## Purple Prisms

This Internet Mathematics Excursion is based on E-example 6.3.2 from the NCTM Principles and Standards for School Mathematics. This is the last activity in a sequence of four lessons designed for students to understand scale factor and surface area of various rectangular prisms. Students manipulate the scale factor that links two three-dimensional rectangular prisms to learn about edge lengths and surface area relationships.

Engage students in a class discussion about their knowledge of surface area, making certain they understand surface area is the two dimensional measurement of a three-dimensional figure. Use several different rectangular prisms for students to demonstrate their understanding of surface area.

*Guiding Questions:*

- How do you find area of a rectangle?
- How many rectangular faces make up a rectangular prism?
- Although each face of the rectangular prism is two-dimensional, together what do they make?
- What is the sum of the areas of the faces on a rectangular prism called?
- Why are square units used when measuring surface area?
- In your own words, give a definition of surface area of a rectangular prism?

Side Length, Volume, and Surface Area of Similar Solids Applet

Go on-line to the Side Length, Volume, and Surface Area of Similar Solids Applet. When the applet opens, students should click on "Measures" under the 3D mode. There are two similar rectangular prisms, one purple and one red. The red prism remains the same. The length is 20 units, the width is 10 units, and height is 7 units. Students should use the slide bar or the red dot to change the size of the purple rectangular prism.

Next students should change the size of the purple prism and observe the change in the ratio of L : 1 (scale factor). Note how the Surface Area of Prism A and Prism B and their ratio change for different length measures.

Using the Rectangular Prisms table on the Purple Prisms activity sheet, students should record the Surface Area of A, the Surface Area of B, the ratio of Surface Area A : B, and the scale factor L : 1 for ten different rectangular prisms.

Ask students the following questions as they complete the activity:

- Why is Surface Area B always 8.93 square units?
- How do you know what the width and height of A are?
- Are these prisms similar? How do you know?
- Which column from the Student Learning Guide represents scale factor? Why?

Students should now look at the L : 1 and Surface Area A : B ratio on the table. Click on Show Surface Area graph. Use the *Questions for Students* (below) to conclude the lesson.

**Assessment Options**

- Individually, students should sketch a rectangular prism, labeling dimensions. Using the sketch, they should describe surface area. Assess students’ understanding that surface area is the sum of the six rectangular faces on the prism measured in square units.
- Students can explain in writing the relationship between scale factor and the ratio of Surface Area A : B.

**Questions for Students**

- What is the relationship between L : 1 and Surface Area A : B?
- Does anyone have a whole number for L : 1 in his or her table? What is the relationship between that whole number and Surface Area A: B?
- Apply that relationship to the other numbers in the L : 1 column and Surface Area A : B column?
- How does the applet graph illustrate this relationship?
- Is this graph linear? Why or why not?

### Blue Squares and Beyond

### Go With Green Rectangles

### Fill'r Up

### Learning Objectives

Students will:

- Explain surface area of a rectangular prism.
- Describe the relationship between the scale factor and the surface area of two similar rectangular prisms.
- Use the applet graph to explain the relationship between the scale factor and the surface area of two similar rectangular prisms.
- Create pattern units of squares, predict how patterns with different numbers of squares will appear when repeated in a grid, and check their predictions.
- Analyze how repeating patterns are generated.

### NCTM Standards and Expectations

- Use visual tools such as networks to represent and solve problems.

- Use geometric models to represent and explain numerical and algebraic relationships.

### Common Core State Standards – Mathematics

Grade 6, Ratio & Proportion

- CCSS.Math.Content.6.RP.A.1

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, ''The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.'' ''For every vote candidate A received, candidate C received nearly three votes.''

Grade 6, Geometry

- CCSS.Math.Content.6.G.A.4

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.