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

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

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

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

#### Specific Teaching Points

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

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

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

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

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

## Areas of Rectangles

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.

Area is the amount of flat surface enclosed within a shape. Commonly used standard units for area are cm

^{2}(square centimetres), m^{2}(square metres), and km^{2}(square kilometres). Squares are used because they iterate, that is they fit together in two dimensions with no gaps or overlaps.Rectangles are the easiest shapes to find the area of because the array structure of iterating units is most obvious. Consider this rectangle filled with square units of area:

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

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

^{2}offers opportunities to apply multiplicative thinking and systematic reasoning. Those opportunities may be lost if students are preoccupied with finding products mentally.The context for this unit can be adapted to suit the interests and experiences of your students. Students could be challenged to find the area of a room in their own home, or the area of a space of interest in your local community, such as a sports ground, skate park, marae or similar. A diagram with measurements could be provided if the area is not readily accessible during school time.

## Session 1

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

How many square tiles will cover this area?Do students recognise the array (rows and columns) structure in the arrangement of units?

^{2}, one square decimetre)Can we count the squares even more efficiently?Look for students to:

Recognise that much will depend on their knowledge of multiplication facts and strategies.

Answers: A (3 x 7 = 21), B (7 x 5 = 35), C (4 x 11 = 44), D (12 x 3 = 36), E (7 x 7 = 49), F (8 x 6 = 48), G (11 x 2 = 22).

What do the answers tell us about these rectangles?Students might measure with a ruler to check that the units are square centimetres.How big are the little squares?

Ask students to include the unit in their answers, e.g. 21cm

^{2}. Recording the notation for each rectangle is good practice.## Sessions 2 and 3

What do we mean by a formula?Do students explain that a formula is like an algorithm for getting the same result each time?

Record W x L = A. This is a mathematical formula written as an equation.

I wonder what the letters W, L and A might represent?For example, Rectangle B had seven rows of five squares.

The row gives the length of the rectangle. In the case of B length equals 5. (rub off L in the formula and write 5 in it’s place)

The number of rows gives the width of the rectangle. In the case of B width equals 7. (rub off W in the formula and write 7 in it’s place).

The formula now reads 7 x 5 = A. I wonder what A equals. What value for area makes the equation true and matches the formula?

^{2}they will need to consider all the factors of 72. Encourage students to find those factors systematically. Some students may need to support of a multiplication basic facts poster.A systematic approach involves starting with 1 as a factor then increasing the smallest factor by one and testing 72 for divisibility.

1 x 72, 2 x 36 (72 ÷ 2 = 36), 3 x 24 (72 ÷ 3 = 24), 4 x 18 (72 ÷ 4 = 18), 5 x (72 is not divisible by 5), 6 x 12 (72 ÷ 6 = 12), 7 x (72 is not divisible by 7), 8 x 9 (72 ÷ 8 = 9).

If the process continues the factors will appear in reverse order, e.g. 9 x 8 = 72. 8 x 9 and 9 x 8 are essentially the same rectangle though they may appear differently if the direction of the label is considered.

Discuss criteria for eliminating labels. For example, a label with width of less than 5cm might be considered too ‘skinny.’

Discuss the best options, cut them out at real size, then use a real jam jar to consider how well each label will work.

^{2}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.^{2}and make estimates of area of the space before they calculate.Do they apply the W x L = A formula?

## Session 4

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

## Session 5

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

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

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

Family and Whanau,

This week at school we have been calculating the area of rectangles using the multiplication formula of length x width. We have been working out the areas of composite shapes by marking the shapes into rectangular shapes. For example,

At home this week your child is to draw 5 different composite shapes that each have a 20cm

^{2}area. The lengths of the sides should be marked. Ask them to explain to you how they worked out the area for each one.## Figure it out

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

## Cuisenaire Rod Fractions: Level 3

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

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

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

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

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

## Specific Teaching Points

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

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

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

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

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

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

Tasks can be varied in many ways including:

^{2}grid paper and coloured felt pens to ease the recording demands (Cuisenaire rods are based on that scale).The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Cuisenaire rods (rakau) are a media often used in introduction of te Reo so some students may already have encountered them. Knowing the relationships between rods of different colours, without having assigned number names to the rods, is very helpful in easing cognitive load. Other contexts involving fractions of lengths might also be engaging for your students. For example, the fraction of a race or journey that has been covered at different points is practically useful. Consuming foods that are linear, such as submarine sandwiches, bananas, or sausages, might motivate some learners. Board games that have a particular number of steps from start to finish provide opportunities to look at a fraction as an operator.

## Prior Experience

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

## Session One

Relative to the orange rod, how long is the yellow rod? How do you know? JustifyThe relationship between the yellow and orange rods can be expressed in two ways:

“The yellow rod is one half of the orange rod.”

“The orange rod is two times the length of the yellow rod.”

So if the orange rod was one then the yellow rod would represent one half. What fraction would the red rod and dark green rod represent? Justify. Convince us you are right.

“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.”

## Session Two

If the b

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

Are ther

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.

If blue is one then what fraction is the orange rod?Thinking that fractions are restricted to less than one is a common constraint students learn so opportunities to name fractions greater than one is important. In this case students will recognise that the white rod fills the gap. Good questions are:

Remember, which rod is one?So what fraction is the white rod?(1/9)How many white rods fit into the blue rod?(nine)How many white rods fit into the orange rod?(ten)So what shall we call the orange rod?(1 1/9 or 10/9)How much more three quarters is than two thirds?How much less one half is than two thirds?

## 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.

What are the size relationships between the yellow and black rods?The students might use the white rod as a reference to say, “The yellow rod is five sevenths of the black rod.” It is more difficult to recognise that “The black rod is seven fifths of the yellow rod.” The key idea is to establish the referent one. If a comparison ‘of a given rod’ is being made then that rod becomes the one.

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)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.

## Session Four

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

If the 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.Dear parents and caregivers,

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

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

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

https://mathsbot.com/manipulatives/rods

## Data cards: Level 3

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

The key idea of statistical investigations at level 3 is telling the

class storywith supporting evidence. Students are building on the ideas from level two and their understanding of different aspects of the PPDAC (Problem, Plan, Data, Analysis, Conclusion) cycle – see Planning a statistical investigation – level 3 for a full description of all the phases of the PPDAC cycle. Key transitions at this level include posing summary investigative questions and collecting and displaying multivariate and time series data.Summary or time series investigative questions will be posed and explored. Summary investigative questions need to be about the group of interest and have an aggregate focus. For example,

What position in the family are the students in our class?What 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 toread beyond the databy asking questions such as: “If a new student joined our class, what reaction score do you think they would have?”The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:

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

Session OnePart One – Introducing Data Set OneWhat do you think the survey questions were?Seek ideas from the class. They may just give you the variable description which is fine.Part Two – Making Class Data CardsWhat would a data card about you look like for these same four survey questions?Part Three – Working with data set oneAre there proportionally more whistling right-handers or whistling left-handers?Is there anything interesting when comparing place in the family and whistling?All the boys in this group who are the youngest can whistle, does this mean every boy who is the youngest in their family can whistle?Session TwoDuring this session, students will be sorting and arranging data cards about themselves, i.e. the students’ own data cards.

What do you expect to find out about the class?Will the things we found out from Data Set One, be different or similar to our class?Session ThreeData Set Three – OptionalA 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 FourToday 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.

Specific instructions will be needed with survey questions like this, so it is clear where to start and finish measuring.

A list of possible favourites to select from is best with survey questions like this.

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.

Session FiveIf 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

SPECIFICvariables and select four variables for their data cards. The first three sections are pretty good to choose from. Then enter sample size – 30 should be enough. Generate sample, then an option to download the sample comes up – select this. Save their sample and then open the spreadsheet and use the information to make their own data cards.## Matchstick Patterns

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.

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

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

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 NumeracyThis 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.

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:

Tasks can be varied in many ways including:

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

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

Session 1: Triangle PathsIn this session we look at a simple pattern created by putting matches together to form a connected path of triangles.

How manyextramatches 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?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). Ask:Kiri’s RuleUsing 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?

What isJamie’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?How many matches will be needed to make a 20-triangle path?How big a path can Jamie make with 201 matches?

How would Kiri and Jamie explain to someone else how they could find the number of matches needed to make a path consisting of any number, say 1000, of triangles?Extension idea:Vey-un has another way to work out the number of matches for a 10-triangle pattern. He writes 10 x 3 – 9 and gets the same number of matches as Kiri and Jamie, 21.

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

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

Session 2: Square PathsHere we look at a simple pattern created by putting matches together to form a connected path of squares.

Where are the additional matches located?What would Kiri, Jamie and Vey-un do for this square pattern?Is there only one possible way to look at the pattern?What might some of the others ways look like?

What is a more efficient way to draw or calculate the total number of matches?Kiri [1 + 20 x 3] Jamie [4 + 19 x 3 Vey-un [20 x 4 – 19]

any numberof 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)]

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?Kiri calculates “One plus three times the term number” to find the number of matches.If Kiri knows the number of matches, how should she undo her rule to find the term number?[Note that order of undoing is important, subtract one then divide by three.]Session 3: House PathsThe ideas learnt in the last two sessions are reinforced here using ‘house paths’.

A new matchstick path is being designed. It is called ahouse 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.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?Session 4:What’s My Path?The ideas of the first three sessions are extended and reinforced in another context. This time the problem gives a rule and the students find the pattern.

My friend made a picture that showed how her fifth matchstick path was made. She named it:

5 lots of 4 and add 2(this was the counting rule used to make the path)She sent it to me via email. However, I was only able to read the name of the path and not see the picture!

Make some possible pictures that she could have sent.

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]

If my friend wanted to build a n-path how many matched would she need?N means any number you give her, say 1000, 53 or 214.“n minus one then multiplied by five then add six”

What might the pattern look like?

One possible answer is:

Session 5: Other Ways of Seeing ThingsIn this session, the concept of a relation is explored with a more complicated spatial pattern.

Find many different ways to work out the total number of matches in Term 10.

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?

Here is a pattern of growing stars made with matches.

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

This week in maths we have been looking at patterns made with matches We looked at the first term, the second term, … the tenth term, … and so on and tried to find a relation between the number of matches and the number of the term. For example, we explored this pattern with matches:

Ask your students to explain how they could predict the numbers of matches in a ten-house path. What else can they share with you about the pattern?

Enjoy your exploration of this algebra problem!

## Figure it Out Links

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

## Symbols and sets

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.

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

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

## Links to the Number Framework

Stages 5- 6

This unit complements the learning activities in

Book 7 Teaching Fractions, Decimals and Percentages.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:

Tasks can be varied in many ways including:

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

Session 1The 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 1Is there a way to look at units fractions like these and know which fraction is larger?Do students recognise the a smaller denominator mean the whole is cut into less parts so the parts are larger?

Activity 2Highlight 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 3Can a triangle be folded into quarter when each part must be equal?Your job now is to reconstruct what the whole looked like for each piece you have.What information do you have to help you?Let students work out the appearance of the original whole for each fraction piece they have. Be aware that there are multiple possible answers pending on how the pieces are arranged. It is an excellent challenge to create as many different wholes as possible for a given fraction piece.Activity 4Why 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)What comes next?How might we write eleven tenths?(11/10 or 1 1/10)If 11/10 is an improper fraction I wonder what that means? (numerator is greater than denominator)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.

What parts would we get if we folded tenths in half?...in quarters? …in tenths?The last fold produces hundredths.Activity 5Conclude this session by writing the words and symbols for common unit fractions and some other proper fractions. Brainstorm on the class chart/book what has been learned about fraction symbols.

Session 2The 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 1Write '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 2Pose and write on a chart the question:

Can the numerator in a fraction be the same as the denominator?Who can make fractions where the numerator and denominator are equal?Explain why any fraction where the numerator and denominator are equal is another name for one.Can the numerator be greater than the denominator?What is true of all fractions where the numerator is greater than the denominator? (All improper fractions are greater than one (whole)).Activity 3Have students play

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

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.

Activity 4Introduce the

Fraction Snapgame. (Copymaster 3).For example:

is a pair, and so is

Session 3The 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 1Distribute 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.

For example if this is 1/4:

the whole shape might look like any of these:

Have students find multiples solutions with the other shapes, writing the fraction in each part and the one whole fraction (4/4, 8/8 etc.) beside the whole shapes.

Activity 2Have the students each take up to 4 beans and place them on their coloured fraction piece.

If this is a fraction of the set, how many beans are in the whole set?For example : 1/4 of a whole set is 3 beans, 4/4 make 1 whole, so 4 lots of 3 beans will make 1 whole set.

“3, 6, 9, 12” or “4 x 3 = 12”

Activity 3Mia’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 4The purpose of this session is to use materials to reinforce the whole to part relationship and to continue to use fractions of regions to build an understanding of fractions of sets. The key connection is made with the operation of division. One view is to see how many equal sets can be made from the starting set, and working out the number of items in each equal set. This is called partitive division.

Activity 1For 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?Activity 2Here 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.

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 3Half of twelve12 shared between 2

12 ÷ 2

12/2

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.

Activity 412 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)

Activity 5Pairs of students can be challenged to write their own fraction problems for their partner to solve.

Session 5The 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 1Make coloured beans,or other objects to be shared, and paper available to pairs of students.

There are 25 beans in a packet. You plant 2/5 of the beans. How many beans do you plant?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?

For example: First you have to find one fifth so you divide 25 by five. You’re asked for two fifths so you have add two fifths together or times one fifth by two. You write this 25 ÷ 5 = 5, 5 + 5 = 10 or 2 x 5 = 10

Notice who uses equal sharing (into regions) and who uses their knowledge of multiplication (and division) to solve the problem.

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?

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 2Have students play

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

Activity 3Conclude this session with a discussion of the game and summary of learning.

Dear parents and whānau,

In class in maths we have been learning to find fractions of sets.

Your child has brought home a game that we have been playing in class. Please return it tomorrow. You might like to write a comment about it and place the comment in the game bag. We hope you enjoy playing the game.

Here are the rules:

Telling the Truth.(Purpose: to identify the correct fractions of sets)

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.

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. Upon Player One’s request for a card, if the Player Two gives an untrue card, Player Two must miss a turn. Player One may immediately make another request.

If Player Two has no suitable cards he tells Player One to pick up from the pile. Each time a player picks up or receives a card they must check it for accuracy.

The player with the most correct matching pairs when all the cards are used, is the winner.