This problem could be used as an introduction to factorising
quadratic expressions, or to develop fluency in this skill. Unlike
standard factorisation exercises, this task is intended to engage
students and motivate them to factorise lots of expressions and
develop efficient strategies in the process.

Relating to this month's
theme, the action of filling in the table given the headers is
straightforward; probing the table and then working out the headers
is the inverse or undoing action. It is challenging to look for the
least amount of information required in general to enable the undoing
action to be completed.

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.

Introduce the class to the problem using the first
interactivity.

Using a Level 1 challenge, reveal ten of the cells to the
class (making sure that at least one cell in each row and column is
revealed). Ask the students to work in pairs or threes, and give
each group one of the revealed cells: "Can you find two expressions
that multiply together to give the expression in your cell?"

After giving them some time to work on this, bring the class
back together. Ask the first group to feed back the expressions
they came up with.

"Can I fill this in on the grid?"

"What other information do I need?"

"Does anyone else have that information?"

Using the class's suggestions, fill in the rest of the grid
and then reveal the headers.

If a computer room is available, students could work in pairs
using the interactivity. Another option, if students have access to
computers outside school, is to ask them to work on the different
challenges for homework and then discuss their strategies in the
next lesson.

If computers are not available, the task can be recreated by
asking each student to create a multiplication grid of their own,
and then draw a blank grid for their partner. As in the
interactivity, the challenge is to ask for as few entries as
possible from the grid in order to work out what the headers
are.

Once students have had a chance to develop and share
efficient strategies, they can move on to the second interactivity
which introduces quadratics with a coefficient of $x^2$ greater
than $1$.

What does the constant term of the quadratic expression tell
you about the headers?

What does the coefficient of $x$ tell you about the
headers?

(For the second interactivity) What does the coefficient of
$x^2$ tell you about the headers?