Satellite Overflight 2
Lesson 2 of 6
Objective: Students will understand how dilating a shape affects the perimeter and area of the figure. Students will be able to describe the relationship between the perimeter or area of an image under dilation and the perimeter or area of the pre-image using an equation or a verbal rule. Students will understand what it means to ‘make sense of problems and persevere in solving them’. Students will be able to explain their reasoning about a non-standard problem and critique the reasoning of others.
Present the warm-up problem (10 minutes).
The warm-up problem for this lesson (found in the slide show) asks students to
find the areas of two similar triangles, then to compare the areas using a ratio. This activity follows the Team Warmup routine, which is described in my Strategy Folder.
While students complete the warm-up problem, I take attendance and circulate around the classroom noting who does not have their homework or their materials.
Motivate the lesson. Review the major learning points of the previous lesson and the class Strategy Talk discussion. Tell the class that today they are going to learn a very
useful strategy that can often help them solve a problem that seems pretty complicated (MP1). The strategy is aptly named, ‘solve an easier problem’. In fact, the
strategy often involves solving several easier problems in order to identify a pattern that can be used to solve the larger problem.
Review learning targets and the agenda for the lesson. Display the agenda and learning goals for the lesson as you distribute materials for the activity. Ask students to read them over and invite
Have students complete the cooperative learning activity (25 minutes). Before class, print out the activity. Make one copy for each team and cut into half-sheets. You may want to make a few extra copies, in order to allow students to start over with a fresh sheet if they go down the wrong path. Distribute the half-sheets to the teams. Assign alternate teams to investigate either the dilation of the letter ‘H’ or the letter ‘S’. Each team needs one of the instruction sheets (either showing dilations of letter ‘H’ or of letter ‘S’) and one of each of the other four half-sheets.
Tell the class that they will have to divide up the work amongst the members of the team in order to complete all the tasks in the time limit. Every student must complete one part of the problem. Tell students that you expect them to ask for or offer help, but no student may let another have his or her part of the problem.
As students are working, circulate around the classroom. Common problems to look for:
- Students may find the perimeter of the shapes by counting every line segment and find the area by counting every square. On the instruction sheet, point out the pattern of shaded squares on the figure showing the block letter dilated with a scale factor of 2. Ask the team if the shading suggests a faster way to find the perimeter and area of the shape (MP7).
- Students may try to draw the dilations of the block letters with unnecessary precision. Or, they may choose a method of drawing the dilation that takes too much time (for example, by drawing rays from a center of dilation through each vertex of the figure). On the instruction sheet, point out the pattern of shaded squares on the figure showing the block letter dilated with a scale factor of 2. Ask the team if the shading pattern suggests a faster way to draw the dilations of the shape (MP7).
- Students may complete the table of area vs. scale factor based on the assumption that the relationship is linear. Ask students to verify their numbers by comparing the areas for scale factors 4 and 5 with the areas obtained using the actual dilations with scale factors 4 and 5.
- Students may not be able to come up with an equation or verbal rule to describe the relationship between area and scale factor. Point to the graph and ask them to describe the curve they see. (Students may think that they recognize a parabola or exponential growth.) Concentrate on the fact that the relationship is not linear. Ask students to look at the pattern of shaded blocks in on the figures to see if that suggests a relationship. What is the area of each of the shaded squares? How does the area of the squares vary with the scale factor?
Select two teams and have them report on their findings (5-10 minutes). Select one team that worked with the block letter ‘H’ and one team that worked with the block letter ‘S’. Use a document camera to display student work.
Have students reflect on the lesson in pairs and teams (5 minutes). Students share in pairs, then in teams. Teams report out by listing three important things they learned on the white board.
I used a Kagan Structure (Timed Pairs Share), with a few modifications. The instructions are in the presentation MTPGeometry_Lesson6_0_5.pptx.
Assign Homework (3 minutes). Homework for this lesson was a review of concepts and skills in Unit 6.