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Missing Multipliers

Age 7 to 14 Challenge Level:

 

Why do this problem?

This problem offers an opportunity for students to consider common factors while gaining fluency in multiplication facts. The interactivity engages students' curiosity and perseverance by challenging them to complete the grid using a minimum number of 'reveals'.

Multiplication tables are often presented with row and column headings filled in, with students challenged to fill in the products. This task inverts that concept, as students can reveal chosen products and work out possibilities for the headings.

 

Possible approach

If computers or tablets are available, students could work in pairs using the interactivity. Students could try a few examples to get the idea, and then work on the challenge of trying to find the grid headings by revealing as few cells as possible. Once they have developed some strategies, they could try the larger grids that include bigger numbers. 

 
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 plenty of time to develop strategies, the key questions below provide a good basis for a plenary discussion, after which students could revisit the interactivity to test out each other's suggestions.
 

Key questions

Which numbers, when revealed, make it straightforward to work out the row and column headings?
Which numbers give lots of possibilities for row and column headings?
 
Is there a strategy for working out the row and column headers in fewer than 10 reveals?

Can you find a way to work out the row and column headers using only 6 reveals?
 

Possible support

Mystery Matrix works in the same way, but some helpful cells have already been revealed.

 

Possible extension

 
There are natural extensions within the problem - working on the 10 by 10 grid provides a real mental workout!

Gabriel's Problem and Product Sudoku would make nice follow-up activities.
 
Finding Factors has a very similar interactivity but the context is factorisation of quadratic expressions.