Late level 3 plan (term 2)

Planning notes
This plan is a starting point for planning a mathematics and statistics teaching programme for a term. The resources listed cover about 50% of your teaching time. Further resources need to be added to meet the specific learning needs of your class, to support your local curriculum and to provide sufficient teaching for a full term.
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Level Three
Integrated
Units of Work
This integrated unit combines measurement of area with multiplication, and algebraic thinking.
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Level Three
Geometry and Measurement
Units of Work
In this unit students use a programming platform, Scratch, to learn mathematics through digital technology.
  1. Create a programme in Scratch for a Sprite to walk out a shape or figure, including:
    • Correct numbers of steps
    • Correct angles
    • Use of repetition
    • Use of blocks
    • Use of variables and operators
  2. Read a programme in Scratch and work out the path that the Sprite will take
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Level Three
Number and Algebra
Units of Work
This unit explores situations that involve multiplication and division using an equal sets model. Students learn to apply the properties of whole numbers under multiplication, to derive new answers from basic facts, and apply inverse operations to division.
  • Derive from basic multiplication facts to solve multiplication problems with equal sets.
  • Apply the commutative and distributive properties of multiplication to solve problems mentally and on paper.
  • Recognise how both measurement and sharing division problems can be solved by ‘building up’...
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Level Three
Statistics
Units of Work
In this unit we experiment with cubes to make predictions about likelihood based on our observations. Students find out that with probabilistic situations there is no certain way to predict exactly what will happen.
  • Make predictions based on data collected.
  • Identify all possible outcomes of an event.
  • Assign probabilities to simple events using fractions (1/2, 1/6 etc).
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Level Three
Integrated
Units of Work
This unit builds the learning of mathematics around celebrations of Matariki, the Māori New Year. The sessions provide meaningful contexts that highlight Māori culture and provide powerful learning opportunities that connect different strands of mathematics.

Session One

  • Recognise a trapezium, right angled triangle, and points in a straight line, in different orientations.
  • Construct a model of all the possible outcomes for rolling a standard dice and use the model to predict the results of an experiment.

Session Two

  • Find the next shape...
Source URL: https://nzmaths.co.nz/user/75803/planning-space/late-level-3-plan-term-2

Fill it up - flat space

Purpose

This integrated unit combines measurement of area with multiplication, and algebraic thinking.

Achievement Objectives
GM3-2: Find areas of rectangles and volumes of cuboids by applying multiplication.
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
NA3-8: Connect members of sequential patterns with their ordinal position and use tables, graphs, and diagrams to find relationships between successive elements of number and spatial patterns.
Description of Mathematics

Area is an attribute, a characteristic of an object. The attribute of area is the space taken up by part of a flat or curved surface. Usually, we begin by helping students to attend to area as an attribute before formally measuring it. Use contexts in which students compare flat spaces by size such as comparing pancakes or footprints. Note that “biggest” may be perceived in different ways. The most common confusion is between area (the space covered) and perimeter (the distance around the outside).

In context students can be focussed on the attribute of area. Suppose some students think that a given pancake is bigger than another because they wrap string around both shapes and find one length longer than the other. “How many bites would it take to eat each pancake?” is an example of an enabling prompt. Partitioning and combining shapes are also useful ways to promote understanding of conservation of area and can lay groundwork for ideas about the areas of triangles, rectangles, trapezia, parallelograms and other polygons. in later years.

Formal measuring of area with units will only make sense to students if they relate their methods to the process of measuring other attributes such as length and mass. Students need to see the need for units and identify the qualities of units that are appropriate. They also need to realise that a number alone does not convey a measure unless the unit is stated as well.

Units require the following properties:

  1. Units are all the same. You can mix units but that makes it harder to be precise and compare measures.
  2. Units fill a space with no gaps or overlaps which explains the convention for using squares that tessellate by equal measure in both dimensions in arrangements of rows and columns.
  3. More smaller units fit into the same space as larger units. Smaller units tend to give a more precise measure. Note that if the smaller units are one quarter of the size of the larger units then four times as many fill the same space.
  4. Units can be partitioned and joined. Note the connection to fractions, e.g. two half units can make a whole unit. 

The standard units of area in real life are the square centimetre (cm2), square metre (m2), hectare (ha.) and square kilometre (km2). While the proportional difference between metres and centimetres is manageable with length, the proportional difference between square centimetres and square metres makes size comparison difficult.

Consider the relationship between square centimetres and square metres. There are 100 x 100 = 10 000 square centimetres in one square metre. That is the same relationship as between square metres and hectares. A hectare is 10 000 m2. Hectares are used to measure areas of land. Think of a hectare as an area that is 100m by 100m. That means that 10 x 10 = 100 hectares are in one square kilometre. Square kilometres are used to measure large areas of land. For example, Stewart Island has an area of 1 746 km2 or 174 600 hectares.

Specific Teaching Points

Sessions One and Two
A suitable unit for measuring area must have these qualities:

  • Be a piece of area (two dimensional)
  • Units must be the same size
  • Units should fit together with no gaps or overlaps
  • Units should be of a size that give adequate precision (accuracy)

Session Three
The area of a flat shape is conserved (stays constant) as parts of it are moved to different places on the shape. Any shape can be ‘morphed’ into a shape with the same area by ‘giving and taking’.

Session Four
Area is the amount of flat space enclosed by a shape. Perimeter is distance around the outside of a shape. Shapes with the same area can have different perimeters, and shapes with the same perimeter can have different areas.

Session Five
A growing pattern can be structured by looking at how the figures are organised. Noticing structure helps with counting the area of a figure, and with predicting further figures in the pattern. Identifying sameness and difference in figures can help in creating a rule (generalisation) for all figures in the pattern.

Observations of students during this unit can be used to inform judgments in relation to the Learning Progression Frameworks. Click for tables of guidelines.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students, by varying the task requirements. Ways to support students include:

  • providing physical materials, such as objects to use as units of area, is essential for all students, but particularly important for students who need to understand the iteration of identical units
  • cutting and moving parts of shapes around to show that shapes can look different but still have the same area (conservation)
  • explicitly modelling of filling a flat space with no gaps or overlaps
  • connecting previous work students may have done with multiplication and division to counting the number of units in arrays
  • making calculators available to ease calculation demands, particularly when factorisation is involved, e.g. finding all rectangle with area of 54m2
  • modelling how to record measurement processes and answers using symbols.

Task can be varied in many ways including:

  • manipulating the complexity of the shapes that students work with
  • reducing number size where factorisation is required
  • allowing flexibility in the way students find areas
  • using collaborative grouping so students share their strategies.

The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Look for everyday examples when your students encounter area. Examples might involve spaces that are meaningful to them, such as their own bedroom, lounge, section at home. Portions of food, such as pancakes or pies, can be compared by area. Students interested in environmental issues might be motivated by contexts such as the areas covered by drift nets or oil spills. Students might find comparing the size of land areas interesting, e.g. How many times does Rarotonga fit into the North Island? Which is larger Upolo (Samoa) or Espiritos Santos (Vanuatu)? High achieving students might be interested in population density.

Required Resource Materials
Activity

Prior Experience

Students are expected to have some experience with measurement of other attributes, such as length, using informal units. They should also have some knowledge of multiplication facts and understanding of how to apply multiplication to finding the number of items in arrays.

Session One: Three Islands

  1. Play this video introducing Three Islands (mp4, 13MB).
  2. Discuss:
    • Size of an island can be measured by coastline (perimeter) or inside space (area).
    • Measurement requires the use of a unit because the islands cannot be directly compared, i.e. brought together to size match. What units are students going to use?
  3. Put the students into small groups of two or three participants. Each group needs an A3 enlarged version of Copymaster 1. Colour is not necessary. Provide the students with a choice of materials. Include items like; string, nursery sticks, dry pasta, beans (different sizes are good, e.g. plastic, lima, red), chick peas, counters, square tiles, transparent 1 cm grid and 5 mm grid made with Overhead Projector Transparencies. Ask students to record their thinking as they work.
  4. Allow the students plenty of time to compare the islands. Look for the following:
    • Do students distinguish perimeter from area?
    • Do students use a single unit consistently with awareness of iteration (copying with no gaps or overlaps)?
    • Do students use sensible number strategies to count the units?
  5. If possible take digital photographs of the students working and play these images as a few groups share their methods with the class. You might select groups to focus on the bullet points above.

Session Two: Measuring Flat Space

  1. Tell students that they are thinking about flat space (area) rather than both area and perimeter. You are interested in how they measure the area of an island.
  2. Work through the slides of Powerpoint 1. It shows other students working on Three Islands. Ask the students what they notice. Particular points to highlight are:
    • Slide Two: The students are using different units. How will they compare their measures?
    • Slide Three: The students are measuring coastline (perimeter) using pasta. Will that tell them about flat space (area)? 
    • Slide Four: The students are using square tiles. Are squares a good unit to use? Why or Why not?
    • Slide Five: The students have filled one island up and moved the lima beans to the other islands. Is this a good strategy or not? Why?
    • Slide Six: The students have used square tile and pasta. Will that work to compare the flat spaces (areas) of the islands? What could the students do?
  3. Provide the students with copies of Copymaster 2, which contains various approaches to measuring two different islands. Ask the students to discuss the measuring strategy that is used. Tell them to think about the questions, What is correct about the strategy? What is incorrect about the strategy?
  4. Look for:
    • Page One: There are gaps and overlaps with the counters. Why are circles hard to use a unit of area?
    • Page Two: There is a mixture of units (square tiles and beans). Could the units be converted to a measure with one unit, e.g. one square for two beans?
    • Page Three: The units are all the same but the Left Island has area missed and Right Island has tiles outside the coastline. How can you allow to missing or outside parts of the area?
    • Page Four: The units are all square tiles but they are different sizes. How many Left Island squares fit into a Right Island square? How could this ratio be used?
  5. After a suitable period of group discussion gather the class to compare their ideas and to decide which island has the most area. Discuss the ’give and take’ of part units combining to a full unit. Record the measures using both number and units, e.g. 46 small squares (Left island) and 11 large squares (Right Island).
  6. Discuss: What is our problem? (Need the same unit). Converting 4 small squares to one large square results in 11½ large squares being the area of Left Island, making it larger.
  7. Discuss: How trustworthy is the result given the ‘give and take’ of part units?
    This will raise issues of precision. Small squares are more precise than large squares. Why?

Session Three: Megabites

  1. Use PowerPoint 2 to tell the story of Yap, the hardworking sheepdog. When the farmer changes Yap’s biscuits he gets suspicious that he has been duped. The key ideas being developed are: 
    • Conservation of area – re-arrangement does not alter the internal space.
    • Partial units can be created for more precision and these partial units can be combined
  2. Ask: How might Yap check to see that the biscuits are the same size?
  3. Students might suggest overlapping the biscuits to directly compare them. That is a useful suggestion.

    Note that by giving and taking, the overlapping triangles can fill the missing space, transforming the trapezium into the square. The biscuits are the same area.
    Students may suggest other strategies involving units. The fourth slide of the PowerPoint 2 has an overlay of square units.
  4. Ask: Why might Yap use squares? (no gaps or overlaps)
  5. Discuss: How will he allow for part squares with the Bonza biscuit?
  6. Read the final slide which has a letter from Yap to the Dog Biscuit Company. The challenge is to create different shaped biscuits that are still 36 squares in area.
  7. Ask the students what shapes they might try for the new biscuits. Make a list of shapes, e.g. rectangle, parallelogram, equilateral triangle, hexagon, octagon, etc. Closed curves such as the circle and ellipse will be very challenging but encourage the students to try. High achievers might look for area formulae online.
  8. Provide students with squared paper, e.g. Copymaster 2 enlarged onto A3, rulers and scissors. In their teams students need to create at least eight different new biscuit designs that are 30 squares in area. At this stage keep the shapes students create separate so they can be sorted later. On the back of any shapes, ask students to record how they checked that the biscuit was 36 squares in area.
  9. After a suitable time of exploration gather the class to look at the different biscuit shapes. Sort the biscuit shapes into categories by their common properties. Visually compare the shapes to see if they look to have the same area. Points to bring out include:
  10. Discuss: What rectangles are possible? Rectangles can be recorded systematically as expressions, i.e. 1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 6. Ask the students to identify what the factors refer to in each rectangle and why 4 x 9 is really the same biscuit as 9 x 4 by the commutative property.
  11. Discuss: What is the relationship between a triangle and the surrounding rectangle? For example, this diagram shows two different triangles.

    The diagrams show that the triangle is one half the area of the surrounding rectangle. For example, if a triangle is 36 squares in area then the rectangle must be twice that area, 72 squares.
  12. New shapes can be made by starting with a ‘parent’ shape that is 36 squares in area and altering the shape by ‘give and take’. For example,a rectangular biscuit might be altered to form an interesting shape with the same area.

Session Four: Yap’s Run

  1. Play the video introducing the square metre (mp4, 38MB). Discuss what kinds of areas are measured in square metres, e.g. house floors, driveways, sports fields and courts. Show the students a square metre made from newspaper and tape. You might choose to construct the square metre in front of the class so they see how it is made.
  2. Ask: How would we figure out the area of our classroom in square metres? We might want to recarpet.
  3. Invite suggestions. Using an array of columns and rows is more efficient than mapping in the square metres one at a time. Link multiplication with the arrangement of rows and columns, e.g. 6 columns of 4 square metres each has an area of 6 x 4m2 = 24 m2. Explain that m2 means the unit, square metre. For homework students might investigate the cost of recarpeting the classroom on line.
  4. Show PowerPoint 3: Yap’s Run.
  5. Stop on Slide 3 to ask students to check that each design has an area of 54 m2. The students will need to partition two of the designs into smaller areas and combine the measures. Also ask what the perimeter of each enclosure is? Does the perimeter matter?
  6. Move on to Slide 4 where the problem is posed. Challenge the students to create an interesting shaped run for Yap that does not exceed a perimeter of 45 metres. Let the students create scale drawings of the enclosures using grids in their mathematics book. Expect them to label each side of the run with appropriate measures and show clearly how the area was calculated. Slide 4 of PowerPoint 3 shows an example with some measures shown. The perimeter of that run is greater than 45 metres. See if the students can work the perimeter out.
  7. Give the students time to create their favourite run. Collect the diagrams at the end as work samples for assessment and display. You may like to go outside with some cones and a trundle wheel to mark out the favourite design in real size. Use the paper square metre as a benchmark and ask the students to calculate the area of parts of the run.

Session Five: Farmer Joe’s Garden

In this lesson students apply their understanding of area to a growing pattern. The task can be used to assess several aspects of mathematics, including multiplicative thinking, measurement, algebraic thinking and equations and expressions.

  1. Show PowerPoint 4: Farmer Joe’s Garden.
    Slide 2 presents the shape of the garden in Year Four. Ask the students what they notice. Look for them to identify properties of the shape and sections of the garden that will be useful structures for finding area. Ask the students to work in pairs to decide on the area of the Year Four garden. You may need to remind them that each small brown square represents one square metre (1m2).
  2. After a suitable time discuss the various ways they structured the year 4 garden to find its area. Highlight the use of multiplication to find the area of arrays within the garden. Slides 3-6 show different ways to find the area of the garden. For each slide discuss how the structure could be recorded using an equation.
  3. Slide 3: 5 x 4 + 2 x 5 + 2 x 4 = 38 m2. Note that brackets are not needed with the order of operations but you might like to record (5 x 4) + (2 x 5) + (2 x 4) = 38 m2. Ask students to identify the connection between each multiplication expression and the diagram on Slide 3.
  4. Slide 4: 7 x 4 + 2 x 5 = 38 m2. How is this equation similar but different to that for Slide 3. Note that 7 x 4 is split into 5 x 4 + 2 x 4 in the equation for Slide 3.
  5. Slide 5: 5 x 6 + 2 x 4 = 38 m2. Compare this equation to that for Slide 3. Note that 5 x 4 and 2 x 5 combine to form 6 x 5 or 5 x 6 using the commutative and distributive properties.
  6. Slide 6: 7 x 6 – 4 x 1 = 38 m2 or just 7 x 6 – 4 = 38 m2.
  7. Slide 7 invites the students to structure successive members of the growing pattern. Encourage the students to represent arrays in each garden using multiplication. For example, the gardens for Years 1-3 might be shown as: 
  8. Structuring is very important if students are to generalise the pattern for later years. Using the same idea Years 5 and 6 would look like this: 
  9. For both years, discuss: What is different and what is the same? How is the area of the middle rectangle related to Year?
  10. Slide 8 requires students to predict the area of the garden for Year 12. This is a challenging task but students can use table based strategies if they cannot generalise the structure. Here is a table of values for the pattern:

    If they look for pattern in the differences students might notice that those differences grow by two each year.  

Starting from Scratch

Purpose

In this unit students use a programming platform, Scratch, to learn mathematics through digital technology.

Achievement Objectives
GM3-3: Classify plane shapes and prisms by their spatial features.
GM3-4: Represent objects with drawings and models.
GM3-5: Use a co-ordinate system or the language of direction and distance to specify locations and describe paths.
Specific Learning Outcomes
  1. Create a programme in Scratch for a Sprite to walk out a shape or figure, including:
    • Correct numbers of steps
    • Correct angles
    • Use of repetition
    • Use of blocks
    • Use of variables and operators
  2. Read a programme in Scratch and work out the path that the Sprite will take
Description of Mathematics

The mathematics in this unit is primarily about geometry though aspects of measurement and number are applied.

Specific Teaching Points

  1. Polygons are planar shapes which means they are two-dimensional. ‘Poly’ is the prefix for many and ‘gon’ means corners or angles. Polygons are bounded (enclosed) by line segment so the sides are straight. The name of a polygon comes from the number of corners (or sides) it has. For example, a pentagon has five corners and five sides since ‘penta’ is the prefix for five.
     
  2. In this unit students need to navigate the boundary of polygons. This requires knowledge of the exterior angles. Imagine beginning at corner A and walking around the outside of the regular pentagon. Regular means that the sides and angles are all equal. Five turns would occur during the journey that has you starting and ending at A and facing the same direction as you started at the end. Since you are facing the way you started you must have completed a full 360° turn. That full turn was divided into five equal small turns which must be 360 ÷ 5 = 72° each.

     
  3. If the polygon was a regular hexagon it would have six corners. The 360° would be equally shared six ways so each external angle would be 360 ÷ 6 = 60°. Therefore, in general, the exterior angles of a regular n-gon are (360 ÷ n)°.
    To have their Sprite facing the correct direction students can use this command:

    To make sense of this command students need to understand bearings, in a simple way. Bearings are measured in a clockwise or anti-clockwise direction from due North (vertical). Here are some examples.

    Note that in Scratch the bearing that is an anti-clockwise turn from the vertical of 40° is written as -40°.
     
  4. Students also need to attend to length, as measured in steps with Scratch. Therefore, some ideas of measurement and proportion are involved. Take the simple situation of navigating from A to B. The first attempt in red was a length of 200 steps.

    To work on the length required students need to apply iterative use of a unit or equal partitioning. One way to solve the problem is to realise that 200 is about two-thirds of the distance. Each third must be 100 steps so the whole distance must be three thirds, 3 x 100 = 300 steps. In working with Scratch students may encounter more difficult proportions. They may need to rely on iteration. In the red line example, they might realise that the line can be equally cut into four parts. One part is 50 steps and can be used as a unit to measure the whole length that is needed. That measurement process of placing units end on end without gaps or overlaps is called iteration.
Opportunities for Adaptation and Differentiation

Students experience with digital technology is likely to vary considerably. The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:

  • physically modelling the movement of a Sprite or robot by students to develop a sense of decisions they make to complete a path
  • developing and interpreting code is small pieces before putting together the complete code. This approach is like modular programming
  • writing and enacting code in a risk-free environment. Simply trial code and accept that sometimes what you get is not what you expect
  • taking a gradual improvement approach by looking at what works and what does not. Refine the existing code then review it again
  • letting students work in pairs to share ideas and expertise.

Task can be varied in many ways including:

  • letting students self-select a level of challenge that they feel comfortable with. The activities have several options that vary in difficulty.

The context for this unit is engaging. Capture the interest that students might have in robotics by showing non-fiction examples, and science fiction stories. There are many videos of films and television shows online. Students will enjoy personalising the movements of a Sprite to create a figure that is of significance to them. Can they train the Sprite to draw a koru, or hibiscus flower, or write their signature? Also use collaborative partnerships to motivate your students. Most coding is completed in teams which reflects the collaborative nature of work in digital technologies.

Required Resource Materials

Access to Scratch software, available for online use or free download at https://scratch.mit.edu/download

Activity

Prior Experience

It is expected that students will have a concept of an angle as a measure of turn and be able to use a protractor to measure angles. Some experience with co-ordinate planes would be an advantage but is not strictly needed. Students will also need to know the properties of simple polygons like triangles, squares and hexagons.

Session One

In this lesson students investigate giving directions with sufficient clarity that a robot or sprite can carry them out. You may like to play a short video from YouTube showing the portrayal of robots in science fiction, e.g. Star Wars or Lost in Space. Begin by discussing this question:

  1. Ask: How do robots know what to do?
    Students need to understand that robots are really computers that are programmed by humans to carry out tasks. This avoids the thorny issue of whether robots will become self-thinking in the future.
  2. Discuss: What is a programme?
    A programme is composed of a set of instructions that the robot can interpret. It only makes sense to tell a robot to “Go forward three steps” if it understands the meanings of “Go forward” and “Steps”. Computers are very good at counting if they understand what “three” means.
  3. Ask the students to work in pairs. One student is the programmer and the other student is the robot. The challenge is:
  4. Give the robot clear instructions to walk the perimeter (outside boundary) of a square.
    You may need to clarify the meaning of perimeter. Let the students work for a few minutes to work through their instructions. Some recording will be necessary. When you gather the class ask a few pairs to model the instructions. These issues should surface.
    • What actions can we assume the robot can do? (Walk forwards/backwards/ etc., turn, count steps, …)
    • Why is precise language important?
    • How important is it to get the sequence of steps correct? What happens if the steps are out of order?
  5. Are some actions repeated? This can be thought of as a ‘block’. If the robot knows a block how can this make a programme easier to write?
  6. Tell the students that they are going to use software called Scratch which is a programming language. Play Video One which shows the creation of a set of commands to draw a square. You might stop the video at certain points to allow students to set up the programme themselves. Video Two shows how to set up a block that can be altered to draw any regular polygon.
  7. Once the students have a workable block for a square introduce the challenge:
    How can the commands be altered to create other regular polygons?
    You may need to discuss the meaning of ‘regular’ as having equal sides and angles. The prefixes of the polygons (many angles) indicate the number of angles, e.g. octa refers to eight, hexa- refers to six. Let the students explore how to create other polygons.
  8. At some point gather the class to see if they have generalised how to create a n-gon (polygon with any number of sides). There suggestions can be tried out using the ‘polygon’ block. In general, it is the combination of number of sides and angle that determines whether the path closes a space.
    You may need to draw a diagram of what the angle is referring to:

    Angle as a turn is one of the most fundamental concepts in Geometry and Measurement. To support students to understand that the sum of exterior angles of a polygon is always a full turn (360°) get a student (robot) to act out walking various polygons.
  9. Notice how our robot ends up facing the direction she/he started with. So, how much has it turned to make a triangle, square, pentagon..?
    In general, to create a regular polygon of n sides the angle needs to be 360° ÷ n. For example, to create a hexagon the angle of turn is 360° ÷ n = 60°.
  10. The final challenge for the class is:
    What values for the variables side length, angle and number of sides do you enter so the beetle draws a circle? Is that possible?
    Maximising the repetitions to 360, and minimising the distance and angle to one, creates a polygon that looks like a circle. In fact, the shape is a 360-gon. Some students might wonder how many sides and angles a circle has.

Session Two

In this session students apply their knowledge of programming in Scratch to create paths. There are seven challenges which vary in difficulty as indicated by the number of stars on the cards (See Copymaster One).

  1. Let students choose a level of difficulty that they think is appropriate for them. Students may wish to work in pairs to work on the challenges.
    The solution codes for the challenges are provided in Copymaster One solutions PDF. Note that there are often many different programmes that produce the same outcome. Also, remind the students that they are trying to get as close as they can to the target paths. It is difficult to get the coding identical to the original.
  2. Gather the class after a suitable time of exploration (this may be a whole maths lesson). Discuss how they went about answering the challenges. Points to bring out are:
    What did you first look for when you saw a path?
    How did you figure out which lengths to put in? (Relative length is important in some paths)
    How did you figure out which angles to put in? (Discuss turns greater than 90°)
    What features of some paths told you that a repeat loop was possible? (This involves partitioning the path into several identical parts)
  3. Students might enjoy the opportunity to create path challenges for others. You might create your own set of challenge cards with solution coding on the back.

Session Three

So far students have worked from a shape to create code. In this lesson they are given the code and expected to anticipate the shape that will be drawn. After anticipating they can always recreate the code on Scratch to see what happens but that is not the aim. Pre-made codes for use in sessions three and four are available in the Scratch Codes zip file.

  1. Show your students PowerPoint One. The first slide gives a piece of code that will draw a letter ‘a’. Several question bubbles appear as you click the mouse.
    The key idea is that students attend to the code in ‘chunks’ and anticipate the effect of each chunk. Recording the information in pieces before assembling the whole figure is very useful. You can find the code saved as ScratchCode-LetterA.sb2.
  2. After you have worked through Slide One ask the students to attempt Slide Two in pairs or threes. The code draws a sand timer figure. Allow the students sufficient time to anticipate the figure that will be drawn. Then gather the class together.
  3. Mouse click reveal highlighted parts of the code that can be discussed before making a collective prediction. You can find the code as ScratchCode-Hourglass.sb2.
  4. Copymaster Two consists of 12 cards with codes. Students are required to anticipate the final figure that will be drawn by the code. Recording the expected figure before testing it is important. There are varying degrees of difficulty, so students can work at a level that is appropriate for them. The number of stars indicates the complexity of the code. Let your students work for at least 30 minutes on the tasks. Allow access to Scratch so students can try out their ideas, including their anticipations for parts of the code. All the codes used are available as files for use with Scratch (See Code Card ###).
  5. Discuss with the students how they worked out what the code did when enacted. You might address:
    How did you separate the whole code into parts?
    What features did you look for first? What did repeats suggest to you?
    How did you deal with blocks and variables?
    What effect did random variables have on the design a code created?

Session Four

In this session students explore the use of Scratch to create directions for movement on a map. Training a robot to move is more complex than the instructions that you might give to another human being. Robots cannot interpret instructions like “Follow Smith Street down to the corner.” They can use their GPS system to travel in a compass direction like North or East but rely on simpler instructions like distance to move and angle to turn.

  1. PowerPoint Two contains a map that is the backdrop to the following activities. Identify the location of four people Zane, Awhina, Ajay and Fatu. Ask the students to imagine telling a robot how to move from Zane’s place to where Fatu lives.
  2. Write down a set of instructions that students give you for the journey. Have a student act out each instruction, one after the other. Ask where the robot’s location, and the direction they face, after the instruction is given.
  3. Open the file ScratchCode-TownRedMove.sb2. Before clicking the flag ask your students to look at the code. Ask what moves they expect the robot to make. Test their predictions by clicking the flag. Tell the students that their task is to complete the set of instructions so the red robot travels to where Fatu lives. The robot must walk along roads or pathways.
  4. After a suitable period bring the class together to discuss their strategies.
    How did you know how many steps to take on each section of the journey?
    How did you know which direction to face?
    Did you use any repetition? Where? How did you know that would work?
  5. A further challenge is to give students access to ScratchCode-Town.sb2. They can choose the person they want to move and their destination. By choosing the Sprite for that person and deleting the others they can code the journey.
    Alternatively pose an even harder challenge by using ScratchCode-City.sb2. The layout is more regular which may enable use of repetition and block, but the journeys are longer.

Eggs and a little bacon

Purpose

This unit explores situations that involve multiplication and division using an equal sets model. Students learn to apply the properties of whole numbers under multiplication, to derive new answers from basic facts, and apply inverse operations to division.

Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
NA3-2: Know basic multiplication and division facts.
Specific Learning Outcomes
  • Derive from basic multiplication facts to solve multiplication problems with equal sets.
  • Apply the commutative and distributive properties of multiplication to solve problems mentally and on paper.
  • Recognise how both measurement and sharing division problems can be solved by ‘building up’ with multiplication.
Description of Mathematics

The simplest form of multiplication problem involves finding the total of a given number of equal sets. Consider this problem:
There are eight cartons of eggs. Each carton contains four eggs.
How many eggs are there altogether?

The problem can be represented mathematically as 8 x 4 = □. Eight represents the number of sets (the multiplier). Four is the number in each set (multiplicand) and represents the unit rate of “four eggs per carton.” The x symbol represents “of” in the sense of connecting eight sets of four. The empty box is the product or total and the equals sign represents sameness of quantity or balance.

Division with equal sets takes two forms depending on which factor is unknown. Sharing division comes for equally distributing a total number of objects, the dividend, into a given number shares (the divisor), which results in an amount per share (the quotient). For example:
There are 32 eggs and eight cartons of the same size.
How many eggs go into each container?

Note that 32 ÷ 8 = 4 represents the sharing of 32 (the dividend) into 8 equal sets (the divisor) which results in a quotient of “4 eggs per carton.” Division also applies to measurement contexts such as:
There are 32 eggs. Four eggs go into each carton.
How many cartons are needed?

Note that the rate is known, “4 eggs per carton”, and that becomes the unit of measure. “How many fours are in 32?” answers the problem. That can be written as 32 ÷ 4 = 8.

Both equal sharing and measuring problems are common in the real world. Developmentally, students tend to build up solutions to these problems using addition at first, progressing to multiplication. With appropriate opportunities to learn, students later come to treat division as an operation in its own right.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

  • Provide physical materials so that students can anticipate actions and justify their solutions. Use materials like cubes and counters, and suitable collection vessels, such as clear plastic glasses, to model situations and connect strategies used by the students to the quantities that are represented.
  • Represent the place value structure of whole numbers using appropriate materials such as place value blocks or bundles of iceblock sticks.
  • Connect symbols and mathematical vocabulary, especially the symbols for multiplication and division (x, ÷) and for equality (=). Explicitly model the correct use of equations and algorithms, and discuss the meaning of the symbols in context.
  • Alter the complexity of the numbers that are used. Multiplication with factors such as two, four, five, ten tend to be easier than factors such as three, six, seven, eight and nine. This classification is also true for divisors. Vary the size and place value structure of the multiplicands to make problems more accessible, e.g. 4 x 15 is easier than 4 x 14 or 4 x 17. A similar classification is true for the choice of dividends, e.g. 63 ÷ 3 is easier that 57 ÷ 3.    
  • Encourage students to collaborate in small groups and to share, and justify, their ideas.
  • Use technology, especially calculators, in predictive, pattern-based ways to estimate products and quotients, e.g. Is the answer to 57 ÷ 3 closer to ten, twenty or thirty? How do you know?

The context used for this unit is bacon and eggs, simply to provide everyday situations that students are likely to be familiar with. You may wish to change the contexts to situations more relevant to your students’ everyday lives, interests, or cultural identities. For example, eggs in cartons might become kumara in kete, players per team, or students in mini-buses. Encourage students to be creative by accepting a variety of strategies, and asking them to create their own problems for others to solve, in contexts that are meaningful.

Required Resource Materials
Activity

Session 1

  1. Begin with a growth problem (see PowerPoint One: Eggs and chickens).
    There are 12 eggs in one dozen.
    How many eggs are in eight dozen?
    What ways can you find to solve this problem?
    Which way is the most efficient? (takes the least effort)

     
  2. Let your students work in small groups to answer the question. Provide place value materials such as connecting cubes, place value blocks, or iceblock sticks and rubber bands for students to access if they wish. Real empty egg cartons are very handy for later.  Encourage them to rehearse the strategy they used so they are ready to share it with the whole class. After an appropriate time bring the class together for discussion. Have eight egg cartons with cubes or counters as ‘make-believe’ eggs available to show what happens to the quantities.

    Some students may use additive thinking. An example might be:
    12 + 12 = 24 (That’s two cartons), 24 + 24 = 48 (That’s four cartons), 48 + 48 = 96 (Possibly using 50 + 50 = 100, that’s eight cartons).

    You might record the strategy using multiplication, like this:
    2 x 12 = 24 → 4 x 12 = 48 → 8 x 12 = 96

    Repeated doubling can be a useful strategy for multiplying by 4, 8, and 16.

    Additive thinking gets the job done but it is an inefficient process. Look for multiplicative strategies to highlight and share. These strategies might include:
    12 is made up of ten and two. 8 x 10 = 80 (That’s the tens) and 8 x 2 = 16 (That’s the twos). 80 + 16 = 96.

    Or (less likely):
    10 x 12 = 120 (That’s two dozen too much) 2 x 12 = 24 (That’s two dozen)
    120 – 24 = 96


    Both strategies use the distributive property which means one factor is ‘distributed’. In this case 12 is distributed into 10 and 2, and 10 is distributed into 8 and 2.

     
  3. Use the egg carton model to show how each strategy works, including the repeated doubling. PowerPoint One can be used if you do not have the physical model of egg cartons. Pose this problem for the students to solve in small groups of two or three (See PowerPoint One). Let students have access to materials if they need them.
    There are 15 rashers of bacon in each pack.
    How many rashers are in six packs?
    What different ways can you find to solve this?

     
  4. Look for students to:
    • Apply multiplicative strategies rather than additive ones.
    • Recognise which strategies are the most efficient.
    • Record their strategies using equations or other written working.
    • Show how their written recording can be represented with physical materials.
       
  5. After a suitable time bring the class together to share ideas. Look for students to think critically about how a multiplier of 6, rather than 8, changes their use of strategies. Six does not lend itself to doubling only strategies though doubling then multiplying by three, or vice versa, works.

    Good examples of using the distributive property are:
    15 = 10 + 5, 6 x 10 = 60 and 6 x 5 = 30. Therefore, 6 x 15 = 60 + 30 = 90.
    6 = 5 + 1, 5 x 15 = 50 + 25 = 75 and 1 x 15 =15. Therefore, 6 x 15 = 75 + 15 = 90.


    An example of using the associative property is:
    6 = 2 x 3, 2 x 15 = 30, 3 x 30 = 90. Therefore, 6 x 15 = 90.
     
  6. Ask students to show what each strategy looks like with materials. Place value-based models are particularly good for this problem. If time permits take each problem (Eggs and Bacon) and change the numbers. For example, you might create a tray of 24 eggs. How many eggs are in three trays?

Session Two

  1. Remind your students of problems they solved the session before. You might show them some examples as a reminder.
    What was the same about these problems? What was different?
     
  2. Students should note that while the stories and numbers were different there was a common structure to the problems they solved. That structure involved a collection or set of some kind, like a packet, and a multiplier that gave how many of the collections were considered. Take one problem and ask your students:
    Where is the collection or set in this problem?
    Where is the multiplier?

     
  3. Provide a different problem and ask the same questions. You might use:
    Eighteen hens lay four eggs each in one week.
    How many eggs is that in total?


    Be aware that students might think that the multiplier is always less than the size of the equal collections. Share some strategies for solving 18 x 4 = 72.
     
  4. Pose this challenge which involves computational thinking.
    Imagine you trained a robot to solve problems like these ones. Make up a set of instructions for the robot. What actions should the robot take? Put the actions in order.
     
  5. Let your students work in small team to devise a set of instructions. Look for them to:
    • Define the equal collections and multiplier as factors, with some criteria
    • Record the factors and steps symbolically
    • Follow a sequence of instructions that work to give the correct product
       
  6. The aim of the follow-up discussion is to create an algorithm, a set of steps that will reliably produce the answer. Act out some instructions that students provide with the example problem. Have place value materials available to check what is going on with quantities as the steps are followed.

    You might end up with a sequence that looks like this:
    Step 1: Find the number of things in the equal collections, call that number [c]
    Step 2: Find the number of equal collections, call that number [n]
    Step 3: Multiply [n] and [c] by:
    • Break up [c] into the number of tens [a] and ones [b]
    • Multiply both [a] and [b] by n
    • Add the two answers together
       
  7. Challenge your students to adapt their algorithm to solve this problem in small groups (see Copymaster 3):
    There are 144 eggs on each pallet.
    You have 4 pallets. How many eggs is that altogether?

     
  8. Provide place value materials to support students to work out the product. Look for students to:
    • correctly establish 144 as the multiplicand - the size of the equal collections
    • correctly identify 4 as the multiplier
    • reecognise that 144 contains hundreds as well as tens and ones
    • multiply hundreds, tens and ones separately then add the partial products.
       
  9. Ask how the set of procedures needs to be changed to accommodate the new problem. Do students recognise the two steps that need minor change?

    Be open to other ways that students might solve 4 x 144, particularly repeated doubling, i.e. 2 x 144 = 288, 2 x 288 = 576. You might try writing an algorithm for the doubling strategy. That will require a first step of deciding whether, or not, the numbers are suitable.
     
  10. Finish the session with Copymaster 1 that is a paper game for practising multiplication of a two-digit number by a single digit number.

    The rules are:
    Players play in pairs. Each person has their own sheet of paper (Copymaster 1)
    1. At the same time they choose two factors by crossing off digits. The first factor is single digit and the second factor has two digits. They do no know what digits their opponent is choosing.
    2. They multiply their factors and record the product on the game sheet.
    3. Both players compare their products. The player with the largest product wins the round. They get the difference between their product and their opponent’s product.
    4. The player with the highest total of differences wins.

      Here is an example of a partly completed scoresheet.


Session Three

In this session students transfer the strategies they developed for multiplication to problems involving division. Understanding that asking themselves the question, “How many x’s fit in y?” structures division problems and supports students’ fluency in calculation. By connecting division to multiplication students learn to apply their multiplication strategies and connect multiplication and division as inverse operations.

  1. Begin with this division problem:
    Henrietta gathers 52 eggs one morning.
    How many half-dozen cartons (six-packs) can she fill?

     
  2. You may need to clarify with the students that a half-dozen carton holds six eggs. Let the students explore the problem in small teams, using materials for support if needed. Ask your students to record their strategies to share with the class. Do your students…?
    • Use additive thinking to ‘build up’ to a solution, e.g. 6 + 6 = 12, 12 + 12 = 24, 24 + 24 = 48, leaves 4 eggs to make 52.
    • Use multiplicative ‘build up’ strategies, e.g. 5 x 6 = 30 so 6 x 6 = 36 … 8 x 6 = 48.
    • Use the distributive property of multiplication, e.g. 10 x 6 = 60 (too many eggs) so 9 x 6 = 54, therefore 8 six-packs are possible with 4 eggs remaining.
    • Apply basic facts knowledge, e.g. 8 x 6 = 48 and 9 x 6 = 54 and 52 lies between the two products.
    • Pay attention to how students handle the remainder of 4 eggs. 
      How much of a six-pack are four eggs? (two-thirds)
       
  3. Gather the class to share strategies. Choose teams that use strategies representative of the points above. Discuss efficiency. Which strategies require the least work?
     
  4. Introduce division equations in the context of the eggs problem.
    52 ÷ 6 = 8 r4
     
  5. Make the meaning of the symbols explicit , i.e. equals as a sign of balance, division as meaning “measured in units of”, 52 as the dividend (total items to be measured), 6 (unit of measure), 8 (number of equal units that fitted), and r4 as the ‘leftovers’ or remainder. You might discuss other ways that the remainder might be recorded, i.e. 4/6 of a carton, 0.66… of a carton.
     
  6. Pose the following ‘bits missing’ problem. Let the students make up the missing data in the problem, i.e. number of eggs collected, and size of the carton.
    Henrietta gathers __ eggs one morning.
    How many __ cartons can she fill?
     
  7. Encourage the students to solve more than one problem with a focus on this question:
    How do your strategies change as the numbers in the problem change?

    Observe how students approach the task. Challenge students who ‘play safe’ to try more difficult numbers.
    What numbers do you think would make the problem more challenging?
    Why do the problems become more difficult with those numbers?
    What strategies do you use with more challenging problems?

     
  8. After a suitable time gather the class together to process their problems. Categorise the problems as easy, medium and hard.
    What makes a problem easy? (small numbers, no remainder, access to basic facts)
    What makes a problem hard? (bigger numbers, remainders, tricky divisors such as three or seven, or outside the range of basic facts)
     
  9. Choose one hard problem and ask the students to solve it. Discuss what knowledge might be handy for solving it?
    How will we use that knowledge?
     
  10. Make a list of strategies such as distributing the dividend, rounding up to a tidy number, halving, etc.

Session Four

  1. Recap the previous session using examples of the problems that students created. Record the problems using division equations, e.g. 72 ÷ 4 = 18. Ask students to be explicit about what the symbols represent in the equation.
    What is the same about these problems? (all the problems are about measurement, i.e. How many x’s fit into y?)
     
  2. Tell the students that they will solve a different type of division. Pose this problem:
    Three hens laid 48 eggs altogether.
    Each hen laid the same number of eggs.
    How many eggs did each hen lay?

     
  3. Let students solve the problem in small teams. Allow them to use materials if needed. Encourage students to record their strategies for sharing later. Look for students to:
    • Recognise that the problem involves sharing the eggs into three equal sets.
      Note that the problem semantics suggest multiplication, i.e. 3 x □ = 48.
    • Use efficient build up strategies that involve multiplication, e.g. 3 x 10 = 30 (18 eggs left), 3 x 5 = 15 (3 eggs left), 3 x 1 = 3, so each hen lays 10 + 5 + 1 = 16 eggs.
    • Possibly use place value with tidy numbers, e.g. 60 eggs would mean 20 eggs each, 48 is 12 eggs less so each hen gets 4 eggs less than 20.
       
  4. Gather the class to share strategies. Look for a team of students that work the problem out as “How many threes are in 48?” If no students do that then introduce the idea as your strategy.
    Can someone explain how that works? The problem is about sharing not measuring.
     
  5. You may need to act the problem out with materials, like cubes, to support students see the connection between the two forms of division. Deal out ‘eggs’ to three students acting as ‘hens’, pausing each time around is complete.
    How many eggs have I shared so far?

    Students should notice that three eggs are used up each time a round of dealing is complete.
    Why are three eggs used up each time?
     
  6. Provide your students with Copymaster 2 which contains variations on the first problem. They can solve the problems individually or in small teams. Encourage them to use their multiplication facts and ask, “How many x’s are in y?” thinking.

Session Five

In this session students explore an open-ended problem that allows them to demonstrate their multiplication and division strategies. The problem requires them to impose or find some information, i.e. the number of hens, how many eggs hens lay per week, and the size of the trays. Invite them to research online if they need to and use their knowledge of everyday context, e.g. trays that eggs commonly come in.

  1. Use Copymaster 4 to introduce the problem and allow students to work on the task independently. Insist that they show their working clearly so someone else can see their reasoning.
    Happy Chook poultry farm there are about 80 free range hens.
    Each Saturday all the eggs for the week are sold at the Farmers’ Market.
    How many trays are needed each week?

     
  2. After a suitable time put the students in pairs to compare their work. Ask some reflection questions such as:
    Was there enough information to solve the problem?
    If not, where did you get your information?
    Were your strategies efficient?
    How might you have solved your problem differently?

     
  3. Gather the work samples as evidence of your students’ current thinking about multiplication and division.

What's in the bag?

Purpose

In this unit we experiment with cubes to make predictions about likelihood based on our observations. Students find out that with probabilistic situations there is no certain way to predict exactly what will happen.

Achievement Objectives
S3-3: Investigate simple situations that involve elements of chance by comparing experimental results with expectations from models of all the outcomes, acknowledging that samples vary.
Specific Learning Outcomes
  • Make predictions based on data collected.
  • Identify all possible outcomes of an event.
  • Assign probabilities to simple events using fractions (1/2, 1/6 etc).
Description of Mathematics

Probability is a measure of chance or likelihood of an event occurring. In this unit bags of cubes are used to provide the sample space, that is the set of all possible outcomes. The chance of selecting a cube of a particular colour obviously depends on what is in the bag. Suppose there are 5 red, 3 blue and 2 yellow cubes in the bag.

The probability of selecting a red cube equals 5 out of 10 which can be written as 5/10, or 1/2, or 0.5 or 50%. Note that there are five outcomes, ways to select one cube, that result in the event of selecting a red cube. The probability of getting a blue cube with one selection equals 3 out of 10 (30%) and the probability of selecting a yellow cube equals 2/10 or 1/5 or 20%.

Probabilities can be used to predict what event is most likely to occur. Selecting a red cube is more likely than selecting a blue cube, which is more likely than selecting a yellow cube. That order assumes the cube is selected randomly which means that each cube has an equal chance of being selected. With probability the prediction, especially for a small samples, cannot be certain. In fact, all three colours might occur if one cube is selected. If enough selections are made, with replacement each time, the distributions of colour will getter closer reflecting the probability fractions.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:

  • physically modelling the sample space. In this unit emptying the bag to reveal the cubes provides a clear picture of all the possible outcomes
  • helping students to record their models of the sample space, particularly through representations such as picture graphs
  • connecting results of experiments with models of all the outcomes (See Session Three)
  • using physical models, such as towers of cubes, to represent the probabilities as fractions such as 5 out of 10.

Task can be varied in many ways including:

  • easing the cognitive load by reducing the number of cubes, and number of colours, in a bag, and using simple fractions like halves and quarters
  • collaboratively grouping students so they can support each other
  • reducing the demands for reporting through templates, e.g. tally marks, pictographs, with less writing and more oral discussion.

The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Many students are interested in kaitiakitanga, guardianship of the environment, so using a context about endangered species may be motivating. Aotearoa has many species in danger of extinction, and scientists conduct regular, ongoing surveys to monitor population numbers, e.g. Maui or Hector’s dolphin, takahe, Chatham Island robin. Cubes in a bag is a metaphor for a wide range of sampling contexts, from predicting the outcomes of games, to the likelihood of the weather being fine for Sports Day.

Required Resource Materials
  • Cubes of different colours
  • Paper bags (preferably opaque)
  • Stickers, or small pieces of tape
  • Calculators
  • Copymaster 1
Activity

Session 1

Today we make predictions about the cubes that are hidden in a bag. We find out that even when we can’t peek in the bag we can still make a good prediction about what is in it. Think of cubes in the bag like all the people in New Zealand or fish in our seas. We cannot know exactly about all of them but we can use statistics to get an idea.

  1. Put four cubes in a paper bag (3 red and 1 blue) without students seeing.
    Here is a bag with four cubes. The cubes are either red or blue and we’re going to try to find out  how many of each colour there are selecting cubes one at a time.
  2. Shake the bag and ask a student to select one cube to show the class. Record the colour on the board and get the student to put the cube back in the bag.
    This is called sampling with replacement. Replacement means we put it back – like a fish.
    (Note: Each time a student takes a cube it must be returned before the next student draws a cube. Otherwise, the probabilities will change.)
  3. Ask another student to select a cube.
    What colour have you got?
    If it is the same colour as the one previously drawn ask: 
    Do you think that it is the same cube? Why or why not?
    If it is a different colour ask:
    Does that mean that half the cubes are red and half are blue?
    The important idea is that students acknowledge that such conclusions are speculative. There is no certainty except that at least one red (same colour drawn), or at least one red and one blue (different colours drawn) are in the bag.
  4. Ask a third student to draw a cube but this time get them to predict what the cube might be.
    Why did you guess that? How certain are you?
  5. Add the third cube colour to the data.
    Has that changed your mind about what is in the bag? Why? Why not?
  6. Ask a fourth student to draw a cube.
  7. Look at the result of the four draws.
    Do you think that we have seen all the cubes?
    Do we know what the colours of the four cubes are? Why or why not?
    Would we find out more if we had more turns?
  8. Let another four students select a cube one at a time with replacement. Add the colour to the data on the board. Before selecting each time ask the student to predict the colour of the cube.
    Record your best prediction about the colours of the four cubes in the bag.
  9. Ask students to justify their predictions. Look for acknowledgement of certainty and uncertainty. For example, if all the draws were red, we cannot say for certain that there is no blue cube. If one or more of each colour have been drawn, we can be certain that there is at least one cube of each colour in the bag.
  10. Before we look in the bag, discuss all the possible combinations for the colours of the four cubes.
    Record these combinations on the board. Student might forget the four of one colour, zero of the other colour possible combinations.
     

     

    Possible combinations for 4 cubes

    Red

    Blue

    0

    4

    1

    3

    2

    2

    3

    1

    4

    0

     

     

  11. Ask the students to decide which combination they think is most likely.
  12. Look inside the bag and check the cubes. Discuss how reliable their prediction was.
    How could we have improved the prediction before checking?
    Students might suggest that more selections might have improved the reliability. Larger samples tend to be more representative than smaller samples
  13. Put the cubes back in the bag and ask:
    I am going to draw a cube. Which colour do you think it will be?Why?
    Can we be sure that I will get that one? Why?
    How could we record your chance of success?
    If students choose red their chance of success is 3 out of 4 or 3/4.
    If they choose blue their chance of success is 1 out of 4 or 1/4.
    Though there is a greater chance that the cube drawn will be red, there is still a 1/4 chance that it might be blue.

Exploring

Over the next three days we work in pairs to make our own bags of cubes. We swap them with our friends to see if they can guess "What’s in the bag?"

Session 2 

  1. Give each pair of students to choose 10 cubes. There should be two colours available. Ask them to put 10 cubes in their bag using any combination of the two colours they want.
  2. Swap bags with another pair of students. Each pair must predict how many cubes there are of each colour in the bag by taking turns drawing cubes from the bag, one at a time with replacement. Remind them to put the cube back in the bag after each draw. Tell students that they have 5 minutes to make as many draws as they can. It is important that they record their results.
  3. Ask the students to make a prediction about the colours of the ten cubes in the bag. How will you use the data to make the best prediction you can?
  4. Gather the class and share predictions. Do the students:
    • acknowledge that their predictions are uncertain?
    • relate fractions to their predictions, e.g. 20 out of 30 trial cubes were red, that’s 2/3 so 6 or 7 of the ten cubes might be red?
    • provide a range of what events might occur, e.g. 5-7 red and 3-5 blue?

Session 3

  1. In this session students predict events from complete knowledge of the set of cubes. In doing so, they consider the likelihood of the colour of the cube selected next.
  2. Show the students paper bag with 24 cubes in it of varying colours, e.g. 10 yellow, 8 blue, 3 red, 2 green, 1 white. Tip the contents of the bag onto the mat or tabletop. Ask some students to sort the cubes by colour.
    I want you to create a data display of the colours. You are free to use whatever display you want. What display might you use? (Students might suggest bar graph, pie chart, frequency table, pictograph, etc.)
    You can make as many trips as you need up here to view the data but you must create your display back at your desk. Give your students adequate time to create their displays.
  3. Explain the rules of the game to your students.
    I am putting all the cubes back into the paper bag. Nothing in the set of cubes has been changed.
    With each turn, one person from the class will take out a cube, tell you what colour it is, show you the cube then put it aside. It will not be put back into the bag.
    Before the cube is taken out you need to make a prediction about what colour it will be. Record your predictions like this:

    Round

    Prediction

    Correct/Incorrect

    Points

    1

    Green

    I

    -1

    2

    Yellow

    C

    3

    3

    Red

    I

    -1

     

     

     

    If your prediction is correct, write C and give yourself 3 points. If your prediction is incorrect give yourself minus 1 point.
    There will be 24 rounds. At the end of the game the +5 and -1 scores will be combined. The player with the highest score wins.

  4. Before the game starts remind your students about the display they created.
    How might your display be useful?
  5. Play the full 24 rounds of the game. As the number of cubes in the bag reduces the students’ chances of correctly predicting increases to the point that one the last draw they should be certain of the outcome.
    Watch for the following behaviour from your students:
    • Do they keep track of the colours of remaining cubes, using their data display?
    • Do their predictions match the likelihoods, as expected from the colour frequencies of the remaining cubes?
    • Do they successfully cope with the plus and minus nature of getting their score?
  6. At the end of the game interview the winner or winners about the secret to their success. Focus on the way they tracked the number of cubes of each colour left in the bag and how they used the frequencies of colours to make predictions.
  7. Pose various scenarios of cubes remaining in the bag and invite students to make predictions about the next cube. For example:
    In the bag are 6 yellow, 4 blue, and 2 red cubes. What coloured cube do you think will be drawn next?
  8. Calculate the probabilities for each colour. For example, there are 12 cubes all together so the chance of yellow equals 6/12 or ½, 4/12 or 1/3 of the cubes are blue, and 2/12 or 1/6 of the cubes are red. Yellow has the greatest chance of being selected.
  9. Trial selecting the next cube in each scenario. Students should come to realise that a colour may have the best chance but still not be selected. A high probability does not guarantee certainty.

Session 4

In this session students explore the impact of ‘tag and release’ methods of sampling. Such methods are common to biological research in which animals need to be returned to their habitat.

  1. Pose the following problem:
    Suppose you are a scientist. The fish in the lake are precious so you want to return each fish after you have tagged it. How can you get an accurate picture of:
    • The total number of fish in the lake?
    • The fraction of all the fish that are each species (type of fish)?
  2. Invite the students to offer ideas. Question b is alike the previous inquiries. Taking fish out of the lake one at a time, noting the species, and returning the fish will allow a reasonable prediction of species as more and more data are gathered.
    What is the advantage of tagging the fish?
    Students will know if a tagged fish is caught that it is already represented in the data.
  3. Produce a bag of cubes (fish), e.g. 8 red, 6 yellow, 4 blue, 2 green.
    I have a lake full of fish here. You are the scientist and I am your fish catching assistant. Our aim is to find out the fraction of each species and get an estimate of the total number of fish.
    Take some time to think about how you will record the data.
  4. Take ten fish from the bag, one at a time, tagging the cube with a sticker before returning it to the bag. Watch how students record the data.
    You might need to suggest using a tally chart and to indicate with an asterisk if the cube is tagged (caught before).
  5. After the sample of ten cubes is complete ask students to predict the fraction of each species and the total number of cubes in the bag. Note that the total number will be impossible to estimate if no tagged fish are caught. Check:
    Do their fractions match the distribution of cube colours in the sample?
    Do they acknowledge that the predictions are very uncertain, given the small sample size?
  6. Carry out another sample of 20 cubes, tagging and replacing. At the end of the sample ask your students to predict the fractions and total number.
    Is it better to think about each sample as separate or treat the combined results as one big sample?
    Pay particular attention to the issue of total number of cubes.
    For example, suppose five of the 20 cubes selected in the last sample is tagged.
    What can this tell us about the total number of cubes in the bag?
    About one quarter of all the cubes in the bag are tagged.
    How might we predict the total number of fish from that?
  7. Draw a pie chart like this to support your students:
     
    If the ten cubes we tagged at first make up one quarter of  of all the cubes in the bag, how many cubes might be in the bag?
    Students might realise that four quarters make the whole population so 4 x 10 = 40 is the best prediction they can make.
    How certain are you? What could we do to improve the reliability?
    Students might say that tagging as many fish as possible would improve the reliability of the prediction. In fact, if we keep tagging fish we might reach a stage when every fish we catch is tagged.
    Why is tag until every fish is tagged not feasible in real life? (Population may be very large, the more animals that you tag the harder it is to find an animal that isn’t tagged.)
  8. Check the actual contents of the bag. Student might realise that the estimate of total number of cubes (population) is much more reliable than the example above (It should be).
  9. Ask students to create a bag for someone else to sample. Limit the colours to a maximum of four but allow students to use any number of cubes between 20 and 50. Provide each student with marking materials like stickers, tape or blutac.
  10. Students exchange bags and carry out an initial sample of ten cubes, tagging and replacing. They then take a sample of 20 cubes, tagging and replacing.
    Tell them to record the data in an organised way and justify the predictions of fractions and total number.
    How do they deal with fish that get selected but are already tagged? Do they tag them again? Does double or triple tagging improve the data? How?
  11. Finish the lesson with a video about tag and release methods used by scientists. Many videos are available online.

Session 5

In this final session, pose the following problem to your students (Copymaster 1). Use their response to consolidate learning and to assess achievement of your students on the probability outcomes.

Look for the following:

Question One: Does the next sample of ten reflect the previous sample but also show variation from that sample? Students should show that the two samples will vary.

Question Two: Can students represent the part-whole relationships as fractions?

Question Three: Do students acknowledge that a green jellybean could still be in the bag, but it has not been selected in the two samples? A sophisticated response will state something about the likelihood of a green being in the bag.

Question Four: Does the sample of five reflect proportions similar to those in the collected samples, with some variation?

After the students complete the task independently gather the class to discuss the ideas above. Students might reflect on how they might change their answers following the discussion.

Attachments

Matariki - Level 3

Purpose

This unit builds the learning of mathematics around celebrations of Matariki, the Māori New Year. The sessions provide meaningful contexts that highlight Māori culture and provide powerful learning opportunities that connect different strands of mathematics.

Specific Learning Outcomes

Session One

  • Recognise a trapezium, right angled triangle, and points in a straight line, in different orientations.
  • Construct a model of all the possible outcomes for rolling a standard dice and use the model to predict the results of an experiment.

Session Two

  • Find the next shape in a tukutuku pattern by ‘chunking’ the pattern spatially or by using numbers organised in a table.

Session Three

  • Use fair testing to create a top the spins.
  • Gather and organise data to improve the performance of a spinning top.

Session Four

  • Create a kowhaiwhai pattern by translating, reflecting or rotating a unit of repeat.

Session Five

  • Use measurement and geometry to construct a field for Ki-ā-Rahi that matches the specifications for the game.
Description of Mathematics

Session one involves students working co-operatively on two puzzles. They will need to think logically to solve both puzzles by evaluating which clues are most important and the best order to deal with the clues. They will need to organise their information, possibly using the Copymaster, and check to see that their solution satisfies all of the clues. 

In the final part of the lesson a dice is used to simulate which stars will be bright and which stars will be dull in order to predict the year ahead. As a class students will collect experimental data and look at which outcomes are most likely. While it might be expected that three bright stars and three dull stars is most likely students should note variation in the number of stars and how brightness and dullness are distributed. This activity may provoke students to consider theoretical probability , for example, all the ways three bright and three dull stars might occur or the likelihood of all six stars being bright.

Session two uses tukutuku designs as a stimulus for continuing and finding rules for linear patterns. Students are encouraged to think spatially to ‘chunk’ each design into parts to structure how a given pattern might be ‘seen’ and how the pattern grows as a new layer is added. Examples of how hypothetical students might see a layer are introduced to encourage flexibility in the way students structure the pattern. They are asked to use recursive thinking (one term to the next) to imagine how the pattern of layers will grow. They apply their structuring to further layers and may even develop a general rule for any layer in the tukutuku design.

Session three uses pōtaka (spinning tops) as a vehicle for thinking about geometry and measurement. Students find ways to locate the centre of a circle to position the spindle of their top correctly. They also use measurement to vary the length of spindle and weighting of their tops and time the spin time. Data is gathered about the tops, including spin time, diameter, spindle length and weighting. The multivariate data is used to look for relationship between these variables and graphs drawn to detect and communicate any relationships.

Session four looks at the mathematics of kōwhaiwhai patterns. Examples of frieze patterns can be seen in some kōwhaiwhai patterns. These are created by combinations of translation, reflection and rotation of a repeating element. Students create an element using ideas like closure of spaces and properties of spirals. They use the element to create a kōwhaiwhai pattern by choosing a transformation to move the element. Other students try to figure out what transformation another student has used to generate their pattern. 

Session five looks at the traditional game of Kī-o-Rahi to consider the properties of circles and scale drawings. Student use geometry and measurement ideas to work out how best to mark out the field. They need to recognise that a circle is a set of points of equal distance from the centre. They also need to use a scale of 1 square to 1 square metre on grid paper to mark the field in correct proportions. The scale is used to work out how far players run in a move within the game.

Activity

Prior Experience

The activities are mostly open ended so they cater for a range of achievement levels. It is expected that students have some experience with naming and classifying basic geometric shapes,  with measuring lengths in centimetres and metres, and with translating, reflecting and rotating shapes.

Session One

In this session students apply their geometry knowledge to solve two puzzles about the stars in the Matariki cluster. They will find out how the brightness of each star was used to predict the coming year, such as when to plant crops, and apply probability to estimate the chances of a ‘perfect year’.

  1. Begin by showing the students a picture of the Matariki cluster. These can be easily found by doing an online image search. Most pictures have the seven stars though as the students will find out there is now a conjecture that nine stars were used to forecast the New Year (see this One News item). Tell the students that two stars Pōhutukawa and Hiwaiterangi are now treated as part of the Matariki cluster. Previously it was believed that only seven stars were visible to the naked eye
  2. Ask: Why is Matariki a special time in Aotearoa (New Zealand)?
    The first new moon after the rising of Matariki in June signals the middle of winter and the beginning of a new year. It is a time to celebrate life and remember those who died in the previous year.
  3. Put the students into groups of three. Each trio will need a set of six puzzle cards (two each) and a diagram of the Matariki star cluster (see Copymaster 1). Tell the students that they need to use the clues on the cards to name each star. They will need to work together as each clue is important. You may need to clarify words like arc and trapezium. Encourage students to find definitions online rather than tell them. After a suitable period of time bring the class back together to name each star.
  4. Use Session one PowerPoint 1A to reveal the star names, working through the set of clues systematically. Tell the students that the stars of Matariki were used to forecast the coming years and make decisions about planting and food gathering. Although the time of forecasting varied from iwi to iwi it was always around the middle of winter. 
  5. Put them back into their trios to solve the second co-operative puzzle (Copymaster 2). There are five cards. If they use the clues correctly they will find out which aspect of life each star was assigned to. Discuss which clues students used to determine each star’s responsibility.  Use Session one PowerPoint 1B to check their answers. The PowerPoint shows how a matrix can be used to organise the information.
  6. You may like to show your students this video of the seven sisters.  The sisters behave like ‘wandering stars’ so are given jobs to do.
  7. Tell the students that one role of a Tohunga (expert) is to read the Matariki stars at dawn of the new moon. If star cluster is bright, clear and appears ‘spread’ this forecasts good news for the year ahead. If the star cluster is dull, fuzzy and appeared ‘close together’ this indicates a year that will be lean. In some iwi the individual stars are read so the long range forecast is more specific. In particular, the stars related to food gathering or growing are looked at closely.
  8. Explain that environmental conditions, like weather, are hard to predict, especially in the long-term. So imagine that these conditions are like a game of chance. There are six stars related to food. Highlight those stars (see slide three). Ask your students to highlight the stars on their copy of Copymaster 2 and put in the direction arrows. Next simulate the chances of the star being bright or fuzzy. Use a standard dice, labelled 1-6. If the dice number is even then the star is bright so highlight it with felt pen or highlighter. If the dice number is odd, colour the star grey as is fuzzy, and hard to see. Work your way around all eight role stars in the direction of the arrows. Look at the completed cluster. 
  9. Ask: What would the Tohunga predict this year? 
    Students might say that the year will be windy and sea fishing poor. However freshwater food, like eels, will be plentiful. The crops will do well and rain will be balanced, not too much and not too little. The forest will have a lean year so don’t count on lots of birds to eat. 
  10. Tell the students to simulate the brightness of the stars using dice and write a prediction for the year ahead. Gather the class together and share some star pictures and predictions. Students should notice the variation is the pictures. Are any patterns the same?
  11. Ask: How many bright stars and how many dull stars did you expect to get? Why?
    Do students notice that there is a one half (50%) chance of getting each option on a dice roll?
  12. “How likely is it that every star will be bright?” This is very unlikely. With only two stars the chances are one in four. With three stars the chances are one in eight. The tree diagram and pattern of probability can be extended if your students understand it.
  13. The chance of all eight stars being bright is one in 256 (1/256). However, it is possible that a picture looks like that or the matching alternative of all six stars being grey. The outcomes are unlikely but not impossible.

Session Two

In this session the students look for patterns in tukutuku patterns. In particular they find ways to count the number of tuinga (cross stitches) in a given pattern and use their counting strategy for one member of the pattern to predict further members. They are also encouraged to look for relationships between variables in each tukutuku pattern.
The session is driven by two resources; 

  • The session two PowerPoint provides coloured slides of each pattern to support students to structure (organise) the pattern
  • Copymaster 3 provides students with a template to record what they see without spending time laboriously copying the patterns
  1. Give each student a copy of Copymaster 3 to record their thinking and follow this sequence:
  2. Slide one: Discuss how tukutuku panels are created by cross-stitching horizontal and vertical rods or wooden panels with square arrays of holes. The rods are called kaho and the stitches are called tuinga.
  3. Slide two: Ask the students to describe how the pattern grows. They should notice the shape is triangular (tapatoru means triangle) and that each layer is requiring more stitches than the one before. Some students might predict how many stitches will be needed for the next layer. There will need to be twelve stitches on each side of the triangle so 33 stitches in total, allowing for the overlaps at the corners.
  4. Slide three: Ask your students to find a way to count the number of stitches in ‘chunks’ rather than one at a time. Get them to share how they saw the pattern with a partner before you have a class discussion. If you have an interactive whiteboard you might draw on the patterns to highlight the ‘chunks’ students are using or have students do so themselves.
  5. Slide four: Finding out if the strategy you applied to one member of the pattern works on further members is an important step in generalisation. Again get them to share their strategies with a partner before discussing possibilities with the class. Encourage you students to find manageable ‘chunks’ that will be in any triangular layer pattern like that. For example, student might count the bottom side then what remains of the right side then finally the inclined side. This produces an interesting sum of 9 + 8 + 7 = 24 crosses. “But 24 equals 3 x 8. Is that important? What would the sum for the next pattern be?”
  6. Slides five to seven provide examples of how students might structure the six crosses across layer. It is also important to consider how each hypothetical student would record what they see.
  7. Slide five: Kiri counts the crosses on one side and knows that the other two sides have the same number of crosses. She also knows that if she calculates 3 x 6 = 18 that will be three crosses too many because the corners are double counted. So she would record 3 x 6 – 3 = 15. “That’s interesting because 3 x 6 – 3 = 3 x 5 again!”
  8. Slide six: Sione tries to avoid double counting the corners. He realises that if he includes only one corner in each side then he counts all the stitches. There are six stitches on each side, less one corner gives five stitches. So he writes 3 x 5 = 15.
  9. Slide seven: Taylor counts the middle crosses on each side, leaving the corner stitches out. That gives her three sides with four on each side. 3 x 4 = 12 but she also needs to add on the three corner stitches to get 15. “But 3 x 4 + 3 = 3 x 5 again!”
  10. Slide eight: Ask students to attempt this task independently and record on the copymaster how they worked it out. The problem is to imagine a tapatoru (triangle) layer being added ‘a long way further on’ in the pattern. This requires a different type of structuring. Students need to apply ‘chunks’ they saw in the examples to create a feasible further member of the pattern. Some may not have observed that the number of crosses on each side is always a multiple of three. Once a bottom side is created the other sides need to have the same number of crosses.
  11. Look for students to apply the strategies used by themselves and others to count the stitches in their ‘further on’ layer. Encourage them not to draw every stitch and to imagine what must be there. Students might just record equations like 99 + 98 + 97 = 294 (for a tapatoru layer with 99 stitches along the base side) or 3 x 300 – 3 = 997 (for a tapatoru layer with 300 stitches on each side).
  12. After a suitable period of investigation ask the students to share their answers in small groups so you can observe the discussion. You might choose to highlight solutions you think are particularly insightful and get the student responsible to share their strategies, particularly for counting the stitches.
  13. Slides nine to eleven: Follow a similar process with the pātiki (flounder) design except expect the ways to structure the pattern to come from your students. Ask them to record how they found the total number of stitches (slide 10) using an equation and link that calculation to ‘chunks’ of the layer. Slide 11 gives them a chance to image how an example of the pattern might look ‘further on’.
  14. Some students might notice that although the pattern looks different it is actually quite similar to the tapatoru pattern and similar thinking can be used. The diamond layers (not a mathematical word) are actually squares that could be rotated so the sides lie vertical and horizontal. Viewed like that the same strategies used to count stitches on three sides can be applied to four sides.
  15. The length of the sides (in crosses) is always a multiple of four, except for the first stitch in the middle. So the pattern in crosses that are added goes like this:
  16. Note that students at Level Three are not expected to write algebraic formulae though some may represent a rule in words of invented symbols, e.g. “If you count the number of crosses along one side of the square you can multiply that number by four then take off four.”
  17. Slide twelve: At this point students might be invited to create their own tukutuku pattern using the grid at the bottom of the copymaster. Alternatively you might give them the challenging task on slide 12. Encourage the students to structure each pattern. For example, the rows of the tapatoru pattern contain 12 + 11 + 10 +…+ 3 + 2 + 1 crosses which totals to 78 crosses. The pātiki pattern has a 5 x 5 array of black crosses and a 4 x 4 array of red crosses, so has a total of 25 + 16 = 41 crosses.

Session Three

Matariki was a time when food was stored, and the weather was not always pleasant outside. So whānau (families) spent time together engaging in cultural pursuits such as storytelling, arts and games. A popular game for tamariki (children) was to make spinning tops. The tops were craved from hardwood like mataī. A whip made from a stick and flax fibre was used to guide the top over obstacles and keep it spinning. A picture of a traditional top can be found at:

There are cheap and accessible materials available to create tops. This video take a scientific approach to the making different tops: 

  1. Any plastic bottle top will work, e.g. milk bottle, yoghurt, marmite jar. Use bamboo skewers for the spindle (vertical part). You can trim the skewers to size, which is a nice variation when students are trying to ‘tune up’ their tops. Small nails or screws are cheap and safe for making the hole in the centre of the top. It is important not to make the hole too big. The skewer needs to be pushed firmly into the hole and checked to see if it is at right angles to the top (students will find that out in the tuning process). Adding weight to the spinner can also increase the spin time. Blutac or plasticine is good for putting on the inside as it keeps the spindle vertical and the position of the weight can be altered. Take care. Not a lot of extra weight is needed and it is very easy to unbalance the top.
  2. The first mathematical challenge is finding the exact centre of a circle. Discuss what is likely to happen if the hole is not in the middle of the plastic bottle top. Since tops rely on balance an off-centre top will not spin for long. Challenge the students to find a way to find the centre of a circle. You might give them some paper circles to experiment with. Rulers and pencils will also be helpful.
  3. Students are likely to suggest the ‘cross method’ of finding two diameters, one vertical and the other horizontal. This method is quite accurate but it is not always easy to ensure the lines are actually diameters.
  4. Draw a large circle on the whiteboard and mark a point on the circumference. Put the zero mark of a metre-long ruler on the point and rotate the rule about it.
  5. Ask: How will I know when the ruler is making a diameter?
    Some students might notice changes to the measurement from the point to where the ruler crosses the other side. 
    If the measurement changes how will I know when I have a diameter?
    The ruler marks a diameter when the measure is at its greatest. The measure declines as the ruler is rotated from that position.
  6. You can now draw a diameter on the circle.
  7. Ask: Where will the centre of the circle be?
    The centre is at the midpoint of the diameter (C).
  8. Students can then make their spinning tops. Since this is a celebration of Māori culture they might like to consider how the inside circle of the top might be decorated. 
  9. Ask: What will happen to the design as you spin your top?
    In fact much specific detail will be lost as the top rotates so encourage your students to keep the design simple. They could decorate directly onto the plastic with spirit based pens by cutting a circle of paper the size of the top, decorating that and putting it inside the top is a safer option.
  10. This beautiful koru spiral top was made from a plastic yoghurt lid and spins for about one minute.
  11. Once you have a good collection of tops emulate the competitive nature of traditional top spinning by having a contest. The game is to have the top that spins the longest. Allow your students to ‘tune up’ their tops. This activity will involve practising the turn of their fingers, trimming the length of the spindle, altering the length of pointy section, and adding little amounts of weight. 
  12. The easiest way to time the tops is to put the students in pairs or threes. So as one student spins the other one or two students observe to check to time the spinning. A top has finished spinning when it no longer turns at all because it is resting on its side. Most students have timers on their watches but you could use stop watches if you have them. Alternatively go for group timing. Count in the start of the spin, “Three, two, one, spin.” Count out the seconds from the start so the students know their top spin time from the last number before it stops.
  13. It is important to record the times of spins. This raises some issues:
    How accurate do we want to be? (Most watches count in hundredths of a second)
    Will each person get only one spin or will they be allowed many spins?
    If many spins are allowed which time is counted? (best, worst, middle, average, total)
  14. Once the class has made these critical decisions give each student a post-it or square of paper to record their spin time. You could include other variables such as origin of the lid (or diameter), length of spindle point, length of spindle handle, weights used? (Yes or No), etc. If you elect multiple variables then creating a data card may be the best way to gather the data.
  15. After the contest is over ask your students to put the post-its or data cards in the centre of a class circle. 
  16. Ask: What could we do with these data? What questions could we ask?”
    Students might come up with summary questions, like “What is our average spin time?”.
    Since most of the variables are numeric you might use dot plots or stem and leaf graphs to display the data. For example, to make a dot plot use a skipping rope to make a long line and ask the students to order their cards along the line by time. Identical times can be arranged vertically on the same spot.

  17. Some students might suggest comparison questions such as:
    Do tops with weights spin longer than tops without weights?
    Comparison questions can be investigated by creating a graph for each group (Yes and No) and comparing the distributions of times.
  18. Some students might suggest relationship questions such as:
    Is a shorter spindle point best for improving spin times?
    You could put the data from both variables into an Excel spreadsheet and graph the relationship using a scatterplot.

Session Four

In this session students investigate kōwhaiwhai patterns. One of the basic units of kōwhaiwhai is the koru or fern frond. Nowadays kōwhaiwhai refers to any figure generated using the koru but your students will investigate some of the designs found in wharenui (meeting houses). These designs are used on the tāhū (ridgepole) and heke (rafters). Tāhū and heke are the spine and ribs of the tipuna (ancestor) embodied in the wharenui so kōwhaiwhai is a way to acknowledge the mana of that important person.

  1. Begin by drawing a simple koru. Spirals, like other stems in nature, usually obey the Fibonacci sequence so that is a good way to start. Slide one of the session four PowerPoint, and Copymaster 4 have a template for the Fibonacci spiral. Before clicking the animation ask the students what they notice about the template.
  2. Some students might notice the circles.
    Ask: How big are the circles? The small circle is located inside the smallest square. Let’s call that a one square.
  3. Ask: How big are the other squares?
    Students might notice that the squares grow in a sequence, 1 x 1, 1 x 1, 2 x 2, 3 x 3, 5 x 5 (this is an interesting leap – Why five?).
  4. Ask: What square would the large circle fit into?” (The 3 x 3 square)
  5. Students can then follow the animation to create a koru pattern. Show the students slide two which shows some examples kōwhaiwhai. Note how the patterns have many koru and how the colours white, red and black are used to convey meaning. Ask, “How do you think the artist made these patterns?” Look for students to recognise that there is an element of repeat in each pattern. Discuss the transformations used. Students should easily be able to identify examples of translation and reflection.
  6. The next part can be completed better using a simple graphics computer packages that allows the element to be copied and pasted repeatedly. Students can create a single element manually and that element can be photocopied or scanned to form the whole pattern. 
  7. Give the students a rectangle of cardboard to create an element (File cards are good). This will form the repeating part of their pattern. Tell your students to create a koru design that starts in the left bottom corner and touches all four sides of the card. It is a good idea to rule up a grid on the card to help students keep the dimensions consistent.
  8. Ask the students to name their elements, e.g. Te Maunga, and collect them for photocopying or scanning. Then look again at slide three of the PowerPoint. Discuss how the element has been moved to create each pattern. Expect students to use mathematical language like ‘translation’, and ‘reflection’ to describe the transformations. 
  9. Provide the students with copies or scans of their element. PowerPoint works well as a computer programme for copying the repeating pattern and performing transformations on it. Alternatively students can cut and paste physical copies onto a strip of art paper. Six elements joined end-on-end by some transformation (translation, reflection, rotation) is a good size and allows other students to figure out which transformations have occurred.
  10. More information about the 7 types of frieze patterns is available in the teachers' notes notes for Kōwhaiwhai, an activity from page 14 of Figure It Out, Level 4, Geometry, Book One.

Session Five

In this session students investigate the geometry and measurement associated with the traditional game of Kī-o-Rahi. The game is based on the intrepid struggles of Rahi-tutaka-hina (Rahi for short) to rescue his wife Ti Ara-kura-pake-wai who was captured by the patupaiarehe (fair-skinned fairies). You can read the legend, and an article about the game, in School Journal, Part 2, Number 3, 2010. Click to download a literature unit based on the legend

The rules for Kī-o-Rahi vary a bit from place to place but those variations are relatively minor. There is now a national championship for the game where teams from all over Aotearoa gather to compete. An instructional video is available from the link below:

  1. The game is played with a soft ball called the kī which was made of flax in older times. Rahi was the hero in the legend. Like netball, Kī-o-Rahi is played in four quarters with the teams taking it in turns to be Kīoma (guardians of the ki) and Taniwha. 
  2. Ask: If we wanted the game to last for 60 minutes, how much time should we allow for each quarter? 
    Quarters of 15 minutes each would allow little time for setting up, rest stops, and packing up, so quarters of ten minutes are the usual practice.
  3. Show the students Copymaster 5. Ask them what they notice about the layout of the field for Kī-o-Rahi. Expect them to notice important features such as the concentric circles (same centre) that mark the zones, the centre target called te tupu which represents the rock Rahi was sheltered on, and the seven pou (yellow cones) that represent the stars of Matariki. 
  4. Challenge the students to work out how they will mark out the field for Kī-o-Rahi. The journal article mentions that the class used a piece of skipping rope and a spray can to mark the circles. “How did they use the rope?” Remind the students that the pou are spread equally around the outer circle. “How are you going to make sure that the pou are in correct positions?”
  5. Give the students a page of grid paper, some string (stand in for the rope), and a protractor to help them map out the field. Each square on the grid paper represents 1 square metre on the real field. Remind them that what they do on paper has to be ‘doable’ with large measures. Students can work in small groups of two or three to devise their plans.
    Look for students to:
    • Work out how to draw a circle by fixing one end of the string in the centre (Te Tupu) and walking around the circumference with the string kept taut;
    • Use the scale on the grid paper to avoid measuring with a ruler;
    • Consider how the pou might be located using either equal distance around the outer circle (string might be looped around the circle to measure the circumference) or using the protractor to divided 360° by seven.
  6. After an appropriate time of investigation bring the class together to discuss their strategies. Focus on the properties of a circle (a set of points the same distance, the radius, from the centre), angles as measures of turn, and use of scale. Pose problems with that scale.
    Imagine a Kioma player who runs in straight lines and touches all the seven pou. How far does she run in metres to do that?
  7. Ask the students to mark a move that a player might make in the game. For example:
    • A kioma player player might touch two pou then run around Te Ao into Te Ara and touch the ball on te tupu (2 goals!). How far does he run?
    • A taniwha player throws the ball to a team member on the opposite side of Te Roto. That player throws the ball at te tupu and scores a tupu wairua (goal!). How far does the ball travel altogether in both throws?
  8. When the class has a well-developed plan for marking the field grab the necessary equipment and play Kī-o-Rahi. 

Printed from https://nzmaths.co.nz/user/75803/planning-space/late-level-3-plan-term-2 at 1:10pm on the 21st January 2022