Cross & Square

Task 196 ... Years 2 - 8

Summary

What you see is not quite what you get with this task. The four shapes can make both a square and a cross, but measuring or calculating area suggests that the areas of the cross and square are different. Pythagoras' Theorem is lurking in the background waiting to be called on to add to the explanation of this paradox.
 

Materials

Content

  • concept of proof
  • decimals, calculations
  • difference between 2 squares
  • measurement, angle
  • measurement, area
  • measurement, length
  • multiplication, array concept
  • multiplication, calculations / times tables
  • multiplication, multiplicative thinking, multiplication principle of counting
  • numbers, square
  • Pythagoras theorem
  • recording mathematics
  • shapes, properties
  • spatial perception, 2D or 3D
Cross & Square

Iceberg

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

First, let's solve the two spatial challenges:

Encourage students to record both solutions in their journal. Tracing the pieces is the easiest way and, as suggested on the card, tracing them first onto the grid paper provided leads into the next step. The grid is made up of 1.5cm squares which is the same size as the construction unit for the wooden pieces.

  • Counting squares to find the area of the Cross is no problem. In fact, students will hopefully show their multiplicative thinking abilities by seeing the Cross as made from 5 squares, each of which is 4 rows of 4 unit squares. A total area of 80 squares.
  • Counting squares to find the area of the Square is possibly even easier. Nine rows of 9, which is a total area of 81 squares.
Huh! The wooden pieces didn't change size, so how can the area be different?

The Cross area must be correct. Looking more carefully at the Square and holding it securely on the grid paper in the top left corner with all its pieces touching shows there is a very thin gap between the edge of the square and the 9 x 9 grid boundary. In this enlargement, the difference is shown in green.

But could the area of that thin green line really be equal to one square??

Let's check it another way, after all, a minute ago we were prepared to believe the area of the Square was 81. However, the other way will require Pythagoras' Theorem, so if you are working with younger children you will be happy that they record something about the Square being just a little shorter than 9 units on each side.

Using Pythagoras

The solutions above show that the sides of the Square are formed from the lengths made by the two cuts on the Cross. Each cut produces two sides of the square. Placing the Cross on the grid paper shows that the cuts are the same length.

Each is the hypotenuse of a right angle triangle with sides 4 units and 8 units. Therefore the length of each side of the Square:
= SQRT (42 + 82)
= SQRT (16 + 64)
= SQRT 80
= 89442719099991587856366946749251
Which is a bit over the top with significant figures of course, but it will help to make a further point.

Firstly though we have proved that the object formed from the pieces of the Cross has equal sides. But this doesn't prove it is a square. To be a square, the 'corner angles' have to be 90. What's the reasoning that would prove they are?

This is not a trivial question, so we leave it to you satisfy yourself geometrically.

Getting back to checking another way that the green area is 1, the green area also represents the difference of two squares which can be calculated using:
= (9 - 89442719099991587856366946749251)
   x (9 + 89442719099991587856366946749251)
= 0.055728090000841214363305325074895
   x 17944271909999158785636694674925
= 1
to the accuracy of the calculator in my Start/Programs/Accessories. How does your calculator perform this calculation?

Here the rule for calculating the difference of two squares has been used on the assumption we know how that rule is derived. If we don't we can:

  • Calculate the green area by seeing it as one long rectangle down the right side and a shorter one across the bottom.
or

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.
   

Turning this task into a whole class lesson involves drawing and cutting with some accuracy. Perhaps that's one additional reason why you might use it. You will need 3 copies of the Recording Sheet for each pair, rulers, pens and scissors for each student. You will also need to click and save Image 1 and Image 2 for display.

Hand out one Recording Sheet, a ruler, pen and scissors to each student. Display Image 1.

In today's lesson we are going to create our own puzzle, but you will need to rule and cut as accurately as you can for it to work. I want you to start by carefully cutting this shape from your paper. Cutting exactly down the middle of the edge lines will help, but the concave corners are still tricky.
When the crosses have been cut, ask for and record the area of the cross.
  • Can you tell me how you know that?
  • Can you check it another way?
Discuss alternatives and then display Image 2.
Now carefully rule these two lines and cut along them. This time try cutting down the middle of your ruled line. How many pieces would expect to make?
Ask for the total area of the 4 pieces and record it.
Your challenge now is to make the Cross pieces into a Square.
When the solution is discovered, ask for the area of the square. No doubt students will say it is 80 squares.
  • Can you tell me how you know that?
  • Can we check it another way?
Hand out another Recording Sheet, this time one between two to encourage discussion, and ask the students to measure the square on the grid paper. Discuss the apparent answer of 81 as appropriate to their level and ask students to record the investigation in their journals.

Continue as above with the calculation investigation if it is appropriate for your students.

At this stage, Cross & Square 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 Cross & Square task is an integral part of:

  • MWA Space & Logic Years 3 & 4
  • MWA Chance & Measurement Years 9 & 10

Green Line
Follow this link to Task Centre Home page.