Intermath | Workshop Support
 Write-up

Title
Combinations

Problem Statement
On a calculator you are allowed to use only these five keys: 3, 4, x, -, =. You can press them as often as you like. Find a sequence of key presses that produces a given number in the display. For example, 3 x 4 - 3 - 3 = will produce 6. Find a way to produce each of the numbers from 1 to 10. Clear your calculator before each new sequence.

Problem setup

The beauty of this solution is that you do not need to solve for the solutions 1-10 in any order.

Plans to Solve/Investigate the Problem

The plan for finding solutions involves starting with the given example and beginning to add and take away subtractions and additions of 3 and 4.  Later we will change the multiplication of 3 * 4 to 4 * 4 when we begin exhaust problem solutions.

Investigation/Exploration of the Problem

Start with the given example, 3 x 4 - 3 - 3 = 6, but eliminate one subtraction of 3:

3 x 4 - 3 = 9

Change the subtraction of 3 to a subtraction of 4, and you have:

3 x 4 - 4 = 8

There seems to be no reason why you cant use the example:

3 x 4 - 3 - 3 = 6

Change one subtraction of 3 to a subtraction of 4:

3 x 4 - 3 - 4 = 5

Change both subtractions of 3 to subtractions of 4:

3 x 4 - 4 - 4 = 4

Now add another subtraction of 3:

3 x 4 - 4 - 4  3 = 1

It would seem that we have exhausted solutions for problems starting with 3 * 4 = 12, so now we shall try 4 * 4 = 16 and subtracting.  (3 * 3 = 9 likely will not give us a large enough number to leave room for sufficient subtractions).  We now have problems for the solutions 9,8,6,5,4, and 1, leaving problems to be written for 2,3,7 and 10.  Lets try to find problems resulting in 2 or 3 figuring we can make a slight change of a 3 or a 4 to get the other solution.

We discover a solution for 10 accidentally:

4 * 4  3  3 = 10

Adding another subtraction of 3 gives us:

4 * 4  3  3  3 = 7

Adding a subtraction of 4 to that problem gives us:

4 * 4  3  3  3  4 = 3

And finally, changing one subtraction of 3 to a subtraction of 4 givens us the final solution:

4 * 4  3  3  4  4 = 2

We found the solutions 1 to 10 using only these five keys: 3, 4, x, -, = in the following order: 9,8,6,5,4,1,10,7,3,2.  The order of solutions was arbitrary; we were concerned with generating new problems.  This is an important idea for students attempting to solve the more challenging extension.

Extensions of the Problem

A good extension of the problem would be to give students the task of finding all solutions from 1 to 25, and to use the numbers 1,2,3,and 4, but in any order.  Each problem should contain each number used only once, and students may use any operation including division.  Solutions to the problem would include use of grouping symbols, and this lesson would be an excellent addition to a unit on order of operations.

Author & Contact

Pat Devane

Memorial Middle School

Conyers, GA
pjdevane@rockdale.k12.ga.us