# Task Maths

One of my favourite resources on the STEM Learning website is the Task Maths series of books written by Barbara and Derek Ball. There are five books aimed at year 7 to year 11 students.

There are many ways to introduce Pythagoras’ Theorem. I use the ‘Tilted Squares’ task (book 4 chapter 10) as it enables students to use a variety of mathematical topics along the way. To start the lesson we discuss the length of this line.

This leads to a discussion about rounding, errors, tolerance…but we agree that the line in 4cm long

We then discus the length of this line.

Is it 4cm long, less than 4cm or longer than 4cm? How do you know? How can we find its length?

A digression: we all agree that the area of this square is 16cm^{2} and discuss how we know.

I then pose the problem, a square has an area of 50cm^{2}, what is the length of one side of the square which leads to a discussion of the use of square roots.

Is it possible to find the area of this square?

If so, how can it be done? There are two usual strategies:

Boxing: in which a larger square is placed around the square. The required area is the difference between area of the larger square and the area of the sum of the four right angled triangles.

Splitting into triangles: the required area is the sum of the areas of the four right angled triangles and the area of the smaller square.

In each case, the area of the required square is 17, allowing us to find the length of the side of the square to be √17 which is 4.1 cm to 1 decimal place.

We denote this tilted square as a square.

Students then explore other tilted squares, finding the area in each case.

We then collect all the data from around the room, display it in an agreed ordered manner and attempt to spot patterns and/or connections and generalise. The connection is made that the area of the tilted square is equal to a^{2} + b^{2}

A little more discussion allows the development of the formalisation of Pythagoras’ Theorem using this diagram based upon areas of squares: the area of the largest square is equal to the sum of the areas of the two smaller squares when the three squares form a right angle.

We then consider the areas of the squares in these two cases: when the squares form an acute angled triangle and when the squares form an obtuse angled triangle.

The Task Maths collection can be found here:

## Pete Thompson

I used the Tilted Squares activity very much as described in this article from Task Maths. As a practical introduction I often used an activity called Take Two Squares [an example attached, although I rarely used it in this exact form - tailoring it to match profile, confidence and previous experiences of my classes, etc]. I always used the basic premise that two squares of unit area cut up will obviously maintain that area when reformed from dissection . . . so (question) "What is the length of each side?" and lead with 4 squares: a² = 4, etc . Then introduce tilted square idea - I would draw on a dotty grid and project for class.

"5-Bar Gates" [attached] is a useful simplified activity that gives practice Pythagoras and opportunities to produce a simple algebraic formula.