Fill It Up, Please – Part III

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MathematicsNATIONALMATH + SCIENCEINITIATIVE

LeveLAlgebra 1 or Math 1 at the end of a unit on linear functions

Geometry or Math 2 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.

MODaLiTyNMSI 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.

P – Physical V – VerbalA – AnalyticalN – NumericalG – Graphical

Fill It Up, Please – Part III

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.

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Mathematics—Fill it Up, Please – Part iii

● 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 1/Math 1 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.

ObjecTivesStudents 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.

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cOMMOn cOre sTaTe sTanDarDs FOr MaTheMaTicaL cOnTenTThis 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 StandardsF-BF.1: Write a function that describes a

relationship between two quantities.★ See questions 1i, 1j, 2c, 5

F-LE.2: 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 1i, 1j, 2c, 5

A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★ See questions 1i, 1j, 2c, 5

F-IF.6: 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 1h, 1j, 6 - 8

F-IF.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).★ See questions 3, 7

Reinforced/Applied StandardsF-LE.5: Interpret the parameters in a linear

or exponential function in terms of a context. See questions 1g, 1j

F-IF.5: 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 1c, 1e, 1j, 2c, 9

N-Q.1: 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 1b, 2a, 9

A-CED.1: 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 3, 5a, 5c

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 1a

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cOMMOn cOre sTaTe sTanDarDs FOr MaTheMaTicaL PracTiceThese 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.1: 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.2: 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 10, students move from working with discrete values to a generic model where the parameters change by different factors of k.

MP.3: Construct viable arguments and critique the reasoning of others. Questions 3 and 5b 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 10, 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 11 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

.

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FOUnDaTiOnaL skiLLsThe following skills lay the foundation for concepts included in this lesson:

● Calculate volumes of rectangular prisms● Write equations of linear functions

assessMenTsThe 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 1 Free Response Questions

● Rate of Change: Average and Instantaneous – Algebra 1 Multiple Choice Questions

● Rate of Change: Related Rates – Algebra 1 Free Response Questions

● Rate of Change: Related Rates – Algebra 1 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

MaTeriaLs anD resOUrces● Student Activity pages ● Colored pencils

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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 low-stakes 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 1 can be used as an exploratory activity or as a classroom example. Question 2 can be used as formative assessment. Question 3 extends the situation to equation solving. Question 4 requires students to consider how changes in the parameters affect the rate of change. Question 5 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 – 11 provide opportunities to apply knowledge in a new situation.

Suggested modifications for additional scaffolding include the following:1-2 Provide models of the boxes, scoops, and sand,

and have students conduct the experiment as they record values in the tables.

1b Modify the table by including additional rows for 0, 2, 5, and 6 scoops.

1c Modify the graph by drawing a dashed horizontal line at 4y = to indicate when the box is full.

1j Add the following to the question: “Hint – (area of base)(height of sand) = 15 in3.”

2a Modify the table by including additional rows for 0-2, 4, 6-7, 9, and 11-12 scoops and including a process column.

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nMsi cOnTenT PrOgressiOn charTIn 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 1 Skills/Objectives

Geometry Skills/Objectives

Algebra 2 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.






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.



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Mathematics

NATIONALMATH + SCIENCEINITIATIVE


Answers

1. a. VhB

=

b. Shaded answers are provided for scaffolding purposes only.

Number of scoops

Process Column for Height

CurrentHeight

0 scoops 0 in. 0 in.

1 scoop

3

2

3in.1 scoopscoop

6in.h

=

1 in.2

2 scoops

3

2

3in.2 scoopsscoop

6in.h

=

1 in.

3 scoops

3

2

3in.3 scoopsscoop

6in.h

=

3 in.2

4 scoops

3

2

3in.4 scoopsscoop

6in.h

=

2 in.

5 scoops

3

2

3in.5 scoopsscoop

6in.h

=

5 in.2

6 scoops

3

2

3in.6 scoopsscoop

6in.h

=

3 in.

7 scoops

3

2

3in.7 scoopsscoop

6in.h

=

7 in.2

8 scoops

3

2

3in.8 scoopsscoop

6in.h

=

4 in.

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Mathematics—Fill It Up, Please – Part III

c.

Hei

ght

inin

ches

Number of Scoops

Box A

s

hH

eigh

tin

inch

es

Number of Scoops

Box A

s

h

0 1 2 3 4 5 6 7 8 9 10 11 120123456

d. 8 scoops

e. 0 8 and whole numberss s≤ ≤ ∈ ; the domain is discrete. The box is initially empty. Sandy adds only whole scoops so the domain must increase by 1 for each scoop. Eight scoops completely fill the box. The domain is discrete because Sandy can use only whole scoops.

f. 1 3 5 70, , 1, , 2, , 3, , 42 2 2 2

; the range is discrete. The box is initially empty. Sandy adds only whole

scoops so the height increases by 12

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.

h.

3 1in. in. 1 in.2 23 scoops 1 scoop 2 scoop

−=

−

i. 1 , 0 8 and whole numbers2

h s s s= ≤ ≤ ∈

j. Initial height of the sand: 3

2

15 in. 5 in.6 in. 2

h = = ; The y-intercept, the domain, and the range will be affected by changing the initial value.

Hei

ght

inin

ches

Number of Scoops

Box A

s

h

Hei

ght

inin

ches

Number of Scoops

Box A

s

h

0 1 2 3 4 5 6 7 8 9 10 11 120123456

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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

by the equation where the restrictions on the domain require that and

s ∈ whole numbers since 3 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

.

k. All remain constant (and should be marked) except “The height of the sand.”

2. a. Shaded answers are provided for scaffolding purposes only.

Number of scoops

Process Column

CurrentHeight


1 scoop

3

2

3in.1 scoopscoop

9in.h

=

1 in.3

2 scoops

3

2

3in.2 scoopsscoop

9in.h

= 2

3in.

3 scoops

3

2

3in.3 scoopsscoop

9in.h

=

1 in.

52

52

12

12

1 52 2

h s= +0 3s≤ ≤

52

5 7, 3, , 42 2

12

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4 scoops

3

2

3in.4 scoopsscoop

9in.h

= 4

3 in.

5 scoops

3

2

3in.5 scoopsscoop

9in.h

= 5

3 in.

6 scoops

3

2

3in.6 scoopsscoop

9in.h

=

2 in.

7 scoops

3

2

3in.7 scoopsscoop

9in.h

= 7

3 in.

8 scoops

3

2

3in.8 scoopsscoop

9in.h

= 8

3 in.

9 scoops

3

2

3in.9 scoopsscoop

9in.h

=

3 in.

10 scoops

3

2

3in.10 scoopsscoop

9in.h

= 10

3 in.

11 scoops

3

2

3in.11 scoopsscoop

9in.h

= 11

3 in.

12 scoops

3

2

3in.12 scoopsscoop

9in.h

=

4 in.

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b.

Hei

ght

inin

ches

Number of Scoops

Box B

s

h

Hei

ght

inin

ches

Number of Scoops

Box B

s

h

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150123456

c.

2 1in. in. 1 in.3 32 scoops 1 scoop 3 scoop

−=

−, so 1 , 0 12 and whole numbers

3h s s s= ≤ ≤ ∈

d. The height of the sand increases by 1 in.3 scoop

, so an initial height of 2 in.3

of sand would

require 2 fewer scoops to fill the box.

3. After 4 scoops, the height of the sand in Box A and Box B will be equal.

Algebraically: 1 1 2 , so 4 scoops2 3 3

s s s= + = . Graphically:

Hei

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inin

ches

Number of Scoops

Box A and Box B

s

h

Hei

ght

inin

ches

Number of Scoops

Box A and Box B

s

h

��

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150123456

Numerically:

Number of scoops Current Height for Box A Current Height for Box B

0 scoops 0 in. 2 in.3

1 scoop1 in.2

1 in.

2 scoops 1 in. 43

in.

3 scoops 3 in.2

53

in.


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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

5. a.

33 in.6 scoopsscoop 34 in.; so in.

(3 in.)( in.) 2x

x

= = ; therefore, the length of the base of the box is 3

2in. 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

2 , 0 6 and whole numbers3

h s s s= ≤ ≤ ∈ .

c.

3

3

in.10 scoopsscoop 94 in.; so in.

3 5(3 in.) in.2

x

x

= =

therefore, the volume of the scoop is 95

cubic inches.

6. a. The following answer uses the notation hs

∆∆

to represent the rate of change in the

height, h, of the sand in the box with respect to the number of scoops, s.

for Box B < for Box A < for Box Ch h hs s s

∆ ∆ ∆∆ ∆ ∆

since 1 in. 1 in. 2 in. .3 scoop 2 scoop 3 scoop

< <

b. The following answer uses the notation B to represent base area.

for Box C < for Box A < for Box BB B B since 2 2 29 in. 6 in. 9 in.2

< < .

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.

7. a.Box D Box E

1 in.2 scoop

; count the change in the height

values and divide by the change in the

corresponding number of scoops.

22 in. 2 in. 1 in.38 scoops 6 scoops 3 scoop

−=

−

Box F Box G

2 in.3 scoop

; the m value of the equation (0,0) and (8,6), so 3 in.4 scoop

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b. The following answer uses the notation hs

∆∆

to represent the rate of change in the

height, h, of the sand in the box with respect to the number of scoops, s.

for Box E < for Box D < for Box F < for Box Gh h h hs s s s

∆ ∆ ∆ ∆∆ ∆ ∆ ∆

since

1 in. 1 in. 2 in. 3 in. .3 scoop 2 scoop 3 scoop 4 scoop

< < <

8. a.

3

2

in.in.scoop

in. scoop=

b.

3in.24 in.scoop

(1.5 in.)(3in.) 9 scoop=

c.

3in.24 in.scoop

3 5 5 scoopin. in.2 3

=

, so 2 scoops of sand will be required to fill the box.

9. a.

3

2

5 in.5 in.scoop

12 in. 12 scoop= The rate of change is the expression that is multiplied times the number of

scoops. It is the volume per scoop divided by the base area.

b. Since the initial height of the sand is 2 inches, the point (0, 2) is on the graph. The equation shows

that after 3 scoops, the height of the sand has increased by 5 in.4

Since the initial height is 2 inches,

the point 53, 24

+ or

13, 34

is on the graph. Other data points that result from this rate of change

are also correct answers.

c.

13 in. 2 in. 5 in.43 scoops 0 scoops 12 scoop

−=

−

d. Sample answer: volume of the scoop is 35 in. and the base area of the box is 212 in. ; the volume of the scoop is 310 in. and the base area of the box is 224 in. In general, the volume of the scoop is 35 in.a and the base area of the box is 212 in.a

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.

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10. 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 2 in. and is increasing at a rate

of , 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 1 cubic inch per scoop, then the base area of the box must be 3 .

If the volume of the sand in the box increases by 6 , then the base area of the box must be 218 in. . In general, for a box modeled by this equation, if the volume of the sand in the box

increases by 1k , then the base area of the box must be 3k .

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 base area of the box, in .

11. 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 3. 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 1. By process of elimination, Cylinder C matches Graph 2.

13

in.scoop

13

2in.3in.

scoop

3in.scoop 2in.

in.scoop

3in.scoop

2in.

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Mathematics

NATIONALMATH + SCIENCEINITIATIVE


1. 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 3 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

Process column for the height of the sand

Current height

of the sand, h

1 scoop

3

2

3 in.1 scoopscoop

6 in.h

=

1 in.2

3 scoops

4 scoops

8 scoops

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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.

Heightininches

Number of Scoops

Box A

s

h

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

1

2

3

4

5

6

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.




j. Suppose Box A initially contained 15 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 3s≤ ≤5 7, 3, , 42 2

12

volume 0 4s≤ ≤1 115, 15 , 16, 162 2

52

real 0 15s≤ ≤1 115, 15 ...21 , 222 2

3 whole 0 3h≤ ≤12

h s=

4 discrete 0 4h≤ ≤1 52 2

h s= +

15 continuous 0 15h≤ ≤1 152

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

__________ .



k. Which of the following remain constant as Sandy fills Box A with sand as described? Mark all correct choices.

0 The base area of the box 0 The height of the box 0 The volume of the scoop 0 The height of the sand 0 The base area of the prism formed by the sand 0 The cross-sectional area of the sand parallel to the base after the sand is leveled 0 The rate of change of the height of the sand with respect to the number of scoops

2. 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 3 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 heightof the sand, h

3 scoops 1 in.

5 scoops8 scoops10 scoops




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.

Hei

ght

inin

ches

Number of Scoops

Box B

s

h

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

1

2

3

4

5

6

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 23

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.

3. 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 23

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.



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.

0 Use a larger scoop 0 Decrease the base area of the box 0 Add the sand faster 0 Add the sand two scoops at a time 0 Begin with the box half full of sand

5. Sandy is using a scoop that holds 3 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 10 scoops? Include units in your work and answer.




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

Hei

ght

inin

ches

Number of Scoops

Box D

s

h

Hei

ght

inin

ches

Number of Scoops

Box D

s

h

1 2 3 4 5 6 7 8 9 10 11 120123456 Total number

of scoops, sCurrent heightof the sand, h

6 scoops 2 in.

8 scoops223

in.

Box F Box G

2 , 0 4 and whole numbers,3

where is the height of the sand in inchesand is the number of scoops of sand

h s s s

hs

= ≤ ≤ ∈Box G has a height of 6 inches and is full of sand when 8 scoops of sand are added.



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. Sandy is using a scoop that holds 2 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 11 in.2

, a width of 21 in.3

, and a height of 31 in.5

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.



9. The following information is known about Box K: ● The box is a rectangular prism and initially contains 2 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.

●

3

2

5 in.5scoop (3 scoops) in.

12 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?




10. 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 1 23

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

13

2in. real height

2 3in. whole discrete

3in.

scoop 1k continuous

62in.

scoop2k only one

543in.

scoop3k 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 1 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 218 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 _________.



11. 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

C

1 2 3

TimeH

eigh

t

Documents

Fill It Up, Please – Part III