Early level 4 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. Getting partial to decimals

Level Four
Number and Algebra
Units of Work
This unit supports students to understand the place value structure of decimals and to carry out addition and subtraction with decimal numbers to three places.
• Understand that decimals are a kind of fraction.
• Add and subtract decimals to three decimal places.
• Find decimal values for fractions through practical modelling. Fitness

Level Four
Geometry and Measurement
Units of Work
This unit examines regular tessellations, that is, tessellations that can be made using only one type of regular polygon, and semi-regular tessellations, where more than one type of regular polygon is involved. Students are required to investigate what properties tessellating shapes must have in...
• Create regular and semi-regular tessellations of the plane.
• Demonstrate why a given tessellation will cover the plane. Tessellating Art

Level Four
Geometry and Measurement
Units of Work
In this unit we apply our understanding of why tessellations work to form our own unique tessellating shapes. We use these shapes to create interesting pieces of art in the style of M.C. Escher, a famous Dutch artist.
• Alter polygons to create unique shapes that tessellate.
• Describe the reflection or rotational symmetry of a shape or tessellation. What are the chances?

Level Four
Statistics
Units of Work
In this unit students investigate the link between experimental estimates of probability and theoretical probability. They also learn about short run variability and independence/dependence of events.
• Explore the theoretical and experimental probabilities of situations involving chance.
• Recognise variability from theoretical expectations, especially with small numbers of trials.
• Estimate and find the relative frequencies of events.
• Distinguish between events and the outcomes that lead to... Down on the farm

Level Four
Number and Algebra
Units of Work
The unit is designed as a simple introduction to systems of linear equations. Students solve problems in which they meet two constraints to find a single solution. Both constraints can be expressed as linear equations.
• Identify and find values for variables in context.
• Identify linear relationships in context.
• Represent linear relationships using tables, graphs and simple linear equations.
• Draw strip diagrams to represent linear equations and ratios.
• Solve problems in which two or more conditions must... Matariki - Level 4

Level Four
Integrated
Units of Work
This unit explores a variety of mathematics ideas in the context of Matariki. Matariki is a significant event in the New Zealand calendar and the Māori New Year is celebrated in many schools. Matariki is an opportunity to engage in activities such as storytelling, astronomy, song, dance, and visual...

Session One

• Recognise the properties of a figure stay constant as the figure is rotated (turned).
• Use compass directions to locate objects.
• Represent the relationship between numeric variables using tables and graphs.

Session Two

• Collect, sort and display multivariate data to...
Source URL: https://nzmaths.co.nz/user/75803/planning-space/early-level-4-plan-term-2

Getting partial to decimals

Purpose

This unit supports students to understand the place value structure of decimals and to carry out addition and subtraction with decimal numbers to three places.

Achievement Objectives
NA4-2: Understand addition and subtraction of fractions, decimals, and integers.
NA4-3: Find fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.
Specific Learning Outcomes
• Understand that decimals are a kind of fraction.
• Add and subtract decimals to three decimal places.
• Find decimal values for fractions through practical modelling.
Description of Mathematics

The decimal system uses repeated equal division into ten parts create smaller units. Dividing one metre into ten equal parts creates deci-metres, a unit that is used in Europe but seldom in New Zealand. A decimetre is one tenth of a metre. If one decimetre is cut into ten equal parts, the parts are called centimetres. That is because 100 centimetres compose one metre (one tenth of one tenth equals one hundredth). A height of 1.78 metres is a combination of 1 whole metre, 7 tenths of one metre, and 8 hundredths of one metre.

Decimal fractions underpin most of the units used in the Standard International (SI) system of measures. For example, one kilogram equals 1000 grams, so one gram is 0.001kg (one thousandth of a kilogram). One litre of water has a mass of one kilogram. Therefore, 1g is the mass of 0.001L (litre) of water, otherwise known as one millilitre. Milli is the Latin prefix for one thousandth.

There are subtle differences in the way decimals behave compared to whole numbers and reliance on whole number thinking is the key cognitive obstacle for students working with decimals. Ordering decimals is a good example. With whole numbers 7<81<657 but it is incorrect to order decimals by ‘longer is bigger’ strategies. 0.81 is greater than 0.7 but 0.657 is not greater than 0.81. Similarly, methods of calculation used with whole numbers must be changed to accommodate decimal fractions. For example, 7 + 28 = 35 but 0.7 + 0.28 ≠ 0.35. The decimal point needs to be thought of as a marker for the ones place rather than as a ‘separator’ that keeps the whole numbers and fractions apart.

Two key concepts must converge for students to understand decimals and how to operate on them. Whole number place value must be extended to tenths, hundredths, thousandths, etc., with recognition that units are nested in other units, e.g. 2.35 has 23 tenths (23.5 tenths to be exact). Equivalent fractions are applied to compare decimals by size, e.g. 0.81 is composed of 8 tenths and 1 hundredth, since 8/10 = 80/100, 0.81 equals 81/100 (81 hundredths).

Students can be supported through the learning opportunities in this unit by differentiating the nature and complexity of the tasks, and by adapting the contexts. Ways to support students include:

• Using physical objects to connect decimals, as numbers, with physicial quantities. Length is used as the model in this unit though area models, such as decimats, and volume models, such as place value blocks, can also be used. The greater the number of dimensions of a model, the more abstract the representation is. Starting with length is a sensible first step.
• Directly modelling how the way a decimal is written reflects the place value structure. Explicitly link the place of each digit with its fraction name, e.g. hundredths, and with the part of the physical model that digit refers to.
• Using calculators to confirm predictions about the results of adding and subtracting decimals. Where differences between predictions and outcomes occur, ask students to work out where their predictions went awry. This strategy helps to cause cognitive conflict and address common misconceptions.
• Encouraging students to work collaboratively in partnerships. Students need time to develop mathematical arguments and to rehearse those arguments with a peer is important for developing clarity and risk taking.
• Restricting the domain of numbers being investigated. Work with tenths initially to support students to see the connectedness of place values across the decimal point, and to rename decimals in ‘nested ways’, e.g. 1.4 equals 14 tenths. Progress as rapidly as possible to hundredths and thousandths.
• Providing formats for recording that scaffold a process. Steps in addition and subtraction of decimals can be complicated, especially where renaming is involved.
• Fold in an out of different levels of abstraction, i.e. materials, images (diagrams), equations. Use symbols as a means to connect across situations. Thinking with patterns in equations, and reasoning why patterns occur, is at the heart of mathematics.

The contexts for this unit are about lengths (heights). Most students find personal data interesting though caution is needed about using measurements such as body weight. Decimals occur through measurement. You might choose contexts that engage your students. For example, Census at school collected data about the weight of school bags, reaction times, and foot lengths. You might also look at populations of countries expressed as millions, e.g. 5.38 million, at distances of planets from the sun in astronomical units, at the number of gigabytes in movie downloads, or the areas of pieces of land in hectares. Choose contexts that engage your students' cultures and interests.

Required Resource Materials
• Adding machine tape (metre long strips) or metre long straws (if available) and pipe cleaners for joining.
• Calculators
• Metre rulers
• Scissors
• Post-it notes
• Paper clips
• Pencils
• Copymasters One, Two, and Three
• PowerPoints One, Two, Three, Four, and Five
Activity

Session One

Students should work through Getting Partial to Fractions before starting this unit as the first activities rely on a length model developed in the earlier unit.

1. Begin with PowerPoint One. Slide One shows Scott Barrett who is 1.98 metres tall. Some students may know that this means 198 centimetres. You might measure the length horizontally with two metre rulers to establish that 1.98 metres is 2cm shorter than 2 metres. A student might lie beside the measure to see how tall Scott is compared is a young teenager.
2. Slide Two shows how the length units in the metric system are derived. One metre is divided into ten equal parts (to form tenths) called decimetres. One decimetre is divided into ten equal parts (to form hundredths) called centimetres. One centimetre is divided into ten equal parts (to form thousandths) called millimetres. Ask predictive questions as you work through the slide, such as:
What will the next place to the right be? Explain why?
How many of this sized unit fit in one whole metre? What will the unit be called?
3. Provide the students with a 1 metre length of receipt machine tape and scissors. A better alternative, if available, is metre long straws which are sold at some dollar shops.
Use your folding skills to create decimetres. Label them as 0.1 (1 tenth) except for the last one. Make the decimetre lengths with the scissors.
Take the last decimetre and fold that into centimetres. Label them as 0.01 and cut them out.
4. Give the students time to create decimetres and centimetres. Challenge them to use the units and metre rulers to measure their own heights and express them as decimals of metres, e.g. 1.53m. Tell them to record their height on a post-it note, and stick the note to the whiteboard.
Watch for students to:
• Recognise that they move to smaller units for greater precision: metres to decimetres to centimetres;
• Record the measure correctly as a decimal;
• Explain the meaning of each digit in the decimal, e.g. 1.62 has six in the tenths place and 2 in the hundredths place.
5. Gather the class and look at the data. Choose heights at random and challenge two students to make the height on the floor. Other students can explain the meaning of the digits. You might ask the class who they think is the height you chose.
How might we sort these data?
6. Student might suggest ordering the decimals from smallest to largest. You might use the data to create a display. The post-it notes can be cut to form a stem and leaf plot. 7. Students might add their height to a dot plot. Graphing packages online allow you to load the data into a CSV (Comma Separated Variable) spreadsheet and import the file for graphing. 8. Discuss the data displays, including:
• The shape of the distribution which should be roughly bell shaped (normal)
• Centre (Median as a measure of average)
• Meaning of decimals like 1.50 and 1.70 and how these numbers are usually not written with the right-hand zero (Why is the zero redundant? Presence of a zero indicates a degree of precision and rounding.)
9. Put the students into groups of four. Set each group the challenge of representing the height of each person in their group in this way:
Make each height using metres, decimetres and centimetres.
Record each height like this: 1.56m = 1 + 5 tenths + 6 hundredths.
Imagine that you group was joined ‘head to foot’ – What is your combined height?
10. Give the students adequate time to complete the task before gathering the class together. Look for students to:
• Correctly name the decimal fractions that make up a decimal number
• Recognise that ten hundredths combine to make one tenth, and ten tenths combine to make one (whole)
11. Gather as a class and find the tallest team. The task requires each group to justify how they know their combined height is correct. The points above should emerge.
12. Finish the session with the last two slides of PowerPoint One.
The first challenge is to order the heights of four teenagers. Can your students correctly name the teenagers by looking at the place value structure of the decimals?
e.g. 1.8m is taller than 1.65m. By how much?
The second challenge has four aspects, including finding decimals with a given difference, and planning an ordering strategy. A key point is that ten hundredths combine to make one tenth, e.g. 1.57 + 3 hundredths = 1.6.

Session Two

In this session students explore addition and subtraction of decimals.

1. Begin the sesion by building the height of the world’s tallest recorded man and woman and the world’s shortest recorded man and woman. Use the metres, decimetres and centimetres made in sesion one.
2. Use PowerPoint Two to introduce the story of Robert Wadlow. Let students estimate Robert’s height using benchmarks in the photograph, such as the doorway or the poster frame.
How high is the doorway?
How tall is Robert?
3. Write 2.72m on the whiteboard. Ask students to explain what the digits mean, i.e. 2 metres, 7 decimetres (tenths), and 2 centimetres (hundredths). Mark Robert’s height on the floor.
Recall that Scott Barrett is 1.98m tall? How much taller is Robert than Scott?
4. Make 1.98m and build onto the model until 2.72m is reached. Watch Slide Two of PowerPoint Two to see how a length model can be used to show the difference. Ask your students to record the jumps on an empty number line and work out the total difference (0.02 + 0.72 = 0.74).
5. Ask your students to work out the difference between their height and Robert’s height. Some students might need to use the decimetre and centimetre pieces from the previous session to support their calculations and check their answers. Encourage students to use empty number lines to track their calculations.
6. Slide Three of PowerPoint Two introduces addition of decimals through combining heights (or lengths). The addition modelled is 1.8 + 1.57 = 3.37m. Rufus’ comment illustrates a common difficulty when students treat the numbers to the right of the decimal point as a set of whole numbers, ignoring the place values of the digits.
Is Rufus correct? Explain.
Why does Barbara say the answer must be more than 3 metres?
How does she know?
7. Build the two heights of Rufus and Barbara using the metre, decimetre, and centimetre lengths. Discuss how the combined height can be found. There are two main approaches; jumping and standard place value. Jumping is illustrated on Slide Four of PowerPoint Two. Standard place value involves combining like units (ones, tenths, hundredths) and is the basis of the written algorithm. The strategy is illustrated on Slide Five. You might illustrate how the steps can be written as a vertical algorithm. 8. Slides Six, Seven and Eight involve the addition of 1.72 + 1.79 = 3.51. The problem involves two lots of renaming, ten hundredths into one tenth, and ten tenths into one (whole). Work through the slides with your students asking them to predict what will happen as much as possible.
9. Put your students into groups of three or four.
Your task is to work out some combined heights. First try all the pairs of people you have in your group, then try the whole group combined. Use the metre, decimetre, and centimetre strips, if you need. Record the answers you get using either a number line or written algorithm.
10. After a suitable time bring the class together.
When two heights are added what is the usual range for the sum?
Why is the sum usually 3.0 – 3.5 metres? (The average height of 11-13 year olds in New Zealand is about 1.6 metres)
11. For further practice students might work from Gentle Giants (Figure It Out, Link, Book 5, Page 18). You might also look at the heights of other students. Graph the heights using a dot plot, and box and whisker. The median height of Year 7-9 students is about 1.59m.
Is the average different for males and females?

Session Three

In this session your students look at the two situations to which subtraction of decimals can be applied, partitioning (separating) and difference. The range of decimals is still constrained to hundredths.

1. Show your students a Barbie Doll and Slide One of PowerPoint Three.
Compare the proportions of the Barbie doll with the normal proportions of a female. What do you notice?
2. Some students may have heard of controversy surrounding the proportions of the Barbie Doll. A lot of criticism focuses on how the Barbie doll proportions create unrealistic expectations around body image, e.g. long neck, narrow waist, leg length compared to body length. Many videos are available about the controversy including attempts to sell more realistically proportioned dolls.
Look at the diagram of normal proportions for men and woman.
What do you notice?
3. The proportions of an adult female and male are based loosely around the length of the head. A full person is about eight times that length. The distance from top of head to navel is about three eighths of the full height. Student should detect other relationships.
4. Look at Slides Two to Four of PowerPoint Three. Ask your students to estimate the measures first and state those measures in decimals of one metre.
What will the rest of Aisla/Barbie measure? Share your strategy.
Subtraction is the most obvious strategy since part of Aisla or Barbie is removed, leaving the rest.  For example, 1.76 - 0.9 = 0.86m gives the length from Aisla’s hip to the top of her head.
5. Next, tell your students to imagine that the length from their navel to their head is removed (see Slide Six).
Measure the length from your navel to the top of your head, and subtract that length from your height. What will the difference tell you? Show your working and answers in decimals of one metre.
6. Let your students work practically on the task in small groups. Only allow them to use the metre, decimetre, and centimetre pieces they created previously to measure the navel to head length. The measures can be checked with a metre rule, and calculations with a calculator, later if necessary. Look for students to apply decimal place value:
• Do they record the measures in correct decimal notation that matches the units they use? e.g. Six decimetres (tenths) and seven centimetres (hundredths) is written as 0.67m.
• Do they subtract correctly to find the differences? e.g. 1.7 – 0.67 = 1.03
7. After a suitable time gather the class to highlight a few good examples from your observations. Ask your students to work through Copymaster One to demonstrate their ability to find differences with decimals.

Session Four

In this session students expand their knowledge of the decimal system to include thousandths. The need for smaller units is inspired by a need for more accuracy. Your students will need access to strips of paper that are exactly one metre long.

1. Being by folding a single one-metre strip in half.
How long is one half?
Some will say 50cm which is correct. Write 50/100 and ask what the fraction is referring to. Note that one half and fifty one-hundredths are equivalent fractions.
2. Ask: How is one half written as a decimal?
Many students know that one half equals 0.5. Performing 1 ÷ 2 = 0.5 on a calculator confirms that.
What does the five mean? (five tenths)
3. Model aligning five decimetre pieces with the one half of one metre. Record the finding like this: 4. Model folding one half in half to get quarters. If one half equals 0.5 then what is the decimal for one quarter?
How long is one quarter?
5. Some will say 25cm which is correct. Write 25/100 and ask what the fraction is referring to. Note that one quarter and twenty-five one-hundredths are equivalent fractions.
What is the decimal for one quarter?
6. Model aligning two decimetres and five centimetres to match the one quarter fold line.
If one quarter equals 0.25 then what is the decimal for three quarters?
7. Progress in the same way to eighths by folding quarters in half. The calculator gives 1 ÷ 8 = 0.125. Challenge your students to discuss in pairs why the decimal is that and what each digit, 1, 2, and 5, refers to. Students are likely to know that millimetres are also on a ruler. Millimetres are one thousandths of one metre.
8. Record the decimal for one eighth like this: 9. Expect students to align one decimetre, two centimetres and five millimetres with the one quarter fold. Note that millimetres are very small units so you might mark them onto a one centimetre length.
10. Set the following challenge: By folding one metre strips and measuring with decimetres, centimetres and millimetres, work out what each of these fractions will be as decimals. 11. Watch as your student work in groups of two or three. Do they…?
• Correctly fold the one metre lengths into unit fractions, such as tenths and fifths;
• Anticipate how many tenths, hundredths or thousandths will be needed to make the unit fraction;
• Check their anticipation by considering how many times their prediction will fit into one, e.g. If 0.125 (125 thousandths) equals one eighth, then 8 x 125 should equal 1000;
• Use multiplication to anticipate what the decimal for the non-unit fraction will be, e.g. If 1/5 = 0.2 then 4/5 = 0.8 because 4 x 2 = 8;
• Recognise that 11/8 is greater than one;
• Notice that one third cannot be made exactly, even with thousandths, since 3 x 333= 999 not 1000.
12. If groups finish early challenge the students to find the decimals for: 13. Gather the class to discuss what they have found out, with a focus on the points above. Answers can be checked by measuring with decimetres, centimetres, and millimetres, and on a calculator using division, e.g. 4/5 = 4 ÷ 5 = 0.8. Students should notice that thirds, sixths and ninths produce recurring decimals. Sevenths also produce recurring decimals, but the element of repeat is harder to find, e.g. 1/7 = 0.142857142857142857
14. PowerPoint Four introduces things that might be measured in millimetres. Discuss how the scale in each picture can be used to estimate the measurement in millimetres then write that measurement in decimals of one metre, e.g. 45 millimetres can be written as 0.045m.
15. Copymaster Two contains problems of anticipating sequences with decimals. Let your students complete the Copymaster without a calculator first. Look to see if they:
• Anticipate further members of each sequence by adding on the increment;
• Notice when renaming between decimal places will occur;
• Explain why the length of decimals increases or decreases due to renaming.

Session Five

In this session students apply their knowledge of decimals to three places to solve addition and subtraction problems. They investigate rainfall patterns for different locations in New Zealand then play a game that involves addition and subtraction of decimals.

1. Look at PowerPoint Five. Rainfall is measured in millimetres (thousandths of one metre). The rainfall for a location is the depth of water that lands on the surface area. Millimetres are used as they provide adequate precision and the numbers for a daily fall tend to be manageable whole numbers. On slide four annual average rainfall for many locations is given. Ask your students:
Is there any pattern to the rainfall in New Zealand? (East Coast tends to be drier than West Coast)
Which locations have the highest/lowest rainfall?
2. Take one location and use a ruler to work out the height of the rainfall. Wellington is a good choice at 1 215mm as that is more than one metre (1.215m).
How would Wellington’s annual rainfall be written in metres?
3. Ask your students to write each annual rainfall in metres. Take particular interest in how students deal with the rainfall of Blenheim (720 thousandths equals 0.72 metres).
4. Pose these problems:
Add the rainfalls of Tauranga, Gisborne, and Napier. Give the answer in metres.
Hemi claims that the Eastern North Island gets nearly 3m of rain per year. Is he right? Explain. (It is not legitimate to add the rainfall of different locations)
5. Look for students to add the decimals correctly, with understanding of renaming place value units, e.g. 1.177 + 0.979 = 2.156m, 2.156 + 0.776 = 2.932m. Some students may add whole numbers of millimetres then convert the answer to metres. Students should realise that the depth of rainfall will be spread across the total land area so the ‘evened out’ rainfall for the region is closer to 1.0m per year.
6. The mayor of Christchurch claims the title of “New Zealand’s driest city”.
“We get more than 10cm less rain than the next driest city, and half the rainfall of Auckland,” she says.
It’s all very confusing. Is she right?
7. Look for your students to identify Christchurch and Blenheim as the two driest locations on Slide Four. The difference in rainfall is 0.72 - 0.618 = 0.102m which is two millimetres more than 0.1 (one decimetre or ten centimetres). The difference between Auckland and Christchurch’s rainfall equals 1.211 – 0.618 = 0.593. Therefore, Christchurch has a little more than half the rainfall of Auckland.
Alexandra is not a city, but it gets only 0.335m of rain per year.
​​​​​​​How much less rain does it get than Christchurch?
8. Challenge your students to calculate other differences between the rainfalls of locations on slide four of PowerPoint Five.
9. Next, introduce the Hare and Tortoise Race (Copymaster Three), a game for practising the addition and subtraction of decimals. Students need a photocopy of the track, a spinner sheet, a paper clip, and a pencil between two players.
• The paper clip makes the spinners. Bend one end of the paper clip out and anchor the other end loop with a pencil tip. Flick the paper clip to make it spin. Spinners need to be centred in each circle of the board.
• One player is the tortoise and the other player is the hare. Players take turns to spin a decimal for the other player. Both spinners give a combination of digit, e.g. 7, and fraction, e.g. tenths, that give the distance to travel, 7 tenths or 0.7. The non-spinning player adds that distance onto their existing destination to reach a new point. The addition should be recorded as a loop on the track sheet, with both the distance and the destination recorded as decimals of one metre.
• The only exception occurs if the spinner lands exactly on a line between sectors. In that case the spinning player nominates one of the fractions either side of the line, e.g. the spinner lands on the line between tenths and hundredths, and the spinning player chooses hundredths. In these situations, the distance is subtracted from the existing destination. The player goes backwards in the race.
• Both the hare and the tortoise start on zero. The first player to cross the finish at ten, wins the game.

Extras

For further development of decimal understanding students should attempt the four e-ako modules that cover addition and subtraction of decimals using a decimat model (AS4.10, AS4.20, AS4.30, AS4.40). That is best done over four days. Each module takes around 20 minutes to complete.

Create 2.8 and Decimal Lineup can be sent home as practice games for students to play with their family.

Fitness

Purpose

This unit examines regular tessellations, that is, tessellations that can be made using only one type of regular polygon, and semi-regular tessellations, where more than one type of regular polygon is involved. Students are required to investigate what properties tessellating shapes must have in order to cover the plane with no gaps or overlaps.

Achievement Objectives
GM4-5: Identify classes of two- and three-dimensional shapes by their geometric properties.
Specific Learning Outcomes
• Create regular and semi-regular tessellations of the plane.
• Demonstrate why a given tessellation will cover the plane.
Description of Mathematics

Tessellations are frequently found in kitchen and bathroom tiles and lino. You can see them in the pattern on carpets and decorative patterns on containers and packaging. Tessellations are a neat and symmetric form of decoration. They also provide a nice application of some of the basic properties of polygons.

To be able to fully understand tessellations using regular polygons, you need to know about their symmetry and about the size of their interior angles. All of the facts that you need to know are accessible to Level 4 students. This unit takes children through the steps that they need in order to establish that there are only three regular polygons that tile the plane. This unit thus follows on from Keeping in Shape from Level 3, where regular tessellations are first discussed.

Moving on from here, the children can consider semi-regular tilings. All that they need to know here is how to sum the interior angles of various regular polygons to 360° . The rest is up to their imagination.

This unit is set for students to learn, and practise, outcomes at Level 4 of mathematics in the New Zealand Curriculum. The geometric focus opens up opportunities for visual reasoning that might prove engaging for students who find numeric reasoning challenging. Here are some approaches to enabling participation.

• Provide physical manipulatives, regular polygons, or virtual equivalents, so that student can experiments with shapes. Encourage anticipation of results, e.g. Will regular hexagons tessellate? How do you know?, and ask students to justify why they believe patterns occur.
• Directly model examples of tessellations, and your reasoning about why the combinations of shapes around each vertex will work.
• Organise the data about regular polygons in a table, especially the measures of internal angles. Make the table accessible to students so they can make predictions about sets of shapes that will, and will not, tessellate.
• Encourage students to work collaboratively in partnerships, and to share and justify their ideas. Share ideas with the whole class regularly.

The difficulty of tasks can be varied in many ways including:

• allow access to calculators and digital tools, so the investigations are more about spatial reasoning than calculation
• restricting the set of shapes at first, e.g. triangles, squares, pentagons and hexagons, until generalisations about angles around a vertex emerge. Then open the investigation to more complex regular polygons
• displaying the work of students as models for others, especially students who provide explanations about why particular tessellations work. Motivate students to add a new, undiscovered tessellation to the class display.

The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. For example, tessellations are prominent in Islamic art traditions, and in tapa cloth designs from Pacific nations. Students might be fascinated by the work of Dutch artist Escher, who built his work on distorting regular polygons to create ‘life-like’ tessellation patterns. Tessellation might fit well with efforts to beautify the school environment. Mosaic tiles can be created from fired clay, or cobblestones created from concrete.

Required Resource Materials
• Isometric dot paper
• Pattern blocks or mosaic shapes
• Protractors, scissors, paper, rulers, pencils
• Copymaster 1 (table showing angles of regular polygons)
• Copymaster 2 (templates of regular polygons from 3-sided to 12-sided)
• Copymaster 3 (semi-regular tessellations)
• Copymaster 4 (regular polygonal tessellations with different vertex arrangements)
Activity

Getting Started

1. Show the students a large cut out equilateral triangle. Mark the vertices (corners) of the triangle with different colours. Say, "I am going to tear off the corners of this triangle and place them around this point (draw point on board). What do you think will happen?" Students may have encountered this before but let them guess what they think will happen. Tear off the corners and place them about the point to confirm that a half turn (or 180°) is created. 2. Ask students whether they think every triangle will produce a 'half turn', no matter what shape it is (isosceles, scalene, obtuse). Is there a way to check this without tearing off the corners? (Measure the interior (inside) angles to see if they total 180° (a half turn).) Tell the students to make a variety of triangles and test this conjecture.
3. If the sum of the interior angles of any triangle is the same - 180°, is it likely that the sum of the interior angles of any quadrilateral is the same? Ask them to predict what will happen if the corners of any quadrilateral are joined about a point. Have them cut out several quadrilaterals of different shapes to check their predictions. Remind them to mark the corners before tearing them off. 4. Once it has been found that the sum of the interior angles of any quadrilateral is 360° (they form a full turn about a point), then students can investigate the sum of interior angles of other polygons by measuring with protractors or tearing corners. The results can be captured in the table below. Encourage the students to look for a pattern to predict what the next angular sum will be.
 Number of Sides Sum of Angles ° Interior Angle of Regular Polygon 3 180 4 360 5 540 6 720

Note that for each side that is added, the sum of the interior angles increases by 180°. This can be explained by the fact that the addition of a side creates another triangle within the shape, and that each triangle has an angular sum of 180°.   Quadrilateral (2 triangles) Pentagon (3 triangles) Hexagon (4 triangles)
5. Challenge the students to use their results to draw a regular triangle (equilateral), a regular quadrilateral (square), a regular pentagon, a regular hexagon, and a regular octagon. Regular means that all sides are the same length and all interior angles are equal.
6. What size are the interior angles of the regular figures we have just been talking about? How can we find out?

For the regular hexagon with six vertices and an angular sum of 720°, we need to divide the sum by the number of angles to find each angle size. So, since 720 ÷ 6 = 120, each interior angle in a regular hexagon is 120°.

7. Get the children to complete the table above for polygons with up to 12 sides. (The regular dodecagon (12-sided polygon) has an angular sum of 1800°, so each interior angle will be 150°.)

Exploring

1. Use a set of pattern blocks to show how equilateral triangles tessellate. Tessellate means that they cover the plane infinitely with no gaps or overlaps. Send the students away in groups with their own set of pattern blocks, or access to an online version, to explore what other tessellations can be discovered using shapes from the set. You may want them to record the tessellations using isometric dot paper.

After a period of exploration bring the class back together to share the tessellations. Note that there are three regular tessellations that can be found, that is tessellations involving use of the same regular polygon. These regular tessellations are 3.3.3.3.3.3 (six triangles about each point or vertex), 4.4.4.4 (four squares about each vertex), and 6.6.6 (three hexagons about each vertex).

2. Focusing on the regular tessellations ask why it is that these patterns work without gaps or overlaps. You may need to remind the students of the angle measures they found in Getting Started. There are two key properties of the shapes involved in regular tessellations:
• side lengths are the same;
• the sum of angles meeting at each vertex is exactly 360° (a full turn). For example, in the case of the tessellation with squares (4.4.4.4), the side lengths are the same and the four angles of 90° add up to 360°. Confirm that the "angles around a point" principle holds for the other tessellations that students have found. 3. Remind the students of the table on angle sums (Copymaster 1) that contains the interior angle measurements for regular polygons. Ask: From the table could we have expected the triangles, squares, and hexagons to have tessellated by themselves? How? (Each of these shapes has interior angles that can be divided into 360° evenly.)
4. Many other tessellations are possible with combinations of pattern block shapes. Get them to experiment with combinations of regular polygons. These tessellations are known as semi-regular tessellations. For example, from the table it looks like 2 squares and three triangles might fit together about a point because 90°+ 90°+ 60°+ 60°+ 60° = 360°.
5. These might be arranged in different ways, e.g. square-triangle-triangle-square-triangle (4.3.3.4.3) and square-square-triangle-triangle–triangle (4.4.3.3.3). Try to produce all possible combinations using pattern blocks.
6. Give the students Copymaster 2 that gives templates for the regular polygons up to the 12-sided shape (dodecagon). By stapling through the centre of each shape onto blank pages underneath students can make multiples of each regular polygon. Ask students to use the table and the cut out shapes to find as many semi-regular tessellations as they can.
7. Share the results as a class to see if all the possible semi-regular tessellations have been found. These are 3.3.3.4.4, 3.6.3.6, 3.3.4.3.4, 3.3.3.3.6, 4.8.8, 3.4.6.4, 3.12.12, 4.6.12 (eight possibilities). Students may wish to create a poster presenting their favourite semi-regular tessellations explaining why each combination of shapes can tessellate the plane. Other tessellations are possible using regular polygons if the constraint of each vertex having the same arrangement of shapes is removed. For example, hexagons, squares and triangles can be used in this way.
Note that with some vertices the arrangement is 6.3.4.4 and at others it is 6.4.4.3 if the shapes are read clockwise about each vertex.
8. Get the students to investigate what other arrangements can be found in this way. Copymaster 4 contains examples of such arrangements. Note that this embodies the principle seen in the square-square-triangle-triangle-triangle arrangements that changing the order of the addends does not affect their sum, in this case 360°.

Reflecting

1. Observation of the table might persuade them that tessellating combinations are not likely for the heptagon (7 sides), decagon (10 sides) and hendecagon (11 sides) since their interior angles measures will not yield combinations to 360°.
2. Ask: Is it possible to show that there are only three regular tessellations?
There are two approaches for this depending on the ability of your students. The first way is to notice that no polygon has an interior angle greater than 180°. They are always less than this. And no polygon has an interior angle smaller than 60°(from the table.) That means that you need at least three polygons to come together at a vertex and no more than six polygons since 6 x 60° = 360°:
• if it is 3, the interior angles have to be 360°/3 = 120°;
• if it is 4, the interior angles have to be 360°/4 = 90°;
• if it is 5, the interior angles have to be 360°/5 = 72°; and
• if it is 6, the interior angles have to be 360°/6 = 60°.
Only three of these are possible for regular polygons. The only tessellation by regular polygons requires an equilateral triangle, a square or a hexagon.
3. How can we find semi-regular tessellations in a systematic way? On way is to start with six triangles around a point. Take away triangles until a different regular polygon will fit in the angle ‘gap’. Remove two triangles leaves a gap of 120° so a regular hexagon will fit. Remove three triangles and there is a gap of 180° that can be filled by two squares. Remove four triangles and two hexagons can fill the gap. Remove five triangles leaves a gap of 300° and two dodecagons, or two squares with a hexagon, can fill the gap. Once all the possibilities with triangles are exhausted, move onto squares around a point and repeat the strategy, ignoring triangles.
Attachments
fitness-1.pdf136.36 KB
fitness-3.pdf235.23 KB
fitness-4.pdf330.27 KB

Tessellating Art

Purpose

In this unit we apply our understanding of why tessellations work to form our own unique tessellating shapes. We use these shapes to create interesting pieces of art in the style of M.C. Escher, a famous Dutch artist.

Achievement Objectives
GM4-8: Use the invariant properties of figures and objects under transformations (reflection, rotation, translation, or enlargement).
Specific Learning Outcomes
• Alter polygons to create unique shapes that tessellate.
• Describe the reflection or rotational symmetry of a shape or tessellation.
Description of Mathematics

This unit is built around the famous artist Maurits Cornelius Escher. Escher was born in Leeuwarden, Netherlands on June 17th, 1898. He studied at the School of Architecture and Decorative Arts in Haarlem but soon gave up architecture in favour of graphic arts at the age of 21.

All M. C. Escher works (C) Cordon Art, Baarn, the Netherlands. All rights reserved. Used by permission.

Escher is famous for two types of engravings. One of these involves impossible situations and the other is his variation on the theme of tessellations. A typical impossible situation shows four men climbing stairs. As you follow the men around and up their particular flights, you realise that they are going round and round. With regard to tessellations, Escher took a tessellation and, by adding and subtracting from the basic unit of the tessellation, turned it into a repeated picture.

There are many web-sites that explore the life and work of M.C. Escher. You can easily find one by entering his name in your search engine.

By emulating Escher and exploring tessellations in this unit, the students will gain a greater appreciation of the way that tessellations work. Hence they will see how mathematics, art and even nature, interact.

Other units that refer to tessellating are Keeping in Shape, Level 3 and Fitness, Level 4. It might be useful to have done Measuring Angles, Level 3 before attempting this unit.

Students can be supported through the learning opportunities in this unit by differentiating the nature and complexity of the tasks, and by adapting the contexts. Ways to support students include:

• providing physical and digital manipulatives so students can experiment with the transformations
• directly modeling examples of altering ‘parent polygons’ and the way in which the new shape is moved to form a tessellation
• discussing the properties of a shape that stay unchanged as it is translated, reflected and rotated
• encouraging students to work collaboratively in partnerships, and to share and justify their ideas. Share ideas with the whole class regularly.

The difficulty of tasks can be varied in many ways including:

• restricting the set of shapes and the complexity of the transformations at first. Rectangles altered by translation are much easier than alteration by rotation with triangles, and hexagons
• displaying the work of students as models for others, especially students who provide explanations about why particular tessellations work. Motivate students to add a new, undiscovered tessellation to the class display.

The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. For example, tessellations are prominent in Islamic art traditions, and in tapa cloth designs from Pacific nations. Tessellation might fit well with efforts to beautify the school environment. Mosaic tiles can be created from fired clay, or cobblestones created from concrete.

Required Resource Materials
Activity

Getting Started

We begin our exploration of tessellating art by altering squares and parallelograms.

1. Show the students the altered square tessellation (Copymaster 1). Why is it a tessellation?
Which of the regular tessellations does it look like it has been made from?
Can you see how this tessellation has been made? Show us.
2. Look at the tessellating shape and predict how it was altered. 3. Give each student a square cardboard tile. Ask them to cut a piece out of one side of the square. Attach the cut out piece to the opposite side of the square with cellotape.
Will your new shape tessellate? Try it and see.
4. Have the students trace their new shapes several times onto blank sheets of paper to see if the altered shape tessellates.
5. Discuss.
Is your altered shape symmetric? What type of symmetry does it have?
Does the new shape have to be symmetric to tessellate?

(Note: When altering a square by translating opposite sides to form a new tessellating figure, the alteration does not have to be symmetrical.)
Describe the movement of the shape as you tessellate with it? (translation or shifting)
6. Ask students to repeat the altering process with other quadrilaterals that tessellate, for example, parallelograms and rectangles. Also include "wonky" (scalene) quadrilaterals. Not all alterations will create tessellating shapes and this a useful discovery. The key to creating tessellating shapes is altering opposite sides in exactly the same way.
7. Allow time to share and discuss at the end of the session.
What did you discover about altering shapes to create new shapes that also tessellate?
Why do you think it is possible to alter shapes in this way and still end up with a shape that tessellates?
Did you create any shapes that do not tessellate? Why do you think that they won’t tessellate?

Exploring

Over the next 2-3 sessions we use our imaginations to create interesting art pieces using altered tessellating shapes.

1. Begin by looking at one of the shapes created yesterday.
2. Brainstorm for ideas about what the shape could be. In the following shape the addition of an "eye" creates a fish-like shape. 3. Have the students select a shape to use for the basis of a piece of tessellating art.
4. After they have decided on a shape, have them trace it across a piece of paper to create a tessellation. To transform it into an Escher style piece let them put details on each of the shapes.
5. Search online for “Creating Escher Tessellations”. Many useful videos exist to show your students how other transformations can be used to alter a ‘parent polygon.’ Useful transformations include rotation of half a side about the midpoint, and rotation of a whole side onto another, You may decide to introduce these more complex transformations gradually so students master one process before beginning another. There are also free digital tools available for creating Escher tessellations.
6. Ask questions as the students work that focus on the symmetries of the shape and the tessellation.
Tell me about the symmetries of your shape? (reflection symmetry, rotational symmetry)
How are you generating the tessellation? (translating? rotating? reflecting?)
Why does your shape tessellate? (Encourage the students to focus on the sum of the angles at any point must equal 360 degrees)
7. Ask the students to make a written record about the process that they used to make the tessellating shape and the tessellation.
8. At the end of each session share the tessellating art.

Reflecting

In this final session we analyse tessellations and attempt to predict the processes that were used to create them.

1. Begin by looking at the bumpy squares tessellation (Copymaster 2).
What shape do you think has been altered?
How do you think it was altered?
2. Next we distribute our tessellating art, created in the previous days, to see if others can determine the tessellating shape and the process by which the tessellation was formed.
3. We check with the artists to see if we predicted correctly.
Attachments

What are the chances?

Purpose

In this unit students investigate the link between experimental estimates of probability and theoretical probability. They also learn about short run variability and independence/dependence of events.

Achievement Objectives
S4-3: Investigate situations that involve elements of chance by comparing experimental distributions with expectations from models of the possible outcomes, acknowledging variation and independence.
S4-4: Use simple fractions and percentages to describe probabilities.
Specific Learning Outcomes
• Explore the theoretical and experimental probabilities of situations involving chance.
• Recognise variability from theoretical expectations, especially with small numbers of trials.
• Estimate and find the relative frequencies of events.
• Distinguish between events and the outcomes that lead to those events.
• Distinguish between dependent and independent events.
Description of Mathematics

Simply put, probability is a measure of the likelihood of an event occurring. Just like 2.3 metres is a measure of length, 50% or 1/2 is a measure of the chance of getting heads with a single coin toss.

Probabilities range from zero to one (0-100%). An event that always occurs is certain and has a probability of 1 or 100%. An event that never occurs is impossible and has a probability of 0 or 0%. There are two ways to estimate or calculate the probability of an event occurring. An experiment consists of trials where the event may occur. For example, a dice might be rolled 100 times and the results used to estimate the probability of heads occurring. The experimental probability is unlikely to be exactly 50% but the results will provide an approximate measure for the likelihood.

The likelihood of some events can be worked out theoretically. A model must find all the possible outcomes and identify which of the outcomes leads to the event occurring. The probability is expressed as a part-whole fraction, Number of outcomes that lead to the event/Total number of possible outcomes. In the simple case of a coin toss there are two possible outcomes, and one of those outcomes leads to the event of heads. Therefore, the probability of heads equals 1/2.

In more complex situations involving chance to models of theoretical probability become more complex. Events are independent if they do not influence each other. Tossing two separate coins involves independent events as the result from the first coin has no impact on the result from the second coin. Selecting two coloured pegs from a bag without replacement involves two dependent events. The result of the first peg draw affects the possible outcomes for the second peg draw.

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:

• Connect drawing diagrams and acting out as strategies to represent how all the possible outcomes can be found. For example, simulate creating three-child families by drawing three cubes of two colours (for boys and girls) from a bag, and drawing the arms of the tree diagram as the outcomes occur.
• Clarifying the language of probability, particularly words like ‘chance’, ‘likelihood’ and more technical terms such as ‘experiment’, ‘trial’, and ‘theory.’
• Encouraging students to work collaboratively so they can share their knowledge and persevere more.
• Modeling efficient ways to record the results of trials, especially tally charts and frequency tables.
• Modeling ways to find all the possible outcomes, such as tables, networks, and systematic lists.
• Easing the calculation demands by providing calculators, particularly for converting among fractions, decimals, and percentages.

Tasks can be varied in many ways including:

• Reducing the complexity of the situations, e.g. two child families before progressing the three and four child possible outcomes.
• Allowing sufficient practical exploration so students develop sense of how the activity works, and a sense of which events are most likely or least likely. Encourage informal prediction about probabilities and test those predictions out by experiment.
• Select sample sizes that make comparison between theoretical and experimental probabilities easier. For example, if there are eight possible outcomes for three child families then use multiples of eight, such as 16, 24, and 40 as the sample size.

The context for this unit can be adapted to suit the interests and cultural backgrounds of your students. Most students are captivated by games of chance and are intrigued when their expectations about fairness do not match what occurs. This unit uses contexts such as traffic lights, fish populations, horse races, and family makeup. These contexts can be adapted to others without losing integrity, such as sampling shellfish populations, or other rare animals such as kiwi or tuatara. Horse races could be replaced by waka racing.

Required Resource Materials
• Counters
• Standard dice
• Paper bags or plastic containers (e.g. icecream)
• Blank wooden cubes (for the optional assessment task)
• Copymasters One, Two, Three, Four, Five, and Six
• PowerPoints One, Two, and Three
Activity

Session One

1. Begin the session by asking: Who in this class is a netball fan?
(Maybe the fans can explain how the game works. Perhaps show a brief highlights video from a recent game)
2. Tell your students that they are going to simulate a game of netball played between two rival netball teams, the Rockets and the Emeralds .
What do you think a simulation is?
A simulation is an imitation of the real thing, so the game will not be real.
3. Discuss: What positions shoot for goal in netball? (Goal Shoot and Goal Attack)
I have looked at a few games played between the two teams to see how accurate the goal shooters are. Here are the data.
Show them this table of results:
 Team Goals Total Shots Taken Rockets 259 298 Emeralds 277 324
4. Tell the students to discuss the data in their groups and to come up with some statements about the shooting performance of the two teams in the last few games. You may wish to lead them with questions such as:
How many games between the two countries do you think are included in these results? Explain. (Teams usually score around 50 goals per game)
How can we compare how accurate the shooters of both teams are?
(Students need to recognise that there were different numbers of shots taken so fractions/percentages are needed. Look online for netball shooting statistics to see that percentages are used – Why? Use a calculator to find the percentages, i.e. 259 ÷ 298% = 87% and 277 ÷ 324 = 85%, and discuss rounding to the nearest percent)
5. Ask: If the goal shoots of each team had ten shots, how many would you expect to be goals? (Eight or nine goals is a good estimate from the percentages. Relate 87% as 87 in every 100 to about 9 in every 10)
What is the purpose of working out the shooting percentage?
6. Share the results of their discussions. Look for statements such as:
• The Emeralds' shooters seem to be slightly more accurate than the Rockets' shooters
• The Rockets' shooters take more shots and get more goals.
7. Ask students to imagine that it is part way through the game and one of the Rockets' shooters is about to take a shot.
What are the chances that this shot will be a goal?
8. Discuss factors that affect the chances, like distance from the goal, how far the shooter has run to get the ball, the attentions of the defence players, who the shooter is, how important the goal is to the game outcome, etc.. Some students may mention that all shooters take bad shots, no matter how reliable they usually are. Introduce the word variable, which means the outcome might be different from what is expected.
9. Ask students to give a word or number that tells us about the chance of the shooter scoring.
Expect words such as likely, almost certain, probably.
Expect numbers such as three quarters (3/4), 85%, 0.8.
10. To simulate the next game between the Rockets and the Emeralds you are going to use standard dice (1-6).
Let’s imagine that rolling the dice is like taking a netball shot. When a Rockets player takes a shot, the numbers 1-5 are goals, and 6 is a miss. When an Emeralds player shoots the numbers 2-6 are goals, and 1 is a miss.
Write the rules on the board:
• Rockets: 1, 2, 3, 4, 5 (Goal); 6 (Miss)
• Emeralds: 1 (Miss);  2, 3,4, 5, 6 (Goal)
11. Ask students to discuss which team is more likely to win. You may see misconceptions such as the Regency Effect, when six is regarded as harder to get than other dice numbers due to the priority it has in board games. Expect students to justify their thinking, e.g.
• Both teams have five out of six chances to score from each shot. Their chances are the same. (Prediction based on theoretical probability for one throw)
• It is impossible to know which team will win. One team might just be luckier than the other. (Acknowledges uncertainty, albeit subjectively)
• Players might throw the dice in a way to get the numbers they want. (Addresses whether the dice outcome is random)
12. Tell the students that we are going to assume that the Rockets' shooters take the same number of shots as the Emeralds' shooters. With their dice the students play a Six-shot game, that is each team has the dice rolled three time. The students look at the data to determine which team wins, or if the result is a draw. Students might use Copymaster One to record the results of their trials.
13. Gather the class to discuss the results.
How many games ended in a draw? In a set of 25 games around seven, eight, or nine will end in draws. There is slightly less than one third chance of a draw.
What are the ways a draw might happen? (Both teams get three goals, two goals, one goal, or no goal)
14. Collate the results (for example):
 Event Rockets win Emeralds win Draw Number of games 8 10 9

Do the results match our predictions? Why? Why not?
Expect students will comment that the results were unpredictable. In probability the word random is used to describe an outcome that is uncertain. However, the number of wins for Rockets and Emeralds are close.

15. Move on to playing a 20-shot game and a 50-shot game. You may decide to use a digital dice simulator online to speed up the process. After each set of games collate the results.
Is a draw more likely with more shots in a game?
Is it more likely that the winning sides are more balanced with more shot games?
Students might note that the results for a small number of shots are erratic and do not appear to match the theoretical idea that both teams have the same chance of winning. This is known as short-run variability, that is, it is the outcomes from a short number of trials often vary from expectations. You might also discuss the difference between an outcome and an event. A win for Rockets, a win for Emeralds, and a draw are events, things that may or may not happen. Many different outcomes contribute to those events. For example, Rockets might win by their three shots all being goals with Emeralds scoring two, one, or no goals.

Session Two

1. In this session we investigate a simpler situation involving chance. The purpose of this session is to look at the relationship between theoretical probability and experimental probability. Students are expected to create a model of all the outcomes and use that model to predict the results of a simulation.
2. Introduce the problem:
A couple I know are expecting their first child. They are not worried about which gender the baby is, but secretly I think they want a girl.

What are their chances of getting a girl baby?
3. Students might think that the probability is 50% or they might call on personal experience that one gender is more likely than the other. Research online with the question “What is the probability of having a female baby?” There is a slight imbalance towards having a male baby. The actual probabilities are 51% male to 49% female.
4. Let’s assume that the chances are 50:50.
Does anyone in the class come from a family that has two children?
What are the genders of the children in your family, starting with the first born?
Record the data you get from students.
MF      FF        MM     FM      MF      MM     FM      FF        FF        MM     MF
If there are not many families with two children, open the data up to other families that your students know about.
5. Discuss the limitations of the data you have collected. This dataset only includes a few families.
Looking at these data, what are the chances of getting two female babies in a two-child family?
6. Express the ‘experimental data’ as fractions, e.g. Probability of two males = 3/10 or 30%, Probability of two males = 2/10 or 20%, etc.
How reliable do you think this data is?
Students might mention issues like a small sample size, and that the sample is drawn only from our families.
7. Tell the students that they are going to investigate having lots of two-child families.
8. Go to Spinner and create a spinner like the one below. Red represents female and blue represents male (or whatever colours you choose). A single spin simulates a single birth, two spins simulates a two-child family. In this case the outcome was FM so the event was a mixed gender pair. Show your students how to reset the spinner each time (Clear results) before creating the next family. Ask them to work in pairs and represent their results using a tally chart. If you prefer you could use two coins to carry out the simulation instead of the spinner (assign a gender to each side of the coin and record the results of flipping).
9. Use a spreadsheet to collate the class results and create a pie chart of the data. What do you notice? (Same gender families are about 1/4 each and the mixed family is about 1/2)
Why do these events have different likelihoods/chances?
Students may explain that there is only one way to get MM, one way to get FF, but two ways to get a mix, MF and FM.
10. Explicitly model how to create a tree diagram of all the possible outcomes (See Slide One of PowerPoint One which is animated on mouse clicks). Ask your students to draw the tree diagram as you work through it. Which outcomes give a mixed two-child family? Circle those outcomes.
Which outcome is a Male-Male family? Circle that outcome.
Which outcome is a Female-Female family? Circle that outcome.
Note that two outcomes out of four give a mix of male and female (MF and FM). 2/4 is equivalent to ½ and 50%.
11. Introduce the next challenge using Slide Two of PowerPoint One. Read the task as a class then let students work on it in pairs.
What mixes of female and male children are possible in a three-child family?
Use the spinner (or coins) to simulate having three-child families, at least 24 times.
Which events happen most frequently?
Which events happen least frequently?
Create a tree diagram to explain your results.
12. Roam the room looking for the following:
• Do your students organise their data using a tally chart?
• Do they notice differences in frequency?
• Do they try to explain why some events are more likely than others?
• Do they construct a tree diagram and calculate probabilities for each gender mix?
13. After a suitable period of investigation gather the class. Slide Three builds the tree diagram for a three-child family. Students might extend the tree diagram they built for the two-child family if they have not already done so.
What is the probability that all the children will be female? (1/8 or 12.5%)
What is the probability that all the children will be male? (1/8 or 12.5%)
What is the probability of a two-female and one-male family? (3/8 or 37.5%)
Where do these two-female and one-male outcomes appear in the tree diagram?
How many possible outcomes result in the event of a two-female and one-male family? (Three: MFF, FMF, FFM)
Why is the probability of a two-male and one-female family the same, 3/8?
14. For assessment purposes ask students to solve the following problem independently.
Jacob tosses two coins. Each coin has a heads side and a tales side.
“There are three possible outcomes, two heads, two tails, or head and tails,” he says.
What does Jacob think is the probability of getting two heads?
Is he correct? Explain.
Do students notice that the situation is the same as the two-child family scenario?

Session Three

In this session students explore the implications of independent and dependent events.

Who likes fish? Who likes fishing?
What species of fish/ika moana can we catch in Aotearoa?
Which species are found in fresh water?
Students might suggest species like trout (brown, rainbow), eel, bullies, etc. Most species in New Zealand lakes were introduced, usually as game fish.
2. Explain that scientists sample fish in our waterways to keep track of numbers and species. That is one way to make sure our sea, lakes, and rivers are not over fished.
You might play a short video about freshwater fish. Forest and Bird has some interesting examples.
3. Show students Copymaster Two that shows pictures of two introduced species, rainbow trout and perch. Both species are common in New Zealand lakes and rivers.
Imagine a tiny lake that has only four fish, two trout and two perch.
4. Cut out two of each card, fold the cards into quarter so the image is obscured, and put the cards into a paper bag, box, or container (a blue icecream container is ideal).
A scientist catches two fish out of the lake, one after the other. The second fish bits so quickly there is no time to return the first one.
What species might the two fish be?
5. Let students discuss the question. Expect them to say that the situation is like two-child families. But is it?
6. Students should say that three different events are possible:
Two Trout (T1 and T2)           Two Perch (P1 and P2)
Trout-Perch (Note that four outcomes can result in this event, T1 and P1, T1 and P2, T2 and P1, and T2 and P2)
7. Ask: What is the probability of each event?
Students may mimic the probabilities from two-child families (1/4, 1/4 and 1/2).
8. Explain that you are going to test out the situation with a simulation:
• Put two cards of each fish in a bag and randomly select two fish. Make sure the cards are folded in quarters so the draw is random.
• Record your results using a tally chart.
• Replace the two cards, shake the bag, and do it all again.
9. Ask each pair of students to take ten samples of two fish.
10. Collate the data using a spreadsheet and create a pie chart. Are the results what we expected?
Students should note that the proportions are a long way off 1/4, 1/4 and 1/2.
Have we got enough data? Do we need to trial the fish catching again?
100 trials represent a reasonable sample size.
Think about the situation. Can you create a model to explain these results? Discuss the model with your partner. Record your thinking.
11. Let students work on the problem for a suitable period then gather the class and s​​​hare students’ ideas. Various models usually emerge, such as:
• Tree Diagram • Network • Table 12. Be aware of the tendency of some students to regard selection of a trout (or a perch) as a single outcome. There are two fish available so selection of a single trout (or a perch) on the first catch can occur in two ways. Notice that the tree diagram and table include the same combination of two fish twice. The actual number of possible outcomes is twelve if order matters (permutations), and six if order is regarded as unimportant (combinations)
13. Whatever representation student select, draw their attention to the set of all possible outcomes.
How many of those outcomes result in a trout and perch combination? (8 out of 12 or 4 out of 6)
Can we express the probability of a mix, as a number? (2/3 or about 67%)
What are the probabilities of two trout and two perch being caught? (1/6 each)
What do the probabilities add up to? (2/3 + 1/6 + 1/6 =1 or 100%)
How is this situation different from the two-child family situation?
14. Draw out the idea that the outcomes for the second fish are affected by which fish is caught first. In probability we say that the outcomes are dependent, one is affected by the other.
15. Pose the following investigation:
Try changing the fish in the lake using Copymaster Two.
Create different mixes of trout and perch in the lake.
What mix gives a probability of ½ that the two fish caught are the same?
What mix gives a probability of 4/10 that the two fish caught are the same?
Try different mixes and work out the probabilities.
16. Look for students to:
• Investigate different mixes of trout and perch.
• Create models for all the possible outcomes.
• Work out the probabilities of the events (Two trout, two perch, trout and perch).
• Recognise the probability of get a same fish combination are increased if the fraction of one species in the lake is increased, e.g. T1, T2, T3, P1 and P2 gives a 1/2 chance that the two fish are the same.

Session Four

In this session students explore a situation in which the probability of each outcome is not equal. To solve the problem, students need to adjust their method of finding all the possible outcomes to balance the different likelihoods.

1. Begin with the scenario on Slides One and Two of PowerPoint Two.
What fraction of the time are the lights on each colour?
1/2 on red, 1/3 on green, and 1/6 on orange.
2. Progress to Slide Three.
How might we simulate Luciana travelling through two sets of traffic light?
Students might suggest creating a spinner and spinning twice. There are other ways to simulate traffic lights, such as using a standard dice:
1-3 represents a ‘stop’ signal, 4-5 represents a ‘go’ signal, and 6 represents a ‘lights changing’ signal.
Talk about how you will organise the data you get from your simulation trials.
3. Let students discuss data collation in pairs. Most students will suggest a table or tally chart. You might decide on a data collection table similar to this (Copymaster Three): 4. Model setting up the traffic light spinner using Spinner. Note that Orange is regarded as a stop because that is what the law expects.
What events might occur when Luciana goes through two sets of lights?
(Apart from saying she might have a crash, students might suggest that she stops zero, one or two times)
Which event do you think is most likely? Why?
Given that Luciana must stop 4/6 or 2/3 of the time at a given set of lights, students should forecast that there will be a lot of stopping.
5. Provide one recording sheet between two students, and ask your students to carry out 16 trials. Collate the results using a spreadsheet. What do you notice?
(The fractions for two stops and one stop are similar. Theoretically both probabilities are the same at p = 4/9.)
Luciana gets through both sets of lights only 11% of the time. What fraction is that? (11/100 or about 1/9)
6. Discuss: Why do these probabilities happen?
Expect students to conjecture that there is more chance of having to stop at a set of light, so both options, stopping once, or twice, are more common.
7. Present this table to the students (Slide Four of PowerPoint Two): What does this table show?
What does the different shading of areas represent? (Dark grey is the ‘stop at both lights’ area, mid-grey is the ‘stop at one light’ area, and light grey is the ‘no stop’ area)
Which area is greatest? (Dark and mid-grey have the same area, 16/36)
What do the areas tell you about the chances of stopping?
8. If students have difficulty interpreting the areas, use Slide Five that shows the much simpler situation of two child families. Students should recall that there was a 1/2 chance of a mixed gender family, and 1/4 chances of both children being of the same gender. Prior experience will support students to recognise that areas can represent the probabilities.
9. Introduce the problem on Slide Six of PowerPoint Two. Ask students to work on the task in pairs. Roam the room. Look for:
• Do your students recognise the fractions for probabilities? (1/2 orange, 1/4 green, 1/4 red)
• Do your students use diagrams to work out the probabilities of no stops, one stop and two stops?
• Do they conclude that Luciana might be worse off, unless she is able to sneak though some lights on orange?
10. Gather the class and discuss strategies that students used. Slide Seven shows a table of the probabilities. Areas show that Luciana has 1/16 chance of getting through both lights without stopping and 6/16 or 3/8 chance of having to stop at one light. Those fractions are smaller than the 1/9 and 16/36 or 4/9 probabilites for her previous route.

Session Five

In this lesson we conclude the unit with a game that involves probability.

1. Introduce the Horse Race game (Copymaster Four) and have three students play a demonstration game for the class. The gameboard explains the rules. you will need the gameboard, two standard dice, and two counters of one colour each (their racing colour).
2. Put the class into groups of three and have them play five games each.
3. While students are playing, roam the room. Look for the following:
• Do students notice that the distance horses need to gallop is different.? For example, Horse One must travel 11 steps but horse Five only needs to travel three steps.
Does that seem fair?
• Do students notice that differences of zero, one and two occur frequently but differences of three, four and five are less frequent?
• Do they notice that a difference of six is impossible?
4. After five games for each trio gather the class to discuss their observations.
Which horse has the best chance of winning? Why?
Students should comment that the horses that have the furthest to go move more often and the horses with the least distance to go move least often.
Why do some horses move more often than others?
How could we find all the possible outcomes, when two dice are rolled, and we find the difference?
5. Invite students to work out the sample space with a partner, using whatever method they like.
Remember that we have used tree diagrams, tables, and networks before in this unit.
Are those methods useful here too?
6. Allow students some time to develop a theoretical model. Gather the class to share their thoughts.
Is 1 on the first dice and 5 on the second dice the same outcome as 5 on the first dice and 1 on the second dice? (No. They are different outcomes, a bit like Trout 1 and Perch 2 being different to Trout 2 and Perch 1)
How many different outcomes are there? (36 possible outcomes)
7. Show Slides One and Two of PowerPoint Three. The tree diagram shows how all 36 possible outcomes occur.
How many outcomes give the event of Horse Zero moving? (Six, (1,1), (2, 2), (3, 3), (4, 4), (5, 5) and (6, 6).
The table shows the differences produced from the set of outcomes.
What patterns can you see in the table? (Students might note the diagonal arrangement of cells with the same differences)
Which horse has the best chance of moving? How do you know?
Can you find the probability of Horse One moving on a single throw? (10/36 ≈ 28%)
8. Discuss the probability of other horses moving on a single throw.
Is the game fair? Does each horse have an equal chance of winning the race?
9. Students might match up the probability of each horse moving on a single throw and the number of steps the horse needs to win.
 Horse 0 1 2 3 4 5 6 Steps to win 6 11 9 7 5 3 2 Probability of moving on one throw 6/36 10/36 8/36 6/36 4/36 2/36 0/36

The table shows that the distances balance the probabilities well except for Horse Six that has no chance of moving.

10. Develop a Horse Race game in which the dice numbers are either added or multiplied. Make the game as fair as possible for the horses.

Assessment

Evaluate students’ understanding of probability using the multiplication basic facts game called Multi-Bet. You will need two dice labelled 4, 5, 6, 7, 8, 9, counters, and a game board for each group of players. In this scenario the students are placed in the shoes of Risky Betts, the Casino owner, who has to determine the pay outs for the game.

1. Introduce Multi-Bet to the class:
Each player starts with ten counters (their loot!).
They place bets in the following way:

The winning number is determined by tossing the two dice and multiplying the numbers that show (e.g. 4 x 6 = 24).
If the winning number is not in those selected by a player, then the casino takes all the counters.
If the winning number is one of those chosen by some students, then the casino must pay out. How much should the Casino pay out for each type of bet?
The odds must be enticing to the players yet ensure in the long run that the casino makes a profit.
2. Once they have allocated odds such as 2:1, which means that \$2.00 is paid out for every dollar placed, students can trial the game to see how their odds work in practice.
Note that there are twenty products on the board in total so that a bet covering four numbers has a four out of twenty (4/20 = 1/5) chance of being successful. The casino will want to offer odds of less than 5:1 if they are to make money in the long run.
3. As a means of assessing their progress in meeting the achievement objectives for this unit, ask the students to record the reasoning they used to decide how they allocated odds.
Some students may note that there are more ways for some numbers to occur than for others. For example, thirty-six can occur in three ways (4,9), (6,6), and (9,4), whereas forty-nine can only occur in one way (7,7).
4. Has the board been designed to separate numbers that have a higher chance of occurring? Could you design the board better?
5. How do the varying probabilities affect the odds that the Casino should paid out?

Down on the farm

Purpose

The unit is designed as a simple introduction to systems of linear equations. Students solve problems in which they meet two constraints to find a single solution. Both constraints can be expressed as linear equations.

Achievement Objectives
NA4-7: Form and solve simple linear equations.
NA4-9: Use graphs, tables, and rules to describe linear relationships found in number and spatial patterns.
Specific Learning Outcomes
• Identify and find values for variables in context.
• Identify linear relationships in context.
• Represent linear relationships using tables, graphs and simple linear equations.
• Draw strip diagrams to represent linear equations and ratios.
• Solve problems in which two or more conditions must be met.
Description of Mathematics

Systems of equations are extremely useful for modelling the real world. Often two or more conditions exist in a situation that must be satisfied. We can often express conditions using representations such as tables, graphs and equations. These representations provide powerful tools for solving problems.

As an example, consider simultaneously meeting these two conditions:

1. There are twice as many chickens as pigs.
2. The number of pigs and chickens equals 90.

Condition (i) can be represented by the following table, graph and equation. c = 2p or p = c/2 where p represents the number of pigs and c the number of chickens.

Condition (ii) can be represented in the same ways. c + p = 90 or c = 90 – p or p = 90 – c

Satisfying both conditions involves finding values for the number of pigs and number of chickens that work. In table form this involves searching for a common pair in both tables. In graph form this involves finding an intersection of both lines. An algebraic method is to solve the two equations simultaneously. There are different ways to do this. Here is a substitution method:

1. 2p = c
2. c + p = 90

Putting 2p in for c in equation (ii) gives:

2p + p = 90 so 3p = 90 and p = 30.

Since c + p = 90 c must equal 60.

The problems in this unit involve using representations to meet common conditions.

Specific Teaching Points

Representing relations in algebraic equations involves two important and connected types of knowledge, related to the language conventions (semiotics), and to the nature of variables. When we write c = 2p, or c = 90 – p + 2 the equations are meaningless to anyone else unless we clearly define what the variables, c and p, represent. Note that both c and p refer to quantities that vary and are not fixed objects, such as a chicken or a pig. Quantities are a combination of count and measurement unit. In this case c expresses many animals. Animals are the unit in this problem. 2p means the number of pigs multiplied by two, not twenty-something.

Semiotics, the meaning of symbols and signs, is central to algebra. Transfer between semiotic forms is difficult at times. For example, a statement such as “there are twice as many chickens and pigs” seems innocuous and it is easy to generate a table of values that satisfy the statement. However, recording the statement as an algebraic equation requires a student to accept letters as variables, not as objects that can be counted. 2p = c or c = 2p is correct but appears ordinally different to the spoken form. Some spoken languages are more consistent with algebra and would express the relation as “To get the number of chickens multiply the number of pigs by two.”

Working with variables also requires acceptance of lack of closure, that is thinking with symbols (c and p in this case) without specifically knowing the values they hold. For example, knowing that c = 2p can be substituted into c + p = 90 while conserving its structure, irrespective of whatever the value of c or p, is itself a generalisation.

The equals sign represents a statement of ‘transitive balance’ meaning that the balance is conserved if equivalent operations are performed on both sides of the equation. Knowledge of which operations conserve equality and those which disrupt it are important generalisations about the properties of numbers under those operations, e.g. distributive property of multiplication.

Students can be supported through the learning opportunities in this unit by differentiating the nature and complexity of the tasks, and by adapting the contexts. Ways to support students include:

1. providing a physical or diagrammatic model, e.g. cubes or counters or icons, so students can experiment with the conditions in the problems
2. modelling how to record relationships as tables of values, graphs, and equations. Videos of modelling these representations are included in the resources
3. explicitly telling students about the conventions of letter symbols, e.g. 5y means five times an unknown y, with a ‘missing’ multiplication sign
4. encouraging students to work collaboratively and share their ideas
5. encouraging students to check their solutions against the conditions of the problem.

Task can be varied in many ways including:

• Alter the complexity of the problems in two ways. Smaller numbers make trial and improvement and physical/diagrammatic strategies viable. Simplifying the conditions makes problems considerably easier. Total conditions, e.g. 12 animals in total, are easiest and ratio conditions are hardest.
• Allow access to calculators so students can experiment with possible values without the burden of calculation. However, expect them to anticipate the results of their calculation changes before pressing the keys.
• Provide or remove examples of a pattern or relationship before trying to generalise the relationship using variables.

The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Animals on the farm provide the contexts for all problems in the unit and appeal to a range of students. Other contexts can also be used, such as buying items at a shop, e.g. socks at \$3 per pair and hats at \$12 each. Catering at a marae or tulaga fale provides a useful context around managing a budget while still providing ample food for manuhiri (visitors). People in an extended whānau also provides an interesting context, with variables such as the number of adults and children.

Required Resource Materials
Activity

Prior Experience

It is anticipated that students at Level 4 and 5 understand, and are proficient with, multiplicative thinking. Students are expected to know about simple ratios though this unit reintroduces some key ideas. Some proficiency at solving linear equations would be beneficial, so working through the units on linear algebra before this unit would be helpful.

Session One

Begin with this new version of a very old problem. The problem solving pathway in e-ako maths includes a version of this problem called "Sheep and chickens". This might be used as extension for students who enjoy the algebraic equation approach.

 On Mr MacDonald’s farm there are only pigs and chickens. He counts 24 heads and 80 legs. How many of each kind of animal is there?

After discussing important conditions in the problem encourage the students to work in small co-operative groups. Allow access to supportive tools such as calculators and computers. Recording on paper will be important.
Look for the following common approaches:

Trial and improvement
This approach typically involves choosing a pair of possible values for the number of pigs and chickens, and making alterations until the other condition is met. For example, 12 pigs and 12 chickens might be tried. This assumption honours the number of heads condition but 4 x 12 + 2 x 12 = 72. So that pair of values fails the number of legs condition. However, systematic adjustment of increasing the number of pigs and reducing the number of chickens will eventually give the answer. Note that this strategy is very protracted if the numbers involved are large.

Figurative diagram Students using this strategy often begin with numbers of pigs and chickens that satisfy the number of heads condition, though sometimes they draw 48 animals of one kind. Some students draw figures that satisfy the number of legs condition. It is important that the drawings are symbolic and not literal (i.e. not life-like pigs or chickens) for two reasons. Life-like drawings are time consuming and indicate that the farmyard context assumes more importance than the conditions of the problem. A drawing might look like this, with 12 pigs and 12 chickens:
Chickens can easily be turned into pigs by adding two extra legs until a solution is found that meets the number of legs condition. Making a table
Often students’ approach the problem using trial and error (rather than improvement). They try combinations of pigs and chickens in an unsystematic way. A table helps them to organise their data but also allows for noticing of patterns that otherwise are missed.
An example of a table based strategy is given below:

 Number of Pigs Number of chickens Number of pigs’ legs Number of chickens’ legs Total number of legs 0 24 0 48 48 1 23 4 46 50 2 22 8 44 52 … … … … …

The table can be extended until a solution is found. Note that use of a spreadsheet makes this strategy highly efficient.

Equations

This strategy is unusual for students at Level 4 unless they have exposure to writing and solving linear equations (link to first algebra learning object unit). First, students need to identify the variables in this problem. While it makes sense to use p and c as symbols, it is very important that students regard these as variables not fixed objects. P is not a pig nor is c a chicken. P represents possible numbers of pigs and c possible numbers of chickens.

Second, students need to write the conditions using these variables. Conventions are involved here, notably that 4p means 4 x p and 2c means 2 x c, and equals means a state of balance or sameness.

1. p + c = 24 (Number of heads condition)
2. 4p + 2c = 80 (Number of legs condition)

Third, solving for p or c involves trusting that these variables can remain ‘unclosed’ and conserved under a sequence of steps. This is a significant shift from arithmetic thinking which aims to ‘close’ the answer as immediately as possible.

Equation (i) can be reorganised as p = 24 – c or c = 24 – p. Either of these equalities can be substituted into equation (ii) so the equation is in one variable:

4(24 – c) + 2c = 80 or 4p + 2(24 – p) = 80

Discussion

1. After a suitable time gather the class to discuss their solution strategies. Focus mostly on students’ thinking and on the relative efficiency of their strategies. It is unlikely that either graphical or equation based approaches will occur naturally. Video One presents a table and graph based solution, should showing that approach seem worthwhile.
2. Ask students to classify the problem.
What kind of problem is this?
You want students to say that the problem involves variables, numbers of pigs and chickens. It also involves two constraints (restrictions) that must both be met for the problem to be solved.
3. Ask: Think about a similar problem where the numbers are much larger. Which strategy is the best to use? Why?
4. Provide the students with Copymaster 1 that contains variations on the original problem. Increasing the difficulty of the problems makes trial and improvement, diagrammatic and table based strategies less viable, and preferences algebraic methods.

Solutions:

Problem One: 21 pigs and 49 chickens
Problem Two: 78 pigs and 48 chickens
Problem Three: 181 pigs and 138 chickens

The "Sheep and chickens" problem solving e-ako provides guidance to developing a general algebraic solution to the pigs and chicken problem.

Session Two

Introduce the next farmyard problem using slide 2 of the PowerPoint.

 On Young Maree MacDonald’s farm the ratio of pigs to sheep to chickens is 2:3:5. Maree has 640 animals in total. How many of each kind of animal are there?
1. Ask: What does a ratio mean?
2. Look for students to recognise that the ratio is a comparison of numbers that applies to all the animals on Maree’s farm. So, the ratio might be expressed as “For every two pigs there are three sheep and five chickens.” You might need to refer the students back to the PowerPoint slide to answer some of these questions.
• What fraction of the total number of animals are the chickens? (one half) How do you know? (⁵/₁₀ is equivalent to ¹/₂)
• Is it true that one fifth of the animals are pigs? (Yes. ²/₁₀ = ¹/₅)
• Is it true that the number of chickens is two and one half times the number of pigs? (Yes. 2 ¹/₂ x 2 = 5)
• How many times more chickens are there than sheep? (1 ²/₃ x 3 = 5 because ⁵/₃ x 3 = 5)
3. Let the students solve the problem of how many of each animal are on Maree’s farm. Then bring the class together to share strategies. Video Two shows how a strip diagram might be used to represent the problem. You can pause the video at points when students are posed a question.
4. Slide 3 of the PowerPoint provides a variation of Maree’s problem is which the part of the collection of animals is given but not the whole. Ask the students to represent the problem as a strip diagram and solve it. Encourage them to work in small groups.
5. Gather the class together to share their solution strategies. Video Three talks through a solution using strip diagrams.
6. Copymaster 2 has a collection of Maree, the farmer, ratio problems for the students to solve. Students might work collaboratively or independently. Look for them to:
• represent the unknowns and unknowns in strip diagrams.
• use the diagrams to identify and calculate missing parts or totals.
• look for common factors in numbers.

Solutions:

Problem One Problem Two Problem Three Problem Four

This problem is easier if you think of parts consisting of 12 animals, and the whole made 45 parts of 12.

 Animal Number Fraction Goats 72 6/45 = 2/15 Llamas 108 9/45 = 1/5 Pigs 240 24/45 = 4/9 Sheep 96 8/45 Cows 24 2/45

Note: Ratio problems like these are like working with parametric equations. In this case the variables, numbers of each animal, are defined in relation to a total number of animals. For example, if n is the total number of animals, then L = n/5.

Session Three

Show the student today’s starting problem on Slide 4 of the PowerPoint. Ask them to identify the important conditions in the problem:

 Jessica Jones buys 60 animals at the market. She only buys cows and pigs. Cows are \$120 each, three times the price of pigs. The total cost is \$4 800. How many of each animal does she buy?
1. Students should recognise that pigs cost \$40 each.
2. Ask: Can we write the conditions algebraically?
First, the variables need defining. Number of cows might be represented by c or any other letter, and the number of pigs by a different letter, possibly p.
Second, equations can be written for the conditions.
c + p = 60 for the condition of number of animals.
3. Ask: How else could this equality be expressed? E.g. c = 60 – p or p = 60 – c
4. The second condition of cost is more difficult. Students may offer p = 3c to represent the cost relationship between cows and pigs. This is incorrect in two ways. P and c are used to stand for the number of each animal not the price of cows and pigs. So, the variables have changed. The relationship is incorrect as well. One pair where the price relationship holds is \$12 for a cow and \$4 for a pig. Is it true that 4 = 3 x 12, which matches the incorrect equation p = 3c?
5. Remind the students of the total legs condition from the pigs and chickens problem.
How did we express that condition?
So, 120c + 40p = 4800 represents the cost condition.
6. Put the students into small co-operative groups to solve the problem. Allow access to tools such as paper and pens, calculators, and computer spreadsheets. As students work encourage them to look for similarities between Jessica’s problem and the pigs and chicken problems. The same strategies that worked for those problem; trial and improvement, diagrams, tables and algebra; should work on this problem.
7. After a suitable time gather the class to share solutions. Discuss the efficiency of the different methods. Video Four shows a spreadsheet supported solution but other methods may be equally efficient. Students may reason that if all the animals were cows then \$4800 would buy 40 cows. Exchanging one cow for three pigs keeps the cost the same but effectively increases the number of animals by two.  So, ten exchanges of one cow for three pigs (a ratio) reduces the cow number to 30 and increases the pigs number to 30, which is the solution.
8. The algebraic solution is also accessible, given the students’ experience in Lesson One:
c + p = 60 so p = 60 – c (total number constraint)
120c + 40p = 4800 (cost constraint)
Putting (i) into (ii) gives:
120c + 40(60 – c) = 4800
120c + 2400 – 40c= 4800
80c = 2400
C = 30

The second problem for this lesson is available on Slide 5 of the PowerPoint.

 Farmer Fred goes to market. He buys 100 animals for \$100. Lambs cost \$10 each. Piglets cost \$3 each. Chicks cost \$0.50 each (50 cents). How many of each animal does he buy?
1. This is an old problem which may explain the cheap prices. It is challenging because there are only two conditions, total number and cost, but there are three unknowns, the numbers of lambs, piglets and chicks. Discuss with the students how they might simplify the search for solutions. Important observations are:
2. Lambs are very expensive at \$10 each, and their total cost is always a multiple of \$10 (\$10, \$20, \$30,…)
3. Chicks are the cheapest animals and the number of them must be even since two of them cost a whole dollar.
4. Putting those clues together can limit the search.
What is the greatest number of lambs that can be bought?
Ten lambs take up all the cost, nine lambs cost \$90 and you can only buy 20 chicks with the remaining \$10, eight lambs cost \$80 and you can only buy 40 chicks with \$20, etc.
By that thinking, the number of lambs it is possible to buy can be one, two, three, four or five. That makes the problem easier to solve.
5. Let the students work at a solution using the strategies they know from previous problems. Look for:
• Do they systematically work through possible solutions for l = 0, 1, 2, 3, 4, 5?
• Do they use diagrams and equations to support their solution finding?
• Do they use spreadsheets and adjust the variables according to the ‘number of lambs’ condition?
• Do they notice when solutions are impossible? For example, fractional or negative numbers of animals are impossible.
6. Bring the class together after a suitable time to share progress towards a solution. Students can be sent away again to work in groups even if they have no found the solution after sharing.

The solution is five lambs, one piglet and 47 chicks.

Session Four

In this lesson students are encouraged to connect their strategies and knowledge of ratios to solve problems.  Slide 6 of the PowerPoint poses the problem of expressing a ratio as an equation.

 The ratio of pigs to sheep is 1:2. If p = number of pigs and s = number of sheep, write an equation for this relationship.
1. Before asking for suggestions make a table of possible numbers for p and s:
 Number of pigs (p) Number of sheep (s) 1 2 10 20 3 6 5 10 7 14 0 0

Students are likely to suggest two equations, s = 2p or p = 2s. Check to see which of the equations works with the table values. An important idea is that p and s refer to numbers of animals not an individual animal, pig or sheep. So, the equation s = 2p works but seems counter-intuitive with the way the ratio is said, “For every pig there are two sheep” or “There are twice as many sheep as pigs.”

2. Slides Seven and Eight give two other ratios for the students to write as equations.
The ratio of goats to cows is 1:5. So c = 5g is the equation.
The ratio of horses to llamas is 2:3. So 2l = 3h is the equation.
In each case creating a table of values helps to verify the correct equation.
3. Next the students work on Copymaster 3. This worksheet presents several problems in a form where ratio statements provide one of the conditions to satisfy. If students can turn the ratio statements into algebraic equations, they can use the strategies learned previously.

Solutions:

Problem One: 183 pigs
Problem Two: 84 chickens
Problem Three: 54 kids
Problem Four: 27 cows
Problem Five: \$220 for the worth of the goats

Matariki - Level 4

Purpose

This unit explores a variety of mathematics ideas in the context of Matariki. Matariki is a significant event in the New Zealand calendar and the Māori New Year is celebrated in many schools. Matariki is an opportunity to engage in activities such as storytelling, astronomy, song, dance, and visual arts that have potential to enrich students’ mathematical experiences in meaningful contexts. New Year is also a chance to honour our ancestors, show care for our natural environment, and celebrate our bi-cultural and multi-cultural heritage.

Specific Learning Outcomes

Session One

• Recognise the properties of a figure stay constant as the figure is rotated (turned).
• Use compass directions to locate objects.
• Represent the relationship between numeric variables using tables and graphs.

Session Two

• Collect, sort and display multivariate data to find patterns and differences.

Session Three

• Find rules for linear relations, and represent those rules verbally, as equations and in graphs and tables.

Session Four

• Scale measures using percentages to create a kite to a given size.
Description of Mathematics

As this is an integrated unit, it ranges over several important mathematical ideas. Here is a summary of the key ideas.

Rotation is a transformation. A rotation is a turn that can be described as an angle about a given point and a direction of that turn. For example, Figure A has the Matariki cluster of stars in its most easily recognised position. Figure B shows the same cluster turned 90° clockwise. Mathematically we are interested in the features of the figure that stay constant as it is rotated. These features allow us to spot the cluster however it is orientated. Distances between stars (as we see them) stay the same as does their position relative to each other. A trapezium connecting four stars will stay the same shape as the figure rotates.

Sessions one and three deal with relationships between variables. Variables are changeable quantities, for example, as year changes so does the date of Matariki. Associating changes in variables is an important idea in mathematics as it is the foundations of functions. Relationships can be represented in a variety of ways, including tables, graphs and rules. At level 4 students are not expected to generate formal algebraic notation for their rules but many are capable of doings so. For example, a tukutuku panel might grow like this: Each kaho (horizontal rod) has three tuinga (cross-stitches) so the pattern is easy. The data could be organised in a table or a graph. The number of tuinga increases by three for each extra kaho so the relation is linear. The graph show points of a straight line. Rules for the pattern take two forms, recursive and direct or function. Recursive rules tell what is done to one term to get the next, in this case “add three”. A direct rule states how to get the value of one variable from the value of the other, in this case ‘multiply by three’. Direct rules tend to be more powerful than recursive rules though they can be hard to find for some patterns.

Session four involves percentages as operators. That means the percentages are used to scale (shrink or increase) the lengths of a template. Suppose we had a simple template like this. You need to put a mark 60% along the line. The location of the 60% mark is dependent on the length of the whole line (100%). If the space between 0 and 100% is 30cm than the 60% mark is at 30 x 60% = 18cm. If the line is 40cm long the 60% mark is at 24cm. Useful strategies to find a percentage mark are to use 10% as a unit, or find the unit rate, that is what 1% is. 10% is found easily by dividing the length by ten and the unit rate is found by dividing the same length by 100.

This unit is an integrated unit aimed at outcomes for Level 4 of the New Zealand Curriculum. As such, the activities range across the strands. All activities can be adapted to cater for the range in current achievement and interests of students in your class. Ways of differentiating instruction are:

• Varying the complexity of the challenges. For example, keep the rotations of the Marariki cluster to quarter and half turns, or constrain the sorting, and display of data about ancestors.
• Varying the level of abstraction. For example, rotations and graphical display might be carried out physically, leading to anticipation of the result and findings about the context.
• Providing tools to support students. Tools might include technology, such as calculators and computers, flowcharts to organise the process of kite making, or templates for creating a table and graph of the tukutuku patterns.
• Explicitly modeling mathematical processes, such as scaling the dimensions on the kite template.

The context for this unit is relevant to all New Zealand students. You may like to invite a local kaumatua to talk to your class about the significance of Matariki to local iwi.  Look for broadcasts that show celebrations around Aotearoa. Your class is likely to contain students from a range of cultures. Ask students to share how New Year is celebrated in their culture and at what time of the year it occurs. Consider why our calendar New Year happens in the middle of summer, rather than winter, due to importing the calendar from the Northern Hemisphere.

Required Resource Materials
Activity

Prior Experience

This unit is targeted at Level 4 so students are expected to have experience at Level 3 including:

• Creating tables and graphs for simple relations
• Knowledge of common graphs for category and numeric data, e.g. stem and leaf, bar graph
• Measurement with metric units
• Simple percentages

Session One

In this session students investigate some of the mathematics of astronomy associated with the rising of Matariki. They learn to recognise the cluster of stars irrespective of orientation. They also learn where to look for the stars at the beginning of Matariki (25 June in 2017) and how the date of the New Year is determined.

Use the PowerPoint for session one to organise the lesson.

1. Slide one: Why is the star cluster Matariki important in Aotearoa/New Zealand at this time?
Matariki is a cluster that wanders the skies in relation to other star formations. For eleven months of the year it is visible as it wanders. In early May it disappears below the horizon and reappears close to the horizon in late May/Early June. The ‘rising of Matariki’ refers to its appearance above the horizon just before the dawn. That is why it was used as a consistent marker to determine the New Year. So the first new moon following the ‘rising of Matariki’ is when the New Year begins, but celebrations occur in the last quarter of the lunar cycle before.
2. Slides two and three: How will you recognise Matariki when you see it?
There are seven stars in the cluster visible to the naked eye, though another two stars can be seen by some with keen eyesight or with binoculars. As a cluster, the seven stars stay in the same formation like a squadron of stunt pilots. However, the cluster appears facing different directions at different times which can make it hard to spot.
3. Give the students Copymaster 1 which includes the star cluster in different orientations. Ask the students to find a way to know if the cluster they see is Matariki. After all, there are billions of stars in the sky.
4. When they return ask the students what methods they use to organise the way they view the cluster. Typically this involves using shapes, relative position and size of the stars are important characteristics. You might put the students in small groups to share their organisations with the aim of creating the easiest way to recognise Matariki. Get the students to apply their organisations to Slides 3-6.
5. Slide four: This is Matariki rotated 120° (⅓ turn) clockwise from the arrangement on Slide 2. Clicking the animation rotates the cluster back to the more easily recognised position. A good question is why the Slide 2 position is easy to recognise. For example, the two right hand stars Tupu-ā-rangi and Tupu-ā-nuku are aligned vertically. Students will have other reasons.
6. Slide five: This is Matariki rotated 210° clockwise or 150° anti-clockwise. An important discussion point is the similarity of effect for these two rotations.
7. Slide six: This is not Matariki as the relative location of Waitā has been changed (moved to the right). If the students have used shapes to organise the cluster then the shift will be obvious.
8. Slide seven: This is a reflection of Matariki so is not how the cluster can appear. Discuss why a reflection (flip) is not a valid transformation while a rotation is.
9. Slide eight: This shows one way to organise Matariki using a rhombus and an isosceles triangles. Share the other ways students used to organise the cluster geometrically. Mention that using figures to describe the arrangement of star clusters is common across cultures.
10. Slide nine: Here you will find instructions about locating Matariki in the sky as it rises an hour or so prior to dawn in the North-Eastern sky. Give you students a Google Map of your local area so they can identify some landmarks to look for to identify East and North-East or let them use the app if you have technology available.
11. Celebrations for Matariki, the New Year, occur after, but not on, the day of the first New Moon after the rising of Matariki in the dawn sky (Late May/Early June). Find out what date Matariki is celebrated this year. You may want to use TimeAndDate.com to help.
The site will show the quarters of the lunar cycle like this: 12. Ask the students, “How many days are in a lunar month?” A lunar month is 29.5 days (New moon to new moon) so it does not match the calendar months of 28-31 days we use. Ask, “So how long does each quarter take?” (About 7 days)
13. You might ask your students to act out the rotation of the moon around the Earth, using a ball and an OHP projector or torch. Acting out the rotation will show if your students understand why the phases of the moon occur. There are videos available online that also demonstrate this. Just search for "Phases of the moon video".
14. Ask the students to find the date of the first full moon in June. You might ask students to find the dates of Matariki for the next few years. TimeAndDate.com is a useful resource for this.
15. Ask: Is there a pattern to the dates? Why do the dates vary when the date of Matariki rising is consistent? (The lunar calendar has a different month to the calendar we use).
You might ask the students to graph the relations, preferably using Excel or another app. The graph reveals a cycle like this that might predict future and previous dates for Matariki. The lines in this graph show the cycle even though the points are discrete (individual). The cycle is erratic, unlike a tide timetable or the pattern of seasons, because the timing of Matariki is dependent on two different patterns. The star cluster varies in the date it first rises each year, and the lunar cycle of 29.5 days does not match our calendar months. Session Two

In this session students investigate the significance of Matariki as a time of remembering ancestors. Ask students to choose an ancestor of their own who has passed in the last year or select a famous New Zealander from an online database. They look at data about the deceased, particularly the nature of the contribution the ancestor has made to the lives of others.

1. Play the section from 31:57 to 35:24 in this inspiring lecture by Dr Rangi Mātāmua about Matariki. https://www.youtube.com/watch?v=hfuEkqz8v3k
Dr Mātāmua shows how the star Matariki is at the bow of the great canoe Te Waka o Rangi. The rising of Matariki signals a time of letting go of the dead from the year before so their souls can be gathered in the trawling net by Taramainuku who casts them into the heavens. In that way our ancestors become stars.
2. Begin with the Figure It Out task “Family Trees” from Figure It Out, Link, Number, Book Four, Family Trees, page 13.
3. Use the PowerPoint for session two to organise the remainder of the lesson.
4. Slide one: Ask, “What is this diagram about?” Some students may know that it shows some of the children of Ranginui and Papatūānuku, the first Māori ancestors. Ask, “What is meant by whakapapa? Why is whakapapa important?”
5. Slide two: This is a picture of Tāne Mahuta, the giant kauri tree in the Waipoua Forest, in Northland (Tai Tokerau). It is named after one of the children, Tāne-mahuta, God of the forest.
6. Slide three: Ask, “What does this picture show?” The diagram shows a bloodline going backwards in generations from a single offspring.
Ask, “Why does it say biological whakapapa?” Children do not always live with their parents and sometime adults find new partners. So the problem has been simplified. Obviously most children have brothers and sisters, and cousins. Great Aunts and Uncles are also referred to as tīpuna.
7. Tell the students to work in small groups to solve the problem on Slide Three. Allow them access to calculators and computer spreadsheets if they are accessible. Look for your students to organise the calculations. A table like this is useful. Encourage students to use formulae rather than repeatedly copying sequences. Some examples of formulae that can be filled down are shown. Formulae allow students to fill down the table for as many generations as they want. So the total number of ancestors in a biological whakapapa is 1023 after ten generations.
8. Ask: Is the second column of numbers familiar? What set of numbers is that?
These numbers are powers of two and can be written in index notation, e.g. 32 = 25. Note that 25 can be written as 2 x 2 x 2 x 2 x 2 (two multiplied by itself five times).
9. Ask: Is there a quick way to find the value of Column C is you know the value of Column D?
Students might notice that the Column C numbers are one less than double Column B, e.g. 1023 = 2 x 512 – 1.
10. The final question asks how many biological whakapapa make up the 100 Billion stars in the Milky Way. Students might interpret this in at least two ways:
• How many generations down the table would it take to reach 100 000 000 000?
The result is surprising as it only takes only 37 generations to get 137,438,953, 471 ancestors. Note that you will need to custom format the cells to take large numbers before you fill down the table columns. If each generation is 25 years apart then there are 4 generations in each century.
• How many centuries equal 37 generations? (37 ÷ 4 = 9.25)
• How many whakapapa of 1023 would make 100 000 000 000?
​1023 is about 1000 so 100 000 000 000 ÷ 1023 ≈ 100 000 000 (one hundred million)
11. Remind the students that Matariki is a time to acknowledge the dead prior to the beginning of the New Year. According to custom, as their names are read out at dawn the souls of the deceased are cast into the heavens to become stars.
12. Invite the students to choose a deceased person to acknowledge. The person might be a member of their whānau (family), someone they knew, or a famous New Zealander who passed recently.
13. Allow students to develop a short acknowledgement of the person, researching on the web if they need to. That acknowledgement should include the contribution the person made, their gender, their location, and possibly their age of death. You might choose to undertake a data based investigation. Students could create data cards about their person for easy sorting: 14. Ask: How might we group the reasons we chose our people?
Students should create categories like family, sport, arts, leadership, business, education, to sort the people into. The person might belong in several categories. For example, Henare might have been a politician as well as a leader.
15. As a class you might create data displays and look for patterns in the data.
• We tend to value our whānau and sports people most.
• Most of our people lived beyond 60 years.
16. To conclude the lesson, show the students the story of the creation of sky and earth, beautifully told by Beth Te Aro (see YouTube links below). According to the legend Tāwhirimātea, God of Wind, was so angered by the actions of his siblings that he pulled out his eyes and threw them into the heavens. Another version says that he threw his tears. His eyes or tears formed the stars of Matariki. For this reason Matariki translates to “The eyes of God” or “Little eyes.”

Session Three

In this session students follow the connection of Matariki as a time to honour the dead and the responsibility of the living to strive for excellence. Matariki occurs in the middle of winter. Traditionally this was a time when adequate food was stored and whānau engaged in cultural pursuits like story-telling, games, creating art works, and singing. So it is appropriate for students to learn about the mathematics of tukutuku panels that adorn the wall of wharenui (meeting houses) of marae. Students look at a traditional design called kaokao. Use the PowerPoint for session three to organise the lesson.

1. Slide one: This kaokao pattern symbolises the strength of the warrior.
2. Slide two: This slide introduces the two components of some tukutuku:These two components form the variables in the pattern when you consider it from an algebraic perspective. The kaokao pattern has reflective symmetry which can also help students to solve the difficult challenges that lie ahead.
• horizontal wooden rods or laths, usually coloured red or black, called kaho;
• cross stitches, made from a variety of coloured fibres, called tuinga.
3. Slide three: This slide introduces the task, completion of which will enable your students to increase their knowledge. Provide students with Copymaster 2 so they can draw over the pattern in an effort to find an easy way to calculate the number of tuinga.
4. Once the task is introduced allow students to work collaboratively in small groups. Look for these things as your students work:
• Do they partition the kaokao into useful ‘chunks’ that mean one-by-one counting is not needed?
• Do they use tables to organise the data they get about the pattern? • Do they notice patterns in the table?, e.g. Four more tuinga are added for each kaho
• Do they use equations to record the way they partition the kaokao into ‘chunks’? , e.g. 3 x 4 + 2 tuinga for 5 kaho.
5. Your ākonga (students) are likely to need a significant time to investigate the pattern. Encourage them to find different ways to ‘see’ the kaokao, that is partition it into useful ‘chunks’. After an appropriate time, bring your ākonga together to share strategies they used to solve the problem.
6. Look at sophistication and efficiency. A strategy that involves adding fours may work for 18 kaho but it will be very cumbersome for 100 kaho. A multiplicative strategy like, “I take two off the number of kaho, multiply that by four then add two”, is much more efficient. A student who invents that would be further up the kaokao than a student who adds. Encourage your ākonga to connect their symbolic recording to the ‘chunks’ they ‘see’. Here is an example, for the multiplicative rule above: 7. Ask students who are presenting to use tables to organise the data. Rather than using recursive (additive) rules encourage them to look for direct (multiplicative) rules as shown below. 8. You might choose to record the general rules in words or equations, such as:
"Take two off the number of kaho, multiply the answer by four, then add two” can be written as , where t is the number of tuinga and a is the number of kaho.
9. Slides 4-7 are aimed at your ākonga ‘seeing’ the kaokao in different ways to create different general rules. These ways of seeing may have already been used by your students.
10. Slide four: Whetu notices groupings of two. She could record: 11. Her challenge is to write the multiplier of two in terms of the kaho number. Notice that 3 is the 2nd odd number, 5 is the 3rd odd number, 7 is the 4th odd number. So for 18 kaho she would find the 17th odd number and multiply that number by two. The 17th odd number is 33 so the number of tuinga is 33 x 2 = 66. She could write her formula as 2[2k-3]=t since 2k-3 gives the k-1th odd number. This formula is quite complex but some ākonga may find it.
12. Slide five: Rawiri sees a collection of six tuinga at the start of the kaokao with three kaho. He notices that four tuinga are added for each new kaho. For 18 kaho he will know that 18 – 3 = 15 sets of four tuinga will be added to the original six. So he would calculate 15 x 4 + 6 = 66. In general is rule is “Take three off the number of kaho, multiply the answer by four then add six” or 4(k-3)+6=t.
13. Slide six: Anikiwa sees two halves to the kaokao and she joins the halves together. She notices that two tuinga overlap. If she created a table for each half of the kaokao it would look like this: 14. If she doubles each ‘half’ (for left and right) and remembers to take off the overlapping two tuinga she will get this table. In general her rule is, “Take one of the kaho number and multiply it be two. Multiply that answer by two then subtract two” or 2[2(k-1)]-2=t.
15. Slide seven: Kahu notices that there are three lots of four in the kaokao for five kaho. So he adds tuinga so there are five lines of four. Five multiplied by four is easy to calculate (5 x 4 = 20) then he takes away the six tuinga he added.
For 18 kaho Kahu will make 18 x 4 = 72 tuinga, then subtract six to get 66 tuinga. In general his rule is “Multiply the kaho number by four then subtract six” or 4k-6=t.
16. Slide eight: The final slide for the lesson gives your ākonga a chance to apply their ‘ways of seeing’ to a new kaokao pattern. In general there are six tuinga per kaho except for the four tuinga that are attached to the top two. You might ask your students to work on this task independently before sharing with others.

Session Four

Matariki was traditionally a time when kites were flown. Some iwi believe that flying kites helps us to get closer to our ancestors whose souls are embodied as stars in the sky. In previous times kites were made from everyday materials, toetoe, raupō and harakeke (flax). 