Squirt Level 3

Working with the learning object with students

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.
Look for students to use an iterative unit in making their estimates. This means that they use a composite, like the result of pouring in three cups, as a new measurement unit. Additively they may step this iterative unit up the bottle to estimate a total capacity, e.g. 3 + 3 = 6, 6 + 3 = 9, etc. 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."
After the practical experience introduce Squirt: two Containers: level 2. This familiarises the students with the icons of Squirt through problems that can be solved through counting and additive reasoning. These problems should not challenge level 3 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 in 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 reversing their thinking. Once students are familiar with Squirt: two containers: level 2, introduce Squirt: three containers: level 1.

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.

Provide no more than three examples for the three container learning object then allow students to explore the problems with a partner.

Students working independently with the learning object

As students work on problems from Squirt: three containers: level 1 discuss how they might record the problems they solve. For example, they may draw diagrams such as:

For students who need support with the multiplicative relationships involved build cube models of several problems to support them.

For example:

When students have completed several problems and recorded their solutions using diagrams, meet together as a group to discuss their solutions.
How did you work out how many a’s were in c?
Why did you do it that way?
Did anybody use a different method?

Students working independently without the learning object

Look for students to develop the generalised strategy of how to solve problems of this type. 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.
For example: I have a blue bottle, a red bottle, a yellow bottle and a green bottle. Three blue bottles fill a red bottle. Two red bottles fill a yellow bottle, and five yellow bottles fill a green bottle. How many blue bottles fill a green bottle?

Pose problems where the unknown comes in a different place or in which the relationships are reversed. For example:
Problem One:  Three red bottles fill one yellow bottle. Nine red bottles fill a blue bottle. How many yellow bottles fill a blue bottle?

Problem Two:   Eight red bottles fill one yellow bottle. Two blue bottles fill one yellow bottle.
How many red bottles fill a blue bottle?

The factor relationships in these problems are:
Problem 1: 3 x ? = 9
Problem 2 : 8 ÷ 2 = ?(This requires students to find the inverse operation, i.e. inverse of x 2 is ÷ 2)

Finally ask the students to pose problems for others to solve. Tell 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.