Reading: Learning to See the Wind, unpublished paper by Sid Rachlin, Pg. 1–9. Please attempt all problems. 2 1. a. Find two fractions in lowest terms with unequal denominators whose difference is . 13 3 1 1 b. What number added to the sum of 5 4 , 1 3 , and 4 4 equals 99? ALGEBRAIC CONCEPTS AND RELATIONSHIPS Problem Set #9—Page #1 2. Two ways to use algebra tiles to represent the expression 3x–2 are: Each diagram could be described by equations. For example, for the first diagram [2x + (–1)] + [x + (–1)] = 3x – 2; for the second diagram 4x + (–x) + 2 + (–4) = 3x – 2 or for the second diagram [2x + 1] + [2x + 1] + [(–x) + (–4)] = 3x – 2. a. Show two ways to represent the expression 2x + 2 b. Write two equations for your representations in part a. c. Show two ways to represent the expression 3 – x. d. Write two equations for your representations in part c. e. If you were to use a tile to represent x2, what would it look like? Why? Problem Set #9—Page #2 Name..................................................................... 3. a. There is a red tag sale at a store. Which deal is better for you? Explain your answer. (1) The store first marks down an item 30% and then marks it down an additional 25%. (2) The store first marks down an item 25% and then marks it down an additional 30%. b. Which deal is better for you? Explain your answer. (1) The store first marks down an item 30% and then marks it down an additional 25%. (2) The store has a half price sale. ALGEBRAIC CONCEPTS AND RELATIONSHIPS Problem Set #9—Page #3 4. a. As a part of the Click-it or Ticket campaign the signs throughout Greenville report seat belt use. Assuming that numbers have been rounded to the nearest per cent, what is the smallest number of cars that could have been sampled to get the record seat belt use of 93%? b. What is the smallest number of cars that could have been sampled to get “last month’s” seat belt use of 88%? c. Suppose the same number of cars are sampled each month. What is the smallest number of cars that could have been sampled to get the percentages indicated on the sign? Problem Set #9—Page #4 Name..................................................................... 5. Draw a diagram representing how you could use base five materials to demonstrate the following operations. After you perform the given operation, represent your answer using the smallest possible number of objects. a. The sum of 412 Base five and 234 Base five. b. The difference of 412 Base five and 234 Base five. Cube Flat Long Unit ALGEBRAIC CONCEPTS AND RELATIONSHIPS Problem Set #9—Page #5 6. Consider the numbers 97, 167, and 391. (Problem is adapted from a task used in a seminar given by Rina Zazkis.) a. List all factors of 97. How did you make sure you found all factors? b. List all factors of 167. c. List all factors of 391. d. Multiply 97 and 167. Call this number J. e. Will 7½J? (This reads, will 7 divide J, meaning will 7 divide J without a remainder?) f. Will 13½J? What method did you use to answer this question? Writing Assignment #9: Authors of mathematics textbooks are sometimes criticized for writing a narrow set of problems that make students appear to be successful, but do not give them a well-rounded conception of a particular idea or operation. For example, students may only think about subtraction as “take-away” and may even use take-away instead of the word subtract. Below is one way to think about categorizing addition and subtraction tasks. Write tasks for the missing sections. Result Unknown (m) Change Unknown (r) Start Unknown (£) £ Æ r = m Deena found 8 books on Write a join task with Norman had some change in JOIN dolphins for her report. Serena change unknown that his desk. He took 83¢ out of gave her 6 more. How many involves fractions. his pocket and put it in the books does Deena have now? desk. He now has $3.52 in the desk. How much money was in the desk to start with? Result Unknown (m) Change Unknown (r) Start Unknown (£) £ – r = m Jorge has a deck plank 8 and Reliable Motors had 127 vehicles Write a separate task SEPARATE 1/2 feet long. If he cuts off a on the lot before their annual with start unknown piece 5 and 3/4 feet long, how inventory reduction sale. They only that involves decimals. long is the remaining piece? had 78 cars at the end of the sale. How many cars did they sell? Difference Unknown (m) Compare Quantity Reference Quantity £ – r = m Unknown (r) Unknown (£) Write a compare task COMPARE with difference unknown Stan is 2 and 1/2” taller than his Marie’s time for 1600m was that involves whole brother. If his brother is 5'10”, how 3.7 seconds better than numbers. tall is Stan? Jonell’s time. If Jonell’s time was 5:02.6, what was Marie’s time? Problem Set #9—Page #6 When I was at the University of Hawaii, a colleague of mine told me a story of an experience he had as a first grader growing up in Hawaii nearly forty years earlier. It was a simpler time, one in which a trip from one side of the island to the other was no small feat, since they traveled over a small mountain road called the Pali that stretched through a pass in the mountain range. At the top of the mountain, his family stopped to enjoy the view. As he watched from the windy edge of the mountain's lookout, he made a discovery he deemed worthy of sharing at show-and-tell the next Monday. Contrary to what he had been taught in class, he could see the wind! On Monday, he anxiously awaited his turn to share his discovery. But to his disappointment, when he told the class of his adventure, rather than share in his excitement the teacher corrected him. “Now David,” she said. “You know that you can not see the wind. What you saw was the dust and dirt being blown around by the wind—the wind is invisible.” There are two things that struck me as important in my colleague’s tale. First, how meaningful an experience this must have been for him to have held it in memory for nearly forty years. What made this a memorable experience in his life? And second, how mathematics is like the wind. As my colleague spoke, my mind wandered to thoughts of the old Claude Raines movies of the invisible man. How did one catch an invisible man? In one Abbott and Costello version, the invisible man was captured in a kitchen by tossing flour in the air. Amidst the flour the invisible man became visible and was caught. But what was it that was seen? After all, the invisible man was by definition invisible, so all that we saw was flour. And what of mathematics? Are the symbols of mathematics really mathematics or merely the bits of flour that we throw about to help us find mathematics. Too often mathematics is taught as if the flour is what is important. Yet all of the symbols, and even the manipulatives, calculators, and the computer software, are only flour. The pages of dittoed ink do not make mathematics. The base ten sets and counting aids do not make mathematics. The texts and technology, manipulatives and worksheets provide opportunities for our students to see mathematics — to see the invisible man. But only if our students are looking for mathematics. Mathematics is like the wind. It is seen, when some would argue it cannot be seen. It is touched, when some would argue it cannot be touched. It is taught, when some would argue it cannot be taught. It is understood, when some would argue it cannot be understood. Learning to See the Wind —1 The challenge of teaching mathematics is to provide an environment in which students can see, touch, understand, and learn. The learning environment we provide, will set the probability of students understanding, re-creating, applying, and retaining the patterns of mathematics. While manipulatives, computer software, cooperative learning and portfolios may be used by students to facilitate their learning, it is the role of instruction to coordinate the alternatives and maximize learning. To this end, instruction and curriculum are inseparable. The curriculum must be organized to enhance the opportunities for students to learn. The instructor must present the curriculum in a manner that increases the learning potential. Mathematics is learned through communication — “students need opportunities not just to listen, but to speak mathematics themselves (Silver, Kilpatrick, & Schlesinger, 1990).” Although it is not uncommon for students to be asked to put homework problems on the chalkboard or the overhead projector so that other students can see their work, students are seldom asked to put into words not only what they did but also how and why they did it. This is particularly true when the tasks they are asked to explain are repetitions of trained algorithms. I am reminded of a third grade classroom that I had an opportunity to visit. The teacher was explaining how to do three-digit subtraction with regrouping. “Suppose I have 568 – 279. 15 1 You can't subtract eight from nine, so you'll have to regroup. Eighteen subtract nine is 4 568 nine.