All lessons in this unit follow the same sequence of phases as given in the diagram above. A poster of the phases is provided as Copymaster One for students to refer to. The notes suggest possible student ideas and teacher reactions to those responses. It is not feasible to anticipate all ideas students might give so you are encouraged to be flexible in how you respond to students rather than ‘teach’ the sample ideas and representations provided.

PowerPoint One gives the equation sets that drive the unit. The sets are labelled in the top left corner of each slide for reference. Slide one has the first pattern to look at. The pattern involves the addition of fractions with the same denominator i.e. *1/**b **+ (**b-1)/**b = **1* which is relatively easy for students to represent. It is used as an example to familiarise students with the approach. Be aware that students’ existing schemes for addition of fractions may be incorrect, e.g. they may simply add across the numerators and denominators:*1/**4 **+ **3/**4 **= **4/**8*.

**Equation Pairs Set One**

**Noticing Consistency**

Use ‘think, pair, share’ by inviting students to look independently at the four examples, work out the missing values, then share their ideas with a partner. In the class discussion expect students to express their observations in ways that are clear to others. Students should re-express their ideas if others do not understand what they are saying. You may need to remind students that the ‘something going on’ relates to all four examples, not just one. Expect responses like:

*S: The first fraction always has one on the top line.*

*T: Can you tell us more? What is the denominator, the bottom number?*

*S: The bottom numbers, denominators are different in each equation.*

Discussion opens the possibility of using correct mathematical terms, like numerator (number above the vinculum), and denominator (number below the vinculum).

*S: The sum (answer) is always one.*

*T: All four equations have a sum of one? What about the top right equation?*

*S: The sum is five fifths. Five fifths make one whole.*

**Making a claim**

Encourage students to state a claim about what is going on with all four examples in Pattern One. They might do so individually at first then work in a small team to refine their ideas and the way they express those ideas.

Expect ideas like:

*If the denominators of the fractions are the same, and the numerators add to the denominator, the sum is one.*
*If the two numerators add to the denominator, then the sum is one.*

Your aim is for students to express their claims in clear, minimal terms, using correct mathematical language. For example, ‘the top numbers’ is not as clear as ‘the numerators’, and ‘add to’ is not as precise as ‘the sum of’.

**Representation**

In this phase students choose representations to show why the pattern holds consistently. Students might choose physical manipulatives, such as fraction strips or circles, draw diagrams such as strips and circles, and use contexts from everyday life, such as sharing cakes. Encourage students to begin with the first two examples of equations then consider how the same relationships might generalise to the last, and other similar, equation pairs.

Examples might be:

*I made **1/**2 **+ **1/**2 **= 1** using fraction strips then I made** 1**/**5 **+ **4/**5 **= 5**/**5** . Five fifths make one, so I put a one strip below.*

*S: I used fraction circles to show that **1/**8 **+ **7/**8 **= 1 **and **1/**3 **+ **2/**3 **= 1**.*

Some representations are less helpful than others in terms of understanding the structure of the fractions that add to one. Circles are useful for showing that a one is complete. Strips possibly make the iteration of unit fractions more obvious. It is important for students to recognise that the denominator in each example defines the size of the equal parts that one is cut into. For example, eighths are created by the equi-partitioning of one into eight parts, so eight of those parts make one. Your questioning is important to support students to connect the symbols and other representations.

*Explain where the eighths are in your representation.*
*How many eighths are needed to make one?*
*Where is the seven in **7/8 *represented in your diagram/model?
*How does your representation show that the sum is always one?*

**Argumentation**

In this phase students are asked to formalise their noticing by creating a statement that generalises to all cases. The discussion may start with a specific equation pair but must be amended so that it accurately represents the general pattern.

*S: With **1/**8 **+ **7/**8 **= 1** the numerators add to eight. Eight eighths equal one.*

*T: So how does that work in the same way with the other equations?*

This might lead to expressing the relationship in general terms.

*S: The first numerator plus the second numerator add up to the denominator.*

*T: If we gave names to the numerator and denominator of the first fraction, like a and b, could we express the property more simply?*

Some students might experiment with algebraic notation such as *a/**b **+ c/b **= 1. *Note that this statement is incomplete since it misses the necessary condition for a + c = b.

*T: Do we need to say something about a and c? *

*S: We could just write a + c = b.*

*T: Give us an example of that.*

*S: If b = 3, the numerators 1 and 2 add to 3.*

Though it is difficult for this level, increased sophistication is evident in a + c = b, and writing the generalisation with only two variables, *a/**b**+ (**b-1)/**b **= 1.* It is also important to note that *b/**b **= 1*, for any whole number value of b except zero. *b* can also be any integer, since rational numbers are of the form *a/**b**, a and b∈ I*, that is a and b must be integers. High achievers should be challenged to express their generalisations using algebra. You may need to support them to develop the coding strategies to do that.

**Equation Pairs Set Two**

If time permits in the first lesson, ask the students to approach the second equation set more independently. From this point each equation set is discussed succinctly using the phases of the approach.

**Noticing consistency**

The four equations apply the addition of fractions with the same denominator. Students need to recognise that the denominators in both addends and the in sum are the same. Watch for students incorrectly applying whole number addition, that is, adding both numerators together and adding both denominators together. The completed equations should be:

Note that ten eighths equals 1 2/8 = 1 1/4

**Making a claim**

In natural language expect the students to use phrases like “you just add the top numbers”. Introduce important vocabulary such as *numerator, denominator, addend,* and *sum* to clarify which numbers are being referred to in the claims. If the claim is restricted to one example, encourage students to see what changes and what stays constant across the other three equations. The aim is to broaden the claim to the equivalent of “If the denominator of both addends is the same, then the numerator of the sum is the sum of the numerators of the addends.”

**Representation**

Expect both physical and diagrammatic representations to be used. A fraction strip representation might look like this:

Diagrams of a ‘sets’ representation can be used but the changing whole can be confusing, since the aim is to generalise a relationship for all for examples.

Whatever representation is chosen by students they need to be clear about the referent one, be it a length, area, or set. The sum must be defined in terms of that original one whole.

**Argumentation**

Look for students to justify that if the denominators of two addends are the same then the sum is the sum of the numerators, over that denominator. Algebraically that is represented as:

*a/**b **+ **c/**b = (**a+c)/**b*

Note that a, b, and c are variables, and the generalised argument becomes clearer, and is confirmed, by translating values from the equations back into the general form.

For example, with *3/**8 **+ **7/**8 **= **10/**8*, a = 3, b = 8, and c = 7.

An interesting discussion point is whether this is any different to addition of whole numbers. For example, 3 + 7 = 10 is the combining of 3 ones and 7 ones, making 10 ones. The ‘ones’ are regarded as the same unit. With *3/**8 **+ **7/**8 **= **10/**8*, eighths are the units being combined.

**Developing the process further**

PowerPoint One contains seven equation sets. The sets might form the basis of a week-long unit. The phases for each equation set are described succinctly below.

**Equations Set Three**

**Noticing consistency**

The four equations both demonstrate the multiplication of unit fractions, that is fractions with one as the numerator. Understanding in general that *1/**b **× **1/**d **= **1/**bd*, e.g. *1/**3 **× **1/**4 **= **1/**12*, is fundamental for understanding equivalent fractions.

**Making a claim**

In natural language expect the students to use phrases like “multiply the two denominators and you get the denominator of the answer”. Expect the use of mathematical vocabulary such as *factor* and *product* to clarify which numbers are being referred to in the claims. Encourage clarity by asking questions like:

*Do the denominators of the factors need to be the same?*
*What can you say about the numerators?*

The aim is to state the claim as something like “The product of two unit-fractions is a unit fraction, with a denominator that is the product of the denominators.”

**Representation**

Expect both physical and diagrammatic representations to be used. Fraction strips are static representations:

Folding a paper strip or a rectangle is a dynamic representation:

A sets model may be used but the product needs to be known to establish the possible numbers of objects in the set. Therefore, identifying the referent one is more complex.

**Argumentation**

Look for students to state the condition that both addends are unit fractions. The starting product might be expressed as *1/**b **× **1/**d*. The product is *1/**bd*. Students should explain why the relationship occurs. Their explanation needs not be in formal algebraic terms, with letters as variables.

The second factor gives the first equal partitioning and creates some parts. The denominator gives the number of equal parts. Then each part is split into equal parts. That number of parts is given by the denominator of the first factor. Therefore, each of the d parts, is equally partitioned into b parts, so the total number of parts equals b x d.

**Equation Pairs Set Four**

**Noticing consistency**

The four equations apply equivalence of improper fractions. The equivalent number is kept at three to limit variation.

**Making a claim**

In natural language expect the students to use phrases like “the answer is always three.” Some students might indicate what they see in a diagram.

Expect the use of mathematical vocabulary such as *denominator, numerator,* *fraction, equivalent, *and* whole number* to clarify which numbers are being referred to in the claims. Encourage clarity by asking questions like:

*What does the equals sign indicate? (both numbers represent the same amount)*
*How is the numerator related to the denominator? (the numerator is the denominator multiplied by three)*
*Why is the number on the right side always three? (For example, with 24 eighths you can make three ones because you need eight eighths to make one.)*

The aim is to state the claim as something like “A fraction that has a denominator that divides into the numerator three times, is equal to three,” or “A fraction that has a numerator that is three times the denominator, is equal to three,”

**Representation**

Fraction circles are useful in demonstrating the completeness of a one.

Eight eighths make one. With 24 eighths you can make 24 ÷ 8 = 3 ones.

Students may develop more schematic ways to represent the connection between numerator and denominator.

Five fifths make one so 15 fifths must make three, since 3 x 5 =15

**Argumentation**

Look for students to justify, using words of diagrams, that the denominator of a fraction specifies the number of equal parts that make one. For example, in *12/**4 *the parts are quarters, four of them are needed to make one. Dividing the numerator by the denominator finds how many ones can be made. With *12/**4*, 12 ÷ 4 = 3, so three ones can be made. The equation *12/**4**=3* represents that equivalence.

For all four equations define the denominator of the first fraction as the variable, say b. The observation that the numerator is always the denominator multiplied by three can be represented as *3b/**b*. The equivalence can be written as *3b/**b**=3*.

Argument may be supported by graphics or other representations.

Students might also generalise that dividing the numerator by the denominator tells how many ones can be made, and the remainder becomes the fraction part of the mixed number, e.g. *13/**5 **= 2 **3/**5*.

**Equation Pairs Set Five**

**Noticing consistency**

The equation set applies equivalence, that is when two fractions represent the same value. Understanding equivalence is essential to ordering fractions by size and to the addition and subtraction of fractions. The completed sets should be:

**Making a claim**

In natural language expect the students to use phrases like “If the numerator is three times the other numerator, then the denominator must be three times as well.” Students tend to notice patterns across the equation like this:

Expect the use of mathematical vocabulary associated with fractions, particularly *numerator* and *denominator*. Draw students’ attention to the equals sign as representing the same value on both sides. Students are most likely to see equals as a signal to work out the answer. Encourage clarity by asking questions like:

*What is the same about all four equations?*
*What is different?*
*How are the numerators related? *
*How are the denominators related?*

The aim is to state the claim as something like “Two fractions are equivalent, if the one numerator is a multiple of the other, and the denominator is the same multiple of the other.”

**Representation**

Static representations might involve length, area (circles or rectangles), or sets.

A more dynamic representation is to fold a paper strip or rectangle into thirds, shade two thirds, then fold the thirds in half. How many sixths are shaded?

**Argumentation**

Look for students to justify, using words of diagrams, that the same multiplication is performed on both the numerator and denominator of a fraction to create an equivalent (same value) fraction. For example, if the numerator and denominator of *2/**3 *are multiplied by six this gives *12/**18,* and *2/**3 **= **12/**18*. Of interest is that the multiplier *6/**6 *is another name for one, the identity element for multiplication.

However, the multiplication means different things for the numerator and denominator. Multiplying the numerator by a number increases the number of parts by that factor. For example, *12/**18* has six times as many parts as *2/**3*. For the denominator the effect is the inverse. Each part in the new fraction is one sixth that of the first fraction, since six times as many of those parts make one. For example, *12/**18* has parts that are one sixth the size of the parts in *2/**3*.

In general, the transformation *a/**b **× **n/**n **= **an/**bn*, gives n times as many parts but each part is one nth the size. *a/**b** * and *an/**bn* represent the same quantity. Note that *n/n* is always equal to one.

**Equation Pairs Set Six**

**Noticing consistency**

The equation set highlights the difference of unit fractions. Note that the denominator of the first fraction is one less than that of the second. This equation set provides an opportunity to apply equivalence. The completed sets should be:

**Making a claim**

In natural language expect the students to use phrases like “the difference has a denominator that is the product of the two other denominators.” Ask students to be more specific in their description.

*What can you say about the two fractions on the left of each equation? (They are both unit fractions and there is a difference of one in their denominators)*
*What does subtraction mean in these equations? (It means "What is the difference between the two fractions? ")*
*Can you make up another equation that would go in this set?*

The aim is to state the claim as something like “If two unit-fractions have denominators that are one different, the difference of the fractions is the unit fraction with the product of the denominators as it’s denominator.”

**Representation**

Students are likely to use specific examples to convince others about how the relationships work. Fraction strips are probably the most useful representation.

It is important for students to consider why the fraction one twelfth fits exactly.

**Argumentation**

Students need to apply equivalence to argue why a given unit fraction is the difference. Their argument might begin for a specific example.

Since one half and one third are involved, the difference must involve sixths, as both one half and one third can be renamed as ‘so many’ sixths. *1/**2 **= **3/**6* and *1/**3 **= **2/**6* so there is a difference of one sixth.

Encourage the students to generalise by naming the denominator of the first fraction as a variable again.

*Can you find a relationship that applies to all four equations?*

If the first fraction is *1/**b*, the second is *1/(**b+1)*, and the product is *1/(**b(b+1))*. Algebraic notation, like this, is not expected at Level Four, but students may invent their own ways to express what they notice. Whatever notation students create, it can be checked by substituting numbers in place of symbols, and seeing if the equations created belong in the set.

**Equation Pairs Set Seven**

Use set seven as an opportunity to see how well students engage in the generalisation process independently. Ask them to record their claims, representations, and arguments in ways that work for them. Some students may prefer writing their work while others may prefer to capture their ideas using a digital recording.

**Noticing consistency**

The equation set highlights the multiplication of fractions, that is a fraction of a fraction. The numerators of the factors are the same, as are the denominators. That is done to reduce variation. The completed sets should be:

**Making a claim**

In natural language expect the students to use phrases like “It is a fraction multiplied by itself. The product is the two numerators multiplied over the two denominators multiplied.” Ask students to be more specific in their description, through questions like:

*What do you notice about the two factors (fractions being multiplied)? (they are the same)*
*Do you know a mathematical term for a number multiplied by itself? (square of the number)*
*Can you use the term square to describe the product fraction in a simpler way?*

The claim, using mathematical language, might be something like, “If a fraction is squared, the product is the numerator squared over the denominator squared.”

**Representation**

An area model is useful since it illustrates the relationship between a number multiplied by itself and a square. Folding and shading squares of paper works well.

Students might also use strip models, but the squaring of numerators and denominators is not obvious.

**Argumentation**

The equations include two variables, that is the value of numerator and denominator in the fraction being squared. Students need to recognise that the denominator names the number of partitions of one. The numerator names the count of those parts.

Students may be able to show how they are thinking with variables (lack of closure), by treating the numerator and denominator as variables in an area diagram.

Algebraically this might be written as *a/**b **× **a/**b **= (**a×a)/(**b×b)* or more succinctly as (*a/b)*^{2} *= **a*^{2}/*b*^{2}.

## Time zones

This unit introduces and explores 24-hour time and time zones.

In this unit students learn that clock time varies around the world. For example, the same moment in time is read as 7:35pm in New Zealand, 5:35pm in Melbourne, Australia, and 6:35am in England, since in summer NZ is on daylight saving time. In fact, what to call a moment in time is arbitrary so nations adopt calibrations (setting of clock times) to suit the lifestyles of citizens. The Earth rotates 360⁰ every 24 hours (roughly) meaning that while it is daylight on one side of the Earth, it is night on the other side, excluding locations close to the North and South poles.

It is generally accepted that 12 noon is midday, halfway between dawn and dusk, the rising and setting of the sun. Therefore, midday occurs at different time points around the globe, based on the longitude of the location. The clocks of time zones are calibrated so that 12 noon is at the midpoint of dawn and dusk in that country. Large countries have multiple time zones, that is points at which the clock reads midday, e.g. Australia has three time zones, Russia has 11 time zones.

With the advance of communication technology and air travel students’ lives are likely to be affected by time zones. Suppose you book a flight from Auckland to Perth. It leaves at 11:10am and arrives at 1:35pm on the same day. That seems like a flight time of only 2 hours and 25 minutes. It is hardly worth buying the entertainment and meal package. It turns out that the flight takes 7 hours and 25 minutes. That is because Auckland time is five hours ahead of Perth time. When you leave Auckland, the time in Perth is 6:10am.

Communicating by email, Skype or phone with someone in another part of the world requires you to know the time zones. England is 13 hours behind New Zealand time. If you Skype a friend at 7:00pm New Zealand time you will wake them up at 6:00am England time.

The International Date line is another complication. The line lies longitudinally between New Zealand and Samoa. It is the line created to separate one calendar day from the next. When it is Wednesday in New Zealand it is Tuesday in Samoa. It is possible to fly three hours from Auckland to Apia and arrive 20 hours before you left.

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

Differentiation is about altering activities to meet the interests of your students to enhance engagement. Since most activities are about international and domestic travel it is easy to choose contexts relevant to your students. For example, students might be more engaged by planning a holiday to Samoa, India, or Korea, than the United Kingdom. Students in your class might have shown an interest in a location, such as Disneyworld in Los Angeles. It is important that you use this unit to further students’ world view as well, and explore locations they may be less familiar with.

Session 1In the first session we introduce the idea of time zones using a globe and by making a phone or internet call.

Imagine that globe is the Earth and the torch is the sun. How do these bodies move in relation to one another?Students should suggest that the Earth orbits the sun.

How long does that take?(a year or 365¼ days)They should also know that the Earth rotates on its axis.

How long does a full turn take?(24 hours or one day)Discuss what you can see with a classmate. What does this mean for people and animals living on the Earth?Out of the discussion students should express the idea that when we are in daylight there are some places in the dark. All places, except the polar regions in summer and winter, undergo day and night in a 24-hour period. You may need to fix a blob of blutac on the globe at the location of your school, and another blob at a location in another country.

What time is it in each place?Stop the rotation in a position when the sun is shining directly at New Zealand.

What time will this be in Aotearoa?(12 noon unless it is daylight saving. With DLS the time will be 1:00pm)Think about the time in these locations:Sydney (two hours earlier)Samoa (one hour ahead which becomes 23 hours behind due to the International Date Line)Shanghai (5 hours behind)London (12 hours behind)New York (6 hours ahead which becomes 18 hours behind due to the International Date Line).time zonesTalk to a classmate about what you notice on this map of the world. Be ready to explain what you see to everyone else.Have you ever tried phoning or Skyping someone in Australia? It’s [time] at our place. They live in [location]. What will the time be over there?How many time zones do you think Australia has?(three)Let students use the map.

Why does Australia have three time zones?(It is a big country from West to East).How many time zones does Russia have?(11 zones)Session 2In this session students learn to express am and pm times in 24-hour time. They use 24-hour time make calculations about time difference between two different locations.

What does a time of 18:45 mean?Some students may know that the time is given in 24-hour time, sometimes referred to as military time.

Why is 6:45pm expressed as 18:45? (It is 18 hours and 45 minutes from the start of the day at Midnight.)Session 3In this session the class practise working with 24-hour time in the context of the commuter train that runs from Waikanae to Wellington. Use of 24-hour time for public transport is quite rare in New Zealand but is common in Asia and Europe.

What does it mean to commute to work?Why might people use public transport to commute rather than drive their car?(Cost, time, and minimising carbon footprint are important reasons)If you were Henare and Leisha what would be the best time to start work? Explain why you would start then.How much time does it take Henare and Leisha to train from Waikanae to Wellington?(Note that there are two services; normal and express)How much longer does it take the normal train than the express train?(3 minutes)Which trains are likely to be the most packed? Why?(Trains arriving in Wellington between 7:45am and 9:00am are likely to be popular)Which train would you advise Henare and Lisa to take? Why?Session 4In this session students investigate departure and arrival times of international flights. Students are given two of three conditions and asked to find the third. Departure time, flight time or arrival time might be omitted. In solving the problems students need to compare time zones using 24-hour time and make calculations with time in hour and minutes.

What is the time difference between Brisbane and Singapore?What do you notice about the clocks at each location as the flight proceeds?From the departure times in each location how could you calculate the flight time?If you are told the flight time how can you calculate the arrival time in the country that you land in?Session 5In this session the class practice using timetables and their knowledge of time zones to plan a trip to London from where they live.

You are going to have a three-week holiday in the United Kingdom (UK). I want you to plan the trip. Both your parents work, so you cannot leave before December 6th and you cannot return later than December 26th. Plan the transport that will give you the longest possible time in the UK. Allow for travel time, stop overs, and minimalising the overall cost. Use the internet to help you but please don’t book any flights by mistake?Family and whānau,

This week we have been working with 24-hour (military) time, discovering how time zones work and why the zones are needed. We have calculated times in other parts of the world when given New Zealand time, and found out how to work out arrival times for long international flights.

Some links from the Figure It Out series which you may find useful are:

## X Marks the Spot

This unit uses games and puzzles to explore co-ordinate systems and their uses. The students are given the opportunity to invent their own systems by locating counters on a sheet of paper and by finding the dead centre of a set of objects. The Cartesian and polar co-ordinate systems are used to locate position on maps. Polar co-ordinates are also used to describe certain loci (sets of points with a common property).

This unit emphasises the value of co-ordinates to locate the position of objects in the plane. Two co-ordinate systems are presented. One is the Cartesian system that uses horizontal and vertical distances from a fixed point (the origin). The other is the polar co-ordinate system that uses angles about a fixed line and distances from a fixed point on that line.

Students may well have seen the Cartesian system before as it is the one commonly used on road maps, and maps generally. It was invented by a French mathematician called René Descartes. You’ll read his story on the way through the unit. The origin of the polar co-ordinate system is less clear. Both systems describe some complicated curves simply though the unit does not go into functions at higher levels of the Curriculum. Students are likely to see polar co-ordinates as an interesting curiosity at this point.However, Cartesian co-ordinates provide the basis of a great deal of work that is undertaken in the senior secondary school. Here algebra and geometry are brought together to form a powerful combination that enable some interesting problems to be solved. Most of the calculus in school is built around the application of algebra to geometry via the Cartesian co-ordinate system.

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

Differentiation involves changing the context of activities to suit the interests of your students. The unit uses quite traditional contexts such as battleships and pirates. Students will find using locations in their local area engaging, but also be prepared to use locations they are less familiar with, such as other towns/cities in Aotearoa. Students might become more engaged in sports they prefer such as basketball, hockey and netball, than golf. Professional sports coaches gather a lot of detail about the locations of their players during games. Traditional settings also provide contexts for study. Māori call the North Island “Te Ika-a-Māui” (the fish of Māui). There were no aeroplanes then. How did they know the island was shaped like a fish? What reference system did pre-European Māori use to navigate?

## Session 1: Four Counters

This lesson introduces the Cartesian co-ordinate system using a game with four counters and "Battleships".

Cartesian co-ordinatesafter René himself. Students will enjoy the story more if they can act it out. Get a student to be the servant who goes to a corner of the room and follows these instructions: "Go three steps across and two steps up. Now splat the fly!" Emphasise the importance of knowing which direction is across and which is up. After all, the spot represented by (3,2) is not the same spot as (2,3).## Session 2: Buried Treasure

X can represent the position of a counter. More traditionally it represents the spot where pirate treasure is buried. We now follow that theme.

5. Each group goes outside and "buries their treasure" marking its location on their map of the school. They return to the classroom and provide another group the five co-ordinates, which describe where the pieces of treasure are located. The other group attempts to find the treasure of the first group using the co-ordinates provided.

## Session 3: Dead Centre

Use the Copymaster Dead Centre in this session.

There is an old joke about the dead centre of a town being the cemetery. Here we get the class to think about the location of the real "dead" centre of a set of objects. In this situation we mean the point that is in some sense in the middle of these objects.

Find a way to work out where that centre is. Point out that the dead centre will be the place that minimises (makes smallest) the total travel distance that the club members need to get there. (You might say that they are looking for a new place to locate their clubhouse.)x) co-ordinates gives (1 + 2 + 4) ÷ 3 = 2.33. The average of the vertical (y) co-ordinates gives (3 + 1 + 2) ÷ 3 = 2. So the point (2.33, 2) gives the centre of the three houses.x-values in one column,y-values in the other, and each column averaged using a function formula. The location given by the averages will define the dead centre of where students in your class live.## Session 4: Polar Co-ordinates

In this lesson we introduce polar co-ordinates.

So polar co-ordinates (45°, 5cm) translate to the location of the point Q.

## Session 5: Co-ordinate Puzzles

This lesson provides an application of polar co-ordinates to revise and strengthen the work of the previous session.

Puzzle One: (5,10), (3,7),(0,5), (3,3), (5,0), (7,3), (10,5), (7,7).

Puzzle Two: (0°, 0cm), (45°, 1cm), (90°, 2cm), (135°, 3cm), (180°, 4cm), (225°, 5cm), (270°, 6cm), (315°, 7cm), (360°, 8cm), (0°, 9cm), (45°, 10cm), (90°, 11cm), (135°, 12cm), (180°, 13cm), (225°, 14cm), (270°, 15cm), (315°, 16cm), (360°, 17cm).

Puzzle Three: (5, 6), (6, 5.9), (7, 5.7), (8, 5.4), (9, 4.8), (10, 3), (9, 1.2), (8, 0.6), (7, 0.3), (6, 0.1), (5, 0), (4, 0.1), (3, 0.3), (2, 0.6), (1, 1.2), (0, 3), (1, 4.8), (2, 5.4), (3, 5.7), (4, 5.9), (5, 6).

Dear family and whānau,

This week we have been looking at co-ordinate systems.

Ask your child to show you how to play golf on paper using polar coordinates. Who is the best shot in your family?

## Figure it Out Links

Some links from the Figure It Out series which you may find useful are:

## Flip and roll

The unit looks at, analyses, and extends, a game of chance in which three coins are tossed. A player wins if two heads and a tail come up.

This unit is based on the following five ideas about probability.

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:

Tasks can be varied in many ways including:

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. Many different cultures have games of chance which can be found by a quick internet search. Research might open discussion about gambling, and how people can easily be tricked into a false hope about winning. For example, Lotto is an interesting game where the chances of winning are low, but the lure of the prize is high.

Session 1In this session we look at the probabilities around flipping coins.

Why is a coin tossed?(A whistle might be hidden in one hand and one captain asked to guess as an alternative).Is tossing a coin a fair way to decide who kicks off, or the way the team face? Why?Students should conclude that there are two possible outcomes, so a coin toss is fair, unless the coin is biased in some way.

What events might happen that cause you to lose?(HHH, TTT, TTH)Does the game seem fair? What does fair mean?(equal chance of winning)Win

Lose

THH, HTH, HHT, …

HTT, TTT, TTH, HHH,…

What did the experimental results show?Why is the game unfair?(Look for attention to the events that might happen – a theoretical idea)Do you agree with Brian and Jo?Discuss their ideas with a partner and come up with a justification for you view.Act out simulations of three tosses of the same coin and locate the outcome at the appropriate branch of the tree.

Which outcomes are wins?How can we write the probability of winning using numbers?(3/8, 0.375, 37.5%)Distinguish between an event (e.g. Three tails or Two heads and one tail) from an outcome, a way that the event might happen (e.g. HTT, THT, HTT).

What are the chances of getting… three heads? Three tails? Two tails and one head?Create rules that make the game fair. That means you have an equal chance to winning and losing.Test your game to see that it is fair.How is the situation similar to the four-coin toss problem?What happens if you add a row?What happens if you change the probability of left and right from 50%?Why does that happen?Session 2In this session, we explore the use of theoretical probabilities to make predictions for different number of trials.

What information do you need to make a prediction?Expect students to talk about probability of boys and girls which might be influenced by number of each gender in the school, which gender tends to leave first, who gets kept after school, etc.

If the chances of males and females are 50:50, how many females do you predict?Are ten females and ten males a likely outcome?How might we act out this problem? (Simulate the situation)Students might suggest tossing a coin twenty time using heads for girls and tails for boys. You might return to a Dalton Board simulation with a single row and 20 marbles to see what happens. Most times a sample of 20 students will not yield a 10:10 result (The probability is only about 18%, which is less than one fifth).

How many tosses will come up heads? Why do you predict that number?How often did you get an outcome of 10 heads and 10 tails? (About 1 trial in every six)How much did your results vary? What was you most unpredictable result?What do you notice about the graph?Students should observe that the outcomes vary considerably but the distribution centres on the predicted value of 10 heads.

To save time we will use this tool to simulate a lot of coin tossing.Each trial will be 100 individual coin tosses.What is your best prediction for the number of heads and tails?Do you expect that the data to look different to the 20 individual coin toss trials?How are the graphs similar? How are they different?Students might notice that the graphs both have a ‘bell’ shape and the distributions centre on the predicted values of ten heads and 50 heads, respectively. They might notice that the shape is smoother for the 100 toss data and more evenly spread among the outcomes. It is unlikely they will notice that the variation is

proportionallyless for the 100 toss data. The spread for the 20 toss data is from 1 to 19. The spread for the 100 toss data is from 35 to 62.The key point is that experimental data varies a lot from the theoretical predictions.

Get a couple of students to try the game, noting the result.

Does the game look fair? How do you know?How could we work out the chances of winning the game?If you trusted this trial, what is the chance that a single coin toss gives you a win?Look for students to express the experimental probability as a fraction, 5/16.

Would you make money playing this game? Explain.What fraction of the trials did you win?Are these fractions close to the same or completely different?Our experimental data show that the chance of winning with a single coin drop is about 25%.With your partner find a way to work out the chance in a theoretical way.Analyse the game.Then change the game in some way so the chance of winning is one half, 50%.Session 3In this session, we work on finding probabilities for two other situations, using theoretical and experimental methods. Two Figure It Out activities are used.

What are the chances that you win the game?Does it matter which option you choose, paper, scissors, or rock?How might be create a model of all the outcomes?In how many outcomes does Player One win? Player Two win?Both players win for 3 out of 9 outcomes with the other 3 outcomes being draws. Therefore, the game is fair, with both players have a 1/3 chance of winning each time.

Let students work on the problems in pairs. Roam the room and look for students to:

What ways did you find to make the game fair?Giving different numbers of points is one method but encourage students to look at the probabilities associated with different options. Possible options might be:

Which option is best? Why?Is there a way to estimate the amount that usually occurs with 52 rolls of the dice?The data should indicate that most times 52 rolls gives a result of about $260, or a bit less. Possibly display the trial amounts using a dot plot.

How might we change the numbers on the dice so the expected amount from 52 rolls equals 52 lots of $5?The easiest solution is to label all the faces $5.

Challenge yourself. Make the number on each face different.Any combination of numbers that average five will work, e.g. 1, 2, 3, 7, 8, 9.

Session 4In this session students investigate the "Top Drop" game.

This is a game for pairs of students. To play the game the players take turns predicting the outcome then dropping a plastic drink bottle top with a smiley face drawn on inside. The first player predicts which way the top will land ‘face up’, ‘face down’ or ‘face side’.

The top must be held on its ‘side’ when it is dropped from a height of approximately 30 cm onto a flat surface.

If the player’s prediction is correct, they are awarded points, as below. If the player is incorrect, they are awarded no points.

Each player must keep track of the outcome of each drop, plus the points awarded.

The first player to be awarded 20 points wins the game.

What is the best strategy to win the game?What did you notice?Expect students to say that Up occurs more frequently than Down, with Side being unlikely.

Ask:

How could we find the probability of each event, Up, Down and Side?Since the situation cannot be modelled theoretically the only way to estimate the respective probabilities is by experiment.

The data might look like this (variation will occur from these results:

Face Up

Face Down

Face Side

159

78

13

What do you notice about the samples of ten tosses?(A lot of variation and Face Up most common)Can we estimate the probabilities of Up, Down, and Side? How?Percentages or decimals might be used to estimate the probabilities, e.g. 159 + 78 + 13 = 250 (total number of tosses), 159/250 = 0.64 = 64%, 78/250 = 0.31 = 31%, 13/250 = 0.05 = 5%.

You might create a pie chart that shows the percentages:

Discuss:

Do these probabilities help work out the best strategy for the Top Drop Game?Students should say that there is twice as much chance of getting Up as Down. That shows in the points allocated.

Is it worthwhile to predict Side, at all?Side only occurs once in twenty tosses, on average, but a win gives to points. A lucky person might get two Sides and win the game.

Students will come up with their own approaches. Look for signs they understand that each toss is an independent event, that is, one toss has no influence over the other. The task might also open up discussion on randomness.

Are the results unpredictable, or is there influence (bias) on what happens?Session 5In the final session use a Figure It Out activity, Left to chance to assess students’ understanding of theoretical and experimental probability. Left to chance also involves coin tosses, so students should bring some understanding from the first session.

How will you record the results of the trial?How will you create a model of the outcomes to find the theoretical probability?For example, they might say that four trials are not enough to provide solid evidence (Steve’s assertion). They might analyse the various ways that each destination in the fourth row might be arrived at (Simon’s false claim). To do so, they recognise that an event, e.g. landing on the second from the left ‘lose’ space, can happen through many different outcomes, e.g. HHT, HTH, THH.

Do students systematically find each set of outcomes for different destinations in the fourth row?

P (Three heads) = 1/8 = 0.125 = 12.5% (This means that the probability of HHH equals 1/8)

P (Two heads and one tail) = 3/8 = 0.375 = 37.5%

P (One head and two tails) = 3/8 = 0.375 = 37.5%

P (Three tails) = 1/8 = 0.125 = 12.5%

Combining the two ‘lose’ events gives a probability of 6/8 or 3/4. The game is definitely not fair!

You might use Slide 13 to discuss all the possible outcomes from three-coin tosses, and the number of those outcomes that lead to each ‘win’ and ‘lose’ destination.

Students might do this by extending the set of possible outcomes to four coin tosses in a systematic way. See Slide 14.

P (Win) = 8/16 = 1/2 = 0.5 =50%

P (Loss) = 8/16 = 1/2 = 0.5 =50%

Dear parents and whānau,

This week we have been looking at events where chance is in play. We are studying probability using simple games with coins and dice.

Ask your student to tell you about the different outcomes that can occur when you toss two coins and what the chances are of getting each of these outcomes. Ask you student to invent a game that uses three coins. Ask them to explain if the game is fair, or not, and try to find out why this is the case.

You will discover that coin tossing creates lots of interesting discussion. Enjoy investigating probability with your student.

## What's going on? Fractions

This unit develops students’ recognition of pattern (consistency) in equations involving equivalence, addition and multiplication of fractions.

This unit develops students’ recognition of pattern (consistency) in equations involving equivalence, addition and multiplication of fractions. The patterns of equations show important characteristics of fractions, such as non-unit fractions as iterations of unit fractions, and equivalence as representing fractions of the same value. Students learn to represent specific examples where the properties are applied then provide convincing arguments about why the properties hold in all circumstances.

An important consequence is that students learn to consider variables as generalised numbers, and express relationships involving fractions, including the significance of considering both numerators and denominators, when adding and multiplying.

In this unit we build on research by Deborah Shifter, and colleagues, about the development of algebraic thinking. Shifter works for The Educational Development Centre, a non-profit research organisation in USA. Her approach follows several steps that can be linked to ‘folding back’ in the Pirie-Kieren model of conceptual development, that is commonly used in New Zealand classrooms.

The phases of approach are as follows:

In this unit claims are developed through equation sets involving fractions. The sets aim at developing students’ understanding of the structure of non-unit fractions, mixed numbers, and addition and multiplication of fractions.

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

The contexts for this unit are strictly mathematical but the materials used can be adapted. Physical items that have significance to your students might be better used than standard mathematical equipment. You may decide to restrict the materials students use, especially if the materials are already familiar to them. A journey might be used as the metaphor for a strip model and could be related to a local story. Whānaungatanga (family) values might involve finding fair and equitable ways to share items. Two families that share harvested potatoes might do so with fractions, to reflect the different number of people or needs of each whānau.

All lessons in this unit follow the same sequence of phases as given in the diagram above. A poster of the phases is provided as Copymaster One for students to refer to. The notes suggest possible student ideas and teacher reactions to those responses. It is not feasible to anticipate all ideas students might give so you are encouraged to be flexible in how you respond to students rather than ‘teach’ the sample ideas and representations provided.

PowerPoint One gives the equation sets that drive the unit. The sets are labelled in the top left corner of each slide for reference. Slide one has the first pattern to look at. The pattern involves the addition of fractions with the same denominator i.e.

1/b+ (b-1)/b =1which is relatively easy for students to represent. It is used as an example to familiarise students with the approach. Be aware that students’ existing schemes for addition of fractions may be incorrect, e.g. they may simply add across the numerators and denominators:1/4+3/4=4/8.Equation Pairs Set OneNoticing ConsistencyUse ‘think, pair, share’ by inviting students to look independently at the four examples, work out the missing values, then share their ideas with a partner. In the class discussion expect students to express their observations in ways that are clear to others. Students should re-express their ideas if others do not understand what they are saying. You may need to remind students that the ‘something going on’ relates to all four examples, not just one. Expect responses like:

S: The first fraction always has one on the top line.T: Can you tell us more? What is the denominator, the bottom number?S: The bottom numbers, denominators are different in each equation.Discussion opens the possibility of using correct mathematical terms, like numerator (number above the vinculum), and denominator (number below the vinculum).

S: The sum (answer) is always one.T: All four equations have a sum of one? What about the top right equation?S: The sum is five fifths. Five fifths make one whole.Making a claimEncourage students to state a claim about what is going on with all four examples in Pattern One. They might do so individually at first then work in a small team to refine their ideas and the way they express those ideas.

Expect ideas like:

If the denominators of the fractions are the same, and the numerators add to the denominator, the sum is one.If the two numerators add to the denominator, then the sum is one.Your aim is for students to express their claims in clear, minimal terms, using correct mathematical language. For example, ‘the top numbers’ is not as clear as ‘the numerators’, and ‘add to’ is not as precise as ‘the sum of’.

RepresentationIn this phase students choose representations to show why the pattern holds consistently. Students might choose physical manipulatives, such as fraction strips or circles, draw diagrams such as strips and circles, and use contexts from everyday life, such as sharing cakes. Encourage students to begin with the first two examples of equations then consider how the same relationships might generalise to the last, and other similar, equation pairs.

Examples might be:

I made1/2+1/2= 1using fraction strips then I made1/5+4/5= 5/5. Five fifths make one, so I put a one strip below.S: I used fraction circles to show that1/8+7/8= 1and1/3+2/3= 1.Some representations are less helpful than others in terms of understanding the structure of the fractions that add to one. Circles are useful for showing that a one is complete. Strips possibly make the iteration of unit fractions more obvious. It is important for students to recognise that the denominator in each example defines the size of the equal parts that one is cut into. For example, eighths are created by the equi-partitioning of one into eight parts, so eight of those parts make one. Your questioning is important to support students to connect the symbols and other representations.

Explain where the eighths are in your representation.How many eighths are needed to make one?Where is the seven in7/8represented in your diagram/model?How does your representation show that the sum is always one?ArgumentationIn this phase students are asked to formalise their noticing by creating a statement that generalises to all cases. The discussion may start with a specific equation pair but must be amended so that it accurately represents the general pattern.

S: With1/8+7/8= 1the numerators add to eight. Eight eighths equal one.T: So how does that work in the same way with the other equations?This might lead to expressing the relationship in general terms.

S: The first numerator plus the second numerator add up to the denominator.T: If we gave names to the numerator and denominator of the first fraction, like a and b, could we express the property more simply?Some students might experiment with algebraic notation such as

a/b+ c/b= 1.Note that this statement is incomplete since it misses the necessary condition for a + c = b.T: Do we need to say something about a and c?S: We could just write a + c = b.T: Give us an example of that.S: If b = 3, the numerators 1 and 2 add to 3.Though it is difficult for this level, increased sophistication is evident in a + c = b, and writing the generalisation with only two variables,

a/b+ (b-1)/b= 1.It is also important to note thatb/b= 1, for any whole number value of b except zero.bcan also be any integer, since rational numbers are of the forma/b, a and b∈ I, that is a and b must be integers. High achievers should be challenged to express their generalisations using algebra. You may need to support them to develop the coding strategies to do that.Equation Pairs Set TwoIf time permits in the first lesson, ask the students to approach the second equation set more independently. From this point each equation set is discussed succinctly using the phases of the approach.

Noticing consistencyThe four equations apply the addition of fractions with the same denominator. Students need to recognise that the denominators in both addends and the in sum are the same. Watch for students incorrectly applying whole number addition, that is, adding both numerators together and adding both denominators together. The completed equations should be:

Note that ten eighths equals 1 2/8 = 1 1/4

Making a claimIn natural language expect the students to use phrases like “you just add the top numbers”. Introduce important vocabulary such as

numerator, denominator, addend,andsumto clarify which numbers are being referred to in the claims. If the claim is restricted to one example, encourage students to see what changes and what stays constant across the other three equations. The aim is to broaden the claim to the equivalent of “If the denominator of both addends is the same, then the numerator of the sum is the sum of the numerators of the addends.”RepresentationExpect both physical and diagrammatic representations to be used. A fraction strip representation might look like this:

Diagrams of a ‘sets’ representation can be used but the changing whole can be confusing, since the aim is to generalise a relationship for all for examples.

Whatever representation is chosen by students they need to be clear about the referent one, be it a length, area, or set. The sum must be defined in terms of that original one whole.

ArgumentationLook for students to justify that if the denominators of two addends are the same then the sum is the sum of the numerators, over that denominator. Algebraically that is represented as:

a/b+c/b = (a+c)/bNote that a, b, and c are variables, and the generalised argument becomes clearer, and is confirmed, by translating values from the equations back into the general form.

For example, with

3/8+7/8=10/8, a = 3, b = 8, and c = 7.An interesting discussion point is whether this is any different to addition of whole numbers. For example, 3 + 7 = 10 is the combining of 3 ones and 7 ones, making 10 ones. The ‘ones’ are regarded as the same unit. With

3/8+7/8=10/8, eighths are the units being combined.Developing the process furtherPowerPoint One contains seven equation sets. The sets might form the basis of a week-long unit. The phases for each equation set are described succinctly below.

Equations Set ThreeNoticing consistencyThe four equations both demonstrate the multiplication of unit fractions, that is fractions with one as the numerator. Understanding in general that

1/b×1/d=1/bd, e.g.1/3×1/4=1/12, is fundamental for understanding equivalent fractions.Making a claimIn natural language expect the students to use phrases like “multiply the two denominators and you get the denominator of the answer”. Expect the use of mathematical vocabulary such as

factorandproductto clarify which numbers are being referred to in the claims. Encourage clarity by asking questions like:Do the denominators of the factors need to be the same?What can you say about the numerators?The aim is to state the claim as something like “The product of two unit-fractions is a unit fraction, with a denominator that is the product of the denominators.”

RepresentationExpect both physical and diagrammatic representations to be used. Fraction strips are static representations:

Folding a paper strip or a rectangle is a dynamic representation:

A sets model may be used but the product needs to be known to establish the possible numbers of objects in the set. Therefore, identifying the referent one is more complex.

ArgumentationLook for students to state the condition that both addends are unit fractions. The starting product might be expressed as

1/b×1/d. The product is1/bd. Students should explain why the relationship occurs. Their explanation needs not be in formal algebraic terms, with letters as variables.The second factor gives the first equal partitioning and creates some parts. The denominator gives the number of equal parts. Then each part is split into equal parts. That number of parts is given by the denominator of the first factor. Therefore, each of the d parts, is equally partitioned into b parts, so the total number of parts equals b x d.

Equation Pairs Set FourNoticing consistencyThe four equations apply equivalence of improper fractions. The equivalent number is kept at three to limit variation.

Making a claimIn natural language expect the students to use phrases like “the answer is always three.” Some students might indicate what they see in a diagram.

Expect the use of mathematical vocabulary such as

denominator, numerator,fraction, equivalent,andwhole numberto clarify which numbers are being referred to in the claims. Encourage clarity by asking questions like:What does the equals sign indicate? (both numbers represent the same amount)How is the numerator related to the denominator? (the numerator is the denominator multiplied by three)Why is the number on the right side always three? (For example, with 24 eighths you can make three ones because you need eight eighths to make one.)The aim is to state the claim as something like “A fraction that has a denominator that divides into the numerator three times, is equal to three,” or “A fraction that has a numerator that is three times the denominator, is equal to three,”

RepresentationFraction circles are useful in demonstrating the completeness of a one.

Eight eighths make one. With 24 eighths you can make 24 ÷ 8 = 3 ones.

Students may develop more schematic ways to represent the connection between numerator and denominator.

Five fifths make one so 15 fifths must make three, since 3 x 5 =15

ArgumentationLook for students to justify, using words of diagrams, that the denominator of a fraction specifies the number of equal parts that make one. For example, in

12/4the parts are quarters, four of them are needed to make one. Dividing the numerator by the denominator finds how many ones can be made. With12/4, 12 ÷ 4 = 3, so three ones can be made. The equation12/4=3represents that equivalence.For all four equations define the denominator of the first fraction as the variable, say b. The observation that the numerator is always the denominator multiplied by three can be represented as

3b/b. The equivalence can be written as3b/b=3.Argument may be supported by graphics or other representations.

Students might also generalise that dividing the numerator by the denominator tells how many ones can be made, and the remainder becomes the fraction part of the mixed number, e.g.

13/5= 23/5.Equation Pairs Set FiveNoticing consistencyThe equation set applies equivalence, that is when two fractions represent the same value. Understanding equivalence is essential to ordering fractions by size and to the addition and subtraction of fractions. The completed sets should be:

Making a claimIn natural language expect the students to use phrases like “If the numerator is three times the other numerator, then the denominator must be three times as well.” Students tend to notice patterns across the equation like this:

Expect the use of mathematical vocabulary associated with fractions, particularly

numeratoranddenominator. Draw students’ attention to the equals sign as representing the same value on both sides. Students are most likely to see equals as a signal to work out the answer. Encourage clarity by asking questions like:What is the same about all four equations?What is different?How are the numerators related?How are the denominators related?The aim is to state the claim as something like “Two fractions are equivalent, if the one numerator is a multiple of the other, and the denominator is the same multiple of the other.”

RepresentationStatic representations might involve length, area (circles or rectangles), or sets.

A more dynamic representation is to fold a paper strip or rectangle into thirds, shade two thirds, then fold the thirds in half. How many sixths are shaded?

ArgumentationLook for students to justify, using words of diagrams, that the same multiplication is performed on both the numerator and denominator of a fraction to create an equivalent (same value) fraction. For example, if the numerator and denominator of

2/3are multiplied by six this gives12/18,and2/3=12/18. Of interest is that the multiplier6/6is another name for one, the identity element for multiplication.However, the multiplication means different things for the numerator and denominator. Multiplying the numerator by a number increases the number of parts by that factor. For example,

12/18has six times as many parts as2/3. For the denominator the effect is the inverse. Each part in the new fraction is one sixth that of the first fraction, since six times as many of those parts make one. For example,12/18has parts that are one sixth the size of the parts in2/3.In general, the transformationand

a/b×n/n=an/bn, gives n times as many parts but each part is one nth the size.a/ban/bnrepresent the same quantity. Note thatn/nis always equal to one.Equation Pairs Set SixNoticing consistencyThe equation set highlights the difference of unit fractions. Note that the denominator of the first fraction is one less than that of the second. This equation set provides an opportunity to apply equivalence. The completed sets should be:

Making a claimIn natural language expect the students to use phrases like “the difference has a denominator that is the product of the two other denominators.” Ask students to be more specific in their description.

What can you say about the two fractions on the left of each equation? (They are both unit fractions and there is a difference of one in their denominators)What does subtraction mean in these equations? (It means "What is the difference between the two fractions? ")Can you make up another equation that would go in this set?The aim is to state the claim as something like “If two unit-fractions have denominators that are one different, the difference of the fractions is the unit fraction with the product of the denominators as it’s denominator.”

RepresentationStudents are likely to use specific examples to convince others about how the relationships work. Fraction strips are probably the most useful representation.

It is important for students to consider why the fraction one twelfth fits exactly.

ArgumentationStudents need to apply equivalence to argue why a given unit fraction is the difference. Their argument might begin for a specific example.

Since one half and one third are involved, the difference must involve sixths, as both one half and one third can be renamed as ‘so many’ sixths.

1/2=3/6and1/3=2/6so there is a difference of one sixth.Encourage the students to generalise by naming the denominator of the first fraction as a variable again.

Can you find a relationship that applies to all four equations?If the first fraction is

1/b, the second is1/(b+1), and the product is1/(b(b+1)). Algebraic notation, like this, is not expected at Level Four, but students may invent their own ways to express what they notice. Whatever notation students create, it can be checked by substituting numbers in place of symbols, and seeing if the equations created belong in the set.Equation Pairs Set SevenUse set seven as an opportunity to see how well students engage in the generalisation process independently. Ask them to record their claims, representations, and arguments in ways that work for them. Some students may prefer writing their work while others may prefer to capture their ideas using a digital recording.

Noticing consistencyThe equation set highlights the multiplication of fractions, that is a fraction of a fraction. The numerators of the factors are the same, as are the denominators. That is done to reduce variation. The completed sets should be:

Making a claimIn natural language expect the students to use phrases like “It is a fraction multiplied by itself. The product is the two numerators multiplied over the two denominators multiplied.” Ask students to be more specific in their description, through questions like:

What do you notice about the two factors (fractions being multiplied)? (they are the same)Do you know a mathematical term for a number multiplied by itself? (square of the number)Can you use the term square to describe the product fraction in a simpler way?The claim, using mathematical language, might be something like, “If a fraction is squared, the product is the numerator squared over the denominator squared.”

RepresentationAn area model is useful since it illustrates the relationship between a number multiplied by itself and a square. Folding and shading squares of paper works well.

Students might also use strip models, but the squaring of numerators and denominators is not obvious.

ArgumentationThe equations include two variables, that is the value of numerator and denominator in the fraction being squared. Students need to recognise that the denominator names the number of partitions of one. The numerator names the count of those parts.

Students may be able to show how they are thinking with variables (lack of closure), by treating the numerator and denominator as variables in an area diagram.

Algebraically this might be written as

a/b×a/b= (a×a)/(b×b)or more succinctly as (a/b)^{2}=a^{2}/b^{2}.## Choices

This unit is about making best option decisions in the real-life situations based on cost. Common examples of such decisions are explored, including the cost of taxis, cooking times, hire cars, and mobile phones.

The mathematics in this unit involves relationships between variables. A variable is a measure that can take up different values. For example, when making decisions about rental cars, customers relate two variables, number of kilometres or days (distance or time) and amount of money (cost).

Most situations define the roles that the variables take. When hiring a car, it is the distance or time, that is the independent, or explanatory variable. An explanatory variable is the variable that explains changes in the other variable, cost. You expect that travelling more kilometres will cost more. Cost is the dependent, or response, variable.

In this unit the relationships are mostly linear. That means there is a constant growth in the response variable, as constant growth occurs in the explanatory variable. For example, Booma Rentals hires cars at $30 per day plus $0.10 per kilometre. The relationship between cost and kilometres is linear, since you pay 10 cents more for every kilometre. If a linear relation is graphed the ordered pairs (co-ordinates) lie on a straight line.

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:

Task can be varied in many ways including:

The context for this unit can be adapted to suit the interests and cultural backgrounds of your students. Most students nowadays own a mobile phone or know someone who does. Rental cars and taxis are a more familiar context for urban students so be aware that other forms of rental or journey may be more appropriate to rural students, e.g. delivery fees for products. Cooking times of meats can easily be changed to include cooking times of roasting vegetables or other foods. There may be contexts involving rates that are more relevant to your students. Examples might include hire to buy schemes, money earned for jobs such as serving at a café or dairy, cost of shipments of hay (delivery charge + cost per bale), water and electricity charges (line + amount per kilowatt hour).

## Teachers’ Notes

Encourage your students to choose appropriate forms of calculation, depending on the complexity of the operations and numbers involved. Mental calculation, recording in written form, and digital devices are all legitimate options. In your discussions ask the class how they got the answers and what methods they used to do the calculations.

Session 1Who has ridden in a taxi recently? Why did you go in a taxi?(Slide One)How did the taxi driver calculate the fare?(Slide Two)Why was there a dollar amount on the meter before you even started off?Tiki Taxis charge $5 flag fall and $3.00 per kilometre. (Slide Three)

Discuss the meaning of “per kilometre” as a rate “for every kilometre travelled.”

Use Slide Four to show the problems:

How much would it cost you to travel 25 kilometres in the cab?How much would it cost you to travel 32 kilometres in the cab?How far could you go for $41?How far could you go for $77?You might use a flowchart like this:

Ask:

Why is the order that we carry out the operations important?How do we find the distance if we are given a fare?You might decide to record equations for the flowchart processes. Define the variables; f represents the fare in number of dollars, and d represents the distance in number of kilometres.

If we start an equation, f=, what do we include to show how the fare is calculated, what fare is equal to? (f = 3d + 5)If we start an equation, d=, what do we include to show how the distance is calculated, what distance is equal to?(d = (f – 5)/3). In this case pay special attention to the vinculum in a fraction also being a symbol for division.Do you see any pattern?(Student might notice that the graph is linear)Why do you think the points lie on a line?(Same rate of $3 per kilometre)What do you notice?Can we work out fares and distances using the graph? How?What would the fare for 30 kilometres be?($95)What is the distance of a fare that costs $65?(20 km)Animating Slide 7 shows how the answers can be read off the axes.

Tiki Taxis find they are not making enough money to cover their costs, and to give their drivers a decent wage.Which option better, if they want to make more money?Raise flagfall to $10.00orIncrease the rate to $3.50 per kilometreWhat does the graph show?(An increase in flagfall of $5 gives more income than an increase in rate of $0.50 per kilometre up to 10 kilometres)How far do you think most taxi trips are?(Create some make-believe trips online using Google Maps and use the distance function)What recommendations would you make to the owners of Tiki Taxis?Session 2What does 43 kilograms per capita mean?(An average of 43 kilograms for every person)Discuss the fact that this might include bones as well as meat, but every person includes little children and elderly people who eat relatively little.

How many whole chickens make up 43 kilograms?(invite predictions then move to Slide Two)What operation should we perform?(43 ÷ 1.2 = 35.83)Therefore, kiwis average nearly 36 whole size 12 chickens per year. That’s three chickens every month!

How long should you cook a size 12 chicken?What calculation should you perform?(1.2 x 55 + 20 = 86)How long should you roast a size 18 chicken for?1.8 x 55 + 20 = 119If a chicken should be roasted for 97 minutes, what mass is the chicken?(97 – 20) ÷ 55 = 1.4If a chicken should be roasted for 75 minutes, what mass is the chicken?(75 – 20) ÷ 55 = 1.0What pattern do you notice?(Linear)Why does this pattern occur?(Constant rate of 55 minutes per kilogram)How is this situation the same as the rental car situation?(A constant rate and fixed time. The fixed time has a similar effect as flagfall in the rental car situation).In the United States of America, it is traditional to cook a turkey to celebrate Thanksgiving.

Commercial turkeys are much larger than chickens and have a mass of between 3.6 and 10.8 kilograms, etc.

Let students work on the problems collaboratively with support from digital graphing tools and calculators. Solutions are below:

(Slide Nine)Is the graph of mass and cooking time linear?The relationship is close to linear.

Find a rule to work out the roasting time for any mass of turkey.The amount of cooking time goes up by a little over 15 minutes for each extra kilo. For example, between the point (8, 210) and (10, 240), mass increases by 2 kilograms and cooking time increases by 30 minutes. That is 15 minutes per extra kilogram. Students may recognise that their rule should begin as t = 15m but a fixed amount of time needs to be added. They might take any point, say (8, 210), and use it to work out the extra fixed amount.

8 x 15 = 120

120 + 90 = 210

The rule is t = 15m + 90 and can be tested using other known masses and times. For example, 15 x 10 + 90 = 240 minutes.

Use your rule to calculate the roasting time for a 7 kilogram turkey.15 x 7 + 90 = 195 minutes

Do turkeys take less time, per kilogram, to roast than chickens?Why might that be?Chickens take an extra 55 minutes per kilogram compared to 15 minutes extra for turkeys. Mostly that is because a much larger bird takes way longer in the oven to reach a cooking temperature (90 minutes vs 20 minutes) but having reached that temperature cooks more quickly. There is little difference in the density of chicken and turkey though turkey is a little leaner, has less fat proportionally.

Kiwis eat about 24 kilograms of pork, per capita, each year.Here are the recommended roasting time details for a shoulder roast.Put in the oven for 30 minutes at a temperature of 220⁰C.Turn the oven down to 180⁰C.Cook the roast for 20 minutes per 500 grams.Create a graph to help people work out the roasting time for pork.Slide Eleven provides a model graph.

Ask students to write an equation for the relation between mass and cooking time. (t = 40m + 30)

## Session Three

Has anyone’s family or friends ever hired a car?Why did you need to hire a car rather than use your own?

How many days did you hire the car for?

How far did you travel?

Do you know how much it might have cost?

What different things did you have to pay for?

Question 1: Suppose that Rent-A-Bomb charges $25 a day and 30c per kilometre.

How much does it cost for a two-day hire if you travel 300 kilometres?2 x 25 + 300 x 0.3 = $160

If you use the car for three days, and the hire costs $291 in total, how many kilometres do you travel?(291 – 3 x 25) ÷ 0.3 = 920km

If you travel 900 kilometres, in total, and the hire costs $650, how many days do you hire the car for?(645 – 900 x 0.3) ÷ 25 = 15 days

What different ways did you use to solve these problems?Which strategies are the most efficient?Note that students are likely to use trial and improvement strategies, particularly to problems b) and c). These strategies can be made more efficient by organising data systematically in tables. For example, b might be approximated by choosing likely numbers of kilometres.

The equations given represent an efficient solution that comes from recognising a rule for working out the number of kilometres from the cost or the inverse of that.

A flowchart for a 3-day hire might support students to see the relationships:

To solve problem 1b put $291 into the cost box and work backwards to find the distance.

Introduce Rent-a-dent, a firm that charges $40 per day and 15c per kilometre. Discuss the parts of problem 2 then ask students to work in their teams. Allow suitable time for students to work on the problems. Then gather the class to discuss their strategies and solutions. Answers are below:

4 x 40 + 400 x 0.15 = $220

{197 – (2 x 40)} ÷ 0.15 = 780km

{480 – (800 x 0.15)} ÷ 40 = 9 days

You might use tables again to organise data and flowcharts to suggest the most efficient ways to calculate the answers.

Let’s compare the two firms, Rent-a-bomb and Rent-a-dent.To simplify things let’s hire the car for one day.Which company is the cheapest to use?Students might offer ideas like:

Rent-a-bomb charges less to hire the car but Rent-a-dent charges less per kilometre.

How might we graph the deals from both companies on the same graph?What are the variables? (Distance and cost)What rule connects kilometres and cost for each company?What is the equation for that rule?Collectively develop a graph that looks like this (PowerPoint Three, Slide One):

Which company belongs to each line? How do you know? Label the lines.Which company is cheapest if you only want to travel only 80 kilometres? How do you know?What does the intersection of the lines mean?Which company is the cheapest to hire from, for one day?Session 4In this session students look at mobile phone plans. For many people mobile phones are also their source of emails, music and internet. Plans are developed by companies to attract consumers who meet patterns of phone use. The best plan for one person is not necessarily the best for another. Variables to consider are time in phone calls, number of texts, and amount of data.

Who has their own mobile phone?What do you use your phone for?How many times a week would you use it to phone a someone?How many text messages would you send in a day?What does it cost to use your phone?Who pays for it?The discussion should show that use of mobile phones is variable and that they cost money.

How do you select the best deal for your phone?What details do you usually look for in mobile phone plans?Generally, plans include the amount of data, minutes of calls, and number of texts. Sometimes they include special deals like extra data, memberships to entertainment providers or music channels, and interest free purchase of phones.

Mobile phone plans contain a lot of variables. Let’s simplify the situation by looking at just one variable, data volume.

What is meant by data?(Pieces of information)Slide Three shows the commonly used file sizes.

A gigabyte equals 1000 x 1000 x 1000 = 1 000 000 000 = 1 billion bytes. To put things into perspective:

1 GB of data is equivalent to about 210 000 text only emails, or 250 3-minute MP3 videos, or 600 high resolution images. A standard movie DVD contains about 4.5 GB of data.

Terabytes (TB) are another common unit of data. For example, if you buy an external hard drive for your computer it is likely to be several terabytes in size One terabyte equals 1 000 GB, which equals 1 000 000 000 000 bytes (1 trillion).

What is the same and what is different about these plans?Students should notice that everything is the same except the volume of data.

Is there a relationship between the amount of data and the price of the plan?

If a plan is exactly the same, but allows 8GB of data, how much should you pay per month?

All of these plans cost the same, but which one is best?J

ustify your choice of plan.Let students discuss the plans. They should notice that the relationship between data and call time is negative. That is, when one goes up the other goes down. Choosing one of the plans is likely to be dependent on usual phone usage, preference for internet browsing versus making calls to friends and family.

Is the relationship linear?How is the relationship different from the others we have seen?Imagine a plan with same cost that has only call time. How many minutes would you get?Imagine a plan with same cost that has only data. How gigabytes would you get?Find a rule that relates call time to data. Let t represent call time, in minutes, and d represent amount of data, in gigabytes.(t = 240 – 40d)Is there a best deal? Explain.Session 5In this session we give the students a chance to operate a hire car company.

The Hire Car Game: Students work in teams of three to operate their own hire car company. Each team has overheads of $20 a day per small car, $25 a day for a medium car, and $35 a day for a large car, whether the car is used or not. Overheads cover the cost of the vehicle, insurance, building rental, cleaning staff, etc. Cost occur each day irrespective of whether the car is hired or not.Small Car $35 per day + 0.25c per kilometre

Medium Car $40 per day + 0.30c per kilometre

Large Car $45 per day + 0.35c per kilometre

Each company choses its stock level of cars from a minimum of 4 cars to a maximum of 10 cars. The company nominates the size of each car and assigns each car an identification code, e.g. Car 1.

Family and whānau,

This week we have been investigating the cost of hiring taxis, renting cars, and operating a mobile phone. We use mathematics to make decisions about the best option to choose, and justify we make the choices we do when selecting a service or plan. Students have been asked to do some research related to the costs of phone plans. It would be helpful if you could discuss your family phone plans and why you have made the choices you have. Ask your student to identify the important considerations when deciding on a phone plan. Looking online for sites that compare phone plans would help to support your discussions. Thanks for your help.

## Figure it out

Some links from the Figure It Out series which you may find useful are: