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1 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Fill It Up, Please Part III Level Algebra or Math at the end of a unit on linear functions Geometry or Math as part of a unit on volume to spiral concepts about linear functions, rates of change, and discrete versus continuous data MODULE/Connection to AP* Rate of Change *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. Modality NMSI emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using these representations to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding. G N P A V P Physical V Verbal A Analytical N Numerical G Graphical About this Lesson In this lesson, students model filling a rectangular prism with sand by using tables, graphs, and equations. They interpret characteristics of the resulting function, including the domain, the range, the y-intercept, and the rate of change, in terms of the situation. Through investigating a variety of situations, students develop an understanding that the rate of change of the height of the sand in the box with respect to the number of scoops of sand added to the box is related to the base area of the box and the volume per scoop of sand added to the box. Throughout the lesson, students are encouraged to share their ideas and understandings with their classmates to clarify and deepen their conceptual base. The activity provides an engaging setting for students to practice and apply their skills with areas, volumes, and writing equations based on data while calculating rates of change. This lesson is part of a series of Fill It Up, Please lessons that includes the following: Fill It Up, Please Part I: In this middle grades lesson, students explore content similar to the content of this lesson including calculating rates of change; however, students are not asked to write the equations for the models. The lesson focuses on linear models and provides additional scaffolding. Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license, i

2 Fill It Up, Please Part II: In this middle grades lesson, students examine volume and rate of change physically, numerically, and graphically. Beginning with a concrete activity, students use nets to build prisms and pyramids, or cylinders and cones, with congruent bases and congruent heights. Students fill the containers they have created in order to examine how the height of the fill material in the container changes as the volume in the container increases. In addition, students begin to explore non-linear rates of change. Fill It Up, Please Part IV: In this Algebra / Math lesson, students perform an experiment in order to examine what happens to the rate of change in the height of fill material in an irregularly shaped container when the fill material is added at a constant rate. The focus of this lesson is on non-linear rates of change. Objectives Students will graph and determine equations for data in a table. determine appropriate domains and ranges and relate them to the quantitative relationship they describe. calculate and interpret rates of change. compare rates of change represented algebraically, graphically, numerically in tables, and by verbal descriptions. modify equations based on changing initial conditions. calculate areas and volumes of rectangular prisms. solve for height, given volume and area of the base of a rectangular prism. ii Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

3 COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENT This lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. The star symbol ( ) at the end of a specific standard indicates that the high school standard is connected to modeling. Targeted Standards F-BF.: Write a function that describes a relationship between two quantities. See questions i, j, c, F-LE.: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). See questions i, j, c, A-CED.: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. See questions i, j, c, F-IF.6: F-IF.9: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. See questions h, j, 6-8 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). See questions, 7 Reinforced/Applied Standards F-LE.: Interpret the parameters in a linear or exponential function in terms of a context. See questions g, j F-IF.: N-Q.: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble an engine in a factory, then the positive integers would be an appropriate domain for the function. See questions c, e, j, c, 9 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. See questions b, a, 9 A-CED.: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. See questions, a, c A-CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. See question a Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license, iii

4 COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICE These standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. NMSI incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice. MP.: Make sense of problems and persevere in solving them. In question 7, students are presented information in four different modalities (graphically, numerically in a table, analytically with an equation, and verbally) and must base their solution pathways on previous information but from different entry points. In question 9, students are presented with a complex problem. While it is based on a similar situation, the volume of the scoop is unknown. They will need to look for entry points based on their previous experiences and apply them in a new situation. MP.: Reason abstractly and quantitatively. Students create equation models based on data and use units to guide their solutions. They write equations that represent the relationship between the number of scoops of sand and the height of the sand in the box, interpret the results in the context of the situation, and consider modifications to the conditions. In question 0, students move from working with discrete values to a generic model where the parameters change by different factors of k. MP.: Construct viable arguments and critique the reasoning of others. Questions and b require students to compare and explain methods for solving questions with other students. Question 6c requires students to write a summary of the features of the prisms which cause the difference in the rate the height changes. MP.8: Look for and express regularity in repeated reasoning. In question 0, students use their experiences from previous questions to complete a question where there is no longer a physical model provided.. Students can quickly answer question by drawing from the connections they made in previous questions between the area of the base and the slope of the height graph even though the shape of the base is a circle rather than a rectangle, the data is continuous rather than discrete, and the independent variable is time rather than number of scoops iv Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

5 Foundational Skills The following skills lay the foundation for concepts included in this lesson: Calculate volumes of rectangular prisms Write equations of linear functions MATERIALS AND RESOURCES Student Activity pages Colored pencils ASSESSMENTS The following types of formative assessments are embedded in this lesson: Students engage in independent practice. Students explain their reasoning to others. Students apply knowledge to a new situation. The following assessments are located on our website: Rate of Change: Average and Instantaneous Algebra Free Response Questions Rate of Change: Average and Instantaneous Algebra Multiple Choice Questions Rate of Change: Related Rates Algebra Free Response Questions Rate of Change: Related Rates Algebra Multiple Choice Questions Rate of Change: Average and Instantaneous Geometry Free Response Questions Rate of Change: Average and Instantaneous Geometry Multiple Choice Questions Rate of Change: Related Rates Geometry Free Response Questions Rate of Change: Related Rates Geometry Multiple Choice Questions Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license, v

6 TEACHING SUGGESTIONS Agroup setting is most appropriate for this lesson and encourages discussion of various approaches to answering the questions. The use of cooperative groups will encourage all students, including ELL students, to share information while working together to complete authentic tasks. The use of cooperative groups will also encourage active engagement in the formation of a conceptual base enhanced by investigating multiple modalities within the problem situation. Incorporating group work is one way to give students space to learn academic language while absorbing content. Students who might be reluctant to talk in whole-class discussions can practice using mathematical language and receive feedback in a relatively lowstakes setting during group work. In this lesson, a student who is beginning to learn English can engage by completing the table of values, plotting points, comparing graphs, and noticing patterns. With some assistance from a bilingual peer, these students can also develop explanations and make generalizations without compromising or simplifying the mathematical language. The questions in this lesson build from an opening investigation to more sophisticated application of the concepts. Question can be used as an exploratory activity or as a classroom example. Question can be used as formative assessment. Question extends the situation to equation solving. Question 4 requires students to consider how changes in the parameters affect the rate of change. Question increases the level of rigor by the addition of a variable for the length of the base of the box. Question 6 provides summary ideas. Question 7 can be used as a formative assessment. Questions 8 provide opportunities to apply knowledge in a new situation. Suggested modifications for additional scaffolding include the following: - Provide models of the boxes, scoops, and sand, and have students conduct the experiment as they record values in the tables. b Modify the table by including additional rows for 0,,, and 6 scoops. c Modify the graph by drawing a dashed horizontal line at y = 4 to indicate when the box is full. j Add the following to the question: Hint (area of base)(height of sand) = in. a Modify the table by including additional rows for 0-, 4, 6-7, 9, and - scoops and including a process column. vi Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

7 NMSI CONTENT PROGRESSION CHART In the spirit of NMSI s goal to connect mathematics across grade levels, the Content Progression Chart demonstrates how specific skills build and develop from sixth grade through pre-calculus. Each column, under a grade level or course heading, lists the concepts and skills that students in that grade or course should master. Each row illustrates how a specific skill is developed as students advance through their mathematics courses. 6th Grade Skills/ Objectives 7th Grade Skills/ Objectives Algebra Skills/ Objectives Geometry Skills/ Objectives Algebra Skills/ Objectives Pre-Calculus Skills/ Objectives From graphical or tabular data or from a stated situation presented in paragraph form, calculate or compare the average rates of change and interpret the meaning. From graphical or tabular data or from a stated situation presented in paragraph form, calculate or compare the average rates of change and interpret the meaning. From graphical or tabular data or from a stated situation presented in paragraph form, calculate or compare the average rates of change and interpret the meaning. From graphical or tabular data or from a stated situation presented in paragraph form, calculate or compare the average rates of change and interpret the meaning. From graphical or tabular data or from a stated situation presented in paragraph form, calculate or compare the average rates of change and interpret the meaning. From graphical or tabular data or from a stated situation presented in paragraph form, calculate or compare the average rates of change and interpret the meaning. Create and analyze geometric and/or numerical models to describe a situation that changes with respect to time. Create and analyze geometric and/or numerical models to describe a situation that changes with respect to time. Create and analyze geometric and/or numerical models to describe a situation that changes with respect to time. Create and analyze geometric, numerical, and/or algebraic models to describe a situation that changes with respect to time. Create and analyze geometric, numerical, and/or algebraic models to describe a situation that changes with respect to time. Create and analyze geometric, numerical, and/or algebraic models to describe a situation that changes with respect to time. Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license, vii

8 viii Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

9 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Fill It Up, Please Part III Answers V. a. B b. Shaded answers are provided for scaffolding purposes only. Number of scoops Process Column for Height Current Height 0 scoops 0 in. 0 in. scoop in. scoop 6in. in. scoops scoops 4 scoops in. scoops 6in. in. scoops 6in. in. 4 scoops 6in. in. in. in. scoops in. scoops 6in. in. 6 scoops in. 6 scoops 6in. in. 7 scoops in. 7 scoops 6in. 7 in. 8 scoops in. 8 scoops 6in. 4 in. Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license, ix

10 c. Height in inches h 6 Box A 4 s Number of Scoops d. 8 scoops e. 0 s 8 and s whole numbers ; the domain is discrete. The box is initially empty. Sandy adds only whole scoops so the domain must increase by for each scoop. Eight scoops completely fill the box. The domain is discrete because Sandy can use only whole scoops. 7 f. 0,,,,,,,, 4 ; the range is discrete. The box is initially empty. Sandy adds only whole scoops so the height increases by inch for each scoop. The box holds 4 inches of sand when it is full. The range is discrete because adding one whole scoop increases the height by one-half inch. g. The y-intercept is 0; the box is empty before any sand is added. in. in. in. h. = scoops scoop scoop i. h= s, 0 s 8 and s whole numbers in. j. Initial height of the sand: = in. ; 6 in. The y-intercept, the domain, and the range will be affected by changing the initial value. Height in inches h 6 Box A 4 s Number of Scoops x Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

11 The y-intercept is which means that for 0 scoops of sand, the height of the sand is inches. The height of the sand in the box changes at a rate of inch per scoop which means that for every scoop of sand that is added to the box, the height increases by inch. The data are discrete because Sandy adds whole scoops. The data can be modeled h= s+ by the equation where the restrictions on the domain require that 0 s and s whole numbers since scoops are required to completely fill the box. Since the box initially contains sand to a height of inches and will hold sand to a height of 4 inches, and since adding one whole scoop increases the height by inch, the range is 7,,, 4. k. All remain constant (and should be marked) except The height of the sand.. a. Shaded answers are provided for scaffolding purposes only. Number Process Current of scoops Column Height 0 scoops 0 in. 0 in. scoop in. scoop 9in. in. scoops in. scoops 9in. in. scoops in. scoops 9in. in. Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license, xi

12 4 scoops scoops in. 4 scoops 4 9in. in. in. scoops 9in. in. 6 scoops in. 6 scoops 9in. in. 7 scoops in. 7 scoops 7 9in. in. 8 scoops 9 scoops 0 scoops in. 8 scoops 8 9in. in. in. 9 scoops h in. = 9in. in. 0 scoops 0 9in. in. scoops in. scoops 9in. in. scoops in. scoops 9in. 4 in. xii Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

13 b. Height in inches Box B h 6 4 s Number of Scoops in. in. in. c. =, so h= s, 0 s and s whole numbers scoops scoop scoop d. The height of the sand increases by in. scoop, so an initial height of in. of sand would require fewer scoops to fill the box.. After 4 scoops, the height of the sand in Box A and Box B will be equal. Algebraically: s = s+, so s = 4 scoops. Graphically: Height in inches h 6 4 Box A and Box B s Number of Scoops Numerically: Number of scoops Current Height for Box A Current Height for Box B 0 scoops 0 in. in. scoop in. scoops in. scoops in. in. 4 in. in. 4 scoops in. in. Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license, xiii

14 4. R Use a larger scoop R Decrease the base area of the box 0 Add the sand at a faster rate 0 Add the sand two scoops at a time 0 Begin with the box half full of sand. a. in. 6 scoops scoop = 4 in.; so x = i ; therefore, the length of the base of the box is ( in.)( x in.) i inches. b. Since there is no sand in the box initially and 6 scoops are required to fill it to a height of 4 inches, the graph contains the points (0, 0) and (6, 4). The equation is h= s, 0 s 6 and s whole numbers. x in. 0 scoops c. = 4 in.; therefore, the volume of the scoop is 9 cubic inches. ( in.) in. 6. a. The following answer uses the notation 7. a. h s to represent the rate of change in the height, h, of the sand in the box with respect to the number of scoops, s. h h h for Box B < for Box A < for Box C since in. < in. < in.. s s s scoop scoop scoop b. The following answer uses the notation B to represent base area. 9 B for Box C < B for Box A < B for Box B since in. < 6 in. < 9 in.. c. As the base area of the box increases, the rate of change of the height of the sand in the box with respect to the number of scoops decreases. In other words, the larger the base area of the box, the longer it takes to fill. Box D in. ; count the change in the height scoop values and divide by the change in the Box E in. in. in. = 8 scoops 6 scoops scoop corresponding number of scoops. Box F Box G in. in. ; the m value of the equation (0,0) and (8,6), so scoop 4 scoop xiv Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

15 h b. The following answer uses the notation to represent the rate of change in the s height, h, of the sand in the box with respect to the number of scoops, s. h h h h for Box E < for Box D < for Box F < for Box G since s s s s in. in. in. in. < < <. scoop scoop scoop 4 scoop 8. a. in. scoop in. in. = scoop b. in. scoop 4 in. = (. in.)(in.) 9 scoop c. 9. a. in. scoop 4 in. =, so scoops of sand will be required to fill the box. scoop in. in. in. scoop in. = The rate of change is the expression that is multiplied times the number of in. scoop scoops. It is the volume per scoop divided by the base area. b. Since the initial height of the sand is inches, the point (0, ) is on the graph. The equation shows that after scoops, the height of the sand has increased by in. Since the initial height is inches, 4 the point, + 4 or, is on the graph. Other data points that result from this rate of change 4 are also correct answers. c. in. in. 4 in. = scoops 0 scoops scoop d. Sample answer: volume of the scoop is in. and the base area of the box is in. ; the volume of the scoop is 0 in. and the base area of the box is 4 in. In general, the volume of the scoop is a in. and the base area of the box is a in. e. If the sand is added at a constant rate (poured in at a steady stream) rather than being added one scoop at a time, the graph would be continuous. Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license, xv

16 0. Since only completely filled scoops are used, the data is discrete, and the values of s must be whole numbers. If the sand is added at a constant rate (poured in at a steady stream) rather than being added one scoop at a time, the graph would be continuous. The height of the sand in the box is initially in. and is increasing at a rate in. of scoop, thus the height of the sand in the box must be a multiple of. This equation models infinitely many box(es). If the volume of the sand in the box increases by cubic inch per scoop, then the base area of the box must be in.. in. If the volume of the sand in the box increases by 6 scoop, then the base area of the box must be 8 in.. In general, for a box modeled by this equation, if the volume of the sand in the box in. increases by k scoop, then the base area of the box must be k in.. The rate of change in the height of the sand in the box with respect to the number of scoops, in in. in. scoop, is the same as the ratio of the increase in the volume of sand in the box, in scoop, to the base area of the box, in in... Cylinder A will take the longest to fill (or have the smallest rate of change) since it has the greatest diameter. Its graph of height versus time is Graph. Cylinder B will take the least time to fill (or have the greatest rate of change) since it has the smallest diameter. Its graph of height versus time is Graph. By process of elimination, Cylinder C matches Graph. xvi Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

17 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Fill It Up, Please Part III. Box A has a height of 4 inches and a base area of 6 square inches. Sandy is filling the box with sand using a scoop that holds cubic inches. To be precise in her measurements, she only adds whole scoops of sand. After each scoop is added, she levels the sand in the box and measures the height of the sand. a. What is the equation for the height of a rectangular prism, h, in terms of the volume, V, and the base area, B? b. Complete the table that indicates the height of the sand, h, in Box A for the total number of scoops of sand, s. Total number of scoops, s scoop Process column for the height of the sand in. scoop 6 in. Current height of the sand, h in. scoops 4 scoops 8 scoops Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

18 c. Plot the ordered pairs, (, s h ), from the table and fill in the additional points that were not determined in the table that indicate that Sandy is filling the box from empty to full. 6 h Box A Height in inches 4 s NumberofScoops d. How many scoops are required to fill Box A? e. What is the domain of the graph of height of the sand, h, versus the number of scoops, s? Is the domain discrete or continuous? Explain the answers in terms of the situation. f. What is the range of the graph of height of the sand, h, versus the number of scoops, s? Is the range discrete or continuous? Explain the answers in terms of the situation. g. What is the y-intercept of the graph of height versus the number of scoops? Explain the answer in terms of the situation. h. What is the rate of change in the height of the sand with respect to the number of scoops? Show how to calculate the answer using two points. Include units of measure in your work and in your answers. i. What is the equation for the height of the sand in the box, h, in terms of the number of scoops of sand, s? Include the domain in the answer. Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

19 j. Suppose Box A initially contained cubic inches of sand. What is the initial height of the sand? Indicate units of measure in your work and answers. What features of the graph will be affected by this new initial value? Using a different color pencil, graph the data on the grid provided in part (c). Use the options in the cells of the table to complete the sentences provided. Some may be used more than once while others may not be used at all. 0 height 0 s volume 0 s 4 real 0 s whole 0 h 4 discrete 0 h 4 continuous 0 h 7,,, 4,, 6, 6,..., h= s h= s+ h= s+ The y-intercept is which means that for scoops of sand, the height of the sand is inches. The height of the sand in the box changes at a rate of inch per scoop which means that for every scoop of sand that is added to the box, the increases by inch. The data are because Sandy adds whole scoops. The height of the sand can be modeled by the equation where the restrictions on the domain require that and s whole numbers since scoops are required to completely fill the box. Since the box initially contains sand to a height of inches and will hold sand to a height of inches, and adding one whole scoop increases the height by inch, the range is. Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

20 k. Which of the following remain constant as Sandy fills Box A with sand as described? Mark all correct choices. 00 The base area of the box 00 The height of the box 00 The volume of the scoop 00 The height of the sand 00 The base area of the prism formed by the sand 00 The cross-sectional area of the sand parallel to the base after the sand is leveled 00 The rate of change of the height of the sand with respect to the number of scoops. Box B has a height of 4 inches and a base area of 9 square inches. Sandy is filling the box with sand using a scoop that holds cubic inches. To be precise in her measurements, she only adds whole scoops of sand. After each scoop is added, she levels the sand in the box and measures the height of the sand. a. Complete the table that indicates the height of the sand in the box after s scoops have been added. Total number of scoops, s Current height of the sand, h scoops in. scoops 8 scoops 0 scoops 4 Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

21 b. Plot the ordered pairs, (, s h ), from the table and fill in the additional points that were not determined in the table that indicate that Sandy is filling the box from empty to full. 6 h Box B Height in inches Number of Scoops s c. What is the equation for the height of the sand in the box, h, in terms of the number of scoops of sand, s? Include the domain in the answer. d. If the initial height of the sand in Box B had been inches, how many fewer scoops would be needed to fill the box? Explain your answer or show your work mathematically. Include units in your explanation, work, and answer.. After how many scoops would the height of the sand in Box A with an initial height of 0 inches of sand and in Box B with an initial height of inches of sand be equal? Justify your answers either algebraically, graphically, or numerically (with a table). Compare your method to someone who used a different method. Include a critique of the reasoning of the other person. Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

22 4. There are several ways that Sandy could modify the conditions of her experiment. What modifications would increase the rate of change in the height of the sand in the box with respect to the number of scoops of sand? Mark all correct choices. 00 Use a larger scoop 00 Decrease the base area of the box 00 Add the sand faster 00 Add the sand two scoops at a time 00 Begin with the box half full of sand. Sandy is using a scoop that holds cubic inches of sand to fill Box C. After each scoop is added, she levels the sand in the box and measures the height of the sand. a. If the box is full when 6 scoops are added, what is x, the length of the base of the box? Include units in your work and answer. b. What is the equation for the height of the sand in the box, h, in terms of the number of scoops, s? Explain the method you used to determine the equation and discuss your method with someone who used a different method. Include a critique of their reasoning. c. If Sandy decides to use a different sized scoop, what size scoop will allow her to completely fill Box C in 0 scoops? Include units in your work and answer. 6 Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

23 6. Compare the boxes using the following information. a. Write an inequality statement comparing the rates of change in the height with respect to the number of scoops of sand for Box A, Box B, and Box C. b. Write an inequality statement comparing the base areas of Box A, Box B, and Box C. c. Write a summary statement relating the base area of the box and the corresponding rate of change of the height of the sand in the box with respect to the number of scoops of sand if the scoop size stays the same. 7. Sandy has four empty boxes whose base areas are unknown. The volume of the scoops she is using to fill the boxes is also unknown. To be precise in her measurements, she adds whole scoops of sand, levels the sand in the box, and measures the height of the sand. Information about Box D, Box E, Box F, and Box G is provided in the following table. Box D Box E Height in inches Box D h 6 Total number of scoops, s 4 0 s Number of Scoops Current height of the sand, h 6 scoops in. 8 scoops in. Box F h= s, 0 s 4 and s whole numbers, where h is the height of the sand in inches and s is the number of scoops of sand Box G Box G has a height of 6 inches and is full of sand when 8 scoops of sand are added. Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license, 7

24 a. Based on the information given in the previous table, determine the rates of change of the height of the sand in Box D, Box E, Box F, and Box G with respect to the number of scoops of sand. In the following table, record your work and explain how to determine each answer in words or show your numerical work. Include units in your work and answers. Box D Box E Box F Box G b. Write an inequality statement comparing the rates of change in the height of the sand in the box with respect to the number of scoops of sand for Box D, Box E, Box F, and Box G. 8 Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

25 8. Sandy is using a scoop that holds cubic inches of sand to fill Box H and Box J. After each scoop is added, she levels the sand in the box and measures the height of the sand. She plans to collect data as she did for Boxes A, B, and C to determine the rate of change of the height of the sand in the box with respect to the number of scoops. Her friend, Crystal, tells her that she knows a short cut. Crystal says, To calculate the rate of change, you divide the volume per scoop by the base area of the box. a. Use dimensional analysis to explain why Crystal s method is correct. b. Use Crystal s method to calculate the rate of change of the height of the sand in the box with respect to the number of scoops for Box H. Include units in your work and answer. c. Box J is a rectangular prism with a length of in., a width of in., and a height of in. Use Crystal s method to calculate the rate of change of the height of the sand in the box with respect to the number of scoops and then determine the number of scoops required to fill Box J. Include units in your work and answer. Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license, 9

26 9. The following information is known about Box K: The box is a rectangular prism and initially contains inches of sand. Sand is being added to the box using a scoop of known volume. The sand is leveled and the height of the sand is measured after each scoop. in. scoop ( scoops) = in. in. 4 a. What is the rate of change of the height of the sand in the box with respect to the number of scoops? Explain how the answer can be determined from the given equation. b. List the coordinates of at least two data points. Explain how you know these are data points. c. Using two of these data points, show another method to determine the rate of change of the height of the sand in the box with respect to the number of scoops. d. There is not enough information to determine the volume of the scoop and the base area of Box K. Provide information about two different scenarios for which the rate of change in the height in inches per scoop is the same. e. How could the process of filling Box K with sand be modified so that the data would be continuous? 0 Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

27 0. Sandy is filling a box with sand using a scoop. To be precise in her measurements, she only adds whole scoops of sand. After each scoop is added, she levels the sand in the box and measures the height of the sand. The equation h= s+ models the height of the sand in the box with respect to the number of scoops. Use the options in the cells of the table to complete the sentences provided. Some may be used more than once while others may not be used at all. 0 in. rational base area in. real height in. whole discrete in. scoop k continuous 6 in. scoop k only one 4 in. scoop k infinitely many Since only completely filled scoops are used, the data is, and the values of s must be numbers. If the sand is added at a constant rate (poured in at a steady stream) rather than being added one scoop at a time, the graph would be. The height of the sand in the box is initially and is increasing at a rate of, thus the height of the sand in the box must be a multiple of. This equation models box(es). If the volume of the sand in the box increases by cubic inch per scoop, then the base area of the box must be. If the volume of the sand in the box increases by, then the base area of the box must be 8 in.. In general, for a box modeled by this equation, if the volume of the sand in the box increases by, then the base area of the box must be. The rate of change in the height of the sand in the box with respect to the number of scoops, in, is the same as the ratio of the increase in the volume of sand in the box, in, to the of the box, in. Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

28 . All of the cylinders drawn below have equal heights but different sized bases. Water is poured into each cylinder at the same constant rate. Match each cylinder with the appropriate height versus time graph. A B Height C Time Copyright 0 National Math + Science Initiative. This work is made available under a Creative Commons Attribution- NonCommercial-NoDerivs.0 United States license,

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