This unit explores how the properties, that hold for multiplication and division of whole numbers, also apply to fractions (commutative, associative, distributive, and inverse). The operations of multiplication and division can be developed using a length model (Cuisenaire rods). While the size of the referent one (whole) is fixed at first, students need to be able to change the referent during the physical process.
‘Fractions as operators’ is one of Kieren’s (1994) sub-constructs of rational number, 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. place one half where 2 1/2 belongs on a number line showing zero to five.
Fractions can operate on other fractions, and the rule for fraction multiplication can be generalised, a/b x c/d = ac/bd. The commutative property holds since c/d x a/b = ac/bd as do the distributive and associative properties, though the former is mostly applied in algebra rather than number calculation. In order that the inverse property holds, i.e. division by a given number undoes multiplication and vice versa, the following must be true:
a/b x c/d x d/c = a/b because c/d x d/c = 1 so division by c/d must be equivalent to multiplication by d/c.
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 for students to develop if they are to master rational numbers and other associated forms of proportional reasoning. By re-unitising it is meant that students can flexibly define a given quantity in multiple ways by changing the units they attend to. By norming it is meant that the student can then act with the new unit. 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.
While the algorithm of ‘invert and multiply’ is easy to learn and enact, understanding why the rule holds requires complex thinking. Yet the ability to re-unitise and norm are central to fluency and flexibility with rational number, and therefore necessary for algebraic manipulation. Consider a simple case of division of a fraction by a fraction, 2/3 ÷ 1/2 = 4/3.
If the dark green rod is defined as one then the crimson rod represents two thirds and the light green rod represents one half.
If we adopt a measurement view of division rather than a sharing view the problem; 2/3 ÷ 1/2 = ? translates into “How many light green rods fit into the crimson rod?” So the light green rod becomes the new referent. The result is that one and one third light green rods make a crimson rod. Note that; 1 1/3 = 4/3.
As a check that the inverse property holds with our answer, we might calculate 4/3 x 1/2 = 4/6 = 2/3 in the same way that we could check 24 ÷ 3 = 8 by calculating 8 x 3 = 24.
This unit is set for students to learn, and practice, outcomes at Level 5 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), whakarea (multiplication) and taurunga (numerator) could be introduced in this unit and used throughout other mathematical learning.
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. However, other units in the Cuisenaire rod fractions collection are available at level 3 and level 4.
Set up this diagram:
Note that your students might not relate multiplication of whole numbers, e.g. 3 x 8 = 24, with ‘of’ as a function. If this understanding is not evidenced, spend some time developing it. See the Level 3 unit Multiplying Fractions for more information.
Identify, and make note of, any gaps in knowledge or misconceptions. Use these as the basis for a teaching session with the relevant students, or for a whole-class review.
The purpose of this session is to consolidate understanding of the multiplication of fractions, and to see if the properties of whole numbers under multiplication hold for fractions as well.
5 x 3 = 3 x 5 4 x 2 x 3 = 3 x 4 x 2 3 x 7 = 3 x 2 + 3 x 5
All three equations are correct as they are examples of the properties of whole numbers under multiplication. Look for students to explain that the first equation is the commutative property, the order of the factors does not affect the product. The second equation is less obvious as it applies the associative property which is about the pairing of three of more factors not affecting the product. Since multiplication is a binary operation only two factors can be operated on at once. Students may recognise that in the third equation seven is distributed into five plus two.
Replacing the blue rod with the light green (one quarter) and dark green (one half) gives this:
Finding two thirds of each rod (light green and dark green) gives:
In this session the purpose is to find the quotient of two fractions by division. A measurement view of division is used rather than a sharing view.
The model should now look like this:
Express both fractions as equivalent fractions:
5/10 = 1/2 (Yellow rod) and 8/10 = 4/5 (Brown rod) so 1/2 ÷ 4/5 = 5/10 ÷ 8/10.
Since the units are all eighths the problem is just like “How many lots of eight somethings fit into five somethings.” Be aware that students might be unaware of the quotient rule for rational numbers, a ÷ b = a/b, for example 5 ÷ 8 = 5/8.
Invert and multiply:
The most common algorithm, and that needed for algebra, derives from, 1 ÷ a/b = b/a. So the answer is the reciprocal. Any change to the dividend, one, is just a scalar so in general, c/d ÷ a/b = c/d x b/a.
The aim of this session is to consolidate students’ connection between multiplication and division of fractions. To do so we keep the representation the same, Cuisenaire rods, but we expect a considerable amount of re-unitising and norming with the units that are created.
All of the above questions treat the orange rod as one.
Students might publish their problems as a word document or PowerPoint so that the sets can be published. Swapping problems among groups is an excellent way for students to check if their reasoning is correct as well as providing practice at re-unitising.
Dear families and whānau,
This week, students will be learning about multiplication and division of fractions, for example “What is two fifths of one half?” We will be using some materials called Cuisenaire rods which are lengths of plastic or wood. Here is a Cuisenaire rod model as an example.
Your son or daughter should be able to identify the orange rod as one and name the fractions for the other rods. The yellow rod is one half and the red rod is one fifth, in this case. Also the red rod is two fifths of the yellow rod.
Feel free to explore and use this online Cuisenaire rods: https://mathsbot.com/manipulatives/rods
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/cuisenaire-rod-fractions-level-5 at 8:37pm on the 26th February 2024