MTE 494 10/07/09Midterm Test #1Instructions: This test contains 4 questions, most of which entail multiple parts. You must complete the test individually and may not consult classmates or anyone else (other than the instructor). You may draw on the course materials and resources, but not in a “copy-and-paste” manner. Instead, paraphrase textual statements and descriptions in your own words in a manner that does not distort or misrepresent their intended meanings. Your exam paper must be composed in a self contained MS Word file, with any diagrams or drawings embedded in the appropriate places within your narrative. Use this file name convention: “MTE494_Midterm1_XYZ”, where X, Y, Z are you first, middle, and last initials, respectively. Submit your file to me as an email attachment no later than 10:00 a.m. on October 12, 2009. Send me a follow-up message asking for confirmation of receipt, to verify that what you sent out was actually submitted to me. If you do not receive a confirmation from me, then try sending via another email account or program._________________________________________________________________________________________________________Question 1SoloClearing is a one-man-owned and -operated landscaping company that specializes in weed clearing. The company uses a high power weed trimmer that consistently clears the same area of weedy land in a given unit of time. One day the company spent a total of 2.5 hours on one job clearing weeds and 3.75 hours on a second job doing the same thing. Suppose the expression 37*(2.5 + 3.75) represents the total area (in square yards) of land cleared with the weed trimmer on that day.(a) The weed trimmer clears __37__square yards of land per hour.If “37*(2.5 + 3.75)” represents the total area and “(2.5 + 3.75)” represents the total number of hours spent clearing that area, then “37” must represent the rate at which area is cleared per hour.(b) Based on order of operations, we perform the operations in the expression in a specific order to calculate the total area of land cleared. In what order do we perform the calculations?First: Add together the number of hours it took to do each job (2.5 + 3.75). This calculates the total number of hours it took to do both jobs.Second: Multiply 37 by the value obtained above: 37*(2.5 + 3.75). This calculates the total amount of area cleared in that number of hours worked. Note: It will be helpful for part (c) to use commutativity of * and think of this value as (2.5 + 3.75)*37.(c) Explain how to interpret the expression in terms of the “making copies” meaning of multiplication: identify which amount is being copied, and how many copies of it are 1MTE 494 10/07/09made, so that it makes sense that the expression represents an amount of area and that it is calculated using the order you specified in part (b).In the “making copies” meaning, the expression 37*(2.5 + 3.75) would denote the amount made by making 37 copies of (2.5 + 3.75). Since (2.5 + 3.75) represents a number of hours, then 37*(2.5 + 3.75) represents 37copies of that number of hours. This means that the resulting value will bea number of hours. But this doesn’t make sense in the given situation, because we are told that the expression 37*(2.5 + 3.75) represents a totalarea. We seem to have a quandary here.We need a way to think about the given expression that both fits the making copies meaning of multiplication AND respects that the expression represents an amount of area. We can use the fact that multiplication is a commutative operation: the value of 37*(2.5 + 3.75) is the same as the value of (2.5 + 3.75)*37, and this latter expression represents the amount made by making (2.5 + 3.75) copies of 37. So we can think of the original expression as the value of the expression (2.5 + 3.75)*37, which according to our meaning represents the amount made by making (2.5 + 3.75) copies of 37 yards [the amount of area cleared in each hour]. In calculating the value of (2.5 + 3.75) first (in part b), we are essentially calculating how many copies of 37 yds to make.Answering this question coherently rests on the subtle yet important fact that the making copies conception of multiplication entails two aspects that must be kept in mind: the expression a*b denotes 1) imagining making a copies of b, and 2) the value obtained by making a copies of b. The commutativity of the * operation in this meaning has to do with aspect 2 and the method of calculating this value, not with aspect 1.Note: Your explanation must describe the meaning of the expression (each of its components) and the operations used to calculate its value explicitly in terms of this model of multiplication. 2MTE 494 10/07/09Question 2Suppose that two students, CJ and Julie, walk the same distance. The figure below shows therelationship between the size of CJ’s step and the size of Julie’s step. Use our meanings of fraction, measure, and multiplication (the ones developed in the course) to reason through this scenario and answer this question:How many steps did CJ take in relation to the number of steps Julie took?Write your reasoning in a way that makes clear to the reader the relevant quantities and relationships involved in the scenario and your solution.1. The figure shows that 1/13th of a CJ step is the same size as 1/8th of a Julie step.2. 1 CJ step is 13 times as much as 1/13th of a CJ step, 3. 1 CJ step is 13 times as much as 1/8th of a Julie step. 4. 1 CJ step is 13 one-eighths = (13/8) Julie steps (by 2 & 3)5. If CJ takes 1 step, then Julie will take 13/8ths Julie steps (to cover the same distance)6. If CJ takes 2 steps, then Julie will take 13/8ths of her steps twice; she will take (2x13/8) Julie steps7. If CJ takes 3 steps, then Julie will take 13/8ths of her steps three times; she will take (3x13/8) Julie steps. ..- In general, if CJ takes n steps, then Julie takes 13/8 times as many steps; she takes (n x 13/8) steps3MTE 494 10/07/09Question 3Consider this scenario:It is 1873 in Territorial New Mexico. The Lone Ranger is chasing a desperate bank robber overthe desert. Both are on horseback, and the bank robber got an 11 mile head start. The bank robber's horse can run steadily at 16 miles per hour. The Lone Ranger's horse can run steadily at 19 miles per hour. The bank robber is heading for the Mexican border, for if he can cross theborder he will be safe. The town that they started from is 49 miles from the border. Will