# Window Frames

### Task 4 ... Years 4 - 10

#### Summary

Students explore the structure of a 0-99 number chart using frames which isolate particular cells of the chart. What emerges is that no matter where the frame is placed on the grid, special relationships exist between the numbers in the window. At one level the problem is about searching for and finding these relationships. At another it is about explaining their existence.

#### Materials

• One 0-99 number chart and a range of plastic window frames like the ones in the photo

#### Content

• basic number facts
• looking for relationships between numbers
• explaining relationships
• algebraic representation #### Iceberg

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

Part of the iceberg is built into this task. For example, there is one frame which allows the numbers from 44 to 48 to show in the windows. A student might notice that, in this case, twice the middle number equals the sum of the two end numbers. But what happens if we move the frame?

The answer is that this property remains true. Our hypothesis becomes: Wherever the frame is put, twice the middle number equals the sum of the end numbers. But how do we test this hypothesis?

Perhaps we reason that the number at one end is the middle minus 2 and the number at the other end is the middle plus 2. Therefore when the end numbers are added the minus and plus 2 will cancel leaving twice the middle.

Or perhaps we might reason that the left end number is a, the middle is (a + 2) and the right end number is (a + 4). Then:
Left + Right = a + (a + 4) = 2a + 4 = 2(a + 2)

But what happens if...

• we change the shape of the window to 3 cells long, or 7 cells, or 4 cells...?
• we use our window frames to explore the times table chart?
Note: There is a strong link between Window Frames and Task 7, Consecutive Sums.

#### 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.

To convert this task to a whole class investigation it is useful to have an overhead projector with a transparent copy of the number chart and the window frames as in the task. Students have a paper copy of the chart and as you move the frame on the OHP from place to place you ask particular students to mark this frame on their paper grid. Every student is then searching for relationships in a different set of numbers governed by the same window frame. If any student thinks they see a relationship, it can be checked by other students. Any hypothesis which results can be recorded on the board so the discussion can shift to the notion of proof.

At this stage, Window Frames does not have a matching lesson on Maths300.

#### 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 Window Frames task is an integral part of:

• MWA Number & Computation Years 3 & 4
• MWA Number & Computation Years 9 & 10 