Modeling with Mathematics - · PDF file21/07/2014 · Session Title: Math – Modeling with Mathematic Description: ELC members will explore how modeling differs in the classroom and

Modeling with Mathematics

Tammy Brown, Math Director National Math and Science Initiative

Charla Holzbog, Program Manager

National Math and Science Initiative

Session Title:

Math – Modeling with Mathematic

Description:

ELC members will explore how modeling differs in the classroom and on the exam.

Objectives:

Participants will

Differentiate between modeling in the classroom and modeling on the PARCC

assessment to inform CCSS implementation and instructional design decision-making

and to increase understanding of PARCC assessment design.

Length:

90 minutes

Materials/Resources:

MP. 4

Sample PARCC questions

Process:

1. Look at MP4 with the K-8 elaborations included. Discuss how modeling in the classroom

may look different that modeling on the PARCC assessments. Today, we are looking at how

modeling looks on the assessment.

2. Participants will sort questions from the EOC into “requires modeling” and “does not require

modeling” based on ideas from the session.

3. Distribute 14 available questions to participants to work in pairs. Have each person share out

the question type (I or III) and how students use modeling in the question by displaying their

work on document camera.

4. Summarize as participants share out on grid paper.

5. Participants will resort questions from the EOC into “requires modeling” and “does not

require modeling” based on ideas from the session.

Note: This portion of the CCSS for Mathematical Practice is verbatim but in bulleted form. The italicized

portions are the exact wording taken from the Standards for Mathematical Practice: Commentary and

Elaborations document.

MP.4 - Model with mathematics.

Mathematically proficient students:

apply the mathematics they know to solve problems arising in everyday life, society, and the

workplace.

o For elementary students, this includes the contextual situations they encounter in the

classroom. When elementary students are first studying an operation such as addition,

they might arrange counters to solve problems such as this one: there are seven animals

in the yard, some are dogs and some are cats, how many of each could there be? They

are using the counters to model the mathematical elements of the contextual problem—

that they can split a set of 7 into a set of 3 and a set of 4. When they learn how to write

their actions with the counters in an equation, 4 3 7 , they are modeling the situation

with numbers and symbols. Similarly, when students encounter situations such as sharing

a pan of cornbread among 6 people, they might first show how to divide the cornbread

into 6 equal pieces using a picture of a rectangle. The rectangle divided into 6 equal

pieces is a model of the essential mathematical elements of the situation. When the

students learn to write the name of each piece in relation to the whole pan as 1

6 , they

are now modeling the situation with mathematical notation.

o In early grades, this might be as simple as writing an addition equation to describe a situation.

o In addition to numbers and symbols, elementary students might use geometric figures,

pictures, or physical objects used to abstract the mathematical elements of the situation,

or a mathematical diagram such as a number line, a table, or a graph, or students might

use more than one of these to help them interpret the situation.

o In middle grades, a student might apply proportional reasoning to plan a school event or

analyze a problem in the community.

o By high school, a student might use geometry to solve a design problem or use a function to

describe how one quantity of interest depends on another.

apply what they know are comfortable making assumptions and approximations to simplify a

complicated situation, realizing that these may need revision later.

identify important quantities in a practical situation and map their relationships using such tools

as diagrams, two-way tables, graphs, flowcharts and formulas.

o For example, if there is a Penny Jar that starts with 3 pennies in the jar, and 4 pennies

are added each day, students might use a table to model the relationship between number

of days and number of pennies in the jar. They can then use the model to determine how

many pennies are in the jar after 10 days, which in turn helps them model the situation

with the expression, 4 10 3 .

o For example, they can roughly fit a line to a scatter plot to make predictions and gather

experimental data to approximate a probability.

analyze those relationships mathematically to draw conclusions.

routinely interpret their mathematical results in the context of the situation and reflect on whether

the results make sense, possibly improving the model if it has not served its purpose.

o As students model situations with mathematics, they are choosing tools appropriately

(MP.5). As they decontextualize the situation and represent it mathematically, they are

also reasoning abstractly (MP.2).

o For example, they can recognize the limitations of linear models in certain situation,

such as representing the amounts of stretch in a bungee cord for people of different

weights.

PARCC Grade 3 Sample Item November 2013

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The Art Teacher's Rectangular Array


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November 2013

1

Grade 4 Three friends’ beads

Type Type III - 3 points

Evidence Statement

4.D.1: Solve multi-step contextual word problems with degree of difficulty appropriate to Grade 4 requiring application of knowledge and skills articulated in the Evidence Statements on the PBA (excludes Reasoning Evidence Statements). Clarification: i) Tasks may have scaffolding if necessary in order to yield a degree of difficulty appropriate to Grade 4.

Most Relevant Standards for Mathematical Content

4.OA.A: Use the four operations with whole numbers to solve problems. 1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. 2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. 3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. This standard is major content in the grade based on the PARCC Model Content Frameworks.

Most Relevant Standards for Mathematical Practice

Students must reason abstractly and quantitatively with the context in

order to understand the relationships given in the question (MP.2). Creating

this multi-step solution path will require students to create an effective

model of the situation (MP.4), but the question allows for students to

create whatever models they are most comfortable employing (e.g., some

students may create a series of expressions, other student may choose to

diagram the situation).

Item Description and Assessment Qualities

This task requires students to apply solution strategies based in 4.OA.

Fourth grade students have a strong foundation in using the four operations

with whole numbers. Students will have to wrestle with these relationships

without much scaffolding so the numbers in the item are not too complex

to ensure the task is still grade appropriate. The procedural skills are an

important element of the solution path; however, the numbers were

carefully chosen to ensure that application was the primary focus of the

task.

2

There are a wide variety of solution processes that students may use to receive full credit. Unlike traditional multiple choice, it is difficult to guess the correct answer or use a choice elimination strategy.

Scoring Information

Scoring Rubric Task is worth 3 points. Task can be scored as 0, 1, 2, or 3. Scoring consists of 3 points for modeling.

2 points are earned for fully describing the steps necessary to find the number of beads that Damian and Trish each received. [Note: 1 point is earned for only describing or showing the steps necessary to find the number of beads Damian or Trish had instead of providing the information for both.]

For example: Elena had 5 beads so Damian has 5 + 8 = 13 beads. Trish has 4 beads. Altogether there are 5 + 13 + 52 = 70 beads. I know that 80 is ten more than 70, so there were 80 total beads.

1 point for calculating the final number of beads correctly.

Total number of beads: 80

Note: Students may receive points for their work in Part B if answers for Part A are correctly added and then added to 10 to find the total number of beads.

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Fraction Model

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Grade 6 Fraction model

Type Type III 6 Points Evidence Statement

6.D.2: Solve multi-‐step contextual problems with degree of difficulty appropriate to Grade 6, requiring application of knowledge and skills articulated in 5.NBT.B, 5.NF, 5.MD and 5.G.A. Clarification: Tasks may have scaffolding if necessary in order to yield a degree of difficulty appropriate to Grade 6.


5.NF.4b: Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. 5.MD.1: Convert among different-‐sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-‐step, real world problems. 5.G.1: Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.


Students must use the technology-‐enabled answer space in order to begin to create a model of the moon exploration area (MP.4). This model requires significant interpretation both in explanation and computation requiring students to reason abstractly and quantitatively (MP.2). Some of these explanations will require precise use of language to differentiate the meaning of the sections in the model.


This six-‐point task asks student to model a situation using a technology-‐enhanced response mechanism. Students use the technology to begin to create a model of the moon exploration area. Then, they interpret the model by identifying the fourth point of the rectangle and describing the length of the sides. Students must thoroughly understand this model in order to calculate the area either by converting the two lengths or finding the area in square kilometers and converting that measurement to square meters. Although this item is based on securely held content from grade 5, plotting points like (3.5, 1) represent the rigor expected at grade 6. In addition, the calculations resulting in an area of millions of square meters reflect the fluency with numbers that are appropriate for grade 6 students. The response for Part A is technology-‐enhanced so that it can be electronically scored. Unlike traditional multiple choice, it is difficult to guess the correct answer or use a choice elimination strategy.

Scoring Information

Scoring Rubric for SAMPLE 6.D.2 (VH043917)

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Task is worth 6 points. Task can be scored as 0, 1, 2, 3, 4, 5, or 6. Scoring consists of 4 points for modeling and 2 points for computation.

Part A, machine-‐scored 1 modeling point for creating a model with three correctly plotted points. Part B, machine-‐scored 1 modeling point for writing the point of the fourth coordinate point.

Part C, machine scored 1 computational point for finding that the horizontal length is 1.5 kilomters. 1 computational point for finding that the vertical length is 3 kilometers.

Part D, hand scored 1 modeling point for creating a solution path that addresses unit conversion. 1 modeling point for creating a solution path to determine the area in square meters.

For example, Part A.

Part B. (2, 1) Part C. 1.5 kilometers and 3 kilometers Part D. I converted 3 kilometers to 3,000 meters and1.5 kilometers to 1,500 meters. Then, I multiplied 3,000 by 1,500 to find that the moon exploration area is 4,500,000 square meters. OR students may solve Part D in this manner: I multiplied 3 kilometers by 1.5 kilometers to get 4.5 square kilometers. One square kilometer is 1,000 meters multiplied by 1,000 meters, which is 1,000,000 square meters. That means that 4.5 square kilometers is 4,500,000 square meters.

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November 2013

HS Popcorn Inventory

Type Type III 6 Points

Evidence Statement

HS.D.1-1: Solve multi-step contextual problems with degree of difficulty appropriate to the course, requiring application of knowledge and skills articulated in 7.RP.A, 7.NS.3, 7.EE, and/or 8.EE.


7.RP.2: Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn . d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. 7.RP.3: Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

These standards are major content in the seventh grade based on the PARCC Model Content Frameworks.


Students creating reasoned estimates must be able to reason abstractly and quantitatively in order to build a model of the situation that is accurate enough for the given situation (MP.2 and MP.4). Working with ambiguity is an important part of the modeling skills expected at high school, and this requires students to productively engage with the quantities given in the context. Modeling is a critical component of the high school standards, and this item requires students to create a model from a complex situation to make a real-world estimate given an unscaffolded situation where a model is a useful tool. To make this model, students will have to reason with the given quantities, their units and the proportional relationships between them. This will require students to understand how to use the numbers mathematically, and then be able to periodically check their own understanding of what those numbers mean.


This application task requires students to create a reasoned estimate in response to

solve a real-world problem. Students must first wrestle with the data displayed on the

Popcorn Inventory page. They should recognize that the amounts in the table are

decreasing because each day boxes and popcorn seeds are used. This means that

students should recognize that they are using about 15 medium boxes each day and

about 25-30 small boxes each day.

In order to address the amount of popcorn sold over the weekend, students must first

create a viable estimate of the number of cups of popcorn sold, then use the ratio

cups of popcorn seed:8 cups of popcorn. Students may choose to use this method to

estimate the amount of popcorn seed used Sunday through Thursday; however, other

methods could be used to determine the amount of popcorn seed used each day.

Students using this method must be sure to account for the amount of popcorn seed

used on Sunday because the original information starts from end of day on Sunday.

The final estimate requires students to use the current amount of popcorn seed, the

amount used Friday and Saturday, and the amount used Sunday through Thursday in

order to estimate the amount of popcorn seeds she should purchase so there are 100-

200 pounds left over next Friday morning.

Note that ratio and proportional relationships are key skills required for college and

career readiness, and this item provides a strong application of that content. Unlike

traditional multiple choice, it is difficult to guess the correct answer or use a choice

elimination strategy.

Scoring Information

Scoring Rubric for HS.D.1-1 Task is worth 6 points. Task can be scored as 0, 1, 2, 3, 4, 5, or 6.

Scoring consists of 2 points for calculation and 4 points for modeling.

Structure (6 points total):

- 2 points for correctly addressing the cups of popcorn seed needed for Sunday-

Thursday

o 1 calculation point for adequate estimate

o 1 modeling point for adequate estimation strategy

- 3 points for correctly addressing the cups of popcorn seed needed for Friday

and Saturday.

o 1 modeling point for adequate estimation strategy for addressing two

sizes of boxes for both days.

o 1 modeling point for accurate use of the proportion of popcorn seed to

popcorn

o 1 calculation point for adequate estimate

- 1 point for correctly estimating the amount of popcorn seed that should be

ordered

o 1 modeling point for adequate estimation strategy (the calculation is

not as important as the strategy)

Example student response: On Friday and Saturday, they will sell about 500 large boxes (250 + 250 = 500). I found

that they sold about 17 medium boxes ( –

) and about 30 small boxes

( –

) each day in the table, so they would sell about 68 (2 x 17 for both

days) medium boxes and 120 (2 x 30 for both days) small boxes on Friday and Saturday

combined.

That means they need to pop:

Large:

Medium (approx.):

Small (approx.):

I added the three amounts of popcorn to find that they will need about 12,400 cups of

popcorn over the weekend.

Since

-cups of popcorn seed makes 8 cups of popcorn, I know that 1 cup of popcorn

seed will make 24 cups of popcorn. That means that they need about

cups of popcorn seed for Friday and Saturday. So, they will need about 525 cups of

popcorn seed for the weekend.

According to the table, they used about 80 cups of popcorn seed each of the remaining

days of the week ( –

). They will need 80 x 5 or 400 cups of popcorn

seed for Sunday-Thursday.

I made this list to make sure she buys enough:

( 69.7 cups currrently)

525 cups of popcorn seed for Friday and Saturday

400 cups of popcorn seed for Sunday-Thursday

+ 100 extra cups to make sure she is between 100 and 200 cups on Friday morning

1,025 cups of popcorn seed to order in the morning

NOTE: There are a wide variety of estimation strategies that can receive full credit.

HS Brett’s race Type Type III 3 Points

Evidence Statement

HS.D.2-5: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-level knowledge and skills articulated in A-CED, N-Q, A-SSE.3, A-REI.6, A-REI.12, A-REI.11-2, limited to linear equations and exponential equations with integer exponents. Clarification: A-CED is the primary content; other listed content elements may be involved in tasks as well.


A-CED Creating Equations A-CED.A Create equations that describe numbers or relationships 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

This standard is major content in the course based on the PARCC Model Content Frameworks.


This item requires students to model the given situation using equations, then students use that model to determine who will win the race and their margin of victory (MP.4). In order to create and interpret these models, students will have to decontextualize and contextualize the information at various points in the solution process to create a mathematical model and then to interpret the meaning and structure of that model (MP.2). Students that choose to use the graph may create another model of the situation, and look for and use structure within that model (MP.7).


This application task requires students to use content from widely applicable algebra standards in order to solve a modeling problem with difficulty expected in high school. Students first create equations that model the situation described in the first paragraph. It is important for students to define their variables when creating equations. Then, students reason with their models, and perhaps the graphing tool, to interpret the model and determine the margin of victory. There are a variety of solution methods that students may use to successfully answer Part B.

Scoring Information

Scoring Rubric for Sample Item HS.D. 2-5 Task is worth 3 points. Task can be scored as 0, 1, 2, or 3. Task has 2 parts. Scoring for Part A – Formulating the Model – 1 point

Student produces two equations to determine the distance in meters from the starting line, of each person as a function of the time x, in seconds since the Olympian starts running. For example, Brett’s distance y, as related to time, x:

𝑦 = 8 13𝑥+ 20. Or y = 100

12 x + 20

The Olympian’s distance y, as related to time, x:

𝑦 = 10𝑥. NOTE: All variables should be defined. The student may choose to define x as time in seconds since Brett starts running. Scoring for Part B Student earns 1 calculation point for stating the correct winner and the correct margin of victory. Students earn 1 modeling point for providing an accurate justification using the equations in Part A. Sample Student Response 1:

• For Brett, 𝑦 = 100 when

100 = 813𝑥+ 20

80 = 813𝑥

𝑥 = 9.6 • For the Olympian 𝑦 = 100 when

100 = 10𝑥 𝑥 = 10.

• So, Brett wins the race by 10 – 9.6 = 0.4 seconds. Sample Student Response 2 :

• When Brett finishes the race at 9.6 seconds, the Olympian is only 10(9.6) = 96 meters from the start. Therefore, Brett was 4 meters ahead of the Olympian when he finished the race.

Note:

• If Part A contains incorrect equations, but Part B is correct based on one or two incorrect equations in Part A, the student is still awarded 1 or 2 points of the 3 possible points.

Task score: The task score is the sum of the points awarded in each component.

November 2013

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Mini-Golf Prices

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PARCC Algebra 1 Sample Item

HS Mini-golf prices

Type Type III 6 points

Evidence Statement

HS.D.2-9: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-level knowledge and skills articulated in F-BF.1a, F-BF.3, A-CED.1, A-SSE.3, F-IF.B, F-IF.7, limited to linear and quadratic functions Clarification: i) F-BF.1a is the primary content; other listed content elements may be involved in tasks as well.


F-BF. 1. Write a function that describes a relationship between two quantities.

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

This standard is major content in the course based on the PARCC Model

Content Frameworks.


Students must model with mathematics (MP.4) in order to create the functions and determine the percent increase of the maximum weekly revenue. The question in Part C is not as scaffolded as the previous parts and will confront students with a novel concept that will take time to ascertain how to begin a correct solution method (MP.1). In order to do this, students must decontextualize the given information and use it to construct a viable solution (MP.2).


This application task requires students to create functions to interpret a

situation. Then, students must use those functions to address application

questions based on this context. This content is complex, the model used in

Parts B and C is quadratic, but the item provides some scaffolding in Part A

so that students can gain familiarity with the underlying structure of the

components. Students first write functions that model various related

quantities in the task. Then, students show that they can use these functions

correctly by calculating a value using the quadratic function. Then, students

use their model to find the maximum weekly revenue, and the percent

increase that change represents over the revenue with no price increases.

Students may choose to solve this problem algebraically or with the

graphing tool. However, students must explain the steps they took to

determine the percent increase.

One of the reasons for modeling with quadratics is that, unlike linear or

exponential functions, quadratic functions can model situations with local

optimums. Optimization is a common and important use of mathematics.

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PARCC Algebra 1 Sample Item November 2013

Graphing technology is available to support student’s work on this item.

Scoring Information

Scoring Rubric for Sample HS.D.2-9 Task is worth 6 points. Task can be scored as 0, 1, 2, 3, 4, 5, or 6. Task has three parts: Part A is worth 2 points, Part B is worth 2 points, and Part C is worth 2 points. Part A: 2 point

1 point is earned for a correct function of p(n):

or an equivalent expression (e.g.,

).

1 point is earned for a correct function of s(n): – or

an equivalent expression (e.g., (

) .

Part B: 2 points

1 point is earned for a correct function of r(n): OR an equivalent expression

(e.g., – ). The student may use quadratic regression to determine their function.

1 point is earned for the correct weekly revenue for a price of $6.25: $656.25.

Possible student work (not scored):

, when n = 5 because 6.25 = 5 + 0.25(5). –

NOTE: Students can receive 1 or 2 points on Part B if they use incorrect functions from Part A to correctly address Part B. Part C: 2 points

1 point is earned for stating that the maximum weekly revenue is 12.5% greater than the weekly revenue with no increases.

1 point is earned for adequate supporting work that has a valid solution method.

Sample Student Response 1 I graphed my function and saw that the vertex is the maximum value. It’s at n = 10, so I calculated p(10) =$7.50 and n(10) = 90, so I know that the maximum weekly revenue will be $675. That would be a $75 increase from

$600,

So, the average weekly revenue would increase 12.5%.

OR I graphed my function and saw that the vertex is the maximum value. It’s at

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n = 10, so I calculated p(10) =$7.50 and n(10) = 90, so I know that the

maximum weekly revenue will be $675.

This shows a 12.5 %

increase. Sample Student Response 2 The vertex form of r(n) can be found by completing the square from the standard form:

–

I know that n = 10 will maximize the value of the equation, and can see that because when n = 10 then n – 10 = 0, so the value of the whole expression is 675. That is a

increase.

NOTE: There are other methods for getting r(n) into vertex form. For

example, one could use the formula for finding the axis of symmetry,

, to

obtain the value of n when at the parabola’s vertex:

Task score: The task score is the sum of the points awarded in each component.

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PARCC Algebra II/Math IIISample Item August 19, 2013 1


Algebra II and Math III Temperature Changes

Item Type Type III – 3 points

Evidence Statement

HS.D.2-10 with content scope of F.BF.A HS.D.2-10: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-level knowledge and skills articulated in F-BF.A, F-BF.3, F-IF.3, A-CED.1, A-SSE.3, F-IF.B, F-IF.7. Clarifications for HS.D.2-10: i) F-BF.A is the primary content; other listed content elements may be involved in tasks as well.


F-BF.A Build a function that models a relationship between two quantities 1. Write a function that describes a relationship between two quantities.

a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. F-IF.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. These standards are major content in the course based on the PARCC Model Content Frameworks.


Students apply their knowledge of functions to construct an accurate model of the situation (MP.4). Students first reason with the context and the potential models (MP.1 and MP.2), then they explain why the other models are inadequate and explain why their model is the best choice (MP.3). Tools and structure are important elements that students can access through the graphing tool (MP.5 – Use appropriate tools strategically and MP.7 – Look for and make use of structure).



This three-point application task requires students to identify and explain which type of function is appropriate to model the situation. The item provides three graphical models and students must identify whether each model is linear, quadratic or exponential. Then, students must reason abstractly with the graph and the context to evaluate each model. Students should explain that the linear model does not fit the data as well and allows for negative values of f(t) within the given range. The quadratic model is also flawed because the temperature begins to increase when t is greater than 165 seconds. The exponential model best matches this data because it has good visual fit and the values of f(t) will never be less than 0. Students may reach these conclusions and base their explanations on the graph and/or the data as provided in the table to evaluate the function types and create a function that models the data. The rubric makes it clear that there are a variety of possible models that could accurately fit this data, and all should receive credit. Note that other versions of this task could require students to use their model to solve for f(t) = 1.

New Scoring Information

Task is worth 3 points. Task can be scored as 0, 1, 2, or 3.

1. Part A: 1 modeling point: Student correctly classifies models as Model A (red): Quadratic Model, Model B (yellow): Exponential Model, and Model C (blue): Linear Model; identifies model B as best; and rejects models A and C for valid reasons. For example: Model A fits the data well, but the temperature of the material should fall to zero, and this model shows that the temperature starts to rise before the temperature reaches zero. Model C doesn’t fit the data as well as model B, and Model C also says that the temperature reaches negative values, which isn’t what the experiment says.

2. Part B: 2 modeling points: Student creates a function that adequately models the data. Not every step has to be justified, but the student’s method should be perceptible with its key steps shown. For example, an idealized solution that does justify each step is shown. (Other approaches besides this approach are possible.) My model is f(t) = A bt. To make my model, I started by finding the ratios of the data points: 141/200 = 0.71, 101/141 = 0.72, 74/101 = 0.73. They are pretty close. So I made the ratio in my model be 0.72 when the difference in time is 40 seconds:


Bt+40/Bt = 0.72

I know the properties so

BtB40/Bt = 0.72 Bt/Bt = 1

B40 = 0.72 Solve for B:

B = 0.721/40 = 0.992. So my model is f(t) = A(0.992)t. To find the A, I made it fit the data at the beginning:

A(0.992)0 = 200 (0.992)0 = 1

A = 200.

So my model is f(t) = 200(0.992)t.

NOTE: Accept other valid methods. If the student uses exponential regression on the calculator, they will get a function of 𝑓(𝑡) =198(0.992)𝑡. Additional notes: A student can earn a maximum of 2 points if they choose an incorrect model and use it correctly. A maximum of 1 point will be deducted if a computation mistake is made. Task score: The task score is the sum of the points awarded in each component.

November 2013

1

Grade 3 Patricia‘s Reading Time

Type Type I - 1 point

Evidence Statement

3.MD.1-2: Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. Clarifications: i) Only the answer is required (methods, representations, etc. are not assessed here). ii) Tasks do not involve reading start/stop times from a clock nor calculating elapsed time.


3.MD.1: Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. This standard is major content in the grade based on the PARCC Model Content Frameworks.


This is a simple modeling situation (MP.4) in which students solve a word problem using content knowledge from the grade.


This application task requires students to solve a grade-appropriate,

real-world application problem. The grade 3 standards show that

students are expected to be able to create an equation to model a

problem situation or solve the problem directly (e.g., 3.OA.8). Standard

3.MD.1 states specifically that these word problems involve adding

and subtracting time intervals in minutes as reflected in this item.

The response box is technology-enhanced so it can be electronically

scored. Unlike traditional multiple choice, it is difficult to guess the

correct answer or use a choice elimination strategy.

Scoring Information

1 point for the correct answer 17

PARCC Grade 3 Sample Item August 19, 2013 1

Grade 3 Vans for a Field Trip Item Type Type I – 1 point

Evidence Statement

3.OA.6: Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. Clarifications for 3.OA.6: i) All products and related quotients are from the harder three quadrants of the times table (a×b where a > 5 and/or b > 5).


3.OA.6: Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. This standard is major content in the grade based on the PARCC Model Content Frameworks.


Students create an equation which is a grade-appropriate model to expect at Grade 3 (MP.4).



This task illustrates the tie between multiplication and division. The Standards list twelve situations for multiplication and this context-based task around equal groups is grade-appropriate for grade 3. Grade 3 students engage deeply with multiplication and division, which requires a firm conceptual understanding to create the foundation needed as they continue their exploration of multiplication, division, and eventually rates and proportional relationships in future grades. Unlike traditional multiple choice, it is difficult to guess the correct answer or use a choice elimination strategy. Students are able to authentically create their own equation and the technology allows for immediate scoring of the result.

Scoring Information Student uses the drop-down menus to create one of the following:

“?” “×” “9” OR “9” “×” “?”.

PARCC Grade 5 Sample Item August 19, 2013

Grade 5 The Area of a Cut Board Item Type Type I – 1 point

Evidence Statement

5.NF.4b-1: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. Clarifications for 5.NF.4b-1: i) 50% of the tasks present students with the rectangle dimensions and ask students to find the area; 50% of the tasks give the fractions and the product and ask students to show a rectangle to model the problem.

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The Area of a Cut Board

PARCC Grade 5 Sample Item August 19, 2013


5.NF.4b: Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. This standard is major content in the grade based on the PARCC Model Content Frameworks.


This is a simple modeling situation (MP.4) in which students solve a word problem using content knowledge from the grade.


Here the area of the board is 1 square foot and students can use the technology to create a diagram that helps them solve the problem.

Note: The grid tool will not be scored in this task. The student types the answer in the space provided and the technology scores the item by checking to see if the value is equivalent to 10

18 .

Scoring Information

1018

or other equivalent expressions

November 2013

Grade 5 Two Aquarium Tanks

Type Type I - 2 points

Evidence Statement

5.MD.5c: Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. Clarification: i) Tasks require students to solve a contextual problem by applying the indicated concepts and skills.


5.MD.5: Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. This standard is major content in the grade based on the PARCC Model Content Frameworks.


This is a simple modeling situation (MP.4) in which students solve a word problem using content knowledge from the grade. Students will make use of structure of combining the volumes of two rectangular prisms to find the total volume of a complex space (MP.7).


This item directly assesses 5.MD.5c using a real-world application of grade-level content. Students work here is scaffolded by first working through the solution using standard steps. Then, in Part B students interact with the content at a higher cognitive level to solve for a missing value. Additionally, students successful on this item demonstrate that they meet the full intent of the standard. If students understand that the volume of solid figures composed of two non-overlapping right rectangular prisms can be determined by adding the volumes, then they should be successful on this question. The response boxes are technology-enhanced so they can be electronically scored. Unlike traditional multiple choice, it is difficult to guess the correct answer or use a choice elimination strategy.

Scoring Information

132 cubic meters; 114 cubic meters NOTE: A student can still receive one point on Part B with an incorrect answer in Part A.


Grade 6 Kelvin’s 100-Meter Dash Item Type Type I – 2 point

Evidence Statement

6.EE.7: Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. Clarifications for 6.EE.7: i) Problem situations are of “algebraic” type, not “arithmetic” type. See ITN Appendix F, Table F.d and the Progression for Expressions and Equations, pp. 3, 4. ii) 50% of tasks involve whole number values of p, q, and/or x; 50% of tasks involve fraction or decimal value of p, q, and/or x. Fractions and decimals should not appear in the same problem. (Cf. 7.EE.3) iii) A valid equation and the correct answer are both required for full credit. iv) The testing interface can provide students with a calculation aid of the specified kind for these tasks.

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Kelvin's 100-Meter Dash



6.EE.7: Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. This standard is major content in the grade based on the PARCC Model Content Frameworks.


MP.4 (Model with mathematics) is the most relevant Standard for Mathematical Practice. Students create an equation to model the information presented through the context.


This application task requires students to write and solve equations as part of the same solution process. Students at Grade 6 are expected to apply equations to problem situations, and then use those equations to find the solution. Here the variable, v, is defined for students. Some items on the PARCC test will allow students to define their own variables. Then, students must use that variable to create an equation relating the information given in the stem. The second part of the task requires students to solve their equation for v. The technology allows students to authentically create their own equation and record the solution. Unlike traditional multiple choice, it is difficult to guess the correct answer or use a choice elimination strategy.

Scoring Information

Task is worth 2 points. Task can be scored as 0, 1, or 2. Part A: 1 point – Student selects “100”, “12.5” and “∙” OR “12.5”, “100”, “÷”. Part B: 1 point – Student types “8”.



Grade 7 Reading Three Books Item Type Type I – 2 points Evidence Statement

7.RP.3-1: Use proportional relationships to solve multi-step ratio problems.


7.RP.3: Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. This standard is major content in the grade based on the PARCC Model Content Frameworks.


This task has some features of Modeling (MP.4) because a mathematical quantity (pages per day) is being defined to capture a real-world notion, “reading rate,” that does not come from the real world with a mathematical definition already associated with it.


This two-point task starts with students engaging in the important procedural skills of calculating and comparing unit rates. Students take information presented through the context to order the unit rates from greatest to least. Then, students use those rates to solve an application problem. Using ratios to solve problems is a critical skill for Grade 7 students. It builds on their earlier work with ratios in Grade 6 to set the stage for important Grade 8 work with functions. The use of technology in this task makes it difficult to guess the correct answer or use a choice-elimination strategy. A calculation aid will be available.

Scoring Information

Task is worth 2 points. Task can be scored as 0, 1, or 2. Part A: This part is worth 1 point. Nancy (160

5 = 32 pages per day)

Barbara (1004

= 25 pages per day)

Colleen (543

= 18 pages per day)

Part B: The part is worth 1 point. Barbara (320−100

25 = 8.8 days)

Nancy (480−16032

= 10.0 days)


Colleen (260−5418

= 11.4 days)

PARCC Algebra I Sample Item August 19, 2013

Algebra 2 Green Tea Observational Study Item Type Type I – 2 point

Evidence Statement

S-IC.3-1: Recognize the purposes of and differences among sample surveys, experiments, and observational studies. Clarifications for S-IC.3-1: i) The "explain" part of standard S-IC.3 is not assessed here; See Sub-claim D for this aspect of the standard.

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Green Tea Observational Study

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PARCC Algebra I Sample Item August 19, 2013

ii) See GAISE report, Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report.


S-IC.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. This standard is major content in the course based on the PARCC Model Content Frameworks.


This is a MP.4 task assessing S-IC.3 which is a modeling standard as indicated by the star symbol (*).


This task requires students to analyze the methods of an observational study and relate the methods to the possible uses of the resulting data. The Standards and the GAISE Report require statistics education so that students can be prepared for colleges and careers that increasingly interact with situations like the one described here. Informed consumers of information need to know the limitations of a study based on methodology so that incorrect assumptions are not made simply based on the size of the sample or prestige of the researchers.

Scoring Information

1 point: Student chooses “There were no treatment groups because it was an observational study.” 1 point: Student chooses “is associated with” and “the women in the study only”.

Documents

Modeling with Mathematics - · PDF file21/07/2014 · Session Title: Math – Modeling with Mathematic Description: ELC members will explore how modeling differs in the classroom and