard is discussed in more detail. To facilitate kids getting to this point, I have been starting grade 3 math each day by having kids do one multiplication fact problem in their head. When they are ready, I have them give me a signal. When all students are ready, I have students share different strategies with the class and as I try to capture some of their thinking in equations and diagrams on the board. From start to finish, this warmup took us 12 minutes. This was time well invested in getting kids to think flexibly and to have them really use what they know about multiplication.
Today's problem was 7 X 8
Below are some pictures and explanations of the strategies kids used to tell me how they know 7 X 8 = 56. They are not aloud to say that they have just memorized it. If they have memorized it, that is great but I still want them to be able to prove it to me.

This student said he saw a picture in his mind of 8 groups of 7 and knew that was the same amount as 4 groups of 14. He then knew four groups of fourteen was the same as 4 groups of 10 and 4 groups of 4. He finds 40 and 16 as the partial products and adds them to get his final answer. 

This student struggles a great deal in math but given time to think, he often comes up with really interesting strategies based on what he knows. Here he said he knew 2 groups of 8 was 16 and 3 groups of 8 was 24. Then he said now I have 5 groups of 8 so I need 2 more groups of 8 which is another 16. Then I just added 16+24+16=56. Click here to read more about strategies for multidigit addition and subtraction. 

This student knew 5 groups of 8 was 40 and 7 groups of 8 is 2 more eights. I find a lot of kids who love this strategy because it builds off from the five facts which kids are so comfortable with. 

This student built off from 6 x 8 by adding another group of 8. This is an example of a student who has the fact 7 X 8 memorized but "proves it" to me by doing something like this. It helps me see that this student understands the concept of multiplication and is not just spitting back memorized facts. 

This is the strategy that our program "teaches" kids to use. Only one student out of the 12 I was working with did it this way even though that is what they have been "taught". It is a fairly efficient strategy but as you can see based on all the other strategies, students invent many ways that make sense to them without ever being taught how to do it. 

This student figured out 9 X 7 by doing the "finger trick" and then they knew 8 X 7 was one group of 7 less.

What kinds of strategies do you see kids using to figure out or prove multiplication facts?
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