Rigor and Relevance
Quadrant Examples
High School
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
Consider this function:
i(t)=10(.95)t
Using different methods and different representations (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. / D
Research medications used to help control diseases. Find data to build functions modeling the amount remaining in the bloodstream at various times. Find the half-life, if appropriate. Discuss some dosage strategies.
A
Consider this function:
i(t)=10(.95)t
Determine i(40) / B
Insulin is an important hormone produced by the body. In 5% to 10% of all diagnosed cases of diabetes, the disease is due to the body’s inability to 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
Solve this equation:
13 = 0.10(x – 200) + 5
Use different methods and different representations (including tables, graphs, analytical methods, symbolic reasoning, using technology, etc.). Analyze and evaluate each method and representation, including advantages and disadvantages of different methods and representations. / D
Research some text-messaging plans available in your area. Find a mathematical model that represents each plan. Given your text-messaging habits and the mathematical models, evaluate these plans, and choose the one that is best for you. Explain your choice and why you think it’s the best plan for you. Include graphs, equations, and tables in your explanation, as appropriate.
A
Solve this equation:
13 = 0.10(x – 200) + 5 / B
Consider this text messaging plan for your cell phone: You pay $5 per 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
Consider this algebraic expression:21.98x – 3.75x – 1.50x – 1.65x – 10x – 365,000. Write four other expressions that are equivalent to the given expression (use expansion and simplifying). Explain how you know the expressions are equivalent and state the properties used. / D
Find the current price of one of your favorite CDs. Profit for the record company that produces the CD is a function of CD sales. Assume that the record company had the following production and distribution costs related to this CD.
- $365,000 for studio, video, touring, and promotion expenses;
- $3.75 per CD for pressing and packaging costs;
- $1.50 per CD for discounts to music stores;
- $1.65 per CD for other discounts.
- $10.00 per CD paid to the band
A
Consider this algebraic expression:
21.98x – 3.75x – 1.50x – 1.65x – 10x – 365,000. Put the expression in simplest form. /
B
Profit for a record company is a function of CD sales. For a given band, the record company had the following production and distribution conditions to consider.$365,000 for studio, video, touring, and promotion expenses;
$3.75 per CD for pressing and packaging costs;
$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 FrameworkGrade Band:9–12
Strand: Algebra
Essential Concept: Understand, analyze, approximate, and interpret rate of change
Essential Skill: Communication
Essential Skill:Problem solving
CConsider this equation:
y = 27
Make a table and a graph, and write an equation using NOW and NEXT. Discuss how the rate of change is shown in each. /
D
Suppose you must compare the elasticity of several different brands of golf balls. Get a variety of golf balls and a tape measure. Begin the comparison by choosing one of the balls. Decide on a method for measuring the height of successive rebounds after the ball is dropped from a height of at least 8 feet. (You may want to use technology to 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
Consider this equation:
y = 27
Describe the rate of change. How is the value of y changed from one integer value of x to the next? /
B
Most popular American sports involve balls of some sort. One of the most important factors in playing with those balls is the bounciness or elasticity of the ball. If a new golf ball is dropped onto a hard surface, it should rebound to about of its drop height. Suppose a new golf ball drops downward from a height of 27 feet and keeps 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 FrameworkGrade 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
Given the following table representing functions f(x) and g(x):
x / f(x) / g(x)
0 / 0 / 0
1 / 16 / 16
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. / D
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 (ignoring air resistance and other complicating factors; focus only on the force of gravity). Determine both the recursive and explicit formulas that model the total distance fallen by a skydiver after each second before her parachute opens. Describe the method(s) you used to find these formulas. What type of function is represented by these formulas? How do you know this? Compare the different representations (table, graph, explicit form, and recursive form) of your function to other types of functions you know.
(See student investigation sheet and problem-based instructional task lesson plan - A Recursive View of Skydiving - in Appendix B)
A
Given this table of function f(x) determine the values of f(6), f(7), and f(10).
x / f(x)
0 / 0
1 / 16
2 / 48
3 / 80
4 / 112
5 / 144
Write a recursive formula for f(x). / B
Below is a table that shows the distance, D(n), a skydiver has fallen during each second when jumping from a plane.
Time
in seconds (n) / Distance Fallenduring each second D(n)
0 / 0
1 / 16
2 / 48
3 / 80
4 / 112
5 / 144
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 FrameworkGrade 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
Describe similarities and differences between using x–y coordinates to locate a point and using latitude and longitude to locate a point. Include at least one similarity and one difference, and give examples to illustrate. / D
Write a brief report on how latitude and longitude are measured on Mars. Describe similarities to and differences from latitude and longitude on Earth. Using images and information from the Internet or other sources, show a map and a give the latitude and longitude coordinates of a mountain on Mars.
A
Given a grid of latitude and longitude lines, plot the following locations on the grid.
(a) N 30, E 60
(b) S 15, W30 / B
A given map of the United States shows latitude and longitude in 5 intervals. A flight from Minneapolis to San Diego recorded the “way points” shown below. Mark the way points as accurately as possible on the 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:TransformationsEssential Skill:Ability to recognize, make, and apply connections
Essential Skill:Ability to construct and apply multiple connected representations
C
Identify composition of transformations that maps the preimage, triangle MLN, to the image, triangle M’’L’’N’’. State the coordinate rule and the matrix rule that would map the preimage to the image.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. /
D
Use a programming language (e.g., LOGO) and your knowledge of transformations to create a computer program that illustrates a rocket launch. Write an explanation for your program to explain the transformations included at each stage.A
Identify composition of transformations that maps the preimage, triangle MLN, to the image, triangle M’’L’’N’’. State the coordinate rule and the matrix rule that would map the preimage to the image./
B
Below is a view of a rocket launch as an observer might see it. Identify the composition of transformations that would map rocket A to A’ to A’’.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
If you wanted to divide a right triangle into two equal parts (equal areas), how many ways are possible? Explain all solutions and any generalizations you can make.
/D
Roger's Farm is a small corn and garden vegetable farm. Roger sells his produce at a local Farmer’s Market. His field is in the shape of 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?
A
Triangle BAC is a right triangle with being the right angle. Where should a line segment that is parallel to the side be located so that the right triangle is divided into two equal areas?
/B
Roger's Market is a small fruit and vegetable stand off of Highway 218 just North of Cedar Falls, Iowa. This year the owner wanted to divide his field, so he could grow equal areas of corn and garden vegetables. He could not figure out how to divide his field accurately. He showed me a sketch of his field that was a right 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 objectsEssential Skill:Reasoning and proof
C
Use dynamic geometry software to investigate the following problem:
Find a point that is the same distance from all three vertices of a right triangle.
Based on your investigation, make a conjecture about the point that is equidistant from all three vertices of a right triangle. Compare your conjecture to those of other students in your class. Discuss and resolve any differences, so that you have a final conjecture.
Prove your conjecture. / D
Mr. Conway has a yard that is in the shape of a right triangle. He wants to put a stake somewhere so that he can attach a leash to the stake and his dog to the 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? Use dynamic geometry software to investigate this question. Make a conjecture for a solution. Compare your conjecture to those of other students in your class. Discuss and resolve any differences, so that you have a final conjecture. [Teacher: Make sure final conjecture agrees with the following: The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices of the triangle.]
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 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
Prove that the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices of the triangle. / B
Mr. Conway has a yard that is in the shape of a right triangle. He wants to put a stake somewhere so that he can attach a leash to the stake and his dog to the 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:9–12Strand: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
If you are given any two angle measurements and a side measurement of a triangle explain how you can find the measures of the other angle and two sides. In your explanation, defend why your method works. / D
A local concrete company is going to pour a concrete parking area. They provide estimates for the amount of concrete needed before starting a project. If the parking area is an irregular shape, the company provides 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
Solve triangle ABC.
/ B
A new library is being built on current city property. Part of the plan for 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?
Rigor and Relevance Framework