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Overturning Fracsum

Solve the system of equations to find the values of x, y and z: xy/(x+y)=1/2, yz/(y+z)=1/3, zx/(z+x)=1/7

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Bang's Theorem

If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.

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Rudolff's Problem

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

What's it Worth?

Stage: 3 and 4 Challenge Level: Challenge Level:1

Why do this problem?

This problem provides a good challenge in reasoning working with multiple unknowns. There are issues of redundancy to think about in the provision of too much information.

This is a pre-algebra task that can introduce students to the sort of manipulations that can be used to solve simultaneous equations.

 

Possible approach

Show the grid below, or hand out this worksheet, or display this PowerPoint slide.

"Each symbol has a numerical value. The total for the symbols is written at the end of each row and column."

"This is a really interesting problem because it can be solved in lots of different ways. Can you find the missing total that should go where the question mark has been put?"

"Can you find more than one way to do it?"

 

Grid containing symbols.

 

After everyone has spent a few minutes on the problem, hand out this worksheet, containing eight different ways of starting to solve the problem.

"Here are the first few lines of the solutions found by some friends working on this problem. Can you complete their solutions?"

You may choose to give all students all eight solutions, or share them out so different groups work on different solutions.

"In a while, I'm going to invite you up to the board to present your favourite method as a completed solution, so be ready to explain every part of your thinking clearly, and to justify why it is your preferred approach."

Here are some prompts that could be offered to students working on the different approaches if they get stuck:

"If you could work out the value of the square/hexagon/circle/triangle, how could you then work out the value of the triangle/circle/hexagon/square?"

"What can you deduce by comparing the bottom two rows?"

For the trial and improvement methods: "If this one didn't work, what value does it make sense to try next?"

Finish off by inviting students to the board to share the completed methods, together with any other methods they thought of for themselves.

The same techniques can be applied to some follow-up tasks. This spreadsheet can be used to generate other versions of the same problem. Here are some grids that could be printed off for students to annotate with the new values.

Here are another three problems that can be solved using similar strategies:

Sweet Shop
Letter Land
Children at Large

 

Possible extension

Using the spreadsheet, can students work out how much information is necessary to solve the problems uniquely, and identify redundant information?

 

Possible support

Students could work in pairs to make sense of the different methods. The trial and improvement methods (1 and 5) are perhaps the most accessible starting point.