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 idea that fractions come from equi-partitioning of one whole. Therefore, the size of a given length can be determined with reference to one whole. When the size of the referent whole varies, then so does the name of 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 (i.e. data that includes many variable, such as gender, age, height, eye colour, bedtime, etc.) that comes from a group of individuals. Data cards hold several pieces of information about individuals, and by sorting and organising a set of...
  • 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.

These Learning Outcomes are covered in every lesson of the unit.

Resource logo
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 whole from a...
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 to introduce this context, because they are an example of a two-dimensional shape that iterates. This means the shape can be repeated over and over again, without any gaps or overlaps.

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

A 15-square rectangle arranged in 3 rows and 5 columns.

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. This is an example of the commutative property - you can multiply numbers (e.g. 3 and 5) in any order and get the same result (15).

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. Consider using these strategies to support students:

  • Manipulate the side lengths of the rectangles you use. Consider the times tables that your students are familiar with and use these as a base of knowledge to build problems from. For example, constructing diagrams of rectangles that all have 3 rows could be an effective way to reinforce your students' knowledge of their 3 times tables, whilst teaching them about the concept of area.
  • Use rectangles with smaller side lengths when introducing the concept of arrays and how the arrangement of rows and columns connects to multiplication equations. In general, rectangles with smaller side lengths make drawing and counting solutions more accessible. However, increasing side lengths promotes the need for more efficient ways to find the area. In that way, students see the efficiency of multiplicative methods.
  • Use diagrams and physical models (e.g. square tiles) to support students in recognising arrays within the boundaries of rectangles. Gradually decrease the use of diagrams and physical supports to encourage imaging and thinking whilst drawing on previously 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 the mental calculations of multiplication.

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

Te reo Māori vocabulary terms such as mehua (measure), mitarau (centimetre), and tapawhā rite (square) could be introduced in this unit and used throughout other mathematical learning.

Required Resource Materials
  • 10cm by 10cm squares of paper or card, e.g. memo pads
  • Large pieces of paper, e.g. butchers’ paper
  • A variety of measuring devices, such as rulers, tape measures, trundle wheels, metre rulers (use whatever is available)
  • Newspaper, recycled cardboard, scissors and tape
  • 1cm square grid paper
  • Copymaster 1
  • Copymaster 2
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 school 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. 

      Modelling these processes on a whiteboard, interactive whiteboard, or with the use of materials could support students to develop their thinking.
       
  3. Ask about how the square units will be arranged. Introduce the terms, rows (across), columns (down), and array (a structure of rows and columns) if students are not familiar with those words. 
    Do students recognise the array structure in the arrangement of square units?
  4. Ask a volunteer to place the squares units side by side on the rectangle. Blu Tac 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.
    Can we count the squares even more efficiently?
  6. Record 3 x 6 = 18 and ask students where they can see representations of six and three in the model (i.e. in the number of rows and columns). Ask where the 18 is found (i.e. it is the 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 with all sides the same length). The rectangles might be cut out of paper, created with play dough, 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 1. 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. Smaller rectangles that utilise simpler times tables could be drawn and used by pairs of learners.
  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. To extend learners, you could ignite discussion around this.
    Answers: A (3 x 7 = 21), B (6 x 6 = 36), C (4 x 11 = 44), D (11 x 3 = 33), E (7 x 7 = 49), F (8 x 6 = 48), G (10 x 2 = 20).
    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. Students may also make connections to playing sports (e.g. a team follows a formula to play well and win), tikanga (correct ways of doing things), or car racing (e.g. in Formula One racing, the “formula” entails a set of rules that all racers’ cars must meet).
    What do we mean by a formula?
    Do students explain that a formula is like an algorithm, or rule, that we can follow to get 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 1). 
    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 its 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 its 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 1. As a group, practise starting with the formula, and substituting the values of length, width, and area for each rectangle. Students may benefit from using materials to model the use of the formula.
  4. Provide students with a group worthy task to work on collaboratively (see Copymaster 2). This could be linked to school events (e.g. make a new sign for our classroom, design a school garden, design the size of a hāngi pit). Students might be given 1cm grid paper, 1cm squares, or work in their exercise books. There are several programmes online that allow students to model the construction of arrays using 1cm squares. Make sure to thoroughly investigate any programme you wish to use, to ensure its use will be appropriate and purposeful for your students.
  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 benefit from the support of a multiplication basic facts poster or list.
    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 a 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 (or object that is relevant to the context of the learning) to consider how well each label/array design will work.
  7. In the jam jar context, students might write a letter to Karly outlining how they investigated her problem and giving their recommendations. Their mathematical thinking could be used as the basis of a persuasive letter in other contexts.
  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 the 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: 
    A composite shape which forms one-half of a capital ‘T’ shape. It is made of two 2 x 4 cm rectangles which are perpendicular to each other. More possible compositions of the shape are described in step 12.
  12. This shape can be seen to be made up 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. Use 1cm square units (e.g. memo pads) to demonstrate the construction of this composite shape. There are different ways to solve composite shapes. However, one of the simplest methods starts with breaking a composite shape down into basic shapes (e.g. 2 rectangles). You could model this with memo pads or tiles. Next, find the area of the basic shapes you have constructed. Finally, add the areas of the basic shapes together. To support the development of this thinking, you could calculate the area of the shape shown above in two different ways. First, calculate the area from 2 rectangles, each with an area of 4cm x 2cm. The total area of each rectangle is 8cm2. Therefore, the total area of the composite shape is 16cm2. Next, calculate the area of the shape as one 2cm x 6cm rectangle (12cm2) added to one 2cm x 2cm (4cm2). Calculating the area of the same shape in different ways will allow for greater student collaboration, and will allow for students to learn from each other.

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 new table for the wharekai in the local marae. The first one they looked at measured 2m by 3m. Sam said if they wanted one with an area twice as big they should get the 4m by 6m size. Is Sam right?
  2. Ask the students to draw pictures of the table 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 the 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. Consider what culturally relevant contexts can be incorporated into this task, to increase the engagement of your learners.

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

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

A rectangular mat design using rectangular pieces of different colours.

Cuisenaire Rod Fractions: Level 3

Purpose

This unit introduces the idea that fractions come from equi-partitioning of one whole. Therefore, the size of a given length can be determined with reference to one whole. When the size of the referent whole varies, then so does the name of 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 the concept 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 (e.g. 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). 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 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 thinking that 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 flexible thinking. Lamon (2007) described re-unitising and norming as two essential capabilities if students are to master fractions. Re-unitising enables students to flexibly define a given quantity in multiple ways by changing the units they attend to. Norming enables students to operate 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:

  • altering 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 often used in the introduction of te reo Māori, meaning they may be familiar to some students. 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. This could be linked to the early journeys of Māori and Pasifika navigators to Aotearoa, or to current journeys your students have experienced (e.g. a bus ride to camp, running a lap of the playground). 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.

Te reo Māori vocabulary terms such as hautau (fraction), hautau waetahi (unit fraction), hautau waetahi-kore (non-unit fraction), rākau Ātaarangi (Cuisenaire rods), hautau ōrite (equivalent fractions), rārangi tau (number line), and the names for individual fractions could be introduced in this unit and used throughout other mathematical learning.

Required Resource Materials
Activity

Prior Experience

Students may have mixed experiences with using Cuisenaire rods. When introducing the Cuisenaire rods, ask students to think about what they could be used to represent in mathematics. Value the contributions of all students.

Session One

  1. Use Cuisenaire rods or the online tool to introduce the relative size of Cuisenaire rods in the following way. Provide whiteboards or paper, or use a large chart or whiteboard, to record students thinking. 
    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.”
    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.

     
  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 collaborate in small groups (mahi tahi). Consider pairing together more knowledgeable students with less knowledgeable students to encourage tuakana-teina (peer learning). 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. Explain 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 there 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. Therefore, opportunities to name fractions greater than one (i.e. mixed or improper fractions) is important. This thinking could be supported by making links to different ‘whole’ items that are different sizes (e.g. two different waka). Linking this learning to contexts that are relevant to your students will increase the level of meaning they can see within this unit. Just like the Cuisenaire rods, they are different sizes, but can still be classified as ‘one whole’. Students might recognise that the white rod fills the gap between the blue and orange rods. 
    Useful 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 groups, roam and 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. As a class, come up with a real-life context that could be represented by a fraction wall. It does not have to be completely realistic, but it should reflect the relevant learning interests and/or an element of the socio-cultural backgrounds of your class. For example, the one whole is the number of students in the class, ½ want to go to the beach for a class trip and ½ want to go to the museum. ¼ of the students who want to go to the beach want to swim, and ¼ want to build sandcastles. As you develop this scenario, use groups of students to represent the division of the “whole” (i.e. the whole class) into different fractions. This will present opportunities for students to compare the sizes of different fractions, in relation to the number of students in your class.
  10. 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? Allow students to discuss this in pairs. Some students may want to use the relevant rods to model their thinking.
    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 1 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. Activities using a number line could be completed digitally (e.g. on a PowerPoint or flipchart), or with the use of a whiteboard or paper chart. 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 question 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. As an extension, you could refer back to the whole-class wall fraction you created, and represent the different fractions on a number line. 
     
  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. Challenge them to frame their problem in relation to the context which framed your whole-class fraction wall.

Data cards: Level 3

Purpose

This unit provides a way of looking at multivariate data (i.e. data that includes many variable, such as gender, age, height, eye colour, bedtime, etc.) that comes 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 wholenumber data and simple time-series data to answer questions; identifying patterns and trends in context, within and between data sets;communicating findings, using data displays.
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 simple 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, meaning the questions can be used to, find out about the whole group. 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
  • altering the type of data collected; categorical data can be easier to manage than numerical data
  • altering the type of analysis – and the support given to do the analysis
  • providing prompts, and examples if necessary, 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. Students will be interested in questions they can ask their classmates that are of significance to them, such as “How fluent are you at speaking te reo?”, “What is your marae, maunga, iwi, etc.?”, “Should fireworks only be seen in public displays?”, or “Does your whānau grow their own vegetables at home?”  Māori can also be used alongside English on the data cards to develop students’ use of te reo Māori. Consider how you can use the data discussed, and found, throughout these lessons to inform learning in other curriculum areas (e.g. persuasive writing).

Te reo Māori vocabulary terms such as tirohanga tauanga (statistical survey), kāri raraunga (data card), taurangi (variable) and kohikohi raraunga (data collection) could be introduced in this unit and used throughout other mathematical learning.

 

Required Resource Materials
Activity

Session One

Part One – Introducing Data Set One

  1. Organise the students into pairs, hand out a set of data cards to each pair, Data Set One Master, and get them to cut out all the data cards. Students should collect the cards together and select one of the data cards each to have a look at. Consider how you can encourage tuakana-teina by pairing more students who are more confident in maths, with students who might benefit from additional support.
  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, such as "that is whether they are the oldest or youngest in their family".
  3. Confirm that the information on these data cards comes from the following four survey questions:
    • Do more students have odd or even birthdates? 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.
    Data card showing four categories: whistle, odd date, right, youngest.
  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 that is 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 set of data cards they cut 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 odd birthdates than even birthdates or more whistlers than non-whistlers, look to see if more odd birthdates than even birthdates 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 different patterns in the data.
    Punnett square arrangement of data cards showing the right and left handedness of of students with odd and even birthdates.
  4. Move around getting the students to explain and show what they have found out. Encourage students to add detail to their observations. This could include thinking proportionally. For example, rather than "One more even date is right handed than odd date", "A larger fraction of even date are right handed, 8 out of 12 even date in comparison to 7 out of 12 odd date 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 more whistling right-handers or whistling left-handers?
    • Is there anything interesting when comparing place in the family and whistling?
    • All the odd dates in this group who are the youngest can whistle, does this mean every odd date 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 from Data Set One in Session One and how they organised the data cards to see discover patterns and differences.
  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 about the unknown group 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 cut out the data cards, then sort and arrange them, and look for patterns of interest. Move around getting students to explain and show what they have found out. Ask them to leave their final sort intact.
  4. Conclude the session by asking the whole class to wander around the final sorts of each group. Encourage ako by asking the displaying pair to explain the patterns and differences they found. Compare the findings to the statements the students made at the beginning of the session and share 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. Tell 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: O, 6, 10 or 13? Seek ideas from the class.
      Data card showing four values: 6, O, 10, 13.
  2. Explain that the four survey questions for these data cards are:
    • Is your birthday an odd or even number? – O (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, in centimatres? – 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 12cm the first time, 15cm the second time, 11cm the third time and 14cm the fourth time, the average is 12 + 15 + 11 + 14 = 52, 52 ÷ 4 = 13, therefore, the reaction score is 13.
    Starting hand position in the reaction-time experiment.Final hand position in the reaction-time experiment.
    Ask students to anticipate what the data may show. Examples might be:
    • Older students will have better reaction scores than younger students
    • Class level will always be five years less than age
    • There should be the same number of odd and even birthdates
       
  4. Get the students to sort and arrange the data cards to look for patterns and differences of interest. Move around getting each pair of students to explain and show what they have found out. Share the patterns and differences as a class.

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 by students. Discuss "issues" that are of current importance to students. For example, students might be interested in whether people think fireworks should be only used in controlled displays, whether buses should be organised to get students to school, or how many different ethnicities or languages are represented in your class. You could frame this with the context of finding out more about make-up of our class, or as learning to inform persuasive writing, speeches or community action. Take care with issues that may be sensitive, particularly those related to body image, gender identity, religion and culture. 
  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) becomes possible.
  3. Discuss and brainstorm suitable survey questions. Discuss, including the measurement unit, when numeric data is gathered, For example:
    • How many centimetres tall are you?
    • How many centimetres across is the span of your right hand? (thumb to pinky finger)
      Specific instructions will be needed with survey questions like this, so it is clear where to start and finish measuring.
    • How many seconds does it take you to run 100m?
    • What is your favourite...?
    • How many languages are spoken in your family?
      A list of possible favourites to select from is best with survey questions like this.
    • What time did you go to bed last night, in hours and minutes?
      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 anticipatory 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. The best way to do this is for each pair to create a "station" for other students to visit. The stations can be labelled with letters of the alphabet to ensure all students visit each station, like a dance card.
    Blank data card. 
  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. Ideally the report includes some display of the data which might be the data cards organised in groups or bars.

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 a 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.

These Learning Outcomes are covered in every lesson of the unit.

Description of Mathematics

This unit develops the concept of a relation by using matches to demonstrate how patterns grow. 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. In this context, 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
  • encouraging students to verbally share their thinking with each other
  • using whiteboards, dot paper, grid paper, and digital drawing tools to represent patterns
  • providing table templates
  • modelling how to create 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.  Look for opportunities to connect learning with the everyday experiences of your students.

Te reo Māori vocabulary terms such as taurangi (algebra), pūtaketake (the base element of a pattern), and ture (formula, rule) could be introduced in this unit and used throughout other mathematical learning.

Required Resource Materials
  • Matches with the heads burnt, or toothpicks, ice-block 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 Smart TV 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.
    This shows how 3, 5, and 7 matchsticks are used to build 1, 2, and 3 triangle paths.
  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.
    This shows Kiri’s method for rearranging the matchsticks. The pattern begins with 1 matchstick. For each term in the pattern, 2 matchsticks are added to complete the next triangle.
    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 shortcut way of counting the number of matches needed to make a 10-triangle path. Get them to write down, using pictures to 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:
    This shows Jamies’ method for rearranging the matchsticks. The pattern begins with 3 matchsticks (one whole triangle). For each term in the pattern, 2 matchsticks are added to complete the next triangle.
    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 explain, using written and/or verbal language, what Jamie's strategy is. Some students may benefit from, or prefer to use, pictures to support their explanations.
    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.]

At this stage, it may be appropriate to revisit or introduce the concept of “BEDMAS”. The acronym BEDMAS signifies the order in which operations should be carried out in an equation: brackets, exponents, division and multiplication in the order that they occur, and then addition and subtraction in the order that they occur. Ask your students to solve 10 X 3 – 9 by doing the multiplication first, which is the correct way (i.e. 30 – 9 = 21), and then by doing the subtraction first (i.e. 10 X -6 = -60). If negative numbers are beyond the knowledge of your students at the time of teaching, then you should adjust the numbers in the equations you provide. The key teaching point is that BEDMAS is used to guide us when solving problems with more than one sign. This is important because the order that we carry out number operations can change the outcome of a problem.

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.
    This shows how 4, 7, and 10 matchsticks are used to build 1, 2, and 3 square paths.
  2. 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?
  3. 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?
  4. 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 provided materials and draw different representations of the pattern. Prompt the students with the following questions: 
    Do you need to draw every square?
    Is there only one possible way to look at the pattern?
    What might some of the other ways look like?
  5. 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 a 20-square path. If there is a wide variety of strategies being presented in the group, ask students to share and justify their strategy with a peer who has developed a different strategy. Share the strategies back to the whole class and validate all thinking. 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?
  6. Have the students use their ‘best method’ to verify that 61 matches are needed to make a 20-square path.
  7. Compare the way the rules might be written:
    Kiri [1 + 20 x 3]                      Jamie [4 + 19 x 3                    Vey-un [20 x 4 – 19]
  8. Students can use these methods or their own ways to predict the number of matches needed to make 14-, 36- and 100-square paths.
  9. 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)]
  10. 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?
  11. Are students able to ‘undo’ their rules to 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 the 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.
    This shows how 6, 11, and 16 matchsticks are used to build 1, 2, and 3 house paths.
  5. 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?

Next, 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 of a pattern found in the local community 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 include the patterns shown on the rest of PowerPoint One (shown below).
    This shows how 6, 10, and 14 matchsticks are used to build 1, 2, and 3 "zero" (rectangle) paths.
    This shows how 6, 10, and 14 matchsticks are used to build 1, 2, and 3 fish paths.
  4. Allow the class time both to report back and discuss their solutions, and to write up what they have discovered.
  5. Olika wanted to make a pattern using the n-rule. N means any number you give her, say 1000, 53 or 214. 
    Can you draw a pattern that matches this rule?
    “n minus one then multiplied by five then add six”
    What might the pattern look like?

One possible answer is: 

This shows how 6, 11, and 16 matchsticks are used to build 1, 2, and 3 hexagon paths.

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.
    This shows how 4, 16, and 28 matchsticks are used to build 1, 2, and 3 plus-sign paths.
  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.
    This shows how 12, 19, and 26 matchsticks are used to build 1, 2, and 3 star paths. The stars are composed of a square (created with 4 matchsticks) with a triangle on each side of the square (created with two additional matchsticks). Each term in the pattern replaces the triangle on the right hand side of the square with a new square surrounded by 3 additional triangles.
    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?
  10. To further engage students in a real-life context asj them to research repeating patterns from other cultural backgrounds. 

 

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. Division involves equal sharing or 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 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.

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
  • easing the calculation demands by choosing fractions and sets that are manageable
  • reducing the demands for a product, e.g. less calculations and words
  • encouraging sharing and discussion of students’ thinking
  • using collaborative grouping so students can support each other and experience both tuakana and teina roles
  • encouraging mahi tahi (collaboration) among students.

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 that 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 your 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.

Te reo Māori vocabulary terms such as tūtahi (whole), hautau (fraction), hautau waetahi (unit fraction), taurunga (numerator), and tauraro (denominator) could be introduced in this unit and used throughout other mathematical learning.

Required Resource Materials
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. Choose 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 if 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 unit fractions like these and know which fraction is larger?
    Do students recognise that a smaller denominator means 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. Can they give clear explanations of the symbolic representation of the fractional part? Model this if necessary.
    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 quarters 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 have a korero 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 depending 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 to 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 recording 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 symbols 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. Explicit links can be made here to Kupu Māori.

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 Fraction Pieces available to the students. 
    Pose and write on a chart the question: 
    Can the numerator in a fraction be the same as the denominator?
  2. Use the fraction 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 Pieces page. 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 game Fraction Snap. (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:
    2/3Two is the numerator.   is a pair, and so is   One sixthArray of sixths, with one-sixth shaded in.
  2. Play Fraction Snap 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 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 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:
    An “L” shape labelled as 1/4.
    the whole shape might look like any of these:
    This shows four “L” shapes being combined to create a rectangle.     This shows four “L” shapes being combined to create an "Ɪ" shape.     This shows four “L” shapes being combined to create a cross shape.
    Have students find multiple 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.
    Three beans placed on an “L” shape in an even arrangement.
     
  2. Explain that these beans, like their shape, are just a fraction of a set of beans. It’s the same fraction as the fraction 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 many 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.
    This shows four “L” shapes being combined to create a cross shape. 3 beans have been placed on each “L” shape.
    “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 to work 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?
    Your whānau were given 4 kūmara. This was one sixth of the total kūmara in the hāngi.
    How many were in the hāngi 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 and emphasise the efficiency of multiplicative strategies that are expected at Level 3.

Activity 2

  1. Have the students collaborate (mahi tahi) 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 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 to 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 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 or she 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.  

Session 6

Remind students of their learning from Activity 1 of Session 5 about finding a non-unit fraction of a set. The difference in this session is that there is no longer a predetermined number in each set. 

It will be helpful to model solving a challenge with the group/class first. As students will need to find a number of cubes/tiles that can be equally split into each fraction in the challenge, a way to support students with this is to explain the following:

For the challenge ‘Build a design that is 1/3 blue and 1/4 yellow’, the number of cubes/tiles in the set has to be able to be split equally into thirds and into quarters. The simplest way to do this is to find a common multiple of the denominators involved, which in this case is 12. 

Note: 12 is not the only number of cubes/tiles that could work for this challenge. In fact, any common multiple of 3 and 4 will work for this challenge. 

 

Fraction Mosaics

You will need a bunch of cubes or square tiles in at least four colours (red, yellow, green, blue), or small square pieces of paper. The task is to build a mosaic design that matches the description in your challenge. Collaborating in pairs (mahi tahi), build your mosaic design and be ready to have a korero about it with the rest of the group/class. These mosaic designs could be combined to create a large siapo or tukutuku design, or a design for a new school garden. Links could be made with designs you have recently seen in your community (e.g. at the marae). Lay a transparent grid over an image of relevant art, and as a class come up with a list of fraction statements that describe the use of colour (e.g. 3/4 of this art uses the colour blue). This could be framed within the context of looking at how early Māori people used pigments derived from natural materials (e.g. clay deposits, iron oxide) to create paint. 

 

Challenge list:

  • Build a design that is 1/3 blue and 2/3 red
  • Build a design that is 1/2 red and 1/4 green
  • Build a design that is 1/5 yellow and 3/10 green
  • Build a design that is 1/3 blue and 1/4 yellow
  • Build a design that is 1/4 red and 1/4 blue
  • Build a design that is 1/5 red, 4/10 green, and 2/5 blue
  • Build a design that is 5/12 blue, 1/6 red, and 2/6 yellow
  • Build a design that is 1/3 yellow, 1/6 red, and 1/2 blue
  • Build a design that is 1/3 red and 1/8 green
  • Build a design that is 2/3 yellow and 1/7 red
  • Build a design that is 3/5 blue and 1/4 yellow

Students could create their own challenges for another pair to solve. What is the same and what is different about the way you solved it and the way another pair solved it?

Questions you could use for group korero:

  • How did you get started?
  • How did you decide what to do?
  • How did you decide how many cubes/tiles to use?
  • Is there another way to complete your challenge? How do you know?
  • Can you prove that your design matches your challenge?

Printed from https://nzmaths.co.nz/user/75803/planning-space/early-level-3-plan-term-2 at 2:55am on the 30th March 2024