# What's It Worth?

### Task 75 ... Years 2 - 8

#### Summary

Value Relations is an important component in developing fraction understanding. Essentially it is about parts making up a whole and assigning a value to either the part or the whole, in order to work out the other. The values in this case are about money, but they could easily be other decimals, fractions, or whole numbers. The triangles provided help some students work out the required values. They also offer spatial challenges in themselves because students are, in effect, asked to work out and record all the pentiamonds and hexiamonds.

#### Content

• spatial patterns & symmetry
• money calculations
• fractions, value relations
• problem solving strategies such as:
• breaking a problem into smaller parts
• trying every possible case #### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

The first question on the card is straightforward calculation. The pentiamond is worth 5 x 55˘ = \$2·75. However even simple questions can be extended.

• How did you work that out?
• Can you check it another way?
• What happens if I tell you that the large shape is worth \$2. What is the value of the triangle?
From here, the task involves considerable spatial problem solving as well as number work. For example, it implies finding all the shapes that can be made with five triangles. Apart from the first one on the card there are:

Four in a row Three in a row All other shapes are rotations or flips of these.

The value of \$3.30 in Question 3 implies shapes made with 6 equilateral triangles at 55˘ each. These are the hexiamonds. The full set can be found by beginning with the four pentiamonds above and finding all the ways of adding one more triangle. There are 12 altogether, one of which is the Sphinx. Students may like to assign names to the others.

The shape in Question 4 uses nine equilateral triangles. Since this is an odd number, one of the triangles must somehow be in the 'middle' to maintain symmetry. Some solutions are:    Rather than individuals searching for all the symmetric shapes as asked for on the card, teachers may prefer to begin a class display which grows as each pair of students finds one or two more. When there is plenty of data on the display, the class's attention could be drawn to looking for ways of classifying the shapes so that it can be decided whether they have all been found.

Extensions
Any of the shapes students make in this task could be seen as a whole thing (the shape) which is divided into equal parts (the triangles). Students can be invited to:

• Make any shape with the tiles:
• Cut it out from the triangle paper and paste it into your journal.
• Write all the fraction stories you can from this shape.
• Make sure you can demonstrate each one with the tiles.
• Make any shape with ten tiles: This is your whole.
• Cut it out from the triangle paper and paste it into your journal.
• Write all the decimal stories you can from this shape.
• Make sure you can demonstrate each one with the tiles.
• Make a whole shape which would allow you to work out:
• 1/2 + 1/3
• What other fraction problems can you work out with this whole?
• How about: 1/2 + 2/3 or 2/3 + 1/6?
• Value Relations work can be extended using other shapes such as Sphinx, Trisquares, or if you want the challenge of three dimensions, Tricubes. Using any of these pieces the process is to limit the students to (say) 4 pieces and ask them to make new shapes from those. Then, to create whole number practice, assign a value to the unit piece or the whole.
• Further extend the value relations work with the software from the Maths300 lesson below.

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

If you can make lots of triangles from card, or wood you can easily use this task as a whole class investigation. The easier way is probably using card and getting the students to cut the triangle pieces from copies. You can also create similar problems of your own based on square tiles which are common in classrooms, but triangles probably give more interesting shapes.

The Maths300 lesson uses the Sphinx as the shape for the lesson (cut-out master is included) and squares and triangles as the shapes for the software.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 99, What's It Worth, which also includes an Investigation Guide.

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The What's It Worth? task is an integral part of:

• MWA Number & Computation Years 3 & 4
• MWA Number & Computation Years 7 & 8

The What's It Worth? lesson is an integral part of:

• MWA Number & Computation Years 5 & 6
• MWA Number & Computation Years 7 & 8
This task is also included in the Task Centre Kit for Aboriginal Students. 