Tangram Teasers

Task 79 ... Years 2 - 8

Summary

The 7-piece tangram is a well known shape puzzle with a long and distinguished history. The challenges in this task are only some of the many hundreds that have been developed over the centuries. None of them are necessarily straightforward and the rocket, in particular, has been known to challenge the spatial ability of teachers as well as students.

This cameo has a two From The Classroom contributions. Stimulated by a superb example of quilting, one uses a limited number of tangram shapes cut from coloured paper squares to create fabulous designs. But it's not a free for all. It's an investigation which comes about by limiting what can be done. The second contribution records a discussion between a student and a mentor about finding a solution to one of the puzzles.

 

Materials

  • 7-piece tangram puzzle

Content

  • 2D spatial perception
  • length, area, perimeter
  • angles and angle sum of polygons
  • informal transformation of shapes
  • value relations
  • problem solving
  • problem posing
  • measurement
Tangram Teasers

Iceberg

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

No solutions are provided for this task. The idea is that as students use the task, the class will build up a set of solutions. As shown in From The Classroom below, it may take some effort to find some of the solutions.

  • How will you record your solutions?
  • Use the pieces to create your own tangram puzzles for others to solve.
  • Find a way to explain to someone else (written report, drawings, comic strip, photos, PowerPoint, poster, etc.) how to make a 7-piece tangram from a 10cm square.
  • Design your own tangram puzzle which has a different odd number of pieces.
  • Invent puzzles made with your design. Try them on others.
  • Investigate the history of tangrams.
  • For the 7-piece tangram:
    • If I tell you that the square you can make with all the pieces is worth $1, can you tell me value of each of the other pieces?
    • If I tell you that the small square piece is worth $1 can you tell me the value of each of the other pieces?
    • Make up some more challenges like this and try them on your friends.
  • What can you tell me about the area of each shape in Question 1 on the card?
    What can you tell me about the perimeter of each shape in Question 1 on the card?
    If the sides that touch have to be multiples of each other (ie: the same, twice as big, three times as big...), make the shape you think has the shortest perimeter. Longest perimeter?
Ulla Öberg, Sweden, has suggested a neat sequence of angle work using the 7-piece tangram. Begin by telling the students that the largest square (the one that is dissected to make the tangram) has 4 right angles.
  • Show me the other shape with 4 right angles.
  • Show me the shape with 0 right angles.
  • Which shapes have some right angles?
  • Put the pieces together to make the largest number of right angles.
Tell the students that because of the way ancient people measured the earth we say that a right angle measures 90o (90 degrees).
  • Find the size in degrees of all the other angles in the shapes.
  • Put pieces together to make polygons of 3, 4, 5, 6, ... sides. They don't have to be regular. Work out the sum of the interior angles for each one.
  • Try to find a connection between the number of sides and the total of the angles.

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.
   

You will need at least one tangram set between two to run the whole class investigation, but you don't necessarily have to buy them. Make the square with the task pieces. This will show you how the 7-piece is designed. Recreate this in a computer drawing program and print multiple copies on card. Students can cut them out and pack them into envelopes as a class set.

The investigation(s) you choose can be guided by any of the directions suggested above.

At this stage, Tangram Teasers 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 Tangram Teasers task is an integral part of:

  • MWA Space & Logic Years 3 & 4

From The Classroom

St. John's Uniting Church

Cowes
Australia
Not so much a From The Classroom contribution as a For The Classroom contribution. This superb example of quilting, photographed at St. John's, was created by Esther Gassmann. Click the image to reveal and save a higher resolution version if required.

Esther has limited the number of shapes she has used. How many different shapes has Esther used? (Don't include the borders.)
  • What mathematical connections can you find between these shapes?

  • What is the area of each panel?

  • Choose one panel. Use square dot paper to sketch the pieces in the panel.

  • The centre square in the panel is worth $1. Inside every shape write its value. What is the value of the whole panel? How do you know? Can you check it another way?

  • Limit yourself to using the same shapes as Esther and cut them from coloured paper squares. You can cut as many as you need but use only two colours. How many different panels like Esther's can you make? Make a display.

  • Think of questions you could ask about your display which can be answered using mathematics.
What happens if you are allowed to use three colours?
Quilting by Esther Gassmann

Editor's Notes:
... We would love to publish photos of student designs and examples of their questions.
... There are connections between this From The Classroom and the contributions in Task 96, Networks, Task 168, Mirror Patterns 3 and Maths300 Lesson 73, Halving Squares.

Tasmanian Primary School

Ruth Court
Upper Primary
An email message received at the Task Centre office read:
To dear Mr Williams,
I am a student for Mrs Court's class and my name is Jessy. I am writting to you to say that I have been working on the Tangram Teasers and I have been on it for 2 days and can not do the Third shape at the bottom it is impossiple! and I think it can't be done. Can you please write back and tell me how to do it PLEASE!
Yours Sincerly Jessy
Dear Jessy,
Thank you for writing to me. I don't have a lot of time to answer a million letters from children, but today you were the only one. Phew!
Now that was one tough puzzle you sent me. I have been struggling with it for about half an hour. But at last I can tell you it is NOT impossible.
Do you want to keep trying or do you need a clue?
Say hello to Mrs. Court for me.
Keep smiling,
Mr. Williams
Dear mr Williams
Thankyou for writting back please can i have a clue.
From Jessy
Hi Jessy,
The first clue is that when a mathematician finds that one strategy doesn't work, she simply tries a different strategy. When I looked at the picture my first strategy was that the biggest triangle should be the bottom piece and the straight side of the rocket should come up from there. But I was wrong! The biggest triangle is not in the base at all!
Keep smiling,
Mr. Williams
Dear Mr Williams
Thanks for the clue. I can't look at it today because I have to go away for my holidays but I will try again when school comes back. I thought I had tried EVERYTHING!!
From
Jessy
Dear Jessy,
Why don't you ask Mrs. Court if you can take the Tangram Teasers task home with you on holiday.
Keep smiling,
Mr. Williams

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