Cups and Cubes
In this unit students explore the use of cups and counters as a model to analyse the effects of operations rather than focusing on specific numbers.
- use a ‘cups and cubes’ model to describe relationships
Queensland researcher, Cyril Quinlan, published the use of cups and counters as a model for algebraic thinking in 1995. Quinlan used the model to teach students about the manipulation of algebraic expressions.
The use of cups filled with “any chosen” number of counters supports students’ conceptual development towards seeing letters as variables rather than as specific unknowns (Kucheman, 1981). Research by Lauren Resnick (1992, 1993) suggested young children could understand the effects of simple operations on protoquantities. Protoquantities refer to assumed quantities that are not measured, like a container of counters or a jug of liquid.
This unit seeks to develop the use of cups and counters as a model for students to analyse the effects of operations rather than focusing on the resulting numbers.
Links to Numeracy
This unit provides an opportunity to focus on the strategies students are using to solve number problems, particularly in the domains of addition and subtraction and multiplication and division.
Session two provides an opportunity for students to consider the reversibility of operations as they devise their own “think of a number” problems. Operations involved may include doubling and halving, addition and subtraction or a combination of these operations. Encourage them to focus on this by asking
Which steps are related in your problem?
Are there parts of your problems that “undo” each other? How do you know?
Tell us about the ways you have combined numbers.
Tell us about the ways you have separated numbers.
As students share their thinking with each other they will be exposed to a variety of different strategies.
In sessions three to five students will be mentally partitioning numbers as they work with cubes and cups. Encourage them to think about the ways they are manipulating numbers and the strategies they are using.
What combinations of numbers are you using as you work with the cups? Why?
What other ways could those numbers be combined? How do you know?
For example,
What numbers of cubes for each colour of cup make each collection equal?
As students consider options probe their thinking
What numbers are the same in both collections?
Which numbers are different in the collections?
What needs to be the same for the totals of each collection to be equal?
How can you arrange the cubes so there are the same numbers in each collection? How do you know?
Are there any other arrangements that will work?
For this problem to be solved blue and green must total yellow plus three. There are many number combinations that will satisfy this condition. Once this has been established students can make a table of these and look at the relationships between the different combinations.
|
Blue |
Green |
Yellow |
|
3 |
4 |
4 |
|
4 |
5 |
6 |
|
5 |
6 |
8 |
|
6 |
2 |
5 |
|
7 |
2 |
6 |
What relationships can you see between these numbers?
How did you work that out?
What other relationships can you see?
relationships, partitioning, combinations, reversibility, inverse operations, table of values, consecutive numbers, equations, equivalence, rules
Session 1
Put two plastic cups, of different colours, on a table. Tell the students that you are going to put 12 cubes into the cups. Ask, “How many cubes might be in each cup?”
Draw up a table of values to organise the students’ responses:
|
Blue |
Green |
|
0 |
12 |
|
1 |
11 |
|
2 |
10 |
|
3 |
? |
|
|
|
Tell the students to draw a graph to show the relation between the number of cubes in the blue cup and how many are in the green cup. Ask them to predict what the graph might look like.
After they have plotted the graph ask the students to explain why they think the points on the graph lie along a downward sloping straight line.
Using a range of different coloured cups and different number of cubes pose similar problems asking the students to come up with a table of values and graph to represent the relation. The students can model the situation with actual cups and cubes if necessary to check the values. Examples might be:
|
|
Each blue cup has the same number of cubes. The number of cubes in the yellow cup equals the number in both blue cups. |
|
|
The number of cubes in the yellow cup equals the total number of cubes in the blue cups. |
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Note that in some of these relations there are an infinite number of values that could be put in the table. For example, in the Y B B B problem, the table could read:
|
Yellow cup |
Blue Cup |
|
0 |
0 |
|
3 |
1 |
|
6 |
2 |
|
9 |
3 |
Ask the students to come up with rules that describe the relation. For the relation above this might be, “There are three times as many cubes in the yellow cup as there are in each blue cup.” (Algebraically this might be recorded as y = 3b, though this is not a learning intention for students at this level).
Session 2
Pose “think of a number” problems for students that result in a magical answer at the end. Encourage the students to reason why the magic occurs. The processes involved can be modelled using cups and cubes. Begin with a problem where the reversing of operations is obvious.
For example:
|
Instructions |
Cups and cubes model |
Algebraically |
|
Think of a number |
|
n |
|
Add five |
|
n + 5 |
|
Take away your starting number |
|
5 |
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Develop more complex examples. Encourage the students to model the steps with cups and cubes to explain the effect of the operations.
For example:
|
Instructions |
Cups and cubes model |
Algebraically |
|
Think of a number greater than 5 |
|
n |
|
Take away four |
|
n – 4 |
|
Double the answer |
|
2 (n- 4) = 2n – 8 |
|
Add ten |
|
2n + 2 |
|
Halve the answer |
|
n + 1 |
|
Take away your starting number |
|
1 |
|
Your answer is one!
|
|
|
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|
Instructions |
Cups and cubes model |
Algebraically |
|
Think of a number |
|
n |
|
Add three to the number |
|
n + 3 |
|
Double the answer |
|
2(n + 3) = 2n + 6 |
|
Take away four |
2n + 2 |
|
|
Halve the answer |
|
n + 1 |
|
What number did you start with? Your final answer was one more than your starting number. |
|
|
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Invite the students to develop their own “think of a number” problems. These can be set as examples for other students to explain.
Session 3
Investigate number properties using cups and cubes to generalise why the properties hold.
Begin with simple properties such as:
In each collection of five cups there are 4 cubes in each red cup and 7 cubes in each blue cup. Which collection will have the most cubes?
Look for students to realise that structurally the collections are the same. This can be provoked by putting different numbers of cubes into the cups, e.g. 3 cubes in each red cup, 8 cubes in each blue cup. The equality of the collections is preserved no matter what numbers are used.
Get students to connect the structural equivalence with the relation ideas from session one. Use problems like:
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Each cup of a given colour, in both collections, must contain the same number of cubes, e.g. all reds hold 4 cubes, all yellows hold 3 cubes, etc.
What numbers of cubes could be put in each cup so that the two collections contain the same total number of cubes?
Students might realise that the number of cubes placed in the red cups is irrelevant to the problem since it is the equivalence of two yellow cups to one blue cup that determines whether or not the collections are equal. They are likely to arrive at this conclusion by experimenting and looking for commonality among the solutions.
Recording their solutions systematically will help:
Solutions for r = 1 Solutions for r = 2 Solutions for r = 3
y = 0 so b = 0 y = 0 so b = 0 y = 0 so b = 0
y = 1 so b = 2 y = 1 so b = 2 y = 1 so b = 2
y = 2 so b = 4 y = 2 so b = 4 y = 2 so b = 4
y = 3 so b = 6 .... ....
Explore other equivalence problems such as:
What numbers of cubes for each colour of cup will make each collection equal?
What is in common with these solutions?
(Two reds must equal one blue plus six)
What numbers of cubes for each colour of cup make each collection equal?
(One green plus one blue must equal one yellow plus three)
Session 4
Connect the students’ understanding of equivalent structures with rules to describe relations. Pose problems like:
Petra and Clive are looking at this matchstick pattern:
They have worked out that 12 triangles take 25 matches to make. Their teacher has challenged them to work out how many matches 100 triangles might take to make.
What rules can you find to help Petra and Clive so they do not have to build 100 triangles?
Students are likely to come up with different direct rules that describe the relation between the number of triangles and the number of matches. For example:
Ask the students to explain where the numbers come from in each person’s rule. Check to see that all three rules give the same answer to the number of matches. Use the explanations to model the structure of each rule using a cup to represent the chosen number of triangles.
Take one off the number of triangles (99), Take the number of triangles (100),
double it, and add three. Double it, and add one.
(99 is the number of triangles after the first one)
Take the number of triangles (100), multiply by three, then take away one less than the number of triangles (99). Note 99 is the number of joins of two triangles and taking away one less than n is like taking n away but leaving one behind.
With the cups and cubes model it is easier for students to appreciate the structural similarity of the rules in that one model can be converted to the others.
Provide the students with a related matches problem and ask them to find as many rules as they can. Tell them to then explain the equivalence of the rules using cups and cubes models. For example:
How many matches would it take to make 20 houses in this pattern?
Rules might include:
19 x 5 + 7 (seven for the first house, five for each house after that)
20 x 5 + 2 (five for each house plus two matches to start)
20 x 7 – 19 x 2 (seven for each house less two matches for each join)
Session 5
Get the students to apply the structural strategies they have to problems that involve proof. Begin with simple properties of consecutive numbers.
Take any three consecutive numbers, e.g. 3, 4, 5.
If you add the numbers (3 + 4 + 5 = 12), this equals three times the middle number
(3 x 4 = 12).
Does this work for any set of three consecutive whole numbers? Why?
Invite students to explain why the rule holds for any set of consecutive whole numbers, and to model their explanation with the cups and cubes model. Let n be the first number:
Students may recognise that both collections total three lots of the first number plus three. Algebraically this could be written as, n + (n + 1) + (n + 2) = 3n + 3 = 3 (n + 1)
Pose another problem involving consecutive numbers and ask the students to explain why the property holds irrespective of which four numbers are chosen.
Choose any four consecutive odd numbers, e.g. 3 + 5 + 7 + 9.
Add the two outside numbers, e.g. 3 + 9 = 12.
Add the two inside numbers, e.g. 5 + 7 = 12.
Are the sums always equals no matter what consecutive odd numbers you choose? Why?
Would this work for consecutive odd numbers? Why?
Pose other more complex problems that can be modelled with cups and cubes.
For example:
Choose any two single digit numbers, e.g. 3 and 8
(modelled by two different cups)
Use the digits to make two different two-digit numbers,
e.g. 38 and 83, 38 = 10 x 3 + 8, 83 = 10 x 8 + 3
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