What Goes Around

Purpose

This unit is made up of five stations which investigate the relationships between the area and perimeters of squares and rectangles. 

Specific Learning Outcomes
  • use a formula to calculate the area of rectangles and squares
  • investigate the relationship between the perimeter and area of rectangles
Description of Mathematics

When the students are able to measure efficiently and effectively using standard units, their learning experiences can be directed to situations that encourage them to "discover" a measurement formula. In area work, the students may realise as they count squares to find the area of a rectangle, that it would be quicker to find the number of squares in one row and multiply this by the number of rows. In the same way, the students might find a formula for calculating the area of a right-angled triangle by seeing it as half of a rectangle.

Dissecting and reassembling parallelograms, trapezia and triangles, allow students to "discover" and understand area formulae.

Summary of Area Formulas

Area formlas

The unit could be taken as a series of sessions with the whole class or the students could circulate around the stations over the five days.

Required Resource Materials

Squared paper (1 cm)

Copymaster of stations

Scissors

Squared paper

Square tiles

30 cm rulers

Calculator

Tape

Coloured pens

Activity

Station 1

In this station we investigate the relationship between the area and perimeters of rectangles. The investigation is posed as two problems for the students to work on independently or with partners.

 Resources

  • Square tiles
  • Squared paper
  • Copymaster of Station 1 

Problem 1:
The landscape gardeners have thirty-six square paving slabs to make a rest area in the middle of a lawn. To make it easy to mow they want the rest area to be rectangular in shape and have the least perimeter (distance around the outside) as possible.
What arrangement of the tiles gives the smallest possible perimeter? (6 x 6)
Can you explain why this happens? (A square is the rectangle with the least perimeter for a given area)

Problem 2:
Use what you have learned from the first problem to work out which rectangular rest area the gardeners would make if they had forty-eight tiles.
(6 x 8 is the closest approximation of a square that can be produced with 48 tiles)

Station 2

In this station you design a house for the Affluent family.

Resources

  • Squared paper (1 cm)
  • Scissors
  • 30 cm rulers
  • Copymaster of station 2 
  1. Mr and Mrs Affluent want you to design their new house. They give you the following sizes for the rooms of the house.
    Living Room: 48 square metres Bathroom: 8 square metres
    Dining Room: 12 square metres Laundry: 6 square metres
    Kitchen : 12 square metres Toilet: 2 square metres
    Bedroom One: 24 square metres Bedroom Two: 16 square metres
    Bedroom Three: 20 square metres Bedroom Four: 12 square metres
    Conservatory/Entrance: 12 square metres
  2. Use squared paper, with a scale of 1 cm = 1m to design the house. You may find it easier to cut each room out of paper first so it can be moved around. Remember to keep passage ways down to a minimum.
  3. When your house plan is complete draw it showing the side lengths and areas of each room. Record the wall lengths of the house and its total area.

Station 3

In this station you investigate the link between the side lengths of a square and its area.

Resources

  • Square tiles
  • Squared paper
  • Calculator
  • Copymaster of station 3 
  1. Make a square with 25 square tiles. What is the side length of the square? Key in √25 on the calculator. What do you notice?
  2. Use square tiles and your calculator (if you need) to complete the entries in this table:

     

    Number of Tiles

    Side Length of Square

    Square Root

    9

     

    √9 =

    36

     

    √36 =

    81

     

    √81 =

    49

     

    √49 =

    100

     

    √100 =

    What does the square root function on a calculator do?

  3. Use what you have found out from part 2 to draw squares with the following areas on squared paper.
    121 square units
    6.25 square units
    12.25 square units
    18 square units
    42 square units
    90 square units
  4. What are the side lengths of these squares?

squares

Station 4

 In this station we investigate the area of rectangles.

Resources

  • Squared paper (1 cm)
  • Scissors
  • Tape
  • Copymaster of station 4
  1. Cut out a 4 cm x 4 cm square from squared paper.
    What are the area and perimeter of the square in centimetres?
  2. By making one straight scissor cut and moving and taping the pieces make a rectangle with a perimeter of 20 centimetres.
    What is the area of the rectangle? Explain how you got your answer.
  3. Cut out a 12 cm x 3 cm rectangle from squared paper. By cutting, moving, and taping (as in part 2) change the rectangle into a square.
    What changes happen to the area and perimeter from the starting rectangle to the square?
    Try to explain why this happens.
  4. Change a 8 x 3 rectangle into a 6 x 4 rectangle by making two straight scissor cuts, moving the pieces and taping. What changes happen to the area and perimeter in this case?
  5. Make up a cut, move and tape rectangle/square puzzle for someone else to solve.
  6. As an extension cut out a 6 x 6 square. By cutting the square in half, moving the pieces and taping change the square into a triangle. Find the height and the longest side length of the triangle. How do these lengths compare to the area of the square?
  7. Investigate changing other rectangles into triangles with one cut. Find the height and longest side length of each triangle and compare it to the area.
    What do you notice?

Station 5

In this station we look for patterns in the perimeters and areas of squares.

Resources

  • Square tiles
  • Squared paper
  • Coloured pens
  • Copymaster of station 5 
  1. Below are a 2 x 2 square and a 3 x 3 square. You may wish to make them with square tiles.
    What is the difference between their perimeters?

    squares 2x2 and 3x3
     
  2. Compare the perimeters of a 3 x 3 square with a 4 x 4 square.
    Compare the perimeter of a square with the next biggest square.
    What do you notice? Why does this occur?
  3. Go back to the 2 x 2 and 3 x 3 squares. Compare the areas of these squares.
    Compare the area of different squares with the area of the next biggest square. What pattern do you notice?
    Organising your results in a table may help:

    Square

    Area

    Difference from next biggest square

    1 x 1

    1

    3

    2 x 2

    4

     

    3 x 3

      

    4 x 4

      

    5 x 5

      

    6 x 6

      

     

  4. Investigate the difference in areas and perimeters between rectangles and the next biggest rectangle, like 1 x 2 and 2 x 3, 2 x 3 and 3 x 4, 3 x 4 and 4 x 5…
    What patterns do you notice?
    Try to explain why each pattern occurs

 


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