4.4 Painted Cubes Goals • Explore the “Painted Cube” situation, which has edge length 10 cm and then have them generalize it to one with edge length of n cm. Let the class work in groups of three to four. linear, quadratic, and cubic functions • Compare linear, quadratic, and exponential functions When a painted cube with edge length n is separated into n3 small cubes, how many of these cubes will have paint on three faces? Two faces? One face? No face? To answer this question, students can consider cubes with smaller dimensions and then look for patterns in the data to make generalizations. Launch 4.4 Explore 4.4 If students are having difficulty getting started suggest that they study the sketches and build cubes like these to find the answers to these painted cube questions. Having a 10 3 10 3 10 base 10 block and a large supply of 1 cm cubes for each group to examine while they work might be helpful. Have one or more groups put their data table on a transparency to use in the summary. Suggested Questions You can use a Rubik’s™ Cube to launch the problem. Note: Having a Rubik’s Cube, a base ten block, or some other large cube to show students while you discuss it would be helpful. Also having small unit cubes or sugar cubes around so students can build some of the smaller cubes of length 2, 3, or 4 will be very helpful. • Where are the unpainted cubes located? (They Suggested Questions Hold up a large cube: • Where are the cubes that have only one face • How many faces, edges and corners does a cube have? (A cube has six faces, 12 edges and 8 corners.) change from cube to cube? (No.) • Suppose we paint a cube with edge length of have? (cube) • What are the dimensions of this cube? (Its dimensions are two less the original cube.) painted? (They are on the faces of the cube but not along the edges.) • Within one face of the large cube, what shape is formed by the set of cubes with only one face painted? (They form a square.) • How many faces are there? (6) • Where are the cubes that have two faces painted? (They form the edges of the cube but not the corners.) • How many edges are there? (12) • Where are the cubes that have three faces painted? (They form the corners of the cube.) • How many corners are there? (8) In this problem you will explore these questions for cubes of edge length 1 to 6 cm and then students will generalize for edge lengths of 10 cm or n cm. Note: you could leave the problem more open by posing the problem with a cube of Investigation 4 What Is a Quadratic Function? 137 4 10 cm and then separate it into 1,000 small centimeter cubes. How many of the smaller cubes will have paint on three faces? Two faces? One face? No faces? (Let students make some conjectures. Some might notice that the cube on a corner will have three faces painted.) • What shape does this set of unpainted cubes I N V E S T I G AT I O N • Do the number of faces, edges and corners are on the inside of the large cube; they are not visible from the outside.) Suggested Questions • What pattern in the differences would suggest a linear relationship? • What pattern in the differences would suggest a quadratic relationship? Summarize 4.4 Cubes painted on one face During the discussion give students an opportunity to verbalize the patterns of change they have found in their tables. This will help them write symbolic statements for Question C. Put up a transparency of a table of data that the students collected to refer to during the summary. Ask students to study the data they collected. Possible student responses are provided. Suggested Questions • What kind of relationship—linear, quadratic, exponential, or other—do you observe in the plotted graph or table of data for the number of cubes with paint on 3 faces? (This one seems to be linear since it is always 8.) Cubes painted on two faces • What kind of relationship—linear, quadratic, exponential, or other—do you observe in the plotted graph or table of data for the number of cubes with paint on 2 faces? (This one seems to be linear since they seem to be in a straight line. Also, the table seems to be increasing by 12 each time.) • What kind of relationship—linear, quadratic, exponential, or other—do you observe in the plotted graph or table of data for the number of cubes with paint on one face? (The relationship seems like it might be quadratic because the second difference is constant.) Cubes painted on three faces If students are having trouble seeing patterns of change for Question C in the tables, ask them to look at the patterns of change. 138 Frogs, Fleas, and Painted Cubes • What kind of relationship—linear, quadratic, exponential, or other—do you observe in the plotted graph or table of data for the number of cubes with paint on 0 faces? [The pattern of change indicates this is not linear, nor quadratic, nor exponential. In fact the relationship is cubic, P = (n 2 2)3, though students will not necessarily see this without some discussion.] • Let’s look closer at the pattern of change in the data for zero faces. • When we look at the differences between y-values in tables where x changes by 1 each time, what tells us that a relationship is linear? (First differences are constant.) • What tells that a relationship is quadratic? (The second differences are constant.) If you asked some groups to make tables of differences, have these students put up their tables of differences or demonstrate finding second differences. (Figures 2 and 3) • Explain that once constant differences of 0 are reached it is pointless to go further. (Figure 4) In this situation, the third differences are constant. • If we continued to take differences in each of these cases, what would happen? Figure 2 Figure 3 Figure 4 2 0 3 12 4 24 5 36 6 48 Edge Length of Large Cube Cubes Painted on 1 Face 2 0 3 6 4 24 5 54 6 96 Edge Length of Large Cube Cubes Painted on 0 Faces 2 0 3 1 4 8 5 27 6 64 First Differences Second Differences 12 0 12 0 12 0 12 First Differences 6 18 30 42 First Differences Second Differences 12 12 12 Second Differences 1 7 19 37 6 12 18 Investigation 4 Third Differences 0 0 Third Differences 6 6 4 Cubes Painted on 2 Faces I N V E S T I G AT I O N Edge Length of Large Cube What Is a Quadratic Function? 139 Suggested Question • We call (n 2 2)3 a cubic expression. What • Let’s use this information to work backward and find the number of cubes painted on 0 faces for an edge length of 7 centimeters. (Figure 5) • What kind of equation do you think describes this relationship? (Students may not guess “cubic” yet.) To help students write the equations, go back to some of the questions from the Explore. For example, to write an expression for the number of cubes with no faces painted: Suggested Questions • Where are the unpainted cubes located? (They are on the inside of the large cube; they are not visible from the outside.) • What shape does this set of unpainted cubes have? (cube) • What are the dimensions of this cube? (Its dimensions are two less than the original cube.) would you conjecture about the first, second, or third differences for a cubic relationship? (First and second differences are not constant. Third differences are constant.) Repeat these questions from the Explore for the pattern of cubes on the faces [6(n - 2)2], on the edges [12(n - 2)], and on the corners (8). Use your calculator to make sketches of the graphs of these relationships. • Describe the shapes of the graphs. • How many different kinds of relationships did we explore in this problem? How can we recognize each relationship from its table, graph and equation? (For cubics they may only be able to say something about the equation containing a variable whose highest exponent is 3. They may also say something about the third difference.) You can end the summary by going back to the cube (10 3 10 3 10) and answering the original questions. • So how can we write this in symbols? [(n - 2)(n - 2)(n - 2) or (n - 2)3] Figure 5 Edge Length of Large Cube Cubes Painted on 0 Faces 2 0 3 1 4 8 5 27 6 64 7 125 140 Frogs, Fleas, and Painted Cubes First Differences 1 7 19 37 61 Second Differences 6 12 18 24 Third Differences 6 6 6 At a Glance 4.4 Painted Cubes PACING 1 day Mathematical Goals • Explore the “Painted Cube” situation, which has linear, quadratic, and cubic functions • Compare linear, quadratic, and exponential functions Launch Having a Rubik’s Cube, a base ten block, or some other large cube to show students would be helpful. Hold up a large cube: • How many faces, edges, and corners does a cube have? • Do the number of faces, edges, and corners change from cube to cube? • Suppose we paint a cube with edge length of 10 cm and then separate it Materials • Base ten thousands blocks (optional; 1 per group) • Rubik’s cube or other large cube (optional) into 1,000 small centimeter cubes. How many of the smaller cubes will have paint on three faces? Two faces? One face? No faces? In this problem you will explore these questions for cubes of edge length 1 to 6 cm and then generalize for edge lengths of 10 cm or n cm. Let the class work in groups of three to four. Explore Materials Suggest students study the sketches and build cubes like these to find the answers to these painted cube questions. To get started ask: • Centimeter or other unit cubes (in four colors or with colored dot stickers, or sugar cubes and colored markers) • Transparency 4.4 • Where are the unpainted cubes located? What shape does this set of unpainted cubes have? What are the dimensions of this cube? Where are the cubes that have only one face painted? • Within one face of the large cube, what shape is formed by the set of cubes with only one face painted? • Where are the cubes that have two faces painted? How many edges are there? Where are the cubes that have three faces painted? How many corners are there? If students are having trouble seeing patterns of change for Question C in the tables, ask them to look at the patterns of change. • What pattern in the differences would suggest a linear relationship? A quadratic relationship? Summarize Give students an opportunity to verbalize the patterns of change they have found in their tables. Materials • Student notebooks • What kind of relationship—linear, quadratic, exponential, or other—do you observe in the plotted graph or table of data for the number of cubes with paint on 3 faces (on 2 faces, on 1 face, on 0 faces)? continued on next page Investigation 4 What Is a Quadratic Function? 141 Summarize continued • When we look at the differences between y-values in tables where x changes by 1 each time, what tells us that a relationship is linear? Quadratic? If we continued to take differences in each of these cases, what would happen? If you asked some groups to make tables of differences, have these students put up their tables of differences or demonstrate finding second differences. Explain that once constant differences of 0 are reached it is pointless to go further. • Notice that when third differences are constant, the expanded form of the equation contains the variable raised to the third power. Or, it is the product of three terms that contain the variable x, like x ? x ? x or x3, or (x 2 2)(x 2 2) (x 2 2). We call such a relationship a cubic relationship. To help students write the equations, go back to some of the questions from the Explore. ACE Assignment Guide for Problem 4.4 used as a factor 3 times, x ? x ? x or x3. The relationship is not linear, quadratic, or exponential. Core 27–30 Other Connections 41–50; Extensions 56, 57; unassigned choices from previous problems Adapted For suggestions about adapting Exercise 4 and otherACE exercises, see the CMP Special Needs Handbook. Connecting to Prior Units 41, 43–45: Covering and Surrounding Answers to Problem 4.4 2. The number of cubes painted on three faces is 8 regardless of the edge length of the large cube. The number of cubes painted on two faces is the edge length of the large cube minus 2 and then multiplied by 12. The number of cubes painted on one face is the edge length of the large cube minus 2, squared, and then multiplied by 6. The number of cubes painted on zero faces is the edge length of the large cube minus 2 then raised to the third power. A. 1. and 2. (Figure 6) B. 1. The number of centimeter cubes in a large cube is the edge length of the large cube Figure 6 Painted Faces of a Centimeter Cube Edge Length of Large Cube Number of cm Cubes Number of cm Cubes Painted on 3 Faces 2 Faces 1 Face 0 Faces 2 8 8 0 0 0 3 27 8 12 6 1 4 64 8 24 24 8 5 125 8 36 54 27 6 216 8 48 96 64 142 Frogs, Fleas, and Painted Cubes 2. 80 60 40 20 0 1 3 5 7 Edge Length (cm) x 9 Edge Length vs. Cubes Painted on 1 Face 600 300 y 200 100 0 1 3 5 7 Edge Length (cm) x 9 400 200 0 1 3 5 7 Edge Length (cm) x 9 8 300 200 100 0 1 3 5 7 Edge Length (cm) x 9 6 4 2 x 0 2 4 6 8 Edge Length (cm) Investigation 4 What Is a Quadratic Function? 143 4 0 We can predict that (edge, 3 faces) and (edge, 2 faces) will be linear because the table indicates first differences that are constant. We can predict (edge, 1 face) will be quadratic because the second differences are constant. I N V E S T I G AT I O N Number of Cubes Edge Length vs. Cubes Painted on 3 Faces 10 y Edge Length vs. Unpainted Cubes 400 y Number of Cubes Number of Cubes Edge Length vs. Total Number of Cubes 800 y Number of Cubes C. 1. For the equations below; let n be the number of cubes and O represent edge length of the large cube in cm. For the relationship between edge length of the large cube and total number of cubes, n = O3. For three faces painted: n = 8 For two faces painted: n = 12(O - 2) or n = 12O - 24 For one face painted: n = 6(O - 2)2 or n = 6O2 - 24O + 24 For zero faces painted: n = (O - 2)3 Edge Length vs. Cubes Painted on 2 Faces 100 y Number of Cubes 3. Three faces: the relationship is linear Two faces: the relationship is linear One face: the relationship is quadratic Zero faces: Not linear, quadratic, nor exponential 144 Frogs, Fleas, and Painted Cubes

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