Filling and Wrapping: Three-Dimensional Measurement Name: Per: Investigation 1: Building Smart Boxes: Rectangular Prisms Date Learning Target/s Classwork Homework Self-Assess Your Learning Mon, Mar. 21 Tues, Mar. 22 Weds, Mar. 23 Thurs, Mar. 24 Inv. 1 Determine the capacity of a container (length, width, height, diagonals, surface area, volume). Given the volume of a box, determine multiple arrangements (length/width/ height) and compare surface areas. Given the volume of a box, determine how to design a box with the least packaging material (surface area). Analyze the impact of scale factor on surface area and volume. Assess understanding of Investigation 1 learning targets. Pg. 2-3: FW 1.1 Finding Volume Pg. 4: FW 1.2 Finding Surface Area (Start) Pg. 5-6: FW 1.3 Finding the Least Surface Area Pg. 8-9: FW 1.4 Scaling Up Prisms Pg. 4: FW 1.2 Finding Surface Area (Finish with Zaption) Pg. 7: FW 1.3 Zaption (Complete/ Correct) Pg. 10: Zelda Puzzle Check Up Pg. 11-12: SBAC Practice Test Zaption (Complete/ Correct) CCSS.MATH.CONTENT.7.G.A.1: Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Inv. 2 Inv. 3 CCSS.MATH.CONTENT.7.G.A.3: Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. CCSS.MATH.CONTENT.7.G.B.6: Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. CCSS.MATH.CONTENT.7.G.B.4: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. Parent/Guardian Signature: Due: 1
FW 1.1: Finding Volume A rectangular prism is a -dimensional shape with a top and bottom ( ) that are congruent and lateral ( ) faces that are. Standard shipping containers are. Some students came up with this list of ways to measure a shipping container: length, width, height, diagonals, surface area, and volume. A. How does each measurement relate to the parts of the container? Use words/diagrams. Length Diagonals Width Surface Area Height Volume B. Match each of the possible measurements in the previous question to the questions they would help answer. Explain your reasoning. How much paint would you use to paint the container in a new color? Could you store a small car in the container? Could you store pipes for a farm sprinkler system in the container? Could you store a long flagpole in the container? How many sacks, of rice, corn, or beans could you store in a container? 2
C. The filled shipping containers are stacked on a ship. The load is built up in layers. 1. How could you calculate the number of containers in each layer of such a load? 2. How could you then calculate the total number of containers in the load? 3. Suppose a ship s load has 10 containers in each row from one side to the other, 15 containers in each row from back to front, and 8 layers of containers. How many containers are in the whole load? D. Basketballs are spheres, but they are often packaged in boxes in the shape of cubes. 1. How many of these boxes would fit into a shipping container that is 6 feet long by 5 feet wide by 4 feet high? Describe the arrangement of the boxes. 2. What are the dimensions of some other containers that would hold the same number of basketball boxes? Describe the arrangement of the boxes in each case. a. Which containers make the most sense? b. Which containers make the least sense? 3
FW 1.2: Finding Surface Area A toy company is planning to market a Wump Family and Imposter Characters collector set. Each character will be sold separately. The Mug Wump character comes in a cube-shaped package with 1-inch edges. The toy company has to ship packages of the Mug Wump characters to many different toy stores. It plans to ship in large boxes, each holding 24 of the cube-shaped packages. A. Find all the ways that 24 unit cubes can be packed into a rectangular prism. Sketch each possibility. Record the dimensions, volume, surface area, and sketches in the table: Length (in) Width (in) Height (in) Volume (in 3 ) Surface Area (in 2 ) Sketch B. Which arrangement of cubes requires the box that can be made with the least material? Which requires the box that needs the most material? C. Which box shape would you recommend for shipping the Mug Wump characters? Explain your reasoning. D. Why do you think the shipping directions called for 24, rather than 26, Mug Wump characters in a box? 4
FW 1.3: Find the Least Surface Area You discovered yesterday that 24 blocks can be packaged in different ways that use varying amounts of packaging material. By using less material, a company can save money, reduce waste, and conserve natural resources. Which rectangular arrangement of cubes uses the least amount of packaging material? Possible Arrangements of 8 Cubes Length Width Height Sketch Volume Surface Area 8 cubic units 8 cubic units 8 cubic units 1. Which arrangement (set of dimensions) used the most packaging material for 8 cubes? 2. Which arrangement (set of dimensions) used the least packaging material for 8 cubes? a. What is the name of this type of prism? 5
Possible Arrangements of 27 Cubes Length Width Height Sketch Volume Surface Area 27 cubic units 27 cubic units 27 cubic units 3. Which arrangement (set of dimensions) used the least packaging material for 27 cubes? a. What is the name of this type of prism? 4. Without solving for the surface area, how could you tell simply by looking at the dimensions which prism will have the least amount of surface area? 5. The dimensions for a box with 12 unit cubes are listed below. 12 x 1 x 1 4 x 3 x 1 6 x 2 x 1 3 x 2 x 2 a. Which set of dimensions will require the least amount of material or have the smallest surface area? How do you know? 6
FW 1.3 Homework Complete/Correct with Zaption How would you design a rectangular box that holds a given volume but uses the least packaging material? A. For each part, do the following: Find the dimensions of the large boxes that require the least packaging material to enclose the given number of cube-shapes boxes for the Mug Wump characters. Explain your strategy and how you know you have designed the box with the least packaging material. Length (in) Width (in) Height (in) Volume (in 3 ) Surface Area (in 2 ) Strategy for Designing Box with Least Packaging Material 12 cubeshaped boxes 30 cubeshaped boxes 64 cubeshaped boxes B. Suppose you need to design a box to hold a liquid, such as juice or milk, or a material, such as rice, cake mix, or pasta. Because the contents of the box are not identical unit cubes, it is possible to consider dimensions other than whole numbers. For each given volume, find the dimensions of the box that uses the least packaging material: 1. 1,000 cubic centimeters 2. 30 cubic centimeters 3. 500 cubic centimeters 7
FW 1.4: Scaling Up Prisms Recipe for a 1-2-3 Compost Box Start with an open rectangular wood box that is 1 foot, 2 feet, and 3 feet. This is a 1-2-3 box. Mix 10 pounds of shredded newspaper with 15 quarts of water. Put the mixture in the 1-2-3 box. Add a few handfuls of soil. Add about 1,000 red worms (about 1 pound). Every day, mix collected kitchen waste with the soil in the box. The worms will do the rest of the work, turning the waste into new soil. A 1-2-3 box will decompose about pound of garbage a day. A. Use grid paper to make a scale model of the 1-2-3 box that will decompose 0.5 pound of garbage each day. Make a sketch of the box and label the dimensions. B. Assume that the number of worms used increases to match the increase in box volume. 1. What changes in the dimensions of the basic design would produce a box that could compost 1 pound of garbage each day? Make a sketch of the new box and label the dimensions. 2. What changes in the dimensions of the basic design would produce a box that could compost 2 pounds of garbage each day? Make a sketch of the new box and label the dimensions. 3. What changes in the dimensions of the basic design would produce a box that could compost 5 pounds of garbage each day? Make a sketch of the new box and label the dimensions. 8
C. The Science Club wants to scale up the basic 1-2-3 design to a larger box that is similar in shape. 1. Complete the following table that shows the cost and capacity of several larger boxes. 2. What growth patterns do you see in the volume and surface area? D. Suppose a large compost box is similar to a 1-2-3 box with scale factor f. 1. How is the surface area of the large box related to that of the 1-2-3 box? 2. How is the volume of the 4-8-12 box related to that of the 1-2-3 box? 3. How is the amount of decomposed garbage related to the volume of the 1-2-3 box? 9
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SBAC Practice Test Part 1 Score: / 7? Question and Answer Correct Answer 1 2 3 4 11
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