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Adding All Nine

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

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Missing Multipliers

Stage: 3 Challenge Level: Challenge Level:1

Why do this problem?

This problem offers an opportunity for students to practise routine multiplication whilst being required to consider common factors and multiples. The task provides an engaging and challenging environment which encourages students to think about efficient strategies for solving a set of related problems.
 
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.  

Possible approach

Introduce the class to the problem using the interactivity.
 
In order to really get to grips with the task, students need to spend quite a long time developing efficient strategies for solving the Level 3 challenges.
 
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.
 
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 discussion.
 
Finally, once students have had a chance to share ideas, they could revisit the interactivity to test out each other's suggestions.

Key questions

Is there a strategy for completing the Level 3 challenge consistently? 
Are there any numbers you could uncover that immediately tell you the row and column headers?
Are there any numbers you could uncover that narrow down the row and column headers to two alternatives?
Does it make a difference if you choose which cells to reveal at the start or if you choose as you go along?

Possible extension

There are natural extensions within the problem - working on the 10 by 10 grid provides a real mental workout.
 
Why was it necessary to include a grey (unavailable) number in one of the headers of the versions of the problem which included negative numbers?
 
Finding Factors uses a similar structure but the context is factorisation of quadratic expressions.

Possible support

This version of the interactivity uses multipliers up to 10 and allows unlimited reveals, so it could be used to allow students to get a grasp of the structure of the problem before moving on to the other challenges.