Prior Experience

Use the sessions to informally assess students' mathematical skills and understanding at the start of the school year.

Session One

1. Welcome your new class to the first week of mathematics adventures. Explain that the goal for the end of the week is to provide a classmate with a poster that will allow them to figure out who you are and learn a bit about you. That is why the name of the unit is Figure me out. A template to fill in is included as Copymaster 1.
    
2. In the first session students will add two clues to their posters; their gender and their date of birth. That sounds easy, but let's make it a bit more difficult for the person getting the poster at the end of the week. 
   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. 
    
3. Ask: 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?
    
4. Another way to look at the problem is to find if two factors, other than 1 x 51, have a product of 51. Let the students discuss this in pairs. Talk about the ways to check if 51 is prime:
   • Get 51 objects and try to sort them into equal sets. That would work but could be time consuming.
   • Go through a times table chart to see if 51 occurs in the products. Why won’t that strategy always work? (51 will not show if a factor is greater than 10 or 12)
   • Use a calculator. Nice idea but how do you use the calculator? You could try all the multiples of two, then three, then four, etc. That would take a long time. Maybe a student might suggest using division. 51 ÷ 2 = 25.5, there is a remainder (0.5) so two is not a factor of 51. But 51 ÷ 3 = 17, there is no remainder so 3 x 17 = 51. Therefore, the person is a female.
      
5. Ask: 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.
    
6. If students are not familiar with divisibility by three you may want to use elements from the Nines and threes activity from the Numeracy Project series.
    
7. In the case of a student who does not want to identify with being either male or female, they could choose a number that is neither prime nor a multiple of three.
    
8. Invite the students to make their first entry onto their poster in the Gender space. The number they put must be greater than twenty but less than 100, so the problem is sufficiently challenging. Allow students to use calculators to test out numbers they select. This time provides opportunities to assess students’ multiplicative thinking.
   • Do they need materials, like counters, to check for factors?
   • Do they use division rather than trial and error to find if a number has factors?
   • Do they recognise that a decimal in the quotient (division answer) indicates a remainder?
   • Do they know that a remainder indicates non-divisibility?
      
9. At this point you need to create a spreadsheet to gather data on your students. You may have a speedy typist in the room who can enter the data or rely on each student to enter their own. The spreadsheet will be needed on the final day of the unit. 
    
10. Move on to the date of birth. If a person wrote 12/04/11 for their DoB, what would that mean? Students should recognise that the date of birth (DoB) is 12^th of April, 2011. Taking out the forward slashes, this date of birth can be written as 120411.
    However, on their poster a person might write one of the following:
    • (12 x 10 000) + (4 x 100) + 11     
    • 333² + 9 522   or (3¹⁰ x 2) – 3⁸ – 3⁶ – 397            
    • (9! ÷ 3) – 549
       
11. Discuss what each expression means. Important points are:
    What is 10 x 10 000? (100 000)? So what is 12 x 10 000? (120 000)
    What does 333² mean? (333 x 333 or two 333’s multiplied together) You may need to show simpler examples of exponents like 4², 5³ and 2⁴.
    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)
     
12. Get the students to use calculators to check that each expression equals 120411. Watch for students to recognise the importance of order of the operations. The expressions included parentheses (brackets) that are unnecessary since exponents precede multiplication and division in operational order, that precede addition and subtraction.
     
13. Let the students come up with an expression that calculates to their DoB and add it to their poster. Encourage them to be creative and use some of the ideas in the three expressions shown. Also update the spreadsheet with the dates of birth.
     
14. For early finishers pose this problem:
    A classmate gives you this clue for their date of birth. Use your calculator and research skills to find their DoB.
    3¹⁰ + 7⁶ – (2 x 5⁵) + 4³          (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 Household

1. Begin with the PowerPoint. Slide One shows a graph. Explain that this graph is something about the homes that students in another class live.
   What might the graph be about? Can we give the graph a title and some labels?
    
2. Let your students discuss what the graph might be about. It is likely they will realise that the size of the numbers prohibits ideas like letterbox numbers or number of cars. The graph is about the number of people that normally live in each student’s house. Discuss a good graph title and labels for the axes. Slide Two provides those additions. The vertical axis shows frequency, the number of data items in each category. Ensure students know this.
    
3. Ask: 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.
    
4. Ask: 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 the 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.
    
5. Ask: 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.
    
6. Use post-it notes to gather the data for your class. Get each student to provide data about how many people normally reside in their household and write the number on a note. You may need to discuss ‘normally’ as some households are very transient. Get students to organise the notes into a bar graph on the whiteboard. Find the mode, median and mean for your class and compare the distributions.
    
7. Finally, ask students to add to their poster by giving a clue about the number of people who live in their household. Encourage them to use what they have learned about averages.
   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 school

In 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.

1. Provide students with a photocopied map of the local area complete with scale. This video gives an example of working out the distance between home and school. Useful questions are:
   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. Once students have established their home to school distance ask them to create an expression for that section of the poster. For example, 1.6 kilometres might be written as 40² metres or as 400² cm

Session Two allows you opportunity to assess students’ understanding of the following concepts:

• Multiplicative thinking – Do they apply multiplication and division to find averages and create expressions?
• Proportional thinking – Do they apply scale correctly to work out distances from a map?
• Measurement system – Do they convert easily among metres, kilometres, and centimetres?

Session Three

In this session students investigate two more clues to add to their poster. First, they investigate the Scrabble total for their given 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 Total

1. Introduce the famous ‘four fours’ puzzle. In the puzzle you find expressions that contain four fours, with any number and operation symbols you like, that represent the numbers 0-100. For example, 4 + 4 – 4 – 4 is an expression for zero and 44 ÷ 4 + 4 = 15. Remind the students that the rules for order of operations must be followed, e.g. multiplication and division before addition and subtraction. Students also need to recognise that division can be expressed as fraction notation, e.g., 44/4 = 44 ÷ 4 = 11.
    
2. Challenge your students to come up with four fours expressions for the numbers 1-10. Expect answers like:
   
    
3. The options for ten bring out an interesting idea about dividing by decimals. Four divided by point four (4 ÷ .4) is equivalent to asking, “How many lots of four tenths fit into four?” Since four equals forty tenths the question can be reworded as, “How many lots of four tenths fit into forty tenths?” Note that the quotient is ten which is larger than the dividend of four. This will be challenging for many students.
4. Copymaster 2 contains a graphic of the tiles used in the game of Scrabble. Some students may not be aware of how the game is played so you might use the paper version to show them. Ask the students to create their given name with scrabble tiles. Here are some examples:
   
   Zoe’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.
    
5. Once they make their name students need to create a ‘four fours’ clue for that total to put on their poster. For example, Kevin or Zoe might write (44 + 4) ÷ 4.

The Icon

1. Give each student 15 connecting cubes to create their personal sculpture. The icon must be able to sit on a desk, so it can be drawn from different viewpoints, and it must be asymmetrical. You may need to discuss with students that an asymmetrical figure has no symmetry, either reflective or rotational. Copymaster 3 has three views of an icon. Invite the students to recreate this icon with their cubes. Look to see if your students:
   • Start with one view and create an icon that fits.
   • Move progressively to the two other views, one at a time, to create an icon that works.
      
2. Next, ask your students to form their own asymmetric icon and draw three different views of it for their poster. The icon should be stored in a desk or tote tray, so it can be used in Session Five.

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.

1. To play the game ‘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.
   1. Each player chooses a digit from 0-9, this could be by drawing numbers from a hat or by randomly picking a digit card. For example Player A chooses 3 and Player B chooses 9.
   2. The players reveal their chosen digits and multiply the numbers, e.g. 3 x 9 = 27.
   3. If the answer is odd, e.g. The ‘odd’ player wins, and the even player loses that round.
       
2. After 25 games each player records their success rate on the poster as a fraction, decimal or percentage. For example, if the player was ‘odd’ they might get 10 out of 25 wins. 10/25 = 2/5 = 0.4 = 40%. They also record their raw score, e.g. 10, in the class spreadsheet.
    
3. After students have played the game gather the class and discuss:
   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?
    
4. Students may suggest that there are three events that might happen, odd x odd, even x odd, and even x even.
   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.
    
5. One way to do that is to create a model. Students may want to create a 10x10 array where they can work out which of the possible products are odd. There are 100 outcomes but only 25 of them produce odd products. The actual probability of getting an odd product is 25/100 = 1/4. Students that are more comfortable with the properties of numbers may identify that they only need to use a 2x2 array identifying whether each factor is odd or even.
    
6. Students might like to compare their success rate with the theoretical chances. They may observe that the results vary a lot from the predictions.
    
7. Finish the lesson with a challenge:
   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?
    
8. Students might suggest that the game will be more even if:
   • Player A wins when the product is less than 20. Player B wins if the product is 20 or more.
   • Player A wins when the product is the result of even and odd digits. Player B wins if the product is the result of odd x odd or even x even.
   • Add the digits instead of multiplying them. Player A wins if the sum is odd. Player B wins if the sum is even.

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?

Prior Experience

Students may have limited experience with using Cuisenaire rods. Their lack of familiarity with the rods is a significant advantage as they will need to imagine splitting the referent one to solve problems. When introducing the Cuisenaire rods, ask students to think about what they could be used to represent in mathematics. Value the contributions of all students. 

Session One

Consider providing a story context to frame the learning in this session (e.g. construction beams, waka lengths, length of a running track). Links will be engaging when they reflect the cultural diversity and/or learning interests of your students. This context could stay consistent in every session. Alternatively, you could introduce each session with a new relevant context to enhance students’ understanding of the mathematical concepts covered in these lessons, to real-world situations.

1. Use Cuisenaire rods or the online tool to introduce equivalent fractions in the following way. Use a digital display, whiteboard, or large sheet of paper to record students' ideas. 
   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:
   • “The crimson rod is one half of the brown rod.”
   • “The brown rod is two times the length of the crimson rod.”
2. Ask: 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? How do you know this is true?
   
   Encourage the students to express the relationships in various ways, such as:
   • “The red rod is one quarter of the brown rod because four of it fit into one (brown rod).”
   • “The brown rod is four times longer than the red rod.”
3. Ask: 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.
4. Ask: What patterns do you see?
   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.
   
5. Ask: 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 - meaning twice as many quarters as halves fit into one, and twice as many eighths as quarters fit into one.
6. Use slide one from the PowerPoint for this unit.
   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. Discuss the pale blue rod and its relation to the brown and grey rods. Equivalent fractions for one half can be found. Record these fractions as equalities:
   1/2 = 8/16 and 1/2 = 3/6
7. Ask: What patterns can you see? Explain why the patterns occur.
   Look for students to use numerator as a count and denominator as the number of equal parts that fit into one.
8. Introduce Investigation One using slide two of the PowerPoint. Provide sets of Cuisenaire rods or access to the online tool. Let the students work in small teams (mahi tahi relationship). It may be necessary for you to work with a small group of students. Look for the following: Do the students refer back to the brown rod as the one? Do they look for a unit fraction to support them to name the green and orange rods? 
   • Can they record the relationships they find as equalities?, e.g. 3/4 = 6/8.
   • Can they conjecture equivalent fractions that they do not have rods for?, e.g. 5/8 = 10/16
9. All of these points can be raised in discussion as a whole class. Continue to ask questions about the meaning of the numerators and denominators in equivalent fractions. For example, us the orange rod as one and find how many tenths equal one half and equal three fifths.
   
10. Finish the session with a reflection question: “Two children are talking (see slide three of the PowerPoint). Millie and Jana have different ideas about two thirds. Who is right? Explain why they are right.”
11. Let the students write an answer individually. You might use your students’ writing as prior assessment and revisit the idea of an infinite number of equivalent fractions sometime later.

Session Two

1. Revise the key points about equivalent fractions from the previous session using the dark blue rod as one.
   
   Ask questions like:
   If the dark 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?
2. Show the students slide four of the PowerPoint for this unit. Create a fictitious fawn coloured rod by joining two dark green rods. On the online tool you can set the rod length to 12 to make the rod. The students might notice that the light green rod is one quarter since it maps into one four times and that the crimson rod is 1/3 and fits the fawn rod three times. If they do not, scaffold this understanding. Use the physical materials to demonstrate the relationship. At this point you might construct this diagram using the online tool with the fawn rod being made up of length 12 (two dark green rods).
   
3. Invite students to describe the total length of the crimson and light green rods combined. Look for ideas like:
   The total is more than one half because two quarters is one half and one third is longer than one quarter.
   
4. Ask: What rod could we use to measure all three rods exactly, light green (one quarter), crimson (one third) and fawn (one)? 
   Different rods might be tried but the white rod is the only unit that measures all three.
   
5. Ask: 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? 
   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 twelve and twelve not added? The units are twelfths so we get “Four twelfths plus three twelfths equals seven twelfths.”
6. Provide the students with a set of Cuisenaire rods and Copymaster 1 for each small group. Let them solve the problems collaboratively as you roam. Look for the following:
   • Can the students name each fraction with reference to the designated one rod?
   • Do they look for equivalent fractions where they are needed?
   • Do they record the addition of each pair of fractions correctly?
   • Look especially for how they deal with non-unit fractions, like three quarters.
7. After a suitable period, bring the class together to discuss their solutions. Bring out the points above. Look for students to justify their renaming of fractions in equivalent form and how they calculated their answers. For example, in question 4 the sum can be written: 1/2 + 3/7 = 7/14 + 6/14 = 13/14.
8. Pose a problem for individuals to solve and record their thinking (see slide six of the PowerPoint for this unit). You may like to use this to assess who among your students understands addition of fractions and who needs further support. 

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.

1. Pose this problem:
   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?
2. Ask the students which Cuisenaire rod they could use to solve the problem. Note that it must be possible to make one half of the rod and two thirds of the rod. You may need to try various rods before dark green is settled on. The difference is the length missing to make the smaller fraction as big as the larger.
   
   Students might recognise that the difference is one sixth of the one (dark green). Scaffold this understanding if it is not demonstrated in discussion. Use physical materials to demonstrate the relationship. 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.
3. Ask: 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.
4. Build this diagram:
   
5. Ask the students to discuss in pairs what fraction difference problems might be made up using this picture. Remind them to think flexibly about which rod is one, assuming the orange rod is one is not the only possibility (dark green and yellow can also be named as one).
6. Record a few possible problems such as:
   • If the dark green rod is one, what fractions are represented by the orange and yellow rods?
   • What is the difference between those two fractions? In this case the orange rod represents one and two thirds or five thirds of the dark green rod. The yellow rod represents five sixths of the dark green rod.
   • So the difference between these two rods represents 5/3 - 5/6. Using white rods to represent sixths then five thirds (orange rod) can be renamed as ten sixths.
     
     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.
7. Let the students attempt Investigation Three (slide five of the PowerPoint) in small groups. They will need a set of rods or access to the online tool, and a means to record their problems, e.g. squared paper. Encourage the students to create differences that they have not seen in the examples you have provided and to use non-unit fractions like two thirds or three quarters. Look for students to use two main approaches, start with the rods they are finding the difference for, or start with a one rod (whole) and experiment with different fractions of that one. Key things to look for are:
   • Approach One 
     Do 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?
     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 Two
     Do 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 onto 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?
8. As you roam choose examples of problems students create. Ask students to record their problems. Also expect them to annotate the document to explain how the model shows a particular difference problem, including words and equations.
9. At the end of the lesson ask several groups to present their problems for others in the class to solve. Record the three examples vertically. You should have equations that look like this:
   1/2 - 3/8 = 4/8 - 3/8 = 1/8
10. Ask the students: 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?
11. Give students time to record instructions/ideas they would give to the new class member. Look at these work samples to assess whether individual students have generalised how to solve difference problems.

Session Four

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

• Locating fractions on a number line requires a fixed length of one
• A point on the number line can have an infinite number of names called equivalent fractions, e.g. 2/3, 4/6, 6/9, … all ‘live’ at the same point
• Between any two fractions are an infinite number of other fractions

1. Begin by building up a number line for thirds in this way. You may want to use the online tool on an interactive whiteboard so the image can be written over.
   If the blue rod is one (mark zero and one on the number line) where would one third be? You could scaffold this further by marking zero and one on a number line below the Cuisenaire rod. 
   
2. Students may recognise that the light green rod is one third of the blue rod. Ask them what fractions could be marked on the number line using one third. Look for them to explain that thirds can be ‘iterated’ (place end on end) to form non-unit fractions, like two thirds. Make sure you push the iteration past one and include the fraction and mixed number ways to represent the amount (see below). Also encourage renaming in equivalent form where this is sensible, e.g. 3/3 = 1, 6/3 = 2.
   
3. Look at the space between zero and one third. Ask, “Are there any fractions that belong in this space?” Students may recognise from previous work that white rods are one ninth of a blue rod. One ninth, and two ninths, will work. Ask students to estimate the exact location of these fractions and check using white rods. Note that three ninths equals one third so can be added to the existing number line in the same position as one third.
   
4. Ask: 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, e.g. 1/6, 1/8).
   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 the rods for these fractions would be is a useful activity in itself.
5. Look at other spaces on the thirds number line and ask students to name fractions they think exist in that space. For example, between two thirds and one are seven ninths, and eight ninths. Imagine cutting each white rod in half.
   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 to 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.
   
6. Show the students slide six (Investigation Four) of the PowerPoint. Choose two of the rods as representing the fractions to compare, for example dark green and orange.
   If I say dark green represents one half, how big is one?
   
   Students should recognise that the fawn rod (12 cm) equals one (whole).
   If dark green represents one half, what fraction is represented by the blue rod?
   
   Students should realise that the blue rod represents three quarters, if the fawn rod equals one.
   What fraction is the difference between one half and three quarters?
   The light green rod is the difference and that rod represents one quarter. Record the difference using a subtraction equation.
   3/4 - 1/2 = 1/4
7. Allow students plenty of time to investigate differences using the four rods on Slide 6. If possible use the online tool to support their thinking. Students might screengrab the model of their fractions. For example the yellow rod might represent one quarter. 
   The dark green rod is six twentieths or three tenths of the one rod. 
   
   The difference between three tenths and one quarter is one twentieth (the white rod).
   The equation for the difference is 3/10 - 1/4 = 1/20 or 6/20 - 5/20 = 1/20

Lesson Five

1. Show the students Investigation Five (Slide 7). You may want to give them paper copies as well.
   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?
2. Let the students investigate the task in small groups. They will need a set of Cuisenaire rods or access to the online tool, and squared paper to record on. The squared paper will help them to maintain scale.
   Look for the following:
   • Do the students realise that the white rod is one fifteenth by finding the difference between two thirds and three fifths?
   • Do they realise that fifteenths can be iterated to the left and right to find the location of zero and one?
   • Having established one as a length of 15cm, can they find the location of other fractions, such as yellow as one third and light green as one fifth? 
3. After a suitable period of investigation, bring the students together to share some solutions. Highlight the correct use of scale using the one rod as a reference unit. Recognise if some students have included fractions greater than one, e.g. 4/3 = 20/15. The work sample might be used as assessment data but you may like to give students the investigations on slides eight and nine of the PowerPoint as individual assessment. Ask students to record how they worked out the location of the numbers. There are many different ways but all strategies require students to show understanding of equivalence.

Session One

1. Show students slides one and two of PowerPoint 1. 
   What 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)
2. Use an image of a thermometer from Slide Two as a focus for the discussion (enlarged on a photocopier or projected onto a screen). To confirm the students' ideas a simple thermometer can be created from a conical flask or bottle, a blob of plasticine, and a transparent straw. Many videos exist online about how to construct a simple thermometer from these materials.
3. Show Slide Three. The first image shows a blank thermometer.
   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.”
4. Ask: 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.
5. Tell students that they are going to investigate how quickly a thermometer adjusts from room temperature to reach the temperature of hot or cold water. Provide each group with two thermometers and two cups of water. One cup should contain tap water with some ice cubes in it, the other should have warm water (no more than 50°C). Ask a couple of students to dip a finger in each cup and estimate the temperature. Ice water usually has a temperature of 4-10°C. Warm water out of a hot tap should not be hotter than 49°C. Invite ideas about what will happen when a thermometer is placed in each cup. (The level of the spirit will rise or sink until the correct temperature is found.)

6. Tell the students to take the temperature reading every 30 seconds once the thermometer is immersed. One student will need to time the intervals with their watch, phone, or the class clock. The results should be recorded in a table like this:

   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	»

   Collect data for about ten minutes.

7. Ask the students to record the results on a line graph showing time in seconds (horizontal axis) and the temperature readings in degrees Celsius. You may want students to construct the graph by hand or use a graphing tool such as a spreadsheet or online graphing tool.
   
8. Gather the class to discuss trends in the time series graphs.
   What trends appear in the graphs?
   What will happen in the next ten minutes? Why?
   Will the lines cross? Why, or why not?
9. After some discussion asks students to return to each cup and check the temperature of the water.
   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.
10. Encourage students to explain what patterns they can see in the data and why they believe it occurs. Look for ideas about the thermometer spirit taking a short time to "heat up" or "cool down" to the temperature of the water and but taking a longer time to close in on the classroom air temperature.
11. The next challenge requires students to appreciate the feel of water temperatures. Each group needs to be provided with plastic cups, thermometers, hot tap water, cold tap water, and some ice cubes. Pose the following problem (see images on Slide Four):
    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?
12. Allow the students time to solve the problem. Note that the 32°C cup of water may be hard to sustain at that temperature and may need to be topped up with hot tap water which is about 45-49°C. Similarly, the 10°C water will need regular addition of ice.
    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.

1. Ask: 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?
2. Invite the students to form conjectures about these questions, encouraging them to explain why they believe things will occur. Set up the following experiments to test the conjectures.
   • Two plastic cups are set up with different amounts of the same hot water. It is important to fill both cups with the same hot water before the test is started. Read the thermometers every minute.
   • Three cups are filled with the same amount of water and covered with cling film. Leave one jar as is and choose different materials to insulate the other two with, ideas could be tinfoil, bubble wrap, fabric, or newspaper. Poke the thermometers through the cling film and read the thermometers every two minutes.
3. Encourage your students to show their findings using graphs and/or tables and to report what their data show. Larger containers of water take longer to cool to room temperature as the ratio of volume to surface area is less, so there is less heat transfer to the air. This phenomenon is also true of animals in adverse climates. Young children are particularly vulnerable in extreme temperatures. Similarly, wrapping containers disrupts heat flow between the water and outside air. Effectively students are insulating two of the containers.
   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.)
4. The Figure It Out activity, Cold Coffee, can be used to provide a dataset for this activity.

Session Three

1. The second day involves explaining what factors influence the temperatures of locations around Aotearoa. Find an online site that provides temperature data. If this unit occurs in New Zealand summer the maximum daily temperatures are likely to be mostly positive. In the New Zealand winter, there will be significant differences between temperatures, including negative values.
   What locations in Aotearoa have you visited when you were on holiday?
   What were these places like? What highlights do you remember?
   How did you know what clothing to take when you visited?
2. That question will lead to a discussion about weather:
   What factors make a difference to the temperature of a location?
   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 location 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.
3. Ask if any students would like to share their iwi, hapū, or marae locations. For example, how do temperatures in Ngā Puhi's land compare with Kai Tahu's? How do temperatures at your local marae compare with another?
   
   Make a list of places to investigate. Each team of three students are allocated five locations from the list. Discuss the idea of average as being a measure of centre or middle, and the need to round decimals to the nearest whole number. Establish a protocol that coastal locations have a lowest altitude of zero metres and a distance to the sea of zero kilometres. Ask your students to find out the following data about each location.
   • Name: Wellington
   • Country: New Zealand
   • Hemisphere: Southern
   • Latitude (distance from equator): -41.2⁰S
   • Altitude (height above sea level): 0m
   • Distance from sea: 0km
   • Average winter temperature: 11.9⁰C
   • Average summer temperature: 20.1⁰C
     
     Additionally, students could investigate cities worldwide and make comparisons between Aotearoa locations and other places around the world.
      
4. Create a class spreadsheet to collate the data. A member of each team can enter the data from each city into the database. You may want to save time by using Google Docs to host the database. Many students can enter data at the same time.

5. Gather the class for a korero.
   We are trying to find out what factors affect the weather in a location over the long term.
   Let’s ask some questions of the data.
   Invite questions such as:
   Are places further from the coast/higher in altitude/further from the equator cooler than those further inland?
   Are locations in Te Ika-a-Māui (the North Island) hotter or colder than those in Te Waipounamu (the South Island)?
   Do inland cities have a bigger average temperature range, between summer and winter, than coastal cities?
   Are cities in the Northern hemisphere hotter or colder than those in the Southern hemisphere?
    

   Encourage sharing and discussion of students’ thinking.
    

6. Encourage your students to use graphing software online to analyse the data. CODAP is a free download and is very intuitive to use. It allows for downloading of the database in CSV form using ‘drag and drop’ functionality. Check to see that students make appropriate choices for the types of graphs they use.
7. After a suitable period of investigation, ask the students to create a report of their findings. The report should include conclusion statements supported by summaries of the data, including graphs.
8. Encourage the students to apply what they have learned to cities in Australia. The range in geography and students’ familiarity with the locations make this an ideal context for study.
   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.

1. Use PowerPoint 2 to introduce the use of negative numbers on the Celsius Scale. Slide One shows a thermometer recording a temperature of -13⁰C, that is, thirteen degrees below zero. The scale is difficult to read as the mark for negative ten degrees is the underline on which -10⁰C is resting, and the negative sign is assumed. Support students to read the scale.
   How cold is negative thirteen degrees?
   Has anyone in the class experienced a temperature like that?
2. Slides Two and Three are about the significance of 100 and zero degrees Celsius, measures of the boiling and freezing points of water at sea level (water boils at a lower temperature at higher altitudes – just ask a mountain climber!)
3. Look online to find a short video about the Celsius scale but avoid anything about converting between degrees Celsius and Fahrenheit (⁰F).
4. Use Slide Four to practise reading a thermometer in degrees Celsius.
5. Provide students with copies of Copymaster 1 that contains several blank thermometers. Slide Five shows how numbers of degrees can be placed on the scale.
   How many degrees is each division on the scale? (10⁰C)
6. Work through Slides Six through Eight and ask you students to create a thermometer that shows each temperature. Ask them to use a fresh blank thermometer each time.
7. Slides Eleven through Fourteen present three problems about differences in temperature. Work through each problem and ask students to work in pairs to solve them. Encourage students to use blank thermometers on Copymaster 1 to show their thinking. The second slide for each problem shows a number line model.
8. Finish the lesson with this problem (Slide Fifteen):
   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.
9. Allow students access to online sites to research their answer. After a suitable time, have a korero about their responses. High temperature is a sign that your body is fighting an infection. Any temperature above 37⁰C indicates a fever, but 40⁰C signals a high fever. Also discuss hypothermia that occurs if a person’s body temperature drops below 35⁰C. You might look up what to do in both cases; fever, and hypothermia.

Session Five

1. In this lesson students investigate mixing of water with different temperatures. Ocean temperatures, particularly those in the main currents, have a significant effect on world climate. Currents act like giant conveyor belts taking warm water from the equator towards the poles, and cool water from the poles towards the equator.
2. Ask: 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.
3. Pose this problem: 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?
4. Invite students’ ideas about the possible solution then allow groups to investigate. They will need cups, thermometer, hot and cold water, and ice cubes for cooling.
   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 appears as the central point between the two measurements.
   
   Simple averaging works when the amounts of water in each cup are the same.
5. Allow the students to investigate other mixtures of equal amounts of water, such as 100mL at 40°C mixed with 100mL at 10°C (Average 25°C). Move to mixture problems where the amounts of water are different, such as 200mL at 40°C and 100mL at 10°C so they can establish the idea of a "weighted" mean. That is twice as much water is at 40°C than 10°C so the mean is worked out by 2 x 40°C + 1 x 10°C = 90°C, 90°C ÷ 3 = 30°C.
6. Copymaster 2 can be used to provide mixing problems.
   How do you work out the mixture temperature when the amounts of water in each cup are different?
7. Students could be asked to write mixture problems for their classmates to solve.

Extra problems

Problem One: Are we good at estimating water temperature?

1. Make up several labelled cups of water at different temperatures up to about 49° C. Ensure each cup is wrapped in newspaper or cloth inside a tin can to limit the loss of temperature. Put an ice cream container beside each cup. The students go to each container and estimate the temperature of the water. Their estimate is written on a small piece of paper that is put into the ice cream container. After all the estimates have been done the cups and containers are distributed among the groups.
2. Ask students to:
   • Find the temperature of the water using a thermometer
   • Display the estimates on a dot plot or stem and leaf graph

Problem Two: Why do people wear light coloured clothing in the summer and dark clothing in the winter?

1. Allow the students to give their reasons and record these on a board or chart.
2. Tell the students that they are going to check the effect of different colours on temperature. Each group makes up the same shallow containers of water (delicatessen or take away containers are ideal) at tap water temperature. Cover each container with plastic wrap to prevent water loss and wrap it with paper or fabric. The papers or fabrics (varying only in colour) used should reflect a range of colours from black to white.
3. Leave the containers in a hot sunny location for a few hours. Record the temperature of the water in each container by poking a thermometer through the plastic wrap. Record the results. Ask: What does this tell us about light and dark clothing?

Problem Three: What times of the day are the hottest? What times are the coldest?

1. Tell students that they are going to take the air temperature at half-hour intervals during the day. Do this in four different locations, direct sunlight (sheltered), direct sunlight (breeze), shade (sheltered), shade (breeze). Ask them what patterns they expect to see in the temperatures. Record their ideas. Carry out the experiments standing the thermometers up. A small cardboard box with one face cut out makes an ideal thermometer holder for many groups. The open face is placed on the ‘sunny side’.
2. Every thirty minutes the students need to check the readings. A period from 9.00 am to 2.00 pm is ideal as it allows time to process the data and suggests that students might like to extrapolate their results for times after 2.00 pm. (e.g. I think the temperature will be 24°C at 3.00 pm.)
3. Ask the students to display their data on a multiple line graph.
4. Invite the students to make statements about what the data shows. They should be encouraged to support their statements by referring to the line graphs.

All lessons in this unit follow the same sequence of phases as given in the phase approach diagram shown. A poster of the phases is provided as Copymaster 1 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. Be flexible in how you respond to students, as opposed to focusing on teaching the sample ideas and representations provided.

PowerPoint 1 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 One

Slide 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.

1. Noticing Regularity
   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 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.
    
2. Articulating a claim
   Encourage students to state a claim about what is going on with all four examples in Pattern One. Initially, they might do this individually, before working in small teams to refine their ideas and the way they express those ideas. Encourage the mahi tahi model where students work collaboratively. Potential ideas may include:
   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.
   
   Aim 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'. You could compare different statements from groups of students, to highlight the importance of such clear language. Links could be made here to explanation writing (i.e. the need to be clear, informative, and concise).
3. Representing
   In 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 cube off each group of five cubes to make one group of seven cubes. I found out I could make five stacks of seven cubes from the seven groups of five cubes.
     
   • 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.
     
4. Some representations are less helpful than others in terms of understanding the structure of the commutative property. For example, jumps on a number line do not show how each item in a set is used to form the new sets. It is important for students to recognise that repeated addition is not multiplication although they arrive at the same number. In New Zealand, when teaching multiplication, the multiplier is the first factor and the multiplicand is the second factor. Other countries teach in reverse so expect some students to be confused.
5. Your questioning is important to support students in connecting symbols and other representations.
   • 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?
      
6. Constructing an argument
   In this phase students are asked to formalise their "noticing" by creating a statement that can be generalised to all cases. The discussion may start with a specific equation pair but must be extended to describe 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, or may use symbols (e.g. emojis) to simplify their thinking. Note that this represents the starting equation pairs. In general, "a sets of b" can be remodelled - meaning 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 Two

Ask the students to approach the second equations set more independently. Consider the needs of your students - it may be more appropriate for you to work with small groups, and to organise students in small groups or pairs for these tasks. From this point each equation set is discussed succinctly using the phases of the approach.

1. Noticing regularity
   The four equations apply doubling and halving, thirding and trebling of the factors in the first equation. This strategy is sometimes 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]
    
2. Articulating a claim
   Expect students to use phrases from their conversational language, such as “when one number doubles, the other halves”. Introduce important vocabulary such as factor  (i.e. all the numbers that evenly divide a whole number -3 and 2 are factors of 6) and product (i.e. the result obtained when the numbers are multiplies together). This will help to 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 a number, n, the other factor is multiplied by n. The product stays constant (the same).”
    
3. Representation
   Expect 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.
   
    
4. Argumentation
   Look 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. Although symbolic representation is restricted at this level, 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÷n can be expressed as a/n. Multiplication can also be represented without the x symbol, e.g. a×b can be represented as ab.

Equation Pairs Set Three

1. Noticing regularity
   The 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]
    
2. Articulating a claim
   Expect students to use phrases from their conversational language, such as “adding on so many lots of the number”. Encourage students to use of mathematical vocabulary such as factor and product to clarify what numbers are being referred to in the claims. You may need to model this for students. 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.”
    
3. Representation
   Expect both physical and diagrammatic representations to be used. Since most of the first equations involve multiples of ten or 100, place value blocks (MAB) might be useful.
   
   
   Diagrams of 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.
4. Argumentation
   Look 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 increases 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 so a×(b+n)=(a×b)+(a×n)
   Multiplication can also be represented without the x symbol, e.g. a×b can be represented as ab, and unnecessary brackets (due to order of operations) can be removed.
   ab so a(b+n)=ab+an

Equation Pairs Set Four

1. Noticing regularity
   The 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]
    
2. Articulating a claim
   In 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.
   
   
   Encourage and scaffold the use of mathematical vocabulary such as factor and product to clarify what numbers are being referred to in the claims. You may need to introduce division terms like divisor (number being divided by), quotient (answer to division) and dividend (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.”
    
3. Representation
   Expect 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.
   
    
4. Argumentation
   Look 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 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.
5. Try to scaffold students on from the use of repeated addition by reinforcing more efficient strategies.

Equation Pairs Set Five

1. Noticing regularity
   The 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
    
2. Articulating a claim
   In 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), and dividend (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 from 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.”
    
3. Representation
   Both 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.
    
4. Argumentation
   Look for students to justify, using words or diagrams rather than symbols. If the divisor stays the same then dividing the dividend by a number results in the quotient being divided by the same number. Similarly, multiplying the dividend by a number results in the quotient also being multiplied by that number.
   Look for students to accept that the dividend and divisor are variables, meaning they can take up any value. 
   Be aware that 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.
   
   
   Algebraically this might be expressed as , "If a amounts of b equal ab, then a/2 amounts of b equal half of ab (ab/2)."

Equation Pairs Set Six

1. Noticing regularity
   The 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]
    
2. Articulating a claim
   Expect students to use phrases from their conversational language, such as “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.”
    
3. Representation
   Students are likely to use specific examples to convince others about how the relationships work.
   
    
4. Argumentation
   Arrays are a useful representation to ‘prove’ what occurs with the difference of perfect squares.
   
   
   In general, a² can be transformed spatially into (a-1)(a+1) by removing one (1 x 1) and moving the unit of (a-1) to create a 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-1)(a+1) = a2-a+ a -1 = a2 -1

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 tool (e.g. Google drawings).

1. Noticing regularity
   The 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]
2. Articulating a claim
   Expect students to use phrases from their conversational language, such as “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.
    
3. Representation
   Given 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.
   
    
4. Argumentation
   Be aware students might go back to the repeated addition view of multiplication to convince others about their claim. Validate the use of this strategy, and encourage students to develop their use of "noticing regularity". A specific example might look like this:
   

Session 1: Measuring the class

PROBLEM: Generating ideas for statistical investigation and developing investigative questions

1. Tell the students that school uniform manufacturers may be interested in information about different body sizes. Introduce differences in sizes by asking for volunteers and standing two students up to discuss what about them could be measured to inform clothing or footwear producers.
   Research online information about sizing guides as well.
2. Brainstorm things that could be measured and compared (height, head circumference, arm span, handspan, foot length). Ideas from this prompt should mostly be numerical (measured) variables.
   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 do not want to be measured could become the measurer for a measurement station.
3. Ask what other things school uniform or sportswear manufacturers may be interested in to help with their creation of clothing or footwear. Brainstorm additional ideas to explore.  Ideas from this prompt may tend to be more categorical variables.
4. Once the initial brainstorming of ideas is done interrogate the ideas by checking them using the following questions:
   • Is this a measurement/idea that the students in our class would be happy to share information with everyone? If not reject the idea [ethics].
   • Can we collect data to answer an investigative question based on this measurement/idea? If not reject the idea [ability to gather data to answer the investigative question].
   • Would you be able to collect the data to answer the investigative question in the timeframe we have (specify)? If not reject the idea [ability to gather data to answer the investigative question].
   • What would be the purpose of asking about the measurement/idea that you have? If it is not purposeful then reject the idea [purposeful or interesting].
   • Would the investigative question we pose involve everyone in the group (e.g. the class)? If not reject the idea [does not involve the whole group].
5. Students to form small groups and select one attribute to measure (each group to select a different attribute). Ensure that one group selects height. Support the students to develop an appropriate investigative question to ask of the 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 question

6. Discuss ways to collect and record the data considering that every group will have data they want to collect. Get students to develop a survey question with instructions to collect their measure (numerical and categorical).
   • Exploring the guide on how to make measures for the CensusAtSchool questionnaire. The guide provides ideas on how to measure numerical data.
   • See session 2 in Crunch the Coach and particularly the data collection stations for additional ideas.
7. Measurement stations could be set up around the room for students to make their measures for the different survey questions designed. A recording card can be used to record individual student responses. See for example the data cards used in the 2019 CensusAtSchool survey. For categorical variables, the station will include the survey question with any response options.
8. A suggested option for the class is to develop an online questionnaire where students can input their responses to all the groups different survey questions (from their individual data card), for example, using Google Forms. This reduces the amount of time needed to collate data and the data can be downloaded into a .csv file for analysis. It is good to think about any demographic data that would be useful as well, e.g. gender (be aware of sensitivity around this also). Names are not needed to be recorded, nor should any identifying demographic data be collected. The teacher needs to be aware of the overall survey and if there are any potential ethical issues. See more about ethics in How much bullying? activity.

Session 2: DATA: Collecting and organising data

1. Students work around the different stations for the collective class survey questions and record the data onto their individual data cards. As the students gather data, the teacher should circulate and provide advice or assistance as required. Ensure that accuracy of measurements is maintained. Any students who were really not keen to be measured could be supervising a measurement station to ensure consistent and valid measures are made at it. For example, they could measure everyone’s height.
2. Students to check in with one another about the measurements they have made. They are checking for errors in recording their results, errors in making measurements or errors in reading measurements. They should share and discuss their thinking.
3. Once they have checked all their measurements and recorded results to any categorical survey questions get the students to input their responses into the class Google Form (or similar).

ANALYSIS introduction: Using an online tool to make data displays

In 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 CODAP

4. Allow the students some time to get familiar with CODAP. Using the Getting started with CODAP example is a good starting point. This has a built-in video that shows the basic features of CODAP and gets you started using the tool. Other support videos can be found here.
5. Note for teachers: Students will use the data collected in this session to make their displays in the next session. Between sessions download the survey data into a .csv file and set up the CODAP document with the data and share a link to this. See the video or written instructions on how to do this. Note the video and the instructions include getting started with CODAP too.

Session 3: ANALYSIS: Displaying and describing the data

Students use data from the previous session to produce graphs in CODAP or similar statistical analysis software.

1. Discuss the data collected in the previous session and explain that the students will be using CODAP or similar software to produce graphs of the data to answer their investigative question(s).
2. Share the link to the CODAP document that has all the class data.
3. Students should first look at the data that is given and decide which variable(s) they need to graph to answer their investigative question.
4. Initially students should be given freedom to experiment with what type of graph they feel best shows the information.
5. Once groups have produced a graph or graphs to answer their investigative question, bring the class together and discuss the graphs produced. Most will have produced a dot plot (default graph for numerical and categorical data). CODAP allows us to look at this data in different ways.
   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.
6. Students will now learn about bar graphs (for categorical data) and histograms (for numerical data) as alternative displays. 
7. Making bar graphs in CODAP for categorical data: If they chose a numerical variable for their investigative question, get them to choose a categorical variable to practice this with. They should display the categorical variable they have chosen.
   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.
        
8. Making histograms in CODAP for numerical data: If they chose a categorical variable for their investigative question, get them to choose the height to practice this with. They should make a display of the height first.
   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 have a korero about 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.

9. Students describe their data displays. A good starter is using “I notice…” as students start to notice features of their displays. For numerical data they might notice:

   • The largest value
   • The smallest value
   • Where most of the values lie e.g. between X and Y
   • Where the data peaks
   • Gaps or clusters
   • Unusual values

   For categorical data they might notice:

   • The most common
   • The least common
   • The majority
   • Combinations of categories
   • Any patterns (depending on the categories)

   All statements in the descriptions need to include the name of the variable and the group, and for numerical data, the value and the units as well.

CONCLUSION: Answering the investigative question and reporting findings

10. Students answer their investigative question using evidence from their analysis. Ensure students support each other and have opportunities to experience both tuakana and teina roles as they do this. 
11. Students can present their findings using a PowerPoint or similar presentation. Restrict to 3-4 slides
    • Their investigative question
    • Display(s) with descriptions (1-2)
    • Answer to their investigative question – linking to the school uniform or sportswear manufacturer purpose

Session 4: Comparing to students like us in New Zealand

The class will use data from CensusAtSchool to compare to their class results.

1. Students from around New Zealand have engaged in the CensusAtSchool questionnaire and measurement data has been collected, some like what we collected. This data is available on the CensusAtSchool site.
2. In this session students will get their own sample of students their age from the CensusAtSchool database to compare with the class data from the previous sessions for measurement variables. The measurements you have taken will determine which variables you can compare. If students do not have an appropriate measurement variable get them to compare heights.
3. Show students the CensusAtSchool random sampler, remembering to accept the conditions of use. Once in there familiarise the students with the tool. There are five parts to the tool.
   • Select database – here we can choose any database from 2005 onwards.  Recommend they use the 2015 or 2017 database as these have a larger range of body measurements.
   • Select subpopulation – because we want to compare with other New Zealander students our age, we want to select specific years. When we select specific years, we get a drop-down list that allows us to select the same year level as the students. Select the year level. The example shows year 8 selected, but you could select any year group.
     
   • Some students may want to also compare data by region. Ask if any students would like to share their iwi, hapū, marae, or location of other whānau connections with the class. They may decide to compare data from their home region with data from a region of Aotearoa they have iwi, hapū or whānau connections with. 
   • Select variables – because we want to look at measurements specifically, we only want to select specific variables. When we select specific variables, we get a drop-down list of all the variables in the survey. Suggest the following variables are selected:
     
   • Select sample type – leave as random sample
   • Enter sample size (Maximum 1000) – suggest they select 100 to give a slightly bigger group to work with.
   • Note: 100 is a good size for a later activity where we find the middle and the middle 50%, also 100 will give a bigger group (sample) size than the class and provides the opportunity to deal with comparing different size groups, something students sometimes think we cannot do.
4. Once they have made the selections in the five parts, they click on generate sample and then download sample.
5. Students save the .csv file and then import into CODAP. (The video here shows how to import data from CensusAtSchool.)
6. Once the data has been collected from the site get the students to discuss how they might make a comparison with our class data e.g. they might suggest that a histogram is easier to compare than the dot plot.
7. Students display the data for any variables we have identified that we can compare using CODAP. If they need to choose, suggest they choose height.
8. They should write “I notice” statements about what they see in their data from CensusAtSchool.
9. Focus the discussion on similarities and differences between the data sets, the class data and the CensusAtSchool sample (we are calling the group from CensusAtSchool, CensusAtSchool sample to describe the group. We are NOT doing sample to population inference – this is in curriculum level 5).
   
   Note also that the different group sizes might be a sticking point for students as they may 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 CODAP

Note that measures of centre are not introduced in The New Zealand Curriculum until level 5. At curriculum level 4 we introduce informal ideas of the middle and the middle 50%

Describing the middle

Students 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 data

1. Make the graph for height using CODAP. Stretch the graph out so that the dots are not over top of one another. This can be done by dragging the bottom corner along so that the graph is wider.
2. Select the ruler (measures) and tick 'Count' and add a movable value.
   

• Ask the students what they see on their graph now.   
• They should see a line with a number in blue with a number on the top and two values, one to the left of the line and one to the right of the line. What do the two values (left and right of the line) represent? This is the count of the number of people in the class (in the given example it is 3+27, 30 students in the class).
• Discuss where the middle value would be, e.g. for this graph it would be in the middle of 30 which is 15, that is 15 on each side of the movable line.  
• Therefore, we want to move the movable value so that the counts are about half each side.   
• The place that we settle the movable value at is the middle value or the median. The median is the technical statistics term for the middle value in a set of data when the data are placed in order from smallest to largest.
   
• Get the students to read the middle value from their graph. In this example the middle value (OR middle height) is 157.5 cm (always include the unit).
  
• They should now write a statement in a text box under their graph that says…
• The median height for students in our class is __________ cm.

Repeat the idea for the data they have from CensusAtSchool.

• Students make the graph for heights for the CensusAtSchool group.
• They click on the ruler and select 'Count' and 'Movable Value'.
• They move the line until they have about half on each side (if they have selected 100 students then there will be 50 on each side).
• They write a statement about the middle value of their graph.
• The median height for students in the CensusAtSchool sample is _________cm.

This special value, the median, can be found using the 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 data

The “signal” for the data is often where the middle 50% of the data is.

• Go back to the class height graph
• Untick 'Median'
• Tick 'Movable Value' (add) so there are two movable values on the graph
  
• Discuss with the students how many people would be in the middle 50% (in this case 15). This would leave 15 outside the middle 50% or 7/8 either side.
• Move the lines so that the counts match this or are very close to this.
  
• Read the values for the middle 50%, from the two movable values. In this case it would be 151.5 cm for the bottom value and 163.5 cm for the top value.
• We would describe this as the middle 50% of heights for students in our class is between 151.5 cm and 163.5 cm.

Repeat idea for the CensusAtSchool sample heights data.

Making comparisons between our class data and the CensusAtSchool sample data

With 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. Ensure students support each other and have opportunities to experience both tuakana and teina roles as they do this.