MTE 494Midterm Test 1DUE: 2 p.m. on 10/21/10(by electronic submission) Instructions: This test contains 6 questions. You must complete the test individually and may not consult classmates or anyone 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 containedMS 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 your first,middle, and last initials, respectively. Use the same file name for the GSP file you will submit. Submit your files to me as email attachments no later than 2:00 p.m. on Thursday, October 21.NOTE: Late submission will not be accepted. Look for a confirmation of receipt message from me. If you do not receive one, then try sending via another email account or program. Please read each question carefully and ensure that your responses address all components you are asked to address; you may ask me for clarification about questions, but I cannot tell you how to answer questions. This exam is intended to evaluate your ability to use core ideas and skills developed in our course thus far, and to pull together resources from the course. You should therefore view it as an opportunity to put forth your best work, thinking, and presentation—aim for thorough, clear, and mathematically accurate explanations, as my evaluation of your responses will reflect this assumption.Question 1 [10 points]1Use only the meanings of multiplication and fraction developed in our course to answer each question. Each response MUST be accompanied by an appropriately labelled array or line diagram that clearly depicts each quantity and the relationship you are asked to explain.Suppose that u is a number represented by the product m x k.(i) Fill in the blanks: u is ________ times as large as __________. Explanation in terms of our meanings:Diagram that depicts each quantity and the relationship(s) in the explanation:(ii) Give another response to part (i): u is ________ times as large as __________. Explanation:Diagram:(iii) Fill in the blank: m is ________ times as large as u.Explanation:Diagram:(iv) Fill in the blank: k is ________ times as large as u.Explanation:Diagram:Question 2 [10 points]2Suppose that two students, CJ and Julie, walk the same distance. The figure below shows the relationship between the size of CJ’s step and the size of Julie’s step (the entire length of the line segment represents 1 CJ step and one segment of it represents 1/8 of a Julie 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 in terms of these meanings.3Question 3 [15 points](a) Write a story about the journey of two boats that could be described by the graphs above. Your story must be in terms of the quantities displayed in each graph. Avoid using catch phrases like “constant speed” or “increasing speed” without explaining the meaning of the ideas to which they point in terms of the quantities displayed in each graph.(b) After completing a curricular unit on speed and graphing in an Algebra II class, a number of students remarked that “Boat 1 and Boat 2 are the same boats because the two graphs look the same, so they describe the same motion”.Imagine that you had the opportunity to interact as a mathematics teacher with one such student. - Outline a conceptual conversation you would have with the student, including the kinds of questions you would ask her/him in order to address her/his reasoning and understanding of the graphs. - Give an ostensible “diagnosis” of the student’s thinking, and state the goalsof your conversation and questions.Question 4 [20 points]Here is a problem scenario:An A-frame barn is to be built so that it is 30 ft high, 40 ft wide and 60 ft long. A rectangular room is to be built inside the barn so that its ceiling abuts the roof.What dimensions will maximize the volume of the room?Consider this scenario from the perspective of modeling quantities and quantitative relationships (see p. 10 of quantitative reasoning module—handout of Sept. 14). 1. Describe as many quantities as you can think of that are involved in this scenario (not just those referenced in the diagram). 2. Which quantities do we know the value of? Which do we not know the value of?3. Which quantities are fixed? Which quantities vary?4. Describe relationships between some of the quantities that are pertinent for solving the problem. In particular use co-variational thinking and language to describe relationships between quantities that vary.5. Work and solve the problem without using calculus, and derive an expression showing the functional relationship between the barn’s volume and its width. Write your solution so that an Algebra II student can understand your reasoning, and why you used the expressions, equations, diagrams, or calculations you used (i.e., explain what these represent and mean in terms of the context of the problem scenario). Keep in mind features of conceptual conversations as a guide for how to write your reasoning, but do not write a script.Question 5 [20 points]Build on your experience in our GSP lab session (and on your own)1 to create a dynamically linked GSP sketch of the A-Frame-Barn with all the features shown in the picture below. Construct your barn (yes, it should be a construction!) so that the figure itself does not move around freely, and only point A can be dragged along the line segment representing the left hand part of the barn’s roof.1 See this site for GSP user resources: http://www.dynamicgeometry.com/Question 6 [25 points]Create a student worksheet containing a sequence of task/question prompts that you would use inconjunction with the A-Frame-Barn sketch to teach a short introductory lesson on the idea of variable quantities and relationships between them.Be clear about what you want your students to do and/or think about, and about your rationale