Work backwards to solve problems Apply exhaustive thinking Use arithmetic operations where calculators are of little help Demonstrate multiple solutions |
Suggested time allotment
One class period
Student social organization
Students working alone or in pairs, after short teacher introduction
Task
Assumed background: The problem assumes that the students have had some experience in using parentheses with arithmetic operations.
Presenting the task: The teacher explains that in bowling the objective is to knock down 10 pins with one or two throws of a bowling ball. Knocking down all 10 pins with one throw is called a strike. Knocking down all the pins in two throws is called a spare. The game of Bowl-A-Fact is similar to bowling, except instead of throwing a bowling ball, one tosses three number cubes (numbered from 1 through 6) and makes as many numbers from 1 to 10 as possible by adding, subtracting, multiplying or dividing the numbers showing on the number cubes. Each number showing on a number cube must be used exactly once on one side of an
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OCR for page 65
Measuring Up: Prototypes for Mathematics Assessment
Bowl-A-Fact
Work backwards to solve problems
Apply exhaustive thinking
Use arithmetic operations where calculators are of little help
Demonstrate multiple solutions
Suggested time allotment
One class period
Student social organization
Students working alone or in pairs, after short teacher introduction
Task
Assumed background: The problem assumes that the students have had some experience in using parentheses with arithmetic operations.
Presenting the task: The teacher explains that in bowling the objective is to knock down 10 pins with one or two throws of a bowling ball. Knocking down all 10 pins with one throw is called a strike. Knocking down all the pins in two throws is called a spare. The game of Bowl-A-Fact is similar to bowling, except instead of throwing a bowling ball, one tosses three number cubes (numbered from 1 through 6) and makes as many numbers from 1 to 10 as possible by adding, subtracting, multiplying or dividing the numbers showing on the number cubes. Each number showing on a number cube must be used exactly once on one side of an
OCR for page 65
Measuring Up: Prototypes for Mathematics Assessment
equation. For each number that can be obtained, the corresponding circle is shaded. (This indicates a pin knocked down.)
Student assessment activity: After explaining the rules of Bowl-A-Fact, the teacher should play one game with the whole class. Pretend that the number cubes on the first toss were 3, 4, and 6. Individual students can devise number sentences to knock down pins that have been drawn on the board, as shown below.
Of course the pins will probably be knocked down in some order different from the one shown. Also, depending on a child's strategy, the pin number could be on the right or left side of the equals sign. For example, one child might ask, "What can I do with 3, 4 and 6?" and come up with "4 + 6-3 = 7," while another might ask, "Is there any way to knock down the 7 pin?" and write the equation "7 = 4 + 6-3.''
Note, too, that there are other ways of getting many of the pin numbers; this should be explicitly pointed out.
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Measuring Up: Prototypes for Mathematics Assessment
Now the teacher should ask the class to pretend that the next roll is 3, 3, and 4. This is sufficient to knock down the remaining pins, thus getting a spare.
Students are given three number cubes and a copy of the following activity sheets.
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Measuring Up: Prototypes for Mathematics Assessment
Name _________________________________________ Date ______________
1. Play a game of Bowl-A-Fact. Roll the three number cubes, and write the numbers you get in these spaces: _____ _____ _____. (Roll again if you get 3, 4, 6 or 3, 3, 4) Use the numbers to knock down as many pins as you can. (For this part and the remaining parts, record your figuring in the spaces provided.)
Did you get a strike? ______
2. If you got a strike, go on to the next page. If you did not get a strike, try for a spare. Roll the three number cubes and write them here: _____ _____ _____. Now use the new numbers to knock down as many of the remaining pins as you can.
____________________________
____________________________
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____________________________
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Measuring Up: Prototypes for Mathematics Assessment
3. Suppose you play again and the numbers you roll are 2, 3, and 6. Knock down as many pins as you can using these numbers.
Did you get a strike?
4. Suppose your next roll is 1, 3, and 5. Try to knock down the rest of the pins (get a spare) using these numbers.
____________________________
____________________________
____________________________
____________________________
____________________________
____________________________
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____________________________
____________________________
____________________________
Did you get a spare?
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Measuring Up: Prototypes for Mathematics Assessment
5. What pins can you knock down with this toss? 1, 2, and 4.
Did you get a strike? ______
____________________________
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Measuring Up: Prototypes for Mathematics Assessment
Rationale for the mathematics education community
Bowl-A-Fact illustrates a purely arithmetic task in a setting that requires students to generate their own equations rather than merely compute answers to a given set of problems. It tests for facility with the number facts for all four operations using a form of exhaustive thinking. Students have to work backwards, asking themselves how they can combine the three given numbers to create the pin numbers.
What is the role of calculators in this activity? Calculators should be available to the students, as always, but it will soon become clear to them that calculators have very limited value in this situation. In fact, using a calculator to generate an equation to knock over pins is an extremely inefficient way to approach the task. Thus one reason for using these kinds of tasks is to sharpen students' perception of when calculators are useful and when they are not.
Another reason for including the task is that it so readily lends itself to subsequent instruction on combinatorics and probability, mathematical content not usually included in the fourth grade curriculum. (See the Variants and extensions section below.)
It is important to note that — with appropriate teacher guidance — the task illustrates that there can be more than one way to solve a problem; for instance, using 3, 4, and 6, one can knock over pin 2 by writing "2 = (3 × 4) ÷ 6" or by writing "2 = 4 -(6 ÷ 3)." This will probably not be apparent to a student who is working alone, because he or she would have little need to consider alternative solutions for a pin value. That is, once a pin is knocked down, there is no point in doing it again. Instead, the idea should be conveyed in the teacher's introduction to the game, or in post-assessment follow-up discussions.
Task design considerations: This task is very much like the kind of game that would be useful as an instructional activity. Two features of the task are designed so that it is useful also as an assessment that could be used for comparative purposes.
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Measuring Up: Prototypes for Mathematics Assessment
One is that the teacher introduction will be exactly the same for all students and that introduction will illustrate all aspects of the game. The teacher specifies exactly what numbers are to be used in the demonstration for the whole class: 3, 4, and 6 for the first roll, followed by 3, 3, and 4. These numbers are carefully chosen to illustrate (a) what a spare and a strike are; (b) what to do if a number appears more than once among the three cubes; and (c) the use of division.
The second technique that is used because the task is intended for assessment is to allow children to play the game using whatever numbers they happen to roll in questions 1 and 2, followed by some predetermined rolls in the remaining questions. The results of questions 1 and 2 could be used by the teacher for diagnostic purposes, but they are not considered to be part of the formal assessment, and they are not discussed in the protorubric. (The reason, of course, is that the difficulty of the task depends on the numbers one happens to roll.)
Variants and extensions: There are many variants of the task, using different numbers on the cubes, different numbers on the pins, different numbers of cubes or of pins, different allowable operations (beyond addition, subtraction, multiplication, and division), and so forth. For example, one might want to try Super Bowl-A-Fact using 15 pins (from 1 through 15), four number cubes (each numbered from 1 through 6, as before), and change the rules so that the player can use the numbers from either three or four of the cubes. This is a good opportunity for students to make up their own variations on the game.
The game can be extended to a consideration of some questions in combinatorics and probability. For instance, "How many different ways are there to make a strike? What's the worst possible roll — or is there more than one? How many different rolls are possible with three number cubes? What's the probability of rolling a strike?" These questions lend themselves to longer projects; groups of students can organize the results of their investigations and present them to the rest of the class.
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Measuring Up: Prototypes for Mathematics Assessment
Protorubric
General notes: For the purposes of this protorubric, questions 1 and 2 are not included, for the reasons cited above.
In questions 3 and 4, here are the possibilities: With 2, 3, and 6, one can knock down the pins 4, 5, 6, 7, and 9. With 1, 3, and 5, one can knock down the pins 1, 2, 3, 7, 8, 9, and 10. Hence it is possible to make a spare, but not a strike, with the given rolls.
In question 5, however, one can knock down all the pins with 1, 2, and 4, so it is possible to get a strike.
Notice that one can obtain useful diagnostic information from children's incorrect equations. Consider, for example, these three incorrect equations (which might be given on question 1 or 2):
The child who writes (a) may not understand how and when parentheses are used; the child who writes (b) may have a problem with basic number facts; he may think either that 6 ÷ 2 is 2 or that 3 + 4 is 6. The child who writes (c) may fully understand that 6 divided by 2 is 3, but be confused about the order in which the operands are conventionally written. Errors such as these can be useful clues in understanding children's mathematical thinking.
Characteristics of the high response:
A high response shows flexibility of thinking. The use of parentheses is accurate, and the numbers obtained are correct. A spare is obtained in questions 3 and 4, and a strike in question 5.
The highest level of response would knock down all five possible pins in question 3, even though pins 7 and 9 could also be knocked down in the second roll.
Note that the questions do not ask for multiple ways of knocking down pins. The fact that
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Measuring Up: Prototypes for Mathematics Assessment
there is often more than one way to knock down a pin may be an interesting and attractive feature of this activity. But from the viewpoint of playing the game, once is enough!
Characteristics of the medium response:
The responses to questions 3, 4, and 5 include at most two incorrect equations, and at most two possible pins are omitted.
Characteristics of the low response:
The low response shows some understanding of basic arithmetic; at least five equations are written correctly. But there is apparently little flexibility of thought, because many of the pins that could be knocked over are left standing.
Reference
Shoecraft, Paul J. (April, 1982). "Bowl-A-fact: A game for reviewing the number facts," Arithmetic Teacher.