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Multiplication Magic

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). The question asks you to explain the trick.

Two Cubes

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]

Common Divisor

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

Finding Factors

Age 14 to 16 Challenge Level:

Why do this problem?

This problem could be used as an introduction to factorising quadratic expressions, or to develop students' fluency in this skill. The 'hook' of an interactive environment draws students in, encouraging them to be resilient as they strive to complete the challenge.
 
Filling in such a table given the headers would be a straightforward task. This problem turns that around and invites students to deduce the headers from the table entries, with the added challenge of trying to reveal as few cells as possible.
 

Possible approach

Students will need to be able to expand pairs of brackets of the form $(x \pm a)(x \pm b)$ before embarking on this problem - Pair Products provides a nice opportunity to practise this.
 
This task works best if each pair of students has access to a computer or tablet so they can work on the interactivity. They may also need pencil and paper so that as they uncover each cell they can work out the factorisation of each quadratic. The first interactivity is a four by four grid where each cell contains a quadratic of the form $x^2+ax+b$ which can be factorised. Give students some time to work on the interactivity. When they think they have correctly identified the column and row headers, they can click "Show the solution" to see if they are right.

Once they are confident at tackling these examples, they could try one of the larger grids mentioned in the problem, where the quadratics are of the form $ax^2+bx+c$.

If individual computers are not available, this could be done as a whole class activity where a few cells are revealed and students are invited to work out as many factors as they can, before requesting the revealing of further cells. Alternatively, students could create their own grids by choosing pairs of brackets for the two columns, multiplying them out, and then revealing certain cells to a partner.
 

Key questions

What does the constant term of the quadratic expression tell you about the numbers in the headers?
What does the coefficient of $x$ tell you about the numbers in the headers?
 
(For the larger grids) What does the coefficient of $x^2$ tell you about the headers?

Possible extension

How Old Am I? invites students to solve a series of problems that can be modelled with quadratic equations, leading to some generalisations.

Possible support

Factorising with Multilink offers a visual representation of the process of factorising quadratics, which some students may find helpful.