Mathematical Thinking 2

Answers to the Quick Test

1.The M should be very close to the 0. Most of us think million and billion are both large. They know a billion is more, but both seem a very long way from zero so they put the M somewhere between one-third and two-thirds the way along the line.

When you tried reasoning, you should have realized that there are a thousand millions in a billion. If you didn’t know this, you could divide a billion by a million (easiest using a simple fraction and reducing it). Thus, your line would need to be divided into 1000 equal spaces and the M placed on the first line past the 0, practically right on top of the zero. You can amaze your friends with this question.  This is an example of Number Sense and Mathematical Reasoning.

2. You should have realized quickly that this is the very same question as how many centimeters are in a meter, etc. It was only the basic unit that was original, not the metric relationships
100 centiKings = 1 King
10,000 square centiKings = 1 Square King (area =  length x width)
1,000,000 (one million) cubic centiKings = 1 Cubic King   (volume =  length x width x height)

I included this question because I had so many students over the years who continued to believe there were 12 inches in a foot, 12 square inches in a square foot, and 12 cubic inches in a cubic foot. Even with paper cut into square feet, and boxes close to a cubic foot, the ideas never seemed to sink in. This is an example of Spatial Sense.

3. The area of the ceiling is 200 SQUARE feet. If you wrote 200 feet or just 200, mark it wrong. Some students think that there isn’t enough information because I didn’t give the dimensions of the ceiling. It should be obvious that the floor and ceiling have the same area.

Through the years, I asked similar questions on tests and students would copy down every number in the question and add them up. One would think, after being “tricked” the first time or two, they would learn, but many continued to use the same strategy. This is an example of Mathematical Communication, understanding what the question asks for. You should use only the relevant data and ignore all irrelevant data.

4. If you say  .3750 yards or .375 yards, you are wrong. If you say 1.1250 feet or 1.235 feet, or 1.23 feet or 1.34 feet, you are wrong. None of these is a good answer.  A good answer would be about 1 foot or 1.2 feet. You could also say .3 yards, or .4 yards. You might also have responded 1 food and about 2 inches. BUT the two inches is not because it is 1.2 feet. It is because two tenths of a foot or ( two tenths of twelve) inches happens to be .24 inches.

The length of the cloth is a measured number, not a counted number.  Measured numbers are never precise. The length of the cloth was probably somewhere between 2.9 and 3.3 yards, usually a little above 3 because the person doing the measurement usually adds a little to be sure there’s enough. The problem is compounded by dividing the cloth into 8 equal pieces. Again measurement is involved. It is impossible to make these 8 pieces precisely even. They will all have the same number of feet, about one foot. They might even be all close to  1.2 feet. They are certainly not going to match to a third, fourth or further decimal point. This is an example of Understanding Measurement or using Significant Digits

5. The reporter’s numbers add up to 148%.

Some would respond, “O.K. So what’s the problem. They don’t understand that 100% includes everyone.  You canmt have more than 100% of the people. I hope you didn’t disagree, thinking that more people read mysteries or something like that. The reporter may have interviewed 148 people and simply added a percent sign to each number. Another possibility is that he interviewed 100 people but allowed some of them to give more than one answer. He should have counted the number of responses in this case, rather than the number of people.  This is an example of Understanding Statistics, Mathematical Reasoning and Mathematical Communication.

6. I imagine some would think the number here is too high or too low. That’s not the real problem. The newspaper refers to “females under 18.” This includes newborn baby girls all the way through seventeen-year-olds. Hopefully most of these girls are not sexually active. It would have made more sense to say 70% of females between age 15 and 17… or whatever the age range happened to be. Another question is “What is meant by sexually active?” If it includes giving or receiving a good-night kiss from their parents, then it could be true, but extremely misleading. This is an example of Understanding Statistics and Mathematical Reasoning.

7. Mrs. Johnson, in spite of having five girls, still has a 50-50 chance of having a boy. The only exception to this is if her husband’s sperm is abnormal.This is an example of Understanding Probability.

8. The gambler is wrong. The Roulette machine, like with Mrs. Johnson’s babies, has a 50-50 chance of landing on an even number. There is no such thing as the “Law of Averages” though many gamblers depend on it. This is an example of Understanding Probability.

9. If  you are thinking of getting a job at this factory, you should not be misled by believing that because the average starting salary is  $56,000 a year, that you have any chance of earning that amount. In fact, there might not have been a single person hired at that salary. You need to ask what most people got as a starting salary who were doing the same job you might be doing.  They could have hired 5 top executives earning an average of $200,000 a year and 20 factory workers each earning $20,000. The average (or mean) starting salary really would be $56,000.  Using an average can very often be misleading. Here, the median starting salary would have been $20,000 and so would the mode.  This is an example of Understanding Statistics.

10. Significant digits are used with measured numbers. When measuring, significant digits include all the numbers you are certain of plus one digit that is estimated. (Check out the next problem.) They are especially important in science. A scientist who says the mass of a small object is 4.6072 grams believes the contents were measured extremely accurately. On most scales the same object might seem to have a mass of anywhere from 4.578256 and 4.632945 grams. The 4 grams is consistent. The second digit varies. Between 4.59 and 4.65, the midpoint would be 4.62 grams.

Using significant digits, the mass should be described as 4.6 grams. It include all numbers we are certain on, (just the 4) and one estimated digit.

11. The graduated cylinder holds  either c. 24.5 ml or  d. 24.6 ml . This includes the digits we are sure of and one digit at the end that is estimated. Since the last number is estimated it could be either answer.

12.  3.56 x 10³ is equal to  c. 3560.  The number, 10³ (or 1000) indicates that we should move the decimal three places to the right…. the equivalent of multiplying by 1000.

13.   What is 45.3769021 x 10°  ?   It is 1.  All numbers to the 0 power are equal to one. It indicates that the number is divided by itself.  If this doesn’t make sense, see the page on special topics.

14.    5x² + 6x² = 11x²  We can’t solve it any further without knowing what x is.
5x² + 6x³ =  5x² + 6x³   We cannot combine the terms because they are not similar.
x² times x³ = x to the fifth power. When multiplying, we can add the exponents.
We cannot add exponents if the problem is   x² times y³. We can only write it as  x²y³ .

15  You may have many example of how our artist uses math. Like all human beings, she uses measurement when she cooks, she should count the dollars and cents when paying for something, or if using a large bill, should count her change. She should be able to balance a checkbook, pay her taxes, estimate what she’s spending in a store if her cash is limited.

But lets consider math in terms of being an artist. Many artists, consciously or not, divide the canvas into thirds. She estimates how long her supplies will last before needing to buy more. When mixing colors, she might have an intuitive feel for how much red and how much yellow should be mixed for a certain shade of orange. If she is wise, she estimates the cost of her supplies and the value of her time before setting a price on her paintings. She might also consider the number of pictures likely be sold for a lower price compared to the number sold for a much higher choice. If painting fairly realistic objects, she needs to use shapes and perspective.  This is an example of Connections to Other Disciplines.

How well did you do?

This isn’t a scientific test. I haven’t tested 1000 students so you can compare your scores with theirs. I’m simply giving you my informed opinion.  If you got all of them right, you either are or could be a math major. 

11-14 correct:  your mathematical reasoning is Excellent
7-10 correct:  your mathematical reasoning should be about average for college students, but well above average for high school grads.
5-6 correct:   below average for college students. You might make good grades in math butyou  need to learn how to think mathematically.
3-4 correct is below average among high school grads. You really need to learn basic math concepts. You might get some help.
0-1-2 correct.  I hope you are kidding, that you didn’t read the questions carefully. You are below the average of  a 7th grader. You definitely need improvement.

To continue:   Nine Ways of Mathematical Thinking

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