Luke is in the 2nd grade, so he has been working on all kinds of addition and subtraction problems, but hasn’t practiced much multiplication yet. Jenny and Luke recently had a conversation about Luke’s math homework. The paper asked him to do an addition task on a grid, and then color in all of the “count by 2” numbers.

Jenny: Count by 2 numbers?? Luke, do you know what an “even” number is?
Luke: Of course! They’re the ones that have a buddy.
Jenny: That’s a great answer!
Luke: I also know my count by 3 numbers! 3…6…9…12…15…18…21…24…27…
Jenny: Wow! You’re pretty good at that!
Luke: And my count by 4 numbers!
Jenny: Count by 4 numbers are hard! Let’s hear them.
Luke and Jenny together: 4…8…12…16…20…24…28…32…
Jenny: I noticed something. 24 is on BOTH your count-by-3 list and your count-by-4 list. Do you think there are other numbers on both lists?
Luke: (thinks for a while) 240?
Jenny: Why’d you say that one?
Luke: Because it’s a lot like 24?
Jenny: That’s true. (Thinks for a minute) I think you’re right about 240. Do you know any others?
Luke: 2,400!
Jenny: Wow!
Luke: 240,000! And really, you could add any number of zeros to the end!
Jenny: Yikes! You didn’t just find one more, you found infinitely many more!

(Warm-up!) Why did Jenny ask Luke about “even” numbers, when the homework asked about “count-by-2” numbers?
Rephrase “count-by-3 numbers” into something about factors and multiples. What is this list of numbers?
Jenny’s question is about numbers on both lists. Rephrase this question into something about factors and multiples. What is Jenny asking Luke to do?
Did Luke give the answer you were expecting? Why or why not?
What do you think Luke means by “240 is a lot like 24”?
Is every odd number also a count-by-three number? Explain why or why not.
Why is Luke right that 240 is on both the count-by-three list and the count-by-four list? Use the rephrased versions of the questions about factors and multiples to answer this question.
Why does a number that’s a multiple of 10 appear on the count-by-3 list? Does it matter that 3 isn’t a factor of 10?
Why is Luke right about 24 with any number of zeros on the end being on both lists? Use the rephrased versions of the questions about factors and multiples to answer this question.
Luke found a lot of numbers on both lists, but he didn’t find them all. Can you find them all? Explain your thinking.
2025-08-13 00:44:24