‘Fractions as measures’ is arguably the most important of the five sub-constructs of rational number (Kieren, 1994) since it represents fractions as numbers, and is the basis of the number line. Fractions are needed when ones (wholes) are inadequate for a given purpose. This purpose is usually some form of division. In measurement lengths are defined by referring to some unit that is named as one. When the size of another length cannot be accurately measured by a whole number of ones then fractions are needed.

For example, consider the relationship between the brown and orange Cuisenaire rods. If the orange rod is defined as one (an arbitrary decision) then what number do we assign to the brown rod?

Some equal partitioning of the one is needed to create unit fractions with one as the numerator. For the size of the brown rod to be named accurately those unit fractions need to fit into it exactly. We could choose to divide the orange rod into tenths (white rods) or fifths (red rods), either would work. By aligning the unit fractions we can see that the brown rod is eight tenths or four fifths of the orange rod.

Note that eight tenths and four fifths are equivalent fractions and that equality can be written as 8/10 = 4/5. These fractions are just different names for the same quantity and share the same point on a number line. This idea, that any given point on the number line has an infinite number of fraction names, is a significant change from whole numbers. Some names are more privileged than others by our conventions. In the case of four fifths, naming it as eight tenths aligns to its decimal (0.8) and naming it as eighty hundredths aligns to its percentage (80/100 = 80%).

‘Fractions as operators’ is another of Kieren’s sub-constructs and applies to situations in which a fraction acts on another amount. That amount might be a whole number, e.g. three quarters of 48, a decimal or percentage, e.g. one half of 10% is 5%, or another fraction, e.g. two thirds of three quarters. Students often confuse when fractions should be treated as numbers and when they should be treated as operators, particularly when creating numbers lines, e.g. they often place one half where 2 1/2 belongs on a number line showing zero to five.

#### Specific Teaching Points

Understanding that fractions are always named with reference to a one (whole) requires flexibility of thinking. Lamon (2007) described re-unitising and norming as two essential capabilities if students are to understand fractions. By re-unitising she meant that students could flexibly define a given quantity in multiple ways by changing the units they attended to. By norming she meant that the student could then act with the new units. In this unit Cuisenaire rods are used to develop students’ skills in changing units and thinking with those units.

Multiplication of fractions involves adaptation of multiplication with whole numbers. Connecting a x b as ‘a sets of b’ (or vice versa) with a/b x c/d as ‘a b-ths of c/d’ requires students to firstly create a referent whole. That whole might be continuous, like a region or volume, or discrete like a set. Expressing both fractions in a multiplication and the answer require thinking in different units. Consider two thirds of one half (2/3 x 1/2) as modelled with Cuisenaire rods.

Let the dark green rod be one, then the light green rod is one half.

So which rod is two thirds of one half? A white rod is one third of light green so the red rod must be two thirds. Notice how we are describing the red rod with reference to the light green rod.

But what do we call the red rod? To name it we need to return to the original one, the dark green rod. The white rod is one sixth so the red rod is two sixths or one third of the original one. So the answer to the multiplication is 2/3 x 1/2 = 2/6 *or* 1/3.

Reunitising and norming are also important when fractions are placed in order of size (magnitude). This is especially true given any fraction has an infinite number of names. Imagine we use the orange rod as one this time and find two fifths. Since five red rods measure one whole (orange rod) then two red rods measure two fifths:

But, what other names does two fifths have? If the red rods were split in half they would be the length of white rods and be called tenths since ten of them would form one. The crimson rod is equal to four white rods which is a way to show that 2/5 = 4/10.

If the red rods were split into three equal parts the new rods would be called fifteenths since 15 of them would form one. The crimson rod would be equal to six of these rods which is a way to show 2/5 = 6/15. The process of splitting the unit fraction, fifths in this case, into equal smaller unit fractions, is infinite. This means that the point on the number line where two fifths exists has an infinite set of number names.

## Figure me out

This unit is designed to engage your class for at least the first week of the school year. It provides students an opportunity to work collaboratively and independently on challenging mathematical tasks. It also provides you, their teacher, with opportunities for you to learn about their current level of achievement.

The mathematics in the unit is varied. However, there is a general requirement for students to think multiplicatively rather than additively. The shift from additive thinking to integration of additive and multiplicative thinking is an essential requirement at Level 4. Indicators of multiplicative thinking are:

12 + 7 x 16 - 20 = 12 + 112 – 20 = 104

^{3}= 6 x 6 x 6 = 216## Specific Teaching Points

Most of the tasks are open ended so students can operate at a level that suits them. Encourage students to experiment with expressions are much as possible rather than operate ‘in the known’.

Also encourage students to work in systematic ways. To identify an unknown student which clues will be most useful? Why?

How will students check to see that they have identified the correct student? What will they do when several students have similar data?

This unit is set for students to learn, and practise, outcomes at Level 4 of mathematics in the New Zealand Curriculum. All tasks can be altered to cater for the range of readiness and interests of students in your class. A main purpose of the unit is for all students in the class is to engage in collaborative inquiry. Many suggestions are given with specific sessions about how to simplify or add more challenge to the tasks. Most activities are open ended so students can engage at a level that is appropriate to their knowledge and interests. Other methods of enabling participation are:

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

The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. In this unit student investigate data about other students in the class, and present that data in informative ways. Students may be interested in other inquiry questions. Examples might include search for the ‘average’ student in the class through investigating:

## Prior Experience

It is expected that you will use the sessions to informally assess students' mathematical skills and understanding at the start of the school year.

## Session One

Figure me out. A template to fill in is included as Copymaster One.For your gender, you might put male or female. Look at the poster template which has a key; If a prime then male, if a multiple of three then female.

Suppose a classmate wrote 51 in that space. Would they be male or female?Some students may know what prime numbers are. You may need to research primes to find they are whole numbers with only two factors, one and themselves. Is a person that puts 51 male or female?

Suppose a classmate wrote 47 in that space. Would they be male or female?Let the students work in pairs to establish if the person is male or female. 47 is prime so the person is male.

^{th}of April, 2011. Taking out the forwards slashes this date of birth can be written as 120411.However, on their poster a person might write one of the following:

^{2 }+ 9 522 or (3^{10}x 2) – 3^{8}– 3^{6}– 397What is 10 x 10 000? (100 000)? So what is 12 x 10 000?(120 000)What does 333(333 x 333 or two 333’s multiplied together) You may need to show simpler examples of exponents like 4^{2}mean?^{2}, 5^{3}and 2^{4}.What does 9! Mean?(9 factorial which is the product of 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9)A classmate gives you this clue for their date of birth. Use your calculator and research skills to find their DoB.

3

^{10}+ 7^{6 }– (2 x 5^{5}) + 4^{3 }(59049 + 117 649 – 6250 + 64 = 170512)## Session Two

In this session students develop two more expressions for the number of people that normally live in their household and the distance of their home from school.

Number of People per HouseholdWhat might the graph be about? Can we give the graph a title and some labels?What is the average number of people that live in Room 5 households?The term average refers to some measure of middle or centrality. Common averages are the mean, median and mode. The mode for Room 5 data is four, the category with the highest frequency.

How might we find the median?(The median is the middle number of people)You could mimic writing all the data points in the graph, like this:

3, 3, 3, 3, 3, 4, 4, 4, 4, ….6, 6, 6, 7, 7, 8+

How will we find the middle number? You could count in one point at a time from the top and bottom. However, halving the total number of data points will give the location of the median. There are 5 + 9 + 7 + 3 + 2 + 1 = 27 data points (adding the frequencies) and 27 ÷ 2 = 13.5 so median lies between the 13

^{th}and 14^{th}data points. Both points lie in the bar for a four-person household, so the median is four.How might we find the mean?(The total of all data points divided by the number of data points)It is easier to calculate the total of scores as (5 x 3) + (9 x 4) + (7 x 5) + (3 x 6) + (2 x 7) + 8 = 126. Note that 8+ is treated as eight though the actual number might be more than eight. 126 ÷ 27 = 4.66…

Therefore, the mean is closer to five people per household which reflects the fact that the median was ‘at the end’ of the four-person bar. It is interesting to reflect on the fact that two-thirds of a person is a mathematical rather than real idea, but it does suggest the centre is closer to five people than four.

For example, if five people live in a student’s household their clue might be:

“The median of 2, 9, 5, 8, 6, 3 and 4”, or “The mean of 1, 3, 4, 7, 7, 8.”

Distance from home to schoolIn the second part of the lesson students provide a clue about the location of their home. To do so they need to work out the distance of their home from school.

How far is 53mm in real life?(53 ÷ 32 = 1.67 and 1.67 x 200 = 334 metres in real life)How far do you estimate the total journey is from Kiwi Street to Hilltop School?(1.6 – 1.8 kilometres)Note that metres can be converted into kilometres. Do your students know how to do that?

^{2}metres or as 400^{2}cmSession Two allows you opportunity to assess students’ understanding of the following concepts:

## Session Three

In this session student investigate two more clues to add to their poster. First, they investigate the Scrabble total for their Christian name. Then they create a personal icon, a sculpture made from connecting cubes. The icon will be used in Session Five to check that the person you identify is who you think they are.

Scrabble TotalZoe’s letter score is 10 + 1 + 1 = 12. Hinemoa has a letter score of 4 + 1 + 1 + 1 + 1 + 1 + 1 = 10 and Kevin has a score of 5 + 1 + 4 + 1 + 1 = 12.

The Iconmust be asymmetrical. You may need to discuss with students that an asymmetrical figure has no symmetry, either reflective or rotational. Copymaster Three has three views of an icon. Invite the students to recreate this icon with their cubes. Look to see if your students:## Session Four

In this session students investigate a game involving chance. They play the game 25 times and express the result of their trial using a fraction, decimal or percentage.

Odds and Evens’students need a partner. One player wins if the product is odd. The other wins if the product is even. This is how play proceeds.What did you notice as you played the game?(Students should notice that the ‘even’ person wins more than the ‘odd’ person)Why does the ‘even’ person win more?(There are more possible outcomes that give an even product)How might we find out the actual chances of an odd or even win?What are the products of these events, even or odd?(odd x odd = odd, even x odd = even, and even x even = even). Students might conclude that the chances of an odd product are one out of three or one third. While some good reasoning is involved to get to this point the conjecture is incorrect since even x odd can occur in two different orders. To establish the probabilities students must consider all the outcomes that contribute to each event.Keep the two players secretly entering two digits, possibly using two calculators or digit cards. However, change the rules so the game is much fairer. How will you do that?

## Session Five

In this session students receive the poster of another student, solve the clues and identify the unknown student in their class. Students will need access to copies of the spreadsheet. Once they think they know who the person is they confirm the identity by matching the three views with the icon in the classmate’s desk or tote tray.

During the session students might identify several students.

You might extend the unit by asking:

What else could we find out about each other?How might we record clues about that information?## Cuisenaire Rod Fractions: Level 4

This unit introduces the fact that fractions come from equi-partitioning of one whole. So the size of a given length can only be determined with reference to one. Usually the one must be defined in context.

‘Fractions as measures’ is arguably the most important of the five sub-constructs of rational number (Kieren, 1994) since it represents fractions as numbers, and is the basis of the number line. Fractions are needed when ones (wholes) are inadequate for a given purpose. This purpose is usually some form of division. In measurement lengths are defined by referring to some unit that is named as one. When the size of another length cannot be accurately measured by a whole number of ones then fractions are needed.

For example, consider the relationship between the brown and orange Cuisenaire rods. If the orange rod is defined as one (an arbitrary decision) then what number do we assign to the brown rod?

Some equal partitioning of the one is needed to create unit fractions with one as the numerator. For the size of the brown rod to be named accurately those unit fractions need to fit into it exactly. We could choose to divide the orange rod into tenths (white rods) or fifths (red rods), either would work. By aligning the unit fractions we can see that the brown rod is eight tenths or four fifths of the orange rod.

Note that eight tenths and four fifths are equivalent fractions and that equality can be written as 8/10 = 4/5. These fractions are just different names for the same quantity and share the same point on a number line. This idea, that any given point on the number line has an infinite number of fraction names, is a significant change from whole numbers. Some names are more privileged than others by our conventions. In the case of four fifths, naming it as eight tenths aligns to its decimal (0.8) and naming it as eighty hundredths aligns to its percentage (80/100 = 80%).

‘Fractions as operators’ is another of Kieren’s sub-constructs and applies to situations in which a fraction acts on another amount. That amount might be a whole number, e.g. three quarters of 48, a decimal or percentage, e.g. one half of 10% is 5%, or another fraction, e.g. two thirds of three quarters. Students often confuse when fractions should be treated as numbers and when they should be treated as operators, particularly when creating numbers lines, e.g. they often place one half where 2 1/2 belongs on a number line showing zero to five.

## Specific Teaching Points

Understanding that fractions are always named with reference to a one (whole) requires flexibility of thinking. Lamon (2007) described re-unitising and norming as two essential capabilities if students are to understand fractions. By re-unitising she meant that students could flexibly define a given quantity in multiple ways by changing the units they attended to. By norming she meant that the student could then act with the new units. In this unit Cuisenaire rods are used to develop students’ skills in changing units and thinking with those units.

Multiplication of fractions involves adaptation of multiplication with whole numbers. Connecting a x b as ‘a sets of b’ (or vice versa) with a/b x c/d as ‘a b-ths of c/d’ requires students to firstly create a referent whole. That whole might be continuous, like a region or volume, or discrete like a set. Expressing both fractions in a multiplication and the answer require thinking in different units. Consider two thirds of one half (2/3 x 1/2) as modelled with Cuisenaire rods.

Let the dark green rod be one, then the light green rod is one half.

So which rod is two thirds of one half? A white rod is one third of light green so the red rod must be two thirds. Notice how we are describing the red rod with reference to the light green rod.

But what do we call the red rod? To name it we need to return to the original one, the dark green rod. The white rod is one sixth so the red rod is two sixths or one third of the original one. So the answer to the multiplication is 2/3 x 1/2 = 2/6

or1/3.Reunitising and norming are also important when fractions are placed in order of size (magnitude). This is especially true given any fraction has an infinite number of names. Imagine we use the orange rod as one this time and find two fifths. Since five red rods measure one whole (orange rod) then two red rods measure two fifths:

But, what other names does two fifths have? If the red rods were split in half they would be the length of white rods and be called tenths since ten of them would form one. The crimson rod is equal to four white rods which is a way to show that 2/5 = 4/10.

If the red rods were split into three equal parts the new rods would be called fifteenths since 15 of them would form one. The crimson rod would be equal to six of these rods which is a way to show 2/5 = 6/15. The process of splitting the unit fraction, fifths in this case, into equal smaller unit fractions, is infinite. This means that the point on the number line where two fifths exists has an infinite set of number names.

This unit is set for students to learn, and practise, outcomes at Level 4 of mathematics in the New Zealand Curriculum. Differentiation involves simplifying or adding more challenge to the tasks in the following ways:

The context for this unit is not real life. However, a story shell such as construction beams, waka lengths, or steps, might be used if there is potential to motivate students. Most students will enjoy the opportunity to work with Cuisenaire rods.

## Prior Experience

Students are unlikely to have previous experience with using Cuisenaire rods since the use of these materials to teach early number has been abandoned. Their lack of familiarity with the rods is a significant advantage as they will need to imagine splitting the referent one to solve problems.

## Session One

What is the size of the crimson rod compared to the brown rod? How do you know?The relationship between the crimson and brown rods can be expressed in two ways:

So if the brown rod was one then the crimson rod would represent one half. What fraction would the red rod and white rod represent? Convince us you are right.Encourage the students to express the relationships in various ways, such as:

How many quarters and how many eighths are the same length as one half (crimson rod)?How might we record these relationships mathematically?

Hopefully students will suggest recording an equality like this: 1/2 = 2/4 = 4/8.

What patterns do you see in the equality?Students should notice the doubling of both numerators and denominators. “Why does this happen?”

It is important to reinforce the idea that the numerator is a count, so the doubling of numerators indicates that there are twice as many parts in the same space.

But why do the denominators double? Does that mean that the parts get twice as big?Look for the students to notice that the denominators double because the parts halve in size, twice as many quarters as halves fit into one, twice as many eighths as quarters fit into one.

Ask:

Imagine we had these new rods, grey and pale blue. What could you say about these new rods in relation to one, the brown rod? How would you write the relationships mathematically? How are these fractions related to one half?Look for students to establish grey as one sixteenth either by halving one eighth or realising that sixteen of the parts fit into one. Look for students to recognise that three pale blue rods fit into one half so six of the rods will fit into one. Therefore, pale blue is one sixth of the one (brown rod). You might also highlight that one sixth is ‘One third of one half.’ From these observations other equivalent fractions for one half can be found (eight sixteenths and three sixths). Record these fractions as equalities:

1/2 = 8/16 and 1/2 = 3/6

What patterns can you see? Explain why the patterns occur.Look for student to use numerator as a count and denominator as the number of equal parts that fit into one. For example, three sixths is made up of three times as many parts (pale blue rods) as one half but three times as many of these parts are needed to make one.

## Session Two

Ask questions like:

If the blue rod is one what do we call the light green and white rods? Justify your answers.What statements can you make about the relative size of the rods?

Can you create an equivalent fraction to two thirds using rods in the picture (2/3 = 6/9)? How might we record this equality?

The total is more than one half because two quarters is one half and one third is longer than one quarter.

What rod could we use to measure all three rods exactly, light green (one quarter), crimson (one third) and grey (one)?Different rods might be tried but the white rod is the only unit that measures all three other rods a whole number of times.

:What fraction of one is the white rod?(one twelfth because 12 of those rods fit into one)How many twelfths are the crimson rod(one third)and the light green rod(one quarter)?How could we record these equalities?(1/3 = 4/12 and 1/4 = 3/12)So what is the total length of one third plus one quarter?(seven twelfths)How might we record this sum mathematically?(1/3 + 1/4 = 7/12)So where do the seven and the twelve in the answer come from?Students might notice that if the sum is written as 1/3 + 1/4 = 4/12 + 3/12 = 7/12 then the origin of the seven and twelve are clear.

Why are the four and three added to make seven but the twelve and twelve are not?Normally when 4 + 3 are added the units are the same, e.g. 400 + 300 (hundreds), 0.4 + 0.3 (tenths), and the answer is seven of those units. In this example the units are twelfths so we get “Four twelfths plus three twelfths equals seven twelfths.”

## Session Three

In this session the purpose is to find the difference between two fractions. Difference is the often neglected context for subtraction though problems can also be solved by adding on.

Lelani got two thirds of a whole Cuisenaire rod and Sala got one half of the same sized Cuisenaire rod. Who got the most and how much more of one rod did they get?Students should recognise that the difference is one sixth of the one (dark green). Ask how this problem might be recorded mathematically. Both addition and subtraction might be used:

1/2 + 1/6 = 2/3 or 2/3 - 1/2 = 1/6.

So why is the answer one sixth when neither of the fractions have sixths?Look for students to recognise from previous sessions that both one half and two thirds can be expressed as sixths. So an extra step in the subtraction equation gives:

2/3 - 1/2 = 4/6 - 3/6 = 1/6

From this recording, students might realise that recording difference as subtraction is a little tidier and certainly more conventional than adding on.

Visually we can see that the difference between the two rods is five sixths which matches the calculation: 5/3 - 5/6 = 10/6 - 5/6 = 5/6.

Approach OneDo the students start with two lengths, made with single rods (unit fractions) or collections of rods (non-unit fractions)?

Are they able to create a one (whole rod) that works for both rods?

Can they express each rod or collection of rods as a fraction of the one rod?

Do they recognise what unit can be the common denominator and rename each fraction?

Can they find the difference and record the result as a subtraction equation?

Approach TwoDo they choose or create a one rod that has a length with many factors?, e.g. 18cm is useful but 19cm is not so useful.

Do they map other rods into the one rod to create unit and non-unit fractions?

Can they name the fractions they create?

Do they use smaller rods to convert each fraction to equivalent forms before finding the difference?

Can they find the difference and record the result as a subtraction equation?

1/2 - 3/8 = 4/8 -3/8 = 1/8

3/4 - 2/3 = 9/12 - 8/12 = 1/12

4/5 - 1/3 = 12/15 - 5/15 = 7/15

Imagine someone new came to class who has not solved problems like this before. What would you tell them about how to solve any fraction difference problem?## Session Four

The aim of this session is to develop students’ mental number line for fractions. Inclusion of fractions with whole numbers on the number line requires some significant adjustments. These adjustments include:

If the blue rod is one (mark zero and one on the number line) where would one third be?

So what unit fractions would exist between zero and one third?(any unit fraction with a denominator greater than three since it will be less than one third).Note that there is no rod in the set that is one quarter, one fifth, one sixth, etc. of the blue rod. However imaging how long rods for these fractions would be is a useful activity in itself.

What fraction will this new rod represent?(eighteenths).Can we express some of these fractions as eighteenths?Can we find locations for numbers of eighteenths that are not showing yet?Students might realise that having eighteenths allows them for find fractions exactly in the middle between numbers of ninths, e.g. 13/18 is exactly half way between 2/3 (6/9) and 7/9.

What information is present?What helpful information is missing?

You need to see this problem as a riddle. There is enough information to put any fraction you want on the number line.What does to scale mean?

Look for the following:

Dear parents and caregivers,

This week your son or daughter will be learning about fractions, for example, three quarters and two thirds. We will be using some materials called Cuisenaire rods which are lengths of plastic or wood. They look like this:

Your son or daughter should be able to name fractions of a given rod. For example, they might say that the light green rod is three fifths of the yellow rod.

There is an online tool that lets you play with Cuisenaire rods on this page:

https://mathsbot.com/manipulatives/rods

## Cool Times with Heat

In this unit we use thermometers to investigate questions about temperature. We explore questions relating to cooling patterns, the effect of location on temperature, and the results of mixing of water with different temperatures

This unit is a useful connection to key science concepts such as experiment design, energy and insulation, and to recording, analysing, and reporting from data.

The focus of the unit is the attribute of temperature and how it is measured. Temperature is the amount of heat present in a substance. The standard unit of measurement for temperature in New Zealand is the degree Celsius (written ⁰C). Two benchmark temperatures ‘anchor’ the scale of degrees Celsius.

0⁰C is the freezing temperature of water at sea level, i.e. changes from liquid to solid (ice).

100⁰C is the boiling temperature of water at sea level, i.e. changes from liquid to gas (steam).

Temperatures below 0⁰C are recorded using negative numbers. For example, a temperature of -15⁰C is fifteen degrees below zero, the freezing point of water.

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

Tasks can be varied in many ways including:

The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Temperatures related to places students are most familiar with, or have interest in, will provide motivating contexts. Use locations that student may have visited on holiday, or have whānau living at. Draw on the everyday experiences of water temperature, such as running a hot bath, swimming in the sea, a lake or river, and making ice-cold drinks on a hot day. Relate temperature to dressing for different weather conditions. Use contexts from history such as traditional cooking in thermal pools, ways that pre-European Māori stayed warm, and importance of fire.

Session OneWhat are these devices?What do they do?When is measuring temperature important?(testing for illness, keeping food cool, knowing what to wear, etc.)How do liquid thermometers work?(As the liquid in the tube is heated it expands and takes up more space inside the tube)What is missing from this liquid thermometer?(Markings)What do the markings show?(Degrees in Celsius, sometimes Fahrenheit)How is zero degrees decided?(Freezing temperature of water)How is 100 degrees decided?(Boiling temperature of water)You might look online to learn how the Celsius scale was developed, and why it is sometimes referred to as “centigrade.”

What is room temperature at present?Invite estimates then check by giving each group a thermometer.

How accurate are our thermometers?You might gather data about the temperature measurements of room temperature. Expect some variation as inexpensive thermometers are less accurate than scientific thermometers.

Time (sec)

0

30

60

90

120

150

180

»

Hot Water

45

43

41

40

39

38

37

»

Cold Water

6

8

10

11

13

14

15

»

What trends appear in the graphs?What will happen in the next ten minutes? Why?Will the lines cross? Why, or why not?Does the data match your prediction?Why do the lines appear to be levelling off?A key idea is that loss or gain in temperature of the water is a result of heat transfer between the water and the surrounding air. If the difference in temperature is great, then the rate of cooling or heating is greatest. As the water gets closer to room temperature the rate of cooling or heating gets less. For example, a hot cup of milo cools fastest straight after it is made.

The average temperature of water in Lake Taupō is ten degrees Celsius (10° C). The water in a bathing pool in the Polynesian Baths (Rotorua) is at a temperature of 32° C. Using the materials you have, mix two containers of water, one at each of these temperatures?If both cups were left for two hours, what temperature would each cup of water be then?(Both will approximate to room temperature.)Why?## Session Two

For the next 3 days the students will use thermometers to investigate questions about temperature. The first questions relate to cooling patterns.

Does a large amount of water cool faster or slower than a smaller amount?Does wrapping a container of water in paper result in it cooling faster or slower?How is the idea of wrapping containers of hot water useful in daily life?(Hot water cylinder cladding; wrapping of pipes in cold climates.)How is the idea of wrapping used by us in everyday life?(Clothing, insulation of houses, etc.),can be used to provide a dataset for this activity.## Session Three

What are cities?What cities have you visited when you were on holiday?What was the city like? What highlights do you remember?How did you know what clothing to take when you visited?What factors make a difference to the temperature of a city?Students will raise the idea that seasons influence temperature. Discuss the rotation of seasons and how that cycle relates for the axis rotation of the Earth as it orbits The Sun. There are many excellent videos depicting the seasons.

Temperature changes with the seasons. Are there permanent features that influence the temperature of a city all year round?

Students might raise factors such as closeness to the equator (measured by latitude), proximity to the coast, and altitude (measured by height above sea level). Make a list of the factors that students believe have a long-run effect on temperature.

We are trying to find out what factors affect the weather in a city over the long term.Let’s ask some questions of the data.Invite questions such as:

Are cities further from the coast/higher in altitude/further from the equator cooler than those further inland?Are cities in the Northern hemisphere hotter or colder than those in the Southern hemisphere.Do inland cities have a bigger average temperature range, between summer and winter, than coastal cities?What do you expect the average temperature of Melbourne, Perth, Darwin, Brisbane, Alice Springs, Brisbane, Broome, Hobart to be like?Will there be much variation between winter and summer in these cities?Students may remember big variations in the temperature of Hobart and Melbourne caused by cold Southerly flows and hot Northerly flows. Temperatures in the tropical cities of Darwin, Broome and Brisbane tend not to vary much from winter to summer but humidity levels change considerably.

You may discuss the impact of humidity, and wind chill, on how temperature feels.

## Session Four

In this session students investigate the use of integers (positive and negative numbers) in the Celsius scale.

How cold is negative thirteen degrees?Has anyone in the class experienced a temperature like that?How many degrees is each division on the scale? (10⁰C)Investigate.Why do parents and doctors often take your temperature when you are unwell?If you have a temperature of 40⁰C should you seek medical advice?Explain.## Session Five

Have you swum in the sea and noticed that it was cold even though the weather was warm?Why was the water so cold when the air was warm?Students will recall experiences when the water was hot and cold.

Today we will investigate what happens when we mix hot and cold water.I have two cups with the same amount of water in each. The water has a temperature of 30° C in one cup and 10° C in the other. If I mix these two cups of water together, what temperature will the mixture be?While some inaccuracies may occur due to the temperature transfer, in general the temperature of the mixture should be the mean (average) of the waters being mixed, i.e. 30° C + 10° C = 40° C, 40° C ÷ 2 = 20° C. On a number line the mean looks like appears as the central point between the two measurements.

Simple averaging works when the amounts of water in each cup are the same.

How do you work out the mixture temperature when the amounts of water in each cup are different?Extra problemsProblem One:Are we good at estimating water temperature?Problem Two: Why do people wear light coloured clothing in the summer and dark clothing in the winter?Problem Three: What times of the day are the hottest? What times are the coldest?Family and whānau,

This week we have been looking at temperatures. We’d like you to do an experiment at home that can be reported to the class.

Temperature ProblemTake three ice blocks that are as close to being the same as possible.

Put one on a plate, one in a glass of cold water from the tap and one in a glass of hot water from a kettle.

Guess:

Bring your results to school by Thursday, and we’ll put all the data together to produce a class graph.

## Figure It Out

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

## What's going on? Properties of multiplication and division

This unit develops students’ recognition of pattern (consistency) in equations involving multiplication and division with whole numbers.

This unit develops students’ recognition of pattern (consistency) in equations involving multiplication and division with whole numbers. The patterns of pairs of equation embody important properties of multiplication and division, such as commutativity, distributivity, and inverse. Students learn to represent specific examples where the properties are used 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 whole numbers under multiplication and division.

In this unit we build on research by Deborah Schifter, 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 the approach are as follows:

In this unit claims are developed through equation sets involving multiplication and division. The sets aim at developing students’ understanding of the properties of multiplication (commutativity, distributivity, associativity, identity and inverse). By expanding the equation sets to include division students will learn how these properties hold or do not hold when the operation is changed.

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. The difficulty of tasks can be varied in many ways including:

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. For example, if you have a big set of shells for environmental studies you might use those shells as the materials Contexts may fall out of the preferred materials. Kaitiakitanga (guardianship over the environment) might be supported by finding clever ways to count the number of toheroa on a beach. Whānaungatanga (family) values might involve finding fair and equitable ways to share shellfish that are harvested. Note that equal shares assumed in the operation of division are not always aligned to values of fair sharing.

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 contains seven equation sets that drive the unit. The sets might form the basis of a week-long unit. The phases for each equation set are described below. The sets are labelled in the top left corner of each slide for reference.

Equation Pairs Set OneSlide one has the first pattern to look at. The pattern involves the commutative property, i.e. a x b = b x a, which students should be familiar with. It is used as an example to familiarise students with the approach.

Noticing RegularityUse ‘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 numbers are just turned around, like 9 x 4 becomes 4 x 9.T: Can you be more specific. Which numbers are turned around?S: The numbers being multiplied each time.Discussion opens the possibility of using correct mathematical terms, like factor (number being multiplied), and product (answer to multiplication).

S: The products (answers) are always the same.T: All four patterns have the same product? What do you mean?S: No. The products are the same when the factors are turned around.Articulating 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 factors are the same, and you turn them around, the product doesn’t change.The first factor changes places with the second factor. The product is the same.Your aim is for students to express their claims in clear, minimal terms, using correct mathematical language. For example, ‘turning around’ is not as clear as order of the factors.

RepresentingIn this phase students choose representations to show why the pattern holds consistently. Students might choose physical manipulatives, such as linking cubes or counters, draw diagrams such as number lines or arrays, and use contexts from everyday life. Encourage students to begin with the first two examples of equation pairs then consider how the same relationships might generalise to the last, and other similar, equation pairs.

Examples might be:

I made 7 x 5 first using cubes. Then I took one off each five to make a seven. I found out I could make five stacks of seven from the seven fives.I drew 7 x 5 as an array. The fives were the columns. When I turned the array around the sevens became the columns, but the total number of cubes was the same.Explain where the 7 and 5 are in your representation.If you start with 7 x 5, why can you only make five sets of seven?What do the 5 and 7 represent in 5 x 7?How does your representation show that the product stayed the same?Constructing an argumentIn 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 to deal with what occurs in general.

S: With 7 x 5, one from each five makes a set of seven. Because there are five in the sets that means exactly five sevens can be made.T: So how does that work in the same way with 9 x 4, 8 x 99, and 5 x 36?This might lead to expressing the property in general terms.

S: The first factor multiplied by the second factor has the same product as the second factor multiplied by the first factor.T: If we gave names to the first and second factors, like a and b, could we express the property more simply?Some students might experiment with algebraic notation such as a x b = c so b x a = c. Note that this represents the starting equation pairs. In general, a sets of b can be remodelled. Taking one object from each set of b creates sets of size, a. This can be done b times, resulting in b x a (b sets of a).

T: Do we need to say both equations have an answer of c? Do we need c?S: We could just write a x b = b x a.Focus on the class of numbers that have been used, i.e. whole numbers. Encourage students to investigate if the commutative property holds for integers and rational numbers, e.g. If ½ x 36 = 18 does 36 x ½ = 18.

Equation Pairs Set TwoAsk the students to approach the second equations set more independently. From this point each equation set is discussed succinctly using the phases of the approach.

Noticing regularityThe four equations apply doubling and halving, thirding and trebling, etc. of the factors in the first equation. This strategy is sometime called proportional adjustment since it underpins the concept of equivalent fractions. The completed sets should be:

8 x 3 = 24 6 x 10= 60

4 x 6 = 24 [Doubling 3, halving 8] 12 x 5 = 60 [Doubling 6, halving 10]

9 x 9 = 81 7 x 6 = 42

27 x 3 = 81 [Trebling 9, thirding 9] 14 x 3 = 42 [Doubling 7, halving 6]

Articulating a claimIn natural language expect the students to use phrases like “one number doubles, the other halves”. Introduce important vocabulary such as

factorandproductto clarify what numbers are being referred to in the claims. If the claim is restricted to doubling and halving draw attention to 9 x 9 = 81 and 27 x 3 = 81. The aim is to broaden the claim to the equivalent of “one factor is divided by n, the other factor is multiplied by n. The product stays constant (the same).”RepresentationExpect both physical and diagrammatic representations to be used. A cube stack representation might look like this:

Diagrams of a ‘sets’ representation might look like this:

Arrays might also be used as a powerful representation. Initially cubes might be used as units of area leading to a more abstract use of side lengths.

ArgumentationLook for students to justify that a given quantity, say 24, can be created by multiplying two factors, say 4 x 6. Keeping our quantity constant, one factor can be divided in equal parts, say each 6 is divided into three equal parts (3 twos). Now there are three times more of those parts making up 24. The number of parts a factor is divided into is a variable. Some students may be comfortable with using a label, like n, to represent the number of equal parts. The factors and product are also variables and might be represented with shapes or letters. Algebraically the relationship might be expressed as:

Look for students to generalise ‘undoing’ nature of the inverse operations, i.e. divided by n, multiplied by n. At this level students are progressing towards the use of letters to represent variables. You might also introduce the quotient interpretation of fractions, e.g.

a÷ncan be expressed asa/n. Multiplication can also be represented without the x symbol, e.g.a×bcan be represented asab.Equation Pairs Set ThreeNoticing regularityThe four equations apply the distributive property. This property is used a lot in the multiplication of multi-digit numbers. The completed sets should be:

7 x 10 = 70 5 x 20= 100

7 x 11 = 77 [Adding 7 x 1] 5 x 22 = 110 [Adding 5 x 2]

9 x 50 = 450 6 x 100 = 600

9 x 53 = 477 [Adding 9 x 3] 6 x 105 = 630 [Adding 6 x 5]

Articulating a claimIn natural language expect the students to use phrases like “adding on so many lots of the number”. Expect the use of mathematical vocabulary such as

factorandproductto clarify what numbers are being referred to in the claims. Encourage clarity by asking questions like:Can you know how much more the second product is than the first? How?What does the first factor mean? What does the second factor mean?The aim is to state the claim as something like “If a number is added to the second factor, then the product increases by the first factor multiplied by that number.”

RepresentationExpect both physical and diagrammatic representations to be used. Since most the first equations involve multiples of ten or 100, place value blocks (MAB) might be useful.

Diagrams of a tens and hundreds can be made schematic to highlight important structure.

Arrays illustrate how the first factor ‘acts’ on the second factor as it is changed.

ArgumentationLook for students to justify that two factors multiply to a given product. The starting product might be expressed as a x b. Adding a number to b results in the second factor becoming b + n (n is the number being added). The product increase by a x n. It is important for students to consider what is happening with all four equation sets, in that n is a variable, and can be ‘any number.’ Students are working towards expressing relationships among variables using letters and equations. Progress can be encouraged by working with the notations that students develop themselves.

Algebraically the relationship might be expressed as:

a×b=c so a×(b+n)=(a×b)+(a×n)Multiplication can also be represented without the x symbol, e.g.

a×bcan be represented asab,and unnecessary brackets (due to order of operations) can be removed.ab=c so a(b+n)=c+anDiscuss the removal of unnecessary variables as well. c is redundant as ab expresses the product. This reduces the property to:

a(b+n)=ab+anEquation Pairs Set FourNoticing regularityThe four equations apply the inverse relationship between multiplication and division. This property is used by students to solve division problems by measurement, e.g. “How many x’s go into y? The completed sets should be:

8 x 6 = 48 7 x 3 = 21

48 ÷ 6 = 8 [expressing as division] 21 ÷ 3 = 7 [expressing as division]

12 x 25 = 300 68 x 9 = 612

300 ÷ 25 = 12 [expressing as division] 612 ÷ 9 = 68 [expressing as division]

Articulating a claimIn natural language expect the students to use phrases like “the factors are being put into a division equation.” Students might indicate what they see in a diagram.

Expect the use of mathematical vocabulary such as

factorandproductto clarify what numbers are being referred to in the claims. You may need to introduce division terms likedivisor(number being divided by),quotient(answer to division) anddividend(the quantity being divided). Encourage clarity by asking questions like:What does the second factor become in the division equation? (divisor)What does the first factor become in the division equation? (quotient)What does the second factor mean?The aim is to state the claim as something like “Two factors multiply to give a product. The product divided by one factor equals the other factor.”

RepresentationExpect both physical and diagrammatic representations to be used. Be aware that students may interpret division in two ways, as equal sharing (most common) or as measuring. Either interpretation can be used to represent the equations. Here is a measurement interpretation since 48 ÷ 6 is seen as “How many sixes are in 48?”

A sharing view interprets 48 ÷ 6 as “48 is equally shared among six parties. How much does each party get?”

Schematic diagrams, like arrays, show the factors as side lengths, and the product as the area. The missing number in an equation can be shown as an empty measure in the diagram.

ArgumentationLook for students to justify that if a product, a x b, is the multiplication of two factors a and b, then the product can be divided into a sets of b or b sets of a. So the product in multiplication can be thought of as a dividend in division. Either factor can be the divisor, but the other factor is the quotient.

Look for students to accept that the factors and product are variables, meaning they can take up any value. The product is dependent on the factors so can always be represented as a x b, or ab.

Some students may use the repeated addition view of multiplication like this:

7 x 3 means seven sets of three, so seven sets of three can be made from 21.

a x b means a sets of b, so a sets of b can be made from ab.

Equation Pairs Set FiveNoticing regularityThe equation set applies proportional adjustment, halving, of the dividend and explores the effect on the quotient. This property can be used by students to solve division problems. The completed sets should be:

24 ÷ 4 = 6 40 ÷ 5 = 8

12 ÷ 4 = 3 [halving of dividend and quotient] 20 ÷ 5 = 4

264 ÷ 11 = 24 72 ÷ 9 = 8

132 ÷ 11 = 12 144 ÷ 9 = 16

Articulating a claimIn natural language expect the students to use phrases like “halving the number being divided halves the answer.”

Expect the use of mathematical vocabulary associated with division, like

divisor(number being divided by),quotient(answer to division), anddividend(the quantity being divided). Encourage clarity by asking questions like:What changes in each pair of equations?What stays the same?Look at equation four. It is different to the others. How?The aim is to state the claim as something like “Multiplying or dividing the dividend by a number, while keeping the divisor the same, results in the quotient being multiplied or divided by the same number.”

RepresentationBoth equal sharing, or measuring interpretations of division can be used to model the relationships in the equation pairs. Sets models, like stacks of cubes, work well but encourage the progress towards schematic diagrams like arrays.

Note that the fourth equation pair shows the effect of multiplying both dividend and quotient by two.

ArgumentationLook for students to justify, using words or diagrams, that if a dividend, ab, is divided by a divisor, b, then the quotient, ab ÷ b or

ab/b, determines now many b’s measure ab. If ab is divided by two, then the number of b’s that can fit intoab/2equals half ofab/b, written asab/2b.Look for students to accept that the dividend and divisor are variables, meaning they can take up any value. The quotient is dependent on those variables, so can always be represented as ab ÷ b or just a.

Some students may transfer the repeated addition view of multiplication to division like this:

8 x 5 means eight sets of five, which totals 40. So, with half that total, 20, it is possible to make half as many fives.

If the factors are regarded as variables, then a more general finding might be ‘proven’ geometrically with arrays.

If a amounts of b equal ab, then

a/2amounts of b equal half of ab (ab/2).Equation Pairs Set SixNoticing regularityThe equation set highlights the difference of squares. This property can be used by students to solve multiplication problems. The completed sets should be:

9 x 9 = 81 5 x 5 = 25

8 x 10 = 80 [One less than 81] 4 x 6 = 24 [One less than 25]

20 x 20 = 400 12 x 12 = 144

21 x 19 = 399 [One less than 400] 13 x 11 = 143 [One less than 144]

Articulating a claimIn natural language expect the students to use phrases like “the first equation has the same factor times itself. The second equation has one more and one less and the answer is one less.” Ask students to be more specific in their description.

Which factor is increased by one?Which factor is decreased by one?Is this true for all four equations?The aim is to state the claim as something like “If a number is chosen, one less than the number, multiplied by one more than the number, equals the square of the number less one.”

RepresentationStudents are likely to use specific examples to convince other about how the relationships work.

ArgumentationArrays are a useful representation to ‘prove’ what occurs with the difference of perfect squares.

In general, a

^{2}can be transformed spatially into (a-1)(a+1) by removing one (1 x 1) and moving the unit of (a-1) to create rectangle with sides of (a-1) and (a+1).Students are not expected to use algebraic notation at this level. High achieving students might like to see that the transformation can be represented as:

a^{2}-1=a^{2}+a-a-1=(a-1)(a+1)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 regularityThe equation set highlights the division equivalent of the distributive property. The dividend is reduced by a multiple of the divisor. This property is used to solve division problems by rounding the dividend up, e.g. 76 ÷ 4 by calculating 80 ÷ 4 first. The completed sets should be:

80 ÷ 8 = 10 60 ÷ 5 = 12

72 ÷ 8 = 9 [One less set of 8] 55 ÷ 5 = 11 [One less set of 5]

270 ÷ 9 = 30 700 ÷ 7 = 100

261 ÷ 9 = 29 [One less set of 9] 686 ÷ 7 = 98 [Two less sets of 7]

Articulating a claimIn natural language expect the students to use phrases like “the first equation has a dividend divided by a divisor. The second equation has one or two times the divisor taken off the dividend, so the quotient is one or two less.” Ask students to be more specific in their description. Diagrams might be useful.

Ask questions like:

Show how the dividend is decreased by one or two times the divisor.What is the effect on the quotient?Is this true for all four equations?The claim using mathematical language might be something like, “If the dividend is reduced by one times the divisor, then the quotient is reduced by one.” Note that the fourth equation pair reduces the dividend by two times the divisor.

RepresentationGiven the dividends are multiples of ten or 100, place value blocks (MAB) might be a suitable representation.

Schematic diagrams, like arrays, provide a clearer view of the structure found in examples.

ArgumentationStudents might go back to the repeated addition view of multiplication to convince others about their claim. A specific example might look like this:

See if students can take the specific examples and turn them into a more generalised form, by using letters or other symbols to present the dividend as the product of two variables:

## Measuring up

In this unit the students will collect statistical data about their own class and school and learn how to compare it to data from students from CensusAtSchool.

By Level 4 students are able to take increasing responsibility for the planning and conducting of statistical investigations. Students should be capable now of incorporating technology into their work.

Informal measures of centre and spread (at curriculum level 4)Formal measures of centre and spread are introduced at curriculum level 5. If your students are ready to explore these then look to curriculum level 5 activities. At curriculum level 4 we introduce the ideas of the middle and the middle 50% of numerical data. We can match the middle of the data on a graph up with the median in an informal sense using technology. We can identify the middle and the middle 50% using technology or marking by eye on a physical graph.

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:

The context for this unit can be adapted to suit the interests and experiences of your students. For example:

Session 1: Measuring the classPROBLEM: Generating ideas for statistical investigation and developing investigative questionsResearch online information about sizing guides as well.

Note: that some students may be sensitive about being measured. It is not appropriate to measure weight.Ideas around the ethics of data collection are attended to in 4. Students who are not wanting to be measured could be put to use as the measurer for a measurement station, thereby detracting from their non-participation in the measuring.interrogatethe ideas by checking them using the following questions:ask ofthe data. They should identify the variable of interest (e.g. height, arm length) and the group of interest (the class). Some groups might want to explore a categorical variable. This is ok as later they will have to do a measurement variable.PLAN: Planning to collect data to answer our investigative questionSession 2: DATA: Collecting and organising dataANALYSIS introduction: Using an online tool to make data displaysIn the remaining time for the session, the students will be introduced to using an online tool for data analysis. One suggested free online tool is CODAP. Feel free to use other tools you are familiar with. This is written with CODAP as the online tool and is assuming students have not used CODAP before. If your students are familiar with CODAP then they can move straight into analysing the data from the class survey.

If you do not want to use an online tool then head to the making displays part and progress with paper versions of bar graphs, dot plots and histograms.

Learning how to use CODAPSession 3: ANALYSIS: Displaying and describing the dataStudents use data from the previous session to produce graphs in CODAP or similar statistical analysis software.

Example of a categorical graph in CODAP:

Example of a numerical graph in CODAP:

We might explore what is missing from the height graph (units) and show students how to include the units in the graph.

Click on the variable in the table or case card view. This gives a pop-up menu.

Select edit attribute properties, and in the pop-up window type in the unit (cm), and click apply.

This updates the graph to show the units.

Students can update all the measurement variables in the table to include the units of measure.

Click on the graph to bring up the tool bar. Select the graph icon and then select Fuse Dots into Bars to make a bar graph.

Click on the graph to bring up the tool bar. Select the graph icon and then select Group into bins (note the different option when numerical data is recognised).

This action groups the dots into bins. Click on the graph icon again and now the options for bin width, alignment and fuse dots into bars comes up. Generally, go with the default settings for bin width and alignment; then select fuse dots into bars.

The resulting graph is:

Students can explore changing the bin width and the alignment. What happens when the alignment is changed?

Students can compare the dot plot with the histogram and notice what is similar and what is different. What are the advantages of the dot plot? What are the advantages of the histogram?

Remind them that they can use multiple displays to show different features of the data to answer their investigative questions.

CONCLUSION: Answering the investigative question and reporting findingsSession 4: Comparing to students like us in New ZealandThe class will use data from CensusAtSchool to compare to their class results.

Note also that the different group sizes might be a sticking point for students as they do not think they are able to compare different sized groups. Get them to focus on the summary information e.g. where is most of the data, where does the data peak, what is the biggest value, the smallest value, the middle value, how do these compare? These were the suggested features from the description of their class data.

Note: If the students have all downloaded their own individual samples from CensusAtSchool the discussions each student makes could be quite different. If you want them all to have the same sample from CensusAtSchool you can download a sample yourself, import into CODAP and then share the CODAP document with your students (see this video on saving and sharing CODAP documents).

Session 5: ANALYSIS: Going deeper - investigating the middle and the middle 50% of our data using CODAPNote that measures of centre are not introduced in

The New Zealand Curriculumuntil level 5. At curriculum level 4 we introduce informal ideas of the middle and the middle 50%Describing the middleStudents to work with their class data initially. The example given is for heights, but the ideas are the same for any numerical data.

Introducing the idea of the median, this is the middle of the datacm(always include the unit).The median height for students in our class is __________ cm.Repeat the idea for the data they have from CensusAtSchool

The median height for students in the CensusAtSchool sample is _________cm.This special value, the median, can be found using CODAP measures tool. Click on the ruler and then select median. They should get a red line showing the median. By hovering over this red line, they can find the median value, e.g. for first height example see picture below. They can also notice how close their guess at the middle was to the actual median (the middle of the values when placed in order from smallest to largest).

Introduce the idea of the middle 50% of the dataThe “signal” for the data is often where the middle 50% of the data is.

cmfor the bottom value and 163.5cmfor the top value.Repeat idea for the CensusAtSchool sample heights data.

Making comparisons between our class data and the CensusAtSchool sample dataWith these two additional pieces of information – the middle (median) and the middle 50% update your discussion around the similarities and differences between the class data and the CensusAtSchool sample data.

Dear parents and whānau,

At school this week we will be measuring the heights of the students in the class and comparing our results to the heights of children overseas. As another comparison we would like to be able to talk about the heights of family members. Could you please help your child measure the heights of their family members and record the information to bring to school?

Can each person in the family answer the following survey question:

How tall are you without your shoes on (answer to the nearest cm)?Thank you.

## Figure it Out Links

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