Innovation Lab Network Performance Assessment Project Subject area/course: Mathematics/ Algebra 2 or Integrated Math Grade level/band: 10-­‐12 Task source: Stanford Center for Assessment, Learning, and Equity (SCALE) Stopping Distance STUDENT INSTRUCTIONS A. Task context: The Departments of Motor Vehicles in California and in New York both publish guidelines for ‘following distance’. Their guidelines are different. Which state’s guidelines do you think are most appropriate, based on your study of the mathematics of stopping distance? B. Final product: Prepare a recommendation to one state’s DMV or the other about changing their published guidelines so that the two states’ guidelines for following distance are consistent. • Support your recommendation with your own mathematical work (calculations, formulas, diagrams, graphs) and with relevant findings from your research (diagrams, graphs, formulas). • Be sure to make clear how any diagrams, graphs, and formulas you use represent the important quantities in the situation. • Your recommendation should be 1-­‐2 pages long, including 2-­‐3 short paragraphs of text and supporting mathematical work. • Use the feedback you receive from your teacher on your individual and group work to inform your final product. ADDITIONAL INFORMATION C. Knowledge and skills you will need to demonstrate on this task: • Identify the quantities that are relevant to the stopping distance of a vehicle. • Interpret given parameters – speed, reaction time, and distance between vehicle and cat – in the context of the situation. • Interpret a given table of data about braking distance in the context of the situation. • Apply your understanding of functions as tools for modeling relationships among real-­‐world quantities. • Recognize the different behaviors of linear, quadratic and exponential functions, and select which would be best for modeling the relationships in the situation. • Interpret and build on other students’ ideas about the relationship between speed and braking distance. • Conduct online research to find relevant graphs, diagrams, additional data and a formula for computing the stopping distance of a car. • Coordinate among representations – graphs, tables, formulas, and diagrams – to identify how the relevant quantities are represented in each representation. This work is licensed under a Creative Commons Attribution-­‐NonCommerical-­‐NoDerivatives 4.0 International License
1 Innovation Lab Network Performance Assessment Project Analyze guidelines for ‘following distance’ using the mathematics of stopping distance. • Synthesize your own mathematical work and the formulas, graphs, and diagrams from your research to support a recommendation about appropriate guidelines for following distance. • Clearly communicate your recommendation in writing, using mathematical representations to support your writing, in a way that your audience will understand. D. Materials needed: • Handout 1: The Situation (Initial Individual Notes) • Handout 2A: The Problem (Initial Individual Work) • Handout 2B: The Problem (Pair Work) • Handout 3: Making Sense of Others’ Ideas (Pair Work) • Handout 4: Building on Others’ Ideas (Individual Writing) • Handout 5: Resource Card (Reference) • Handout 6: Culminating Product (Individual Writing) • Handout 7: Following Distance Guidelines for NY and CA (Reference) • Handout 8: Three Scenarios to Consider (Reference) E. Time requirements: You will have approximately 1-­‐2 weeks to complete this task. Your teacher will provide details regarding the timeline and due dates. F. Scoring: Your work will be scored using SCALE Math Performance Assessment Rubric (Grades 9-­‐12). You should make sure you are familiar with the language that describes the expectations for proficient performance. •
This work is licensed under a Creative Commons Attribution-­‐NonCommerical-­‐NoDerivatives 4.0 International License
2 Innovation Lab Network Performance Assessment Project Subject area/course: Mathematics/Algebra 2 or Integrated Math Grade level/band: 10-­‐12 Task source: Stanford Center for Assessment, Learning, and Equity (SCALE) Stopping Distance TEACHER'S GUIDE A. Task overview: This task begins with an animated video clip showing a cat jumping into the road, and a car nearly hitting it, which motivates a framing question: Can the driver stop in time? This sets the context for a whole-­‐class brainstorm about what quantities might need to be considered in order to answer the question. The brainstorm creates some common ground for students to develop an approach to a well-­‐specified problem, which is to figure out if they would be able to stop in time, given certain parameters and data about the situation. Students work on the problem individually first and then in pairs or small groups. The central mathematics of the problem includes figuring out that there is a linear relationship between time and distance that can be modeled with the given speed and ‘reaction time’, and then figuring out how to model the more complicated relationship between speed and distance, given a table of data; this can be approached in several mathematically valid ways. While students work on the problem, the teacher circulates to listen for student comments and ideas that offer partial insights or suggest productive approaches or questions about the relationship between speed and distance; the teacher records selected comments/ideas on a ‘Student Ideas’ sheet in preparation for a whole-­‐class discussion. The teacher then facilitates a discussion organized around the student comments/ideas they have recorded. The purpose of this discussion is to give students a chance to refine and develop their own ideas and their peers’ ideas. Following work on the problem and the discussion about students’ ideas, students then consider three ideas from fictitious students about the table of data given in the problem. They choose from among these three ideas and any ideas from classmates shared by the teacher, and correct and develop 1-­‐2 of these ideas into coherent arguments. This is individual written work that should be assessed formatively. Students will use feedback from the teacher on their work to inform their work on the culminating product. Students then conduct research online about stopping distance and engage in a discussion about how a given formula relates to diagrams/graphs from their research. This discussion provides an opportunity to coordinate across different representations of the situation, with attention to representations of linear and quadratic relationships. The culminating individual product is to write a recommendation about guidelines for ‘following distance’. Students first analyze the guidelines published by the California Department of Motor Vehicles (DMV) and the New York DMV, which are inconsistent, and use the mathematics of stopping distance to justify their recommendation about which state’s guidelines are more appropriate. They are provided three scenarios to consider as they develop their recommendation. This work is licensed under a Creative Commons Attribution-­‐NonCommerical-­‐NoDerivatives 4.0 International License
1 Innovation Lab Network Performance Assessment Project B. Aligned standards: 1. Primary Common Core State Standards CCSS.MATH.CONTENT.HSF.IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. CCSS.MATH.CONTENT.HSF.BF.A.1 Write a function that describes a relationship between two quantities. CCSS.MATH.CONTENT.HSA.SSE.A.1 Interpret expressions that represent a quantity in terms of its context. CCSS.MATH.CONTENT.HSN.Q.A Reason quantitatively and use units to solve problems. CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively. CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP4 Model with mathematics. 2. Secondary Common Core State Standards CCSS.MATH.CONTENT.HSA.CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically. CCSS.MATH.PRACTICE.MP6 Attend to precision. CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning. 3. Critical abilities Analysis of Information: Integrate and synthesize multiple sources of information (e.g., texts, experiments, simulations) presented in diverse formats and media (e.g., visually, quantitatively, orally) in order to address a question, make informed decisions, understand a process, phenomenon, or concept, and solve problems while evaluating the credibility and accuracy of each source and noting any discrepancies among the data. Communication in Many Forms: Use oral and written communication skills to learn, evaluate, and express ideas for a range of tasks, purposes, and audiences. Develop and strengthen writing as needed by planning, revising, editing, and rewriting while considering the audience. Use of Technology: Present information, findings, and supporting evidence, making strategic use of digital media and visual displays to enhance understanding. Use technology, including the Internet, to research, produce, publish, and update individual or shared products in response to ongoing feedback, including new arguments or information. This work is licensed under a Creative Commons Attribution-­‐NonCommerical-­‐NoDerivatives 4.0 International License
2 Innovation Lab Network Performance Assessment Project C.
D.
E.
F.
Interpersonal Interaction and Collaboration: Develop a range of interpersonal skills, including the ability to work with others, to participate effectively in a range of conversations and collaborations. Modeling, Design, and Problem Solving: Use quantitative reasoning to solve problems arising in everyday life, society, and the workplace, e.g., to plan a school event or analyze a problem in the community, to solve a design problem or to examine relationships among quantities of interest. Plan solution pathways, monitoring and evaluating progress and changing course if necessary, and find relevant external resources, such as experimental and modeling tools, to solve problems. Interpret and evaluate results in the context of the situation and improve the model or design as needed. Time/schedule requirements: This task will take approximately 1-­‐2 weeks. See the outline below for a suggested implementation timeline. • Part A: The Situation and The Problem – 2-­‐3 days • Part B: Building on Others’ Ideas – 1-­‐2 days • Part C: Research and Discussion – 1-­‐2 days • Part D: Culminating Product – 2-­‐3 days Materials/resources: • Video of Tibbles the Cat: https://www.youtube.com/watch?v=R3yFku2L6V4 • Handout 1: The Situation (Initial Individual Notes) • Handout 2A: The Problem (Initial Individual Work) • Handout 2B: The Problem (Pair Work) • Handout 3: Making Sense of Others’ Ideas (Pair Work) • Handout 4: Building on Others’ Ideas (Individual Writing) • Handout 5: Resource Card (Reference) • Handout 6: Culminating Product (Individual Writing) • Handout 7: Following Distance Guidelines for NY and CA (Reference) • Handout 8: Three Scenarios to Consider (Reference) • Student Ideas Discussion Tool (Teacher tool; optional for students) Prior knowledge: Students must have a solid understanding of using functions to model real world phenomena, and some experience using quadratic functions to model real world phenomena. Students must have experience graphing linear, quadratic, and exponential functions. Connection to curriculum: None listed. This work is licensed under a Creative Commons Attribution-­‐NonCommerical-­‐NoDerivatives 4.0 International License
3 Innovation Lab Network Performance Assessment Project G. Teacher instructions: Part A: The Situation and The Problem [Whole Class, Individual, and Pair Work: 2-­‐3 days] Watch video of Tibbles the Cat [Whole class: 5 mins] • Play first 32 seconds of video: https://www.youtube.com/watch?v=R3yFku2L6V4 Handout 1: The Situation [Individual or Pairs: 5-­‐10 mins] • Provide Handout 1 for students to make some initial notes on about what they would need to know in order to begin to develop an approach to the problem. • Students are not to start solving the problem, but to just brainstorm the quantities that might be important to consider. Discussion [Whole class: 5-­‐10 mins] • Students share ideas about what quantities might be important to consider in the situation. • Record students’ ideas on the board, and as ideas accumulate, do some tentative grouping into categories: related to the driver (human), related to the vehicle, and related to driving conditions. Keep a record of these ideas for reference in Part D. Handouts 2A & 2B: The Problem [Individual: 10 mins, Pairs/Groups: 15 mins] • Provide Handout 2A to each student, and ask them to get started on the problem on their own for 10 minutes. • Provide Handout 2B to each pair of students, and ask them to continue working on the problem together, adding to their individual work on Handout 2A as needed. Student Ideas Discussion Tool [Use as students work on Handouts 2A and 2B] • Circulate and notice patterns in students’ approaches as they work. Use a blank Student Ideas sheet (discussion tool with speech/thought bubbles) to record students’ comments and questions. • Look for: Initial evidence of understanding the concept of a function, understanding the concept of covariation, use of variables, use of coordinate axes, attempts to determine type of function, recognizing possibilities of linear vs. quadratic vs. exponential functions and behaviors of each. • Look for: Examples of students exhibiting math practices, especially SMP 2, SMP 4, and SMP 5. • Listen for: Partial insights, initial steps forward, and common approaches. Record/paraphrase comments from students on Student Ideas discussion tool. • Review individual work and prepare feedback questions and prompts. Suggested feedback questions and prompts for use during individual and pair work 1. What types of function did you consider? How did you choose your function? 2. How do you know that it is the best model for the situation? Show two ways of justifying your choice. 3. How might you need to revise your model? What are the limitations of your model? This work is licensed under a Creative Commons Attribution-­‐NonCommerical-­‐NoDerivatives 4.0 International License
4 Innovation Lab Network Performance Assessment Project Discussion [Whole class: 10-­‐20 mins] • Facilitate a discussion about different approaches to modeling the relationship between speed and braking distance, in particular, and different ways of accounting for all of the given parameters in the situation as a whole. • Use student ideas you have recorded on the Student Ideas discussion tool to call attention to particular approaches or partial insights that shed light on common issues in student work. • Invite students to elaborate on their own ideas when presented for discussion, and to answer questions from classmates about their reasoning. Part B: Building on Others’ Ideas [Pair & Individual Work: 1-­‐2 days] Handout 3: Making Sense of Others’ Ideas [Individual: 5-­‐10 mins, Pairs: 10-­‐20 mins] • Provide Handout 3 for some initial individual think time for students to begin to make sense of each student’s idea, and then discuss the ideas in pairs and make some notes. Handout 4: Building on Others’ Ideas [Individual: 20-­‐30 mins] • Provide Handout 4 for students to develop 1-­‐2 ideas into more coherent arguments. • Students can choose from among the ideas on Handout 3 as well as any ideas you have shared for discussion in Part A (using Student Ideas tool with speech/thought bubbles). Part C: Research and Discussion [Group Work: 1-­‐2 days] Conduct research on stopping distance [Whole Class/Groups/Pairs: 10-­‐20 mins] • Students look up ‘stopping distance’ online and begin to interpret the graphs and diagrams that come up when clicking ‘Images’. Handout 5: Resource Card [Whole Class/Groups: 30-­‐60 mins] • Use Handout 5 as a focus for discussion, and as a primary reference if online research is not possible (or introduces too much for students to process). • Refer to Figures 1 and 2 on Handout 5 to focus a class discussion about how the two terms of the formula relate to the parts of the diagram. Ensure that students are attending to how the linear term and the quadratic term are represented in the diagram. • You may want to have students work in groups to do the same exercise of relating the formula to a diagram with a different diagram from their online research. • The purpose of this discussion is to give students an opportunity to coordinate across multiple representations of stopping distance, with a focus on how the linear term and the quadratic term are each represented. Part D: Culminating Product [Individual Work: 2-­‐3 days] Discussion [Whole class: 10-­‐20 mins] This work is licensed under a Creative Commons Attribution-­‐NonCommerical-­‐NoDerivatives 4.0 International License
5 Innovation Lab Network Performance Assessment Project •
Facilitate a discussion about what quantities might matter for developing guidelines for following distance. Suggested discussion prompts: 1. What quantities might matter for determining a safe distance between cars on the road? [Refer back to ideas from opening discussion for Part A as much as possible.] 2. What are some possible benefits of a shorter following distance? [From a traffic management perspective, a shorter following distance allows more cars to fit on the road at once] 3. How far do you travel in two seconds if you are traveling 1 mph? How far do you travel in three seconds if you are traveling 1 mph? What about if you are traveling 10 mph? 4. Do the type of car and tires matter? Do road conditions matter? In the formula on Handout 5, these were accounted for in the “friction coefficient,” µ. 5. What about the time it takes for a driver to apply the brakes once they realize then need to stop or slow down (In the formula on Handout 5, this was represented by tp-­‐
r, the “perception-­‐reaction time”)? Does this vary depending on the age of the driver, or depending on whether the driver is tired or distracted? Handout 6: Culminating Product • Provide Handout 6 for students to read the description of what is expected. • Individually, students compose a recommendation that synthesizes the relevant mathematics from all work on the task so far. Their recommendation should consist of a written argument and explanation supported by graphs, formulas, diagrams, and calculations. Handout 7: Following Distance Guidelines from CA and NY • Provide Handout 7 for students to review the guidelines published by the New York and California Departments of Motor Vehicles. (Online research is also an option, to extend comparison to other states’ guidelines.) Handout 8: Three Scenarios to Consider (Optional) • Provide Handout 8 to support further thinking about the mathematics of the situation as students begin to develop their recommendations. • The purpose of these scenarios is to provide a structure for students to investigate the effects of different parameters and variables: speed, following distance, friction coefficient, and perception-­‐reaction time. • This handout should be considered optional; some students may perform better when relying on their own approach to the mathematics of the situation. H. Student support: As students work in pairs and groups on Handout 2B, provide each pair/group with a copy of the Student Ideas sheet (empty speech/thought bubbles) for their own use. This provides students with an optional tool for recording comments/questions from their group-­‐mates This work is licensed under a Creative Commons Attribution-­‐NonCommerical-­‐NoDerivatives 4.0 International License
6 Innovation Lab Network Performance Assessment Project and/or their own ideas. Instruct them to use the sheet as a listening tool for hearing things that make sense to them (or a thinking tool to record their own ideas) as they work together on the problem. Tell students you will collect any sheets students want to submit, and may use them to help focus a whole-­‐class discussion about the relationship shown in the table. Then collect and review any sheets students want to submit, and present these to the class during the closing discussion for Part A to motivate the development of students’ thinking as they head into Part B. I. Extensions or variations: Ask students to explore the consequences of using a linear, quadratic, and exponential function to model the relationship between speed and distance using the braking distance data in the table or using data on braking distance for other vehicles found through research. Make a prediction first: which model will yield the greatest braking distance for a speed of 25mph? 65 mph? J. Scoring: Student work can be scored using the SCALE Math Performance Assessment Rubric (Grades 9-­‐12). This work is licensed under a Creative Commons Attribution-­‐NonCommerical-­‐NoDerivatives 4.0 International License
7 STUDENT HANDOUT
The Situation You are driving home one day when Tibbles the cat darts into the road and freezes in fear directly in your path. Can you stop in time before hitting him? What do you need to know to begin to develop an approach (not a solution) to this problem? Make some notes and be prepared to share your ideas. Handout 1: Initial Individual Notes STUDENT HANDOUT
The Problem Can you avoid hitting Tibbles? •
You are a conscientious driver, and are driving 25 mph.
•
Tibbles is 63 feet ahead of you when you see him.
•
It takes 0.7 seconds from the moment you see Tibbles to the moment you hit the brakes
(your ‘reaction time’).
The table below shows the relationship between your speed and the average braking distance for your car.1 Speed (mph) 10 20 30 40 50 60 Braking distance 5.87 23.44 52.71 93.69 146.38 210.77 (feet) Get as far as you can in solving this problem on your own. Represent your solution strategy as clearly as possible. You will have further opportunities to work on your solution after today. 1
https://www.georgiastandards.org/resources/Lexile_in_Action/MM2D2_The-­‐Mathematics-­‐of-­‐Stopping-­‐Your-­‐
Car.pdf Handout 2A: Initial Individual Work STUDENT HANDOUT
The Problem, Again Can you avoid hitting Tibbles? •
You are a conscientious driver, and are driving 25 mph.
•
Tibbles is 63 feet ahead of you when you see him.
•
It takes 0.7 seconds from the moment you see Tibbles to the moment you hit the brakes
(your ‘reaction time’).
The table below shows the relationship between your speed and the average braking distance for your car.2 Speed (mph) 10 20 30 40 50 60 Braking distance 5.87 23.44 52.71 93.69 146.38 210.77 (feet) Discuss possible approaches to this problem with a partner, and get as far as you can in solving it together. Represent your solution strategy as clearly as possible. 2
https://www.georgiastandards.org/resources/Lexile_in_Action/MM2D2_The-­‐Mathematics-­‐of-­‐Stopping-­‐Your-­‐
Car.pdf Handout 2B: Pair Work STUDENT HANDOUT
Making Sense of Others’ Ideas These students are trying to use the table to write a function that models the car’s stopping distance. Try to understand each student’s idea on your own first. Then discuss with your partner: Which of these ideas are correct, and which are incorrect? How do you know? Make some notes about which ideas make sense to you and which do not. Handout 3: Pair Work STUDENT HANDOUT
Building on Others’ Ideas Choose one or two ideas to develop into more coherent arguments. Pick the ideas from Handout 3 or from your teacher. For each idea you develop, begin by identifying any errors in reasoning. Then develop each idea by correcting it as needed, and adding explanation, examples, diagrams and/or graphs to support it. Idea #1 Idea #2 Handout 4: Individual Writing STUDENT HANDOUT
Diagram Source: http://www.cyberphysics.co.uk/graphics/graphs/highwaycodeSD.png Formula Source: http://en.wikipedia.org/wiki/Braking_distance Handout 5: Resource Card STUDENT HANDOUT
Culminating Product: A Recommendation The Departments of Motor Vehicles in CA and in NY both publish guidelines for ‘following distance’. Their guidelines are different. Which state’s guidelines do you recommend, based on your study of the mathematics of stopping distance? Prepare a recommendation to one state’s DMV or the other about changing their published guidelines so that the two states’ guidelines for following distance are consistent. o Support your recommendation with your own mathematical work (calculations, formulas,
diagrams, graphs) and with relevant findings from your research (diagrams, graphs,
formulas).
o Be sure to make clear how any diagrams, graphs, and formulas you use represent the
important quantities in the situation.
o Your recommendation should be 1-­‐2 pages long, including 2-­‐3 short paragraphs of text
and supporting mathematical work.
o Use the feedback you have received so far from your teacher on your individual and group
work to inform your final product.
Handout 6: Culminating Product – Individual Writing STUDENT HANDOUT
California DMV Three Second Rule Do not be a tailgater! Many drivers follow too closely (tailgate) and are not able to see as far ahead as they should because the vehicle ahead blocks their view. The more space you allow between your vehicle and the vehicle ahead, the more time you will have to see a hazard, and stop or avoid that hazard. Most rear end collisions are caused by tailgating. To avoid tailgating, use the “three-­‐second rule”: when the vehicle ahead of you passes a certain point such as a sign, count “one-­‐
thousand-­‐one, one-­‐thousand-­‐two, one-­‐thousand-­‐three.” Counting these numbers takes approximately three seconds. If you pass the same point before you finish counting, you are following too closely. You should allow a four-­‐second or more cushion when: •
•
•
•
•
•
•
•
Being crowded by a tailgater. Allow extra room ahead, do not brake suddenly. Slow
down gradually or merge into another lane to prevent being hit from behind by the
tailgater!
Driving on slippery roads.
Following motorcyclists on wet or icy roads, on metal surfaces (e.g., bridge gratings,
railroad tracks, etc.), and on gravel. Motorcyclists can fall more easily on these surfaces.
The driver behind you wants to pass. Allow room in front of your vehicle so the driver
will have space to move in front of you.
Towing a trailer or carrying a heavy load. The extra weight makes it harder to stop.
Following large vehicles that block your view ahead. The extra space allows you to see
around the vehicle.
You see a bus, school bus, or a placarded vehicle at railroad crossings. These vehicles
must stop at railroad crossings; so, slow down early and allow plenty of room.
Merging onto a freeway.
If you follow too closely and another driver “cuts” in front of you, just take your foot off the gas. This gives you space between your vehicle and the other driver, without having to slam on your brakes or swerve into another lane. Handout 7: Following Distance Guidelines for NY and CA STUDENT HANDOUT
New York DMV Two Second Rule ALLOW YOURSELF SPACE http://dmv.ny.gov/about-­‐dmv/chapter-­‐8-­‐defensive-­‐driving#all-­‐spc Four of every 10 crashes involve rear-­‐end collisions, normally because a person is following too closely (tailgating). Leave enough room between your vehicle and the one ahead so you can stop safely if the other vehicle stops suddenly. Brake early and gently when you prepare to stop or turn. It gives drivers behind you plenty of warning that you plan to decrease your speed. For a good "space cushion," use the two-­‐second rule: Select an object near or above the road ahead like a sign, tree or overpass. As the vehicle ahead passes it, count slowly, "one thousand one, one thousand two." If you reach the same object before you finish the count, you are following too closely. bad weather and when following large trucks, increase the count to at least three or four seconds for additional space. In If a driver follows you too closely (tailgates) move to another lane if possible, or reduce speed and pull off the road to let the driver go by you. Make sure to signal when you drive off the road and when you return to it. Do not press your brakes to warn the driver behind you -­‐ this could make a difficult condition become even more dangerous. In case you must change lanes quickly or pull over to avoid a hazard, leave some "escape" room to your left and right. Handout 7: Following Distance Guidelines for NY and CA STUDENT HANDOUT
Three Scenarios to Consider Scenario #1 Imagine you are driving behind a car that suddenly brakes to avoid hitting Tibbles the cat. •
Both you and the driver you are following are driving 22 mph.
•
Tibbles is 55 feet ahead of the car you are following.
•
You are using the “two-­‐second rule” to determine the distance between your car and the
car ahead of you. (You learned to drive in New York.)
•
It takes 0.7 seconds for the driver ahead of you to hit the brakes once she sees Tibbles.
•
It takes 0.7 seconds for you to hit the brakes once you see the brake lights ahead of you.
•
You can use the formula1 below to compute the total stopping distance for your car and
the car ahead of you.
This formula shows the total stopping distance as the sum: distance traveled during perception-­‐reaction time + distance traveled while braking In order to use this formula, suppose the following: §
Both you and the driver ahead of you have a ‘perception-­‐reaction time’, tp-­‐r, of 0.7 seconds. §
For both cars, the friction coefficient, µ, is 0.6. §
For both cars, the standard acceleration due to gravity, g = 32 ft/sec. 1 Source: http://en.wikipedia.org/wiki/Braking_distance Scenario #2 Just like Scenario #1, except: • You and the driver ahead of you are both driving 55 mph.
• You are using California’s ‘three-­‐second rule’ to determine your following distance.
Scenario #3 Just like Scenario #1, except: • The friction coefficient, µ, for your car is 0.5 instead of 0.6.
• You are distracted for a moment by a friend in your car, and your perception-­‐reaction
time, tp-­‐r, is 1.5 seconds.
Handout 8: Three Scenarios to Consider Math Performance Assessment Rubric (Grades 9-12)
The ability to reason, problem-solve, develop sound arguments or decisions, and create new ideas by using appropriate
sources and applying the knowledge and skills of a discipline.
EMERGING
PROBLEM SOLVING
What is the evidence
that the student
understands the
problem and the
mathematical
strategies that can be
used to arrive at a
solution?
REASONING AND
PROOF
What is the evidence
that the student can
apply mathematical
reasoning/procedures
in an accurate and
complete manner?
CONNECTIONS
What is the evidence
that the student
understands the
relationships between
the concepts,
procedures, and/or
real-world
applications inherent
in the problem?
COMMUNICATION
AND
REPRESENTATION
What is the evidence
that the student can
communicate
mathematical ideas to
others?
E/D
DEVELOPING
PROFICIENT
P/A
ADVANCED
• Does not provide a model
• Ignores given constraints
• Uses few, if any, problem
solving strategies
• Creates a limited model to
simplify a complicated situation
• Attends to some of the given
constraints
• Uses inappropriate or inefficient
problem solving strategies
• Creates a model to simplify a
complicated situation
• Analyzes all given constraints,
goals and definitions
• Uses appropriate problem solving
strategies
• Creates a model to simplify a
complicated situation and identifies
limitations of model
• Analyzes all given constraints, goals and
definitions and implied assumptions
• Uses novel problem solving strategies
and/or strategic use of tools
• Provides incorrect
solutions without
justifications
• Results are not interpreted
in terms of context
• Provides partially correct
solutions or correct solution
without logic or justification
• Results are interpreted partially or
incorrectly in terms of context
• Constructs logical, correct,
complete solution
• Results are interpreted correctly in
terms of context
• Constructs logical, correct, complete
solution with justifications
• Interprets results correctly in terms of
context, indicating the domain to which a
solution applies
• (Monitors for reasonableness, identifies
sources of error, and adapts
appropriately)
• Little or no evidence of
applying previous math
knowledge to given
problem
• Applies previous math knowledge
to given problem but may include
reasoning or procedural errors
• Applies and extends math
previous knowledge correctly to
given problem
• Applies and extends previous
knowledge correctly to given problem;
makes appropriate use of derived
results
• (Identifies and generalizes the
underlying mathematical structures of
the given problem to other seemingly
unrelated problems or applications)
• Uses representations
(diagrams, tables, graphs,
formulas) in ways that
confuse the audience
• Uses incorrect definitions
or inaccurate
representations
• Uses representations (diagrams,
tables, graphs, formulas), though
correct, do not help the audience
follow the chain of reasoning;
extraneous representations may
be included
• Uses imprecise definitions or
incomplete representations with
missing units of measure or
labeled axes
• Uses multiple representations
(diagrams, tables, graphs,
formulas) to help the audience
follow the chain of reasoning
• With few exceptions, uses precise
definitions and accurate
representations including units of
measure and labeled axes
• Uses multiple representations
(diagrams, tables, graphs, formula) and
key explanations to enhance the
audience’s understanding of the
solution; only relevant representations
are included
• Uses precise definitions and accurate
representations including units of
measure and labeled axes; uses formal
notation
©2013 Stanford Center for Assessment, Learning and Equity (SCALE) and Envision Schools
D/P
Adapted by New Tech Network