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:

Session 1

For this session you will need plenty of bottles and containers of a range of sizes, including several that hold 1 litre. Fruit juice bottles, shampoo bottles, and yoghurt containers are particularly good containers for this task. You could either ask students to bring bottles and containers to school with them or collect them yourself. To ignite interest in this session, begin with a discussion around why it is important to know the amount that can be held in a container/ Possible contexts for framing this discussion could include looking at the ways in which people travelled to New Zealand (e.g. by ship, waka, plane etc.) or looking at planning the amount of food and drink needed for a school camp.

1. Begin by selecting 5 or 6 containers of various sizes and shapes.
2. Ask students which one they think has the least space in it.  Introduce the word 'capacity' to mean the space within a container, and 'volume' as the amount of liquid or gas a container holds. Explain that although these terms mean different things, they are often used to talk about the same thing. It would be wise to choose one term to use with your class, throughout the sessions in this unit. Ask them to explain why they made their choice of container with the smallest capacity.
3. Explain that we are going to order the containers from those that hold the least, to those that hold the most.
4. Ask for suggestions for how to compare the size of the containers. Ensure that students understand that they are comparing the space inside the containers.
5. Gather suggested strategies then trial strategies to establish an effective way to order the containers by volume.  The most effective strategy will probably be to pour water from one container to another. If the water that fits in one container does not fit into another then the first must have been larger. Discuss how to organise the containers, given that only two can be compared at one time in that way.
6. Group students and provide them several containers for each group.  Ask each group to order their containers by capacity, from 'holds the least' to 'holds the most'. Watch to see that your students can organise the ordering of many containers, when the comparisons are two at a time. Consider grouping students together that have a range of mathematical abilities to encourage tuakana-teina (peer learning) and mahi tahi (collaboration)
7. Share the techniques and strategies used by each group to order the containers.
8. Ask 2 groups to pair up to combine their containers on one continuum of least to most volume. Check that they understand that volume is conserved (i.e. that it is the same quantity of water, even though its appearance may change in a different shaped bottle) and that the order of each group’s containers will not change by adding another group’s containers.
9. Establish an order for all the containers available. This task raises efficiency and estimation. Suppose ten containers are already ordered by capacity.
   
   What is the most efficient way to find the place of this (new) container amongst the others?
   
   Students might suggest estimating first to get a ‘ballpark’ idea of where the new container might go. Next, compare the capacity of the new container to the others by pouring. How many pourings are needed?
10. An additional challenge can be to anticipate the water level if water is poured from a smaller container into a larger container. Rubber bands can be used to mark the predicted levels. Look for students to discuss strategies for anticipating the levels, such as considering the cross sectional area of the container.

Session 2

The following activities are to provide students with experiences to compare volumes/capacities of different objects and to create a benchmark for a container that holds one litre.

1. Make a 1 litre container available for students to use to give them a ‘feel’ for one litre.
   Compare it to a large Place Value Block cube.
   Which object takes up the most space, that is, has the greatest volume?
   
   The visual appearance of the large cube makes it look smaller than most other objects with the same volume of 1 litre (1000 cm3). It is fun to fill a bucket of water to the brim and ‘dunk’ the containers one at a time. The water that overflows is equivalent to the volume of the container or cube.
2. Group students and provide a variety of containers for each group.  Ensure items that hold 1 litre (like a 1 litre measuring jug or a 1 litre container of milk or water) are included as such items will become useful benchmarks. 
3. Ask each group to draw and label the following buckets on large sheets of paper. 
   
4. Students work together to place the containers in the most appropriate bucket, then check their estimates using a 1 litre container. Be aware that interpretation of "lee than", "about", and "more than" is a bit subjective.

5. Gather the class and discuss the strategies students used to make their estimates. Consider the following points:

   Do taller objects have more volume than shorter objects?

   How does the cross-section affect the volume of the object?

   If you have an object that you know is 1 litre, how do you compare its volume to that of an object that has different height and cross-section?

6. Provide students with this open challenge. They need scrap cardboard, scissors, rulers and strong tape.

   Create a cuboid (rectangular prism), cone, or cylinder shaped container that can hold exactly 1 litre of water.

   You may need to support students with creating nets, rolling pieces of card to form cones or cylinders, and applying their understanding of the fact that 1 litre equals 1000 cubic centimetres.

7. Ask students to locate items from around their home that they believe would make good benchmarks for 1 litre and, if possible, bring them to school.

Session 3

In this session students compare their benchmarks for one litre and try to estimate one litre.

1. Share the containers that have been brought to school as good benchmarks for 1 litre and identify which are closest to 1 litre in volume.
2. Discuss which of the benchmarks are the most useful. For example, objects which you don’t usually see are not particularly good benchmarks as you will not be familiar with their volume. Common objects are easier to visualise.
3. Give students a plastic bag and ask them to put one litre of water in it.  Vary the size of the bags you use. You may prefer to do this activity outside.
4. Compare the bags and discuss differences in appearance.
   We know that all the bags hold 1 litre but they look different. Why is that?
   Compare the bags to reliable benchmark objects.
5. Introduce millilitres as a unit that is helpful for measuring containers that hold less than a litre. 
   • What does milli stand for? 
   • How many millilitres equal 1 litre?
   • How many millilitres equal 2.5 litres?
   • How many litres equals 1500 millilitres?
     You might use a small place value block to give students a sense of the size of 1 millilitre.
     How many millilitres will fill a teaspoon? (5mL)
     ....a dessert spoon (10 mL)?
     ....a tablespoon (20 mL)?
     Show the students a Place Value Block flat.
     How many millilitres is this? (10 x 10 = 100)
     How many lots of 100 millilitres make 1 litre? (10 since 10 x 100 = 1000)
     Stack ten flats to form a large 1 litre cube to prove the result.
6. Take several containers, measure the capacities, and express the measurements using both millilitres and litres, e.g. 750ml = 0.75 L. Discuss why 750 mL is the same as 750/1000 of 1 litres and is written as 0.75

   • Provide the students with some conversion examples between millilitre and litre measures, such as:

     Millilitres	Litres
     500 mL	1 L
     250 mL
     750 mL
     300 mL	0.3 L
     900 mL
     1200 mL
     456 mL	0.456 L
     685 mL
     903 mL
     	0.728 L

      

Session 4

In this session students work with volume as the amount of space that an object takes up.

1. Provide a range of familiar objects of different volumes (preferably things that will sink in water). Make sure all items are waterproof. Bath toys make good objects.
2. Ask students which of the objects has the largest volume.  If there is confusion, explain that volume does not just mean the amount that a container can hold, it also means the amount of space an object takes up.
3. Show students how they can find the volume by displacement. Place a container full of water inside an empty container or tray.  Submerge the object in the container of water and measure how much water is displaced (overflows) into the empty container. This is equal to the volume of the object – discuss why this is so with the class.  If you can find a copy, read ‘Mr. Archimedes' Bath ’ by Pamela Allen.
4. Allow students some time to experiment with this concept and to order objects by volume. Discuss the importance of considering all three dimensions, not just one dimension such as height.
5. If you have plasticine available pose open challenges like:
   Make a blob that has a volume of 48 cm³ which is the same volume as 48 mL.
   Change the volume to provide more challenge, e.g. 0.124 L.

Session 5

1. Bring this unit to a conclusion by asking students to share the benchmarks they are going to use for 1 litre. 
2. List the various benchmarks on a large sheet of paper to be displayed as a reference. 
3. Share the various strategies and techniques students have developed to establish near estimates for objects they are asked to estimate the volume of.
4. Ask students to think about other possible accessible items that could be used as benchmarks to measure items that are less than 1 litre in volume. 
   What is the volume of a can of soft drink? 
   Why might that volume be a good ‘size’?
   What is the volume of your lunchbox?
   Why might that volume be a good ‘size’?
   What would be a good volume for a chilly bin?
5. School bags and backpacks are often measured in litres to indicate the capacity of the bag. Research standard backpack sizes online to find out the usual dimensions. 
   Why is the capacity of a backpack important?
   How many litres is your backpack in capacity?
   Use the large Place Value Block cube as the benchmark of 1 litre to estimate the students' backpacks. 
    

Before starting this unit discuss with children what is meant by the terms volume and capacity and what measurement they will be using for volume. Capacity is a measure of how much the container can hold whereas the volume is the measure of the amount of space that a 3-dimensional object occupies. The measurement for volume in this unit is mL (millilitres). All copymasters will need to be carefully introduced so children will be confident when using them.

Session 1

Rainbow jellies are individual cups of jelly, striped different colours like a rainbow.

1. Introduce students to the topic of rainbow jelly and check that all students know what they are.
   Ideally, bring in one that you have made at home and ask students how they think it has been made.
2. Describe how the layers need to be made one at a time, being left to set between each layer.
3. Explain that this week they are going to work in groups to plan and make their own rainbow jelly.
4. Divide students into groups and give out three different shaped plastic containers to each group, at least one with parallel sides.
5. Have students experiment with measuring spoons and water in their cups to investigate the effects of the same volume in the different shaped containers.
6. Note that the shape of the cup will affect the way the final jelly looks. For example, if the cup has parallel sides and equal volumes of jelly are used for all colours the stripes will be the same height. Encourage students to investigate this and discuss their findings.

   How could we make the stripes of jelly wider?
   How could we make the stripes of jelly thinner?
   How could you make all the colours the same height?

The task of making the stripes even will be more complicated in containers where the sides are not parallel.

Exploring

Over the next three sessions have each group of students complete plans for several different jellies, working with a different container each session.
Session 2: container with parallel sides
Session 3: container with sloping sides
Session 4: unusual shaped container, encourage students to bring a container from home for this purpose

During each session students can use water to fill containers to different levels as they plan designs for their jellies. They may choose stripes of the same depth, alternating thick and thin stripes, stripes that get progressively narrower. The plans they make need to specify the colours and volumes for each stripe alongside a sketch of the design.

Each different plan can be recorded on Copymaster 1.

As different plans are drawn up share these with the class and discuss the volumes of jelly used. Encourage conversion between teaspoons, tablespoons and cups.

How many tablespoons of red jelly have you used?
How many teaspoons would that be? How much of a cup?
Which colour have you used most of? Which colour have you used least of?
How much more red than yellow have you used?

Once planning is complete get the students to choose which of their plans they will make. Ask them to calculate how many packets of each colour jelly they will need if there are 8 people at the party. How many for 20 people?

Copymaster 2 can be used to guide and record these calculations.

Reflecting

Groups of students make their jellies. It would be simplest to mix the jelly required in bulk rather than have each group mix each colour individually. Note that the hot water needed could present a safety issue. To overcome this a small amount of boiling water could be used to dissolve the crystals first, then  cold water added before students use the jelly.

Once all the groups have their jellies complete and they have set, get students to estimate the volume of each colour jelly used in the different designs. Their estimates can then be compared to the actual volumes used. Copymaster 3 can be used for this.