Making & Breaking Rectangles Years 2 - 7

Summary The rectangle shape is critical to heaps of mathematics and it is almost always associated with multiplication. This activity develops the factors aspect of the array model - a model well supported through other activities listed in Content Finder. This activity is built around the children making rectangles (first with discrete objects, then with graph paper and eventually 'in their head') then breaking them and remaking them in an organised way, but keeping the total number in the array / area of the rectangle constant. Suitable for threading. Materials One Poly Plug per pair One calculator per pair Graph paper

Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.

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Acknowledgement

In an address to the 2013 annual conference of the Mathematical Association of Victoria, Mike Askew showed a little work from classrooms on making and breaking rectangles 'on paper'. Putting that together with elements from a video produced by Paul Haenen, Loxton North Primary School, South Australia for one of our six day professional development programs and adding a touch of working like a mathematician, generated this activity.

Our challenge today is to split the rectangle and move sections around to create a new rectangle shape. The number of children stays the same each time. Of course we have to record on the calculator and in our journal too.

The screen captures below from Paul's video give an idea of how this plays out when the children move from the introduction to working with their own materials and numbers. Red Poly Plug are excellent for this activity, and a bit less noisy than bottle caps scraping on the table. These children were in Years 3/4. Keep your eye on the child in the background who is using 24 tops. The girl in the foreground is explaining the activity to the camera.

Recording Notice that 1x... and ...x1 are usually the last ones to be recorded to match the language being used. Given the position of the tummy being implied by the orientation of the writing above, the drawing showing 1 x 24 is actually showing 24 rows of 1. Similarly for the related drawing. Also, it is just as important to be sure about what doesn't work as well as what does. The boy in this image is trying very hard to make 24 into rows of 5, but even though he has told the camera that he doesn't think it will work, he keeps moving caps into the empty space for a little longer before deciding it really can't be done. How could this arrangement be recorded on the calculator?

Next Steps
This is a perfect activity to thread because the children can spend a few minutes exploring and recording a new number three or four times a week for two or three weeks. The word factors can be introduced as appropriate.

However using concrete materials can limit the size of the numbers explored, so the next step is to begin with a rectangle drawn on graph paper. The starting rectangle sets the number for the day, and children try to find all other rectangles with the same number of squares inside (same area) and record the factor pairs of each one. The subsequent rectangles don't have to be drawn, but any proposed factors do have to be checked on the calculator to make sure they equal the 'area of the day'.

A later development is to sketch a rectangle on plain paper, rather than graph paper, and mark its dimensions. Then list the factors of all the rectangles that can be derived from it by making and breaking, again checking with the calculator. Keep encouraging children to imagine the rectangle that matches the factor pair. Set challenges like:

• Arrange your factor multiplications in a pattern.
• What happens if the first number in the multiplication is one half (or another fraction, or a decimal)? Use your calculator to help you find the second number. Sketch the rectangle.
• Investigate starting with squares rather than rectangles.
• Investigate making and breaking cuboids to create factor triples.

With older children you might try graphing the lengths and widths (factor pairs) of the rectangles derived from the number of the day. This will produce points that appear to lie on an interesting path. It should also provide another way to check predictions for decimal and fraction widths or lengths.

Option 3 of the software for Maths300 Lesson 97, Tackling Times Tables, relates to and reinforces these later challenges.

Further, making and breaking rectangles is the basis of factorisation in algebra. Introducing Algebra is therefore a Calculating Changes activity related to Making & Breaking Rectangles.

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