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
If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
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?
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.
Show the grid below, or hand out this worksheet, or display this ppt 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?"
After everyone has spent a few minutes on the problem, hand out this worksheet, containing six different images, suggesting different starting points. The images are also available on these ppt slides.
"There's a lot of information in the grid, so you had to make some choices about what to focus on first. Each image on the worksheet draws your attention to particular parts of the grid. What can you deduce from each? How can each help you to solve the problem?"
We've set out the worksheet so if you don't want everyone to work on every image, you can give half the class the odd numbered images and half the class the even numbered images.
"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?"
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:
Using the spreadsheet, can students work out how much information is necessary to solve the problems uniquely, and identify redundant information?
Students could work in pairs to make sense of the different methods. The problem could also be solved by trial and improvement, by inviting students to choose a value for a particular shape and explore the consequences.