Early level 3 plan (term 2)

Planning notes
This plan is a starting point for planning a mathematics and statistics teaching programme for a term. The resources listed cover about 50% of your teaching time. Further resources need to be added to meet the specific learning needs of your class, to support your local curriculum and to provide sufficient teaching for a full term.
Resource logo
Level Three
Geometry and Measurement
Units of Work
In this unit students learn to use the multiplication formula to find the area of a rectangle. Using proportional reasoning students explore what happens to the area when the length and/or height of a rectangle is doubled.
  • Use multiplication to calculate the area of a rectangle.
  • Measure the length of a side using a ruler.
  • Use proportional reasoning to find the area of a rectangle.
Resource logo
Level Three
Number and Algebra
Units of Work

This unit introduces the fact that fractions come from equi-partitioning of one whole. So the size of a given length can only be determined with reference to one. When the size of the referent whole varies then so does the name given to a given length.

  • Name the fraction for a given Cuisenaire rod with reference to one (whole).
  • Find the one (whole) when given a Cuisenaire rod and its fraction name.
  • Create a number line showing fractions related to a given one (whole).
  • Identify equivalent fractions
Resource logo
Level Three
Statistics
Units of Work
This unit provides a way of looking at multivariate data from a group of individuals. Data cards hold several pieces of information about individuals, and by sorting and organising a set of data cards, things can be found out about the group. This unit uses secondary data (data collected by others)...
  • Recognise what the variables are in a secondary data set.
  • Sort the given data into categories.
  • Answer investigative questions by sorting, organising and arranging data.
  • Make sensible statements about the data with supporting evidence.
Resource logo
Level Three
Number and Algebra
Units of Work
The unit investigates patterns made using matches and tiles. The relation between the number of the term of a pattern and the number of matches that that term has, is explored with a view to finding a general rule that can be expressed in several ways.
  • Predict the next term of a spatial pattern.
  • Find a rule to give the number of matches in a given term of the pattern.
  • Find the member of the pattern that has a given number of matches.
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Level Two
Level Three
Number and Algebra
Units of Work
This unit builds upon the students’ experiences of making, naming and recognising common fractions using different physical representations. Its purpose is to develop understanding of fractions of sets, and the formal language and symbols associated with simple fractions and their representations.
  • Read and write words and symbols for fractions.
  • Introduce the terms ‘unit fraction’ and ‘proper fraction’, numerator and denominator.
  • Make and understand different ways to represent 1 (whole).  
  • Use regional representations to find fractions of sets.
  • Solve problems that involve finding the...
Source URL: https://nzmaths.co.nz/user/75803/planning-space/early-level-3-plan-term-2

Areas of Rectangles

Purpose

In this unit students learn to use the multiplication formula to find the area of a rectangle. Using proportional reasoning students explore what happens to the area when the length and/or height of a rectangle is doubled.

Achievement Objectives
GM3-2: Find areas of rectangles and volumes of cuboids by applying multiplication.
Specific Learning Outcomes
  • Use multiplication to calculate the area of a rectangle.
  • Measure the length of a side using a ruler.
  • Use proportional reasoning to find the area of a rectangle.
Description of Mathematics

Area is the amount of flat surface enclosed within a shape. Commonly used standard units for area are cm2 (square centimetres), m2 (square metres), and km2 (square kilometres). Squares are used because they iterate, that is they fit together in two dimensions with no gaps or overlaps.

Rectangles are the easiest shapes to find the area of because the array structure of iterating units is most obvious. Consider this rectangle filled with square units of area:

The units are arranged in three rows of five squares. The total number of units can be found by multiplication, 3 x 5 = 15. Similarly, the rectangle contains five columns of three squares, so 5 x 3 = 15 also gives the total area.

    Opportunities for Adaptation and Differentiation

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

    • Manipulate the side lengths of the rectangles you use. Rectangles with smaller side lengths make drawing and counting solutions more accessible. In general, use rectangles with smaller side lengths when introducing the concept of arrays and how the arrangement of rows and columns connects to multiplication equations. However, increasing side lengths promotes the need for more efficient ways to find area. In that way, students see the efficiency of multiplicative methods.
    • Use diagrams and physical models to support student recognise arrays within the boundaries of rectangle. Gradually withdraw the diagrammatic and physical support to encourage imaging and thinking with established results. Refer to pages 11-13 of Teaching number through measurement, geometry, algebra and statistics for further ideas.
    • Allow access to calculators where calculation is not the primary purpose of the lesson. For example, finding all rectangles with areas of 72cm2 offers opportunities to apply multiplicative thinking and systematic reasoning. Those opportunities may be lost if students are preoccupied with finding products mentally.

    The context for this unit can be adapted to suit the interests and experiences of your students. Students could be challenged to find the area of a room in their own home, or the area of a space of interest in your local community, such as a sports ground, skate park, marae or similar. A diagram with measurements could be provided if the area is not readily accessible during school time.

    Required Resource Materials
    • 10cm by 10cm squares of paper or card, e.g. memo pads
    • Large pieces of paper, e.g. butchers’ paper
    • Tape measures, trundle wheels, metre rulers (whatever is available)
    • Newspaper, recycled cardboard, scissors and tape
    • 1cm square grid paper
    • Copymasters One and Two
    Activity

    Session 1

    In this session students are introduced to the idea of using multiplication to find the area of a rectangle.

    1. Show the students a large rectangular piece of paper measuring 30cm by 60cm and a pile of smaller squares each measuring 10cm by 10cm (like memo squares). Tell the students you want to know how many of these small squares are needed to cover the large paper rectangle. You can set a context such as this is the back garden and these are the concrete tiles we will be using to cover it. 
      How many square tiles will cover this area?
    2. Let students briefly discuss how they might estimate an answer, then share the ideas. Look for students to explain two main processes:
      • Iteration – repeated copying of the unit of measurement (memo square) along a side, with no gaps or overlaps.
      • Equi-partitioning – equally splitting a side until the divisions are about the same length as the sides of the memo square.
    3. Ask about how the units will be arranged. Introduce the terms, rows (across) and columns (down) if students are not familiar with those words. 
      Do students recognise the array (rows and columns) structure in the arrangement of units?
    4. Ask a volunteer to place the squares side by side on the rectangle. Blutac can help to secure the units in place.
    5. Ask the students for ways to work out the total number of units. One by one counting, or skip counting/repeated addition (6, 12, 18 or 6 + 6 + 6 = 18) are legitimate strategies given the small number of units. Explain that the area of the rectangle is 18 squares. (Actually, 10cm x 10cm squares are one dm2, one square decimetre)
      Can we count the squares even more efficiently?
    6. Record 3 x 6 = 18 and ask student where the six and three reside in the model (rows and columns). Ask where the 18 is found (total number of square units).
    7. Model the same process with different sized rectangles, e.g. 20cm x 80cm, 50cm x 40cm, 100cm x 100 cm (A square is a special rectangle). The rectangles might be cut out of paper, drawn on the whiteboard, or drawn on the carpet/concrete with chalk.
      Look for students to:
      • Recognise the array structure.
      • Use multiplication as an efficient method to calculate the area.
    8. Provide the students with copies of Copymaster One. Tell them to work with a partner to find out the area of each rectangle in small squares. As students work look for their calculation strategies. Are they using additive or multiplicative methods?
      Recognise that much will depend on their knowledge of multiplication facts and strategies.
    9. Gather the class and share solutions. It is interesting that Rectangle E, a square has the greatest area, though other rectangles may look larger.
      Answers: A (3 x 7 = 21), B (7 x 5 = 35), C (4 x 11 = 44), D (12 x 3 = 36), E (7 x 7 = 49), F (8 x 6 = 48), G (11 x 2 = 22).
      What do the answers tell us about these rectangles?
      How big are the little squares?
      Students might measure with a ruler to check that the units are square centimetres.
      Ask students to include the unit in their answers, e.g. 21cm2. Recording the notation for each rectangle is good practice.

    Sessions 2 and 3

    1. Discuss the idea of a formula. You might find a funny video online about someone using a formula to make something. A recipe is a type of formula.
      What do we mean by a formula?
      Do students explain that a formula is like an algorithm for getting the same result each time? 
      Record W x L = A. This is a mathematical formula written as an equation. 
      I wonder what the letters W, L and A might represent?
    2. Apply the formula to the examples students worked on in the previous lesson (Copymaster One). 
      For example, Rectangle B had seven rows of five squares.
      The row gives the length of the rectangle. In the case of B length equals 5. (rub off L in the formula and write 5 in it’s place)
      The number of rows gives the width of the rectangle. In the case of B width equals 7. (rub off W in the formula and write 7 in it’s place).
      The formula now reads 7 x 5 = A. I wonder what A equals. What value for area makes the equation true and matches the formula?
    3. Ask students to use the examples from Copymaster One and rehearse starting with the formula, and substituting the values of length, width, and area for each rectangle.
    4. Provide students with a group worthy task to work on collaboratively (see Copymaster Two). Students might be given 1cm grid paper or work in their exercise books.
    5. Look for students to apply the W x L = A formula to construct appropriate rectangles. For example, if they choose an area of 72cm2 they will need to consider all the factors of 72. Encourage students to find those factors systematically. Some students may need to support of a multiplication basic facts poster.
      A systematic approach involves starting with 1 as a factor then increasing the smallest factor by one and testing 72 for divisibility.
      1 x 72, 2 x 36 (72 ÷ 2 = 36), 3 x 24 (72 ÷ 3 = 24), 4 x 18 (72 ÷ 4 = 18), 5 x (72 is not divisible by 5), 6 x 12 (72 ÷ 6 = 12), 7 x (72 is not divisible by 7), 8 x 9 (72 ÷ 8 = 9).
      If the process continues the factors will appear in reverse order, e.g. 9 x 8 = 72. 8 x 9 and 9 x 8 are essentially the same rectangle though they may appear differently if the direction of the label is considered.
    6. Gather the class to discuss solutions and look at real sized diagrams of the possible labels. Some options are mathematically correct but unworkable as a label option.
      Discuss criteria for eliminating labels. For example, a label with width of less than 5cm might be considered too ‘skinny.’
      Discuss the best options, cut them out at real size, then use a real jam jar to consider how well each label will work.
    7. Students might write a letter to Karly outlining how they investigated her problem and giving their recommendations.
    8. Another good investigation is to tile a large rectangular area with 1m2 carpet tiles. A hall or gymnasium is an ideal area though a classroom is also viable. Tiles of that size are commonly found at hardware stores. You will find an advertisement easily online. 
    9. Get students to construct a unit square using newspaper or recycled boxes. They can use the unit to get a sense of the scale of 1m2 and make estimates of area of the space before they calculate.
    10. Ask students to work in small teams to calculate the number of tiles that will be needed for the rectangular space. Look for them to measure the side lengths of the rectangular area using tape measures, trundle wheels, or metre rulers.
      Do they apply the W x L = A formula?
    11. Students can find the area of composite shapes by finding the area of the rectangles. For example: 
      Composite shape
    12. This shape can be seen to be comprised of two 2cm by 4 cm rectangles, or a 2cm by 6cm rectangle and a 2 cm by 2 cm rectangle, or 4 cm by 6 cm rectangle with a 4 by 2 rectangle missing. There are different ways to solve composite shapes.

    Session 4

    In this session students explore using proportional reasoning to find areas of rectangles.

    1. Pose the problem: Sam’s family was shopping for a ground sheet to take camping. The first one they looked at measured 2 by 3m. Sam said if they wanted one with an area twice as big they should get the 4 by 6m size. Is Sam right?
    2. Ask the students to draw pictures of the ground sheets and to help them decide if Sam is correct.
    3. Work with students to establish that doubling the area only involves doubling one side of the rectangle. Doubling both sides of the rectangle increases the area by four times.
    4. Using this proportional reasoning students will be able to solve problems without recalculating from side lengths. Here are some example problems:
      • The recipe made enough icing to cover the top of a 20cm by 20cm cake, what size cake can you ice if you double the amount of icing?
      • The birthday card had a front cover measuring 15cm by 10cm, what is area of the piece of cardboard used to make it?
      • The marae had two areas that needed paving. Each area measured 5m by 8m. What is the total area to be paved?
      • The gardener charged his customers by the area of their lawn. If the bill was $20 to mow a lawn that was 6m by 20m, what should the bill be for a 20m by 12m lawn?

    Session 5

    In the session students demonstrate their ability to apply measurement of area independently.

    The following links provide pages from Figure It Out books that are suitable:

    Students might also create a mat design and provided the dimensions and areas of the rectangular pieces that compose it. An example is given below:

    Colourful  rectange

    Cuisenaire Rod Fractions: Level 3

    Purpose

    This unit introduces the fact that fractions come from equi-partitioning of one whole. So the size of a given length can only be determined with reference to one. When the size of the referent whole varies then so does the name given to a given length.

    Achievement Objectives
    NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
    NA3-5: Know fractions and percentages in everyday use.
    Specific Learning Outcomes
    • Name the fraction for a given Cuisenaire rod with reference to one (whole).
    • Find the one (whole) when given a Cuisenaire rod and its fraction name.
    • Create a number line showing fractions related to a given one (whole).
    • Identify equivalent fractions
    Description of Mathematics

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

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

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

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

    Specific Teaching Points

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

    Consider this relationship between the dark green and blue rods. Which rod is one? Either could be defined as one and the other rod could be assigned a fraction name.

    If the blue rod is one then the dark green rod is two thirds, as the light green rod is one third. If the dark green rod is one then the blue rod is three halves since the light green rod is now one half.

    Re-unitising and norming are not just applicable to defining a part to whole relationships like this. In this unit students also consider how to use re-unitising to find the referent one and to name equivalent fractions. For example, below the crimson rod is named as two fifths. Which rod is the one (whole)? If the crimson rod is two fifths, then the red rod is one fifth. Five fifths (red rods)form the whole. Therefore, the orange rod is one.

    What other names does two fifths have? If the red rods were split in half they would be the length of white rods, and be called tenths since ten of them would form one. The crimson rod is equal to four white rods which is a way to show that 2/5 = 4/10. If the red rods were split into three equal parts the new rods would be called fifteenths since 15 of them would form one. The crimson rod would be equal to six of these rods which is a way to show 2/5 = 6/15. The process of splitting the unit fraction, fifths in this case, into equal smaller unit fractions, produces an infinite number of fractions for the same quantity.

    Opportunities for Adaptation and Differentiation

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

    • providing Cuisenaire rods for students to manipulate when solving problems
    • modelling how to record fraction symbols and drawing attention to the meaning of numerator and denominator
    • drawing diagrams to clarify the unit of comparison and the one (whole) in problems
    • encouraging students to work collaboratively, especially where some students are affected by colour blindness.

    Tasks can be varied in many ways including:

    • alter the complexity of the rod relationships that students work with. Working with halves and quarters tends to be easier than with thirds and fifths
    • providing 1cm2 grid paper and coloured felt pens to ease the recording demands (Cuisenaire rods are based on that scale).

    The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Cuisenaire rods (rakau) are a media often used in introduction of te Reo so some students may already have encountered them. Knowing the relationships between rods of different colours, without having assigned number names to the rods, is very helpful in easing cognitive load. Other contexts involving fractions of lengths might also be engaging for your students. For example, the fraction of a race or journey that has been covered at different points is practically useful. Consuming foods that are linear, such as submarine sandwiches, bananas, or sausages, might motivate some learners. Board games that have a particular number of steps from start to finish provide opportunities to look at a fraction as an operator.

    Required Resource Materials
    Activity

    Prior Experience

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

    Session One

    1. Use Cuisenaire rods or the online tool to introduce the relative size of Cuisenaire rods in the following way.
      Relative to the orange rod, how long is the yellow rod? How do you know? Justify

      The relationship between the yellow and orange rods can be expressed in two ways:
      “The yellow rod is one half of the orange rod.”
      “The orange rod is two times the length of the yellow rod.”
      So if the orange rod was one then the yellow rod would represent one half. What fraction would the red rod and dark green rod represent? Justify. Convince us you are right.

       
    2. Encourage the students to express the relationships in various ways, such as:
      “The red rod is one fifth of the orange rod because five of it fit into the whole (one)”
      “The orange rod is five times longer than the red rod.”
      “So the dark green rod must be three fifths of the orange rod because three red rods make one dark green rod.”
      ​A more complex question is “How many dark green rods (three-fifths) fit into the orange rod (one)?” While the correct answer is five-thirds, or one and two thirds, students will be unlikely to name the relationship that precisely. Expect answers like “Almost two but not quite.”
       
    3. Introduce Investigation One using Slide 1 of the PowerPoint. Encourage students to record both their names for each rod (relative to the brown rod) and their reason for naming it that way. Provide sets of Cuisenaire rods or access to the online tool. Let the students work in small teams. Look for the following:
      • Do the students refer back to the brown rod as the one?
      • Do they name each rod with reference to how many times it fits into one?
      • Do they use the relationship between rods to name them? (For example, if pink is one half then red must be one quarter and white must be one eighth).
      • Can they name a rod larger than one as an improper fraction or mixed number? (For example, the orange rod now represents one and one quarter (1 1/4 or 5/4).
         
    4. All of the points above can be raised in discussion as a whole class. Extend the conversation to which rods were hardest to name and why that was so. For example, the light green rod does not fit into the brown rod an exact number of times but the white rod (one eighth) can be used as a reference.
       
    5. Also discuss equivalence. The diagram below shows 1/2 = 2/4 = 4/8. Note that equivalent fractions are different names for the same quantity.

    Session Two

    1. Revise the key points from the previous session using the blue rod as one.

       
    2. Ask questions like:
      If the blue rod is one what do we call the light green and white rods? Justify your answers.
      What statements can you make about the relative size of the rods?
      Are ther
      e equivalent fractions in the picture (1/3 = 3/9)? So what fraction is equivalent to… two thirds? (2/3 = 6/9), … to three thirds? (3/3 = 9/9).
      Students might notice some patterns in the symbols such as the same multiplier between numerators and denominators in the equalities.
       
    3. Reflect back on fractions where the rod was larger than one. Ask: If blue is one then what fraction is the orange rod?

      Thinking that fractions are restricted to less than one is a common constraint students learn so opportunities to name fractions greater than one is important. In this case students will recognise that the white rod fills the gap. Good questions are:
      • Remember, which rod is one?
      • So what fraction is the white rod? (1/9)
      • How many white rods fit into the blue rod? (nine)
      • How many white rods fit into the orange rod? (ten)
      • So what shall we call the orange rod? (1 1/9 or 10/9)
         
    4. Ask students to attempt Investigation Two of the PowerPoint (Slide 2). Remind them of the necessity for recording their solutions and justifications.
       
    5. As they investigate in small teams look for:
      • Do they accept the new imaginary rod as one?
      • Do they name the other rods as unit fractions in terms of how many of that rod fit into one?
      • Do they know how to name non-unit fractions using copies of unit fractions? E.g. Three quarters (blue rod) is three copies of one quarter (light green).
      • Do they realise that equivalent fractions are different names for the same quantity?
         
    6. Share the results as a class attending to the points above.
       
    7. Construct a fraction wall with the gold rod as one. Name each unit fraction (1/2 ,1/4 ,1/12 ,1/3 ,1/6). Ask if these are the only unit fractions that are possible and why that is so. Students may note that the denominators are all factors of 12. Look for equivalence in the fractions within the wall. Encourage students to find non-unit fraction equivalence as well, e.g. 2/3 = 8/12 and 3/4 = 9/12.

       
    8. Use the wall to create a number line as shown. Ask:
      How much more three quarters is than two thirds?
      How much less one half is than two thirds?

       
    9. Let students work on Investigation Three from the PowerPoint. Their work will extend into Session Three. Look for the following:
      • Can students name the fractions for the rods that are being joined?
      • Can they record the combinations as sums like, 1/2 + 1/3 + 1/6 = 1?
      • Can they use equivalence, particularly referring to twelfths, to explain why the combinations add to one?

    Session Three

    In this session the purpose is to reconstruct the one rod. Students connect from part to whole as opposed to whole to part.

    1. Begin by going over previous ideas in the context of this model.

       
    2. Ask: What are the size relationships between the yellow and black rods?
      The students might use the white rod as a reference to say, “The yellow rod is five sevenths of the black rod.” It is more difficult to recognise that “The black rod is seven fifths of the yellow rod.” The key idea is to establish the referent one. If a comparison ‘of a given rod’ is being made then that rod becomes the one.
       
    3. Ask: So if you were told that the yellow rod was five sevenths of the one rod, what colour would the one rod be? (black).
      If you were told that the black rod was seven fifths of the one rod, what colour would the one rod be? (yellow)
       
    4. Provide another scenario. If you were told that the pink rod was one half of the one rod, what colour would the one rod be?

      Students might easily recognise that two halves make one so the rod colour of one is brown. This is an easy scenario as a unit fraction is given. Therefore, ask a harder problem like this:
      If you were told that the dark green rod was two thirds of the one rod, what colour would the one rod be?
      The dark green rod does not fit exactly into the mystery one but half of it does. That half of the green rod is the light green rod (one third). So the one rod must be blue.

       
    5. Ask the students to complete Copymaster One in pairs or threes. Point out the need to justify their decisions about which rod is one in each case. Students need to use Cuisenaire rods or the online tool for this activity. They should not rely on the pictures being to scale. Look for:
      • Do they adjust to the variable one in each case?
      • Do they use the given rod as a unit of measure?
      • Do they subdivide the visible rod to find a unit fraction they can measure with? For example if told the rod is two thirds, do they divide the rod equally into two parts to create a one third measure?
         
    6. If students complete Copymaster One, ask them to create similar part to whole problems for other students. 

    Session Four

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

    • A point on the number line can have an infinite number of names called equivalent fractions, for example, 2/3 ,4/6 ,6/9 … all ‘live’ at the same point.
    • Between any two fractions are an infinite number of other fractions (this is known as ‘density’ of the number line).
    1. Begin by building up a number line for quarters in this way.
      If the brown rod is one (mark zero and one on the number line) where would one quarter be?

      Students may now know that the red rod is one quarter of the brown rod. Ask: What fractions could be marked on the number line using one quarter? Look for them to explain that quarters can be ‘iterated’ (place end on end) to form non-unit fractions. 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, for example, 2/4 = 1/2, 4/4 = 1.

       
    2. Look at the space between one quarter and one half. Ask, “Are there any fractions that belong in this space?” Students may recognise from previous work that white rods are one eighth of a brown rod. Three eighths will work. Note that three eighths measure exactly half way between one quarter and one half. Ask, “What fraction would belong half way between one quarter and three eighths?” (five sixteenths). The last questions requires students to use their imagination, as there is no rod that is half the length of a white rod.

       
    3. Show the students the diagram on Slide 4 of the PowerPoint. It shows zero and two fractions (orange rod as two thirds and blue rod as three fifths placed on a number line. Ask, “How could we find the length of the one rod?” From the part-whole task in the previous session students should reply that one third or one-fifth need to be located by equally partitioning the orange rod in two parts to get the yellow rod or equally partitioning the blue rod into three parts to get the light green rod. These unit fractions can then be iterated to get the referent one (three yellow rods or five light green rods).
    4. Ask students to create a number line with the orange rod as two thirds and the blue rod as three fifths. They must include one and any other fractions they can find. In the event of early finishers to this infinite task, provide the following challenge fractions to locate on the number line:  4/3, 9/5, 3/15, 14/15, 2/9. Look for the following:
      • Do the students use fifths and thirds as unit fractions to locate other non-unit fractions, like four fifths?
      • Do the students find fractions greater than one by iterating unit fractions?
      • Do the students record equivalent fractions in the same location, for example five fifths and three thirds at one?
      • Do the students subdivide unit fractions to form other units, for example divided thirds into two equal parts to form sixths?
      • Do the students attempt to identify fractions between fractions, for example, which fractions lie between two thirds and four fifths?
         
    5. Bring the class together after a suitable period of investigation to share results. You will need a large number line on the whiteboard. By the time the discussion is over you should saturate the number line with fractions. You may like to ask if it ever possible to complete this task. Students may already realise that there are an infinite number of fractions that could be located.
       
    6. After discussing the fractions that can be located on the above number line ask the students to make up a similar number line problem for someone else. The problem must include enough fractions already placed to locate the referent one and at least six other fractions to be located on the line. They are free to choose whatever rod they want as the one and may even create a rod that is not in the set.

    Data cards: Level 3

    Purpose

    This unit provides a way of looking at multivariate data from a group of individuals. Data cards hold several pieces of information about individuals, and by sorting and organising a set of data cards, things can be found out about the group. This unit uses secondary data (data collected by others) as well as primary data (data collected by the class).

    Achievement Objectives
    S3-1: Conduct investigations using the statistical enquiry cycle: gathering, sorting, and displaying multivariate category and whole-number data and simple time-series data to answer questions; identifying patterns and trends in context, within and
    Specific Learning Outcomes
    • Recognise what the variables are in a secondary data set.
    • Sort the given data into categories.
    • Answer investigative questions by sorting, organising and arranging data.
    • Make sensible statements about the data with supporting evidence.
    Description of Mathematics

    The key idea of statistical investigations at level 3 is telling the class story with supporting evidence. Students are building on the ideas from level two and their understanding of different aspects of the PPDAC (Problem, Plan, Data, Analysis, Conclusion) cycle – see  Planning a statistical investigation – level 3 for a full description of all the phases of the PPDAC cycle.  Key transitions at this level include posing summary investigative questions and collecting and displaying multivariate and time series data.

    Summary or time series investigative questions will be posed and explored.  Summary investigative questions need to be about the group of interest and have an aggregate focus.  For example, What position in the family are the students in our classWhat are the reaction times of students in our class?

    Data displays build on the frequency plots from level two and can be formalised into dot plots and bar graphs.  Students should have opportunities to work with multivariate data sets, data cards are a good way to do this. Data cards allow students to flexibly sort their data and to correct errors or make adjustments quickly.

    Students will be making summary statements, for example, the most common reaction score for our class is 13 cm, five people have a reaction score of 12 cm (read the data), or most students (16 students out of the 27 in our class) have a reaction score between 13 and 14 cm (read between the data). Teachers should be encouraging students to read beyond the data by asking questions such as: “If a new student joined our class, what reaction score do you think they would have?”

    Opportunities for Adaptation and Differentiation

    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:

    • setting up the plan for data collection for students to follow
    • 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
    • providing prompts for writing descriptive statements.

    The context for this unit can be adapted to suit the interests and experiences of your students. Preparing data cards with other information on them (sourced from Census At School New Zealand) that will be of interest to your students is one way to do this.

    Required Resource Materials
    Activity

    Session One

    Part One – Introducing Data Set One

    1. Organise the students into pairs, hand out to each pair a set of data cards, Data Set One Master, and get them to cut out all the data cards. They should collect them together and select one of the data cards each to have a look at.
    2. Tell the students that this is a data card. A data card is a piece of paper containing information or data about a person.  The data for these data cards has been collected by asking four survey questions. What do you think the survey questions were? Seek ideas from the class. They may just give you the variable description which is fine.
    3. Confirm that the information on these data cards comes from the following four survey questions:
      • Are you a boy or girl? Gives the top variable in the data card
      • Can you whistle? Gives the left variable in the data card
      • Are you the oldest, youngest or a middle child in your family? (Only children are classified as oldest) gives the bottom variable in the data card
      • Which hand do you write with to produce your neatest work? Gives the right variable in the data card.
    4. Get the students to put the data set to one side, we will come back and use it again soon.

    Part Two – Making Class Data Cards

    1. Show the data card below.
    2. Ask the class to tell you something about this student.
      • Does anyone in the class fit this data card?
      • Do you know someone that fits this data card not in this class?
      • How many people could this data card be correct for?
    3. Discuss the importance of knowing exactly what each piece of data is about, i.e. the importance of specific survey questions. Discuss how some students could answer the same survey question differently, e.g. "Are you right handed or left handed?" could give two different answers for the person who throws a ball with one hand and writes with the other. Survey questions need to be specific, with no ambiguous answers. Hence the survey question about handedness specifically asked about which hand your write with to produce your neatest work and not just are you left-handed or right-handed.
    4. Ask: What would a data card about you look like for these same four survey questions?
    5. Hand out a blank data card for each student to fill out. Once completed collect all data cards.
    6. After this session the teacher needs to photocopy all the data cards onto a piece of paper, one set for each pair of students in the class. Photocopying onto coloured paper is suggested to make it easy to recognise the class’s data set. This data set will be used during the next session.

    Part Three – Working with data set one

    1. Ask the students to get the data cards we cut out earlier back out. Have the students sort and organise the data cards to find out things about this group of students.
    2. Encourage the students to look for multi-dimensional interesting things. This means looking for interesting things within different categories rather than simply counting the number in categories. For example, rather than seeing if there are more girls than boys or more whistlers than non-whistlers, look to see if more boys than girls are left handed or if there is a link between place in family and the ability to whistle.
    3. Arranging the data cards like below, is one way to help see things in the data.
      arranging data squares
    4. The teacher is to move around getting the students to explain and show what they have found out. The teacher is to encourage students to add detail to their observations. This could include thinking proportionally. For example, rather than "One more girl is right handed than boys", "A larger proportion of girls are right handed, 8 out of 12 girls in comparison to 7 out of 12 boys are right handed." More able students are to be encouraged to think proportionally when the number in comparing groups is not the same, e.g. 8 out of 20 is a smaller proportion than 7 out of 9.
    5. The following questions could be asked to encourage thinking:
      • Are there proportionally more whistling right-handers or whistling left-handers?
      • Is there anything interesting when comparing place in the family and whistling?
      • All the boys in this group who are the youngest can whistle, does this mean every boy who is the youngest in their family can whistle?
    6. On a large piece of paper write up what the students discover or get each pair of students to write down what they found out about this group. Keep this information, as it can be used in the next session to compare with the class data set.

    Session Two

    During this session, students will be sorting and arranging data cards about themselves, i.e. the students’ own data cards.

    1. Before the class data set is handed out, remind the students about what they found out about Data Set One in Session One and how they organised the data card to see things.
    2. Briefly discuss what they expect to find out about their class:
      1. What do you expect to find out about the class?
      2. Will the things we found out from Data Set One, be different or similar to our class?
    3. Hand out a set of class data cards to each pair of students. The pairs are to cut out the data cards, sorting and arranging them to look for things of interest. The teacher is to move around getting students to explain and show what they have found out.
    4. Conclude the session by considering the statements the students made at the beginning of the session and sharing other things of interest.

    Session Three

    1. Hand out a set of data cards, Data Set Two, to each pair of students. The pairs are to cut out the data cards for use in class. Ask the students to select one or two of the data cards they have just cut up, telling them it is information from students in a class like ours, then ask them the following questions:
      • What do you think the letter and numbers mean?
      • Why are letters and numbers used instead of words?
      • What specific survey questions could give the answers: B, 6, 10 or 13? Seek ideas from the class.
    2. Explain that the four survey questions for these data cards are:
      • Are you a boy or girl? – B (top)
      • What year level are you at school? – 6 (left)
      • How many years old are you? – 10 (right)
      • What is your reaction score for catching a ruler? – 13 (bottom)
    3. The reaction score is the average length a ruler falls, before being caught, when it is dropped four times. To work out the reaction score, one student holds a ruler vertically above the test student’s first finger and thumb; the bottom of the ruler is in line with the top of the thumb. The ruler is released and the test student closes their finger and thumb as quickly as they can to catch the ruler. The number of centimetres the ruler falls through the finger and thumb is the score. This is repeated four times, with the scores averaged to give the reaction score. For example, if the ruler fell 12 cm first time, 15 cm second time, 11 cm third time and 14 cm fourth time, the average is 12 + 15 + 11 + 14 = 52, 52 ÷ 4 = 13, therefore, the reaction score is 13.
      reaction startreaction end
    4. Get the students to sort and arrange the data cards to look for things of interest. The teacher is to move around getting each pair of students to explain and show what they have found out.

    Data Set Three – Optional
    A third data set has been included for teachers wishing to repeat the activity in this session. The data for this set was obtained from 
    www.censusatschool.org.nz/.

    Data Set Three is a data set of 24 students. The data is: top – male/female, left – arm span in cm, right – height in cm, bottom – age in years.

    Session Four

    Today the students, in small groups, will design and compile their own data card set. Each small group of students will design three survey questions to ask the students in the class.

    1. Provide students with blank data cards and explain that each group will be collecting their own data based on four survey questions. The first survey question will be "Are you the oldest, middle or youngest child in your family?" and three new survey questions will be added.
    2. Discuss how to define position in the family? Oldest – no siblings older than you; Middle – have older and younger siblings; Youngest – you have older sibling(s), but no siblings younger than you. An only child would be the oldest; if there are two children then there is an oldest and a youngest, from three children onwards a middle child (or children) become possible.
    3. Discuss and brainstorm suitable survey questions, for example:
      • How many centimetres tall are you?
      • How many centimetres is your right hand?
        Specific instructions will be needed with survey questions like this, so it is clear where to start and finish measuring.
      • How fast can you run 100m?
      • What is your favourite...?
        A list of possible favourites to select from is best with survey questions like this.
      • What time did you go to bed last night?
        When organising the data from survey questions like this, categories may be needed, e.g. before 8 pm, 8 to 9 pm, 9 to 10 pm, and later than 10 pm.
    4. Students could also look at the Census At School questionnaires for ideas of survey questions they could ask. On the explore the data page the questionnaires are available on the right hand side.
    5. Before starting to collect data each small group of students needs to write three statements about what they expect to find out about the class.
    6. Each small group of students needs to collect information and make data cards from students in the class.
       
    7. Once the data cards are completed, students are to sort and arrange them to look for things of interest. The small groups of students are to prepare a brief report of the things they have found out.

    Session Five

    If a further session is required, the ideas from session 4 can be repeated, or students can make up their own set of data cards by selecting a small sample of students from Census At School. To do this they would need to go to the random sampler and agree to the terms.  Then select SPECIFIC variables and select four variables for their data cards. The first three sections are pretty good to choose from. Then enter sample size – 30 should be enough. Generate sample, then an option to download the sample comes up – select this. Save their sample and then open the spreadsheet and use the information to make their own data cards.

    Matchstick Patterns

    Purpose

    The unit investigates patterns made using matches and tiles. The relation between the number of the term of a pattern and the number of matches that that term has, is explored with a view to finding a general rule that can be expressed in several ways.

    Achievement Objectives
    NA3-8: Connect members of sequential patterns with their ordinal position and use tables, graphs, and diagrams to find relationships between successive elements of number and spatial patterns.
    Specific Learning Outcomes
    • Predict the next term of a spatial pattern.
    • Find a rule to give the number of matches in a given term of the pattern.
    • Find the member of the pattern that has a given number of matches.
    Description of Mathematics

    This unit builds the concept of a relation using growing patterns made with matches. A relation is a connection between the value of one variable (changeable quantity) and another. In the case of matchstick patterns, the first variable is the term, that is the step number of the figure, e.g. Term 5 is the fifth figure in the growing pattern. The second variable is the number of matches needed to create the figure.

    Relations can be represented in many ways. The purpose of representations is to enable prediction of further terms, and the corresponding value of the other variable, in a growing pattern. For example, representations might be used to find the number of matches needed to build the tenth term in the pattern. Important representations include:

    • Tables of values
    • Word rules for the nth term
    • Equations that symbolise word rules
    • Graphs on a number plane

    Further detail about the development of representations for growth patterns can be found on pages 34-38 of Book 9: Teaching Number through Measurement, Geometry, Algebra and Statistics.

    Links to Numeracy

    This unit provides an opportunity to focus on the strategies students use to solve number problems.The matchstick patterns are all based on linear relations. This means that the increase in number of matches needed for the ‘next’ term is a constant number added to the previous term.

    Encourage students to think about linear patterns by focusing on the different strategies that can be used to calculate successive numbers in the pattern. For example, the pattern for the triangle path made from 9 matches can be seen as in a variety of ways:
    3 + 2 + 2 + 2
    1 + 2 + 2 + 2 + 2
    3 + 3 X 2
    1 + 4 X 2

    Questions to develop strategic thinking:

    • What numbers could you use to describe the way the pattern is made and how it grows?
    • What do the numbers and operations tell you about the pattern?
    • In what order do we perform the calculations like 3 + 3 x 2? (Note order of operations)
    • Are the expressions the same in some way? For example, How is 3 + 2 + 2 + 2 the same as 3 + 3 x 2?
    • Which expressions are the most efficient ways to calculate the number of matches?

    Strategies for representation and prediction will support students to engage in the more traditional forms of algebra at higher levels.

    Opportunities for Adaptation and Differentiation

    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:

    • providing matchsticks so students can build the growth patterns
    • using colour to highlight repeating elements in diagrams of the growth patterns
    • easing the calculation demands by providing calculators
    • modelling creating tables and other ways for students to record their working and ease demands on their working memory.

    Tasks can be varied in many ways including:

    • reducing the ‘distance’ of the terms involved, particularly predicting the number of matches for terms that are easy to build and check
    • reducing the complexity of the patterns, e.g. increasing in twos, threes, and fives rather than sixes, twelves, etc
    • collaborative grouping so students can support others
    • reducing the demands for a product, e.g. oral presentation rather than a lot of calculations and words.

    The context for this unit can be adapted to suit the interests and cultural backgrounds of your students. Matches are a cheap and accessible resource but may not be of interest to your students. They might be more interested in other thin objects such as leaves or lines on tapa (kapa) cloth.  You might find growth patterns in friezes on buildings in the community.  Be aware of opportunities to learn that connect to the everyday experiences of your students.

    Required Resource Materials
    • Matches with the heads burnt, or toothpicks, iceblock sticks, nursery sticks, trimmed bamboo skewers, etc.
    • Dot paper as an alternative to using matches
    • PowerPoint One 
    Activity

    Note: All of the patterns used in this unit are available in PowerPoint 1 to allow easy sharing with data projector or similar.

    Session 1: Triangle Paths

    In this session we look at a simple pattern created by putting matches together to form a connected path of triangles.

    1. Introduce the session by telling the students that Kiri made the following matchstick paths using 1, 2, and 3 triangles – she called them a 1-triangle path, a 2-triangle path, and a 3-triangle path. Note that 1, 2, and 3 are the term numbers in Kiri’s pattern.
    2. Ask the students to use Kiri’s method to make a 4- and then a 5-triangle path.
      How many extra matches would be needed to make a 6-triangle path? A 7-triangle path?
      How many matches would Kiri need to make a 20-triangle path?
    3. Let students work out the number of matches needed for the 20th term. Use think, pair, share to allow students to compare their strategies.
    4. Kiri noticed that if she rearranged the matches, she could count them quite quickly. The following picture shows how she rearranged them.

      How does Kiri’s method work?.
      How would Kiri rearrange a 7-triangle path?
      What expression would she write to show her calculation? (1 + 7 x 2 or 1 + 2 + 2 + 2 + 2 + 2 + 2 + 2)
    5. Tell the class that Kiri says that using her method, she can see a short cut way of counting the number of matches needed to make a 10-triangle path. Get them to write down, using pictures support their explanation, what Kiri’s short cut method might be.
    6. Let’s call Kiri’s method, Kiri’s Rule. Ask:
      Using Kiri’s Rule, how many matches will be needed to make a 20-triangle path?
      Reverse the problem by asking: How big a path can Kiri make with 201 matches?
    7. Allow students time to develop an answer and compare their strategies.
      • Do students that relied on repeated addition change to multiplicative strategies with increased demand?
      • Are students able to recognise that the term number is required, not the number of matches?
      • Can students ‘undo’ their previous rules to find the term number?
    8. Kiri’s friend Jamie arranged his matches differently. His pictures looked like this:

      What is Jamie’s Rule?
      What is Jamie’s picture for a 12-triangle path?
      What expression could Jamie write for the 12-triangle path (Term 12)
      How are 3 + 2 + 2 + …+ 2 and 3 + 11 x 2 the same?
    9. Jamie says that using his method, he can see another short cut way of counting the number of matches needed to make a 10-triangle path. Get the class to write down, using pictures to support their explanations, what Jamie’s strategy is.
      How many matches will be needed to make a 20-triangle path?
      How big a path can Jamie make with 201 matches?
    10. Get the students to explain how Kiri’s Rule is different and the same compared to Jamie’s Rule.
    11. Ask the class: How would Kiri and Jamie explain to someone else how they could find the number of matches needed to make a path consisting of any number, say 1000, of triangles?

    Extension idea:

    Vey-un  has another way to work out the number of matches for a 10-triangle pattern. He writes 10 x 3 – 9 and gets the same number of matches as Kiri and Jamie, 21.

    Ask students to explain how Vey-un’s strategy works. What do the numbers in his calculation refer to?

    [Vey-un imagines ten complete triangles that require 10 x 3 = 30 matches to build. He imagines that the ten triangles join and that creates nine overlaps. He subtracts nine from 30 to allow for the overlapping matches.]

    Session 2: Square Paths

    Here we look at a simple pattern created by putting matches together to form a connected path of squares.

    1. Following the same general procedure as above, allow the students to explore ways of counting the number of matches that are needed to make square paths. Present the students with the following picture.

    1. Have your students construct a 4-square and 5-square path with matches or by drawing. Focus on how many extra matches were added each time. Where are the additional matches located?
    2. Ask your students how they could develop a quick and easy way of finding the number of matches needed to make a 20-square path.
      What would Kiri, Jamie and Vey-un do for this square pattern?
    3. Let the students work in groups of two or three. Ask the groups to make a picture showing how the 20-square path is made. They can experiment with the matches, and record their pictures. Do you need to draw every square?
      Is there only one possible way to look at the pattern?
      What might some of the others ways look like?
    4. Some pictures will be very helpful in counting the number of matches needed to make a 20-square path – some will not. Have the students choose the picture that they think best explains how successive square paths are made up AND gives a quick and easy method for counting the matches needed for an 20 -square path. Note the cumbersome nature of repeatedly drawing squares and repeatedly adding three matches.
      What is a more efficient way to draw or calculate the total number of matches?
    5. Have the students use their ‘best method’ to verify that there are 61 matches are needed to make a 20-square path.
    6. Compare the way the rules might be written:
      Kiri [1 + 20 x 3]                      Jamie [4 + 19 x 3                    Vey-un [20 x 4 – 19]
    7. Students can use these methods or their own ways to predict the number of matches needed to make 14-, 36- and 100-square paths.
    8. Ask them to write down how they would use their method to count the number of matches needed to make a square path consisting of any number of squares, say 1000 squares. Depending on the comfort of students with their rules you might use algebraic notation to represent the word rules:
      Kiri [1 + 3n]                                  Jamie [4 + 3 (n-1]                   Vey-un [4n – (n-1)]
    9. Reverse the problems so students must work out the term number for a given number of matches.
      How many squares are in a square path with 31, 304 and 457 matches?
      How many matches will be left over if you make the biggest square path that you can with 38, 100 and 1000 matches?
    10. Are students able to ‘undo’ their rules of find missing terms?
      Kiri calculates “One plus three times the term number” to find the number of matches.
      If Kiri knows the number of matches, how should she undo her rule to find the term number? [Note that order of undoing is important, subtract one then divide by three.]

    Session 3: House Paths

    The ideas learnt in the last two sessions are reinforced here using ‘house paths’.

    1. Use the techniques developed in the last two sessions to explore the following problem:
      A new matchstick path is being designed. It is called a house path. The first three terms are shown below. Develop a counting rule, that is, a short-cut way of counting the number of matches needed to make a 1000-house path.
    2. Have the students illustrate how they developed their counting rule. They could do this, by using pictures, words or numbers (or some combination of these).
      Do you need to draw every house?
      Do you need to add on 999 times?
      What do you think Kiri, Jamie and Vey-un might do with this pattern?
    3. Get the class to discuss the various approaches that were used and methods that were obtained.
    4. Allow time for the class to write up its conclusions about the most efficient strategies.
    1. Latitia has 503 matches. How many houses are in her path if she uses all the matches? Will she have any matches left over?

    Session 4: What’s My Path?

    The ideas of the first three sessions are extended and reinforced in another context. This time the problem gives a rule and the students find the pattern.

    1. Give students the following problem:
      My friend made a picture that showed how her fifth matchstick path was made. She named it:
      5 lots of 4 and add 2 (this was the counting rule used to make the path)
      She sent it to me via email. However, I was only able to read the name of the path and not see the picture!
      Make some possible pictures that she could have sent.
    2. It is worth noting that there are many answers to this. So even if two groups get a different answer, they may still both be correct.
      We have many different pictures that match the word rule. How are they different and how are they the same?
      [The common property is that the pattern starts with two matches and build on using four matches for each additional shape]
    3. Examples might be: 

    4. Allow the class time both to report back and discuss their solutions, and to write up what they have discovered.
    5. Ask: If my friend wanted to build a n-path how many matched would she need? N means any number you give her, say 1000, 53 or 214.
    6. Another friend sent this n-rule. Can you draw a pattern that matches his rule?
      “n minus one then multiplied by five then add six”
      What might the pattern look like?
      One possible answer is: 

    Session 5: Other Ways of Seeing Things

    In this session, the concept of a relation is explored with a more complicated spatial pattern.

    1. Show the class the pattern below that is made up of matches. The 1st, 2nd, and 3rd terms of the sequence are shown.
    2. Challenge the students with this problem:
      Find many different ways to work out the total number of matches in Term 10.
    3. Remind students about the ways that Kiri, Jamie and Vey-un represented their patterns, including rules that work for any term.
    4. Let students work in pairs or threes. Ensure they record their thinking using diagrams and expressions. Do your students:
      • Look for the growth between terms, i.e. 12 matches.
      • Create tables of values to represent the number of matches for each term
      • Use multiplicative strategies to predict the number of matches for term 10
    5. Gather the class to process the ideas. Highlight the efficiency of multiplicative strategies such as 10 x 12 – 8 and 4 + 9 x 12 compared to additive strategies like 4 = 12 + 12 + …
    6. Ask students to connect the numbers and operations in their expressions to the figural pattern of matches.
      Why is the number of matches increasing by 12 each term?
      How many groups of 12 matches will be in the 10th term?
      Why does Kiri subtract 4 at the end?
    7. How could our rules be used to predict the number of matches needed for Term 23? Term 101? Term n?
    8. If Taylor uses 604 matches to build a figure in this pattern, what Term does she make?
    9. To assess the ability of students to personally make predictions and create general rules pose this assessment task. 
      Here is a pattern of growing stars made with matches.

      How many matches are needed to make Term 15, that has 15 stars?
      Can you write a rule for the number of matches needed to make Term n, any term?
      If you have 244 matches, what is biggest number of stars you can make in this pattern?
    Attachments

    Symbols and sets

    Purpose

    This unit builds upon the students’ experiences of making, naming and recognising common fractions using different physical representations. Its purpose is to develop understanding of fractions of sets, and the formal language and symbols associated with simple fractions and their representations.

    Achievement Objectives
    NA2-1: Use simple additive strategies with whole numbers and fractions.
    NA2-5: Know simple fractions in everyday use.
    NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
    Specific Learning Outcomes
    • Read and write words and symbols for fractions.
    • Introduce the terms ‘unit fraction’ and ‘proper fraction’, numerator and denominator.
    • Make and understand different ways to represent 1 (whole).  
    • Use regional representations to find fractions of sets.
    • Solve problems that involve finding the whole from a part.
    • Find fractions of sets showing solutions in multiple ways including connecting fractions of sets with division.
    Description of Mathematics

    Fractions arise from the need to divide. That division make involve equal sharing of measuring. Many equal sharing situations can be solved without needing fractions. For example, 1/3 of 15 or 15 ÷ 3 can be accomplished by putting five objects in each of the three shares. However, other equal divisions of sets and objects require partitioning ones, e.g. 1/3 of 16 or 1/3 of a pie. Measurements in which the units do not fit into a space a whole number of times demand the use of fractions of that unit. For example, if a length of 13 cubes is measured with a unit of 4 cubes, 13 ÷ 4 = 3 ¼ units fit.

    In this unit students learn about fractions as numbers and as operators. Fractions are symbols in two parts, the numerator and denominator. In the fraction 3/4 , three is the numerator and 4 is the denominator. The numerator, 3, is the number of parts being counted, and the denominator, 4, gives the size of those parts. Quarters are of a size that four of them make one (whole). When fractions operate on other quantities the meaning for numerator and denominator is consistent. For example, finding 3/4 of 20 involves finding 1/4 of 20 first,by equally dividing 20 objects into four equal parts. Three of those parts are counted, so 3/4 of 20 = 15. Note that the symbol for ‘of’ is x so the operation might be written as 3/4 x 20 = 15.

    Links to the Number Framework

    Stages 5- 6

    This unit complements the learning activities in Book 7 Teaching Fractions, Decimals and Percentages.

    Opportunities for Adaptation and Differentiation

    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:

    • providing equipment such as shapes/regions, and sets of objects, so students can physically enact the operations
    • helping students to record their working with diagrams, and equations, to ease demands on working memory.

    Tasks can be varied in many ways including:

    • easing the calculation demands by choosing fractions and sets that are manageable
    • using collaborative grouping so students can support others
    • reducing the demands for a product, e.g. less calculations and words.

    The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Capitalise on the interests of your students. Food is appealing to most students but it is important the it is used as a ‘story shell’ not as a piece of equipment. Selection of equal teams for sports, and other activities, is a useful context for fractions of sets. Pastimes that currently engage you students, such as collectables, favourite toys, and earning money from jobs, will offer opportunities to engage them. Art and design often provide situations where shapes need to be equally partitioned

    Required Resource Materials
    • Play dough
    • Plastic knives
    • Paper shapes
    • Scissors
    • Sets of Fraction Circles or similar materials (Material Master 4-19)
    • Dice
    • Plastic beans or other sharing objects like counters or cubes
    • Copymasters One, Two, Three, Four, Five
    Activity

    Session 1

    The purpose of this session is to learn how to make and represent equal parts and to read and write words and symbols for fractions. The formal language of ‘unit fraction’ and ‘proper fraction’ is also introduced.

    Activity 1

    1. Begin by placing a length of play dough and a plastic knife in a place where all students in the group (class) can see. Distribute word cards (only) from Copymaster 1 to each pair of students.
    2. In their pairs ask students to discuss how they would make the equal part on their word card, if they were to use the play dough. Chose several students to explain their strategy to the group. Look for two main strategies:
      • Equal partitioning by halving (symmetry), e.g. Halving halves to get quarters
      • Iterating, estimating the part size and mapping it end on end to see it it works
    3. Listen for and highlight a description of the number of cuts that will be made and the language of equal parts, e.g. Three cuts creates four equal parts called quarters.
    4. Make sure each play dough length is the same so there is a uniform whole. Use the fraction cards to compare two fractions.  Discuss which fraction is largest. You might cut play dough lengths to confirm predictions.
      Is there a way to look at units fractions like these and know which fraction is larger?
      Do students recognise the a smaller denominator mean the whole is cut into less parts so the parts are larger?
    5. Use the play dough fractions to make non-unit fractions as well, such as three quarters and five eighths. Non-unit fractions have a whole number more than one as the numerator.

    Activity 2

    1. Randomly distribute symbol cards (Copymaster 1) to the student pairs. You will need to split the class into two groups of pairs and provide each group with a full set of cards. Have students discuss the symbol and, if possible, agree about how to read it.
    2. Pairs display, and read, their word cards aloud to the group, one at a time. As they do so, the pair that has the matching symbol card offers it to the holders of the word card. The donor pair must explain why they are giving the symbol to the word holder pair. Praise those who give clear explanations of the symbolic representation of the fractional part.
      Highlight the fact that each of the symbols is known as a unit fraction because it has 1 as the top number and it tells that there is just 1 of the equal parts being referred to[ Highlight the fact that one quarter and one fourth are different names for the same part. ]

    Activity 3

    1. Make a range of shapes, made from coloured paper, available to the students. Copymaster 2 could be used if you prefer. 
    2. Explain that they are to select a paper shape each and fold it to make a fractional part matching their word and symbol, then write the word and symbol on each of the equal parts. Encourage them to consider which shape would be the best to choose, given their fraction. For example thirds of equilateral triangles are easy to fold. Fifths of pentagons, eighths of octagons, etc. are easy to find after the centre of the polygon is located. Seventh and ninth parts are challenging. Choosing a paper strip might be wise and a strip in halves can be used to simplify the challenge. 
    3. After students select, fold, and label their fractions of a shape, ask them to consider what they would do if the shape was different.
      Can a triangle be folded into quarter when each part must be equal? 
    4. Encourage those who finish quickly (for example those with 1/2 or 1/4) to complete the task with a different shape, or, try a more difficult unit fraction.
    5. Let your students pair share their results, then talk as a class/group about why some fractions were easier to fold than others.Discuss how students approached the ‘trickier’ fractions. Highlight halving as a useful way to simplify the challenge. 
    6. Ask your students to cut the shapes into their fraction parts. Put the pieces into an empty box and give the pieces a solid shake. Use monitors to deliver a collection of pieces to each pair.
      Your job now is to reconstruct what the whole looked like for each piece you have.
      What information do you have to help you?
      Let students work out the appearance of the original whole for each fraction piece they have. Be aware that there are multiple possible answers pending on how the pieces are arranged. It is an excellent challenge to create as many different wholes as possible for a given fraction piece.

    Activity 4

    1. Ask all your students fold a paper strip into tenths, and write 1/10 on each of the equal pieces. Cut the strips into tenths.
      Why are these pieces called tenths? (ten equal parts make one whole)
      Where have you encountered tenths? (Students may connect to money and decimals in general)
    2. Count in tenths with students laying the pieces down end on end as they count, ”One tenth, two tenths, … ten tenths.”
      What comes next?
      How might we write eleven tenths? (11/10 or 1 1/10)
    3. Write ‘unit fraction’ on the class/group chart/. Repeat and write the explanation that any fraction with a top number of 1 is called a unit fraction because it is a single piece. Add the words ‘proper fraction’ to the group chart, explaining and record that a fraction in which the top number is smaller than the bottom number is called a ‘proper fraction’. Record “Improper Fraction”.
      If 11/10 is an improper fraction I wonder what that means? (numerator is greater than denominator)
    4. Write 3/10 on the class chart. Ask students to take 3 of their tenth pieces and write the words and symbol for these three parts. Discuss and model the fraction symbol as appropriate, highlighting the fact that the bottom number tells us how many equal parts make one (ten), and the top number (3) tells us how many of those equal parts we have chosen.
      Also point out that often we see the flat line separating these numbers (this line is called the vinculum) shown with a horizontal or sloping line like this, 1/2 , 1/4. 
    5. Write other non-unit fractions with tenths on the chart, having students take that many tenths and recording the appropriate words and proper fraction.
      What parts would we get if we folded tenths in half?...in quarters? …in tenths? The last fold produces hundredths.

    Activity 5

    Conclude this session by writing the words and symbols for common unit fractions and some other proper fractions. Brainstorm on the class chart/book what has been learned about fraction symbols.

    Session 2

    The purpose of this session is to introduce the language of numerator and denominator and practise using and interpreting fraction symbols. Students work from whole to part and part to whole. An understanding of different ways of showing one whole is also developed.

    Activity 1

    1. Distribute the cards from Copymaster 1 and have each student read out their card. As they listen they should identify the person with a matching card. This person becomes their partner for this session. 
    2. Introduce a Fraction Dictionary.
      Write 'numerator' in the dictionary. Explain that it is the top number in a fraction. Have several students come up and write their favourite fraction and circle in a different colour the top number.
      Ask student pairs to discuss what the job of the top number is and to suggest a definition of 'numerator'. Numerator means the counter so that number represents the number of parts that are chosen
      Write 'denominator'. Read the word together, and discuss the meaning. It is important that students know that denominator represents the size of parts, how many of those parts make one.
      Have several students again write their favourite fractions, this time writing over the denominator number in a different colour (not the same as that used for the numerator).

    Activity 2

    1. Make sets of fractions circles available to the students. (see Material Master 4-19)
      Pose and write on a chart the question: 
      Can the numerator in a fraction be the same as the denominator?
    2. Use the fraction circle pieces.
      Who can make fractions where the numerator and denominator are equal?
    3. Let students make fractions like 2/2, 3/3, 4/4, etc. Do students notice that the fractions are equivalent to one (whole)?
      Explain why any fraction where the numerator and denominator are equal is another name for one.
      Can the numerator be greater than the denominator?
    4. Ask students to make improper fractions, such as 3/2, 5/3, 7/4, etc…
      What is true of all fractions where the numerator is greater than the denominator? (All improper fractions are greater than one (whole)).

    Activity 3

    Have students play Roll for 3 in pairs.
    (Purpose: To make one from equal parts and recognise the equivalent fraction notation) Students each need a fraction circles page (see Material Master 4-19). The game is played in pairs or threes.

    1. The players take turns to roll a standard dice and colour in that many parts of one of the circles they have selected. For only one roll they can nominate the number they get on the dice (1-6). It is best to keep that option until the last roll as they must complete each circle exactly.
    2. The winner is the player who is first to complete their circles and has correctly recorded each one (whole) as a fraction, e.g.3/3, 8/8, once each circle is complete.
      The important rule is that they can colour fewer parts and keep building to make one whole, but they must, at some point, roll the exact number needed to complete a whole.
      For example, Player One rolls 6. She colours 6/8 of her circle divided into eighths.  On her next turn she rolls 3. She cannot use this to complete her eighths circle because 6/8 + 3/8 is more than 8/8 (1 complete circle). She must roll 2 or two 1s in different rolls to compete 1 exactly. She can however work on her tenths circle and take 3/10 and add this to her 1/10.
    3. Emphasise that the students should record their fraction additions.

    Activity 4

    Introduce the Fraction Snap game. (Copymaster 3).

    1. Hold up selected pairs of cards from Fraction Snap asking the students to decide for themselves if the two cards match. If so they slap the ground as if playing Snap. Discuss examples of correct ‘Snaps’ so the game is well understood.
      For example:
          is a pair, and so is    
    2. Play Fraction Snap cards with a full set of cards (Copymaster 3).
    3. Each player takes a turn to turn up a card from their pile and place it face up in the centre of the group. As a student adds their card to the pile of face up cards, all players watch closely and are ready to say ‘Snap’, quickly putting their hand on the pile if the played card matches the one that was top of the face up pile. The first player to say ‘Snap” collects the pile and adds it face down to the bottom of their existing pile.
    4. The game continues until one person has all the cards or until players decide to stop.

    Session 3

    The purpose of this session is to use materials to develop an understanding of fractions of sets. Equal sharing of sets is linked to regions models of fractions. Children make connections with equal sharing experiences in their own lives.

    Connections between repeated addition and multiplication are made as part-to-whole fraction problems are explored.

    Activity 1

    1. Make recording material available to the students.
      Distribute a shape from Copymaster 4 to each student.
      This activity requires students to work from part to whole rather than the usual whole to part requirement.
    2. Explain that each shape is a fraction or part of a whole shape. Ask students to show with drawings what the each whole shape might look like. Model an example and show they can cut out each shape then draw around it. (Alternatively attribute blocks or foam geometric shapes can be used. The students will need to be told what fraction of the whole they are working with.)
    3. Give the students the opportunity to explore the problems before prompting with questions like, “If that is 1/4 of the shape, how many of pieces like that will be in the whole shape?” or, “Do you think that there might be another way to show the whole shape?”
    4. Have students complete their drawings, writing the unit fraction in each part and an equivalent fraction for 1 beside their drawing (eg. 4/4, 2/2). Have them buddy share their results. Challenge the students to see how many ‘1’ shapes they can make for any single fraction piece.
      For example if this is 1/4:

      the whole shape might look like any of these:
                
      Have students find multiples solutions with the other shapes, writing the fraction in each part and the one whole fraction (4/4, 8/8 etc.) beside the whole shapes.

    Activity 2

    1. Make plastic beans available to the students.
      Have the students each take up to 4 beans and place them on their coloured fraction piece.

       
    2. Explain that these beans, like their shape, are just a fraction of a set of beans. It’s the same fraction as the fractions shape they have (1/4, 1/3, 1/2 etc.). Pose the question:
      If this is a fraction of the set, how many beans are in the whole set?
       
    3. Give the students the opportunity to explore the problem before prompting with questions like, “How quarters are in a whole set?” “How can you use your shape pictures to help you work out how many beans would be in the whole set?” If the students exploration is unsuccessful, stop the class/group and model an example, by putting the same number of beans on each of the fraction parts in the drawing of the whole shape and skip counting (or if appropriate multiplying) to reach a total.
       
    4. Model and record several examples on the class/group chart.
      For example : 1/4 of a whole set is 3 beans, 4/4 make 1 whole, so 4 lots of 3 beans will make 1 whole set.

      “3, 6, 9, 12” or “4 x 3 = 12”
    5. Discuss efficiency, that is which method of calculation requires the least amount of work.Have students explore and record at least 3 more examples using different fractions and shapes and different small amounts of beans.

    Activity 3

    1. Conclude this session by asking students to give examples of when someone has shared with them and they had received an equal part of a whole set of something.
    2. In the class book record some of the students’ story examples: For example:
      Mia’s friend Amy gave her 1/2 of her jellybeans. Mia had 5. How many did Amy have altogether before she shared?
      Tony received 6 pretzel sticks from Tama who told him he’d given him 1/3. How many did Tama have to start with?
      Encourage the students to picture these fractional amounts and what the whole amount might look like. Ask students to describe what they pictured in their minds.

    Session 4

    The purpose of this session is to use materials to reinforce the whole to part relationship and to continue to use fractions of regions to build an understanding of fractions of sets. The key connection is made with the operation of division. One view is to see how many equal sets can be made from the starting set, and working out the number of items in each equal set. This is called partitive division.

    Activity 1

    1. Begin this session by reviewing Session 2, posing some fractional part to whole contextual problems.
      For example: You were given 6 cherries. This was one third of the total in the bag. How many were in the bag to start with?
    2. Encourage students to image the problem and solution, but if appropriate, have a student model with an appropriate drawing.
    3. Repeat with several examples, highlighting repeated addition and multiplication as strategies for reaching a solution. Discuss the efficiency of multiplicative strategies that are expected at Level 3.

    Activity 2

    1. Have the students working in pairs with recording materials available.
    2. Pose the problem: Here is a container of strawberries. There are 12 berries in the container. You share the 12 berries equally between the two of you. How many strawberries will you each get? What is 1/2 of 12?
      Show and write how you work out your share, using pictures words and symbols.
    3. Pose several more examples with different numbers in the container: What is half of: 14, 20, 21, 25?
    4. Give the students time to draw, record and write about their sharing. Explain that these will be shared with other students and displayed.
      Change the number of shares and starting number of strawberries to add more challenge. For example:
      1/3 of 24          1/5 of 30          1/8 of 64          1/6 of 42          1/7 of 42

    Activity 3

    1. Have students share their work and comment on any examples where they use fractions of regions (a shape divided into halves) to support their calculations. Imaging of sharing, or simply calculating with numbers are signs of increasing sophistication.
    2. On the class chart write:
      Half of twelve
      12 shared between 2
      12 ÷ 2
      12/2
    3. Take time to discuss the important connections between these ways of recording. Highlight:
      • half of twelve is a two-way equal share of 12
      • when 12 is shared with two people we can write 12 ÷ 2 = 6
      • ÷ is the sign for division. It looks a bit like a fraction itself 
      • 12/2 also means twelve shared between two. It also means 12 halves which makes a total of 6 ones.
    4. In sharing both 21 and 25 between them, the student pairs will have had to share 1 of the strawberries and will have written 1/2. Highlight that we read this as ‘one half’ and that this symbol is also showing 1 ÷ 2, or 1 shared between 2.
      The symbol is an expression of both the problem itself and the quotient (resulting share).
      These key ideas about mathematical notation should be regularly reviewed.
    5. Transfer these ways of recording to the more complex example, such as 1/3 of 21 or 1/8 of 24

    Activity 4

    1. Remind students how they used shapes (regions) in Session 3 to help solve problems. They may find the shapes  useful now.
    2. Now pose equal sharing word problems such as:
      12 strawberries shared between 3 people, 1/3 of 12 (or 12/3)
      16 shared between 4, or 1/4 of 16 (or 16/4)
      20 ÷ 5, or 1/5 of 20 (or 20/5)
      21 ÷ 4, or 1/4 of 21 (or 21/4)
    3. Have students use pictures, words and symbols to record their solutions to the problems. Have students pair share their work.

    Activity 5

    Pairs of students can be challenged to write their own fraction problems for their partner to solve.

    Session 5

    The purpose of this session is to develop a conceptual understanding of finding a non-unit fraction of a set. The language introduced in this unit is consolidated.

    Activity 1

    1. Begin this session by highlighting strengths of some of the student work from Session 4, noticing the way they have drawn their diagrams and recorded their ideas using words and symbols.
      Make coloured beans,or other objects to be shared, and paper available to pairs of students.
    2. On the class chart write and pose this problem:
      There are 25 beans in a packet. You plant 2/5 of the beans. How many beans do you plant?
    3. Ask students to solve this problem in pairs, using equipment as required.Require them to record what their strategies and share their methods and answers (solutions) with another pair.
      Do students recognise that finding one fifth of 25 is needed first?
      Do they build two ‘iterations’ (copies) to make two-fifths
      Are their strategies only equipment based or do they use numbers and operations to anticipate the result?
    4. Record on a chart students’ methods for solving the problem.
      For example: First you have to find one fifth so you divide 25 by five. You’re asked for two fifths so you have add two fifths together or times one fifth by two. You write this 25 ÷ 5 = 5, 5 + 5 = 10 or 2 x 5 = 10
      Notice who uses equal sharing (into regions) and who uses their knowledge of multiplication (and division) to solve the problem.
    5. Ensure that equipment is available and pose further problems, having students show and record their solutions in their preferred ways.
      For example:
      There are 16 beans in a packet. You plant 3/8 of the packet. How many beans is that?
      There are 18 beans in a packet. You plant 5/9 of the packet. How many beans is that?
      There are 18 beans in a packet. You plant 5/6 of the packet. How many beans is that?
    6. Summarise finding a non-unit fraction of a set to support students to generalise the operation. Encourage explanations like “When you are finding more than a unit fraction of a set, you divide the number in the set by the denominator of the fraction. This gives you the unit fraction of the set. Then you multiply by the numerator of the fraction because this tells you how many of these equal parts are needed.”
      For example: To find 3/8 of 16:
      Find 1/8 first by solving 16 ÷ 8 = 2.
      Find 3/8 by solving 3 x 2 = 6.
      Give the answer as 6 is 3/8 x 16 = 6.

    Activity 2

    Have students play Telling the Truth (Copymaster 5) in pairs.
    (Purpose: to identify the correct fractions of sets)

    1. The aim of the game is to be the player to collect the most pairs of questions with correct answers. Five cards are dealt to each player who must firstly decide which of the cards in their hand do not tell the truth. They discard these cards, turning them upside down and placing them to one side (they may need to be checked later in the game). They then find any matching pairs in their hand and place these face up in front of them.
    2. The players then take turns to ask for an answer card to any of the question cards in their hand, or to ask for a question card that matches an answer card in their hand.
    3. Upon Player One’s request for a card, if the Player Two gives an untrue card, Player Two must miss a turn. Player One may immediately make another request.
    4. If Player Two has no suitable cards he tells Player One to pick up from the pile. Each time a player picks up or receives a card they must check it for accuracy.
    5. The player with the most correct matching pairs when all the cards are used, is the winner.

    Activity 3

    Conclude this session with a discussion of the game and summary of learning.  

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