4 Arm Shapes

Task 154 ... Years 3 - 7

Summary

Tiling patterns of various sorts can be found in many buildings, perhaps even in your school. The pattern used here is straightforward. The shape is an addition sign and the central tile is highlighted by a change of colour. The essence of the challenge is:
If I tell you any length for each arm can you tell me how many tiles are needed?
This cameo has a From The Classroom section which shows unique ways of 'seeing' the generalisation that is the basis of the algebra in this problem.

4 Arm Shapes also appears on the Picture Puzzles Pattern & Algebra B menu where the problem is presented using one screen, two learners, concrete materials and a challenge.

 

Materials

  • 16 tiles of one colour and one of another
  • marker pen & cloth

Content

  • basic number skills
  • seeking & seeing patterns
  • generalisation
  • equivalent algebraic expressions
  • symbolic representation
  • substituting into equations
  • solving equations
  • graphing ordered pairs
  • relationship to gradient and y intercept
4 Arm Shapes

Iceberg

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

Answers are:

Length of arm 1 2 3 4 ... 10 ... 100
Number of tiles 5 9 13 17 ... 41 ... 401

and students should keep this record in their journal.

The iceberg begins with the last challenge: Can you explain how you worked it out?
There are at least four ways and it is important to realise that the way the student 'sees' is the one that makes sense to them. The way we 'see' the generalisation may be different, but it is not more correct.

  • Generalisation A
    To find the number of tiles double the arm length and add one, then add two arm lengths.
  • Generalisation B
    To find the number of tiles double the arm length and add one, then multiply this answer by two and subtract one.
  • Generalisation C
    To find the number of tiles you need five the first time you build then four each round after that.
  • Generalisation D
    To find the number of tiles multiply the arm length by four and add one.
Encourage students to record their view and at least one other. Record in words as here once the generalisation has been made orally. The written words are the genesis of symbolic representation as an equation:
  • Generalisation A
    To find the number of tiles (T =) double the arm length and add one (2A + 1), then add two arm lengths (+ 2A).
  • Generalisation B
    To find the number of tiles (T =) double the arm length and add one (2A +1), then multiply this answer by two and subtract one [2(2A + 1) - 1].
  • Generalisation C
    To find the number of tiles (T =) you need five the first time you build (5) then four more each round after that [+ 4(A-1)].
  • Generalisation D
    To find the number of tiles (T =) multiply the arm length by four and add one (4A + 1).
which become:
  • Generalisation A ... T = 2A + 1 + 2A
  • Generalisation B ... T = 2(2A + 1) - 1
  • Generalisation C ... T = 5 + 4(A - 1)
  • Generalisation D ... T = 4A + 1
These are all equivalent algebraic expressions and, by reference to the tiling pattern, students will be able to tell you what each symbol means and why particular operations and numbers are there.

Extend further with questions such as:

  • Do these different ways of seeing the pattern give the same answers for arm lengths of 5, 17, 26?
  • Suppose I tell you a number of tiles I have. Can you tell me the length of the arm of the tiling pattern I could build?
  • Can I tell you any number for the number of tiles? Discuss.
  • Choose any five numbers for the arm length. Work out the number of tiles in each case and make pairs of numbers like this (arm, tiles). If these pairs were plotted on a graph what would you expect to see. Plot them to check your hypothesis.
  • If you joined up these dots with a pencil line, how could you measure the slope (gradient) of the line? Which number does it go through on the vertical axis?
  • What happens if we change the tiling pattern?

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 begin this lesson you will need a number of cards about 20cm square, one of which is different from the others. Hand out the cards and use a central floor or table space to invite the students to tile the area with the pattern above. Explain that the problem is:

If I tell you any length for each arm can you tell me how many tiles are needed?
and more importantly:
Can you explain it to me in more than one way?
Provide small tiles (or Poly Plug) to allow group exploration. Gather the students back at the floorboard model and invite them to use it to explain how they see the problem. Proceed further with questions such as those above.

It might help you to picture how this lesson could play out in your classroomas you watch two colleagues explore Teaching Craft with 4 Arm Shapes in this video on the You Tube channel of the Association of Teachers of Mathematics (ATM), UK. You can also find it through Mathematics Centre Cube Tube.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 40, 4 Arm Shapes, which includes an Investigation Guide.

Visit 4 Arm Shapes in Menu Maths Pack A.

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 4 Arm Shapes task is an integral part of:

  • MWA Pattern & Algebra Years 3 & 4
  • MWA Pattern & Algebra Years 7 & 8

The 4 Arm Shapes lesson is an integral part of:

  • MWA Pattern & Algebra Years 3 & 4
  • MWA Pattern & Algebra Years 7 & 8

4 Arm Shapes task is also included in the Task Centre Kit for Aboriginal Students.

From The Classroom

Doug. Williams
Consultant
My classrooms are usually with teachers because my work is 100% professional development. however, the great thing about my work is that the professional learning is often mine. I often use 4 Arm Shapes in algebra courses; it's seems so obvious and text-book like, but has so much more. In fact, at least four text book chapters are built into the one activity. I used it recently when working overseas and that's when I learnt a little more about how you could look at this problem.

A teacher in one of my workshops at the ATM conference in England suggested that perhaps someone was ripping up an old courtyard to leave the four arm shape:

The old courtyard would be a square, each side of which is as long as 2 arms plus 1. Four squares would be ripped out and each one would have a side length of 1 arm. So the number of tiles in the 4 arm shape could be calculated using:

(2A + 1)2 - 4A2

Which makes the 4 arm shape, physically and algebraically, the difference between two squares!

(Note: If this teacher reads this, please make contact so you can be given due credit.)

Johan Olsson, a teacher in training at one of my workshops at Högskolan Malmö, Sweden, used transformations to explain his way of seeing the algebra. Move the left and top arm as shown:
Three arms will always form a rectangle 3 units wide with a depth of 1 arm plus 1. The remainder of the fourth arm will always be two units less than 1 arm. So the number of tiles in the 4 arm shape could be calculated using:

3(A + 1) + (A -2)

Algebra is concrete, visual and makes sense! This principle is explored further in the video Teaching Craft with 4 Arm Shapes, stored in Cube Tube.

Green Line
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