This unit introduces the fact that fractions come from equi-partitioning of one whole. Therefore, the size of a given length can only be determined with reference to the size of one whole. 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 (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 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 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 conceptual change compared to understanding of 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. Some examples are:
A confusion students sometimes have is when fractions should be treated as numbers and when they should be treated as operators. A particular case us when creating numbers lines, e.g. they often place 1/2 where 2 1/2 belongs on a zero to five number line.
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. By re-unitising she meant that students could flexibly define a given quantity in multiple ways by changing the units they attended to. "Norming" is the process of operating with 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 requires 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 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 exist has an infinite set of number names.
This unit is set for students to learn, and practice, outcomes at Level 4 of mathematics in the New Zealand Curriculum. Differentiation involves simplifying or adding more challenge to the tasks in the following ways:
The context for this unit is not real life. However, a story shell such as construction beams, waka lengths, or steps, might be used if there is potential to motivate students. Most students will enjoy the opportunity to work with Cuisenaire rods.
Te reo Māori vocabulary terms such as hautanga (fraction), hautau ōrite (equivalent fraction), tauira (patterns), tauraro (denominator) and taurunga (numerator) could be introduced in this unit and used throughout other mathematical learning.
Students may have limited experience with using Cuisenaire rods. Their lack of familiarity with the rods is a significant advantage as they will need to imagine splitting the referent one to solve problems. 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.
Consider providing a story context to frame the learning in this session (e.g. construction beams, waka lengths, length of a running track). Links will be engaging when they reflect the cultural diversity and/or learning interests of your students. This context could stay consistent in every session. Alternatively, you could introduce each session with a new relevant context to enhance students’ understanding of the mathematical concepts covered in these lessons, to real-world situations.
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.
The aim of this session is to develop students’ mental number line for fractions and develop their capacity to use the number line to find differences between fractions. Inclusion of fractions with whole numbers on the number line requires some significant adjustments. These adjustments include:
Dear parents and caregivers,
This week your son or daughter will be learning about fractions, for example, 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 son or daughter 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:
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/cuisenaire-rod-fractions-level-4 at 8:36pm on the 26th February 2024