Pair Products

Problem | Teachers' Notes | Hint | Solution | Printable page |
Stage: 4 Challenge Level: Challenge Level:1

Why do this problem?

This problem is for students who are working numerically, this is an excellent context for observing, conjecturing and thinking about proof. It can be a good introduction to the power of algebra.

For students who are algebraically fluent, this problem involves the construction of an algebraic model, some manipulation, and interpretation back into the context of the problem.

Possible approach

Establish that the group knows the meaning of 'consecutive' - consecutive days, consecutive letters in the alphabet...

Choose four consecutive numbers and tell your students that you will multiply the outer ones and the inner ones.Ask students to pick their own sets of four consecutive numbers and do the same. Record all the results on the board. What do they notice?
Will this always happen? Even with consecutive negative numbers?
How could we explain it? Encourage algebraic and geometric reasoning.

Much of what will follow will depend on the arithmetical and algebraic confidence of the group. Use the extension and support remarks below to indicate the best way to use this resource beyond the questions in the problem.
 
You can read about one teacher's experience of using this task in the classroom.

Key questions

    Why is it important that the numbers are consecutive?

    Can you explain the number patterns that you have noticed by representing the multiplications as the areas of rectangles?

    How can you express consecutive numbers algebraically?

      Possible extension

      This problem only operated on the end numbers and the 'end but one' numbers. Could you make a more general statement and justify it?

      If you have an odd number of consecutive numbers, what's the difference between the product of the end numbers and the square of the middle number?
       

      Possible support

      This problem could also be approached purely numerically, as an exercise in developing fluency with multiplication tables while looking for pattern and structure.

      A multiplication grid could be used for recording results, with the pair products highlighted according to how many consecutive numbers were being used.

      Visualisation through blocks of dots or rectangle areas may help students explain why their pattern must work in every case.



       

      Published May 2004.