Mathematical Inquiry
If you think math is mainly about getting the same answer that someone else has already figured out, you are missing the wonder of it. Math is a language that can explain the world for you, and you can use mathematical inquiry in life.
Begin Your Inquiry
Imagine, for instance, that you are stuck in traffic on a freeway. Rather than just sitting there and getting angry, you could do a little mathematical inquiry.
Ask a question or identify a problem.
Why are we suddenly creeping along? A minute ago, it was free flow, and now it’s stop and go. What’s up with that?
Identify variables and constants.
What’s changed? It’s been raining the whole time. We haven’t lost a lane. There’s no accident. Sure, people have been merging, but people have also been exiting. I wonder if more people are merging than exiting?
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Exit 36: |
Entering: 12 Exiting: 11 |
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Exit 37: |
Entering: 19 Exiting: 0 (no exit) |
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Exit 38: |
Entering: 8 Exiting: 16 |
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Totals: |
Entering: 39 Exiting: 27 |
Research Your Inquiry
You decide to find out whether your hunch about more people merging is true. At home, watching a traffic camera, you count the number of vehicles that enter in one minute and the number that exit in one minute. You find that 39 vehicles enter while 27 exit, and now traffic has stopped. So the problem seems to be too many vehicles sharing the road.
Design a formula to explain the phenomenon.
You discover that, in an eighth-mile stretch of road, each lane on your side has an average of 19 vehicles, while the other side has an average of 9. In subsequent samples, you find that the other side can get up to 15 vehicles in an eighth-mile stretch of lane and still flow. At 17, they slow. And at 19, they stop, like you.
Free-Flow Traffic Formula
If F is free-flow density, then . . .
F < 17 vehicles per lane/eighth mile
F < 136 vehicles per lane/mile
(the < means less than)
Create and Improve Your Inquiry
Next, you devise a formula to explain the density of vehicles at which free flow ceases. You simplify the formula to account for flow over a mile of roadway (more than you could count).
Run data through the formula to get results.
Check results against observations and adjust formula.
Free-Flow Traffic Formula
If F is free-flow density, then . . .
F < 22 vehicles per lane/eighth mile
F < 180 vehicles per lane/mile
Present Your Inquiry
Your calculation was off by 44 vehicles! Even with your error rate, you didn’t expect a 44-vehicle difference. If math were all about getting the same exact answer as someone already got, you might be feeling pretty bad right now. But mathematicians in the real world don’t have answer keys. They are playing with numbers. If their results are different, they try to figure out the reason for the difference. Then you remember: It was raining that day. People slow down in the rain. And that’s only one of the possible factors that could have caused your variant observation.
Present your formula for others to use.
Your Turn School Traffic: You probably encounter “traffic jams” every day in school: hallways, entrances, exits, the cafeteria, the gym, the auditorium. Use a similar math inquiry process to come up with a flow-rate formula for traffic in these places. After you establish your formula, compare it to those of other classmates. Discuss any differences and variables that you notice.
Patterns of Flow: Think of other things that flow: liquids, gasses, people, particles, ideas, videos, emotions. In your current class, make a list of “Things That Flow.” Choose one idea from your list and explore it using a similar mathematical inquiry process.
Mathematical Estimating
As you saw on previous pages, mathematical inquiry often involves estimating. When you don’t have the time or means to count or measure something exactly, you need to be able to make an educated guess that is close enough.
An eighth of 21 gallons is more than 2 gallons, and 2 gallons times 23 miles per gallon is more than 39 miles, so I can keep driving.
Your car has a 21-gallon tank that is 1/8 full, it gets 23 miles per gallon, and the next gas exit is 39 miles down the road—do you exit now?
If you don’t estimate well, you may run out of gas, order too little pizza, pay too much at the register, or report ridiculous statistics. Estimation is a key math skill.
Estimating Techniques
The best way to become skilled at estimating is to practice. Here are some specific strategies that work well.
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4,683 + 7,336 is about 12,000. |
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183 ÷ 58 is about 3 (183 ÷ 58 is like 180 ÷ 60, or 18 ÷ 6) |
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0.92 × 5/9 is about 0.5 (0.92 is almost 1 and 5/9 is a little over half) |
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44 + 95 + 63 is about 200 (44 and 63 are about 100, and so is 95) |
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723 + 695 + 733 + 701 + 692 is about 3,500 (700 × 5) |
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7,000 × 2,000 is 14,000,000 (not 14,000) 7,000 ÷ 2,000 is 3.5 (not 3,500) |
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Your Turn Estimate the number of jellybeans in the jar to the right. Indicate how you came up with your estimate. Compare your answer with a classmate’s.
Mathematical Problem Solving
Think of word problems as verbal versions of the real-world problems you’ve solved before. Use the inquiry process you are familiar with, but apply it to the given question and problem.
Word Problem: A man has three sons—Eli, Aidan, and Gabe—and the younger two have cats, Merlin and Gato. Eli is two years older than Aidan, and Aidan is twice the age of Gabe. Gabe is the same age as Merlin. The boys’ ages add up to 17. How old is each boy?
Ask a question or identify a problem.
What is the age of each boy? Eli (E) is two years older than Aidan, and Aidan (A) is twice the age of Gabe (G). The boy’s ages add up to 17. (It doesn’t tell how old Merlin is, so it doesn’t matter that Gabe is the same age.)
Identify variables and constants. (Get rid of unimportant information.)
E = A + 2
A = 2G
G = ?
What age is each boy?
Run calculations through the formula to get results.
If A = 2G, then
E + 2G + G = 17
If E = A + 2 and A = 2G, then E = 2G + 2, and
2G + 2 + 2G + G = 17
Isolate the variable. (Add the G’s.)
Design a formula to explain the phenomenon.
5G + 2 = 17
5G + 2 – 2 = 17 – 2
5G = 15
5G/5 = 15/5
G = 3
Check results against observations and make adjustments.
Use the numbers in the original equations.
G = 3
A = 2G so A = 6 (Aidan is twice Gabe’s age)
E = A + 2 so E = 8 (Eli is two years older)
8 + 6 + 3 = 17
Present your results.