Topics: Multiplication, patterns,
Materials: Pencil & paper
Common Core: 4.NBT.1, 4.NBT.5, 5.OA.1, 5.OA.2, 5.NBT.5, MP1, MP2, MP3, MP6, MP7, MP8.
How does multiplying by 37 have this strange effect? This problems in multiplication is designed to give students the practice they need once they understand the basic idea of multiplication, with a context to provide motivation and deeper understanding.
If you feel like acting, you can pretend to “discover” this trick in the course of demonstrating a 2-digit multiplication problem. Ask students to pick a single digit. Then you follow the next steps:
Add the number to itself 3 times.
Multiply the result by 37.
(The answer is a surprise: the digit repeated three times.)
For example, say we pick the digit 7. We add 7 + 7 + 7 to get 21, and then multiply 21 x 37. When you work that out, the answer is 777. We might have written (7 + 7 + 7) x 37 = 777. And that’s the power of 37.
Once you demonstrate one example, pose the question of whether this will always work. Will (3 + 3 + 3) x 37 = 333? Will (9 + 9 + 9) x 37 = 999? Can the students find a counterexample? If not, why does this trick work?
Let students work, alone or in pairs. Have them check this trick for all the digits. They’ll likely need about 15 minutes.
For students who finish early, have them try the trick for two-digit numbers: does (10 + 10 + 10) x 37 fit in with the earlier pattern? What about (12 + 12 + 12) x 37?
Discuss what students found, and why they think it’s true. If students did the multiplication correctly, they’ll have noticed that every digit works. But why?
There are a few good explanations I can think of, and students may have more. Here are two examples:
Explanation 1: (1 + 1 + 1) x 37 = 3 x 37 = 111. If replace the ones by twos, that would double the answer, so you’d get 222; and so on for threes, fours, etc. up to nines.
Explanation 2: (7 + 7 + 7) x 37 = (7 x 3) x 37 = 7 x ( 3 x 37) = 7 x (111) = 777, and likewise for other digits.
Students may or may not have found these solutions. If not, you don’t have to give it to them. They’ll have gotten the multiplication practice they need, and the mystery can be something to continue thinking about later.
Tips for the Classroom
- Once students know that they’re trying to confirm or disprove this pattern, use that to get them double-check their work when they get an answer that doesn’t fit with the pattern. Emphasize that it would be a really important result to break the pattern, so they should make doubly sure they’ve got the calculation right.
- Don’t let them know that you believe this trick will always work, or students won’t be motivated to solve it for every digit.
- You can follow this question with the division version: if you repeat digit three times to get a 3-digit number, will it always be divisible by 37? For example, does 777 ÷ 37 always have an even answer? If you understand the relationship between multiplication and division, the answer is clearly yes, but most students won’t intuitively understand that connection, and will need to check that the division works too.