Finding factors
The interactivity below is similar to the one used in Missing Multipliers so you may wish to work on that first.
In the multiplication grid below, the headings and the answers have been hidden. Each of the headings is an expression of the form $x + a$ where $a$ is an integer between $-5$ and $5$. By revealing some of the answers, can you work out what each heading must be?
Drag the green and purple labels onto the headers to make the correct expressions.
What is the smallest number of answers you need to reveal in order to work out the missing headers?
Once you've developed strategies for finding the factors in the interactivity above, you might like to have a go at solving some bigger grids which include coefficients of $x$ greater than 1 in the row and column headings:
Six by six grid
Eight by eight grid
Ten by ten grid
You may also be interested in the other problems in our Working backwards, leaping forwards Feature.
What does the coefficient of x tell you about the headers?
Aryan, Zain, Nilukshan, Gharan, Sajuthan, Ibraheem, Tianah and Salome from Northgate Primary in the UK focused on how to find the headings having revealed the answers.
This is the beginning of Sajuthan and Ibraheem's work:
Aryan and Zain focused on the signs of the numbers in the answer:
If in the [product] the top digit is negative (e.g. $-5$ and the bottom is positive (e.g. $+6$) both numbers that make the [product] are negative.
Nilukshan and Gharan continued this idea:
We found H.C.F. out of the 4 numbers (in this case, the bottom numbers, $+6$) in each row. Then we found if the H.C.F. would be positive or negative. After that we put the negative or positive number on the box of the row.
Tianah and Salome described how to find the two factors from the answer. This is their work. Read down and then across.
Tanya from Bangkok Patana School in Thailand thought about how many answers you have to reveal in order to find the solution. This is Tanya's work:
It is actually possible to do it by revealing $2(a-1)$ answers:
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.
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.
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?
Possible support
Factorising with Multilink offers a visual representation of the process of factorising quadratics, which some students may find helpful.
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.