B59 — Connecting Multiplication to Division using the Array

Julia (the teacher): Here is another problem. (It's the sixth problem in the string.) Now my new array has 27 squares and it has 3 columns. How many rows does the array have? Ok, thumbs up when you are ready to share how you came up with your answer. (When most thumbs are up, Julia starts discussion.)

Fiona: 9. I knew 10 x 3 was 30 so 9 x 3 would be 27. I took a row away.

Julia: Let's see if we can see that on the array. You said you used a 10 x 3 array and you knew that had 30 squares in it. Who can continue to explain what Fiona did next?

Christina: Since the new array only had 27 squares, she took a full row away and that was a 9 x 3.

Julia: That's pretty cool the way that works. What equations could we write for this? Turn and talk to a partner. (Julia moves around and listens to pairs as they talk to determine her next move. After a few minutes she goes back to the whole group.) Christina and Jack, you were having an interesting discussion. Would you please start? 1

Christina: We were thinking that we could write 10 x 3 = 30 and 9 x 3 = 27.

Jack: We also wrote 27 ÷ ? = 3.

Julia: Great. Any others?

Fiona: Maybe 27/3 = 30/3 – 3?

Julia: Oh…that's an interesting one. Let's see if it works. What's 30 divided by 3? (A chorus of "10" resounds.) Okay, minus 3 makes 7, hmm… 27/3 doesn't equal 7. Why doesn't Fiona's equation work… turn and talk about this. How could we fix it? (After several minutes of pair talk, Julia resumes.) Christina, what are you thinking? 2

Christina: 30/3 = 10 and the answer for 27/3 needs to be 9, so I think we need to subtract 1, not 3.

Julia: 1 row, or 1 column, or 1 square? Let's try to explore this on the array. (Julia draws an open array as shown below.)

Christina: It is 30-3 to get to 27, but it is just 1 row.

Julia: What if we tried imagining the array turning, and we wrote it like this: 27/3 = 30/3 – 3/3?