Grades 6-8, Claim 4 Task Model 1

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1 Task Model 1 Task Expectations: A student can be expected to solve problems that involve extracting relevant information from within the problem, finding missing information through research or the use of reasoned estimates, or identifying extraneous information. Apply mathematics to solve problems arising in everyday life, society, and the workplace. Target A Example Item 1 (Grade 6): Primary Target 4A (Content Domain NS), Secondary Target 1B (CCSS 6.NS.A), Tertiary Target 4D, Quaternary Target 1A (CCSS 6.RP.A) Darcy likes to eat peanut butter and raisins on apple slices. On each apple slice she puts butter and 8 raisins. cup of peanut Darcy has cup of peanut butter and 80 raisins. She eats a whole number of apple slices until the peanut butter is all gone. What fraction of the 80 raisins did she eat? Enter the fraction in the response box. Rubric: (1 point) Student enters the correct fraction (e.g., ). Response Type: Equation/Numeric Example Item 2 (Grade 7): Primary Target 4A (Content Domain NS), Secondary Target 1B (CCSS 7.NS.3) A store is having a sale. Each customer receives either a 15% discount on purchases under $100 or a 20% discount on purchases of $100 or more. Kelly is purchasing some clothes for $96.60 before the discount. She decides to buy the fewest packs of gum that will increase her purchase to over $100. The price of each pack of gum is $0.79. After the discount, how much less will Kelly pay buying the clothes and the gum instead of buying only the clothes? (Assume there is no sales tax to consider.) A. $1.05 B. $1.67 C. $3.69 D. $3.87 Rubric: (1 point) The student selects the correct amount (e.g., B). Response Type: Multiple Choice, single correct response 16 Version 2.0

2 Task Model 1 Apply mathematics to solve problems arising in everyday life, society, and the workplace. Example Item 3 (Grade 7): Primary Target 4A (Content Domain RP), Secondary Target 1A (CCSS 7.RP.A) Elias is a produce manager at a grocery store. He buys fresh vegetables from local farmers each week. Based on previous sales, he has identified the following ideal ratios (in pounds) to keep in stock for certain vegetables. The ratio of tomatoes to onions is 3:2. onions to peppers is 2:1. peppers to cucumbers is 2:5. This table shows the amount, in pounds, of each vegetable a local farmer has available to sell to Elias. Target A Vegetable Amount (lbs) Cucumbers 50 Onions 55 Peppers 30 Tomatoes 85 Elias buys all 50 pounds of the farmer s cucumbers. He then buys the remaining vegetables according to the ideal ratios shown above. Enter the amount of peppers, in pounds, Elias buys in the first the response box. Enter the amount of tomatoes, in pounds, Elias buys in the second response box.. Rubric: (2 points) The student correctly enters the amounts for both peppers and tomatoes (e.g., 20, 60). (1 point) The student enters the correct amount for one vegetable, but not both. Response Type: Equation/Numeric (2 response boxes) 17 Version 2.0

3 Task Model 1 Example Item 4 (Grade 8): Primary Target 4A (Content Domain G), Secondary Target 1I (CCSS 8.G.C), Tertiary Target 1A (CCSS 7.RP.3) Apply mathematics to solve problems arising in everyday life, society, and the workplace. An empty corn silo in the shape of a cylinder is being filled with corn. Target A The silo is filled at a constant rate for a total of 10 hours. The table shows the amount of corn, in cubic feet, in the silo at the given number of hours after filling started. Number of Hours Amount of Corn (cu ft) Enter the percent of the silo that is filled with corn at 10 hours. Rubric: (2 points) The student enters the correct numerical value for the percent of volume that is filled at the end of 10 hours (accept the range ). (1 point) The student gives the amount filled after 10 hours (8250) but forgets to find the percentage of the filled amount to the volume OR the student finds the volume of the silo only ( ). Response Type: Equation/Numeric 18 Version 2.0

4 Task Model 2, 4 Construct, autonomously, chains of reasoning to justify mathematical models used, interpretations made, and solutions proposed for a complex problem. Task Expectations: The student either: justifies the mathematical model(s) used justifies the interpretation(s) shown and/or justifies the solution(s) given to a complex problem. Example Item 1 (Grade 8): Primary Target 4B (Content Domain SP), Secondary Target 1J (CCSS 8.SP.1), Tertiary Target 1F (CCSS 8.F.4) This scatter diagram shows the lengths and the widths of the eggs of some American birds. Target B Part A What does the graph show about the relationship between the lengths of birds eggs and their widths? Part B Fossils show that dinosaur eggs closely resemble the shape of bird eggs. One such dinosaur (sauropods) grew from eggs that were 180 millimeters in length. Assume that sauropod eggs were the same shape as bird eggs. Approximate the width of sauropod eggs. Explain how you determined your answer. Label your answers to both parts of the problem in the response box. Be sure to use information from the 19 Version 2.0

5 scatterplot to support your answers. Exemplar 3 : Part A: Typically, the greater the length of the egg, the greater the width. Part B: The width is approximately 126 mm (accept values between 115 and 135 mm). I multiplied by about 0.7 or The width is a little less than of the length or I doubled the width of the egg that is 90 mm long. Rubric: (2 points) The student is able to answer both parts correctly and provide sufficient explanation/support for the answer to Part B. (1 point) The student only answers one part correctly. Response Type: Short Text (handscored) 3 An exemplar response represents only one possible solution. Typically, many other solutions/responses may receive full credit. The full range of acceptable responses is determined during rangefinding and/or scoring validation. 20 Version 2.0

6 Task Model 3 1, 2 Task Expectations: Tasks ask students to use stated assumptions, definitions, and previously established results in developing their reasoning. In some cases, the task may require students to provide missing information by researching or providing a reasoned estimate. State logical assumptions being used. Target C Example Item 1 (Grade 6): Primary Target 4C (Content Domain EE), Secondary Target 1G (CCSS 6.EE.C) Justin predicts that the temperature change from Friday to Saturday will be twice as much as it was from Thursday to Friday. Use the Add Point tool to plot a point on the graph that could represent Justin s predication for Saturday s temperature. Temperature ( F) Temperatures for the Week Sun Mon Tue Wed Thu Fri Sat Days of the Week Rubric: (1 point) The student graphs a point on Saturday that represents the temperature that represents a change of 10 degrees either direction (e.g., 0 or 20). Response Type: Graphing 21 Version 2.0

7 Task Model 4 Task Expectations: Tasks should ask students to link their response back to the problem s context, e.g., a judgment by the student of whether to express an answer to a division problem or a rationalization for the domain of a function being limited to positive integers. Interpret results in the context of a situation. Example Item 1 (Grade 8): Primary Target 4D (Content Domain F), Secondary Target 1F (CCSS 8.F.5) This graph shows the average number of words in a child s vocabulary from birth to 36 months. Target D Which statement is the most reasonable explanation for the shape of the graph? A. Children begin to show significant growth in vocabulary after 12 months. B. Children begin speaking around 26 months and stop learning new words at 36 months. C. Children are constantly adding new words to their vocabulary from the moment they are born. D. Children do not begin talking for several months, but then begin to pick up words very quickly. Rubric: (1 point) The student chooses the best interpretation of the graph (e.g., D). Note: To distinguish from Claim 1 items, interpretations should extend beyond simply looking at the graph and should help to evaluate whether students understand which interpretations are defensible. Item authors should be careful with language not to overstate a particular conclusion since all data based interpretations are subject to some error. Response Type: Multiple Choice, single correct response 22 Version 2.0

8 Task Model 4 Interpret results in the context of a situation. Example Item 2 (Grade 8): Primary Target 4D (Content Domain F), Secondary Target 1F (CCSS 8.F.B) Cory is buying copper for a construction project. He pays $1.85 per pound of copper for the first 100 pounds. He pays $1.75 per pound of copper for every pound over 100 pounds. Cory calculated that it would cost $ to purchase 125 pounds of copper. He writes an algebraic equation that will allow him to determine the cost of copper for any number of pounds of copper over 100 pounds. Let y be the amount of money, in dollars, Cory pays for x total pounds of copper when x is greater than 100. y = a(x 100) + b Target D Determine the values of a and b that Cory used to calculate the total cost for any purchase over 100 pounds of copper. Enter the value of a in the first response box. Enter the value of b in the second response box. Rubric: (1 point): The student enters the correct values for a and b (e.g., a = 1.75 and b = 185). Response Type: Equation/Numeric 23 Version 2.0

9 Task Model 4 Example Item 3 (Grade 8): Primary Target 4D (Content Domain SP), Secondary Target 1J (CCSS 8.SP.2), Tertiary Target 1J (CCSS 8.SP.3) This scatter plot and line of best fit show the relationship between animal weight and heart rate. Interpret results in the context of a situation. Target D The table shown lists the animal weights and heart rates used to construct the scatter plot. Select two rows containing data that demonstrate an error greater than 30 beats per minute between the heart rate estimated by the line and the animal s actual heart rate. 24 Version 2.0

10 Task Model 4 Interpret results in the context of a situation. Target D Average Animal Weight (pounds) Average Heart Rate (beats per minute) Rubric:(1 point) The student selects the described 2 rows of data (e.g., weights of 3 and 20 pounds). Response Type: Hot Spot 25 Version 2.0

11 Task Model 4 Example Item 4 (Grade 8): Primary Target 4D (Content Domain F), Secondary Target 1F (CCSS 8.F.B), Tertiary Target 1F (CCSS 8.F.4) The relationship between Jack s distance from home and time since he left home is linear, as shown in the table. Interpret results in the context of a situation. Target D Time (hrs) Distance (mi) Based on the table, determine whether each statement is true. Select True or False for each statement. Statement True False Jack s initial distance from home is 7.5 miles. Jack s distance increases by 5 miles every 1 hour. Jack s distance from home at 3 hours is 23.5 miles. Rubric: (1 point) Student determines each statement as being either true or false (e.g., T, T, F). Response Type: Matching Tables 26 Version 2.0

12 Task Model 4 Example Item 5 (Grade 8): Primary Target 4D (Content Domain EE), Secondary Target 1D (CCSS 8.EE.C) This table represents the cost of renting a truck from Moving Company A and Moving Company B. Each company charges a one-time rental fee plus a charge for each mile driven. Interpret results in the context of a situation. Moving Company One-time Rental Fee Charge per Mile A $150 $0.25 B $ 50 $0.75 Target D Part A Use the Add Arrow tool to graph a system of two linear equations to model the cost of using each moving company. Part B Select the moving company that will be the least expensive for the specified number of miles. Number of Miles Company A Company B Rubric: Each part of this item is scored independently for a total of 2 points. Part A (1 point): The student correctly graphs both lines. Part B (1 point): The student selects the correct cells in the table. 27 Version 2.0

13 Task Model 4 Exemplar: Interpret results in the context of a situation. Target D Number of Miles Company A Company B Interaction: The Add Arrow tool will be available (with one arrow) to graph the lines, as well as Hot Spot to select the correct cells in the table. Response Type: Graphing and Hot Spot 28 Version 2.0

14 Task Model 5 3, 4 Analyze the adequacy of and make improvements to an existing model or develop a mathematical model of a real phenomenon. Task Expectations: Focus on developing a mathematical model of a real phenomenon. Any of the following scenarios can be used to assess this target. Given a situation, the student will identify or create a symbolic or graphical model to represent the situation (includes equations, diagrams, and graphs). Given data (table of values, scatterplot, etc.) the student will identify the type of function that might best model the situation. The student will assess the fit of a particular model being used, including models used in two and three-dimensional geometry. May use a simulation that mirrors the functioning of a formula-based online calculator. Example Item 1 (Grade 8): Primary Target 4E (Content Domain F), Secondary Target 1F (CCSS 8.F.B) (Source: Adapted from Illustrative Mathematics 8-F Modeling with a Linear Function) Select all situations that can be modeled by the linear equation y = 2x+5. Target E A. There are initially 5 rabbits on the farm. Each month thereafter the number of rabbits is 2 times the number in the month before. How many rabbits are there after x months? B. Joe earns $2 for each magazine sale. He also earns $5 for each hour he spends trying to sell magazines. How much money will he earn after selling magazines for x hours? C. Sandy charges $2 an hour for babysitting. Parents are charged $5 if they arrive home later than scheduled. Assuming the parents arrived late, how much money does she earn for x hours? D. Sneak Preview is a members-only video rental store. There is a $2 initiation fee and a $5 per video rental fee. How much would Laney owe on her first visit if she becomes a member and rents x videos? E. Andre is saving money for a new CD player. He began saving with a $5 gift and will continue to save $2 each week. How much money will he have saved at the end of x weeks? Rubric: (1 point) The student identifies all situations modeled by the equation (e.g., C and E). Response Type: Multiple Choice, multiple correct response 29 Version 2.0

15 Task Model 5 3, 4 Analyze the adequacy of and make improvements to an existing model or develop a mathematical model of a real phenomenon. Example Item 2 (Grade 8): Primary Target 4E (Content Domain F), Secondary Target 1F (CCSS 8.F.5) The table shows the relationship between the average number of hours students study for a mathematics test and their average grade. Hours Studying Average Grade Target E Which type of function is most likely to model these data? A. linear function with positive slope B. linear function with negative slope C. non-linear function that decreases then increases D. non-linear function that increases then decreases Rubric: (1 point) The student recognized the function most likely to model the data (e.g., D). Response Type: Multiple Choice, single correct response 30 Version 2.0

16 Task Model 6 1, Task Expectations: The mapping of relationships should be part of the problem posing and solving related to Claim 4 Targets A, B, E, and G. Data is presented in a table or graph, or extracted from a context. The student may be asked to determine conclusions that are plausible based on the data. Identify important quantities in a practical situation and map their relationships (e.g., using diagrams, two-way tables, graphs, flowcharts, or formulas). Example Item 1 (Grade 6): Primary Target 4F (Content Domain EE), Secondary Target 1F (CCSS 6.EE.8), Tertiary Target 4F (Content Domain EE), A boat takes 3 hours to reach an island 15 miles away. The boat travels: at least 1 mile but no more than 6 miles during the first hour at least 2 miles during the second hour exactly 5 miles during the third hour Use the Connect Line tool to show the range of miles the boat could have traveled during the second hour, given the conditions above. Target F Number of Miles Interaction: The student uses the Connect Line tool to create a line segment that shows all possible values for the inequality. Rubric: (1 point) The student graphs the correct solution set (e.g. a line segment from 4 to 9). Response Type: Graphing 31 Version 2.0

17 Task Model 6 1, Example Item 2 (Grade 6): Primary Target 4F (Content Domain EE), Secondary Target 1F (CCSS 6.EE.7) Megan has $2500. She spends money on the following: Identify important quantities in a practical situation and map their relationships (e.g., using diagrams, two-way tables, graphs, flowcharts, or formulas). Target F $800 on housing $400 on food $200 on utility services $250 on loan payments x on other expenses Write an equation to represent the amount of money Megan has left, based on how much she spends on other expenses. Let y represent the amount of money Megan has left, and let x represent how much Megan spends on other expenses. Rubric: (1 point) The student computes Megan's spending and represents the remaining money with an equation (e.g., y = 850 x). Response Type: Equation/Numeric 32 Version 2.0