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### Multiplication Table Sums

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Topics: Multiplication, summation, patterns and structure
Materials: Multiplication table, pencil and paper
Common Core: 4.OA.1, 4.OA.3, 4.OA.5, 4.NBT.5, 5.OA.2, 5.OA.3, 6.RP.1, 6.EE.1, MP1, MP2, MP3, MP6, MP7, MP8.

This is a great puzzle for kids who have some comfort with multiplication but still need to deepen their understanding.
It gets them looking at the multiplication table with purpose, and they’ll find some serious structure there.

What is the sum of all the numbers on a multiplication table?

Simple to pose, this problem leads to numerous discoveries about the structure of multiplication, and finally, a truly surprising and powerful solution that allows us to quickly find a sum of the numbers on a multiplication table of any size.

1. Students should start small. A four by four multiplication table turns out to be very suggestive, and smaller ones are good too. Fortunately, it’s easy to write out a multiplication table that large.
Another good organizational tactic is to try adding up the rows (or columns) first. Teachers can suggest this to kids who get stuck or overwhelmed.
2. Once students have solved the smaller sizes, expand to larger tables—10 by 10, 12 by 12, or bigger. For students who are ready can try to write an algebraic formula for the sum of all the numbers.
3. Compare this problem to the extensions below.

Here are some very closely related problems, though the nature of the relationship is not at all obvious at first.

How many rectangles are on a 4 by 4 grid?

What is the sum of the consecutive cube numbers?

$1^3 + 2^3 + 3^3 + 4^3 + \ldots + n^3$

Here’s a PDF of the Dot Array Multiplication Table 5 by 5 for easy printing.