Reverse Arithmetic

I like to start tutoring with what I will call reverse arithmetic problems. The basic problem is to find 3 numbers that add up to 10. Or 28, or 100. Easy start:

1. Tell me two numbers that add up to 10.

Harder start:

2. Tell me three numbers that add up to 10.

In normal arithmetic, the numbers take control -- given the problem 4 + 3 = ?, the numbers control the answer. In these reverse arithmetic problems, you control the numbers. You can be creative, or not, and for some of these you do have to learn to use the numbers like tools, including finding the right tool for the right job.

3. Do problems like this until they become boring. Then you can move on to some of the variations.

Discovering zero

4. 10 numbers that add up to 5.
5. A thousand numbers that add up to 0.

This forces the child to use zeroes. (The child could use negative numbers. If your student is good enough to do that, don't worry about zero.)

From a mathematical point of view, the most elegant solution to the problem of three numbers that add to 100 is 0 + 0 + 100. If your student starts there, fine. To most people, zero is a second-class number, so they won't use it to solve these problems until they are forced to. To a mathematician, zero is as good a number as any other.

The important principle is that zero is a useful tool for adding to a number and not changing it. To get this, your student has to learn that any number added to zero is that number. Sure, you can say it, but learning it in a problem is different.


Do the same thing except multiplication.
5. Find two numbers that multiply to 6.
6. Find two numbers that multiply to 18.
7. Find three numbers that multiply to 8.
8. Find three numbers that multiple to 18.

Multiplying by 1

9. Find two numbers that multiply to 5.
10. Find two numbers that multiply to 7.
11. Find two numbers that muliply to 13.
12. Find two numbers that muliply to 1971.

This is the same as the zero variation for addition -- the number 1 is a useful tool for multiplying without changing a number, and the underlying concept is that 1 times a number is that number. On this first lesson, a bright third-grader could easily do the first two. #11 was difficult but he got it with insight. I think he had to be shown the answer to #12, but then he immediately grasped the concept.

Multiplying by 0

13. Find 10 numbers that multiply to 0.


When adults do the basic problem, they usually fall into a pattern very quickly. Children are less inclined to do this. Or else, children who like math have formed the habit of trying to make boring problems needlessly complex so that they will be interesting. But anyway, if your student falls into a pattern, that is a good segue into the next problem, identifying and naming patterns.


This teaches creativity, the property of numbers, and using numbers as tools.

Ode to the View from the Window in My Algebra Class
Sun-warmed concrete benches
next to tables with built-in checkerboards
and the graffiti left by hundreds
before us in
Day-Glo spray paint:

Joanne Loves Richie
Punx Rule
Nuke Fags and Lesbos
Amber Is a Slut

The dead leves and plastic bags scatter
in the breeze from the park
and men in business suits try to keep the
last few remaining strands of hair covering
their pink bald spots.
Cigarette packets and used-up chewing gum
coat the gray sidewalk.

And I think
What does it matter
that it is not a linear equation if any variable is raised
to a power?
We're all just going to die anyway.

From "The Princess Diaries (Vol 2.), by Meg Cabot.

And, not to spoil the poem, but if you are trying to solve a black box problem, it makes a big difference if the equation inside is linear or not.