Essential Concepts and Skills of a World-Class Curriculum



Rigor and Relevance

Quadrant Examples

High School

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|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Algebra |

|Essential Concept: Understand, analyze, represent, and apply functions |

|Essential Skill: Problem solving |

|Essential Skill: Ability to construct and apply multiple connected representations |

|C |D |

|Consider this function: |Research medications used to help control diseases. Find data to build |

|i(t)=10(.95)t |functions modeling the amount remaining in the bloodstream at various |

|Using different methods and different representations |times. Find the half-life, if appropriate. Discuss some dosage strategies.|

|(tables, graphs, symbolic reasoning, and technology), | |

|determine i(40) in as many ways as possible. Analyze | |

|and evaluate each method and representation used. | |

|Include advantages and disadvantages of the different | |

|methods and representations. | |

|A |B |

|Consider this function: |Insulin is an important hormone produced by the body. In 5% to 10% of all |

|i(t)=10(.95)t |diagnosed cases of diabetes, the disease is due to the body’s inability to|

|Determine i(40) |produce insulin; therefore requiring people with the disease to take |

| |medicine containing insulin. Once insulin gets to the bloodstream, it |

| |begins to break down quickly. After 10 units of insulin are delivered to |

| |a person’s bloodstream, the amount remaining after t minutes might be |

| |modeled by the following function: i(t)=10(.95)t. Find the half-life of |

| |the insulin. Describe the practical and theoretical domain and range of |

| |the function i(t). |

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Algebra |

|Essential Concept: Understand, analyze, solve, and apply equations and inequalities |

|Related Skill: Communication |

|Essential Skill: Ability to construct and apply multiple connected representations |

|C |D |

|Solve this equation: |Research some text-messaging plans available in your area. Find a |

|13 = 0.10(x – 200) + 5 |mathematical model that represents each plan. Given your |

|Use different methods and different representations |text-messaging habits and the mathematical models, evaluate these |

|(including tables, graphs, analytical methods, symbolic |plans, and choose the one that is best for you. Explain your choice |

|reasoning, using technology, etc.). Analyze and evaluate each|and why you think it’s the best plan for you. Include graphs, |

|method and representation, including advantages and |equations, and tables in your explanation, as appropriate. |

|disadvantages of different methods and representations. | |

|A |B |

|Solve this equation: |Consider this text messaging plan for your cell phone: You pay $5 per|

|13 = 0.10(x – 200) + 5 |month for 200 text messages, then you are charged $0.10 for each |

| |additional message either sent or received. Find an equation that |

| |models this text messaging plan. Use your equation to determine how |

| |many text messages you can send or receive in a month if you are |

| |willing to spend $13 that month on text messages. |

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Algebra |

|Essential Concept: Understand, analyze, transform, and apply algebraic expressions |

|Essential Skill: Problem solving |

|Essential Skill: Ability to recognize, make, and apply connections |

|C |D |

| | |

|Consider this algebraic |Find the current price of one of your favorite CDs. Profit for the record company that produces |

|expression: |the CD is a function of CD sales. Assume that the record company had the following production |

|21.98x – 3.75x – 1.50x – 1.65x –|and distribution costs related to this CD. |

|10x – 365,000. Write four other |$365,000 for studio, video, touring, and promotion expenses; |

|expressions that are equivalent |$3.75 per CD for pressing and packaging costs; |

|to the given expression (use |$1.50 per CD for discounts to music stores; |

|expansion and simplifying). |$1.65 per CD for other discounts. |

|Explain how you know the |$10.00 per CD paid to the band |

|expressions are equivalent and |Given this information and the selling price of the CD, write a formula for a function that |

|state the properties used. |shows how the record company’s profit depends on the number of CDs sold. Using this formula, can|

| |the record company make a profit on this CD? Can they make a profit if the CD sells for $25? Can|

| |they make a profit if the CD sells for $12? For the selling price(s) for which the record |

| |company can make a profit, how many CDs must be sold before they begin making a profit? How many|

| |CDs must they sell to make a profit of $250,000? For the selling price(s) for which they cannot |

| |make a profit, how would you suggest they modify their costs so that they can make a profit? |

|A |B |

| | |

|Consider this algebraic |Profit for a record company is a function of CD sales. For a given band, the record company had |

|expression: |the following production and distribution conditions to consider. |

|21.98x – 3.75x – 1.50x – 1.65x –|$365,000 for studio, video, touring, and promotion expenses; |

|10x – 365,000. Put the |$3.75 per CD for pressing and packaging costs; |

|expression in simplest form. |$1.50 per CD for discounts to music stores; |

| |$1.65 per CD for other discounts. |

| |$10.00 per CD paid to the band |

| |The CD sells for $21.98. Using this information, write a formula for the function that shows how|

| |the record company’s profit depends on the number of CDs sold. Explain your formula. Make the |

| |formula as simple as possible. |

Activities in quadrants B and D adapted from Contemporary Mathematics in Context, Course 3, Everyday Learning Corporation, 1999.

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Algebra |

|Essential Concept: Understand, analyze, approximate, and interpret rate of change |

|Essential Skill: Communication |

|Essential Skill: Problem solving |

|C |D |

|Consider this equation: |Suppose you must compare the elasticity of several different brands of golf balls. Get |

|y = 27[pic] |a variety of golf balls and a tape measure. Begin the comparison by choosing one of the|

|Make a table and a graph, and write an |balls. Decide on a method for measuring the height of successive rebounds after the |

|equation using NOW and NEXT. Discuss how |ball is dropped from a height of at least 8 feet. (You may want to use technology to |

|the rate of change is shown in each. |gather the data, such as a motion detector.) Collect data on the rebound height for |

| |successive bounces of the ball. Describe the change in consecutive rebound heights. |

| |Write an equation using NOW and NEXT that relates the rebound height of any bounce to |

| |the height of the preceding bounce. Write an equation y = …to predict the rebound |

| |height after any number of bounces. Use a different type of ball and repeat the process|

| |two more times. Compare the results of the three data sets. Write a brief report |

| |summarizing your findings. |

|A |B |

|Consider this equation: |Most popular American sports involve balls of some sort. One of the most important |

|y = 27[pic] |factors in playing with those balls is the bounciness or elasticity of the ball. If a |

|Describe the rate of change. How is the |new golf ball is dropped onto a hard surface, it should rebound to about [pic] of its |

|value of y changed from one integer value |drop height. Suppose a new golf ball drops downward from a height of 27 feet and keeps|

|of x to the next? |bouncing up and down. Make a table and plot of the data showing the expected heights |

| |of the first ten bounces. How does the rebound height change from one bounce to the |

| |next? How is that pattern shown by the shape of the data plot? What equation relating|

| |NOW and NEXT shows how to calculate the rebound height for any bounce from the height |

| |of the preceding bounce? Write an equation y = ….. to model the rebound height after |

| |any number of bounces. Discuss how the rate of change is shown in each equation. |

Activities in quadrants B and D adapted from Contemporary Mathematics in Context, Course 1, Janson Publications, 1997

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Algebra |

|Essential Concept: Understand and apply recursion and iteration |

|Essential Skill: Ability to recognize, make, and apply connections |

|Essential Skill: Ability to construct and apply multiple connected representations |

|C |D |

|Given the following table representing functions f(x) and |Skydiving is an exciting but dangerous sport. Many precautions are |

|g(x): |taken to ensure the safety of the skydivers. The basic fact underlying |

| |these precautions is that acceleration due to the force of gravity is |

|x |32 feet per second per second (written as 32 ft/sec2). Thus, each |

|f(x) |second that the skydiver is falling, her speed increases by 32 ft/sec |

|g(x) |(ignoring air resistance and other complicating factors; focus only on |

| |the force of gravity). Determine both the recursive and explicit |

|0 |formulas that model the total distance fallen by a skydiver after each |

|0 |second before her parachute opens. Describe the method(s) you used to |

|0 |find these formulas. What type of function is represented by these |

| |formulas? How do you know this? Compare the different representations |

|1 |(table, graph, explicit form, and recursive form) of your function to |

|16 |other types of functions you know. |

|16 |(See student investigation sheet and problem-based instructional task |

| |lesson plan - A Recursive View of Skydiving - in Appendix B) |

|2 | |

|48 | |

|64 | |

| | |

|3 | |

|80 | |

|144 | |

| | |

|4 | |

|112 | |

|256 | |

| | |

|5 | |

|144 | |

|400 | |

| | |

| | |

|Determine both the explicit and recursive formulas that | |

|represent f(x) and g(x). What type of functions are f and | |

|g? Explain how you know this. Compare the different | |

|representations (table, graph, explicit formula, and | |

|recursive formula) for f and g. Describe similarities and | |

|differences in the representations. | |

|A |B |

|Given this table of function f(x) determine the values of |Below is a table that shows the distance, D(n), a skydiver has fallen |

|f(6), f(7), and f(10). |during each second when jumping from a plane. |

| | |

|x |Time |

|f(x) |in seconds (n) |

| |Distance Fallen during each second D(n) |

|0 | |

|0 |0 |

| |0 |

|1 | |

|16 |1 |

| |16 |

|2 | |

|48 |2 |

| |48 |

|3 | |

|80 |3 |

| |80 |

|4 | |

|112 |4 |

| |112 |

|5 | |

|144 |5 |

| |144 |

| | |

|Write a recursive formula for f(x). | |

| |Determine the distance fallen during 6, 7, and 10 seconds. Write a |

| |recursive formula for the distance fallen during each second, D(n). |

Activities in quadrants B, C, and D adapted from Navigating Through Discrete Mathematics in Grades 6-12, NCTM, 2008.

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Geometry |

|Essential Concept: Represent and solve geometric problems by specifying location using coordinates |

|Essential Skill: Problem solving |

|Essential Skill: Ability to recognize, make, and apply connections |

|C |D |

| | |

|Describe similarities and differences between using x–y |Write a brief report on how latitude and longitude are measured on Mars. |

|coordinates to locate a point and using latitude and |Describe similarities to and differences from latitude and longitude on |

|longitude to locate a point. Include at least one |Earth. Using images and information from the Internet or other sources, |

|similarity and one difference, and give examples to |show a map and a give the latitude and longitude coordinates of a mountain|

|illustrate. |on Mars. |

|A |B |

| | |

|Given a grid of latitude and longitude lines, plot the |A given map of the United States shows latitude and longitude in 5( |

|following locations on the grid. |intervals. A flight from Minneapolis to San Diego recorded the “way |

|(a) N 30(, E 60( |points” shown below. Mark the way points as accurately as possible on the |

|(b) S 15(, W30( |map. |

| |(a) At 3:46 GMT, N 42( 1.675’, W 101( 2.590’ |

| |(b) At 4:20 GMT, N 40( 40.125’, W 106( 18.641’ |

Activity in quadrant B adapted from Navigating Through Geometry in Grades 9–12, NCTM, 2001.

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Geometry |

|Essential Concept: Transformations |

|Essential Skill: Ability to recognize, make, and apply connections |

|Essential Skill: Ability to construct and apply multiple connected representations |

|C |D |

|Identify composition of transformations that maps the |Use a programming language (e.g., LOGO) and your knowledge of |

|preimage, triangle MLN, to the image, triangle M’’L’’N’’. |transformations to create a computer program that illustrates a rocket |

|State the coordinate rule and the matrix rule that would |launch. Write an explanation for your program to explain the |

|map the preimage to the image. |transformations included at each stage. |

|[pic] | |

|Decide whether this composition is commutative or not. | |

|Justify your decision why it is commutative or is not | |

|commutative through a graph, matrices, and coordinate | |

|rules. | |

|A |B |

|Identify composition of transformations that maps the |Below is a view of a rocket launch as an observer might see it. Identify|

|preimage, triangle MLN, to the image, triangle M’’L’’N’’. |the composition of transformations that would map rocket A to A’ to A’’.|

|State the coordinate rule and the matrix rule that would | |

|map the preimage to the image. | |

|[pic] |[pic] |

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Geometry |

|Essential Concept: Understand and apply properties and relationships of geometric objects |

|Essential Skill: Reasoning and proof |

|Essential Skill: Problem solving |

|C |D |

|If you wanted to divide a right triangle into two equal parts |Roger's Farm is a small corn and garden vegetable farm. Roger sells |

|(equal areas), how many ways are possible? Explain all |his produce at a local Farmer’s Market. His field is in the shape of|

|solutions and any generalizations you can make. |right triangle with the two legs of length 1295 feet and 405 feet, |

| |pictured below. He wants to divide his field into two equal areas by|

| |creating a dividing line parallel to AC. Divide the field according |

| |to these requirements. Prove that your solution is correct. What is |

| |the area in each of the two field sections? One section of the field|

| |will be planted with sweet corn. Search the Internet to find |

| |estimates for the yield of sweet corn. How much sweet corn can Roger|

| |produce? |

| |[pic] |

|A |B |

|Triangle BAC is a right triangle with [pic] being the right |Roger's Market is a small fruit and vegetable stand off of Highway |

|angle. Where should a line segment that is parallel to the |218 just North of Cedar Falls, Iowa. This year the owner wanted to |

|side [pic]be located so that the right triangle is divided |divide his field, so he could grow equal areas of corn and garden |

|into two equal areas? |vegetables. He could not figure out how to divide his field |

| |accurately. He showed me a sketch of his field that was a right |

|[pic] |triangle with the two legs 1295 feet and 405 feet respectively. He |

| |wanted to separate his field so that the dividing line was parallel |

| |to one of the legs. How should he divide the field? |

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Geometry |

|Essential Concept: Understand and apply properties and relationships of geometric objects |

|Essential Skill: Reasoning and proof |

|C |D |

|Use dynamic geometry software to investigate the |Mr. Conway has a yard that is in the shape of a right triangle. He wants to put a|

|following problem: |stake somewhere so that he can attach a leash to the stake and his dog to the |

|Find a point that is the same distance from all |leash in such a way that the dog can reach all three corners of the yard and the |

|three vertices of a right triangle. |shortest leash is used. Where should the stake be placed? Use dynamic geometry |

| |software to investigate this question. Make a conjecture for a solution. Compare |

|Based on your investigation, make a conjecture |your conjecture to those of other students in your class. Discuss and resolve any|

|about the point that is equidistant from all |differences, so that you have a final conjecture. [Teacher: Make sure final |

|three vertices of a right triangle. Compare your |conjecture agrees with the following: The midpoint of the hypotenuse of a right |

|conjecture to those of other students in your |triangle is equidistant from the three vertices of the triangle.] |

|class. Discuss and resolve any differences, so | |

|that you have a final conjecture. |Now you need to prove the conjecture. Consider the four diagrams below (see |

| |Diagram 7.37 in Appendix B), each of which illustrates a different proof of the |

|Prove your conjecture. |conjecture. Work in your groups to write a complete proof related to each |

| |diagram. You will be asked to write one proof on chart paper to display and |

| |explain to the whole class. |

| | |

| |After completing and discussing each of the four proofs, discuss these questions:|

| |• Describe the general strategy used in each proof. |

| |• How are the strategies and proofs similar and different? |

| |• What are some advantages and disadvantages of each proof method? |

| |• Are some proofs easier or more convincing to you than others? Why? |

| |• What mathematical ideas are used in each of these proofs? |

|A |B |

|Prove that the midpoint of the hypotenuse of a |Mr. Conway has a yard that is in the shape of a right triangle. He wants to put a|

|right triangle is equidistant from the three |stake somewhere so that he can attach a leash to the stake and his dog to the |

|vertices of the triangle. |leash in such a way that the dog can reach all three corners of the yard and the |

| |shortest leash is used. Where should the stake be placed? Prove your answer. |

Activities in quadrants B, C, and D adapted from Principles and Standards for School Mathematics, NCTM, 2000.

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Geometry |

|Essential Concept: Use trigonometry based on triangles and circles to solve problems |

|Essential Skill: Problem solving |

|Essential Skill: Ability to recognize, make, and apply connections |

|C |D |

|If you are given any two angle measurements and a side |A local concrete company is going to pour a concrete parking area. They |

|measurement of a triangle explain how you can find the |provide estimates for the amount of concrete needed before starting a |

|measures of the other angle and two sides. In your |project. If the parking area is an irregular shape, the company provides |

|explanation, defend why your method works. |estimates by dividing the area into quadrilaterals and triangles and then |

| |finding measurements. In order to save time, it is helpful to measure the |

| |least number of sides and angles by hand and to calculate mathematically the |

| |remaining measurements. |

| |Create several different non-quadrilateral designs, divide the areas into |

| |quadrilaterals and triangles, identify the needed measurements, and decide |

| |how to calculate the remaining measurements. Use the measurements to estimate|

| |the amount of concrete needed. Be sure to take into account the depth of the |

| |concrete. |

|A |B |

| |A new library is being built on current city property. Part of the plan for |

|Solve triangle ABC. |developing the property is to include a new bridge connecting the library and|

| |the existing play area. Approximately how long will the bridge need to be? |

| |[pic] |

|[pic] | |

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Geometry |

|Essential Concept: Use diagrams consisting of vertices and edges (vertex-edge graphs) to model and solve problems |

|Essential Skill: Ability to recognize, make, and apply connections |

|Essential Skill: Ability to construct and apply multiple connected representations |

|C |D |

|A complete graph is a graph in which there is exactly one edge|You have been hired by the city as a deputy street supervisor. Part |

|between each pair of vertices. Does a complete graph with 2 |of your job is to inspect city streets for potholes. Your area of |

|vertices have an Euler circuit? A complete graph with 3 |inspection is shown on the given street map. Devise a plan for |

|vertices? With 4 vertices? With 5 vertices? With n vertices?|street inspection that starts at your office, inspects each block at|

| |least once, ends at your office, and takes the least amount of time |

|Investigate and summarize your findings. Explain your process |(time is money). Assume it takes five minutes to walk a block for |

|and your reasoning. |inspections including corners and one minute to just walk a block |

| |without inspecting (called deadheading). (See attached lesson – |

| |Street Inspection – in Appendix B.) |

|A |B |

|Determine if the following vertex-edge graph has an Euler |The street network of a city can be modeled with a graph in which |

|circuit or path. If there is an Euler circuit or path, find |the vertices represent the street corners, and the edges represent |

|it. If there is not an Euler circuit or path, explain why not.|the streets. Suppose you are the city street inspector and it is |

|[pic] |desirable to minimize time and cost by not inspecting the same |

| |street more than once. |

| |[pic] |

| |a. In this graph of the city, is it possible to begin at the garage |

| |(G) and inspect each street only once? Will you be back at the |

| |garage at the end of the inspection? |

| |b. If not, find a route that inspects all streets, repeats the least|

| |number of edges possible, and returns to the garage. |

B and C quadrant activities adapted from Crisler, Nancy, Patience Fisher, and Gary Froelich, Discrete Mathematics through Applications 2nd Edition. Quadrant D lesson cited in appendix.

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Statistics and Probability |

|Essential Concept: Understand and interpret descriptive statistics |

|Essential Skill: Ability to recognize, make, and apply connections |

|C |D |

|Draw a scatterplot of the given (X,Y) data. Describe|Engage in the following example of statistical thinking: |

|patterns you see in the data. Discuss the strength |• Consider this question: Is size a useful predictor of price for houses in |

|of association between X and Y. Can you predict Y if|Des Moines – city-wide and in specific neighborhoods? |

|you know X? Explain your reasoning using graphs, |• Gather data that will help you answer this question. |

|regression, and correlation. |• Analyze the data using summary statistics, graphs, regression, and |

|(See data below.) |correlation. |

| |• Draw conclusions. Write a report based on your data and analysis that helps|

| |answer the initial question. Provide all necessary details. Explain and |

| |justify your conclusions. |

|A |B |

|Draw a scatterplot of the given (X,Y) data. Describe|Consider the data below, where |

|the relationship between X and Y. (See data below.) |X = size (in hundreds of sq. ft.) of a home sold in a particular neighborhood |

| |in Des Moines last spring and |

| |Y = selling price of the home (in thousands of dollars) |

| |Is there a relationship between size of house and price of house in this |

| |neighborhood? Can you predict price if you know size? Explain your reasoning. |

| |(See data below.) |

Data for quadrants A, B, and C:

|X |

|C |D |

|Each student should roll a regular die 20 times and record|In a trial in Sweden, a parking officer testified to having noted the |

|the number that comes up after each roll. Consider the |position of the valve stems on the tires on one side of a car. Returning|

|following events: |later, the officer noted that the valve stems were still in the same |

|• A: the number that comes up is even |position. The officer issues a ticket for overtime parking. However, the|

|• B: the number that comes up is a factor of 6 |owner of the car claimed he had moved the car and returned to the same |

|• C: the number that comes up is at most 4 |parking place. Who was right? Use probability to justify your answer. |

|• A[pic]B |(See attached task – Compound Events in a Trial in Sweden – in Appendix |

|• A[pic]B |B.) |

|• A|C | |

|Each student should find the relative frequency for each | |

|event based on his or her outcomes. Then, all students | |

|should combine their results. Find the class relative | |

|frequencies. Then, determine the theoretical probability | |

|for each of these events. Discuss the connection between | |

|the experimental relative frequencies and the theoretical | |

|probabilities. | |

|A |B |

|Suppose you roll a regular die and see which number comes |The diagram below shows the results of a two-question survey |

|up. |administered to 80 randomly selected students at Highcrest High School. |

| | |

| |(See attached diagram —Highcrest High School Survey—in Appendix B.) |

|List all the elements in the sample space. | |

| | |

|List the elements in each of events A, B, and C, below: |• Of the 2100 students in the school, how many would you expect to play |

| |a musical instrument? |

|A: the number that comes up is even |• Estimate the probability that an arbitrary student at the school plays|

|B: the number that comes up is a factor of 6 |on a sports team and plays a musical instrument. How is this related to |

|C: the number that comes up is at most 4 |estimates of the separate probabilities that a student plays a musical |

| |instrument and that he or she plays on a sports team? |

|Find the probability of these events: |• Estimate the probability that a student who plays on a sports team |

|P(A), P(B), P(C), P(A[pic]B), P(A[pic]B), P(A|C). |also plays a musical instrument. |

Activities in quadrants B and D adapted from Principles and Standards for School Mathematics, NCTM, 2000.

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Statistics and Probability |

|Essential Concept: Understand and interpret inferential statistics |

|Essential Skill: Reasoning and proof |

|C |D |

|Suppose a population is such that the population |M&M’S® Milk Chocolate Candies have been a popular treat ever since they were |

|percent for “success” is 0.13. Consider drawing |first manufactured in 1940. The candies come in different colors. Red candies |

|random samples of size 100 from this population, |were discontinued in 1976 due to concerns about food coloring, but by popular |

|and counting the number of “successes” in the |demand the color red was brought back in 1987. Today, M&M’S® Milk Chocolate |

|samples. Using technology, design and carry out a|Candies are produced so that there are 13% reds. Suppose you are a quality |

|simulation that will produce a simulated sampling|control manager at the M&M® factory. Part of your job is to make sure the factory|

|distribution in this situation. Based on your |produces the correct percentage of reds, that is, 13% reds. If this correct |

|simulated sampling distribution, what sample |percentage is not produced then the machines need to be shut down and reset, |

|outcomes would you consider to be rare events? |which is a very expensive process. Suppose you pull a random sample of 100 M&M’S®|

| |from the production line and you count 19 reds. Should you order the machines to |

| |be shut down and fixed? Explain and carry out the analysis and reasoning you |

| |would do to answer this question, including simulating or otherwise constructing |

| |a sampling distribution. (In fact, a quality control manager would probably pull |

| |several samples over time and use Statistical Process Control to help make a |

| |decision. In this problem, just assume that a single sample is drawn and use |

| |appropriate statistical reasoning to help make a decision.) |

|A |B |

|The given sampling distribution (provided) is |M&M’S® Milk Chocolate Candies are produced so that there are 13% reds. Suppose |

|based on counting the number of “successes” in |you are a quality control manager at the M&M® factory. Part of your job is to |

|random samples of size 100 drawn from a |make sure the factory produces the correct percentage of reds, that is, 13% reds.|

|population in which the population percent for |If this correct percentage is not produced then the machines need to be shut down|

|“success” is 0.13. Suppose you draw a random |and reset, which is a very expensive process. Suppose you draw a random sample of|

|sample of 100 and count 19 “successes.” Show |100 M&M’S® and count 19 reds. Compare this sample outcome to a given sampling |

|where this sample outcome would appear in the |distribution for this situation. Should you order the machines to be shut down |

|sampling distribution. Would you consider this |and fixed? Explain. |

|sample outcome to be a “rare event?” Explain. | |

Information from the M&M’S® company website:

Red percentage from:

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Quantitative Literacy |

|Essential Concept: Understand and apply number operations and properties |

|Essential Skill: Ability to recognize, make, and apply connections |

|Essential Skill: Problem solving |

|C |D |

| | |

|A four digit number (in base 10) |Research the change of the ISBN system from the former 10-digit format to the new 13-digit |

|aabb is a perfect square. Discuss |format. Explain the reasons for the new system and determine the benefits and any problems |

|ways of systematically finding this |that may occur. Also determine what types of entities are affected by this major change and |

|number. |how they are affected. |

| | |

|(“Novemberish” from |(“Check that Digit” prepared by Doug Schmid on |

|) |) |

|A |B |

| | |

|Find the congruent value of 100 in |Many codes including UPC product codes, ISBN book numbers, and credit card numbers have a |

|mod 2, mod 7, and mod 12. |“check digit” as the last digit of the code. This allows the companies and computers to |

| |determine if account numbers and identification numbers are valid by combining digits using an|

| |algorithm and checking to see if the result is divisible by a certain number or meets some |

| |criterion based on modular arithmetic. Determine if the following UPC numbers are valid using|

| |the given algorithm. |

| | |

| |(See “Check that Digit” prepared by Doug Schmid on |

| |) |

Activity in quadrant C adapted from the nrich. website. Activities in quadrants B and D adapted from the NCTM Illuminations website.

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Quantitative Literacy |

|Essential Concept: Understand, analyze, apply, and evaluate mathematical aspects of civic literacy. |

|Essential Skill: Communication |

|Essential Skill: Problem Solving |

|C |D |

|Determine the plurality, Borda, runoff, and |Research a variety of voting methods including (but not limited to) plurality, Borda, |

|sequential runoff winners for the following |runoff, and sequential runoff. Create a proposal for how to fairly conduct an |

|set of preferences. Create a situation for |election for Homecoming King and Queen. In your proposal identify the current method |

|the set of preferences and decide which |and compare it to what you believe would be the most fair. Create mock ballots |

|method of voting would be the most fair. |depending on the voting method and conduct a mock election with these different |

|Justify your choice. |ballots. Write a short paper reporting the results and why you propose the voting |

|1st |method you have chosen. |

|A | |

|B | |

|C | |

|C | |

| | |

|2nd | |

|D | |

|D | |

|B | |

|D | |

| | |

|3rd | |

|C | |

|A | |

|D | |

|A | |

| | |

|4th | |

|B | |

|C | |

|A | |

|B | |

| | |

|Total Number of Voters | |

|16 | |

|20 | |

|12 | |

|7 | |

| | |

|A |B |

|Determine the plurality, Borda, runoff, and |You have been chosen to serve on the committee that decides who this year's Homecoming|

|sequential runoff winners for the following |King and Queen will be. As a committee, you have already determined the three sets of |

|set of preferences. |finalists to be, in no particular order, Alan and Alice, Bob and Betty, and Carl and |

| |Cathy. All finalists are seniors. You have already held elections in each class |

|1st |through class meetings and have collected the following results: |

|A | |

|B | |

|C |Freshmen |

|C |Sophomores |

| |Juniors |

|2nd |Seniors |

|D | |

|D |1st |

|B |Alan/Alice |

|D |Bob/Betty |

| |Carl/Cathy |

|3rd |Carl/Cathy |

|C | |

|A |2nd |

|D |Bob/Betty |

|A |Alan/Alice |

| |Bob/Betty |

|4th |Bob/Betty |

|B | |

|C |3rd |

|A |Carl/Cathy |

|B |Carl/Cathy |

| |Alan/Alice |

|Total Number of Voters |Alan/Alice |

|16 | |

|20 |Class size |

|12 |60 students |

|7 |50 |

| |students |

| |40 students |

| |30 students |

| | |

| | |

| |Look at the election information above. In your opinion, which couple should reign as |

| |Homecoming King and Queen? Who would finish 2nd and 3rd? Write an explanation |

| |explaining your methodology for determining first, second and third. |

Quadrant A adapted from Discrete Mathematics Through Applications, 2nd Edition, W.H. Freeman and Company, New York: 2000, p. 13. Quadrant B activity retrieved and adapted on 4/20/06 from a lesson from the The Discrete Math Project:

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Quantitative Literacy |

|Essential Concept: Understand, analyze, apply, and evaluate mathematical aspects of civic literacy. |

|Essential Skill: Ability to recognize, make, and apply connections |

|Essential Skill: Communication |

|C |D |

| | |

|Suppose that C = {1, 3, 4, 6, 8} and D = {2, |Logical operators (sometimes called Boolean operators) are used by most Internet |

|3, 4, 8, 10, 12}. |search engines and the search feature of most websites. Research at least two |

| |search engines or search features on websites for information about how they explain|

|(a) Find C ∩ D. |and use logical operators. (Hint: Check the Advanced Search link.) Write a brief |

|(b) State and solve a problem involving a |report that addresses the following questions: |

|different set operation on these sets. |• What information is provided about the logical (or Boolean) operators used? |

|(c) Illustrate the problems in Parts (a) and |• How are the operators related to set operations like intersection and union? |

|(b) using Venn diagrams. |• How can the operators be illustrated with Venn diagrams? |

| |• For each of the operators, describe an Internet search that would involve that |

| |operator. |

| |• Is one logical operator used as the “default”? If so, explain why you think this |

| |is done. |

|A |B |

| | |

|Suppose that C = {1, 3, 4, 6, 8} and D = {2, |Alice conducted a survey of her friends and found that Sam, Juan, Hannah, Ryan, and |

|3, 4, 8, 10, 12}. Find C ∩ D. |Jesse liked using Google for Internet searches. Holly, Sam, Jesse, Finley, and Alex |

| |liked using Yahoo as their search engine. How many of Alice’s friends like using |

| |both search engines? Use set notation to show and explain your solution. |

|Rigor and Relevance Framework |

|Grade Band: 9–12 |

|Strand: Quantitative Literacy |

|Essential Concept: Understand and apply the mathematics of systematic counting |

|Essential Skill: Ability to recognize, make, and apply connections |

|Essential Skill: Problem solving |

|C |D |

|Find the value of P(7, 3) and |A ribonucleic acid (RNA) is a messenger molecule associated with deoxyribonucleic acid (DNA). RNA |

|C(7, 3). Describe the |is made up of a “chain” (sequence) of smaller molecules called nucleotides. The nucleotides |

|difference between permutations|contain the bases: U(uracil), C(cytosine), G(guanine), and A(adenine). It is difficult to observe|

|and combinations. Write a |exactly what an entire RNA chain looks like, however it is possible to observe fragments of a chain|

|formula, using P(n, k) and C(n,|by breaking it up with certain enzymes. Knowledge about these fragments can sometimes determine |

|k) that shows the relationship |the makeup of an entire chain of RNA. The “G-enzyme” will break an RNA chain after each G(guanine)|

|between these two numbers. |link. The “U-C-enzyme” will break an RNA chain after every U(uracil) and every C(cytosine). |

| |Consider the RNA chain AGUGGAUUGUCAUGA. A G-enzyme will break this chain into the fragments AG, |

| |UG, G, AUUG, UCAUG, and A. While the U-C-enzyme will break the same chain into the fragments AGU, |

| |GGAU, U, GU, C, AU, and GA. Unfortunately, the fragments of a broken-up chain may be mixed up and |

| |in the wrong order. |

| | |

| |Suppose an RNA chain is broken by a G-enzyme into the fragments AUG, AAC, CG, and AG. While the |

| |U-C-enzyme breaks the same RNA chain into the fragments GC, GAAC, and AGAU. What is the complete |

| |RNA chain of 10 bases? |

|A |B |

|Given the 4 letters A, B, C, |A deoxyribonucleic acid (DNA) molecule is made up of a “chain” (sequence) of smaller molecules |

|and D. How many different |called nucleotides. The nucleotides contain the bases A(adenine), C(cytosine), G(guanine), and |

|5-letter sequences are possible|T(thymine). In 1952, building on their predecessors’ research in genetics, James Watson and |

|if letters can be repeated? |Francis Crick realized that the DNA molecule was too thick to be a single strand. After trying |

| |several models, they made one in which two strands were wrapped around each other (a twisted |

| |ladder). Today this twisted-two-strand model (called a double helix) is accepted as the correct |

| |structure for DNA. In addition, in the early 1950s American scientist Edwin Chargoff made an |

| |important discovery about the four nitrogenous bases. Chargoff’s work led to the discovery that |

| |across the rungs of the DNA twisted ladder Adenine always pairs with Thymine and Cytosine always |

| |pairs with Guanine. |

| | |

| |Given this information, how many different 5 molecule sequences are possible? |

Student Tasks, Lesson Plans, Resources, Data, and Diagrams

for

Selected Quadrant Examples

Student Investigation – High School – Algebra – Recursion and Iteration

See Rigor and Relevance Quadrant D Example from Appendix A

(Adapted from Navigating Through Discrete Mathematics in Grades 6-12, NCTM, 2008.)

A Recursive View of Skydiving

Skydiving is an exciting but dangerous sport. Many precautions are taken to ensure the safety of the skydivers. The basic fact underlying these precautions is that acceleration due to the force of gravity is 32 feet per second per second (written as 32 ft/sec2). Thus, each second that the skydiver is falling, her speed increases by 32 ft/sec. (Throughout this investigation we ignore air resistance and other complicating factors; we focus only on the force of gravity.)

In this investigation, look for answers to these four questions:

• What are recursive and explicit formulas for the total distance fallen by a skydiver after each second before her parachute opens?

• What methods can you use to find these formulas?

• How are quadratic functions involved and why?

• How do the formulas, tables, and graphs for quadratic functions compare to those for other functions you have studied?

To help answer these questions, consider the following table, which you will complete in the problems below. [You may need to write your solutions to problems on a separate sheet of paper.]

A Skydiver’s Speed and Distance Fallen Before the Parachute Opens

|Time |Instantaneous |Average Speed |Distance Fallen |Total Distance Fallen |

|in seconds |Speed |during each second |during each second |after n seconds |

|n |at time n | |D(n) |T(n) |

|0 |0 |0 |0 |0 |

|1 sec |32 ft/sec |16 ft/sec |16 ft |16 ft |

|2 sec |64 ft/sec |48 ft/sec |48 ft |64 ft |

|3 sec | | | | |

|4 sec | | | | |

|: |: |: |: |: |

|n sec | | | | |

1. Explain each entry in the row corresponding to Time = 1 sec in the table above. (The basis for computing all entries is the fact that acceleration due to gravity is 32 ft/sec2.)

2. Explain each entry in the row corresponding to Time = 2 sec.

3. Complete the table for Time = 3 sec and Time = 4 sec. Compare your table entries to those of some of your classmates. Discuss and resolve any differences.

Distance Fallen During Each Second – Now you will use the completed table to help find formulas for distance fallen. First, consider distance fallen during each second.

4. Find recursive and explicit formulas for D(n), Distance Fallen during the nth second, as follows. (For this problem, ignore the row in the table for Time = 0 sec.)

a. If NOW is the Distance Fallen during any given second and NEXT is the Distance Fallen during the next second, write an equation for NEXT in terms of NOW.

b. Rewrite the NEXT/NOW equation using D(n) and D(n-1).

That is, if D(n) = Distance Fallen during the nth second, and D(n-1) = Distance Fallen during the (n-1)st second, write an equation for D(n) in terms of D(n-1). (This is a recursive formula since D(n) is expressed in terms of a previous value, D(n-1).)

c. If D(n) = Distance Fallen during the nth second, write an equation for D(n) in terms of n. Explain how you got your equation and why it is correct.

(A formula like this, where D(n) is written as a function of n, is called an explicit or closed-form formula.)

Total Distance Fallen After n Seconds – The main goal of this investigation is to find formulas for T(n), the Total Distance Fallen after n seconds. You will use several methods to do this:

• use a general analysis

• use an arithmetic sequence

• use the method of Finite Differences.

Each of these methods is carried out in the next three problems.

5. Find formulas for Total Distance Fallen after n seconds, as follows. Let T(n) = Total Distance Fallen after the nth second.

a. As part of Problem 3, you computed T(3). Describe how you computed T(3).

b. Describe all the methods you can think of for how to compute T(n).

c. Write a formula for T(n) in terms of T(n-1) and D(n). (This is a recursive formula since T(n) is expressed in terms of the previous value, T(n-1).)

6. Find an explicit formula for T(n) by summing an arithmetic sequence, as follows. (This is optional – for those who have studied arithmetic sequences.)

a. One way that you may have described in Problem 5b for finding T(n) is to sum all the terms up to D(n) in the D(n) column. If you didn’t already describe this in Problem 5b, explain here why this is a valid method for computing T(n).

b. You found in Problem 4c above that D(n) = D(n-1) + 32 ft (ignoring the row for Time = 0 sec). This formula shows that you add a constant, 32, each time to get the next value of D(n). Thus, the terms D(n) form an arithmetic sequence. Therefore, T(n) = the sum of the arithmetic sequence: D(1) + D(2) + … + D(n). Compute this sum to find an explicit formula for T(n) in terms of n.

7. Another way to find an explicit formula for T(n) is to use a finite differences table. Here’s how it works.

a. Complete the three remaining entries in the bottom of the table below.

Finite Differences Table

|n |T(n) |1st Differences |2nd Differences |

| | |(entry in the previous column) |(entry in the previous column) |

| | |– (entry just above it) |– (entry just above it) |

|1 |16 |--------------- |--------------- |

|2 |64 |64 - 16 = 48 |--------------- |

|3 |144 |144 – 64 = 80 |80 – 48 = 32 |

|4 |256 |256 – 144 = 112 | |

|5 |400 | | |

b. Describe the pattern in the 2nd differences column.

c. Now we apply a key fact: If the nth differences in a finite differences table are constant, then the formula for T(n) is an nth-degree polynomial. In this case, the 2nd differences are constant, so the formula for T(n) is a 2nd-degree polynomial, that is, the formula is quadratic. (Proving this key fact is not too hard, but it will take too long to do it now. If you are interested in the proof, you can ask your teacher for guidance or search for some references.)

So we know that T(n) is quadratic and thus it looks like:

T(n) = an2 + bn + c

Now we need to find the coefficients, a, b, and c. One way to find a, b, and c is to generate and solve a system of three linear equations. To help us do this, we know the value of T(n) for several values of n. Thus we get:

T(n) = an2 + bn + c

n=1 ( 16 = a + b + c

n=2 ( 64 = 4a + 2b + c

n=3 ( 144 = 9a + 3b + c

Explain the details of how these three equations are generated.

d. Now you need to solve this system of three linear equations. One way to do so is by using matrices. (For another method, see Problem 8 below.) To begin, a system of linear equations like this can be represented using matrices, as follows:

[pic]

Explain where all the entries in the matrices come from, and why this matrix equation is equivalent to the linear system in Part c.

e. You can solve this matrix equation by multiplying both sides of the equation on the left by the inverse of matrix [pic].

This gives you: [pic].

Find this inverse matrix and carry out the multiplication to solve the matrix equation. (You may want to use your calculator to carry out these computations.) What are the values for a, b, and c?

f. Using the values for a, b, and c that you just found, what is the formula for T(n)?

g. Check the formula you found in Part f by evaluating it for some values of n, and verifying that you get the same values for T(n) as in the tables above.

8. [Optional] Another way to solve systems of linear equations like the system in Problem 7 is to use algebra without matrices. To do this, you need to combine and manipulate the three equations in Part c until you can solve for a, b, and c. The combining and manipulating is similar to what you do for a system of two linear equations, but more complicated since there are more equations. Try this method. Check that you get the same solution as in Part f. (For an example of how this is done in a similar problem, see Mission Mathematics, Grades 9-12, NCTM, p. 23.)

9. Summary

a. What are recursive and explicit formulas for T(n), the total distance fallen by the skydiver?

b. Describe the methods you used for finding the formulas in Part a.

c. The explicit formulas that you found for T(n) are quadratic functions. Compare patterns of the quadratic functions that you have worked with in this investigation to patterns of linear and exponential functions you have studied previously, as follows:

• Examine the list of values for T(n) shown in the table at the beginning of this investigation (the last column in the table on page 1). How is the pattern of change shown in the list of values for T(n) different from the pattern of change in tables for linear and exponential functions?

• How is the recursive formula you found for T(n) in Problem 5c different from recursive formulas for linear and exponential functions?

• How are the explicit formulas you found for T(n) in Problems 6 and 7 different from the explicit formulas for linear and exponential functions?

Adapted from Navigating Through Discrete Mathematics in Grades 6-12, NCTM, 2008.

A Recursive View of Skydiving

Problem-Based Instructional Task Lesson Plan

________________________________________________

Objectives/Benchmarks:

• Mathematical modeling – Use mathematical modeling to find recursive and explicit formulas for distance fallen as governed by the force of gravity alone.

• Recursive and explicit formulas – Develop and apply formulas.

• Arithmetic sequences – Review and apply the sum of an arithmetic sequence.

• Finite differences tables – Learn and apply finite differences tables.

• Quadratic functions – Represent quadratic functions explicitly and recursively, and compare to previous work with linear and exponential functions.

Title: A Recursive View of Skydiving

(Adapted from Navigating Through Discrete Mathematics in Grades 6-12, NCTM, 2008.)

Grade Level/Course: Grade 10-12, depending on students and curriculum

Pre-requisite Knowledge:

• Arithmetic sequences (although can be skipped if necessary)

• Quadratic functions (some initial study)

• Linear and exponential functions

NCTM Standard(s): (shaded)

|NCTM Content |Number & Operations |Algebra |Geometry |Measurement |Data Analysis & Probability|

|Standards | | | | | |

|NCTM Process Standards |Problem Solving |

|A |B |

Materials Needed:

• Copies of the student investigation sheet, “A Recursive View of Skydiving,” one for each student.

• Calculator or computer with inverse matrix capability

MAIN LESSON DEVELOPMENT

LAUNCH:

Purposeful introduction to the lesson, typically teacher led

Develop a situation, discussion, questions that:

• set the stage for the lesson

• motivate

• engage

• find out what students know (note formative assessment)

• foreshadow the big ideas of lesson

• keep it relatively brief

For this problem-based instructional task, ask students:

• Have you ever been skydiving, or know someone who has?

• What do you think influences how fast and how far a skydiver falls?

• Of the types of functions you have studied, which one do you think models the total distance fallen by a skydiver?

After a brief discussion of these questions, tell students that they will consider this last question carefully as they work through the investigation. They will have a good answer at the end. Then start students on the Explore part of the lesson - see below.

EXPLORE:

Get students engaged in investigating important mathematics, typically in teams.

Characteristics:

• effective guiding questions – in the task and by the teacher

• student-to-student communication

• high level of student engagement

• Note opportunity for formative assessment.

Teacher will:

• guide the students to work out the examples themselves, instead of the teacher and text presenting completed examples

• be prepared for and carry out mini-summaries as needed

• be prepared for and help students and teams with key points and trouble spots

For this problem-based instructional task, use the attached student investigation sheet, entitled “A Recursive View of Skydiving.” This investigation will likely work best if you put the students into teams, so that they can work together and discuss. Students will answer the structured set of questions and solve the problems in the attached investigation sheet.

Circulate and check on student teams. Be prepared to guide their work with questions. If many groups are having trouble on the same problem, you could bring the class together for a teacher-led discussion and resolution of that problem, then put them back into their teams to continue.

In particular, be sure to check all teams’ work on Problems 3, 5c, and 7e.

SUMMARIZE

Closure and summary for the lesson, typically teacher led:

• Identify the 2-4 main points of the lesson.

• Ask 2-4 questions that get students to review, synthesize, and explain the main points.

• Note opportunity for formative assessment.

For this problem-based instructional task, students should answer the questions in the Summary section of the investigation, namely, Problem 9. After students have answered these questions, you could bring the class together and have randomly chosen teams explain their answers to each of the 3 questions. Discuss and resolve any errors and confusions.

Modifications/Extensions:

• If students have not studied sums of arithmetic sequences, they can skip Problem 6, or do some preview meaningful distributed practice on arithmetic sequences before this lesson.

• For some students, you may ask them to solve the system of equations in Problem 7 using the optional method in Problem 8.

Checking for Understanding (Note formative assessment, in addition to above.)

• What will you assess?

o This problem-based instructional task has clear focus questions for the students at the beginning. You should make sure students are focused on those questions and can answer them at the end.

o Also, see the objectives at the beginning of this lesson plan.

o Also, see the Summary problem, Problem 9.

• How will you assess it?

o The engaged student discussion and teacher questioning will provide ongoing formative assessment throughout the lesson. See above.

o You can use student work on Problem 9 as one way to assess students’ learning in this problem-based instructional task. You might ask them to write answers to the questions in Problem 9, which are turned in and graded; or you might ask student teams to give brief reports on Problem 9.

o You might create a checklist, and make checks as the student teams are working and you are circulating, to record students’ successful completion of the key problems identified in the Explore section, namely, Problems 3, 5c, and 7e.

o You might create a quiz with a similar problem.

--------------------(REFLECTION AFTER TEACHING THE LESSON)------------------

• How did the students perform? How do you know?

• What parts of the lesson went well? Not so well? How do you know?

• How will you use this information to guide future instructional decisions, about this lesson and more generally?

Student Investigation/Project – High School – Vertex-Edge Graphs

See Rigor and Relevance Quadrant D Example in Appendix A

Street Inspection

You have been hired by the city as a deputy street supervisor. Part of your job is to inspect the city’s streets for potholes. Your area of inspection is shown on the given street map.

Devise a plan for street inspection that starts at your office, inspects each block at least once, ends at your office, and takes the least amount of time (time is money). Assume it takes five minutes to walk a block for inspections including corners and one minute to just walk a block without inspecting (called deadheading).*

Begin by completing the Traveling Networks Group Project. Next devise inspection plans for a variety of rectangular (both square and non-square) streets. Look for patterns in your plans. You should now be ready to complete the original task of inspecting the city streets for which you were hired. Explain why your plan is the most cost effective.

Can you think of other occupations that might want to use your plan or a variation of your plan?

*This problem is a variation of the Chinese postman problem first studied in 1962 by Meigu Guam, a Chinese mathematician.

[pic]

[pic]

Traveling Networks

Group Project

1. Determine which of the thirty networks can be traveled. Indicate where you started and where you ended your travel.

2. For the networks that can be traveled indicate whether or not your trip was a path or a circuit.

3. Complete the following conjectures.

The degree of a pass through vertex must be …

A network can be traveled whenever ...

A network can be traveled via a path whenever ...

A network can be traveled via a circuit whenever ...

4. Write a convincing argument that your conjectures are true. Your argument should convince someone that the conjectures are true for any network, not just the thirty you were given. Your argument should be written so that a non-math person would be convinced. You may use diagrams as part of your argument. You may assume that the person has the list of vocabulary words and that you have fully explained the terms to the person.

5. Return to the original street inspection problem for which you were hired. Why does this problem require an Eulerization of the street map? Investigate several other rectangular (both square & non-square) street maps and look for patterns in Eulerizing these street maps.

6. Find the most cost effective inspection plan for the street map you were hired to inspect and explain why your plan is the most effective.

Network Vocabulary

Network: A network is a collection of vertices (points) connected by edges (segments).

[pic]

Traveling a Network: A network can be “traveled” if the network can be drawn with a pencil without lifting the pencil off the paper and without retracing any edges. Vertices can be passed through more than once.

Odd Vertex: An odd vertex has an odd number of edges with that vertex as an endpoint.

Even Vertex: An even vertex has an even number of edges with that vertex as an endpoint.

Loop: A loop is an edge (street – called a cul-de-sac) with only one endpoint. Note: A loop is counted twice at its endpoint. See Figure 1.

Euler Path: Traveling a network such that the starting point and ending point are different.

Euler Circuit: Traveling a network such that the starting and ending points are the same.

Note: An Euler path or circuit can be recorded by listing the vertices in the order in which they are passed through. The first vertex listed is where the path or circuit starts and the last vertex listed is where the path or circuit ends. For example, a possible Euler path for the network in Figure 1 can be recorded as A,E,E,D,A,B,C,D. When a loop is traveled it is indicated by listing its endpoints twice.

Eulerizing a Network: The process of revising a network by adding edges so that the revised network has an Euler circuit.

Networks

A “•” indicates that there is a vertex joining two or more edges. You may only move from one edge to another at a vertex.

1. 2. 3.

[pic] [pic] [pic]

4. 5. 6.

[pic] [pic][pic]

7. 8. 9. 10.

[pic] [pic] [pic] [pic]

11. 12. 13.

[pic] [pic][pic]

14. 15. 16.

[pic] [pic][pic]

17. 18. 19.

[pic] [pic][pic]

20. 21.

[pic] [pic]

22. 23.

[pic] [pic]

24. 25.

[pic] [pic]

26. 27.

[pic] [pic]

28. 29.

[pic] [pic]

30.

[pic]

The Paths and Circuits

|Network |Ordered Listing of Vertices for Path or Circuit |

|1 | |

|2 | |

|3 | |

|4 | |

|5 | |

|6 | |

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|28 | |

|29 | |

|30 | |

Can a Network be Traveled?

|Network |# of odd vertices* |# of even vertices** |Can it be traveled? |Path or Circuit?*** |

|1 | | | | |

|2 | | | | |

|3 | | | | |

|4 | | | | |

|5 | | | | |

|6 | | | | |

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|24 | | | | |

|25 | | | | |

|26 | | | | |

|27 | | | | |

|28 | | | | |

|29 | | | | |

|30 | | | | |

| | | | | |

| * An odd vertex has an odd number of edges with that vertex as an endpoint. |

| ** An even vertex has an even number of edges with that vertex as an endpoint. |

| *** An Euler path is a traveled network such that the starting vertex and ending vertex are different. |

| An Euler circuit is a traveled network such that the starting and ending vertices are the same. |

Streets Project – What to hand in.

The Group

1. An ordered listing of vertices for the paths and circuits

2. Data on number of odd and even vertices.

3. Complete statements of the conjectures from Group Project #3.

4. Convincing arguments why your conjectures are true, which should include:

a. A complete explanation why pass through vertices in paths and circuits must be even,

b. A complete explanation why in a circuit the vertex that is both the starting and ending vertex must be even, and

c. A complete explanation why in a path both the starting and ending vertices must be odd.

5. Diagrams indicating an Eulerization of various street networks such as: 2x2, 3x3, 4x4, 1x2, 1x3, 1x4, 2x3, 2x4, 2x5, 3x4, 3x5.

6. An Eulerization of the original street inspection problem with an indication of the least amount of time needed to complete the inspection.

Each Individual

1. An evaluation of your participation in the group.

2. An evaluation of the participation of the other members of your group.

Grading (Included in Unit Test Category)

75 points: Correctness of all work

5 points: Completion of all of the required work

5 points: Effective group member

5 points: Organization of hand-in materials

5 points: Clarity of hand-in materials

5 points: Neatness of hand-in materials

Copyright © 2006, J. Maltas, Price Lab School, University of Northern Iowa

Resources – High School

See Rigor and Relevance Quadrant Examples in Appendix B

Highcrest High School Survey

| | |Do you play a musical instrument?|

| | |Yes |No |

|Do you play on | | | |

|a sports team? |Yes |14 |32 |

| | | | |

| |No |20 |14 |

(From NCTM’s Principles and Standards for School Mathematics, p. 331)

[pic]

(From NCTM’s Principles and Standards for School Mathematics, p. 356)

Compound Events in a Trial in Sweden

In a trial in Sweden, a parking officer testified to having noted the position of the valve stems on the tires on one side of a car. Returning later, the officer noted that the valve stems were still in the same position. The officer noted the position of the valve stems to the nearest "hour." For example, in figure 7.26 [p. 332] the valve stems are at 10:00 and at 3:00. The officer issued a ticket for overtime parking. However, the owner of the car claimed he had moved the car and returned to the same parking place.

[pic]

The judge who presided over the trial made the assumption that the wheels move independently and the odds of the two valve stems returning to their previous "clock" positions were calculated as 144 to 1. The driver was declared to be innocent because such odds were considered insufficient—had all four valve stems been found to have returned to their previous positions, the driver would have been declared guilty (Zeisel 1968).

Given the assumption that the wheels move independently, students could be asked to assess the probability that if the car is moved, two (or four) valve stems would return to the same position. They could do so by a direct probability computation, or they might design a simulation, either by programming or by using spinners, to estimate this probability. But is it reasonable to assume that two front and rear wheels or all four wheels move independently? This issue might be resolved empirically. The students might drive a car around the block to see if its wheels do rotate independently of one another and decide if the judge's assumption was justified. They might consider whether it would be more reasonable to assume that all four wheels move as a unit and ask related questions: Under what circumstances might all four wheels travel the same distance? Would all the wheels travel the same distance if the car was driven around the block? Would any differences be large enough to show up as differences in "clock" position? In this way, students can learn about the role of assumptions in modeling, in addition to learning about the computation of probabilities.

Students could also explore the effect of more-precise measurements on the resulting probabilities. They could calculate the probabilities if, say, instead of recording markings to the nearest hour on the clockface, the markings had been recorded to the nearest half or quarter hour. This line of thinking could raise the issue of continuous distributions and the idea of calculating probabilities involving an interval of values rather than a finite number of values. Some related questions are, How could a practical method of obtaining more-precise measurements be devised? How could a parking officer realistically measure tire-marking positions to the nearest clock half-hour? How could measurement errors be minimized? These could begin a discussion of operational definitions and measurement processes.

(From: NCTM’s Principles and Standards for School Mathematics, pp. 332-333)

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