Prepare to Read Story. “Cipher Force!” is a silly story about a team of four superheroes and their sidekick. The superheroes—Multiply by Zero, Divide by Zero, Add Zero, and Subtract Zero—embody basic operations with zero. Several adventures show students what happens when you add, subtract, multiply, and divide with zero.
In the story, the superhero Divide by Zero attempts to carry out two divisions by zero. To make these episodes more accessible, discuss similar problems that do not involve zero before reading the story.
Tell students you will be reading a story about superheroes whose special strengths involve performing math problems with zero. In the story, Divide by Zero invents a roller coaster. Pose the following question and ask students to solve the problem using any method they wish.
- 24 people need to ride the roller coaster. How many roller coaster cars are needed if each car can carry 4 people?
Have students share their solutions and strategies. Then link this problem with the number sentence 24 ÷ 4 = 
. Explain that just as we can use repeated addition to solve multiplication problems, we can use repeated subtraction to solve division problems. Show students a solution using repeated subtraction as shown in Figure 4.
Tell students that they are going to think about how many hops various constant hoppers need to hop to get from 0 to 100 because the superhero invents a constant hopper in the story.
Record all number sentences students generate when you ask:
- How many hops does a +10 constant hopper need to travel from 0 to 100? Show us on a number line.
(10 hops forward; see Figure 5.)
- What is an addition number sentence for this hopper? A multiplication number sentence?(10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 = 100; 10 × 10 = 100)
- How many hops does a −10 constant hopper need to travel from 100 to 0? Show us on a number line. (10 hops backwards; see Figure 6.)
- What is a subtraction number sentence for this hopper? A division sentence? (100 − 10 − 10 − 10 − 10 − 10 − 10 − 10 − 10 − 10 − 10 = 0; 100 ÷ 10 = 10.)
- How many hops does a +5 constant hopper need to travel from 0 to 100? Show us on a number line. (20 hops forward; see Figure 7.)
- What is an addition number sentence for this hopper? A multiplication number sentence? (5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 100; 20 × 5 = 100)
- How many hops does a −5 constant hopper need to travel from 100 to 0? Show us on a number line. (20 hops backwards; see Figure 8.)
- What is a subtraction number sentence for this hopper? A division sentence? (100 − 5 − 5 − 5 − 5 − 5 − 5 − 5 − 5 − 5 − 5 − 5 − 5 − 5 − 5 − 5 − 5 − 5 − 5 − 5 − 5 = 0; 100 ÷ 5 = 20)
- What is the same about all of these number sentences? (Possible response: They all use the same numbers, 5, 20, and 100. We are trying to get from 0 to 100 or 100 to 0 on the number lines.)
- What is different? (In some of them we add, some we subtract, some we multiply, and some we divide.)
- We talked about the addition and multiplication sentences together and then the subtraction and division sentences together. Why does that make sense? (Possible response: Multiplying is like repeatedly adding. For 20 × 5, we are putting together 20 groups of hops of 5 until we get to 100. Dividing is like repeatedly subtracting. For 100 = 5, we are taking 5 away, or hopping backwards 5 twenty times from 100 until we get to 0.)
In “Cipher Force!” Divide by Zero attempts to determine how many hops his 0 constant hopper will need to take to get from 100 to 0. He tries to use repeated subtraction to solve 100 = 0. Students will see that his attempts are futile in this illustration of why division by zero is impossible.
Read Story. Read “Cipher Force!” in the Student Guide and use the following prompts to guide discussion.
You may want to read the story twice: once to clarify the plot and again to draw out more of the mathematics.
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- What does cipher mean? (Our word cipher comes from the Arabic word sifr meaning empty (as in an empty column on a counting board or abacus). Zero comes from the same root. Many mathematical terms derive from the Arabic language (e.g., algebra, algorithm) because of the work of medieval Arabic mathematicians.)
- Why is the Cipher Force's hideout called Null House? (Null is another word for zero.)
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- What does the Cipher Force insignia (
) stand for? (the empty set; nothing)
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- How would you explain to Mult why any number times zero is zero? (Possible response: No matter how many sets or groups of zero you have, you still have zero.)
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- What's wrong with Div's “zero coaster” idea? (There is no true answer—he would never have enough cars.)
- What does Div's design have to do with dividing by zero? (Division can be accomplished by repeatedly subtracting the divisor to make groups. See the Background for further discussion.)
- Show how Conrad could use his method to find how many baskets on a Ferris Wheel are needed if 15 people want to ride and each basket holds 3 people. What number sentence could go with this situation? (15 − 3 − 3 − 3 − 3 − 3 = 0. Three was subtracted five times, so five baskets are needed. 15 ÷ 3 = 5.)
Try more problems with different divisors and dividends. Link each problem situation to a division sentence.
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- How would you explain to Div that there will never be enough cars for the 24 students? (Possible response: No matter how many cars the roller coaster has, if there are zero students in a car, there will always be 24 students left.)
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- Add Zero and Subtract Zero say they are completely different. Is that true? (No, not only do they look quite similar, but their properties—addition with zero and subtraction with zero—always yield the same results: the original number.)
- What happens when Subtract Zero takes nothing from the rich? (Nothing. The rich still have the same amount of money.)
- What does this have to do with subtracting zero? (They have exactly what they had before—this is the effect of subtracting zero.)
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- What happens when Add Zero gives nothing to the poor? (Nothing.)
- What does this have to do with adding zero? (They have exactly what they had before—this is the effect of adding zero.)
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- Is Conrad speaking real English? (Yes and no. Some of these words are made up, and some have to do with space travel or computers.)
- How do you think Cipher Force will fight the aliens? (Answers will vary. Ask students to think about what each character usually does.)
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- What will happen when Subtract Zero takes nothing from the aliens? (Nothing.)
- What is a number sentence that might go with this? (Aliens − 0 = Aliens)
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- What will happen when Add Zero adds nothing to her forces? (Nothing.)
- What is a number sentence that might go with this? (Forces + 0 = Forces)
- What does Div mean when he says “I'll finish them off none by none”? (We usually say one by one meaning one at a time. He will divide by using repeated subtraction. He'll subtract zero at a time.)
- Who defeats the aliens? Why? (Mult does because multiplying anything by zero results in nothing or zero. She made nothing out of the aliens.)
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- Explain to Div why his 0 constant hopper won't ever get to 0. (Possible response: He is repeatedly subtracting 0 from 100 so his 0 hopper is not going to move.)
- The multiplication fact 20 × 5 = 100 helped us think of 100 ÷ 5 = 20 in the same fact family. Is there a multiplication sentence that can help us solve Div's problem, 100 ÷ 0 = ? What number times 0 equals 100? (There is no number times zero that will equal 100 because any number times zero equals 0. Division by zero is not possible.)
