The purpose of this activity is to engage the students to use proportional reasoning and their knowledge of volume to solve a problem involving fractions.
This activity assumes the students have experience in the following areas:
- Measuring volumes and capacities with metric units.
- Expressing part-whole relationships as fractions.
- Finding fractions of whole number amounts.
- Using simple scale maps to describe movement and location.
The problem is sufficiently open ended to allow the students freedom of choice in their approach. It may be scaffolded with guidance that leads to a solution, and/or the students might be given the opportunity to solve the problem independently.
The example responses at the end of the resource give an indication of the kind of response to expect from students who approach the problem in particular ways.
A 600 mL jar is ⅓ full of water.
If all that water is poured into a 300 mL jar, what fraction of the smaller jar will it fill?
The following prompts illustrate how this activity can be structured around the phases of the Mathematics Investigation Cycle.
Make sense
Introduce the problem. Allow students time to read it and discuss in pairs or small groups.
- Do I understand the problem? (Students might act out the problem using arbitrary, or real containers and water.)
- What are the important words and symbols? (The meaning of measures is required, such as 600 mL meaning “600 milllitres.” Students need a sense of the size of the units. They also need to know what the symbol 1/3 represents, “one of three equal parts” in this situation.)
- What will my solution look like? (The solution will state the fraction of the small jar that the water fills with some evidence to support the answer.)
Plan approach
Discuss ideas about how to solve the problem. Emphasise that, in the planning phase, you want students to say how they would solve the problem, not to actually solve it.
- What are the maths skills I need to work this out? (Measurement and fraction knowledge and skills are needed.)
- Do I know the maths that I need or do I need to find things out?
- What tools might be useful? (Real jars, water, and rubber bands will be useful.)
- What strategies might I use? Act it out? Draw a picture? Write an equation?
- How might I work with others? What roles will we take?
Take action
Allow students time to work through their strategy and find a solution to the problem.
- Is my strategy working or should I try something else?
- Am I recording in a way that helps me solve the problem?
- Does my answer seem correct? How can I check my answer? Could I work backwards?
- Is my answer the same as others?
- Does my answer match the question?
Convince yourself and others
Allow students time to check their answers and then either have them pair share with other groups or ask for volunteers to share their solution with the class.
- What is the answer? Does it seem reasonable?
- Is my working clear for someone else to follow?
- How would I convince someone else I am correct? What drawings, models or symbols might I use to convince them?
- Could I have solved the problem in a more efficient way?
- How would my answer help me to solve other problems?
Examples of work
Work sample 1
The reasons with measures mentally. They calculate one third of 600 mL then compare the answer, 200 mL, to 300 mL. Finally they express 200/300 as an equivalent fraction, 2/3.
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Work sample 2
The student creates a regions model on grid paper. The draw the 600 mL container two thirds full. The same amount (number of squares) is mapped into the 300 mL container. They express the part-whole relationship in the diagram as 2/3.
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Work sample 3
The student uses proportional reasoning to solve the problem. As the small container is half the volume of the big container then the fraction in the small container must be twice that in the large container. This might be expressed as 1/3 x 600 = 2/3 x 300.
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