Session 1

In this session students are introduced to the idea of using multiplication to find the area of a rectangle.

1. Show the students a large rectangular piece of paper measuring 30cm by 60cm and a pile of smaller squares each measuring 10cm by 10cm (like memo squares). Tell the students you want to know how many of these small squares are needed to cover the large paper rectangle. You can set a context such as "this is the school garden and these are the concrete tiles we will be using to cover it". 
   How many square tiles will cover this area?
2. Let students briefly discuss how they might estimate an answer, then share the ideas. Look for students to explain two main processes:
   • Iteration – repeated copying of the unit of measurement (memo square) along a side, with no gaps or overlaps.
   • Equi-partitioning – equally splitting a side until the divisions are about the same length as the sides of the memo square. 
     
     Modelling these processes on a whiteboard, interactive whiteboard, or with the use of materials could support students to develop their thinking.
      
3. Ask about how the square units will be arranged. Introduce the terms, rows (across), columns (down), and array (a structure of rows and columns) if students are not familiar with those words. 
   Do students recognise the array structure in the arrangement of square units?
4. Ask a volunteer to place the squares units side by side on the rectangle. Blu Tac can help to secure the units in place.
5. Ask the students for ways to work out the total number of units. One by one counting, or skip counting/repeated addition (6, 12, 18 or 6 + 6 + 6 = 18) are legitimate strategies given the small number of units. Explain that the area of the rectangle is 18 squares.
   Can we count the squares even more efficiently?
6. Record 3 x 6 = 18 and ask students where they can see representations of six and three in the model (i.e. in the number of rows and columns). Ask where the 18 is found (i.e. it is the total number of square units).
7. Model the same process with different sized rectangles, e.g. 20cm x 80cm, 50cm x 40cm, 100cm x 100 cm (A square is a special rectangle with all sides the same length). The rectangles might be cut out of paper, created with play dough, drawn on the whiteboard, or drawn on the carpet/concrete with chalk.
   Look for students to:
   • Recognise the array structure.
   • Use multiplication as an efficient method to calculate the area.
8. Provide the students with copies of Copymaster 1. Tell them to work with a partner to find out the area of each rectangle in small squares. As students work, look for their calculation strategies. Are they using additive or multiplicative methods?
   Recognise that much will depend on their knowledge of multiplication facts and strategies. Smaller rectangles that utilise simpler times tables could be drawn and used by pairs of learners.
9. Gather the class and share solutions. It is interesting that Rectangle E, a square, has the greatest area, though other rectangles may look larger. To extend learners, you could ignite discussion around this.
   Answers: A (3 x 7 = 21), B (6 x 6 = 36), C (4 x 11 = 44), D (11 x 3 = 33), E (7 x 7 = 49), F (8 x 6 = 48), G (10 x 2 = 20).
   What do the answers tell us about these rectangles?
   How big are the little squares? Students might measure with a ruler to check that the units are square centimetres.
   Ask students to include the unit in their answers, e.g. 21cm². Recording the notation for each rectangle is good practice.

Sessions 2 and 3

1. Discuss the idea of a formula. You might find a funny video online about someone using a formula to make something. A recipe is a type of formula. Students may also make connections to playing sports (e.g. a team follows a formula to play well and win), tikanga (correct ways of doing things), or car racing (e.g. in Formula One racing, the “formula” entails a set of rules that all racers’ cars must meet).
   What do we mean by a formula?
   Do students explain that a formula is like an algorithm, or rule, that we can follow to get the same result each time? 
   Record W x L = A. This is a mathematical formula written as an equation. 
   I wonder what the letters W, L and A might represent?
2. Apply the formula to the examples students worked on in the previous lesson (Copymaster 1). 
   For example, Rectangle B had seven rows of five squares.
   The row gives the length of the rectangle. In the case of B length equals 5. (rub off L in the formula and write 5 in its place)
   The number of rows gives the width of the rectangle. In the case of B width equals 7. (rub off W in the formula and write 7 in its place).
   The formula now reads 7 x 5 = A. I wonder what A equals. What value for area makes the equation true and matches the formula?
3. Ask students to use the examples from Copymaster 1. As a group, practise starting with the formula, and substituting the values of length, width, and area for each rectangle. Students may benefit from using materials to model the use of the formula.
4. Provide students with a group worthy task to work on collaboratively (see Copymaster 2). This could be linked to school events (e.g. make a new sign for our classroom, design a school garden, design the size of a hāngi pit). Students might be given 1cm grid paper, 1cm squares, or work in their exercise books. There are several programmes online that allow students to model the construction of arrays using 1cm squares. Make sure to thoroughly investigate any programme you wish to use, to ensure its use will be appropriate and purposeful for your students.
5. Look for students to apply the W x L = A formula to construct appropriate rectangles. For example, if they choose an area of 72cm² they will need to consider all the factors of 72. Encourage students to find those factors systematically. Some students may benefit from the support of a multiplication basic facts poster or list.
   A systematic approach involves starting with 1 as a factor then increasing the smallest factor by one and testing 72 for divisibility.
   1 x 72, 2 x 36 (72 ÷ 2 = 36), 3 x 24 (72 ÷ 3 = 24), 4 x 18 (72 ÷ 4 = 18), 5 x (72 is not divisible by 5), 6 x 12 (72 ÷ 6 = 12), 7 x (72 is not divisible by 7), 8 x 9 (72 ÷ 8 = 9).
   If the process continues the factors will appear in reverse order, e.g. 9 x 8 = 72. 8 x 9 and 9 x 8 are essentially the same rectangle though they may appear differently if the direction of the label is considered.
6. Gather the class to discuss solutions and look at real sized diagrams of the possible labels. Some options are mathematically correct but unworkable as a label option.
   Discuss criteria for eliminating labels. For example, a label with a width of less than 5cm might be considered too ‘skinny.’
   Discuss the best options, cut them out at real size, then use a real jam jar (or object that is relevant to the context of the learning) to consider how well each label/array design will work.
7. In the jam jar context, students might write a letter to Karly outlining how they investigated her problem and giving their recommendations. Their mathematical thinking could be used as the basis of a persuasive letter in other contexts.
8. Another good investigation is to tile a large rectangular area with 1m² carpet tiles. A hall or gymnasium is an ideal area though a classroom is also viable. Tiles of that size are commonly found at hardware stores. You will find an advertisement easily online. 
9. Get students to construct a unit square using newspaper or recycled boxes. They can use the unit to get a sense of the scale of 1m² and make estimates of the area of the space before they calculate.
10. Ask students to work in small teams to calculate the number of tiles that will be needed for the rectangular space. Look for them to measure the side lengths of the rectangular area using tape measures, trundle wheels, or metre rulers.
    Do they apply the W x L = A formula?
11. Students can find the area of composite shapes by finding the area of the rectangles. For example: 
    
12. This shape can be seen to be made up of two 2cm by 4 cm rectangles, or a 2cm by 6cm rectangle and a 2 cm by 2 cm rectangle, or 4 cm by 6 cm rectangle with a 4 by 2 rectangle missing. Use 1cm square units (e.g. memo pads) to demonstrate the construction of this composite shape. There are different ways to solve composite shapes. However, one of the simplest methods starts with breaking a composite shape down into basic shapes (e.g. 2 rectangles). You could model this with memo pads or tiles. Next, find the area of the basic shapes you have constructed. Finally, add the areas of the basic shapes together. To support the development of this thinking, you could calculate the area of the shape shown above in two different ways. First, calculate the area from 2 rectangles, each with an area of 4cm x 2cm. The total area of each rectangle is 8cm². Therefore, the total area of the composite shape is 16cm². Next, calculate the area of the shape as one 2cm x 6cm rectangle (12cm²) added to one 2cm x 2cm (4cm²). Calculating the area of the same shape in different ways will allow for greater student collaboration, and will allow for students to learn from each other.

Session 4

In this session students explore using proportional reasoning to find areas of rectangles.

1. Pose the problem: Sam’s family was shopping for a new table for the wharekai in the local marae. The first one they looked at measured 2m by 3m. Sam said if they wanted one with an area twice as big they should get the 4m by 6m size. Is Sam right?
2. Ask the students to draw pictures of the table and to help them decide if Sam is correct.
3. Work with students to establish that doubling the area only involves doubling one side of the rectangle. Doubling both sides of the rectangle increases the area by four times.
4. Using this proportional reasoning students will be able to solve problems without recalculating from side lengths. Here are some example problems:
   • The recipe made enough icing to cover the top of a 20cm by 20cm cake. What size cake can you ice if you double the amount of icing?
   • The birthday card had a front cover measuring 15cm by 10cm, what is the area of the piece of cardboard used to make it?
   • The marae had two areas that needed paving. Each area measured 5m by 8m. What is the total area to be paved?
   • The gardener charged his customers by the area of their lawn. If the bill was $20 to mow a lawn that was 6m by 20m, what should the bill be for a 20m by 12m lawn?

Session 5

In the session students demonstrate their ability to apply measurement of area independently. Consider what culturally relevant contexts can be incorporated into this task, to increase the engagement of your learners.

The following links provide pages from Figure It Out books that are suitable:

• Square Skills
• Adobe Bricks

Students might also create a mat design and provide the dimensions and areas of the rectangular pieces that compose it. An example is given below:

Station 1

At 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

At this station students design a house for the Affluent family. There are three problems to be investigated.

 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 = 1 m 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 passageways 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. That area is 156 m² (“156 square metres”)

Station 3

At this station students investigate the link between the side lengths of a square and its area. There are four parts to this problem: making a square with a given area, finding the side length of the square (by finding the square root of the area), drawing squares with given area measurements, and finding the area of square diagrams.

 Resources:

• Square tiles
• Squared paper
• Calculator
• Copymaster of station 3

Problem:

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?
   √25 gives 5 which is the side length of a square with area of 25 cm².

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	3	√9 =
   36	6	√36 =
   81	9	√81 =
   49	7	√49 =
   100	10	√100 =

   What does the square root function on a calculator do?
   Gives you the square root of that number (the number that multiplies by itself to give that number). In this context, it tells you the side length of the square with that area.

3. Use what you have found out from part 2 to draw squares with the following areas on squared paper.
   121 square units (11 x 11)
   6.25 square units (2.5 x 2.5)
   12.25 square units (3.5 x 3.5)
   18 square units (4.24 x 4.24)
   42 square units (6.48 x 6.48)
   90 square units (9.49 x 9.49)
    
4. What are the side lengths of these squares?
   
   The left square has an area of 2 cm² so the side lengths are √2.
   The right square has an area of 4.5 cm² (1/2 x 9) so the side lengths are √4.5.

Station 4

At this station we use several different problems to investigate the area of rectangles.

 Resources:

• Squared paper (1 cm)
• Scissors
• Tape
• Copymaster of station 4

Problems:

1. Cut out a 4 cm x 4 cm square from squared paper.
   What are the area and perimeter of the square in centimetres? (area = 16 cm² and perimeter = 16 cm.)
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.
   
   Area remains unchanged (8 x 2 = 16 cm²)
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.
   
   The area remains constant (36 cm²).
   The perimeter decreases from 30 cm to 24 cm.
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?
   Here is one solution. There are others.
   
   The area remains constant (24 cm²). The perimeter changes from 22 cm to 20 cm.
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?
   
   The longest side of the triangle is 12 cm and the height is 6 cm.
   The area of both the square and triangle is 36 cm².
   The area of the triangle is ½ x 12 x 6 = 36 cm².
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?
   Students should notice that ½ x b x h gives the area of a triangle.

Station 5

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

 Resources:

• Square tiles
• Squared paper
• Coloured pens
• Copymaster of station 5 

Problem:

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?
   
   The 2 x 2 square has a perimeter of 4 x 2 = 8 units.
   The 3 x 3 square has a perimeter of 4 x 3 = 12 units.
2. Compare the perimeters of a 3 x 3 square with a 4 x 4 square.
   The 3 x 3 square has a perimeter of 4 x 3 = 12 units.
   The 4 x 4 square has a perimeter of 4 x 4 = 16 units.
3. Compare the perimeter of a square with the next biggest square.
   What do you notice? Why does this occur?
   The difference is always 4 units because 1 unit is added and there are four sides.
4. Go back to the 2 x 2 and 3 x 3 squares. Compare the areas of these squares.
5. 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:

6. 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.
   Just like the square, 1 unit is added to each side and there are four sides. So the differences in perimeter are four each time.
   The difference in area increases by even numbers.
    

Activity

This activity is based on the perimeter and area of rectangles. As a general introduction, have your students look at this rectangle where the side lengths are given as l and w.
 

The area can be found by: area = l x w.
The perimeter can be found by: perimeter = 2 x l + 2 x w.
Problems that involve maximising or minimising one measurement while either holding the other constant or minimising it are common in the real world. Fred’s fence is typical of constrained maximisation or minimisation problems.
Students exploring question 1 are likely to try different side lengths that will result in an area of 80 square metres. The problem requires a systematic approach, so encourage your students to organise their results in a table or organised list:

In this way, the students can find all the solutions with whole-number side measurements and calculate the perimeters at the same time. They may notice that the closer the side measurements become to each other, the smaller the perimeter becomes.
Encourage your students to explore the minimum perimeters for rectangles with the areas 16, 36, and 64 (square numbers). They will find that the perimeter is minimised when the rectangle is a square. In this situation, the length of each side is the square root of the area. They can then go back to question 1 with the knowledge that the solution is the closest whole number to √80 = 8.944 (to 4 significant figures). Students are likely to argue that the question asked for a rectangle and that a rectangle is not a square. It is worth stopping to discuss this reasonable view. In everyday use, a rectangle and a square are different shapes, but in mathematics, a square is just a special case of a rectangle.
Provide the students with a set of rectangles and squares and ask them to describe the attributes of these shapes. Encourage them to come up with minimal definitions, listing just the attributes that are absolutely necessary to define the shape. Students will typically say that a rectangle has:
• 4 sides
• 4 right-angled corners
• 2 pairs of parallel sides.
If you ask them to draw a 4-sided polygon that has right-angled corners but does not have 2 pairs of parallel sides, they will find that this is impossible. So it is not necessary to state that opposite sides must be parallel. This gives us the minimal definition for a rectangle. The minimal definition of a square is “a 4-sided polygon with right-angled corners and equal sides”. Squares are therefore a subclass of rectangles.
In the Investigation, students try to find rectangles that have the same number for the measurement of their perimeter as they do for the measurement of their area.
One solution is a square with sides of 4 metres. Its perimeter is 16 metres, and its area is 16 square metres. If they are systematic, students should be able to establish the existence of two other whole-number solutions.
They could begin by setting the length (at, say, 2 metres) and exploring what widths might work. They will discover that no whole-number solution will work for a side length of 2. But if they then try 3, they will find that a 3 x 6 rectangle has an area of 18 square metres and a perimeter of 18 metres. 6 x 3 is a third solution, but this is not a genuinely different rectangle.
Having got this far, your students may guess that there are other rectangles that meet the requirement but that they do not have whole-number sides. There are in fact an infinite number of such rectangles. In the table below, there are six rectangles that happen to have a whole-number measurement for one of their two dimensions. You could give your students the length of side b and challenge them to find the length of side a (in bold in the table), using a trial-and-improvement strategy.

There is an algebraic relationship between the pairs of values of a and b that satisfy the requirement that the number of perimeter units must be equal to the number of units of area. The relationship can be expressed in this way:
(To find the length of the second side, double the length of the first and divide by its length less 2.) Students who are developing an understanding of symbolic notation may like to try using this formula to find other pairs for a and b with the help of a calculator or spreadsheet program such as that shown.

Links

Numeracy Project materials (see https://nzmaths.co.nz/numeracy-projects)
• Book 9: Teaching Number through Measurement, Geometry, Algebra and Statistics Investigating Area, page 11
Figure It Out
• Number: Book Three, Years 7–8, Level 4 Orchard Antics, page 23
• Number Sense and Algebraic Thinking: Book One, Levels 3–4 Tile the Town, Tiny!, pages 20–21

Answers

1. a. 5 different rectangular shapes are: 1 m by 80 m, 2 m by 40 m, 4 m by 20 m, 5 m by 16 m, and 8 m by 10 m. Only the last two shapes would suit the dodgems (the other three would be too narrow).
b. The 8 m by 10 m rectangle would use 36 panels and cost $108, which is cheaper than the other options. It is one of the shapes that would suit the dodgems.
2. a. There are 12 different-sized rectangles that could be made.

b. The largest option is 12 m wide and 12 m long, which gives an area of 144 m². The shape of this area is a square.
3. 50 m. The length must be 15 m because 10 x 15 = 150. 2 x (10 + 15) = 2 x 25 = 50 gives the perimeter.

Investigation
Answers may vary. There are three whole-number solutions: 4 x 4, 3 x 6, and 6 x 3 (which is the same as 3 x 6). There is an infinite number of solutions if rectangles with only one whole-number side or no whole-number sides are included.

Session 1

In this session students are introduced to the idea of using multiplication to find the area of a rectangle.

1. Show the students a large rectangular piece of paper measuring 30cm by 60cm and a pile of smaller squares each measuring 10cm by 10cm (like memo squares). Tell the students you want to know how many of these small squares are needed to cover the large paper rectangle. You can set a context such as "this is the school garden and these are the concrete tiles we will be using to cover it". 
   How many square tiles will cover this area?
2. Let students briefly discuss how they might estimate an answer, then share the ideas. Look for students to explain two main processes:
   • Iteration – repeated copying of the unit of measurement (memo square) along a side, with no gaps or overlaps.
   • Equi-partitioning – equally splitting a side until the divisions are about the same length as the sides of the memo square. 
     
     Modelling these processes on a whiteboard, interactive whiteboard, or with the use of materials could support students to develop their thinking.
      
3. Ask about how the square units will be arranged. Introduce the terms, rows (across), columns (down), and array (a structure of rows and columns) if students are not familiar with those words. 
   Do students recognise the array structure in the arrangement of square units?
4. Ask a volunteer to place the squares units side by side on the rectangle. Blu Tac can help to secure the units in place.
5. Ask the students for ways to work out the total number of units. One by one counting, or skip counting/repeated addition (6, 12, 18 or 6 + 6 + 6 = 18) are legitimate strategies given the small number of units. Explain that the area of the rectangle is 18 squares.
   Can we count the squares even more efficiently?
6. Record 3 x 6 = 18 and ask students where they can see representations of six and three in the model (i.e. in the number of rows and columns). Ask where the 18 is found (i.e. it is the total number of square units).
7. Model the same process with different sized rectangles, e.g. 20cm x 80cm, 50cm x 40cm, 100cm x 100 cm (A square is a special rectangle with all sides the same length). The rectangles might be cut out of paper, created with play dough, drawn on the whiteboard, or drawn on the carpet/concrete with chalk.
   Look for students to:
   • Recognise the array structure.
   • Use multiplication as an efficient method to calculate the area.
8. Provide the students with copies of Copymaster 1. Tell them to work with a partner to find out the area of each rectangle in small squares. As students work, look for their calculation strategies. Are they using additive or multiplicative methods?
   Recognise that much will depend on their knowledge of multiplication facts and strategies. Smaller rectangles that utilise simpler times tables could be drawn and used by pairs of learners.
9. Gather the class and share solutions. It is interesting that Rectangle E, a square, has the greatest area, though other rectangles may look larger. To extend learners, you could ignite discussion around this.
   Answers: A (3 x 7 = 21), B (6 x 6 = 36), C (4 x 11 = 44), D (11 x 3 = 33), E (7 x 7 = 49), F (8 x 6 = 48), G (10 x 2 = 20).
   What do the answers tell us about these rectangles?
   How big are the little squares? Students might measure with a ruler to check that the units are square centimetres.
   Ask students to include the unit in their answers, e.g. 21cm². Recording the notation for each rectangle is good practice.

Sessions 2 and 3

1. Discuss the idea of a formula. You might find a funny video online about someone using a formula to make something. A recipe is a type of formula. Students may also make connections to playing sports (e.g. a team follows a formula to play well and win), tikanga (correct ways of doing things), or car racing (e.g. in Formula One racing, the “formula” entails a set of rules that all racers’ cars must meet).
   What do we mean by a formula?
   Do students explain that a formula is like an algorithm, or rule, that we can follow to get the same result each time? 
   Record W x L = A. This is a mathematical formula written as an equation. 
   I wonder what the letters W, L and A might represent?
2. Apply the formula to the examples students worked on in the previous lesson (Copymaster 1). 
   For example, Rectangle B had seven rows of five squares.
   The row gives the length of the rectangle. In the case of B length equals 5. (rub off L in the formula and write 5 in its place)
   The number of rows gives the width of the rectangle. In the case of B width equals 7. (rub off W in the formula and write 7 in its place).
   The formula now reads 7 x 5 = A. I wonder what A equals. What value for area makes the equation true and matches the formula?
3. Ask students to use the examples from Copymaster 1. As a group, practise starting with the formula, and substituting the values of length, width, and area for each rectangle. Students may benefit from using materials to model the use of the formula.
4. Provide students with a group worthy task to work on collaboratively (see Copymaster 2). This could be linked to school events (e.g. make a new sign for our classroom, design a school garden, design the size of a hāngi pit). Students might be given 1cm grid paper, 1cm squares, or work in their exercise books. There are several programmes online that allow students to model the construction of arrays using 1cm squares. Make sure to thoroughly investigate any programme you wish to use, to ensure its use will be appropriate and purposeful for your students.
5. Look for students to apply the W x L = A formula to construct appropriate rectangles. For example, if they choose an area of 72cm² they will need to consider all the factors of 72. Encourage students to find those factors systematically. Some students may benefit from the support of a multiplication basic facts poster or list.
   A systematic approach involves starting with 1 as a factor then increasing the smallest factor by one and testing 72 for divisibility.
   1 x 72, 2 x 36 (72 ÷ 2 = 36), 3 x 24 (72 ÷ 3 = 24), 4 x 18 (72 ÷ 4 = 18), 5 x (72 is not divisible by 5), 6 x 12 (72 ÷ 6 = 12), 7 x (72 is not divisible by 7), 8 x 9 (72 ÷ 8 = 9).
   If the process continues the factors will appear in reverse order, e.g. 9 x 8 = 72. 8 x 9 and 9 x 8 are essentially the same rectangle though they may appear differently if the direction of the label is considered.
6. Gather the class to discuss solutions and look at real sized diagrams of the possible labels. Some options are mathematically correct but unworkable as a label option.
   Discuss criteria for eliminating labels. For example, a label with a width of less than 5cm might be considered too ‘skinny.’
   Discuss the best options, cut them out at real size, then use a real jam jar (or object that is relevant to the context of the learning) to consider how well each label/array design will work.
7. In the jam jar context, students might write a letter to Karly outlining how they investigated her problem and giving their recommendations. Their mathematical thinking could be used as the basis of a persuasive letter in other contexts.
8. Another good investigation is to tile a large rectangular area with 1m² carpet tiles. A hall or gymnasium is an ideal area though a classroom is also viable. Tiles of that size are commonly found at hardware stores. You will find an advertisement easily online. 
9. Get students to construct a unit square using newspaper or recycled boxes. They can use the unit to get a sense of the scale of 1m² and make estimates of the area of the space before they calculate.
10. Ask students to work in small teams to calculate the number of tiles that will be needed for the rectangular space. Look for them to measure the side lengths of the rectangular area using tape measures, trundle wheels, or metre rulers.
    Do they apply the W x L = A formula?
11. Students can find the area of composite shapes by finding the area of the rectangles. For example: 
    
12. This shape can be seen to be made up of two 2cm by 4 cm rectangles, or a 2cm by 6cm rectangle and a 2 cm by 2 cm rectangle, or 4 cm by 6 cm rectangle with a 4 by 2 rectangle missing. Use 1cm square units (e.g. memo pads) to demonstrate the construction of this composite shape. There are different ways to solve composite shapes. However, one of the simplest methods starts with breaking a composite shape down into basic shapes (e.g. 2 rectangles). You could model this with memo pads or tiles. Next, find the area of the basic shapes you have constructed. Finally, add the areas of the basic shapes together. To support the development of this thinking, you could calculate the area of the shape shown above in two different ways. First, calculate the area from 2 rectangles, each with an area of 4cm x 2cm. The total area of each rectangle is 8cm². Therefore, the total area of the composite shape is 16cm². Next, calculate the area of the shape as one 2cm x 6cm rectangle (12cm²) added to one 2cm x 2cm (4cm²). Calculating the area of the same shape in different ways will allow for greater student collaboration, and will allow for students to learn from each other.

Session 4

In this session students explore using proportional reasoning to find areas of rectangles.

1. Pose the problem: Sam’s family was shopping for a new table for the wharekai in the local marae. The first one they looked at measured 2m by 3m. Sam said if they wanted one with an area twice as big they should get the 4m by 6m size. Is Sam right?
2. Ask the students to draw pictures of the table and to help them decide if Sam is correct.
3. Work with students to establish that doubling the area only involves doubling one side of the rectangle. Doubling both sides of the rectangle increases the area by four times.
4. Using this proportional reasoning students will be able to solve problems without recalculating from side lengths. Here are some example problems:
   • The recipe made enough icing to cover the top of a 20cm by 20cm cake. What size cake can you ice if you double the amount of icing?
   • The birthday card had a front cover measuring 15cm by 10cm, what is the area of the piece of cardboard used to make it?
   • The marae had two areas that needed paving. Each area measured 5m by 8m. What is the total area to be paved?
   • The gardener charged his customers by the area of their lawn. If the bill was $20 to mow a lawn that was 6m by 20m, what should the bill be for a 20m by 12m lawn?

Session 5

In the session students demonstrate their ability to apply measurement of area independently. Consider what culturally relevant contexts can be incorporated into this task, to increase the engagement of your learners.

The following links provide pages from Figure It Out books that are suitable:

• Square Skills
• Adobe Bricks

Students might also create a mat design and provide the dimensions and areas of the rectangular pieces that compose it. An example is given below:

In this activity, the students will need to apply their knowledge of area and how to calculate the area of rectangular shapes. This may be the first time they have found area from a diagram drawn to scale. You should discuss why it is not possible to do a drawing of the actual park in the book and why it has to be represented by a diagram drawn to scale. In this case, the scale is 1 : 1 000 or 1 centimetre : 10 metres.
The students will have had previous experiences finding area by counting non-standard units and concrete units, such as 1 centimetre cubes and squared diagrams (or pieces of fudge, as on page 4). From this, you will need to check that they understand and can express in their own words that the area is found by “multiplying the measurement along the length by the measurement across the width” as they may have done with a multiplication array. Some visual models of 1 metre square
would also be helpful for imagining dimensions involved in the task. The students will have worked with 1 metre square in page 4 of Measurement, Figure It Out, Levels 2–3. The students will probably find it easier to begin by calculating the area of this year’s gala.


Encourage the students to use a problem solving approach to investigate different
ways of finding the area of last year’s gala.
This may include:
• taking the smaller rectangle away from the larger

During the activity, encourage the students to stop to discuss and share the strategies that the groups are using before they continue to work out other strategies. Students having difficulty with formal measurement could compare the two shapes using concrete materials and non-standard units.
Don’t forget to compare the measurements of the two diagrams to answer the original problem. As an extension, the students could attempt to draw an irregular shape with the same area as this year’s gala field.

Answers to Activity

Yes. The total area of the park used for last year’s gala is (75 x 60) + (30 x 25) = 5 250 m². The area available for this year’s gala is 90 x 60 = 5 400 m².

Activity

This activity is based on the perimeter and area of rectangles. As a general introduction, have your students look at this rectangle where the side lengths are given as l and w.
 

The area can be found by: area = l x w.
The perimeter can be found by: perimeter = 2 x l + 2 x w.
Problems that involve maximising or minimising one measurement while either holding the other constant or minimising it are common in the real world. Fred’s fence is typical of constrained maximisation or minimisation problems.
Students exploring question 1 are likely to try different side lengths that will result in an area of 80 square metres. The problem requires a systematic approach, so encourage your students to organise their results in a table or organised list:

In this way, the students can find all the solutions with whole-number side measurements and calculate the perimeters at the same time. They may notice that the closer the side measurements become to each other, the smaller the perimeter becomes.
Encourage your students to explore the minimum perimeters for rectangles with the areas 16, 36, and 64 (square numbers). They will find that the perimeter is minimised when the rectangle is a square. In this situation, the length of each side is the square root of the area. They can then go back to question 1 with the knowledge that the solution is the closest whole number to √80 = 8.944 (to 4 significant figures). Students are likely to argue that the question asked for a rectangle and that a rectangle is not a square. It is worth stopping to discuss this reasonable view. In everyday use, a rectangle and a square are different shapes, but in mathematics, a square is just a special case of a rectangle.
Provide the students with a set of rectangles and squares and ask them to describe the attributes of these shapes. Encourage them to come up with minimal definitions, listing just the attributes that are absolutely necessary to define the shape. Students will typically say that a rectangle has:
• 4 sides
• 4 right-angled corners
• 2 pairs of parallel sides.
If you ask them to draw a 4-sided polygon that has right-angled corners but does not have 2 pairs of parallel sides, they will find that this is impossible. So it is not necessary to state that opposite sides must be parallel. This gives us the minimal definition for a rectangle. The minimal definition of a square is “a 4-sided polygon with right-angled corners and equal sides”. Squares are therefore a subclass of rectangles.
In the Investigation, students try to find rectangles that have the same number for the measurement of their perimeter as they do for the measurement of their area.
One solution is a square with sides of 4 metres. Its perimeter is 16 metres, and its area is 16 square metres. If they are systematic, students should be able to establish the existence of two other whole-number solutions.
They could begin by setting the length (at, say, 2 metres) and exploring what widths might work. They will discover that no whole-number solution will work for a side length of 2. But if they then try 3, they will find that a 3 x 6 rectangle has an area of 18 square metres and a perimeter of 18 metres. 6 x 3 is a third solution, but this is not a genuinely different rectangle.
Having got this far, your students may guess that there are other rectangles that meet the requirement but that they do not have whole-number sides. There are in fact an infinite number of such rectangles. In the table below, there are six rectangles that happen to have a whole-number measurement for one of their two dimensions. You could give your students the length of side b and challenge them to find the length of side a (in bold in the table), using a trial-and-improvement strategy.

There is an algebraic relationship between the pairs of values of a and b that satisfy the requirement that the number of perimeter units must be equal to the number of units of area. The relationship can be expressed in this way:
(To find the length of the second side, double the length of the first and divide by its length less 2.) Students who are developing an understanding of symbolic notation may like to try using this formula to find other pairs for a and b with the help of a calculator or spreadsheet program such as that shown.

Links

Numeracy Project materials (see https://nzmaths.co.nz/numeracy-projects)
• Book 9: Teaching Number through Measurement, Geometry, Algebra and Statistics Investigating Area, page 11
Figure It Out
• Number: Book Three, Years 7–8, Level 4 Orchard Antics, page 23
• Number Sense and Algebraic Thinking: Book One, Levels 3–4 Tile the Town, Tiny!, pages 20–21

Answers

1. a. 5 different rectangular shapes are: 1 m by 80 m, 2 m by 40 m, 4 m by 20 m, 5 m by 16 m, and 8 m by 10 m. Only the last two shapes would suit the dodgems (the other three would be too narrow).
b. The 8 m by 10 m rectangle would use 36 panels and cost $108, which is cheaper than the other options. It is one of the shapes that would suit the dodgems.
2. a. There are 12 different-sized rectangles that could be made.

b. The largest option is 12 m wide and 12 m long, which gives an area of 144 m². The shape of this area is a square.
3. 50 m. The length must be 15 m because 10 x 15 = 150. 2 x (10 + 15) = 2 x 25 = 50 gives the perimeter.

Investigation
Answers may vary. There are three whole-number solutions: 4 x 4, 3 x 6, and 6 x 3 (which is the same as 3 x 6). There is an infinite number of solutions if rectangles with only one whole-number side or no whole-number sides are included.

Session 1

In this session we make our own centimetre rulers and metre tapes so that we can measure the perimeter of objects in the classroom and playground. 

1. Gather the class on the mat and tell them that your two pet ants, Anton and Antonia, are having a race. Anton will race around the perimeter of this book and Antonia will race around this book. (Choose two books that are of different shape but similar perimeter). 
   Is the race fair? 
   Is it longer around the perimeter of this book or this book?
   Can anyone tell me what perimeter means? (The perimeter of an object or shape is the measure around its edges.)
   How could we find out which perimeter is longer?
2. Tell the students that they are going to make their own centimetre rulers.
   What is a centimetre?
   What things can you think of that are 1 centimetre in size? 
   Can you show me with your fingers how big a centimetre is?
   Can you find me something in the classroom that is one centimetre long? (Centimetre cube, width of a pencil, thickness of a book, width of a fingernail, point to markings on a ruler)
   Why are they called centimetres?
   Students may know that centi- is the prefix for ‘one hundredth’ so one metre is divided up into 100 centimetres, just like one dollar is divided into 100 cents.
3. Use a metre ruler to explain the relationship between metres and centimetres.
4. Give the students a 10 centimetre long strip of card.
   Can you think of anything in our classroom that is similar to this strip in length?
   How long do you think this strip is?
   How could we check? (Use a ruler, line centimetre cubes along the strip.)
5. Discuss how they can use the centimetre cube (unit place value block) to make centimetre markings on the strip.
   What will the strip look like then? 
   
6. Ask: Should we put the marks at the end or in the middle of the centimetres? (The markings are at the end of the unit and those units may need subdivision into smaller units, such as millimetres).
7. Talk about where the number markings are positioned and why they are at the “end” of each centimetre. Talk about where the “0” would go and how this relates to the “0” on commercially produced rulers.
   Why is having a zero mark important? (Zero is the conventional baseline to start measuring from)
   Why are numbers put on the ruler at all? 
   
8. Remind the students that the problem is to compare the perimeters of the two books. Anton and Antonia are awaiting a decision.
   How can you keep track of the measurement? (Record and then add the length of the sides later.) 
   Do you need to measure all of the sides?
   What’s the smallest number of sides you need to measure?
9. When the measurements have been made ask the students to add up the side lengths. This provides a good opportunity to practice and share strategies for adding and multiplying numbers. Value the strategies your students demonstrate and, if necessary, model and encourage the use of relevant strategies. 
   How did you combine the numbers?
10. Repeat with the second book and then make the comparison. This provides further opportunities to share strategies for adding and subtracting numbers. 
11. Ask students to select two books from around the room. Can you find two books that are not the same rectangular shape but have very similar perimeters?
12. Draw two or three rectangles that look different but have the same perimeter. Let students investigate the problem in pairs using their personal rulers. After a suitable time share the results.
    Is there a way to estimate whether or not two books have the same perimeter before you measure?

Session 2

We begin this session by posing a perimeter problem that is “too large” for our 10 cm rulers.  We discuss the need for a larger measuring instrument and then construct and use a 5 metre tape.

1. Gather the class on the mat and tell them that you would like to know if the perimeter of our classroom is larger or smaller than the perimeter of half a netball court.
   How could we find out which is larger? 
   Are our centimetre rulers appropriate for this? Why not? 
   What could we use? (metres) How many of our centimetre rulers make a metre? How do you know?
2. Line 10 of the rulers end to end to show a metre.
   
3. Place a strip of adding machine tape, or other long paper strip, alongside the rulers and discuss how it can be used to make markings on the tape.
4. Give each pair of students a 3 metre strip of paper and tell them that they are going to use their 10 cm ruler to create a metre measuring tape.
5. Circulate as the students construct their tape and ask questions to support their thinking: 
   How many centimetres in a metre? 
   How do you write centimetre? 
   What does this line mean on your tape? Is it at the end of the metre or in the middle? 
   How many centimetres are there in 2 metres?  
   What objects would you choose to measure with the tape rather than the ruler? 
   How accurate is your tape? Why?
6. Remind the students of the question that was posed at the start of the session (is the perimeter of our classroom is larger or smaller than the perimeter of half a netball court?) and ask students to compare the perimeters of the two spaces.
   How can you keep track of the measurement? (Record and then add the length of the sides later)
   Do you need to measure all of the sides?
   How accurate do you need to be? (As the question posed was one of comparison the level of accuracy will depend on how close the perimeters of the two objects are.)  
7. If the measurement requires parts of metres to be used then discuss ways of recording these. 
   1 m 20 cm             120 cm
8. When the measurements have been made ask the students to add up the side lengths. If necessary, share and revisit addition strategies.
   How did you combine the numbers?
9. Repeat with the second space and then make the comparison.  
10. Provide other opportunities to use the metre strip to measure other perimeters: 
    • What are the perimeters of tables in this room? (Comparing perimeters is very interesting when the tables are different shapes, e.g. circles, trapeziums.
    • What is a good perimeter if four students are going to work at a table?
    • Does the shape of the table matter?

Session 3

In this session the students are involved in a Perimeter Hunt. This involves them finding objects whose perimeter is of a certain length (in either metres or centimetres). You could adapt this to better suit the context of your class by picking items for students to find that relate to another area of learning. Consider how you might integrate the perimeter hunt with a 'fact finding hunt' related to your chosen objects.

1. Give each pair of students a simple folded booklet in which they are to record the results of their perimeter hunt.
   Ask them to select 10 perimeter ranges to find. Head up each page with a different perimeter range.  
   • 0 - 10 cm
   • 10 - 20 cm
   • 20 - 30 cm
   • 30 - 40 cm
   • 40 - 50 cm
   • 50 – 60 cm
   • 60 – 70 cm
   • 70 – 80 cm
   • 90 – 100 cm
   • 1 – 2 metres
   • 2 – 3 metres
   • 5 – 10 metres
   • 10 – 20 metres
   • Over 20 metres
2. Pose the perimeter hunt challenge: Students are to find objects that have perimeters which match the measure on each page. Students should work in pairs to estimate the perimeter of the object, and then and use their handmade measuring tools to measure the different objects. After measuring the object, students should draw the object, label the measurements for each side, and calculate the perimeter, before recording their findings in their booklet.
3. Discuss methods students use to estimate perimeters before they are measured. Focus on the use of benchmarks like, “I know 10 centimetres is the width of my hand,” or “Two of my steps is about one metre.”
4. Also discuss the impact of the shape on what the perimeter looks like. Why are circles tricky to estimate compared to rectangles? Why do long skinny rectangles look small but have large perimeters?

Session 4

In this session the students investigate open ended problems involving perimeter. Students may work in pairs to answer the following problems.

1. How many different shapes can you draw that have a perimeter of 24 centimetres?
2. Explain using words or a diagram why the perimeter of a rectangle can be found by using a combination of addition and multiplication.
3. How could you find the perimeter of a curved shape? (High achieving students might specifically focus on finding the perimeters of circles. That might lead to informal discovery of pi)
4. Imagine you have a rectangle. For one pair of opposite sides, you add one centimetre to each side. You take one centimetre off each of the other two sides. What happens to the perimeter? Why? Does the area stay the same?? Draw some examples to support your answer.

Session 5

In this session we compare the objects we found during the Perimeter Hunt and discuss the findings in our investigations.

1. Show the students an object and ask them to estimate which “page” of the perimeter book it belongs on.
2. Check the predictions.
   What length do you think this side would be?
   What do you use to make your estimate?
   What about the length of this side? (pointing to another side)
3. Using the agreed estimates ask the students to calculate the perimeter. Share the strategies used for summing the side lengths.
   How did you combine the measurements to estimate the perimeter?
   The conversation might lead to recording a rule for finding the perimeter of any rectangle.
4. Next ask the students to consider which perimeter was the easiest to find objects for.
   Why do you think that [perimeter] was the easiest?
5. Share the objects found for that perimeter length.
6. Ask one pair to read out the side measures for their object while the rest of the students check the calculation mentally. Share the strategies used to sum the numbers mentally.
7. Repeat with other perimeter measures.