Mobile Math Teachers Circle The Return of the iclicker

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1 Mobile Math Teachers Circle The Return of the iclicker June 20, Dr. Spock asked his class to solve a percent problem, Julia set up the proportion: 4/5 = x/100. She then cross-multiplied to solve for x. Which mathematical reason best explains why crossmultiplication works? (Select ONE answer.) A. In a proportion, the product of the means equals the product of the extremes. B. You are actually multiplying both sides of the equation by 5 and by 100 and then simplifying. C. Cross-multiplication is the rule for solving proportions. You multiply the numerator of one times the denominator of the other, and set them equal. D. You are actually multiplying 4/5 by the identity for multiplication, 20/20, to get x = 80. E. 100 divided by 5 equals 20 and 4 times 20 is 80. F. You do it because Dr. Spock told you so. 2. Al is 5 feet tall and has a shadow that is 18 inches long. At the same time, a tree has a shadow that is 15 feet long. Al sets up and solves the proportion as follows: Al claims the solution is, x = 54 in. 5 ft 18 in = 15 ft x in. Which of the following statements is correct? (Select only ONE.) A. The units work out correctly, that indicates that Al s solution is correct. B. Al s solution is wrong because he did not convert all units to inches. C. Al set up the ratios correctly but he did not properly solve for x. D. Shadow problems are tricky and cannot be solved with ratios. E. Al s answer is incorrect. A more appropriate ratio would have been 5 ft 18 in 1.5 ft = = x ft 15 ft 15 ft.

2 3. Dr. Seuss asked his class to find 16% of 25. The Grinch decided to calculate 25% of 16 instead. To Dr. Seuss surprise he got 4 as the answer. What can we say about the Grinch s method? (Select ONE answer.) A. The Grinch s method is not correct and gave the wrong answer to this problem. B. The Grinch s method never gives the correct answer. C. The Grinch was just lucky. The method works because 25% is just a quarter, a special case. The method would not work in general. D. The Grinch s method gives the correct answer and could be used for any numbers. E. The Grinch stole Christmas. 4. Students in Mr. Bieber s class were discussing the average speed of two moving objects. The graph below shows the position of a train and the Batmobile, over ten seconds. The tic marks on the horizontal axis are located at 1 second intervals, and the tic marks on the vertical axis are at 20 meter intervals. distance in meters Train 20 Batmobile time in seconds Which of the following statements is correct? (Select only ONE.) A. The average speeds of the train and the Batmobile are the same over this interval. B. The average speed of the train is greater than the average speed of the Batmobile because the graph for the train is always above that for the Batmobile. C. There isn t enough information to determine the average speed of either vehicle. D. You can determine the average speed of the train because it is a straight line, but cannot determine the average speed of the Batmobile from this graph. E. The Batmobile is always faster.

3 5. After visiting a local machine shop on a field trip Ms. Potter s class is given the following problem: The machine shop just replaced an older machine with a modern one. For every 3 parts that the old machine produced, the new machine will produce 7 parts. If the old machine produced 27 parts in one hour, how many will the new machine produce? Students calculated the solution in different ways. Of the following examples of student work, which is least likely to represent correct thinking? (Select ONE answer.) 7 27/3 = 63 (3 7) + (3 7) + (3 7) = = /3 = = Mr. Dumbledore introduced the notion of two quantities being inversely proportional to his class. In order to gage whether his kids understood the concept he asks several of them to give some examples where inverse proportions appear. Which of the following scenarios contain examples of quantities that are inversely proportional? Select one of the following: Inverse proportions appear Inverse proportions do not appear I m not sure (a) Hermione notes that if a charity wants to distribute $1, 000 of aid in equal-sized amounts. It must decide whether to distribute small amounts to many people, or larger amounts to fewer people. (b) Ron just came from physics class. He reports that a radioactive substance loses half its mass each 10 years due to radioactive decay. How is the mass of the substance measured as a function of time? (c) Harry says: Today in economics class we discussed so-called zero-sum-games. Suppose two of us start with $100 each, and we are going to play a zero-sum-games, so that, as much as one of us wins, the other loses. (d) Dudley suggests the following: Inverse proportions appear if we want to compare the bases and heights of all rectangles whose area is exactly 100 square feet.

4 7. Professor Voldemort is trying to figure out whether his students really understand rational and irrational numbers. He gave them a few statements about such numbers and asked his students to decide whether the statements were true or false. Which of the statements is true? For each item below, select True False I m not sure (a) If you add a rational number to an irrational number, the answer is always irrational. (b) Between any two different rational numbers, there is at least one irrational numbers. (b) Between any two different rational numbers, there are infinitely many rational numbers. (c) A decimal number with a repeating pattern is rational only if the pattern starts immediately to the right of the decimal point (e.g., is rational but is irrational). (d) The only way of getting an irrational number is by taking the square root of a number that is not a perfect square. (e) If you divide an irrational number by a rational number the result can be a rational number. 8. Ms. Babbling wants her students to explain the meaning of 1/8 2/5. For each of the following below, decide whether it is an mathematically valid interpretation of 1/8 2/5. (Select VALID, NOT VALID, or I M NOT SURE for each.) (a) How many 2/5 s are in 1/8? (b) How much is 1/8 of 2/5? (c) What number is 1/8 two-fifths of? (d) How many 2/5 s can you subtract from 1/8 before reaching 0?

5 9. Susan s asked to work the problem She explains her solution as follows: First I totally ignore the decimal point and just do the multiplication, which gives me 2, 328. Then I use estimation to place the decimal point. I know that 0.24 is about a quarter and 97 is close to 100. Now a quarter of 100 is 25, so my answer should be Which of the following is most appropriate to say about Susan s approach? answer.) (Select ONE It happens to work in this case, but will not work for most problems. It only works if one of the numbers is a whole number. It works for any numbers, but some examples are harder to estimate. It works equally well for all problems. I m not sure 10. Mr. Binns asks his colleagues to suggest word problems that he could use on his quiz on division with decimals. Which of the following suggestion(s) could he use as a word problem for ? (Select YES, NO or I M NOT SURE for each.) (a) You have 30 cents, but need 70 cents to buy some candy. amount do you already have? What fraction of the total (b) My pet lizard eats 0.3 ounces of bugs per day. If I purchase 0.7 oz of bugs, how many days will the bugs last? (c) One inchworm races 0.3 yards. Another inchworm only makes it 0.7 the distance the first inchworm raced. How far did the second inchworm go? (d) A snail can crawl 0.7 meters per hour. If the snail needs to move 0.3 meters to find food, how long will it need to crawl to get to the food?

6 11. Professor Quirrell gave his students the problem Later the students were asked to share their solutions. Draco Malfoy said, Because there s one decimal place in 0.7, I need to move the decimal point in both numbers over one place and then do the division. Then he wrote the following on the board: = 2.7 Ron Weasley commented, Hey, you cheated. You changed the problem to That does not explain that 2.7 is the answer to ? Of the following, which provides the best basis for explaining why this procedure works? (Select ONE answer.) (a) It works because you can t divide by a number with a decimal point, so the divisor must be converted to a whole number. (b) It works because converting the decimals to fractions and dividing produces the same quotient. (c) It works because the quotient stays the same as long as you move the decimal point the same number of places in both the divisor and the dividend. (d) It works because multiplying the dividend and the divisor by the same number is really multiplying by 1, so the quotient will remain the same. (e) It works because moving the decimal points maintains a constant difference between the divisor and dividend, which leaves the quotient unchanged.