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. Usually the one must be defined in context.
‘Fractions as measures’ is arguably the most important of the five sub-constructs of rational number (Kieren, 1994) since it represents 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 do we assign 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 that 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. This idea, that any given point on the number line has an infinite number of fraction names, is a significant change from whole numbers. 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 = 80%).
‘Fractions as operators’ is another of Kieren’s sub-constructs and applies to situations in which a fraction acts on another amount. That amount might be a whole number, e.g. three quarters of 48, a decimal or percentage, e.g. one half of 10% is 5%, or another fraction, e.g. two thirds of three quarters. Students often confuse when fractions should be treated as numbers and when they should be treated as operators, particularly when creating numbers lines, e.g. they often place one half where 2 1/2 belongs on a number line showing zero to five.
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 understand 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 units. In this unit Cuisenaire rods are used to develop students’ skills in changing units and thinking with those units.
Multiplication of fractions involves adaptation of multiplication with whole numbers. Connecting a x b as ‘a sets of b’ (or vice versa) with a/b x c/d as ‘a b-ths of c/d’ requires students to firstly create a referent whole. That whole might be continuous, like a region or volume, or discrete like a set. Expressing both fractions in a multiplication and the answer require thinking in different units. Consider two thirds of one half (2/3 x 1/2) as modelled with Cuisenaire rods.
Let the dark green rod be one, then the light green rod is one half.
So which rod is two thirds of one half? A white rod is one third of light green so the red rod must be two thirds. Notice how we are describing the red rod with reference to the light green rod.
But what do we call the red rod? To name it we need to return to the original one, the dark green rod. The white rod is one sixth so the red rod is two sixths or one third of the original one. So the answer to the multiplication is 2/3 x 1/2 = 2/6 or 1/3.
Reunitising and norming are also important when fractions are placed in order of size (magnitude). This is especially true given any fraction has an infinite number of names. Imagine we use the orange rod as one this time and find two fifths. Since five red rods measure one whole (orange rod) then two red rods measure two fifths:
But, 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, is infinite. This means that the point on the number line where two fifths exists has an infinite set of number names.
Students are unlikely to have previous experience with using Cuisenaire rods since the use of these materials to teach early number has been abandoned. Their lack of familiarity with the rods is a significant advantage as they will need to imagine splitting the referent one to solve problems.
Use Cuisenaire rods or the online tool to introduce equivalent fractions in the following way.
“What is the size of the crimson rod compared to the brown rod? How do you know? Justify”
The relationship between the crimson and brown rods can be expressed in two ways:
“The crimson rod is one half of the brown rod.”
“The brown rod is two times the length of the crimson rod.”
So if the brown rod was one then the crimson rod would represent one half. What fraction would the red rod and white rod represent? Justify. Convince us you are right.
Encourage the students to express the relationships in various ways, such as:
“The red rod is one quarter of the brown rod because four of it fit into one (brown rod).”
“The brown rod is four times longer than the red rod.”
Ask “How many quarters and how many eighths are the same length as one half (crimson rod)?”
“How might we record these relationships mathematically?”
Hopefully students will suggest recording an equality like this: 1/2 = 2/4 = 4/8.
Ask, “What patterns do you see in the equality?” Students should notice the doubling of both numerators and denominators. “Why does this happen?”
It is important to reinforce the idea that the numerator is a count, so the doubling of numerators indicates that there are twice as many parts in the same space.
Ask, “But why do the denominators double? Does that mean that the parts get twice as big?” Look for the students to notice that the denominators double because the parts halve in size, twice as many quarters as halves fit into one, twice as many eighths as quarters fit into one.
Use slide one from the Powerpoint for this unit. Ask, “Imagine we had these new rods, grey and pale blue. What could you say about these new rods in relation to one, the brown rod? How would you write the relationships mathematically? How are these fractions related to one half?”
Look for students to establish grey as one sixteenth either by halving one eighth or realising that sixteen of the parts fit into one. Look for students to recognise that three pale blue rods fit into one half so six of the rods will fit into one. Therefore, pale blue is one sixth of the one (brown rod). You might also highlight that one sixth is ‘One third of one half.’ From these observations other equivalent fractions for one half can be found (eight sixteenths and three sixths). Record these fractions as equalities:
1/2 = 8/16 and 1/2 = 3/6
Ask, “What patterns can you see? Explain why the patterns occur.” Look for student to use numerator as a count and denominator as the number of equal parts that fit into one. For example, three sixths has three times as many parts (pale blue rods) as one half but three times as many of these parts are needed to make one.
Introduce Investigation One using slide two of the PowerPoint. Provide sets of Cuisenaire rods or access to the online tool. Remind them that there is no grey rod in the set but it can be made up by joining two rods. Let the students work in small teams. Look for the following: Do the students refer back to the grey rod as the one? Do they look for a unit fraction to support them to name the blue and brown rods? For example, to name the blue rod they might notice that four light green rods make one. So the light green rod is one quarter. Three light green rods make blue so the blue rod is three quarters.
Can they record the relationships they find as equalities?, e.g. 3/4 = 9/12.
Can they conjecture equivalent fractions that they do not have rods for?, e.g. 3/4 = (4 1/2)/6.
All of these points can be raised in discussion as a whole class. Continue to ask questions about the meaning of the numerators and denominators in equivalent fractions. For example:
Finish the session with a reflection question: “Two children are talking (see slide three of the PowerPoint). Millie and Jana have different ideas about two thirds. Who is right? Explain why they are right.”
Let the students write an answer individually. You might use your students’ writing as prior assessment and revisit the idea of an infinite number of equivalent fractions sometime later.
Revise the key points about equivalent fractions from the previous session using the blue rod as one.
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?”
“Can you create an equivalent fraction to two thirds using rods in the picture (2/3 = 6/9)? How might we record this equality?”
Show the students slide four of the Powerpoint for this unit. They should remember the fictitious grey rod from Session One that can be made by joining two dark green rods. The students should notice that the light green rod is one quarter since it maps into one four times and that the crimson rod is one thirds since it maps in three times. At this point you might construct this diagram using the online tool with the grey rod being made up of two dark green rods.
Invite students to describe the total length of the two rods combined. Look for ideas like:
“The total is more than one half because two quarters is one half and one third is longer than one quarter.”
Ask: “What rod could we use to measure all three rods exactly, light green (one quarter), crimson (one third) and grey (one)? Different rods might be tried but the white rod is the only unit that measures all three other rods a whole number of times.
Ask, “What fraction of one is the white rod? (one twelfth because 12 of those rods fit into one)”
“How many twelfths are the crimson rod (one third) and the light green rod (one quarter)?”
“How could we record these equalities? (1/3 = 4/12 and 1/4 = 3/12)”
“So what is the total length of one third plus one quarter? (seven twelfths)”
“How might we record this sum mathematically? (1/3 + 1/4 = 7/12)”
“So where do the seven and the twelve in the answer come from?”
Students might notice that if the sum is written as 1/3 + 1/4 = 4/12 + 3/12 = 7/12 then the origin of the seven and twelve are clear.
“Why are the four and three added to make seven but the twelve and twelve are not?”
Normally when 4 + 3 are added the units are the same, e.g. 400 + 300 (hundreds), 0.4 + 0.3 (tenths), and the answer is seven of those units. In this example the units are twelfths so we get “Four twelfths plus three twelfths equals seven twelfths.”
Provide the students with a set of Cuisenaire rods and Copymaster 1 for each small group. Let them solve the problems collaboratively as you roam. Look for the following:
Can the students name each fraction with reference to the designated one rod?
Do they look for equivalent fractions where they are needed?
Do they record the addition of each pair of fractions correctly?
Look especially for how they deal with non-unit fractions, like three quarters.
After a suitable period bring the class together to discuss their solutions. Bring out the points above. Look for students to justify their renaming of fractions in equivalent form and how they calculated their answers.
For example, in question 4 the two fractions are one half (black rod) and three sevenths (red rods) since seven red rods fill the one (grey rod). Using white rods each fraction can be renamed as fourteenths, 1/2 = 7/14 and 3/7 = 6/14. The combined total is 13 fourteenths so the sum can be written: 1/2 + 3/7 = 7/14 + 6/14 = 13/14.
Pose a problem for individuals to solve and record their thinking (see slide six of the PowerPoint for this unit). You may like to use this to assess who among your students understands addition of fractions and who needs further support.
In this session the purpose is to find the difference between two fractions. Difference is the often neglected context for subtraction though problems can also be solved by adding on.
Pose this problem:
“Lelani got two thirds of a whole Cuisenaire rod and Sala got one half of the same sized Cuisenaire rod. Who got the most and how much more of one rod did they get?”
Ask the students which Cuisenaire rod they could use to solve the problem. Note that it must be possible to make one half of the rod and two thirds of the rod. You may need to try various rods before dark green is settled on. The difference is the length missing to make the smaller fraction as big as the larger.
Students should recognise that the difference is one sixth of the one (dark green). Ask how this problem might be recorded mathematically. Both addition and subtraction might be used:
1/2 + 1/6 = 2/3 or 2/3 - 1/2 = 1/6.
Ask: “So why is the answer one sixth when neither of the fractions have sixths?” Look for students to recognise from previous sessions that both one half and two thirds can be expressed as sixths. So an extra step in the subtraction equation gives:
2/3 - 1/2 = 4/6 - 3/6 = 1/6
From this recording, students might realise that recording difference as subtraction is a little tidier and certainly more conventional than adding on.
Build this diagram:
Ask the students to discuss in pairs what fraction difference problems might be made up using this picture. Remind them to think flexibly about which rod is one, assuming the orange rod is one is not the only possibility (dark green and yellow can also be named as one).
Record a few possible problems such as:
If the dark green rod is one, what fractions are represented by the orange and yellow rods?
What is the difference between those two fractions? In this case the orange rod represents one and two thirds or five thirds of the dark green rod. The yellow rod represents five sixths of the dark green rod.
So the difference between these two rods represents; 5/3 - 5/6 =?. If the white rods are used to represent sixths then five thirds (orange rod) can be renamed as ten sixths.
Visually we can see that the difference between the two rods is five sixths which matches the calculation: 5/3 - 5/6 = 10/6 - 5/6 = 5/6.
Let the students attempt Investigation Two (slide seven of the PowerPoint) in small groups. They will need a set of rods or access to the online tool, and a means to record their problems, e.g. squared paper. Encourage the students to create differences that they have not seen in the examples you have provided and to use non-unit fractions like two thirds or three quarters. Look for students to use two main approaches, start with the rods they are finding the difference for, or start with a one rod (whole) and experiment with different fractions of that one. Key things to look for are:
Do the students start with two lengths, made with single rods (unit fractions) or collections of rods (non-unit fractions)?
Are they able to create a one (whole rod) that works for both rods?
Can they express each rod or collection of rods as a fraction of the one rod?
Do they recognise what unit can be the common denominator and rename each fraction?
Can they find the difference and record the result as a subtraction equation?
Do they choose or create a one rod that has a length with many factors?, e.g. 18cm is useful but 19cm is not so useful.
Do they map other rods into the one rod to create unit and non-unit fractions?
Can they name the fractions they create?
Do they use smaller rods to convert each fraction to equivalent forms before finding the difference?
Can they find the difference and record the result as a subtraction equation?
As you roam choose examples of problems students create. Ask students to record their problems. Also expect them to annotate the document to explain how the model shows a particular difference problem, including words and equations.
At the end of the lesson ask several groups to present their problems for others in the class to solve. Record the three examples vertically. You should have equations that look like this:
1/2 - 3/8 = 4/8 -3/8 = 1/8
3/4 - 2/3 = 9/12 - 8/12 = 1/12
4/5 - 1/3 = 12/15 - 5/15 = 7/15
Ask the students, “Imagine someone new came to class who has not solved problems like this before. What would you tell them about how to solve any fraction difference problem?”
Give students time to record instructions/ideas they would give to the new class member. Look at these work samples to assess whether individual students have generalised how to solve difference problems.
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:
Begin by building up a number line for thirds in this way. You may want to use the online tool on an interactive whiteboard so the image can be written over.
If the blue rod is one (mark zero and one on the number line) where would one third be?
Students may now know that the light green rod is one third of the blue rod. Ask them what fractions could be marked on the number line using one third. Look for them to explain that thirds can be ‘iterated’ (place end on end) to form non-unit fractions, like two thirds. Make sure you push the iteration past one and include the fraction and mixed number ways to represent the amount (see below). Also encourage renaming in equivalent form where this is sensible, e.g. 3/3 = 1, 6/3 = 2.
Look at the space between zero and one third. Ask, “Are there any fractions that belong in this space?” Students may recognise from previous work that white rods are one ninth of a blue rod. So one ninth, and two ninths, will work. Ask students to estimate the exact location of these fractions. Their estimates can be checked using the white rods. Note that three ninths is one third so can be added to the existing number line in the same position as one third.
Ask, “So what unit fractions would exist between zero and one third?” (any unit fraction with a denominator greater than three since it will be less than one third). Note that there is no rod in the set that is one quarter, one fifth, one sixth, etc. of the blue rod. However imaging how long rods for these fractions would be is a useful activity in itself.
Look at other spaces on the thirds number line and ask students to name fractions they think exist in that space. For example, between two thirds and one are seven ninths, and eight ninths. Imagine cutting each white rod in half. “What fraction will this new rod represent?” (eighteenths). “Can we express some of these fractions as eighteenths?” “Can we find locations for numbers of eighteenths that are not showing yet?” Students might realise that having eighteenths allows them for find fractions exactly in the middle between numbers of ninths, e.g. 13/18 is exactly half way between 2/3 (6/9) and 7/9.
Show the students slide eight of the PowerPoint. You may want to give them paper copies as well. Introduce the investigation. Ask students what they notice about the diagram. “What information is present? What helpful information is missing? You need to see this problem as a riddle. There is enough information to put any fraction you want on the number line.”
“What does to scale mean?”
Let the students investigate the task in small groups. They will need a set of Cuisenaire rods or access to the online tool, and squared paper to record on. The squared paper will help them to maintain scale.
Look for the following:
Do the students realise that the red rod is one twelfth by finding the difference between two thirds and three quarters?
Do they realise that twelfths can be iterated to the left and right to find the location of zero and one?
Do they realise that the yellow rod being greater than one fifth suggests that one is slightly less than five yellow rods?
Having established the one as equivalent to 24 white rods (twenty fourths) do they position fractions using equivalence? e.g. 3/8 = 9/24, 1/8 = 4/24.
After a suitable period of investigation bring the students together to share some solutions. Highlight the correct use of scale using the one rod as a reference unit. Recognise if some students have included fractions greater than one, e.g. 4/3 = 32/24. The work sample might be used as assessment data but you may like to give students the problem on slide nine of the PowerPoint as individual assessment. Ask them to record how they worked out the location of the numbers. There are many different ways to locate these numbers but all strategies require students to show understanding of equivalence.