3.1: Dilating, Again (10 minutes)
Students dilate a triangle from a center of dilation. Then, they think of things they notice and wonder about their completed drawings. The purpose of the activity is to elicit the idea that the triangle and its dilation can be viewed as representations of the cross sections of a three-dimensional pyramid. This will help prepare students for the next activity in which they create a representation of a pyramid by suspending several dilations of a rectangle. While students may notice and wonder many things about the figures they draw, the three-dimensional visualization is the most important discussion point.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
Tell students that they will draw a dilated figure, then think of at least one thing they notice and at least one thing they wonder about the resulting drawing.
Dilate triangle \(BCD\) using center \(P\) and a scale factor of 2.
Look at your drawing. What do you notice? What do you wonder?
Choose a student’s drawing and display it for all to see. Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the displayed drawing. After all responses have been recorded without commentary or editing, ask students, “Is there anything on the list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If the idea that the drawing resembles a pyramid does not come up during the conversation, ask students to discuss this idea. Remind students that pyramids, like prisms, are named for their bases, so this pyramid should be called a “triangular” pyramid.
3.2: Pyramid Mobile (25 minutes)
In this activity, students dilate rectangles and suspend them to resemble cross sections of a pyramid. This work connects to later lessons in which students use areas of cross sections in the development of the pyramid volume formula.
This task reinforces the idea that scaling by a factor of \(k\) adjusts all distances in a figure by a factor of \(k\), and foreshadows the upcoming concept that if a figure is scaled by a factor of \(k\), its area changes by a factor of \(k^2\).
Monitor for groups who choose different lengths of string, therefore creating pyramids of different heights.
Arrange students in groups of 4.
Draw a triangle and a square on the board. For each shape, hold a ball or another small object representing point \(A\) off the surface of the board, projected directly out from the center of the shape. Invite students to imagine the three-dimensional objects created by connecting \(A\) to each vertex of the shape: a triangular pyramid and a square pyramid.
Display this image for all to see. Tell students that it is a top-down view of the three-dimensional shapes they just imagined.
Ask students how they could dilate each two-dimensional figure by a scale factor of \(\frac12\) using \(A\) as the center of dilation. With students’ guidance, complete the dilations. Be sure students see that the dilation is created by measuring the distance from \(A\) to each vertex of the “base” and multiplying that distance by \(\frac12\) to find the halfway point. Ask students what the dilations could represent: cross sections of each solid.
Finally, explain to students that a mobile is a kind of sculpture in which materials are suspended in the air. Tell them they’ll be using the concepts just discussed to make a pyramid mobile.
Design Principle(s): Support sense-making
Supports accessibility for: Organization; Attention
Your teacher will give you sheets of paper. Each student in the group should take one sheet of paper and complete these steps:
- Locate and mark the center of your sheet of paper by drawing diagonals or another method.
- Each student should choose one scale factor from the table. On your paper, draw a dilation of the entire sheet of paper, using the center you marked as the center of dilation.
- Measure the length and width of your dilated rectangle and calculate its area. Record the data in the table.
- Cut out your dilated rectangle and make a small hole in the center.
|scale factor, \(k\)||length of scaled rectangle||width of scaled rectangle||area of scaled rectangle|
Now the group as a whole should complete the remaining steps:
- Cut 1 long piece of string (more than 30 centimeters) and 4 shorter pieces of string. Make 4 marks on the long piece of string an equal distance apart.
- Thread the long piece of string through the hole in the largest rectangle. Tie a shorter piece of string beneath it where you made the first mark on the string. This will hold up the rectangle.
- Thread the remaining pieces of paper onto the string from largest to smallest, tying a short piece of string beneath each one at the marks you made.
- Hold up the end of the string to make your cross sections resemble a pyramid. As a group, you may have to steady the cross sections for the pyramid to clearly appear.
Are you ready for more?
Is dilating a square using a factor of 0.9, then dilating the image using scale factor 0.9 the same as dilating the original square using a factor of 0.8? Explain or show your reasoning.
If students struggle to create the dilations, remind them of the demonstration from the activity launch. Suggest that they draw lines connecting the center of the paper to each vertex, and measure the lines. Then invite them to think about how the scale factor they’re using will apply to these distances.
The goal of the discussion is to make observations about the pyramid structure and about the relationships in the table students created. Here are some questions for discussion:
- “For the student who chose the scale factor \(k=1\), what did you have to do to the rectangle?” (Nothing! A scale factor of 1 yields the exact same figure.)
- “Where is the rectangle with scale factor \(k=1\) located in the pyramid?” (It is the pyramid’s base.)
- “When we did the dilations, we used the center of the rectangle as the center of dilation. If we imagine doing the dilations in 3 dimensions instead of 2, like in the activity launch, what is the center of dilation?” (It is the vertex at the top of the pyramid.)
- “Staying in 3 dimensions, what would a dilation by the scale factor \(k=0\) look like? Where would it be located in the pyramid?” (It would be at the center of dilation, the vertex at the top of the pyramid.)
- “How do the length and width of your cross sections relate to the scale factor you chose?” (The length and width of the dilated rectangles were changed by a scale factor of \(k\), to the limit of precision of the tools we’re using.)
- “Was the area of the dilated rectangle also changed by a factor of \(k\)?” (No. At this point, students do not need to draw any further conclusions about the area; this will be analyzed in an upcoming lesson.)
If time permits, display the applet for all to see.
Demonstrate the “hide pyramid” and “collapse layers” tools. Ask students to describe how the cross sections are related to the pyramid.
The goal of the discussion is to extend students’ understanding of cross sections and dilation. Here are some questions for discussion:
- “For 2 groups with pyramids of different heights, how did the placement of the scale factor \(k=0.5\) triangle differ in each pyramid?” (It was exactly halfway up the pyramid in each case.)
- “What is the difference between using the center of the rectangle as the center of dilation, and using the top vertex of the pyramid as the center?” (When we use the center of the rectangle, the dilation stays in the same plane. That’s what we did, but then we assembled those into a pyramid. The top vertex of the pyramid can be viewed as the center of dilation in 3 dimensions.)
- “What would happen if we dilated the rectangle by a scale factor of \(k=0.1\) or \(k=0.9\), using the top vertex of the pyramid as the center of dilation?” (The first would be a cross section near the base and the second would be a cross section near the top vertex of the pyramid.)
- “What is the range of scale factors that create cross sections of the pyramid?” (The scale factors between 0 and 1 create cross sections. Larger scale factors extend outside of the pyramid.)
3.3: Cool-down - Circle Dilation (5 minutes)
Student Lesson Summary
Imagine a triangle lying flat on your desk, and a point \(P\) directly above the triangle. If we dilate the triangle using center \(P\) and scale factor \(k=\frac12\) or 0.5, together the triangles resemble cross sections of a pyramid.
We can add in more cross sections. This image includes two more cross sections, one with scale factor \(k=0.25\) and one with scale factor \(k=0.75\). The triangle with scale factor \(k=1\) is the base of the pyramid, and if we dilate with scale factor \(k=0\) we get a single point at the very top of the pyramid.
Each triangle’s side lengths are a factor of \(k\) times the corresponding side length in the base. For example, for the cross section with \(k=\frac12\), each side length is half the length of the base’s side lengths.