B75 — Mystery Numbers: Part/Whole and Equations
Problem Strings for Fluency and Beyond · Mystery Numbers: Part/Whole and Equations
The final strings in this section are designed to encourage students to generate equivalent expressions using strategies they have been learning, without needing to do any calculations. An important algebraic idea is to see expressions on either side of the equal sign as equivalent and know why they are interchangeable. For example, when looking at the first problem in the string and noticing that the factor of 4 in the first expression, has been doubled in the second, means the unknown must be a double of 2. Doubling and halving explains this: 4 x 4 = 2 x 8. In the second problem simplification is the justification for why 20/4 is interchangeable with 10 /2. This justification explains the next problem also: 40/4 = 80/8.
Notice the numbers chosen and the placement of the unknowns. This will likely generate scaling and simplifying strategies, and possibly even partial products and/or partial quotients as well. This will be the case throughout this section for finding unknowns on both sides of the equal sign. Students may choose to use the previous problems in the string to find the solutions to the new problems but more importantly they will begin to use the strategies from their earlier work in the division section to solve for the unknowns without needing to do any calculations. These strategies to find unknowns and equivalent expressions are important understandings as students move up into Algebra.
Inside One Classroom: A Portion of the B75 Minilesson
Problem Strings for Fluency and Beyond · Grade 3
Julia (the teacher): Here is another problem, 20 ÷ 4 = 10 ÷ ?, (the second problem in the string). Remember we’re trying to find the number to replace the question mark so that both sides of the equal sign are equivalent. Ok, thumbs up when you are ready to share how you came up with your answer. (When most thumbs are up, Julia starts discussion.)
Maya: I think it’s 2. I know that 20 ÷ 4 is 5 because I know that 4 x 5 is 20. So, I knew the other side had to equal 5. Then I thought what divided by 10 would equal 5 and I knew 5 x 2 is 10.
Julia: Wow, you said a lot there, let’s try to unpack it a little.
Christina: Maya used multiplication facts she knew and said what x 4 is 20; the something for that was 5. Then she used division facts to get 10 divided by something was also 5 and she knew that was 2.
Julia: Is that what you were thinking? (Maya nods.) Nice. Did anyone do it a different way?
Jack: I also got 2. I saw that 10 was half of 20 so I knew that I had do something with the 4 to make the sides of the equation equal. So, since I halved the 20, I halved the 4.
Julia: Let’s see if we can see that on the array. Does this look like what you said? 20/4 = 10/2
Jack: Yes.
Julia: That’s cool thinking, Jack! You didn’t have to do any arithmetic! Let’s try to explain why Jack’s strategy works. 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 whole group.) Bella and Nick, you were having an interesting discussion. Would you please start?
Bella: It’s like cutting off the bottom of the array. The 5 columns stay the same, whether it is half the array or the whole array, and that is the answer. Since he divided the inside in half, he had to divide the number of rows in half to 2.
Julia: So, 20/4 = 10/2. Such great thinking. If we ever forget 20/4, we can always use 10/2 to solve it. Let’s continue to think about powerful strategies like this that will be helpful for division problems.
- ? x 4 = 2 x 8
- 20 Ă· 4 = 10 Ă· ?
- 40 Ă· ? = 80 Ă· 8
- 48Ă· 6 = 24 Ă· ?
- 36 Ă· 6 = ? Ă· 3
