Session 1

SLO:

• Use plan views to represent three-dimensional models made from cubes

In this context, you might draw on the knowledge of community members (e.g. builders, architexts) and have them show architectural plans to your students.

1. Use slides one and two of PowerPoint 1 to introduce the idea of architectural plans.
   What are these pictures used for?
   Discuss the idea of flat (two-dimensional) drawings used to show the structure of three-dimensional structures. 
   Sometimes more than one drawing is needed. Why?
   Tell the students that today they are becoming architects.
2. Give a student ten multilink cubes and ask them to create a building for you. The only stipulation is that faces meet exactly.
   Show the class the building laying flat on your hand. After a few seconds of viewing time, place the building flat on a desktop. Move to the front of the desk, crouch down and take a digital photograph with the house at lens level.
   Draw what you think the photograph looks like.
3. Let students sketch their idea of the viewpoint. Show the photograph on an interactive whiteboard, television, or using a projector.
   How do the cubes show in the photograph? (as squares)
   Why do they appear that way? (Only one face of each cube is visible)
   What strategies did you use to get the viewpoint correct? (layers or columns, relative position, etc.)
   Be aware that many students are likely to have attempted to show depth in their pictures. Point out that the camera can only capture what it sees. 
4. Repeat the exercise for the right-hand and birds-eye views.
   Are students understanding that information is lost when a 3D object is represented in 2D diagrams?
5. Show slide three that depicts a correct layout for plan views. Plan views are often called orthogonal because they are at right angles to faces of the model.
   How many cubes make up this building? How do you know? (9 cubes)
6.  Ask students to take nine or ten cubes and make their own building.
   Put your building on a desktop.
   Draw your building from the front, right side and top.
   Provide grid paper (Copymaster 1) to support students with drawing squares.
7. Roam the room to support students. Taking digital photographs of their models and showing the image is useful for students who find it hard to minimalise the information they show.
8. After a suitable time, ask students to bring their plan views to the mat, and to leave the model on a desktop somewhere in the classroom.  Gather all the plans, shuffle them, and deal out one per student.
   Can you find the model that goes with the plan?
9. Let students have a suitable time to locate the models. Some plans may have more than one appropriate model.
10. Discuss the features they looked for in locating the model.
    Which plan was the most useful? (A key point is that one viewpoint, often the top, is a good screening tool for possible models. Other views can be used to confirm the correct model).
11. Return to Slide Three. Ask the students to use the cubes from their previous model to build a structure that matches the views. Once they believe they have a correct model, students can justify their answer to a partner. Animate Slide Three to reveal the correct answer.
    Discuss the use of the top view to organise the information from the other views.
12. Slides Four and Five have two other plan view puzzles. Animating each slide provides a model answer.
13. Students might also work on the Figure It Out activities called Building Boldly and X-ray vision. PDFs are available for the student pages.

Session Two

SLOs:

• Coordinate different views of the same structure to form a model of it.
• Represent cube models with isometric drawing.

Before class gather at least five different shaped objects from around the classroom. The objects might be mathematical models (e.g. cube, pyramid, sphere, etc.) or common objects (book, cone, bottle, box, etc.) or a combination of things. It is better if the objects are different heights.

1. Using a large sheet of paper placed on a desktop, draw a grid of squares. 10cm x 10cm squares are a good size. Arrange the objects at different locations on the grid. Take digital photographs with the grid at lens level. Capture views from all four compass points.
2. Place the grid on the mat or on a central table.
   A spy took these photographs of an enemy city. She took four pictures, one from each of the compass points. After returning to her base she emailed the images.
   Show the students all four views of the ‘city’ on an electronic whiteboard or using a data projector. 
   Your job back at Kiwi Intelligence is to construct a plan map of the city. You know there are these buildings (objects). Look carefully at the photographs to work out where to put each building.
3. Let students sketch a birds-eye view of the city. They might name the buildings (object) on their plan rather than draw the shapes. After a suitable time, gather the class to decide where to position each object. Look for students to coordinate views to do so.
4. It is common for travellers to create optical illusions of places they visit. Images appear impossible, such as someone holding up the leaning tower of Pisa. Slide One of PowerPoint 2 has an illusion like that.
   How do these tricks work? (Objects that are further away look smaller, even to a camera. That is called perspective)
   How do artists adjust what they draw to allow for perspective? (Show Slide Two)
5. One way to represent cube models is to use isometric drawing. That method does not have the vanishing points of perspective drawing, but it does partly show that the object is three dimensional. In isometric drawings things that are further away do not get smaller, but all parallel edges remain parallel (Show Slide Three)
6. Slide Four shows how to draw a model made with five cubes. Step through the process of drawing slowly with students copying each step on a sheet of isometric paper (Copymaster 2). You might like to go to an online isometric drawing tool (https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Isometric-Drawing-Tool/) to show how the drawing is built up from the cube that is hidden.
   
7. Ask students to create a model made from interlocking cubes. A maximum of ten cubes is wise. Students sketch their models on isometric paper. The sketches can be given to a partner who makes the model. Sometimes different models can be made for the same drawing.
8. Discuss the strategies students used to create correct isometric drawings, such as:
   • Begin with the front-most cube.
   • Hold the model so the leading edges face up and down.
   • Build the drawing across and up first.
   • Create an L shape for cubes that come out at right angles.
   • Watch for parts of faces that might be visible.
   • Imagine a light shining on the model from behind (to shade faces)
9. Create some isometric drawings using the online tool. For each, discuss:
   • How many cubes are needed to build this model?
   • Is that the smallest possible number of cubes?
   • What is the largest number of cubes that could be in the model? 
10. Challenge students with the Figure It Out activity, Cube Creations. In the task students firstly build models from isometric drawings and join the models to create cubes. The second challenge is for them to create a cube puzzle of their own and draw the pieces (models) using isometric paper.

Session Three

SLO:

• Connect plan views and isometric drawings for the same three-dimensional cube model

1. Show students Slide One of PowerPoint 3. Provide students with another sheet of isometric dot paper (Copymaster 2).
   Draw what this model looks like on isometric dot paper. Before you start, you will need to decide whether to draw the front of the model angled to the left or right. Model looking at the model angled to the left and to the right.
   Let students attempt to draw the model. There are two possible perspectives depending on which direction the front is angled. Both drawings are shown on slide two.
   • Discuss the strategies that were helpful to producing a correct drawing. Ideas might include:
   • Identifying which direction is the front.
   • Starting with the front most stack of cubes.
   • Building the ground layer first before building up.
   • Considering the cubes that cannot be seen.
   • Erasing unwanted lines.
   • Shading faces as you go so the blocks look solid.
2. Engage in reciprocal partnerships (tuakana teina) again. Both partners draw a model they create. They choose three plan (front, top, side) or isometric views to draw. The other partner creates a different drawing of the same model, then builds it to check.
3. Ask students to work on the Figure It Out activity called A Different View. In this activity students match isometric views, with directional arrows, to the corresponding plan views from those perspectives. They also draw 2D representations of everyday objects such as cups and paper clip holders. A PDF of the student page and answers are provided. An extension activity can also be found in Missing Anything, a Level 4+ Figure It Out page. 

Session Four

SLO:

• Create nets for simple solids (prisms and pyramids).

1. Begin this session by showing your students some graphics of simple solids. Slides One to Three of PowerPoint 4 show three types of prism, triangular, hexagonal and rectangular (cuboid). Show each solid in turn, and ask:
   Where are you likely to see a shape like this?
   What is the shape in this picture called?
   What are the shapes of its faces?
   How many vertices (corners) and edges does it have?
2. Discuss: What do all three solids have in common?
   The common properties that define a prism are, a solid that has two identical parallel faces and all other faces are parallelograms.
   Slide Four shows a loaf of bread being sliced.
   How are a loaf of bread and a prism the same?
3. Prisms are sometimes defined as solids with constant cross section. Slices of bread are a similar shape. It is the cross-section that determines the name of a prism. Slide Five shows a pentagonal prism as the cross section is a five-sided polygon.
4. Show Slide Six. 
   Here is a rectangular prism shaped box that holds soap powder.
   Imagine that I open out the packet to form the flat pattern that makes it.
   Sketch what you think the net will look like.
5. You may give students copies of Copymaster 1 to make measurement of length, and creating right angles, easier. Encourage students to create a net (flat pattern) that folds exactly to form the packet. Roam the room and look for:
   • Do students attend to the shape of faces in constructing the net?
   • Do they visualise the effect of folding up faces?
   • Do they consider which sides of the net will need to be the same length for edges to form correctly?
   • Do they consider tabs needed for gluing the net together? (Usually every second side of the net.)
      
6. Provide Slides Seven and Eight for students to create their own nets. The triangular and hexagonal prisms are more challenging than the cuboid, particularly getting the angles and side lengths correct. You may need to support some students to create 60⁰ internal angles for equilateral triangles and 120⁰ angles for regular hexagons. Use protractors to get accurate measures.
7. Show Slide Nine that shows the three nets.
   What is the same about all three nets? (Rectangular faces in a line)
   What is different about the three nets? (Parallel faces that create the cross section)
   How can you tell how many rectangular faces the prism needs? (The number of rectangular faces equals the number of sides on one of the parallel faces)
   Visualise the net for the pentagonal prism. What does that net look like? (Five rectangles in a line, with two pentagonal faces).
8.  Slide Ten gives images of three pyramids; tetrahedron (triangular based), square based, and hexagonal based pyramid.
   What are these three-dimensional shapes called?
   In what way are the solids related?
   Look for students to discuss the properties of a pyramid; a base of a given shape, triangular faces that meet at an apex.
9.  Challenge your students to create nets for the three pyramids. Blank A3 paper, a protractor and ruler are useful. Roam the room. Look for students to:
   • Construct a base that is a regular polygon (same side lengths and angles)
   • Arrange the triangular faces so they emanate from each side of the base shape
   • Construct isosceles triangles with two equal sides for the lateral faces.
10. Constructing a pentagonal, octagonal or dodecagonal based pyramid is an excellent challenge for students who are competent.
    Let students investigate the problem in pairs and record their ideas.
11. Use the models of prisms and pyramids to look at the number of faces, edges, and vertices in each solid. Discuss systematic ways to count. For example, to count the edges of a prism, count around each parallel face and add the lateral edges.
12. Create tables for the solids you have models for.

13. Look at the table for prisms together (see below):

    	Number of faces	Number of edges	Number of vertices
    Triangular Prism	5	9	6
    Rectangular Prism	6	12	8
    Pentagonal Prism	7	15	10
    Hexagonal Prism	8	18	12

14. Ask: If you know the number of sides of the cross section, can you predict the number of faces, the number of edges, and the number of vertices?
    Students might notice that:
    • The number of faces is two more than the number of sides in the cross section. Why?
    • The number of edges is three times the number of sides in the cross section. Why?
    • The number of vertices is two times the number of sides in the cross section. Why?
       

15. ​​​Look at the table for pyramids together (see below):

    	Number of faces	Number of edges	Number of vertices
    Triangular Pyramid	4	6	4
    Square based Pyramid	6	8	5
    Hexagonal Pyramid	8	12	7

16. Ask: If you know the number of sides of the base shape, can you predict the number of faces, the number of edges, and the number of vertices?
    Students might notice that:
    • The number of faces is one more than the number of sides of the base shape. Why?
    • The number of edges is double the number of sides in the base shape. Why?
    • The number of vertices is one more than the number of sides of the base shape. Why?

Session Five

SLOs:

• Establish whether, or not, a given net for a simple solid is viable.
• Visualise which sides and corners of a given net will meet when the net is folded.

1. Begin with Slide One of PowerPoint 5 that shows a viable, though unconventional, net for a triangular prism.
   Will this net fold to form a solid?
   Which solid will it create?
   How do you know? (Consider the number and shapes of faces, the result of folding)
2. Mouse click and a single corner of the net will be highlighted.
   Imagine this net is folded. What other corners of the net will meet?
3. Mouse clicks reveal the other corners that meet.
4. Another mouse click shows a side of the net.
   Which other side meets this one when the net is folded?
   How do you know?
5. Mouse click to see the other side that connects.
6. Discuss how many corners meet to form a vertex (three) and how many sides form an edge (two).
7. Ask similar questions for Slides Two and Three that show other nets.
8. Provide students with Copymaster 3 that contains a set of similar folding puzzles for different solids. Students might work in collaborative, small groups and justify their solutions to each other. Tell them that their first task is to decide whether, or not, the net folds to make a solid.
9. Gather the class after an appropriate time and discuss the strategies students used. Ideas that might emerge are:
   • Imagine the net is folded and tracking the destination of corners and sides as they form vertices and edges of the target solid.
   • Consider the properties of the target solid, e.g. parallel faces of a prism, corners of a pyramid converging apex.
   • Eliminate obvious corners and sides first.
   • Recognise when the positioning of shapes in a net, results in overlaps, or omissions of faces, in the target solid.
10. Challenge students to create similar puzzles for their classmates. The net may be possible or impossible and they should choose a corner and a side that challenges the solver.
11. Extra activities related to nets can be found using these Figure It Out resources: 
    • Caught in the nets
    • Loads of lamingtons
    • Nutting out nets

Session 1

Make up a set of three containers by trimming down 1.5 litre plastic soft drink bottles. It is preferable that these containers are of the same type. Straight-sided bottles are better than curved ones as they make it easier for the students to predict the relationships. The three cut down containers should hold different amounts of water when full. Make their capacity no more than 500 ml and ensure that these capacities are not multiples of each other, e.g. 120 ml, 230 ml, and 400 ml would be better than 100 ml, 200 ml, and 400 ml. Label the containers p, q, and r respectively.

1. Introduce the containers to the class and use them to set representation problems for the students by pouring from the p, q, and r containers into the large container, preferably the original bottle type that p, q, and r were created from.
   With each problem students write an expression for the water level. Here are some examples. You can extend these as appropriate for your class.
   1.  Pour one lot of p and one lot of r into the big bottle:
      
       
   2. Pour two lots of p, one lot of q and one lot of r into the big bottle:
      
       
   3. Pour four lots of p into the big bottle. 
      How can you write an expression for what is in the bottle? (p + p + p + p or 4p, noting that 4p has an ‘assumed’ multiplication sign, 4 x p) 
      Next pour from the bottle into container r and container q until the mark is reached.
      What is left in the bottle? How might you write an expression for that amount? (This gives 4p – r – q which can also be written as 4p – (r + q))
       
   4. Pour five lots of r into the large bottle (5r).
      Pour out two lots of p (5r – 2p).
       
   5. Pour three lots of r and three lots of q into the large bottle (3r + 3q or 3(p + q))
      Pour out two lots of p.
      (This gives 3r + 3q – 2p which could be written as 3(r + q) – 2p).
       
   6. Pour two lots of r and one lot of p into the large bottle (2r + p)
      Repeat pouring 1, that is add two lots of r and one lot of p into the large bottle         
      (2r + p + 2r + p, or 2(2r + p) or 4r + 2) 
       
2. Change the activity from representing pouring with an expression to imaging the pouring that matches a given expression:
   1. What does p + q – r mean? What pouring happens?
   2. What does 3r + q – 2p mean? What pouring happens?
   3. What does 3(p + q) mean? What pouring happens? (Put p + q in the bottle. Do that three times.)
   4. What does 3(r + q) – 2p mean in terms of pouring? (Put r and q lots together 3 times and then pour out p twice.)
       
3. Pose problems that give opportunities for students to discover equality or inequality of expressions. You will need two identical bottles to pour into.
   Are these expressions equal or unequal? Why?
   1. q + 2p and p + 2q (only equal if p and q are equal)
   2. q – r + p and p + q – r (yes they are equal)
   3. 3r + 2p and 3p + 2r (only equal if p and r are equal)
       
4. Explore equality of expressions, particularly in the cases where order does or does not make a difference to the result, e.g. 2r + p = p + 2r = r + p + r. 
   Can students simplify the equality using inverse operations?
   p + 2r = r + p + r
   2r = r + r (subtracting p from both sides)
    
5. Provide students with applications of equality and inverse operations.
   Which of these are correct and which are false? Explain your answers.
   1. 4(r + q) = 4r + 4q (equal)
   2. 3(r + q) – 2p = q + r + 2(r – p) + 2q (equal)
   3. 4p – r – q = p + 2(p – r – q) + q + r + p (equal)
   4. 2r + p + q = 2(p + q) + 3r – q – p – 2r (unequal)

Students could make up their own set of containers, label them with algebraic letters, and develop challenges for each other. Alternatively, the class containers could be left as a station for students to use independently.

Session 2

In this session students look for patterns within each equation set and use these patterns to predict further equations in the set. They may do this using recursion, that is finding a relation between consecutive equations, rather than by looking for relationships within the equations across the equals sign. Highlight relationships that might be found between the numbers in each set of equations and encourage the students to look for ways to describe these relationships. It is important that students find the unknowns using mental calculation rather than with calculators, as their attention needs to be on the relative size of numbers.

Some learners will need support from physical materials to notice and describe the patterns. Connecting cubes in stacks make a useful representation.

Below are some suitable equation patterns:

1. 1 – [   ]= 1
   2 – [   ] = 1
   3 – [   ] = 1
   4 – [   ] = 1
   ...
   456 – [   ] = 1
   Why is the right-hand side of the equation always one?
   What is the rule for all equations in this pattern?
   Use this pattern to solve:
   2000 – [   ] = 1
   1001 – [   ] = 2
    
2. 0 + 1 + 2 =    [   ]
   1 + 2 + 3 =    [   ]
   2 + 3 + 4 =    [   ]
   3 + 4 + 5 =    [   ]
   Is there anything in common with the numbers in the boxes?
   Why do you think this happens? (The number in the  [   ]  is three times the middle number on the left-side of the equation.)
   a + b + c = 300 , a, b and c are different numbers.
   What numbers could they be?
   What values for a, b and c would fit the pattern?
   
   A physical representation of the pattern, using connecting cubes, would look like this:
   
    
3. 42 – 8 = [   ] – 10
   52 – 8 = [   ] – 10
   62 – 8 = [   ] – 10
   ...
   92 – 8 =    [   ] – 10
   712 – 8 =    [   ] – 10
   What rule can you find for equations in this pattern?
    
4. 1 x 9 + 1 = [   ]
   2 x 9 + 2 = [   ]
   3 x 9 + 3 = [   ]
   4 x 9 + 4 = [   ]
   ...
   In [   ] x 9 + [   ] = 100, what is [   ]  if both [   ] are the same number?
   What rule can you find for all the equations in this pattern?
   
   A physical model of this pattern, made with cube towers, looks like this
   (example of 4 x 9 + 4 = 40 (4 x 10)):
   
    
5. 1 = [   ] ÷ 11
   2 = [   ] ÷ 11
   3 = [   ] ÷ 11
   4 = [   ] ÷ 11
   ...
   In [   ] ÷ 11 = 7,  what is [   ]?
   In 9 = [   ] ÷ 11, what is [   ]?
   What rule can you find for equations in this pattern?
   Would these equations be further down in the pattern? 
   
   Give an explanation. [   ] = 121 ÷ 11
   [   ] = 594 ÷ 11
   [   ] = 682 ÷ 11

Session 3

Students solve “What’s my Number?” problems and record how they found the final answer. At this stage trial and improvement are legitimate strategies though the problems encourage students to attend to structure, and apply their understanding of inverse operations.

Flowcharts can support students to apply inverse operations. Consider the problem; “Take my number, multiply it by three then subtract 4. The answer is 20. What is my number?” The problem is equivalent to 3x - 4 = 20.  
A flowchart looks like this:

Applying inverse operations gives:

1. If you take my number, multiply it by three then add seven, you get fifty-two.
   What is my number? 
   (15 since 15 x 3 + 7 = 52, or 3 x [   ] + 7 = 52, so 3 x [   ] = 45, and [   ] = 15)
   
    
2. If you take my number, subtract ten from it then divide it by two, you get sixteen.
   What is my number? 
   (42 since 16 x 2 = 32 and 32 + 10 = 42)
   
3. If you take my number and add twenty-four to it, the answer is three times my number.
   What is my number? 
   ([   ] + 24 = 3 x [   ], so 24 = 2 x [   ], and [   ] = 12)
    
4. If you take my number and divide it by three, the answer is the same as my number minus forty.
   What is my number? 
   ([   ]  ÷ 3 = [   ] - 40, so 40 = 2 x [   ], and [   ] = 60)
    
5. If you take my number and multiply it by itself, the answer is my number added to itself.
   What is my number? 
   (2 since 2 x 2 = 4 and 2 + 2 = 4 and 0 for the same reason; 
   or [   ]  x  [   ]  = [   ]  + [   ],  so [   ] x [   ] = 2 x [   ] , and [   ] = 0 or [   ] = 2.)
    
6. Ask students to make up “What’s my number?” problems for others to solve. They should provide hints to other students about solving the problem. For example, “My number is multiplied by six. How do you undo multiplying by six?”

Session 4

In this session students work out the functional rule for given input and output numbers. The functions increase in complexity as the week progresses. Each example offers the students three input/output pairs. to the right of these three pairs are three other pairs (typed in bold) that could be used if needed. Copymaster 1 can be used to make stimulus cards by cutting and folding the output of the three additional pairs behind so they can not be seen unless needed. 

1. 
   The rule is two times the input number less one gives the output number.
    
2. 
   The rule is times three plus one.
    
3. 
   The rule is times three take away nine.
    
4. 
   The rule is divide by two plus one.
    
5. 
   The rule is the input number multiplied by itself (squared).

Discuss how a rule might be found.

If the output numbers are always more than the input numbers what types of rules do you try first? 
Students are likely to suggest addition or multiplication. That strategy is useful but needs to be qualified. Multiplying by a fraction less than one or an integer will result in smaller output numbers.
How can you check if only addition is involved?
The difference between input and output numbers must be constant. That generalisation is true of subtraction as well.
How can we check if only multiplication is involved?
The differences between input and output numbers change. That will not work as a strategy. Students will need to estimate what the multiplier is, possibly from two pairs, then check to see if the multiplier applies to the other pairs.
If division is in the rule, how can you tell?
Students usually say that division, like subtraction, makes the output number smaller. That generalisation is naïve since division with a fraction between zero and one makes the output number larger. However, provided the divisor is larger than one the claim that “division makes smaller” holds. As with multiplication, the divisor will need to be estimated first then tried on the input-output pairs.

Challenge students to come up with their own tables of values. You might set up a spreadsheet of tables as a template, with hidden rules. Alternatively create a paper template as seen in Copymaster 1.

Session 5

These activities involve students in working out the number of counters or cubes that are in each cup of a given colour. Several clues are provided, and students must combine these clues to find a solution. Use opaque coloured plastic cups, that are readily available in supermarkets, warehouses, and dollar stores. For each problem all students should solve it, then record their reasoning, before the solution is “revealed.”

Encourage students to check their solutions by putting the values they assign to each cup back into the original clues. Do all the clues work? If not, try again!

Examples of cups and counters problems follow. The first sentence is the instruction for the teacher and gives the answer. It does not appear on (Copymaster 2) as students use the clues to work out how many cubes are in each coloured cup.

1. Into each yellow cup put four cubes, into each blue cup put five cubes.
   Clues:
   
   14 cubes in total                16 cubes in total
    
2. Into each yellow cup put three cubes, into each blue cup put six cubes.
   Clues:
   
   Four yellow cups have the same number of cubes as two blue cups.
   
   Three yellow cups and one blue cup have 15 cubes in total.
    
3. Into each yellow cup put five cubes, into each blue cup put three cubes.
   Clues:
   
   Three yellow cups have the same number of cubes as five blue cups.
   
   One yellow cup and two blue cups have 11 cubes in total
    
4. Into each yellow cup put two cubes, into each blue cup put four cubes, and into each red cup put six cubes.
   Clues:
   
   Six yellow cups have the same number of cubes as three blue cubes that have the same number of cubes as two red cups.
   
   Two blue cups and one red cup have 14 cubes in total
    
5. Into each yellow cup put seven cubes, into each red cup put seven cubes, and into each blue cup put five cubes.
   Clues:
   
   Five yellow cups have the same number of cubes as seven blue cups.
   
   Five red cups have the same number of cubes as seven blue cups.
   
   One red cup and one blue cup contain 12 cubes.
   Is there more than one possible answer? (y = 6, r = 6, b = 6 also works)

Session one

This session begins with a lesson created by Thomas Kieren, a distinguished Canadian mathematics educator. You will need many strips of paper that are the same length as a standard ruler (30cm). Also have a strip that is 48cm long but has the same width.

1. Begin by asking the students how a strip might be folded in half to make two halves. This will seem trivial as symmetry is natural.
   How do I know that these parts are halves? (Equal parts, all the strip is used, two parts)
2. Provide a couple of non-examples by folding other strips into parts that are not equal, or not all of the strip is used. 
   What is true if the parts are truly halves?
   Look for students to say that one is cut into two equal parts. Fold a 48cm strip into halves.
   Are these parts halves? Why are they a different size to the other halves?
   The important point is that the unit of one determines the size of the halves. Use the half strips to draw two different number lines to represent the relationship of one half to one (whole).
   
3. We are working with fractions as numbers. To do this we need to think about all ones as being the same size. That is very important.
   Where would the fraction 2/2 (or two halves) be on the number line? 
   Where would fraction 5/2 (or five halves) be on the number line? 
   The important point is that non-unit fractions are located by repeatedly joining unit fractions. Five halves are made of five units of one half.
4. Set your students the task of investigating fractions using the paper strips:
   You all know how to fold a strip of paper in two equal parts and call those parts halves. How can you fold one strip into three equal parts? How can you fold one strip into five equal parts?
5. Let your students experiment with developing an equal partitioning method for thirds and fifths. Share their strategies. Students might:
   • Use a ruler to measure the length of the strip then use division to create equal parts. (Is it possible to fold the parts without measuring?)
   • Use one half as a benchmark, knowing that thirds are smaller than halves and fifths are smaller again. (Why are thirds smaller than halves?)
   • Overlap the strip as they fold it to produce three or five equal parts.
6. Discuss: If we can fold into halves, thirds and fifths what other fractions can we make by combining these folds?
7. Demonstrate folding into half then half again.
   What fraction parts have I made?
8. Let your students explore combinations of folds for a prolonged time. Stress the importance of labeling the parts that are made and recording the folds that were used.
9. After a suitable time, share the result of the investigation. Of most interest is how the result of repeated folding can be anticipated. For example:
   If you halve first, then third, what fraction parts do you make? (sixths)
   Does it matter if you change the order of folding to thirding then halving? (No)
   How can you record your folding? (e.g. 1/3 x 1/2 = 1/6) Note that x means "of" as in “one third of one half".)
10. Demonstrate how strip fractions can be used to create a number line: line up several strips that have been folded to show different fractions and mark the folded fractions onto one number line. Ask your students to form their own number line locating fractions of their choice. Encourage them to include some fractions greater than one. Make sure that the students collect their fraction pieces in an old envelope for later use.

Session two

1. Today we will explore equivalent fractions.
   What do you think the word equivalent means?
   From everyday life, students may know that equivalents are the same, e.g. equivalent amounts, equivalent methods, equivalent ingredients or medications.
2. Have students put these fractions in order from smallest to largest.
    
   Do your students recognise that the fractions are all names for one (whole)?
   You may need to make each fraction using a student’s strips from the previous session.
3. Ensure that students understand that these fractions are all on the same spot as one on the number line.
   What other fractions also equal one? 
   What is the same about all fractions that equal one? (The numerator and denominator are the same)
4. Provide the students with copies of page one of Copymaster 1. Alternatively, you could use commercial fraction strips if you have them.
   What equivalent fractions can you see on this page?
   Students should notice that there are three fractions equivalent to one half. Record their observations as an equality, 1/2 = 2/4 = 3/6 = 4/8. 
   Imagine that tenths and twelfths were on the sheet. What equivalent fractions for one half could you find?
   Students might notice other equivalents to one third and two thirds (1/3 = 2/6 and 2/3 = 4/6) and one quarter and three quarters (1/4 = 2/8 and 3/4 = 6/8).
   The fifths do not seem useful to find equivalent fractions. Why is that? 
   Do you notice any patterns in the equations?
   Students are likely to notice doubling of both numerator and denominator without fully appreciating why that occurs, e.g. 1/2 = 2/4 = 4/8. 
5. Provide the students with page two of Copymaster 1 1 and get them to work through it:
   Find as many examples of equivalent fractions as you can using this Copymaster. You may need to cut out some strips so you can move them around.
   Record what you find as equations and drawings.
6. Give your students an extended amount of time to find examples and record. Look for your students to:
   • Show understanding that non-unit fractions are combinations of unit fractions.
   • Identify fraction pairs that might be equivalent by using their multiplication knowledge.
   • Notice relationships between the numerators and denominators of equivalent fractions, e.g. doubling/halving, trebling/thirding, etc.
7. After a suitable period, bring the class together to discuss how all the possible equivalent pairs can be found. A systematic method is to begin with the smallest denominators (largest pieces) and work down Copymaster 1 in sequence. Record the results in order:
   • Halves:
     
   • Thirds:
     
   • Quarters (excluding fractions with halves and thirds):
     
   • etc...
8. Students might forget to include equivalent fractions for zero and one:
   
   Is there a way to anticipate if fractions are equivalent, without using fraction pieces?
9. Students are most likely to identify the relationships between numerators and denominators. You might record some examples:
   
10. It is important for students to understand the reason why the patterns occur. Make sure that a strip model of each equivalence is available, and support students to relate symbols to quantities. For example, for the equivalence 4/5 = 8/10:
                  
     
    When four is doubled to give eight, what does this mean? (There are twice as many parts in eight tenths, than there are in four fifths)
    When five is doubled to give ten, what does this mean? (There are twice as many tenths in one as there are fifths. Tenths are half the size of fifths)
11. Look at Videos One, Two, and Three. For each video, ask your students to fold and shade paper strips, as directed. Discuss the final questions:
    What fraction part can you see?
    How many of those parts are shaded?
    Which two fractions have you found to be equivalent?
12. Finally, begin with folding and shading a non-unit fraction, like three fifths or two thirds.
    If you fold the fifths/thirds in half, what sized parts do we make?
    How many of the pieces are shaded? 
    How do you write an equation to represent what we’ve found?
    What happens if you fold the fifths/thirds into other numbers of parts? What equivalent fractions can you make?
    Can you make fifths into hundredths? How? What about thirds?
13. Finish the session with this challenge:
    Put these fractions in order from smallest to largest.
                         
    Ask students to solve the problem first then check their answer using fraction strips. Have them explain their strategy to a classmate.

Session three

In this session students explore ordering fractions and finding the difference between two fractions. Difference can be found either by subtraction or adding on. Students should come to understand that both strategies yield the same result. The length model is used again.

1. Begin with this problem (Using PowerPoint 1):
   Which is more, one third or one half of the same submarine sandwich?
   How much more?
2. Ask students to anticipate the difference between the fractions, one half and one third. They can check their answer using fraction strips, or you can use slide two of the PowerPoint to illustrate the difference of one sixth.
   Why is the difference one sixth?
   Students should notice that 2 x 3 = 6, the connection between the denominators. However, the important understanding is that sixths work because both fractions can be renamed into equivalent fractions with that common denominator.
   How many sixths is one half equivalent to?
   How many sixths is one third equivalent to?
   Could we use twelfths instead? What would be the difference then?
   The difference can be one sixth or two twelfths. Is it strange to get different answers?
   Students should notice that one sixth and two twelfths are equivalent fractions so both answers are different names for the same number. Slide three shows different ways the problem can be expressed as an equation. Discuss what each equation means, e.g. “One third and how much more equals one half?”
3. Move on to the other problems in PowerPoint 1. The problems get more complex as the difficulty of renaming increases. Let your students work in small groups to solve the problems in the most efficient ways they can. Allow access to fraction strips to check answers, and to justify why the answers are correct.
4. After the four examples, hold a short plenary about finding the differences by subtraction.
   I have a new problem. 
   What is the difference between two thirds and five eighths of the same submarine sandwich?
   Where should I start?
   Students should suggest that a common denominator is needed because the parts are different sizes.
   What common denominator will work?
   Hopefully, students suggest 24 using 3 x 8.
   Can two thirds be renamed as so many twenty fourths? How many?
   Can five eighths be renamed as so many twenty fourths? How many?
5. Capture their ideas in symbolic form to create a ‘paradigm example’ of how to subtract fractions.
   
6. Provide your students with Copymaster 2 and ask them to work individually or in pairs to solve the problems. Access to fraction strips will provide some support for students, though some differences are outside the range of the fraction pieces. Look for your students to:
   • Recognise that differences can be found by subtraction, or by adding on;
   • Know that common denominators are needed if the denominators of the two fractions are not equal;
   • Rename one or both fractions as equivalent fractions to enable subtraction;
   • Systematically record their calculations, as above;
   • Represent part-whole relationships as fractions in Question 2, e.g.  120/160 = 3/4;
   • Recognise that pie charts are proportional representations of frequency data, and that comparison of categories can be made using fractional differences.
7. After a suitable time, gather the class to discuss their solutions. Emphasise the relationship between difference and subtraction, and the importance of applying equivalent fractions with common denominators.

Session four

In this session students explore how to add fractions. They learn to recognise when one or both fractions in an addition operation need to be renamed.

1. Use PowerPoint 2 to introduce this problem:
   Tieri’s parents pay for two credits each week so she can play her favourite computer game.
   She is careful not to use all her credits up in one day.
   On Saturday Tieri used up three quarters of a credit. She used another three quarters of a credit on Sunday.
   How much credit does she have left for the rest of the week?
2. Let your students work out the answers in pairs then share strategies with the class. Use the fraction strips made in earlier sessions to model strategies students suggest. Possible strategies might include:
   • Subtracting three quarters from each full credit leaving two lots of one quarter of a credit. The two quarters can be added to make one half.
   • Adding three quarters to three quarters to get six quarters. Six quarters equals one and one half so there will be one half left from two whole credits.
   • Naming two as eight quarters and subtracting 8 – 3 = 5, 5 – 3 = 2, to work out that two quarters of a credit is left.
3. Ask: What was it about the fractions in this problem that made it reasonably easy to solve?
   Students might note that only quarters were involved. Change Tieri’s computer credits scenario to five eighths plus one quarter (slide three of PowerPoint 2). Let your students solve the problem in pairs. Watch out for incorrect strategies such as adding numerators and denominators, e.g. 5/8 + 2/8 = 7/16. Use a strip model to represent the problem and use that model to address problems your students have.
   
4. Discuss how changing the fractions affected students’ strategies. Look for students to highlight that the fractions had different denominators so could not be combined in the same way that three quarters and three quarters were. In this case one quarter can be renamed as two eighths then the fractions can be added. Record and ask:
   How many credits does Tieri have left? (nine eighths, which equals one and one eighth)
   Knowing that fractions can only be added if they refer to the same sized parts is very important. The denominators must be the same.
5. Progress to slides four and five of PowerPoint 2 which have these problems:
   Imagine a week in which Tieri uses up two thirds of a credit on Saturday and one half of a credit on Sunday.
   How much credit does she have left for the rest of the week?
   In a week of the school holidays Tieri uses up four fifths of a credit on Saturday and three quarters of a credit on Sunday.
   How much credit does she have left for the rest of the week?
6. Tell the students that they need to be able to justify why their answer is correct using fraction strips as evidence. Look for students to:
   • Recognise that the fractions have different denominators so cannot be added in that form, e.g.  2/3 + 1/2 ≠ 3/5;
   • Rename each fraction with a denominator that is in common, preferably sixths for the first problem, though twelfths will work as well;
   • Model the equivalence of 2/3 = 4/6 and 1/2 = 3/6 with materials for the first problem;
   • Model the equivalence of 4/5 = 16/20 and 3/4 = 15/20 with materials for the second problem;
   • Record the operation in an efficient way, e.g. 4/5 + 3/4 = 16/20 + 15/20 = 31/20.
   • Find the credits Tieri has left by renaming two in terms of the common denominator.
7. To practise addition of fractions students can work cooperatively on Copymaster 3. They may use fraction strips to support them, if needed, though ask predictive questions to encourage anticipation of the result, such as:
   • Will the credits used add to less than one, or more than one? How do you know?
   • What common denominator/s will work? How do you know?

Session Five

In this session students learn to distinguish between problems in which fractions are treated as measures (numbers), and problems in which the fraction operates on another number. Rational numbers (which fractions are a subset of) conform to the same properties as whole numbers when operated with (added, subtracted, multiplied and divided). Recognition of that conformity is a critical transfer for students at Level Four.

1. Begin with the problem on slide one of PowerPoint 3. The Elise and Harry scenario is, "Two thirds of Elise’s pocket money equals three quarters of Harry’s pocket money." 
   How is this possible? 
   Who gets more pocket money each week, Elise or Harry? How do you know?
2. Challenge your students to find many amounts of pocket money for Elise and Harry that would make the scenario true. Allow the students time to explore and record possible amounts. For example, if Elise and Harry both spend $12, how much do they earn each? $12 is two thirds of what Elise earns so she gets $18. $12 is three quartes of what harry earns so he gets $16.
3. As a class, discuss a generalised way to find a fraction of another quantity. For example, three quarters of $12 can be written as 3/4 x 12 = 9. Students often do not connect the meaning of x in whole number multiplication with its meaning, ‘of’, as in “three sets of four.”

4. Organise the solution data in a table, such as:

   Elise’s weekly pocket money (e)	Harry’s weekly pocket money (h)
   $18	$16
   $81	$72
   $45	$40
   $9	$8
   etc.	etc.

5. Ask: Do you see any patterns in the table?
   Students might notice that Elise’s amounts are multiples of nine and Harry’s amounts are multiples of eight.
   What operations change Elise’s amounts into Harry’s amounts?
   Try dividing Elise’s amounts by nine. What do you notice?
   Students might see that dividing Elise’s amounts by nine then multiplying the result by eight gives Harry’s amount. High achieving students can be encouraged to express that relationship algebraically (8/9 e = h or e = 9/8 h).
6. Work through the other slides in PowerPoint 3 to support students to develop an algorithm for multiplying a fraction by a whole number. With each problem record the calculation, e.g. 4/5 x 15 = (15 ÷ 5) x 4.
7. Provide the students with paper strips and rulers. Cutting an A3 sheet lengthwise with a width of 2 cm is a good size of strip. The final slide of PowerPoint 3 has three challenges for students to solve, individually or in small groups. Look for students to:
   • Recognise that the common denominator for thirds and eighths is twenty fourths, so make the strip a multiple of 24cm long (Challenge One);
   • Recognise that equivalent fractions of the same length are at the same location (Challenge Two);
   • Accept that neither the length of the whole or what fraction of this length is at 18cm are known, but work with the scenario to identify these variables. FOr example, if 18cm is one quarter then the whole strip is 4 x 18 = 72 cm long.

Session 1

In this session we discuss the attributes of an orange that could be measured.

1. Present the class with an orange.
2. Ask: How can you measure an orange?
3. Focus the discussion on the attributes of an orange that could be measured.  List the possibilities on the board. For example:
   • Amount of juice
   • Length or area of the skin of the orange
   • Circumference or diameter of the orange
   • Mass of whole orange, skin, and flesh
   • Number of pips or segments
   • Angles of segments
   • Volume
   • Time it takes to peel or eat
   • Temperature of an orange (Peel, flesh, centre)
   • Rolling distance - time to stop
   • Acidity and sweetness of the juice (might be measured on a scale or using tools like litmus paper)
   • Fraction of the orange that is water
   • Density of an orange (Does it float?)
   • Colour of an orange (Look up spectrophotometer)

You might go online to look up “fascinating facts about oranges” that might yield interesting attributes to measure.

4. Invite the students to select one of the attributes to measure in small groups.  Try to ensure that a variety of attributes are being investigated.  More than one group could measure each attribute. Consider the knowledge make-up of these groups. Is there a mix of more mathematically-confident students and less mathematically-confident students?
5. Send the groups away to discuss how they are going to measure their selected attribute.  If multiple groups are measuring the same attribute encourage them to devise different approaches.  Instruct them to write up a proposal for their investigation, including details on:
   • attribute being measured
   • equipment required
   • method (step by step)
   • recording and presenting their results
6. Discuss the idea of precision.
   How can we make our measurement as precise as possible?
   Students might mention that oranges vary a bit so multiple oranges might be needed. An average might be used.
   How can we make sure we are consistent in the way we measure?

Session 2

In small groups we attempt to measure one of the attributes of an orange identified in Session 1.  Our measurements will be compiled into a class report on The Orange.

1. Share the groups' proposals with the class.
2. Discuss whether proposals are reasonable to carry out, or what improvements need to be made. You might put teams together to peer review proposals, and encourage tuakana-teina.
3. Emphasise the importance of accuracy.
   What units will you use? (cubic centimetres (cm³) or millilitres (mL) for volume, centimetres or millimetres for length, grams for mass)
   How will you ensure that your measurements are accurate?  (repeat where possible, double check when reading scales, read the scale correctly, take three different measurements then average, etc.)
4. Once proposals have been checked by the teacher, students have the rest of the session to carry out their measurements and begin writing them up. Distinguish the measurements that damage the orange from those that do not. Discuss why measurements that do not damage the orange would be preferable. For example, measuring the circumference and diameter does not damage the orange, but measuring the skin does (unless students look up and use the rule for surface area of a sphere). Limit damaging measures but encourage groups that are not damaging the orange to measure many different oranges. Volume can be found by formula or by immersing the orange in water to see the volume it displaces. Comparing the accuracy of volumes from formula and measurement is interesting (opens up the idea of measurement error).

Session 3

In this session students complete the write up of their Orange investigations, that are then compiled into a class report on The Orange.  Students then select a different attribute which they will investigate as a group over the next two sessions.

1. Students are given time to complete their group reports on their orange or oranges as required.
2. When students measure many oranges, they sort the data and display that data in a suitable way, e.g. stem and leaf graph for masses, volumes, areas, and lengths.
3. Bring all groups together and discuss findings.
   What are each group’s results?
   What could each group have done differently?
   How accurate are each group’s results?
   How could they have been more accurate?
   What attributes were easiest/hardest to measure?
   Why?
   What were successful/unsuccessful approaches to measuring each attribute?
   Why were the approaches successful or unsuccessful?
4. Compile the class results into a report on The Orange.
5. Explain that students will have the next two days in small groups to investigate a different attribute.

Session 4

This session is given to students to work in their groups measuring their oranges.

1. Circulate and provide assistance to students, where required, and ensure that students are using good measurement techniques. Can your students?

   • Read the scales on measurement tools to the nearest unit
   • Round appropriately where needed
   • Record the measurement with correct numbers and units
      

   Where necessary provide mini-tutorials for groups of students about using measurement tools.
    

2. Emphasise accuracy:
   What can you do to get your results to be at a sensible level of accuracy?
3. Tell groups that they should have finished most of their measurements by the end of the session.  There will be some time next session to finish off, but they will need to leave time for writing up their results.
4. Teams of students might compare their results to those of other teams.
   
   Why might results be different? (Different oranges, different choice of tools and units, measurement error, etc.

Session 5

In this session students complete their measurements of their oranges, and write up their results to share with the class.

1. Allow time as required for students to complete their investigations and write up results.
2. Bring groups together to share results.
3. Class discussion – What have you learned?
   What have you learned about your orange?
   Were any of the results surprising?
   What have you learned about measurement?
   Were there attributes that you measured that you didn’t previously know how to?
   What was the most interesting thing that anyone in the class measured? What made it interesting?
   Who might be interested in our results? Why would they be interested? (e.g., Marmalade makers might be interested in mass of oranges)
   What other objects could we measure?
4. Ask for suggestions on what other objects around the classroom, or school, that could be measured. Try to create a list of objects which could be measured in a variety of ways.  Consider how these objects could reflect the cultural diversity, and/or current learning interests of your students (e.g. culturally significant items of clothing, culturally significant buildings, native plants). This learning could be linked to the idea of ko wai au? (Who am I?) and be used to develop connections between students. It could also be used as a context to look at how different items have changed over time. Some examples include:
   • Pieces of clothing, such as hats, shoes, jerseys
   • A school feature such as pool, shed or hall
   • A plant, tree, flower, or other living object
   • A piece of furniture, or another inanimate object
   • Cars in the carpark or street
   • Small objects such as raisins, seeds, or acorns.
5. Allow groups some time to choose an object which they want to investigate.
6. Instruct groups to write up a proposal of all the attributes they intend to measure and how they will measure them.  Encourage groups to find at least three distinct attributes to measure.  If they cannot, suggest that maybe they should choose a different object.

The following six stations provide a range of problems for students to apply and interpret measurement of mass. Consider what would be the most effective method for introducing these to your class. You could work on them as a whole class and provide support to groups of students. Alternatively, you could use another relevant igniting activity to introduce the context for learning, before directing students to work on one or more of these stations independently, or in small groups. These stations could serve as the basis for learning in different sessions, or could be used as one session. At the conclusion of these stations, students draw on the problems presented and create their own stations to be used in other lessons.

Station 1: A kilo of coins

You have won a prize which can be just one of the following:

• 1 kilogram of $1 coins
• A 1.5 metre long trail of $2 coins (lying flat and touching)
• A 0.5 metre high stack of 50c coins

What is your choice?

Answers:

1 kg of $1 coins (1000 ÷ 8 = 125 coins, so $125)

1.5 metre of $2 coins (1500 ÷ 26.5 = 57 coins, so $114)

0.5 metre stack of 50c coins (500 ÷ 1.7 = 294 coins, so $147)

Station 2: Largest Lasagne

This problem could be adapted to reflect food that is meaningful to your students (e.g. the largest tray of pani popo).

The world’s largest lasagne was made in 2012 at a restaurant in Wieliczka, Poland.
It weighed 4865 kg and measured 25 m x 2.5 m.

The ingredients were:

2500kg of pasta, 800kg of mince, 400kg of mozzarella cheese, 100kg of peas, 100kg of carrots, and equal amounts of white sauce and tomato sauce.

1. How much did the white sauce and tomato sauce weigh?
2. What would be the size of a 500g piece from the lasagne?
3. How many people could be fed with the whole lasagne?
   Show how you arrived at your estimate.

Answers:

1. The other ingredients total 3900kg so the sauces must weigh 4865 – 3900 = 965 kg.
   500L of each sauce was used. Does that sound right?
2. A 500g piece would be about 1/10 000 of the whole lasagne. One way is to cut both the length and width into 100 parts, since 100 x 100 = 10 000. A single piece would measure 25cm x 2.5cm. That’s a bit skinny so 12.5cm x 5cm might work better.
3. The lasagne was actually cut into 10 000 pieces so that’s how many people were fed. Each piece had a mass of 0.486 kg or 486 grams. That is a good serving of lasagne.

Station 3: Weighing Tonnes

Konsihiki was the largest active sumo wrestler in the world with a mass of 287 kg. Now he is retired.

How many Konishikis weigh as much as 1 tonne?

Make a table of tonne weights using objects in the classroom. Remember that 1000 kg is a tonne.

Answers:

The number of Konshihikis in 1 tonne equals 3.48, about 3 and ½ of him.

To find how many of any object make 1 tonne, divided 1 000 by the weight of the object in kilograms. For example, if a schoolbag weighs 5kg then 1 000 ÷ 5 = 200 make 1 tonne.

Station 4: Jumbo facts

Find out facts about the mass of very large animals and make a report about these animals for the class. To get you started here are some facts about the Blue Whale, which can be seen in New Zealand waters.

The blue whale is the largest animal living on Earth. It can reach up to nearly 30 metres in length and weigh up to 180 tonnes (t). Their tongues alone can weigh as much as some elephants and their hearts are huge, weighing a whopping 180kg. They have the largest babies on Earth. When they are first born they can be 8 metres (m) in length and weigh 4000kg. Imagine a jet engine that registers at 140 decibels. A blue whale, when it calls, registers at 188 decibels. Compare the facts about the Blue Whale with the large African elephant

The African elephant is the biggest animal on land. Fully grown the male can be 7 metres long, 3.2 metres tall at the shoulder and have a mass of 6500kg. Its tusks can weigh as much as 100kg each. The largest pair of tusks on record are in the British Museum and weigh 133kg each. 

What combination of animals could be equal to the elephant's weight?

For example, it takes 6500 ÷ 5 = 1300 big domestic cats to weigh 1 elephant or 130 big dogs.

How many rhinoceroses, lions, giraffes, or hippopotamuses weigh the same as an elephant?

Answers:

Answers will vary depending on what other animals your students research.

Station 5: Mass of water

Measure out one litre (l) of water.

1. What is the mass of one litre of water?
   If 1L = 1000ml, what is the mass of 1mL of water?

2. For each container, estimate the capacity of the container, measure it to check, estimate the mass of water when the container is full, and find the mass of the water using scales.

   Record your results like this:

   Container	Estimate capacity	Measure actual capacity	Estimate mass of water	Measure actual mass
   A
   B
   C
   D
   E

3. How many drops of water are needed to fill each container?
4. What is the mass of a single drop?

Answers:

1. 1 litre (l) of water has a mass of 1 kilogram (1000 grams). 1 millilitre (mL) of water has a mass of 1 gram.

2. Answers depend on the size of the containers. Here is an example:

   Container	Estimate capacity	Measure actual capacity	Estimate mass of water	Measure actual mass
   A	400 mL	450 mL	390 g	450g

3. About 20 drops make 1 ml of water. Find the capacity of the container in mL then multiply by 20 to get the number of drops.
4. A single drop has a mass of 1/20 of 1g, that’s 0.05g.

Station 6: Frank’s arms

Counting on Frank by Rod Clement (1990; Harper Collins Publishers: Sydney) has some great ideas for measurement investigations. You can view readings of the book on YouTube if you cannot source a copy of the book. One of the ideas introduced in the story is about Frank carrying a trolley load of cans to the supermarket.

1. Trolleys measure 60 litres or 80 litres.
   What do those measures mean?
2. How heavy do you think Frank’s load of cans is?
3. How many cans are you able to carry in a reusable supermarket bag?

How did you work that out?

Here is another task based on Counting on Frank that you may choose to use.

Answers:

1. 1 litre is a unit of capacity, but it is also used as a unit of volume. One litre measures 10cm x 10cm x 10cm. 60 litres is 60 times that size.
2. Frank has 47 cans. They could be 420g cans so they would weigh 47 x 420g = 19 740g or 19.740 kg (about 20 kg). If the cans are bigger, say 820g, then the mass equals 47 times the mass of one can.
3. You could get 24 x 420g cans in a supermarket bag, or 36 cans if you add another layer. The mass of the bag would be 24 x 420g = 10.080 kg or 36 x 420g = 15.12 kg.

Session 1: Introduction to the mathematics of maps

This lesson provides an introduction to the context of the mathematics of maps, and the idea of reading maps. You could start this session with investigation into the traditional navigation methods used by Māori and Pasifika peoples. Consider inviting local iwi members to your classroom to discuss where the local iwi heralded from. These locations, and the locations of other iwi relevant to your students, could be identified on a map.

1. Present the class with a map of the local area (Google Maps might be used).
2. Ask them to identify what it is.
3. Ask them if they can explain why you would be looking at a map in mathematics time.
4. Encourage students to identify the maths involved in reading and drawing maps (directions, scale, distances etc.)
5. Challenge students to name things about maps that are part of mathematics.
6. Hand out some examples of maps of Aotearoa or the local area and ask students to find features that they think involve mathematical ideas.
7. Encourage and support students to identify the following features:
   • Scale
   • Coordinates
   • Compass bearings
8. Discuss each of these features, and explain that you will be looking at all three over the next few sessions. You will invite a guest speaker to come in to tell the class about how they use maps in their life and work.

Session 2: Scale

In this session we look at the scale of maps, what they mean, and how to interpret and apply them. We draw a scale map of the classroom.

1. Show the students a map with a scale clearly labelled. Ideally use a map of the local area or somewhere the students are familiar with. You might use a source that provides free maps, such as Request a Map (print or request large touring maps), Google Maps (free online maps), or the AA (you can collect maps from your local AA site or order them online). Ensure that you have enough copies of the map to be shared in small groups. A digital version of the map could make it easier for all students to see.
2. Ask students to tell you what the scale of the map is.
3. There are two distinct types of scale you may find on maps, a ratio, such as 1:100 000; and a graphical bar that shows how far 1km, for example, is on the map. Ensure that students have the opportunity to see both.
4. For this session we will be working with a ratio type scale.  Ask students to describe what the scale means. For example, a ratio of 1:50 000 means each centimetre represents 50 000 centimetres (500 metres in real life). If the scale is 1:100 000, then every cm on the map represents 100 000 cm, or 1000 m or 1 km in reality.
5. Ensure that all students understand that the scale tells how many times larger distances are in reality than they are in the map.  For instance, you might model this by showing a range of maps, and explaining what their scales means in reality.
6. Pose some questions around the map you have given the class to check that they understand:  How far is it from …? Model calculating this first, and then support students to independently figure out the distances. Calculators can be used to help students make accurate calculations with larger numbers. 
7. Explain to the students that they are going to draw a 1:20 scale map of the classroom (Check that the class will fit onto whatever paper you intend to use, and adjust the scale accordingly). At 1:20 a 6m x 5m classroom will be drawn the size of a 30cm x 25cm. As an alternative or extension you could draw a scale map of the whole school if you think your class could manage this. More knowledgeable students could work independently, or in small groups, to draw scale models of specified buildings, and you could work with less knowledgeable students to draw scale models of specified buildings in a scaffolded environment.
8. Ask students what this scale will mean. (1m in the real classroom will be 5cm on the map)
9. Start by measuring the dimensions of the class and drawing the outline of the room to scale to make sure that it fits on the paper.  (Possibly you could do this as a class to ensure that students understand the process.)
10. Allow students the rest of the session to draw the remainder of the room in as much detail as possible. Ensure that they include landmarks, such as desks and doorways in the correct places.
11. Students might work from the Figure It Out resource, who lives where? to practise using the scale on a map.

Session 3: Coordinates

In this session we will draw a grid on the map of the classroom and use it to identify places within the class. Note that there are two ways in which location is represented on maps, grid references and coordinates. In both cases the representation is an ordered pair, e.g. F4 as a grid reference, and (6,4) as a pair of coordinates. The convention is to state the horizontal reference first, and the vertical reference second.

In the grid system (left image) the location of the person is an area. In the coordinate system (right image) the location of the person is a specific point at the intersection of the lines. 

1. Refer back to the map the whole class looked at in the session 1.
   What are the vertical and horizontal lines on the map there for?
   What do the numbers at the tick marks (ends of the lines) represent?
2. On larger scale maps, they are likely to be lines of longitude and latitude, but on smaller scale maps they will be grid lines, which are used for finding and indicating specific positions on the map.
   How do they work?
3. See if students can explain how they would describe a given point on the map using the coordinates.
4. Once students have given suggestions teach them how coordinates work: First give the number on the x-axis (along the top or bottom of the map), and then the number on the y-axis (up the sides) that lines up with the point chosen. "X is a cross, Y is in the sky" may be a helpful mnemonic for students to remember the order in which to read coordinates.
5. Allow students to try a few examples.
6. Now explain that we want to be able to do this with our maps of the class, so we will need to draw a grid over the maps that we drew last session.
7. Ensure that students use a consistent grid such as 1 line per cm, starting in a specific corner of the room. As the maps are all to the same scale this should make the coordinates match up between maps.
8. There is now the opportunity to play a game of eye spy, with the format “I spy an object at coordinates (3,4)”, either as a class or in pairs. This will reinforce students' understandings of coordinates.
9. You may give students the Figure It Out activity Map Mysteries to practise finding coordinates in a pirate context. If you have a large world map with coordinates on the x-axis and y-axis, then you could use this as an opportunity for students to identify countries they whakapapa to. In turn, this could help establish connections and cultural understanding between your students. If your students are ready to be extended in their use of coordinates, they could complete this activity independently using a printed world map, and compare their maps to those completed by their peers. They could also investigate the places their ancestors travelled from to arrive in New Zealand (e.g. from Samoa) and record this information using coordinates.

Session 4: Polar Coordinates

In this lesson we draw a north arrow on our maps, and introduce polar coordinates.

1. On the map of the local area used in the previous sessions point out the North arrow and ask what it is for.
2. There may be two arrows, one for Magnetic North and one for True North.
   • True North is the direction towards the North Pole, the point where the axis that the earth spins around passes through.
   • Magnetic North is the direction that a compass points to, and is very close to but not exactly the same as True North. Magnetic North moves slowly and inconsistently.
3. Explain that we are going to draw a north arrow on our maps, using Magnetic North.
4. Use a compass, or phone, to find Magnetic North, and ensure that all students accurately draw a North arrow on their maps. Include the other three cardinal directions, east, south and west, as well.
5. As a few questions such as:
   What is north of the teacher’s desk?
   What is east of the door?
6. Now explain that this idea can be used more accurately, if we break down the spaces between the compass points:
   What is north-east of the mat?
   What is south-west of the bookshelf?
7. This can be made more accurate yet by breaking the gaps one more time:
   What is north-north-west of the middle of the whiteboard?
   What is west-south-west of the doorway?
8. If you think your students will be able to cope you can introduce direction given as an angle, and include a distance. Ensure that students know that the angle is called ‘a bearing’, and that the number of degrees is measured clockwise from north.
   What object is 5m on a bearing of 130⁰ from where I am standing?
9. This could again be made into a game with students in pairs challenging each other. Ideally you should use compasses to find the bearings, but a protractor used on the maps could also be used.
10. To apply the idea of bearings and distance further show students how to use orienteering compasses or the compass app on their phone. In small groups students prepare a set of up to ten clues for another group to locate a position. For example, a clue might be “Walk 45 metres at a bearing of 76°. You might use a local legend about a lost person, usually a tale of love, such as the story of Matakauri and Manata from Lake Wakatipu or Hinemoa and Tutanekai from Rotorua. The clues are given to another group who try to find the lost person.

Session 5: Guest speaker

In this session we invite a member of a local orienteering club, or possibly a local surveyor to come and talk to the class about how they use maps.

1. Before this session, organise a guest speaker to come in and talk about maps and how they use them. A member of a local orienteering club or Land Search and Rescue would be ideal, or possibly a local surveyor. You can get a contact for a local orienteering club from http://www.nzorienteering.com/
2. Have students think about questions they might want to ask, and possibly write them down so the guest can be given some idea in advance of what they might be asked.
3. Have a few selected students show the guest the maps of the classroom and explain what the class has been working on.