The purpose of this series of lessons is to develop understanding of the connection between division and fractions. In the unit both types of division, sharing and measurement, are explored to establish a need for fractions and to develop generalisations about division and fractions.
Students need to understand and use the appropriate mathematical language for the numbers and symbols in division equations. Students need to understand and apply addend plus addend equals the sum for addition, and factor (multiplier) times factor (multiplicand) equals product for multiplication. Mathematical language for division allows for clear definition of the symbols, i.e., the meaning of those symbols, and allows for succinct expression of generalisations. In division, the dividend is partitioned by the divisor and this results in the quotient.
a (dividend) ÷ b (divisor) = c (quotient)
Division models the operation of equal partitioning in two different, but connected, types of situations. The first situation is sharing or partitive division, which often involves answering the question, “If a objects are equally shared among b parties, how many objects does each party get?” The second form of division is the measurement interpretation (sometimes referred to as quotative division). Here the number in the group, or size of each measure is known. That group or quantity becomes the unit of measure. The unknown in measurement division, is the number of those units that can be made from a given amount. This interpretation is often associated with repeated subtraction, as one way to solve this kind of problem is to keep removing the given equal groups (measures) from the whole amount, until nothing is left. Counting each repeated subtraction gives the solution to the question.
Measurement division situations are the easiest contexts for division by fractions. Problems with division by a fraction involve finding how units of a given (i.e. fractional size) ‘fit within’ another fraction quantity. For example, 1 1/2 ÷ 1/4 is interpreted as how many units of one quarter measure 1 1/2?
When students carry out the operation of division with whole numbers, their expectation is that the quotient will be smaller than the dividend, for example, 20 ÷ 2 = 10, and sometimes smaller than both the dividend and the divisor, for example, 20 ÷ 5 = 4. It is a conceptual shift for students to come to understand that when they are dividing a fraction by a fraction the quotient may be larger than both the dividend and the divisor, for example 1/2 ÷ 1/4 = 2.
There are many real-life occasions when we divide fractions by fractions, however, the fractions are frequently presented as decimals in measurement situations. For example, if you pay $15.00 for three quarters of a kilo of prime lamb and you want to know the price per kilo, that will involve you dividing a whole number by a fraction, i.e. 15 ÷ 3/4 =$20. Providing simple fractional problems in context, and asking students to consider and create their own contexts, is helpful to their connection of mathematics to the real world.
Links to the Number Framework
Stages 7 - 8
This unit supports and complements the learning activities in Book 7 Teaching Fractions, Decimals and Percentages.
Students can be supported through the learning opportunities in this unit by differentiating the nature and complexity of the tasks, and by adapting the contexts. Ways to support students include:
Task 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. Fractions can be applied to a wide variety of problem contexts, including making and sharing food or land from inheritance, constructing items, travelling distances, working with money, and sharing earnings. The concept of equal shares and measures is common in collaborative settings. Be conscious that fair sharing is not necessarily equal in whānau or community groups, e.g., bigger families might get more of a share. In te ao Māori, the sharing of food with visitors reflects manaakitanga. Hākari (feasts) are often included as a part of the celebrations of special events. In Samoan culture, food is an important part of events (e.g. weddings, funerals, church celebrations). Food, like other forms of wealth, is often shared evenly amongst the aiga (extended family). The learning in these sessions could be linked to learning about the tikanga (customs) of sharing food in the different cultures reflected in your classroom. In turn, this could foster greater connections and understanding amongst your learners.
Te reo Māori vocabulary terms such as hautau waetahi (unit fraction), hautua (fraction, part of a whole), whakarea (multiply), whakawehe (division, divide), toharoha (share, distribute), rautaki tohatoha ōrite (equal sharing strategy) and ōrite (equal) could be introduced in this unit and used throughout other mathematical learning.
Consider adapting this problem to suit the context of your class. It could be reframed as 84 students are taking part in Polyfest this year. They stand in 6 rows, with an equal number of students in each row. How many students stand in each row?
Note that most of the fractions in these teachers’ notes are displayed as typed text, meaning that the vinculum (the line between the numerator and the denominator) is diagonal rather than horizontal. When writing fractions on the board or in students’ books it is recommended that you use a horizontal vinculum.
Activity 1
Activity 2
In this session students explore division as sharing into equal parts. They learn that sharing situations can also be answered using measurement division.
Activity 1
Activity 2
In this session, division as equal sharing is applied to problems with continuous whole, rather than the sets situations used in the previous two sessions. Students come to understand that quotients can sometimes be fractions, the result of division in which ones (wholes) are not sufficiently accurate.
Use Slide One of PowerPoint 4 to present this sharing (partitive) problem:
Problem | dividend | divisor | quotient |
I have this much pizza (4½). I want to share the pizza equally among the three people at my party. How much pizza should each person get? |
Record the discussion using the table used in problem one.
Problem | dividend | divisor | quotient |
I have 4½ metres of cloth. Each bag is made from ¾ of one metre of cloth. How many bags can I make? | 4 ½ or 18/4 | 3/4 | 6 |
In this session students explore division where the dividend is a whole number and the divisor is a fraction, e.g. 1 ÷ 2/5 = 5/2 = 2 ½. Fraction strips (See Copymaster 3) are used as the physical representation since length is the simplest attribute for students to use. The problems involve measurement division since a fractional number of shares makes no sense. For example, 1 ÷ 2/5 means “How many quantities of two fifths equal one?”
Dear parents and whānau,
We have been exploring the connection between division and fractions in class.
You might enjoy solving these two problems together to see what they have been learning.
Problem One: Four people equally share three doughnuts. How much of one doughnut does each person get?
Problem Two: If each person needs three eighths of a big pizza, how many people can be fed with one and one half pizzas?
Printed from https://nzmaths.co.nz/resource/dividing-fractions-0 at 9:25am on the 29th April 2024