Squirt Level 4
This unit explores how the suite of learning objects, "Squirt", can be used to support students’ development of multiplicative thinking. Squirt encourages students to anticipate multiplicative measurement relationships, e.g. three measures of A fit in B, by partially filling a container and imaging how many more squirts will be needed. Since the containers are not always cylindrical it also develops ideas about conservation of volume.
use proportional thinking to find the relationships between three or more containers and represent the relationships using equations involving specific unknowns.
The most difficult learning object in this suite (Squirt: Complex Proportional Relationships) poses problems where the multiplicative relationships involve fractions. For example, 5/3 lots of container a fills container c. A difficult concept for students in interpreting these relationships is the concept of inverse. An inverse operation is one that undoes another, as subtract four undoes add four, so divide by four undoes multiply by four. Translating this into relationships between containers, if four container a’s fill container b this means that container a contains one quarter of container b, the capacity of container b divided by four.
It also shows how the object can be used to develop the transitive thinking that is the basis for the use of measurement devices. Transitive thinking applies in situations where you know a measurement relationship between two things, e.g. B is twice as long as A, and you also know a measurement relationship between one of those objects and another, e.g. C is three times longer than B. From this i nformation you determine a relationship between A and C without bringing the objects together, e.g. C is six times as long as A. When you use a measurement devise, e.g. a jug, you accept that it acts as a "third party" in that it tells you how many capacity units an object measures without the units themselves being present.
In Squirt students are also required to express the measurement relationship between the containers in equations form, e.g. 5/3 a = b. This sets some foundations for the conventions of writing equations and expressions.
Relevant Stages of the Number Framework
This suite of learning objects is suitable for students working at Stages 7-Advanced Multiplicative and Stage 8-Advanced Proportional of the Number Framework. To solve most problems in these learning objects students can rely on additive strategies. However, as the complexity of the relationships develops additive thinking becomes less efficient and multiplicative thinking becomes more compelling. To solve problems in the most difficult learning object, Squirt: three containers: complex proportional relationships, students must reason multiplicatively.
For example: Three squirts of a into b have resulted in this situation:
Students thinking additively will image the measurement of b as 3 + 3 + 3 + 3 = 12 squirts. An early multiplicative strategy would reason that 2x 3 = 6 squirts half fills b, so 2 x 6 = 12 squirts fills b completely.
In the learning objects of Squirt that involve 3 containers additive thinking is less productive. For example, given this situation:
Determining the relationship between a and c involves multiplicative thinking, e.g. 4 x 5 = 20.
In the situation where the relationships a to b and b to c are not always smaller to larger or whole number multiples of one another then more complex multiplicative relationships are required.
For example, given this situation:
To determine the relationship between a and c students must apply the concept of inverse. The relationship of a to b is the inverse of times four, i.e. divided by four. So the relationship of a to c is the combined effect of divided by four then multiplied by three, ? ÷ 4 x 3 is the same effect as ? x 3/4.
Working with the learning object with students
1. Begin with a practical problem such as "How many cups of water will fill this bottle (e.g. 2.25L)?" Encourage the students to make estimates. Pour a cup of water into the bottle and ask the students to revise their estimates. Repeat this by pouring in more cups of water and discussing why students might have changed their estimates.
2. Look for students to use an iterative unit multiplicatively in making their estimates. This means that they use a composite, like the result of pouring in three cups, as a new measurement unit. Students applying the iterative unit in a multiplicative way will create a factor by measuring how many times the iterative unit measures the bottle and using multiplication, e.g. "I found that you need five lots of three cups. That’s 15 cups."
The bottle estimate problem also raises issues of conservation of volume. As the bottle is not cylindrical and narrows at the top the uniform stepping of the iterative unit up the bottle will create error. Some students will acknowledge the smaller cross-section at the top of the bottle by adjusting the number of iterative units that will fill the top section, e.g. "The bottle is skinnier at the top so I thought it would fill more quickly. So the last part took two cups not three."
3. After the practical experience introduce Squirt: three Containers: level 2. This familiarises the students with the icons of Squirt through problems that cannot be easily solved through counting and additive reasoning. These problems should not significantly challenge level 4 students so their focus will be on the function that each icon performs (i.e. tap, plug, arrows) and on how to express the relationship by typing a measurement into the matching multiplication equation.
Explore how each problem can be solved in two ways, squirting from the smaller container into the larger container or vice versa. Squirting from the larger container into the smaller container develops fractional whole to part relationships, e.g. What fraction of container b fills container a? To express the relationship in the multiplication equation requires the students to relate part to whole, thereby applying inverse thinking.
Challenge the students to solve the given problems in the shortest number of keystrokes and mouse clicks possible. This will encourage estimation from partial filling and dissuade random guessing.
4. Explain that the goal of each problem is to find out how many times container a will go into container c, but the learning object will only let the a to b and b to c relationships be established.
Illustrate how these relationships can be found using the same strategies as for Squirt: two containers: level 2 then challenge the students to find the a to c relationship. Encourage the use of multiplicative strategies although you may need to show this through repeated addition. For example given these relationships:
a to b goes four times and b to c goes three times, additive strategies would involve squirting a four times to fill b, squirt b into c, squirting a four times to fill b, squirt b into c, and squirting a four times to fill b, squirt b into c. (yes, that is repetitive!) Multiplicative strategies would involve finding four a’s fill b, three b’s fill c, so 3 x 4 a’s fill c.
5. Provide students with no more than three examples for the three container learning object working with a partner. Introduce Squirt: three containers: complex proportional thinking by working through a few problems collaboratively. If necessary address the concept of inverse through simplifying the problems to two container relationships. The most likely cause of difficulty is likely to be expression of the multiplicative relationships.
Consider this situation in which the a to b and b to c relationships have been established.
6. Students may struggle with the idea that the quantities involved are not specified. Lack of closure like this presents a cognitive obstacle for many students. One way to approach this is to allocate a quantity to container a and look at the relationships that occur from that. Since a holds five times as much as b choose a multiple of five, say ten litres.
Students can then establish what containers b and c hold from the known relationships, i.e. b must hold 2 litres and c must hold 6 litres. Making bars of these amounts from cubes might help students to see the a to c relationship.
Ask students what fraction of container c is held by container a. Encourage students to express this in the simplest fraction possible, i.e. as .
7. After the students have worked through sufficient examples collaboratively allow them to work independently in pairs on Squirt: three containers: complex proportional relationships.
Students working independently with the learning object
As students work on problems from Squirt: three containers: complex proportional relationships discuss how they might record the problems they solve. For example, they may draw diagrams such as:
This will encourage the students to consider the inverse relationships involved.
For students who need support with the multiplicative relationships involved build cube models of several problems to support them.
For example, model this problem:
The learning object only requires the students to express the a to c relationship in one way, e.g. a = c. Ask the students to record the relationship using the inverse, i.e. a = c.
Students working independently without the learning object
Look for students to develop the generalised strategy of how to solve problems of this type.
The generalised strategy should involve acceptance of a, b, and c, as unknowns, requiring the students to accept lack of closure. This begins with looking for similarity and difference in three examples solved from the learning object. Students should note that the numbers involved differ but the similarity is that multiplying the a to b and b to c factors gives the a to c relationship.
Get them to extend this to situations involving three or more containers without the use of the learning object.
1. Problem One: Five red containers fill a yellow container. Three blue containers fill a yellow container. What fraction of a red container fills a blue container?
Pose connected problems that have the same a to c relationship but provide inverse a to b and b to c relationships. Students need to generalise that the effect of ? X 5 ÷ 3 has the same effect as ?÷ 3 X 5.
2. Problem Two: Three yellow containers fill a red container. Five yellow containers fill a blue container. What fraction a red container fills a blue container?
3. Ask the students to pose similar problems for others to solve. Ask them to provide possible solutions on the back of the problem page. This will motivate them to solve the problems themselves and will constrain the difficulty of the problems. Share the problems in small groups so students can discuss their strategies. The learning objects supports students in determining the fractional relationship between a and c by displaying the relative size of the containers. These problems are more difficult if the relationships are given in words or, even more difficult when the relationships are given as equations.
Consider the above this problem:
a = 3b, 4b = c, What is the relationship between a and c expressed as an equation?
Encourage the students to work from the equations to models or diagrams that will support their reasoning:
Strip diagrams are useful representations for problems of this sort.