Early level 4 plan (term 3)

Planning notes
This plan is a starting point for planning a mathematics and statistics teaching programme for a term. The resources listed cover about 50% of your teaching time. Further resources need to be added to meet the specific learning needs of your class, to support your local curriculum and to provide sufficient teaching for a full term. Areas and volumes

Level Four
Geometry and Measurement
Units of Work
This unit is about finding the areas of triangles, and parallelograms, using what students already know about the areas of rectangle. Students will practice multiplication and division strategies as they calculate areas. The ideas are extended to finding the volumes of cuboids (rectangular prisms)
• Recognise that two identical triangles can be partitioned and joined to make a rectangle.
• Recognise that a triangle has half the area of a rectangle with the same base and height lengths.
• Apply the rule ‘area of triangle equals half base times height’.
• Connect the area of a parallelogram to... Equivalent Fractions

Level Four
Number and Algebra
Units of Work
In this unit we are exploring ways to find equivalent fractions. We use the concept of equivalent fractions to convert fractions to the benchmark fractions of halves, quarters, thirds, fifths and tenths. From these benchmark fractions it is easier to convert fractions to decimals and percentages. We...
• Explore and know equivalent fractions including halves, thirds, quarters, fifths, tenths and hundredths.
• Use equivalent fractions to convert fractions to decimals and percentages.
• Use equivalent fractions to order fractions with different denominators. Travel to school

Level Four
Statistics
Units of Work
In this unit students investigate methods of travel to school, using technology to produce data displays and investigate distributions.
• Pose investigative questions for statistical enquiry.
• Plan an investigation.
• Use spreadsheets to collate data.
• Use technology to display data.
• Discuss features of data display.
• Compare features of data distributions. Getting partial: Fractions of sets

Level Four
Number and Algebra
Units of Work
This unit develops the concept of a fraction as an operator, or multiplier, acting on an amount, e.g. two-thirds of 24. Using fraction multipliers to represent the relationship between different amounts is also explored.
• Find a unit and non-unit fraction of a set, e.g. two thirds of 24 (2/3×24).
• Use a fraction to represent the relationship between part of a set and the whole set.
• Recognise when the part-whole fractions of two different sized sets are equivalent, and when one fraction is greater than another.
• ... Marble Roll

Level Four
Statistics
Units of Work
In this practical unit students make ramps, roll marbles down them, record the distance the marble travels from different starting positions, graph these distances, predict other distances, and make statements based on the data they create.
• Construct a stable ramp to meet conditions.
• Follow and repeat a procedure to replicate exactly the same conditions (fair testing).
• Measure and record distances accurately.
• Graph the starting positions and distances travelled on scatter graphs.
• Talk about distinctive features of scatter...
Source URL: https://nzmaths.co.nz/user/75803/planning-space/early-level-4-plan-term-3

Areas and volumes

Purpose

This unit is about finding the areas of triangles, and parallelograms, using what students already know about the areas of rectangle. Students will practice multiplication and division strategies as they calculate areas. The ideas are extended to finding the volumes of cuboids (rectangular prisms)

Achievement Objectives
GM4-3: Use side or edge lengths to find the perimeters and areas of rectangles, parallelograms, and triangles and the volumes of cuboids.
Specific Learning Outcomes
• Recognise that two identical triangles can be partitioned and joined to make a rectangle.
• Recognise that a triangle has half the area of a rectangle with the same base and height lengths.
• Apply the rule ‘area of triangle equals half base times height’.
• Connect the area of a parallelogram to the area of the rectangle from which it can be created.
• Find the volumes of cuboids with whole number dimensions.
Description of Mathematics

Area is a two-dimensional concept, that is it relates to flat space. Area is the internal space enclosed by some boundary. With polygons, like rectangles and triangles, the boundaries are lines or line segments. Area is defined in the maths curriculum as the size of a surface expressed in square units. Investigations of the size of an area should begin with comparisons between different surfaces and progress to calculating with standard, units, such as square centimetres (cm2), square metres (m2), and square kilometres (km2). The use of formulae to calculate the areas of common polygons is an advanced stage of the learning sequence.

Units of area are squares because squares fit together with no gaps or overlaps, and tile infinitely in both dimensions. At the early stages it is important for students to realise that a simple shape, like a rectangle, can be filled with squares arranged in rows and columns to create arrays. Multiplication is an efficient way to count the number of square units. The idea of tiling must become more imaginary (abstract) as students deal with two things:

• Shapes, like triangles and circles, where squares do not fit exactly. Part units, and smaller square units increase precision of the measurement.
• Sides that are not whole numbers, e.g. Find the area of a rectangle that is 3 ½ x 4 ½.

Volume is the three-dimensional equivalent of area. Volume is the amount of space occupied by an enclosed solid. The boundaries might be flat planes, or curved surfaces. The simplest enclosed solid to find the volume of is a cuboid (rectangular prism). Cube shaped units are used for volume for the same reason that squares are used for area. Common units of volume are cubic centimetres (cm3), cubic metres (m3), and cubic kilometres (km3). Cubes fit together with no gaps or overlaps in three dimensions (length x width x depth). Since the units are arranged in three-dimensional arrays multiplication can be used to count the number of units in a cuboid shaped space. If the cuboid measures 6cm x 4cm x 5cm then 6 x 4 x 5 = 120 cm3 gives the volume.

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

• using grid paper with square centimetre dimensions so students can ‘see’ the units, and similarly using 1cm cubes to fill cuboids in an array structure.
• cutting up shapes made with paper, moving the parts, and taping, to show that the areas stay the same, e.g. transform a rectangle into two similar triangles, or a rectangle into a parallelogram.
• directly modelling measurement with scales, like rulers, with opportunities for students to copy correct use of tools.
• clarifying the language of measurement units, such as “metre square” as an area that is 1m x 1m.
• clarifying the meaning of symbols, particularly 45cm2 as 45 square centimetres, and 45m2 as 45 square metres; 45cm3 as 45 cubic centimetres, and 45m3 as 45 cubic metres.
• easing the calculation demands by providing calculators where appropriate. The purpose of the lessons is to understand measurement concepts, not to practise calculation strategies.

Tasks can be varied in many ways including:

• reducing the complexity of the numbers involved, e.g. whole number versus fraction dimensions for side and edge lengths.
• allowing physical solutions with manipulatives before requiring abstract (in the head) anticipation of measures.
• working in two-dimensions before three-dimensions, then connecting the rules for area and volume.
• reducing the demands for a product, e.g. less calculations and words, and more diagrams and models.

The context for this unit can be adapted to suit the interests and cultural backgrounds of your students. Area is useful to many everyday pursuits, such as painting a bedroom, creating a vegetable patch, tiling a kitchen or bathroom, interpreting house plans, and working out the size of a playground, marae or field. Volume of cuboids applies to contexts like comparing the size of schoolbags, chilly bins, car boot spaces, or refrigerators, choosing the correct air conditioning unit for a room, looking at packaging (watch for wasted space), and estimating the amount of firewood that has been delivered.

Required Resource Materials
• Cardboard boxes (cuboids) from home, e.g. cereal boxes, cracker packets, dishwasher tab boxes, etc.
• 1 cm3 cubes, e.g. place value blocks, centi-cubes, or similar
• Multilink, unifix, or connecta-cubes (2cm x 2cm x 2cm)
• Rulers
• Calculators
• Grid paper
• Copymasters OneTwo and Three
• PowerPoints One, Two, Three and Four
Activity

Session 1

In this session students revise the rule for the area of a rectangle. This session may not be required if this unit is being taught following a unit on area of rectangles.

1. Use PowerPoint One to illustrate finding the area of a rectangle using multiplication of length by width.
Apart from chocolate blocks, when do we use area in real life?
2. Discuss how area in used in contexts like carpeting a house, planning a vegetable patch, stocking farms, comparing the size of countries, checking moles on our skin for growth, ordering the paint for a wall, etc.
3. Discuss the square units that are used to measure area, such as square centimetres (cm2), square metres (m2), and square kilometres (km2). You might use google Maps to illustrate the size of the larger units using your local area.
4. Provide students with grid paper (Copymaster One) and pose the following challenge:
Draw four different rectangles that each have an area of 36 square centimetres.
Measure the perimeters of the rectangles.
Are the perimeters equal? Explain.
Students should create 2 x 18, 3 x 12, 4 x 9, and 6 x 6 rectangles though they may not consider a square as a type of rectangle (It is!) Adventurous students might consider fractional side lengths such as 4 1/2 x 8, 5 x 7 1/5. Students should note that the perimeters vary with the square having the shortest perimeter.
5. Tell your students to create a triangle within each rectangle in the following way: 6. Mark a spot on the top side. Draw a triangle with that spot, and the two bottom corners of the rectangle as the corners.
Find the area of each triangle. What do you notice?
Students should notice that the areas are all the same, 18 square centimetres.
How does the area of the triangle compare to the area of the rectangle?
Will that still be true if I cut up the rectangle like this? Students should note that the triangle has half the area of the rectangle. The rectangles and triangles can be cut up confirm the relationship. Show students that the formula for area of a triangle can be written as area=1/2 × base × height or a=1/2(b×h).
7. Give your students Copymaster Two to work on in pairs or as individuals. The worksheet requires finding the areas of triangles. Look for students to:
• Find the base and altitude of the triangle, irrespective of orientation.
• Recognise where the surrounding rectangle could be drawn.
• Calculate the area using the formula, including recording correct units (cm2)
8. Gather the class and compare answers.

Session 2

In this session students investigate the relationship between areas of rectangles and related parallelograms. They recognise that a parallelogram can be partitioned into two similar right-angled triangles plus a rectangle. This composition is used to develop a formula for the area of a parallelogram.

1. Begin with Slide One of PowerPoint Two.
Imagine each small square is 1cm2.
What is the quadrilateral called? (parallelogram) Why is that a sensible name?
Find a way to work out the area of the quadrilateral.
2. Let students explore the problem. They might benefit from using grid paper (Copymaster One) but encourage visualisation as much as possible.
3. Discuss the strategies that students use. Sketch diagrams of their partitions but suggest that counting individual squares and parts of squares is very time consuming, and error prone. Animate Slide One to see how the parallelogram can be partitioned into two triangles and a rectangle. The total area is 8 +8 + 80 = 96cm2.
4. Slide Two shows how the three shapes can be combined to form a single rectangle that has dimensions of 12 x 8 = 96cm2. Slides Three and Four contain two similar problems.
5. Work in reverse, starting with a rectangle and partitioning it to form a parallelogram. Slide Five shows how.
What is the area of this parallelogram? (12 1/2 x 10 = 125)
6. Challenge for the lesson:
Use your quad exercise book (or Copymaster One). Draw six parallelograms that each have an area of 48cm2.
7. Let your students experiment to find suitable parallelograms.
Do they begin with a rectangle they know has area of 48cm2?
8. Share students’ answers as a class. Try to bring out the following points:
• Any rectangle with an area of 48cm2 can be used to create a set of different parallelograms.
• 48 has many factors, (1 x 48, 2 x 24, 3 x 16, etc.) that give rectangles that can be the base for many parallelograms.
9. Show your students Slide Six.
Do all the parallelograms look like they have the same area?
Are his/her parallelograms correct? Do they all have areas of 48cm2?
How do you know?
How could we write the rule for finding the area of a parallelogram? (length x height = area)
10. Slide Seven has a final puzzle that might be used for assessment. Watch to see how students solve the problem.
• Do they find the area of each piece separately?
• Do they work out the area of the whole square (576cm2), then find the fraction of that area for each piece?
For example, the parallelogram is one eighth of the initial square. 1/8×576=72cm2.
11. You might look up the Tangram puzzle online and find some target problems for students to solve. Alternatively students can create their own set of shapes by partitioning a shape, like a square, rectangle, or equilateral triangle.

Session 3

In this session students connect the l x w formula for rectangles to finding the volume of cuboids (rectangular prisms).

1. Use Slide One of PowerPoint Three or a packet from home to introduce the cuboid.
Where is the world would you find cuboids?
Students should provide examples such as packets, boxes, buildings, shipping containers, etc.
2. Discuss the features of the cuboid.
What shapes make up the surface of a cuboid? (rectangles which is the reason a cuboid is also called a rectangular prism)
What shapes do we get if we cut a cuboid with a straight cut?
Students will probably think firstly of rectangles, including squares, but other cross sections are possible with non-parallel cuts.
3. Introduce the concept of volume.
Who might be interested in the amount of stuff this cuboid holds?
How do we measure the space inside a container?
Students will have ideas of possible measures, such as cubic centimetres. Introduce a small place value block cube which is 1cm3 in volume. Point out the 3 dimensions of 1cm.
4. Ask: How many cubes of this size could I fit into this cuboid, with no gaps or overlaps?
5. Let students discuss their ideas with a partner then share.
What information would you like to know?
Students might suggest measuring the edges of the cuboid.
How do those measurements help?
6. Discuss how the 1 cm3 cubes might be arranged in the cuboid box. If you have enough cubes create a layer that covers one rectangular face.
How many cubes make one layer?
Students should connect this question to area using l x w.
How many layers can I make? (You may need to physically step out the layers)
How do I work out the total number of cubes?

Try to arrive at a length x width x depth rule.
7. Show the students the cuboids on Slides Two, Three and Four. Challenge them to work out the volumes in cm3. It is fine for some students to use a calculate if the calculation is too difficult for them.
8. Discuss each cuboid and notice how less information is provided each time. Ask students to explain what the multiplications mean in terms of layers in the cuboid.
9. Provide each pair of students with at least one cuboid shaped packet. Ask them to find the volume of the box then exchange boxes with another pair to cross-check the calculations. Gather the class.
Is it easy to compare the volumes of cuboids just by looking at them?
You might find two or three boxes with similar volumes but quite different appearance.
How did the designer of the box know how big to make it?
Students might suggest that the designer knew what the box had to hold. For example, they might know that a 750g box holds 48 weetbix.
What would they do then to design the box?
Students might suggest that the designer arranges the weetbix in rows and works out the edges of the box from there.
10. Suppose that you used these cubes (unifix or multilink) to fill your packet. You know the volume in cm3. Can you use that result to find how many of these cubes will fit in?
11. Give each pair multilink cubes and ask students to solve the problem.
• Do students recognise that each multilink cube has a volume of 8cm3?
• Do they divide the existing volume in cm3, by eight?
• Do they recognise that fraction length sides occur?
• How do they deal with the idea that bigger cubes will be harder to fit in?
12. Discuss questions above. Larger cubes leave more empty spaces than smaller cubes if the problem is regarded as a physical challenge.
How might we deal with empty spaces?
Can we still use length multiplied by width multiplied by depth?
Students might realise that fractions are needed, and those fractions are usually represented as decimals in measurement.
13. Provide each pair of students with 36 multilink cubes, Copymaster One (grid paper), scissors, and tape. Search online for nets of a cuboid” so students recall the flat pattern. Discuss the shapes in the net and how those shapes relate to the faces of the cuboid.
Make three different cuboid shaped boxes that will each hold 36 cubes with no gaps or overlaps.
Give students ample time to build boxes before convening the class.
14. Share the boxes that student pairs create. Find a way to record the dimensions, such as:
 Designers Length Width Depth Number of cubes Hone and Bex 4 3 3 36 Tatia and Fatu 2 2 9 36 Mei and Kylie 6 6 1 36 Sid and Leon 9 2 2 36

Is each box unique, that is different from the others? (Students might note that the same dimensions occur but agree that the location of the top makes the box unique)
How will we know when we have found all the boxes? (Encourage a systematic approach based on finding all the factors of 36)

Session 4

In this session students draw on their knowledge of area and volume to solve problems. There is emphasis on surface area of 3 dimensional solids with links to difference between area and perimeter.

1. Show the students PowerPoint Four.
Simone is estimating the area of Lake Taupō. She has two ways,using a triangle and using a rectangle. (Show Slides One and Two).
What units of measurement is she using? (kilometres so the area will be in square kilometres, km2)
Which way do you think is the most accurate? Why?
Students should discuss the pieces of area that are missing from and outside each shape.
Do the outside parts make up for the missing parts?
2. Ask your students to calculate the estimate of area on each slide. Calculators may assist some students, if needed.
Triangle: 12×42×30=630km2
Rectangle: 26×25=650km2
Check the estimates against the actual area of 616km2.
You could challenge students to estimate the areas of other lakes or cities in Aotearoa or leave that work for another time.
3. Use Slide Three to introduce the idea of surface area. Painters and plasterers are dependent on measuring surface area correctly when quoting for jobs. Take one of the cuboid boxes used for measuring volume.
Surface area is the combined area of all the faces of this box.
How would I figure that area out?
Students might realise that each rectangular area can be worked out separately then the areas can be added.
Do any of the faces have the same area? (There are matching parallel faces)
4. Use the dimensions of the chosen box to calculate the area of each face then find the sum. Be sure to use the correct unit of measure for area (cm2 in this case).
5. Ask students to work in pairs again to find the surface area of cuboid box. Again, they should exchange boxes with another pair to check calculations.
6. After a suitable time gather the class and discuss efficient ways to find the total surface area of a cuboid.
If a cuboid has a larger surface area than another cuboid, then the first cuboid has the largest volume. True or false?
Do students realise that a long skinny cuboid may have a large surface area but have little volume?
Do students connect this idea to perimeter and area of rectangle? (A large perimeter does not mean large area)
7. Use Slide Four to pose a challenge.
What is the surface area of this cuboid?
Each edge is involved in four rectangles, combining twice with each other edge. With this cuboid the rectangles are:
10 x 6 (Twice)             10 x 4 (Twice)             6 x 4 (Twice)
(10 x 6) + (10 x 6) + (10 x 4) + (10 x 4) + (6 x 4) + (6 x 4) = 248cm2.
8. Show students Slide Five. You can make the slide available as a Copymaster if you want.
What shape will this net make if you fold it up?
Let students visualise the complete solid. The name for the solid is an oblique prism. You will find images for it online using the search “building oblique prism”.
Discuss: What would be the advantages/disadvantages of a building that is shaped like an oblique prism?
Challenge: Make the solid by cutting, folding, and taping.
The building has glass walls. Calculate the surface area of the glass in square centimetres. (You can pretend the squares are square metres in you want).
Use the task for assessment: Can your students calculate the areas of rectangles and parallelograms?

Session 5

In this session students are given a range of problems related to areas of triangles, rectangles and parallelograms, and to the volume of solids (see Copymaster Three). Let students work on the problems individually before sharing their work with a partner. Ensure students check their solutions against those provided below.

1. Rectangle (4 x 9 = 36 cm2)
Triangle (1/2 x 8 x 4 = 16 cm2)

Parallelogram (4 x 8 = 32cm2)
Note that orientation of the shapes may affect students’ ability to identify the base and height.
2. If you stick to whole number side lengths then the area of the square might be 1, 4, 9, 16, 25, 36, etc. What are these numbers? (Square numbers)
Suppose you chose 4 x 4 = 16 cm2 for the square, then the rectangle might be 1 x 16, 2 x 8, or 4 x 4 (since a square is also a rectangle). The parallelogram is just a ‘pushed’ version of one of those rectangles. It must have the same base and height as one of the rectangles. The triangle can also be made from a rectangle. Keep the base the same but double the height or double the base and keep the height the same.
3. You could calculate each volume separately, but two edges are the same on each box. The left box has half the volume of the middle box since it is half as tall. It holds more than half as much powder, so it has less air space.
The right box is three quarters the volume of the middle box since 18 is three quarters of 24. The right box holds three quarters as much powder as the middle box since 600 is three quarters of 800. The middle and right boxes have the same ratio of air space to powder space. However, since the middle box has more volume it has the most air space.
4. Tasmania has an area of 68 401km2. How close did you get? Which shape gave the best estimate.
5. The rectangle and parallelogram have the same area because they have the same base and height. Measure if you would like to check.

Equivalent Fractions

Purpose

In this unit we are exploring ways to find equivalent fractions. We use the concept of equivalent fractions to convert fractions to the benchmark fractions of halves, quarters, thirds, fifths and tenths. From these benchmark fractions it is easier to convert fractions to decimals and percentages. We use equivalent fractions to compare fractions.

Achievement Objectives
NA4-4: Apply simple linear proportions, including ordering fractions.
NA4-5: Know the equivalent decimal and percentage forms for everyday fractions.
Specific Learning Outcomes
• Explore and know equivalent fractions including halves, thirds, quarters, fifths, tenths and hundredths.
• Use equivalent fractions to convert fractions to decimals and percentages.
• Use equivalent fractions to order fractions with different denominators.
Description of Mathematics

Fractions are an extension of whole numbers and integers. Fractions are needed when wholes (ones) are not adequate for a task. Division often requires equal partitioning of ones. Sharing two chocolate bars equally among five people requires that the bars be cut into smaller equal parts. The operation might be recorded as 2 ÷ 5 = 2/5. Note that the number two fifths, is composed of two units of one fifth. In practical terms the equal share can occur by dividing each of the two bars into fifths, then giving each person one fifth from each bar.

If the bar was made up of ten pieces then each person might be given two tenths from each bar, giving them four tenths in total. Four tenths are the same quantity of chocolate as two fifths. Any fraction can be expressed as an infinite number of equivalent fractions that represent the same quantity and occupy the same position on the number line.

Fractions are important to measurement, especially where whole units are not precise enough for the purpose. The symbolic expression does not explain why equivalent fractions represent the same amount. Consider these equivalent fractions: 2/3 = 4/6 = 8/12 . A fraction strip (length) model of the relationships looks like this: Sixths are half the size of thirds so twice as many sixths fit into the same length as thirds.

Twelfths are quarter the size of thirds so four times as many twelfths fit into the same length as thirds.

The relationship between two thirds and eight twelfths can be represented in this equality. Four times as many twelfths comprise one as thirds. Therefore, each third can be divided into four twelfths.

Understanding equivalent fractions is critical to making sense of decimals and percentages. Consider the names for 3/4 = 0.75 = 75%. Both 0.75 and 75% represent 75/100 which is an equivalent fraction to 3/4. If each quarter is equally partitioned into 25 parts, those parts are called hundredths since 4 x 25 of those parts fit into one.

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

• vary the level of abstraction
• alter the complexity of the numbers involved, or the relationships between numerators and denominators
• allow use of scientific calculators that can process fractions.

Adaptation involves changing the contexts used for problems to meet the interests and cultural backgrounds of your students. Where contexts such as food and ratios of orcas and dolphins may not be appropriate for your students, find other situations likely to engage them. Birthday Cakes might be replaced by areas of land, dart boards, or gold coins. Orcas and dolphins might be replaced by other animals that need conservation. Linear models are easily applied to journeys that students make, or physical objects such as tape and rope.

Required Resource Materials
• Scissors
• Cubes, counters, or other discrete materials of different colours.
• Calculators
• Plastic Fraction Strips (if available) or use Copymaster Four
• Copymasters One, Two, Three, Four, and Five
• PowerPoints One, Two, and Three
Activity

Session One

1. Begin by writing the fractions 2/3 and 3/4 on the whiteboard.
How do we read these fractions? (two thirds and three quarters)
What picture do you see when you think …about two-thirds? ...about three quarters?
Invite individual students to draw representations of the fractions.
Main points to bring out are:
• Two thirds consist of two copies of one third and three quarters is made of three copies of one quarter.
• The numerators (top numbers) are counters of the number of parts, two and three, respectively.
• The denominators give the size of parts, three in 2/3 indicates that the part is one of three equal parts that form one, and four in 3/4 indicates that the part is one of four equal parts that form one.
• Both fractions can be represented as areas (e.g. squares or circles), lengths, sets, and a variety of other ways.
2. Use Slide One of PowerPoint One to introduce this problem:
Nia and Ashanti are identical twins.
Each birthday they each get identical birthday cakes.
Nia eats two thirds of her cake and Ashanti eats three quarters of her cake.
Who eats the most?
How much more cake does she eat than her sister?
3. Let students solve the problem in pairs. Expect them to record how they solved the problem using drawing, symbols, or a combination. As they work, roam the room. Look for students’ understanding that:
• Three quarters is more than two thirds (Why?)
• Twelfths are needed to find the difference between 2/3 and 3/4.
4. After a suitable time gather the class to discuss their solutions. You might use a fraction circle manipulate online, or fold squares or rectangular pieces of paper to find the difference between the two fractions.  5. Discuss why the difference is one twelfth (more obvious with the pre labelled circle pieces). Students might notice that the denominators multiply to give 12 (3 x 4 = 12).
6. Ask each student to take two rectangular pieces of paper to represent the birthday cakes. Fold one lengthways into thirds, and shade two thirds. Fold the other rectangle in quarters lengthways and shade three quarters. 7. Fold the pieces of paper back to the unit fractions thirds and quarters. 8. Fold the thirds into four equal parts lengthways.
What fraction of the whole cake is each piece? (twelfths – Why?)
Two thirds equals how many twelfths?
9. Fold the quarters into three equal parts lengthways.
What fraction of the whole cake is each piece? (twelfths – Why?)
Three quarters equals how many twelfths?
10. Open the pieces of paper up to align the partitions. How much greater is three quarters than two thirds? (one twelfth)
You might record 3/4 > 2/3 because 9/12 > 8/12 and ask students to explain the meaning of the symbols.
11. Slides Two and Three of PowerPoint One provide other examples of comparing fractions in the same way.
The Po and Mangu story compares 3/5 and 5/8. Students should realise that both fifths and eighths can be equi-partitioned into fortieths.
3/5 = 24/40 and 5/8 = 25/40 so 5/8 is 1/40 greater than 3/5. Some students might use division on a calculator to check the comparison. 3/5 = 0.6 and 5/8 = 0.625 . This means of comparison might open up a conversation about fractions as decimals.
The Thalia and Andreas story compares 2/3 and 7/10. Both thirds and tenths can be equi-partitioned into thirtieths. 2/3 = 20/30 and 7/10 = 21/30 so 7/10 is greater by 1/20. The decimal conversions are 2/3 = 0.6666... and 7/10 = 0.7.
12. Let students work on Copymaster One, either individually or in pairs. Look to see that students can make comparisons symbolically and diagrammatically. Roam the room to see that students are converting fractions to equivalent forms to solve the problems and creating diagrams that match their calculations. After a suitable time gather the class and discuss the solutions.

Session Two

1. Write the words “equivalent fractions” on the board.
When we say that two or more fractions are equivalent, what do we mean?
2. Invite ideas with the aim that students recognise that equivalent fractions are different names for the same quantity.
Do you know any pairs of equivalent fractions already?
3. Make a list of three pairs offered by students. For example: What patterns are true for all three pairs?
Students might notice that there is always and equals sign. What does that symbol mean?
Some might notice that the numerators and denominators are multiplied by the same number, e.g. By four in the pair 2/3 = 8/12.
What does the multiplication mean? In the example, four times as many twelfths fit in one as thirds, so four times as many twelfths fit into the same space as two thirds.
4. Give your students practice at fraction conversions using the examples on Slide One of PowerPoint Two. You might copy that page for students who find board to page translation difficult. Allow calculator use for students who do not know their basic facts. Answers are on Slide Two. Note that the last four examples have a range of solutions. One correct answer is given.
5. Slide Three shows the following “Hay There” problem. The type of division is quotative or sharing.
Three goats share two hay bales equally. Each goat gets the same amount.
Six sheep share four hay bales equally.
6. Which animal gets more hay, a goat or a sheep?
7. Let students attempt the problem in pairs and discuss what they notice. Bring the class together at a suitable time and share answers. Discuss:
How could we record the problem using a diagram? (Students often draw lines connecting animals and bales)
How could we record the problem using symbols? Sharing can be represented by division; 2 ÷ 3 = 2/3 (goats) and 4 ÷ 6 = 4/6 (sheep). Do your students recognise that two thirds of a bale is equivalent to four sixths of a bale? The shares are equal since 2/3 = 4/6.
8. Slides Four and Five of PowerPoint Two provides other quotative contexts.
Work through each problem with your students. Let them attempt the problem first before sharing strategies.
Students may get confused by which number is the divisor in the kiore and heihei problem. Three kiore sharing four kumara should be represented as 4 ÷ 3 = 4/3 = 1 1/3.
Nine heihei sharing 12 kumara should be represented as 12 ÷ 9 = 12/9 = 1 3/9.
Since 1/3 = 3/9 the shares are equal.
Shares for the kotare and kiwi are as follows:
4 ÷ 5 = 4/5 (kotare) and 9 ÷ 10 = 9/10 (kiwi)
A qualitative judgment is needed to establish that kiwi get more worm each.
4/5 = 8/10 so kiwis get 1/10 of a worm more than kotare.
9. Provide students with Copymaster Two to work from in pairs. Roam the room and look for students to:
• Use division to work out equals shares
• Compare the shares using equivalent fractions
10. After a suitable time gather the class to share solutions. The answers are:
1. 3 ÷ 6 = 3/6 = 1/2         4 ÷ 8 = 4/8 = 1/2
The shares are equal.
2. 10 ÷ 6 = 10/6 = 1 4/6 = 1 2/3             15 ÷ 9 = 15/9 = 1 6/9 = 1 2/3
The shares are equal.
3. 12 ÷ 8 = 12/8 = 1½   8 ÷ 5 = 8/5 = 1 3/5
Albatrosses get more, 1/10 of an oyster more.
4. 9 ÷ 12 = 9/12 = 3/4  5 ÷ 9 = 5/9
Goats get more, 7/36 of a bale more.
5. Any fraction equivalent to 4/5 works, if the denominator is greater than 15.
4/5 = 16/20 = 20/25 = 24/30 etc.
6. 5 ÷ 8 = 5/8 so any fraction equivalent to 5/8 works.
5/8 = 10/16 = 15/24 etc.

Session Three

1. Begin the session by creating a number line using equal lengths of tape (adding machine tape is ideal). Start by creating the space between zero and one, then continue to include two and three on the whiteboard. Where does the number one half live on the number line?
Be aware that some students may think you mean one half of the whole line, i.e. one half of 3 or 1 ½. This common issue is about confusing 1/2 as an operator with 1/2 as a number.
How can we locate one half exactly?
2. Let students estimate the location first then confirm their estimate by folding a strip in half lengthwise and marking where the fold comes to when one end is located at zero.
3. Mark the location of other fractions by estimating first then folding strips to locate the fractions exactly. Check that students remember that the numerator is a count of how many parts iterate (copy end on end) to create the fraction. Good fractions to use are:
Three halves (3/2)          three quarters (3/4)              five quarters (5/4)          ten quarters (10/4)
Five eighths (5/8)           eleven eighths (11/8)              two thirds (2/3)              seven thirds (7/3)
4. Pose this challenge:
You will get a set of cards (Copymaster Three: Set One).
Your job is to draw a line and organise the cards in order along the line.

Students might use a large sheet of paper and glue stick to create their number lines in pairs.
5. If you have commercial fraction strip sets available provide them to students. If not have Copymaster Four (Paper strips) available for students who need them.
6. Roam the room as students work on Set One. Provide Set Two for students who complete the initial challenge.
Look for students to:
• Recognise where two fractions are equivalent.
• Locate equivalent fractions in the same location (arrange vertically).
• Recognise that a fraction with a numerator of zero equals zero.
• Recognise that a fraction with the same numerator as denominator, e.g. 5/5, equals one.
• Work on the fraction set systematically, starting with the most familiar fractions.
7. After sufficient time gather the class to share number lines.
Which fractions are the hardest to locate? Why?
Is there a way to simplify those fractions so the task is easier?
8. Discuss looking for a common factor in the numerator and denominator. For example, in 8/12 both 8 and 12 share a factor if four. Dividing both numbers by four gives the equivalent fraction 2/3.
9. Finish the lesson with a riddle.
I am a fraction.
I am between two thirds and three quarters.
My denominator is 24.
Who am I?

Session Four

1. Use PowerPoint Three to introduce the context of orca and dolphin numbers. Slide Three shows these data:
 January July Orcas 8 3 Dolphins 16 9

What do you notice about the data?
Students might comment that the numbers of creatures is much less in July compared to January. Why?
Compared to January, what fraction of the total number of creatures were in July?
12/24 = 1/2 so there are half as many creatures in July.
Are the fractions the same for both months?
8/24 (8 out of 24 for orcas) and 16/24 (16 out of 24 for dolphins) in January
3/12 (3 out of 12 for orcas) and 9/12 (9 out of12 for dolphins) in July
Can we simplify these fractions to make them easier to compare?
8/24 = 1/3 and 16/24 = 2/3
3/12 = 1/4 and 9/12 = 3/4
The fraction of orcas is slightly less in July than in January. Perhaps orcas prefer cooler water.

2. Slide Four shows how the January data can be grouped to form different fractions.
What fractions can you see? Explain where you see those fractions.
3. Slide Five shows the July data.
Draw a diagram to show how the fractions ¼ and ¾ can be seen in the orca and dolphin data.
Let students draw their own diagram before animating the slide.
4. Slide Six shows a survey in a different location, Otago Harbour.
Which month has the greatest fraction of orcas in the whole group?
Do students recognise that the fraction of orcas equals three fifths for both months?
5. Provide your students with Copymaster Five to work on individually or in pairs. The worksheet applies equivalent fractions of sets. Roam the room as students work to see that they can:
• Express the proportions for each creature as fractions
• Simplify the fractions by re-unitising (finding common factors)
• Compare the fractions for orcas and dolphins using fractions as numbers.

Kawhia Harbour

January                                                                     July

Fraction of orcas = 1/2                                             Fraction of orcas = 1/2

Fraction of dolphins = 1/2                                       Fraction of dolphins =1/2

Is there a change in the fraction for each creature comparing July to January? Same

Whitianga Coast

January                                                                    July

Fraction of orcas = 10/25 = 2/5                              Fraction of orcas = 4/10 = 2/5

Fraction of dolphins = 15/24 = 3/5                        Fraction of dolphins = 6/10 = 3/5

Is there a change in the fraction for each creature comparing July to January?       Same

Queen Charlote Sound

January                                                                          July

Fraction of orcas =  6/24 = 1/4                                     Fraction of orcas = 4/16 = 1/4

Fraction of dolphins = 18/24 = 3/4                              Fraction of dolphins = 12/16 = 3/4

Is there a change in the fraction for each creature comparing July to January?       Same

Kaipara Harbour

January                                                                 July

Fraction of orcas = 10/20 = 1/2                          Fraction of orcas = 4/10 = 2/5

Fraction of dolphins =10/20 = 1/2                     Fraction of dolphins = 6/10 = 3/5

Is there a change in the fraction for each creature comparing July to January?

The fraction of dolphins increases and the fraction of orcas decreases.

Akaroa Harbour

January                                                                  July

Fraction of orcas = 4/24 = 1/6                             Fraction of orcas = 2/16 = 1/8

Fraction of dolphins = 20/24 = 5/6                     Fraction of dolphins = 14/16 = 7/8

Is there a change in the fraction for each creature comparing July to January?

The fraction of dolphins increases and the fraction of orcas decreases.

Hawke Bay

January                                                                 July

Fraction of orcas = 12/40 = 3/10                         Fraction of orcas = 9/21 = 3/7

Fraction of dolphins = 28/40 = 7/10                   Fraction of dolphins = 12/21 = 4/7

Is there a change in the fraction for each creature comparing July to January?

The fraction of orcas increases and the fraction of dolphins decreases.

Session Five

1. Use the following Figure It Out pages to set independent work for your students. You could use the work samples as evidence of student progress on the Achievement Objectives of the NZC and on the Multiplication and Division aspect of the Learning Progressions Framework.
2. The links take you to teachers’ guide pages with a PDF of the student page/s and answers.

Extension

• Number Level 3-4, Book 1: Pages 4-5, Page 9, Bean Brains.

Travel to school

Purpose

In this unit students investigate methods of travel to school, using technology to produce data displays and investigate distributions.

Specific Learning Outcomes
• Pose investigative questions for statistical enquiry.
• Plan an investigation.
• Use spreadsheets to collate data.
• Use technology to display data.
• Discuss features of data display.
• Compare features of data distributions.
Description of Mathematics

Arnold’s (2013) research identified six criteria for what makes a good investigative question.  At curriculum level 4, students should be introduced to the criteria, potentially through “discovering” them.  See for example, the following lesson on CensusAtSchool New Zealand https://new.censusatschool.org.nz/resource/posing-summary-investigative-questions/ .

The six criteria are:

1. The variable(s) of interest is/are clear and available or can be collected
2. The group of interest is clear
3. The intent is clear (e.g. summary, comparison, relationship, time series)
4. The investigative question can be answered with the data (e.g. question is specific, data can be collected, ethics)
5. The investigative question is one that is worth investigating, that it is interesting, that there is a purpose
6. The investigative question allows for analysis to be made of the whole group.

Categorical variables

Categorical variables classify individuals or objects into categories.  For example, the method of travel to school; colour of eyes.

Numerical variables

Numerical variables include variables that are measured e.g. the time taken to travel to school, and variables that are counted e.g. the number of traffic lights between home and school.  Measured numerical variables are called continuous numerical variables.  Counted numerical variables are called discrete numerical variables.

Measures

We “measure” both categorical and numerical variables. For example, if we were to ask about how students carry their school bag, we would have to decide what categories we will offer as options for carrying a school bag.  Fortunately, this is one of the survey questions from CensusAtSchool so we can use their wording.  Additionally, if we want to measure the distance from our house to school, we need to plan to help students work this out.  For example, we might choose to map on google maps and take the distance from google maps.

Key to deciding about measures is to support getting valid and reliable measures.  Valid measurements measure what they claim to measure, and reliable measurements are those that give you or someone else approximately the same result time after time when taken on the same individual or object. For example, using google maps to find the distance from home to school is both valid and reliable. It measures the distance from home to school (validity) and we will get the same result regardless of if the student or someone else was to get the information from google maps (reliability).

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

• the type of data collected; categorical data can be easier to manage than numerical data
• the type of analysis – and the support given to do the analysis
• setting up blank CODAP documents with the data already in and some graph blanks ready to use for students
• providing prompts for writing descriptive statements
• teacher support at all stages of the investigation.

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

• the statistical enquiry process can be applied to many topics and selecting ones that are of interest to your students should always be a priority
• this investigation focuses on travel to school and comparing across the different CensusAtSchool databases from other countries, the ideas for comparing using CensusAtSchool data can be adapted to other data that is available, see additional related exploration at the end.  Variables include reaction time and memory game score; importance of … ideas; height, foot length and arm length.
Required Resource Materials
• Computer(s) with Internet access
• Copymaster One
Activity

Session 1: Getting our travel data

PROBLEM: Generating ideas for statistical investigation and developing investigative questions

We are going to explore daily travel to school.

1. Discuss what information the class could collect about their daily travel to school. Collect ideas on the board.
2. For each idea identify potential variables that they could “measure”. E.g. method of travel; number of passengers per vehicle; time to travel to school; distance travelled to school; number of traffic lights/roundabouts/stop signs between their house and school
3. For each potential variable identify if it is categorical or numerical.  For numerical variables identify if they are discrete (counted) or continuous (measurement).
4. In groups select an idea/variable to explore further.  Each group should have a different idea/variable.
5. As a group the students interrogate their ideas/variable by answering the following questions:
• Is this an area that the students in our class would be happy to share information with everyone? If not reject the idea [ethics] (refer criteria 4).
• Can we collect data to answer an investigative question based on this area of interest? If not reject the idea [ability to gather data to answer the investigative question] (refer criteria 4).
• What would be the purpose of asking about the idea that you have, do you think it will provide meaningful information about our topic on daily travel to school? If it is not purposeful then reject the idea [purposeful or interesting] (refer criteria 5).
• Would the investigative question we pose to explore this idea involve everyone in the class? If not, then reject the idea [involving the whole group] (refer criteria 6).

If the students end up rejecting their idea, they select a new one and repeat the process.

1. Each group develops an investigative question to explore their idea/variable. To help them get started ask the groups to identify what the variable is (this is what we want to measure, refer criteria 1) and to describe the group (this is who we will measure, the group is likely to be “our class”, refer criteria 2).
2. Groups to interrogate their investigative questions by asking the following questions:
• Is the variable clear? (refer criteria 1)
• Is the group we are investigating clear? (refer criteria 2)
• Is the question purposeful?  (refer criteria 5)
• Is the question about the whole group? Check that the question is not just finding out about an individual or smaller group of the whole class (refer criteria 6).
• Is the question one that we can collect data for? (refer criteria 4)
• Is it clear whether the question is a summary investigative question (about a single variable) or a comparison investigative question (a single variable compared across two or more groups)? (refer criteria 3)

Session 2: Planning to collect travel data

PLAN: Planning to collect data to answer our investigative question

1. Each group decides how they will collect their data by defining the variable and working out how to measure it so that they get valid and reliable measures. They develop survey questions to ask the class.
2. Groups interrogate their plan using the following interrogative questions:
• Can a single measurement capture the ideas needed to answer the investigative question?
• Is there a usual way of measuring this variable?
• Does this measurement really capture the variable I want to measure (validity)?
• If I were to measure the same units again, would I get very similar results (reliability)?
• If different people were to make the same measurement will they get very similar results (reliability)?
• Do I have the equipment to make this measure?
3. Groups update their plan as required.
4. Groups decide on how they are going to record their data, encourage electronic versions such as google or excel spreadsheets.
5. As a class you might decide to use a google form (or similar) to collect the data.
6. Make sure that a survey question about method of travel to school and time taken to travel to school are included if none of the groups have chosen to do these. The survey questions from the CensusAtSchool questionnaire is recommended to use. The survey questions are:
What is the main way you usually travel to school? (Choice from) walk | car | bus | train | bike | boat | scooter | other
How long does it usually take you to get to school? Answer to the nearest minute.
___________ minutes

DATA: Collecting and organising data

1. The groups share their survey questions that they will use to collect data from the class at the next session.  Any data that requires students to take an action to be able to give a response the next time needs to be highlighted.  For example, if students are required to know how long it takes to get to school, they need to know this so that the next day they can be conscious of recording the time taken to get to school.

Session 3: Collecting and organising our travel data

In this session the students will be 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 session is written with CODAP as the online tool and is assuming students are familiar with CODAP.

If your students are unfamiliar with CODAP see:

The main features that students need to be familiar with are how to draw a graph and how to import their data. More on importing data into CODAP can be found here.

1. Allow students time to collect their data for their survey questions.
• It is possible that a single spreadsheet could be set up and the students individually put their data into the spreadsheet for each of the survey questions.
• A google form (or similar) could be created for all the survey questions and the students complete this.
• Alternatively, students create a paper table to collect the data into and then input this into an electronic spreadsheet.

The aim is to have the data in an electronic form so they can use technology to make the displays.

ANALYSIS: Using an online tool to make data displays

1. Share the spreadsheet with the students (if they do not already have access).  This should be shared as a .csv file.
2. Students import the data into CODAP (see here for information on how to do this).
3. Students explore different ways in CODAP to graph their data to answer their investigative question.
4. Students present their graph to the rest of the class.  This could be through showing their CODAP working document, through sharing using Power Point or google slides, or in a google or word document.
5. As each graph is presented ask the students what “they notice…” about the data. Encourage groups to add the “I notice…” statements to their display.
• In CODAP they can write I notice statements using a text box.
• On a google slide/Power Point, word/google doc they can type the I notice statements below the graph.
6. Get the students to share their graphs and statements with you.  Find the method of travel graph and the time taken to travel to school graph and have it ready for the next session. If no groups chose these variables make the graphs ready for the next session.
7. Ask the students what they notice about the graphs for the different types of variables, for example how does a graph of categorical data compare to a graph of numerical data.
8. Homework activity for students: In a couple of sessions we will compare how we travel to school with how our parents and caregivers travelled to school. Overnight can you ask your parents and caregivers the following two survey questions (Copymaster One):
• What was the main way you usually travelled to school when you were my age? (Choose from) walk | car | bus | train | bike | boat | scooter | other__________
If they select other, record how they travelled to school.
• How long did it take you to travel to school (your best guess) in minutes?

Session 4: Making comparisons

1. Before the lesson set up a google form or similar to collect the data from the students about their parents and caregivers that they filled in overnight.
2. Ask the students to complete the two-question survey with the data from their parents and caregivers.
3. Collect in their sheets (so you can check data input if needed).
4. In preparation for session 5, download the parent and caregiver data and import into a CODAP file.  Share the CODAP file with your students.

How do we compare with other New Zealanders our age?

1. Students from around New Zealand have also been asked how they travel to school, and the time taken to travel to school. This data has been collected and 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 these two variables.
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.
1. Select database – here we can choose any database from 2005 onwards.  Recommend they use the latest database (this is the pre-selected option).
2. Select subpopulation – because we want to compare with other New Zealanders 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. 3. Select variables – because we want to look at method of travel and travel time to school, we only want to select specific variables. When we select specific variable, we get a drop-down list of all the variables in the survey. Ask the students which variables we should select if we want to compare with our class data about method of travel to school and time taken to get to school. Select these two variables.
4. Select sample type – leave as random sample
5. Enter sample size (Maximum 1000) – suggest they select 30 as this is similar in size to a class.

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 their own data.
7. Students display the data for the two variables using CODAP. They should write “I notice” statements about what they see in their data from CensusAtSchool.
8. If students have not yet learnt how to convert the categorical data into a bar graph show them how to do this. Select the graph and the tool bar comes up.  Select the bar graph icon and select fuse dots into bars. Students can then select the ruler and choose percent.  This will show the percentage in each bar allowing for comparisons between data sets of different sizes more easily.
​​​​​  9. Ask the students how they might compare the numerical data. For example, having the same scale on the x-axis is important to help with visual comparisons.
10. Students then compare their CensusAtSchool sample with the class data noticing what is similar and what is different.
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).
11. Reflect on which of the data types (categorical or numerical) was easiest to compare and why.

How do we compare with the USA?

1. Other students from around the world have also been asked how they travel to school. For example, the USA. This data has been collected and is available on the CensusAtSchool USA site.  The USA data is messy, but fortunately the New Zealand CensusAtSchool team has tidied it up and the tidied up data is available at this link
• As with the New Zealand data, confirm the conditions of use
• Select the CAS USA database
• Make the total sample size 30
• Import into CODAP

Note: the filters might not be working on the USA database, therefore it might not be possible to get an age appropriate data set. Including all ages should be fine for the comparison ideas for the travel to school. 1. Once the data has been collected from the site get the students to discuss how they might make a comparison with their own data.
What do they notice is different about this data set compared with the one we did for the New Zealand data? We have all of the variables in the USA data set whereas in the New Zealand data set we just had the two variables we wanted to explore.
Which variables do we want to graph from this data set to compare with our class data and the New Zealand data?  Students can convert the data from a table view to a case card view (in CODAP), this makes it easier to see the variables. Students identify that travel_to_school and travel_time_to_school are the variables we are interested in. 2. Conclude with a reporting time where the students are given the chance to show their graphs and to discuss what they have found out. They will probably have compared each mode of travel between the three sets of data and the time taken to get to school between three sets of data. Can they give reasons for any differences they have found?

Session 5: Has the method of travel to school changed?

1. Explain that today we will look at how our parents and caregivers travelled to school.  Ask the students to predict what they think will be the same as our class and what will be different.  Capture the ideas on the board or a large sheet of paper.
2. Share the parents and caregivers’ data set with the students (share the CODAP document).
3. Get the students to make displays and discuss how the data set is the same and different from the class data and then compare with the New Zealand sample and the USA sample.
4. What conclusions have they reached? What factors mean that this may not be a totally valid comparison? (e.g. the differing ages of the parents and caregivers, the fact that parents and caregivers may not have lived in this area when they were young, etc).

CONCLUSION: Answering the investigative question and reporting findings

1. Students develop a short report to share with their parents and caregivers that compares the class data with the parents and caregivers’ data.

Exploring across five countries

Using the old random sampler https://new.censusatschool.org.nz/tools/random-sampler/ on CensusAtSchool, students can select the CAS international database.  If they select subpopulation and then country, boxes will show so they can select 30 from each country. The data set contains 20 variables, two of which are the travel variables we have been looking at. Encourage students to explore the full data set using CODAP and developing further their graphing and describing skills.

Attachments

Getting partial: Fractions of sets

Purpose

This unit develops the concept of a fraction as an operator, or multiplier, acting on an amount, e.g. two-thirds of 24. Using fraction multipliers to represent the relationship between different amounts is also explored.

Achievement Objectives
NA4-1: Use a range of multiplicative strategies when operating on whole numbers.
NA4-3: Find fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.
Specific Learning Outcomes
• Find a unit and non-unit fraction of a set, e.g. two thirds of 24 (2/3×24).
• Use a fraction to represent the relationship between part of a set and the whole set.
• Recognise when the part-whole fractions of two different sized sets are equivalent, and when one fraction is greater than another.
• Use a fraction to represent the relative size of two sets, e.g. 12 is two thirds of 18.
Description of Mathematics

Fractions are an extension of whole numbers and integers. Fractions are needed when wholes (ones) are not adequate for a task. Division often requires equal partitioning of ones. Sharing two chocolate bars equally among five people requires that the bars be cut into smaller equal parts. The operation might be recorded as 2÷5=2/5. Note that the number two fifths, is composed of two units of one fifth. In practical terms the equal share can occur by dividing each of the two bars into fifths, then giving each person one fifth from each bar.

Many contexts involve relating the relative size of discrete quantities, that is, quantities that are collections of individual items. According to Watanabe there are two types of relationship, part to whole, and whole to whole.

1. Part to whole

Fractions can represent the relationship between part of a set and the whole set. In the diagram below one quarter or four eighths of the whole set is grey. 1. Whole to whole

Fractions can represent the relationship between two independent sets or amounts. To complicate matters the sets can be within a whole set, as in a ratio situation.

In the left-side diagram, the grey set has three quarters or six eighths of the number of items that the white set has. In the right-side diagram, there are three quarters as many grey circles than there are white circles, within the same set.  Set to set comparisons are common in the real world, in situations like comparing prices with those some time ago, calculating lambing or calving rates, measuring population growth. Part-whole comparisons are particularly important in situations like comparing groups in statistics, finding the most accurate goal shooter, or sharing sets equally.

Set models tend to be more complex than continuous models like lengths, areas, and volumes, though measurement attributes vary in perceptual difficulty. Set models have the added complexity of competing wholes (ones). Suppose that a box has 12 chocolates, and three of them are strawberry flavour. To establish that one quarter of the set is strawberry requires imagining the 12 chocolates as one whole set, when each chocolate is a whole in it’s own right. Establishing quarters requires imagining the three wholes as one quarter, relative to the newly established whole set.

Comparison of fractions, and equivalence of fractions is particularly challenging with sets models. The fraction of grey circles in the left-side set can be expressed as 1/4 or 3/12. The fraction of grey circles in the right-side set can be expressed as 1/4,  2/8, 3/12 or 6/24. Each expression requires you to imagine the set as a whole and create different partitions of that whole. Note that both sets have the same fraction of grey although the size of the sets are different. The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

• Simplifying the fractions involved in problems to those directly related by halving, such as halves, quarters, and eighths, and to those easily accessible through knowledge of basic multiplication facts, like fifths and tenths.
• Varying to scaffolding of support for students, particularly through using physical materials, and diagrams, progressing to more abstract symbolic representations.
• Connecting materials, words, and symbols through meaningful contexts. The meaning of numerator as counter, denominator as size of parts with reference to one, and multiplication as “of” are significant in this unit.

The context for this unit is toppings put of pizza. While most students are familiar with pizza as a common food, other contexts may be more meaningful to your ākonga (students). Activities can be adapted to suit the interests and experiences of your students. For example:

• A pizza might be exchanged for a plot in a community garden. The plot is shared into equal parts for different numbers of people to farm it. Pizza toppings can be replaced by different seedlings, such as lettuce, spinach, cabbage, and tomato.
Required Resource Materials
• Unifix cubes, counters
• Blank cubes or dice and stickers (to make dice)
• Scissors
• Calculators (optional)
• Copymasters One, Two, Three, Four, Five, and Six
• Note that some Copymasters are designed to be copied, laminated and cut up as cards or puzzle pieces.
• PowerPoints One, Two, Three, and Four.
Activity

Getting Started

In this lesson students connect circular region models of fractions with finding fractions of a set. By the end of the lesson students should be able to unitise a given set into different parts, and name the part-whole relationships in terms of those parts.

Begin with Powerpoint One that presents the scenario of distributing toppings, like pieces of salami, evenly around a pizza. Students are often familiar with a circular region model so the connection to a discrete model is useful.

For each scenario (topping) ask questions like:

• How many (toppings) do you think will go in each piece (unit fraction)?
• How did you work that out?
• How might the equal sharing be written as an equation?

Note that the problems are about finding a unit fraction of a set, so the fraction behaves as a multiplier. Therefore, the equation might be written as multiplication or division, e.g. 1/4×36=9 or 36÷4=9. Students who use additive build up to anticipate the result should be encouraged to use multiplicative thinking with questions like:

• I see you gave four (toppings) to each person, and went 4 + 4 = 8, 8 + 8 =16, to work out how many were used up. What multiplication fact could you use instead?
• Could you give each piece ten (toppings)? Why or why not?
• If ten is too much/little then how could you fix that up?

The final question is about finding a multiplication or division fact that approximates the solution, from which the student can derive the solution. For example, 40 ÷ 5 = 8 can be derived from 10 ÷ 5 = 2 (See Slide Four).

After working through PowerPoint One ask students to work on Copymaster One in pairs. The problems all involve finding a unit fraction of a set, in the same pizza scenario. You might make materials, like counters, available to some students but encourage them to anticipate the result of the equal sharing first. Look for your students to:

• Apply multiplicative strategies rather than counting or additive build up;
• Record symbols or diagrams to support their thinking;
• Record the situation correctly as a multiplication equation;
• See connection among the problems, that is, use one result to get another.

After completing Copymaster One, students play The Pizza Game. Cards for the game are made from Copymaster Two. Print two sets of the Copymaster to make a total of 32 cards. The game is played with two or three players. Each team of players needs two dice made from blank wooden cubes, or by using stickers over normal dice. The dice are labelled:
Easy Game: 14, 15, 16, 18, 20, 24, and 25, 27, 30, 32, 36, 40.
Hard Game: 35, 36, 40, 42, 45, 48, and 50, 54, 56, 64, 72, 81.

Spread all the cards in the middle, face up so all players can see them.

How to play

Players take turns to:

• Roll the two dice. Choose one of the dice numbers.
• Choose a pizza card so that the number of toppings on the dice can be shared equally onto the pizza. The player must explain how they know each piece will have the same number of toppings.
• If the player is correct, they keep that pizza card and it cannot be used again.
• If they are challenged by the other players, and are incorrect, the pizza card stays in the middle.

The winner is the person with the most pizza cards when play finishes, or when all pizza cards are gone.

Exploring

In the following two sessions students extend their knowledge of fractions as multipliers (operators) to non-unit fractions, that is fractions with whole numbers greater than one as the numerator, e.g. 2/3 and 4/5. Then students explore the part-whole relationships in ratios, using fractions to represent relationships.

Session Two

Begin with PowerPoint Two that shows how to find a non-unit fraction of a set. Progress to Slide 11. Students will notice that the first four slides are the same as PowerPoint One.

• What is the same about the problems so far?
• How do we solve problems like this?

Students should suggest that the number of toppings is divided by the number of pieces. Why? (To establish the unit fraction). Then the unit fraction is multiplied by the number of pieces that are eaten.

Take at least two of the examples from PowerPoint Two and write equations for the situations.

• 2/3×30=20    (Two thirds of 30)
• 3/4×32=24    (Three quarters of 32)
• 4/5×35=28    (Four fifths of 35)

Discuss the meaning of the x symbol as ‘of’ and how the numerator and denominator of the operating fraction impact on the calculation, e.g. 4/5 is found by dividing by five then multiplying by four.

Put the students into groups of four and provide the puzzles made from Copymaster Three. There are five puzzles. Each student gets one card for each puzzle. They must work with their team-mates to find a solution that matches all the clues. Each player must ‘own’ their clue, that is, ensure that they retain it and that the solution matches their clue. Encourage the students to record their strategies in some way on paper.

• Identify the information that is missing. The information might be numerator or denominator of the operating fraction, the whole number of toppings, or the part of the whole consumed.
• Record the problem symbolically or diagrammatically in ways that are helpful. For example, expressing Puzzle C as 2/3×?=32.
• Use multiplicative strategies to solve the puzzles, rather than resort to drawing and share by ones or composite groups (skip counting) strategies. Note that access to a basic facts chart may help some students to think multiplicatively.

After an appropriate time bring the class back together. Focus on one or two puzzles that brought out interesting strategies. Choose groups to share their solutions. Conclude with slides 12-14 of PowerPoint Three. An important idea is that either order of multiplying by the numerator and dividing by the denominator works. Usually dividing first is more efficient.

Session Three

Begin with the game “3 in a line.” Gameboards can be printed and laminated using Copymaster Four. The game requires students to practise finding non-unit fractions of whole numbers. An important feature of the game is that equivalent fractions operating on the same whole numbers produce the same product, e.g. 3/4×24=18 and 6/8×24=18.

3 in a line

You need:

• Gameboard for each pair (Version D is harder than the others)
• Two colours of counters and two paperclips

To play:

Players take it in turns to choose a paper clip on their choice of fraction and on their choice of whole number. They find that fraction of the number and, if it is not yet taken, place a counter of their colour on it. The winner is the first player to have three counters of their colour in a row.

• Fluently find non-unit fractions of whole numbers using multiplication and division
• Make strategic choices about which fractions and whole numbers to choose
• Recognise that equivalent fractions operating on the same whole number give the same product

Next return to the pizza context to introduce the concept of part whole fractions within ratios. Use PowerPoint Three to introduce the concept and provide important questions. Emphasise these key points:

• Fractions can be used to represent the relationships of parts to the whole
• By re-unitising single objects into sets, different fraction names for the same part-whole relationships can be found
• Ratios can be used to represent the relationships among parts that make a whole set.

Let students work on Copymaster Five individually or in small groups. You might use counters or cubes of various colours to represent the toppings. Manipulation of the ‘toppings’ in physical form can help students as long as they are invited to anticipate the results of division before acting. Look for your students to identify the whole, use fractions to represent the part-whole relationship for each topping, and simply the fraction, if possible, using common factors. The second page of problems requires students to organise a complex set of clues to establish a mix of toppings. Trial and error approaches may be common but encourage your students to organise the clues by usefulness. Physical objects can be useful to try swap ‘toppings’ as clues are considered.

Reflecting

The final session of the units develops the use of fractions to describe the relationship between different wholes and between parts in the same whole.
Begin with the starting problem on Powerpoint Four. This task is very rich as it involves ideas about area and comparative price, as well as the comparison of toppings.

If you like pepperoni pizza, what are you better to buy, two small pizzas or one large pizza?

Let your students investigate the problem in small groups. Encourage them to develop an argument about which option is best, and their reason for choosing it. After a suitable time gather the class to discuss their answers. Groups should balance at least two characteristics, from area, price, number of pieces of salami, convenience, amount of crust, etc.

Each option is okay because the large pizza is twice the price but is twice as big the little pizza (attending to area though the large pizza is actually 2(1 )/4 times as big).

The small pizza is better because the larger pizza is only one and one half times as big (attending to diameter) but twice the price as the small pizza.
Note that capable students might calculate the areas (Small: π×132= 530.93cm2; Large: π× 19.52= 1194.59cm2) and provoke a discussion about comparative areas.

Encourage different kinds of comparison, such as:
If you just looked at the number of slices of salami, which option should you go for?

The small pizza has eight pieces of salami while the large pizza has 14 pieces. Buying two small pizzas gives two more pieces of salami for the same price.
Slides 2-9 focus on using fractions to describe the relationships between different numbers of salami pieces (different wholes). The large pizza has 14/8=7/4=1 3/4 as many pieces of salami as the small pizza. Conversely, the small pizza has 8/14=4/7 times as many pieces of salami as the large pizza.

Lining up the collections of pizza pieces makes the comparison easier, but student still need to do the following:

• Establish which collection is the unit to ‘measure’ with
• Compare the other collection to the measurement unit
• Unitise the two collections into common parts
• Express the whole to whole comparison as a fraction operator

Using Copymaster Six students create a puzzle that involves fractional multipliers. The four pizza cards are put at the corners of a square. Then students place the arrow cards between the pizza cards to represent the correct relationships between the numbers of pieces of salami. For example, the multiplier card ×4/3 goes between the six-piece pizza and the eight-piece pizza, since 6×4/3=8. Let the students solve their puzzle in pairs or threes of mixed ability. Challenge students to consider how the multipliers might be expressed as mixed numbers as well, e.g. ×4/3 can also be expressed as ×1 1/3. Note that most of the operators require simplification using common factors, e.g. ×8/6 is simplified to  ×4/3.

As a final challenge students, can used the blank sheets 3 and 4 of Copymaster Six to create their own puzzle for others to solve.

Marble Roll

Purpose

In this practical unit students make ramps, roll marbles down them, record the distance the marble travels from different starting positions, graph these distances, predict other distances, and make statements based on the data they create.

Specific Learning Outcomes
• Construct a stable ramp to meet conditions.
• Follow and repeat a procedure to replicate exactly the same conditions (fair testing).
• Measure and record distances accurately.
• Graph the starting positions and distances travelled on scatter graphs.
• Talk about distinctive features of scatter graphs.
• Make statements, backed up by reference to graphs, about possible actions.
Description of Mathematics

Scatter graphs are introduced in this unit. To understand scatter graphs students need to comprehend the relationship between the horizontal and vertical axes, i.e. x-axis and y-axis, and how one mark on the graph displays two variables. A standard convention is to represent the independent, or explanatory, variable on the x-axis. The dependent, or response, variable is represented on the y-axis. A given point, say (30, 22), represents a single occurrence such as a marble released from a height of 30cm rolls for 22cm on the carpet. A collection of single instances is represented by a collection of points on a scatter graph.

Making predictions based on information from graphs and from student’s own experimenting is a feature of this unit. The nature of predicting, the risks of predicting based on a small number of trials, handling unexpected results, looking for patterns and trends and making sense of the results are areas you need to be aware of, and specifically teach to, if needed.

The practical nature of the activities will make the unit accessible to most students. The learning opportunities in this unit can be differentiated by providing or removing support to students, or by varying the task requirements. Ways to support students include:

• scaffolding the steps of an investigation in a diagram to support some students to work through the process
• limiting the variables being investigated, e.g. relate only angle and roll length
• explicitly model measuring lengths and the drawing of scatter graphs in a ‘I do then you do’ way. Draw students’ attention to ordered pairs as representing one trial within a big experiment
• using graphing technology to support students who find fine motor tasks difficult.

The context of the unit, marble rolling, is a contrived setting. The implications of the experiment for everyday life are profound. Relate the rolling of a ball to riding a bicycle, or driving a car, downhill.

As you go faster how much harder is it for you to stop?
How does the surface alter your stopping distance? How does the surface relate to cycling or driving downhill, e.g. wet weather?

These discussions might lead to a broader investigation of safety related to moving bodies. For example:

Why are avalanches so dangerous?
If a runaway sheep is racing downhill, should you try to stop it?

Required Resource Materials
• Material to raise one end of the ramp, e.g. books, bricks, blocks of wood
• Instrument to measure distances, i.e. ruler or tape measure
• Material to make a stable ramp, e.g. wooden metre rulers with a central groove, long cardboard tubes from material, polythene rolls, plastic tubing, lengths of recycled guttering
• One marble
• Paper and pencil
• Copymaster One: Scatter Graph - Blank
• Copymaster Two: Scatter Graph Samples
Activity

Session One

During this session students are to make ramps, roll marbles down the ramps, and record the distance the marbles travels.

1. Organise the students into pairs and have them make a stable ramp. The ramp could be made using a single wooden metre ruler with a grove down the middle of it, by joining two metre rulers, long cardboard tubes, plastic tubing, recycled plastic guttering, or other suitable material. The ramp needs to remain consistent throughout the session, with only the elevation being altered. Place one end of the ramp on books to make a slope and the other end positioned to allow the marble to roll until it stops without hitting anything. Some experimentation will be needed to get an appropriate height and position to allow the marble to roll freely to a stop. A carpeted surface is better at this point that wood, concrete, or linoleum.  2. Once a stable ramp has been made and tested, ask the students to record the features of their ramp so it can be put away, and rebuilt in exactly the same way in later sessions. Digital photographs might be taken as a record of the setup.
3. Demonstrate rolling a marble down on of the ramps. Tease out the idea of fairness in testing. For example, the roller should have no influence on the marble so releasing rather than dropping the marble at the top of the run is kept consistent.
4. Show how the distance from the end of the ramp to the resting place of the marble is measured and recorded.
What unit of measure is best? (centimetres may not be accurate enough so millimetres might be used).
What is the best way to record the results? (Creating a table on a digital platform will allow for quick display and analysis of the data)
5. Each pair of students needs to create a sample of 20 rolls and to record the data. 6. Time might permit looking at the sample data.
Why does the marble stop? (Discuss friction)
Did the marble roll the same distance each time? (Unlikely)
Why did the distance vary? (Consider factors like marble release, variations in the surface of the run and carpet floor, dust on the marble or run, interference from movement or wind, or any other factors that might cause variation)
What is the best way to get a measure of ‘usual’ run from the top of the run? (Students might suggest finding an average or simply recording the range. Some distances might be removed as they are outliers)
7. Once completed, pack the ramp and marbles away, reminding the students that they will need to set up their ramps in exactly in the same way for the next session. Make sure students save the 20 distances they recorded.

Session Two

1. Ask the students to rebuild their ramp in exactly the same way as it was in Session One. Once the ramps are completed, students should use the same marble used in Session One.
Will the starting point of the marble make any difference to the distance the marble travels?
Students should expect that the marble will roll further if released from higher on the ramp.
Why will the marble roll further when released further up?
Students will have ideas about why variation in distance occurs. Avoid giving the scientific explanation. The higher the point that the marble is released from, the greater is its potential energy. That energy converts to greater speed when the end of the ramp is reached. Friction takes longer to slow a faster marble to a stop.
2. Set up the following experiment.
Release a marble from each of the five starting positions, listed below. Carry out five trials (marble releases) from each starting position. Record the distance travelled from the end of the ramp for each trial.
• Task 1: Roll the marble from the top of the ramp (release height is 100 cm from the bottom of the ramp)
Start the marble at the top of the ramp and record the distance it travels from the bottom of the ramp.
• Task 2: Start 90 cm from the bottom of the ramp, i.e. 10 centimetres from the top of the ramp.
• Task 3: Start 80 cm from the bottom of the ramp, i.e. 20 cm from the top of the ramp.
• Task 4: Start 20 cm from the bottom of the ramp, i.e. 80 cm from the top
• Task 5: Start 10 cm from the bottom of the ramp, i.e. 90 cm from the top.
3. Ask the students to leave the ramps in place and shift their focus to the recording of the distances onto a scatter graph. Explain to the class what a scatter graph is and how the two axes are used to show the release height and the distance rolled using a single cross. Show the students the completed scatter graph below. Could this graph be helpful in predicting other starting positions?
What distance would the marble travel if the release height was 50 cm?
5. Discuss the distinctive features of the above graph:
• the distances that do not follow the pattern, i.e. outliers
• the clusters or groupings of the distances starting from the same place
• the overall pattern or trend of the data, i.e. a straight line
• other pattern or trends that could be possible
• maximum and minimum distances 6. Hand out a blank scatter graph for the students to use to display the data from their trials for today.
7. Ask the students look at their own scatter graphs and predict the distance the marble will travel when started 40 cm, 50 cm and 60 cm from the bottom of the ramp. Before rolling the marble on their ramps, have each pair explain, or write down, why they think it will roll the distances they have predicted.
8. After making their predictions, students can roll the marble down their ramp at the various starting points. Record the rolling distances. Discuss these results in relationship to their predictions.
How did you use the data from your trials to make predictions? (Students should reveal an understanding of a trendline)
9. Pack up the ramps and marbles reminding the students they will need to use the ramps again during Session Four.

Session Three

This session has the students looking at a range of scatter graphs of data from marble rolling, with the task of predicting distances the marble is likely to roll. The discussion about making sense and using information from scatter graphs started in Session Two continues and is developed during this session.
There are six scatter graphs for the students to look at and predict from. The teacher needs to work out the best way to get all the students discussing the graphs, i.e. handed them out altogether, one at a time or set up six stations for groups of students to visit. Discussing and explaining their thinking is a very important part of this session. The discussing and predicting allows teachers to assess student understanding and the amount of teaching needed.
Scatter Graph Samples, Copymaster Two, has the six scatter graphs for students to look at.
The following questions could be used as part of the discussion to answer the two questions on the sheet.

• What distance is the marble likely to roll if it was started from the top of the ramp? If the release height was 50 cm?
• Is the pattern or trend linear, i.e. a straight line?
• Describe the overall trend of this ramp.
• Is there a starting position that will have the marble stop at 180 cm? 80 cm?
• Why are some distances clustered together closer than others?
• What is the range of distances are you absolutely sure the marble will stop at, given a set starting position?
• What is the range of distances you think the marble will stop at?
• On a scale from 1 to 10, how confident are you in your prediction? Explain why. Scale: 1 = not confident at all, 10 = extremely confident.
• Do scatter graphs with non-linear trends, i.e. curves, result from curved ramps?

Sessions Four and Five

During this session the students are to build their ramps again with one aspect of it changed. The students could change the height of the ramp, a golf ball could be used instead of the marble, the surface the marble rolls on could be different, etc.

1. Once their ramps have been changed they are to roll and record the distance the marble travels. Roll the marble:
• 5 times, release height 30 cm
• 5 times, release height 50 cm
• 5 times, release height 70 cm
2. Graph the results on a scatter graph, then predict the distance the marble will travel when rolled from the release heights of 100 cm, 90 cm, and 80 cm. Predict the distance and state how accurate they think they are before starting the experiment. Adjustments to predictions are acceptable as the rolling and measuring continues, as long as they are accompanied by explanations.
3. This activity could be repeated several times with different changes made.
"What if . . ." questions could be posed to challenge students:
What if the ramps from Session One were two metres long, how long would the marble roll from the top of the ramp?
What if the ramps were curved? 4. Students should be asked to choose one of their experiments and produce an A4 poster describing what they did and what they found. Discuss an appropriate layout for the poster:
• Title
• Description of the experiment
• Scatter graph with trend line
• Description of results
5. At the end of Session Five students could present their posters to the rest of the class
Attachments