Chapter 77 APostalProblem Grades 4–8 Ⅺ × Total group activity Ⅺ × Cooperative activity Ⅺ × Independent activity Ⅺ × Concrete/manipulative activity Ⅺ × Visual/pictorial activity Ⅺ × Abstract procedure Why Do It: Students will review concepts from geometry and apply math- ematical skills, including logical thinking, and computation with a calculator, to an everyday problem-solving situation. You Will Need: This activity requires pencils; paper; and a calculator (rec- ommended, but not required). Also, if some of the boxes are to be constructed, large pieces of cardboard or tagboard, scissors, and tape will be required. HowToDoIt: 1. Share the following U.S. Post Office shipping problem with the students and ask how they would attempt to solve it: U.S. Post Office regulations note that packages to be shipped must measure a maximum of 108 inches in length plus girth. What size rectangular box, with a square end, will allow you to send the greatest volume of goods? 297 2. After the students have shared various ideas, they should diagram (or physically construct with tagboard and tape) one or two of the boxes they proposed and determine their volumes. When students are calculating the volumes, be certain that they understand how to relate the Length + Girth measurements to the formula Volume = Length × Width × Height (volume of a rectangular solid). It may help to refer to the Example in order to do this. Example: Three boxes of different dimensions, but each totaling 108 inches in length plus girth, are shown on page 299. The width and height mea- surements have, in each case, been derived from the initial girths (at the square ends of the boxes). The computed volumes for each are different, and students may find a calculator very helpful in finding these. Have students determine whether the square-ended box shown will yield the greatest volume, or whether another will be better. (Hint: The best arrangement is two cubes piled one on top of the other.) 298 Investigations and Problem Solving Box Dimensions Length × Width × Height = Volume (in Cubic Inches) 108 inches (total) −40 inches (girth) = 68 inches (length) 68 10 10 10 10 68 inches ×10 inches ×10 inches = 6,800 cubic inches 108 inches (total) −48 inches (girth) = 60 inches (length) 60 12 12 12 12 60 inches ×12 inches ×12 inches = 8,640 cubic inches 108 inches (total) −80 inches (girth) = 28 inches (length) 20 20 20 20 28 28 inches ×20 inches ×20 inches = 11,200 cubic inches Extensions: 1. Some students may need to physically compare the volumes. It might be useful therefore to build boxes of tagboard or cardboard to specified dimensions (such as those in the Example above). Then have students compare the volumes of the different-shaped boxes by pouring the contents of one into another; Styrofoam packing chips work well when doing this comparison. APostalProblem 299 2. Inform students that the United Parcel Service (UPS) allows boxes up to 130 inches in length plus girth. Have them determine what size rectangular box, with a square end, will allow them to send the greatest volume of goods through UPS. 3. Ask students to consider either the 108-inch limit or the 130-inch limit on girth and to determine what shape box or boxes will provide a greater volume than a square-ended rectangular box. Have them show their diagrams and calculations to demonstrate their solutions. 300 Investigations and Problem Solving Chapter 78 Build the ‘‘Best’’ Doghouse Grades 4–8 Ⅺ × Total group activity Ⅺ × Cooperative activity Ⅺ × Independent activity Ⅺ × Concrete/manipulative activity Ⅺ × Visual/pictorial activity Ⅺ × Abstract procedure Why Do It: This activity provides a real-life investigation experience that may be solved in a variety of ways. Students will draw plans, from which they will construct their own ‘‘best’’ doghouses. You Will Need: Required for each participant are a piece of graph paper with 1-inch squares, a 4- by 8-inch piece of tagboard (old file folders may be cut up), tape, scissors, a ruler, and a pencil. A bag of rice, or other dry material, may be used when determining volume. 301 How To Do It: 1. Begin by posing the following problem: You have a new dog at home, and it is your job to build a doghouse. You have one 4- by 8-foot sheet of plywood, and from it you want to build the largest doghouse possible (having the greatest interior volume). You also decide that it will have a dirt floor, that any windows or doors must have closeable flaps, and that it will have a roof that rain will run off (see drawing below). Prior to construction, you must first draw a plan showing how you will cut the pieces from the plywood, and you will also need to construct a doghouse model using a 4- by 8-inch piece of tagboard. 2. Provide each student with a piece of graph paper and have them use their rulers and pencils to mark a 4-inch by 8-inch border. Explain that this bordered area will represent (as a scaled version) the 4- by 8-foot sheet of plywood. Then provide time to investigate where best to draw the lines so that the cut-out pieces will allow them to create the largest doghouse. Remind them that plywood does not bend and that all walls and the roof must be filled in. They may, however, splice together some sections of the doghouse, but very small slivers are not allowable. 3. When students have finished their plans, provide each with a 4- by 8-inch piece of tagboard, scissors, and tape. Have them mark their tagboard, cut out the pieces, and tape them together to form doghouse models. (Note: Any material excess should be taped inside the doghouse, and the student’s name should also be written inside.) 4. When a number of the investigators have finished, allow them to compare and contrast the doghouse models that they built by discussing, for example, whether it was better to build a 302 Investigations and Problem Solving long doghouse or a square one, or whether a tall doghouse provided more inside space (volume) than a short one. Advanced investigators may use mathematical formulas to determine the outcomes, but young learners may need to take a more direct approach in deciding which doghouse has the greatest volume. To do so, they may simply tape any doors and windows shut, turn the doghouse models upside down, fill one with rice (or other dry material), and pour from one to another until it is determined which model holds the most. A discussion noting the attributes of the best doghouse should follow. Example: The doghouse models shown below have both been built from 4- by 8-inch pieces of tagboard, but their shapes and volumes are quite different. Extensions: 1. Younger students will likely need to complete this task in an intuitive manner; it may be helpful to relate it to the activity Building the Largest Container (p. 288). In any event, help them understand that shape does affect volume. 2. Students who are somewhat advanced should be expected to make use of area and volume formulas as they attempt to determine the optimal shape for their doghouses. 3. Advanced students might be challenged to complete this activity using a 4- by 8-foot piece of bendable material, such as aluminum. Build the ‘‘Best’’ Doghouse 303 Chapter 79 Dog Races Grades 4–8 Ⅺ × Total group activity Ⅺ × Cooperative activity Ⅺ Independent activity Ⅺ × Concrete/manipulative activity Ⅺ × Visual/pictorial activity Ⅺ × Abstract procedure Why Do It: Students will learn about probability while enjoying a ‘‘sta- tistical’’ dog race game. You Will Need: Dog Races requires dice, crayons or markers, and copies of the ‘‘Dog Race Chart’’ for each student (chart provided). How To Do It: This activity provides a fun way to think about some basic probabilities and what they mean. Students will toss two dice and record the number they get by adding the dice. By performing this experiment many times, students will see a pattern develop and be able to answer some interesting questions. Beginning at the top of the chart, each student is to number the dogs 1 through 13 and circle the dog that he or she thinks will win. Working in groups of four or five, students will take turns rolling the dice. After each roll, all students in the group will add the numbers on the faces of the dice. Then each student in the group will color in a square for that numbered dog on his or her own chart. This continues until one dog has won the race. 304 Example: BEAGLE BOXER ST. BERNARD FOX TERRIER BASENJI In the Example above, only a portion of the chart is shown. The following sumsonthedicehavebeenrecorded,2,5,4,6,7,7,5,5.Itlookslike the St. Bernard is winning so far. After the game is finished and one dog has won, or there is a tie, ask the students to answer the following questions. 1. Did you pick the winner? 2. How many times did the winning dog move forward? 3. If this race were run again, would the outcome probably be the same? (Encourage students to check by running the race again three or four times, using extra copies of the ‘‘Dog Race Chart.’’) 4. Which dog in this race lineup is likely to win most often? Why? 5. Are there any dogs in this race that can never win? Why? Extensions: 1. Have the students make a chart and list the ways they can get each of the numbers 1 through 13 when using the dice. Next, ask the students to find the probabilities that each dog will win. They should write their probabilities in fraction form. For example, because there are 36 different outcomes for the sum of the numbers on two dice, then the probability that a St. Bernard would win is 4/36 or 1/9. 2. Use some different types of dice, like an octahedron (8-sided die). These can often be purchased at a school supply store, or online. Extend the ‘‘Dog Race Chart’’ to have some more dogs racing (for octahedron dice you would have to have 3 more dogs on the chart). Have students repeat the game above and repeat Extension 1. Dog Races 305 3. In a real dog race, which of these dogs would be likely to win? Which might come in second, third, and so on? (You might find out about the different breeds of dogs at your library or from an expert who raises dogs.) 306 Investigations and Problem Solving [...]... a difference What they found was that the tape needed for the side-by-side arrangement measured almost 1 4-3 /8 inches + 1 inch overlap = 1 5-3 /8 inches, whereas the triangle configuration was approximately 1 2-3 /8 inches + 1 inch overlap = 1 3-3 /8 inches TAPE TAPE (SIDE BY SIDE WE NEED 1 5-3 /8 INCHES OF TAPE.) (AS A TRIANGLE WE NEED 1 3-3 /8 INCHES OF TAPE.) Because most of the mail orders were for more than... the average (mean) value of the outcomes for many repetitions of the experiment The answer to this problem is 1/2(−$2) + 1/4(−$2.50) + 1 /8( $1) + 1/16($7) + 1/16($9) = 1/2(−2) + 1/4(−5/2) + 1 /8( 1) + 1/16(7) + 1/16(9) = − 1 − 5 /8 + 1 /8 + 7/16 + 9/16 = −1 − 4 /8 + 1 = −4 /8 = −1/2 = −$0.50 4 Change the game board to 3 Change the value of each section on the square board look like the one shown (and on the... Chapter 80 Four-Coin Statistics Grades 4 8 × Ⅺ × Ⅺ Ⅺ × Ⅺ × Ⅺ × Ⅺ Total group activity Cooperative activity Independent activity Concrete/manipulative activity Visual/pictorial activity Abstract procedure Why Do It: Students gain understanding of statistical data-gathering processes and learn how such information can be used to make predictions You Will Need: Four coins and a duplicated copy of the ‘‘Four-Coin... Solving Four-Coin Chart 3 Heads 1 Tail 2 Heads 2 Tails 3 Tails 1 Head 4 Tails Copyright © 2010 by John Wiley & Sons, Inc 4 Heads Four-Coin Statistics 311 Chapter 81 Tube Taping Grades 4 8 × Ⅺ × Ⅺ × Ⅺ × Ⅺ × Ⅺ × Ⅺ Total group activity Cooperative activity Independent activity Concrete/manipulative activity Visual/pictorial activity Abstract procedure Why Do It: Students will investigate a real-life problem... U T Example: Outcomes for trial = N, N, U, N, N, U, N, T; Number of marbles = 8 Trial Outcomes # of Marbles 1 2 3 4 5 6 7 8 9 10 Find the mean number of marbles: Did your average number of marbles match your guess at the top of the page? Winning a Prize Spelling ‘‘NUT’’ 331 Chapter 85 Building Toothpick Bridges Grades 4 8 × Ⅺ × Ⅺ × Ⅺ × Ⅺ × Ⅺ × Ⅺ Total group activity Cooperative activity Independent... needed for each student or group of students (you can use any different-colored objects as long as they are the same shape), as well as paper (or photocopies of the chart provided) and pencils 3 28 How To Do It: 1 Begin by posing the following problem (This problem is also on the reproducible handout for this activity.) The local health-food store is promoting eating nuts as a source of protein At the cash... surveyors, forest-service personnel, and others Extension: Transits and levels (which are similar to the hypsometer) are frequently used to make accurate land, architectural, or other measurements Invite someone who uses these devices, such as a highway surveyor, an architect, or a forest-service timber cruiser, to give a class demonstration 320 Investigations and Problem Solving Chapter 83 Fairness at... activity Abstract procedure Why Do It: Students will investigate a real-life problem with multiple solutions, incorporating hands-on experiences, visual mapping, and the use of formulas You Will Need: A collection of paper or plastic tubes of the same diameter are needed (Tubes of 2-inch diameter match the situation in this activity, but another size will work if the story is modified.) Also required are... arrangement for each number of tubes that uses the least amount of tape and is therefore the most cost-effective way to send the packages Tube Taping 313 Example: Shown below are possible arrangements for 4 to 7 tubes Ask students to determine which is the best arrangement (Note: See Extension 2 for a more in-depth explanation.) 4 TUBES 5 TUBES 6 TUBES 7 TUBES 314 Investigations and Problem Solving Extensions:... activity and again They can also comrepeat the simulation pute the expected value as shown in Extension 2 $2 50¢ $20 $5 50¢ $8 $10 $3 $1 $4 $2 5 Players can make up their own carnival game and use the blank circles provided to make their own spinners (The blank circles are marked in 10-degree increments.) They can then perform the simulation with their new spinners 6 Players can also change the price to . for the side-by-side arrangement measured almost 1 4-3 /8 inches + 1inch overlap = 1 5-3 /8 inches, whereas the triangle configuration was approximately 1 2-3 /8 inches + 1inchoverlap= 1 3-3 /8 inches. (SIDE. Volume (in Cubic Inches) 1 08 inches (total) −40 inches (girth) = 68 inches (length) 68 10 10 10 10 68 inches ×10 inches ×10 inches = 6 ,80 0 cubic inches 1 08 inches (total) − 48 inches (girth) = 60 inches. doghouse model using a 4- by 8- inch piece of tagboard. 2. Provide each student with a piece of graph paper and have them use their rulers and pencils to mark a 4-inch by 8- inch border. Explain