13.1: Three Figures (5 minutes)
The purpose of this activity is to notice that rectangles can have the same diagonal length but different areas. The concept of aspect ratio is also introduced in the activity synthesis.
Monitor for students checking whether the diagonals are all the same length by using a ruler, compass, or edge of a piece of paper to compare their lengths. Also monitor for students mentioning that the rectangles have different areas.
Provide access to rulers, or suggest students use the edge of a blank piece of paper if they would like to compare lengths.
How are these shapes the same? How are they different?
Ask selected students to share their observations. Ensure that these ideas are mentioned:
- The diagonals of the rectangles are all the same length.
- The rectangles have different areas. The leftmost rectangle has a relatively small area compared to the others, and the rightmost rectangle has a relatively large area.
- Each rectangle has a different ratio of height to base. Students might mention the slope of the diagonals, which is related to this idea.
Tell students that in photography, film, and some consumer electronics with a screen, the ratio of the two sides of a rectangle is often called its aspect ratio. In the rectangles in this activity, the aspect ratios are \(5:1\), \(2:1\), and \(1:1\).
Demonstrate how the length of one side is a multiple of the other, on each rectangle. Students may be familiar with selecting an aspect ratio when taking or editing photos. Some common aspect ratios for photos are \(1:1\), \(4:3\), and \(16:9\). Also, from ordering school pictures, 5 by 7 and 8 by 10 may be common sizes they've heard of.
13.2: A $4:3$ Rectangle (20 minutes)
The purpose of this activity is to really understand what an aspect ratio means when one side is not a multiple of the other, and to think about how you can figure out the side lengths if you know the rectangle’s aspect ratio and some other information. This problem is a simpler version of the type of work needed for the more complicated activity that follows.
Provide access to calculators that can take the square root of a number.
Make sure that students understand what it means for the rectangle to have a \(4:3\) aspect ratio before they set to work on figuring out the side lengths. Give them time to productively struggle before showing any strategies. It may be necessary to clarify that the rectangle's diagonal refers to the segment that connects opposite corners (which is not drawn).
Supports accessibility for: Language; Conceptual processing
A typical aspect ratio for photos is \(4:3\). Here’s a rectangle with a \(4:3\) aspect ratio.
- What does it mean that the aspect ratio is \(4:3\)? Mark up the diagram to show what that means.
- If the shorter side of the rectangle measures 15 inches:
- What is the length of the longer side?
- What is the length of the rectangle’s diagonal?
- If the diagonal of the \(4:3\) rectangle measures 10 inches, how long are its sides?
- If the diagonal of the \(4:3\) rectangle measures 6 inches, how long are its sides?
Invite students to share their solutions. If any students solved an equation such as \((3x)^2+(4x)^2=6^2\) for the last question, ensure they have an opportunity to demonstrate their approach.
13.3: The Screen Is the Same Size . . . Or Is It? (20 minutes)
The purpose of this activity is to give students an opportunity to solve a relatively complicated application problem that requires an understanding of aspect ratio, the Pythagorean Theorem, and realizing that a good way to compare the sizes of two screens is to compare their areas. The previous activities in this lesson are meant to prepare students to understand the situation and suggest some strategies for tackling the problem.
Monitor for students using different approaches and strategies. Students may benefit from more time to think about this problem than is available during a typical class meeting.
Provide access to calculators that can take the square root of a number. The task statement is wordy, so consider using the Three Reads protocol to ensure students understand what the problem is saying and what it is asking.
We chose not to provide diagrams drawn to scale in the student materials, since it makes it somewhat obvious that the new phone design has a smaller screen area. However if desired, here is an image to show or provide to students:
Design Principle(s): Support sense-making
Before 2017, a smart phone manufacturer’s phones had a diagonal length of 5.8 inches and an aspect ratio of \(16:9\). In 2017, they released a new phone that also had a 5.8-inch diagonal length, but an aspect ratio of \(18.5:9\). Some customers complained that the new phones had a smaller screen. Were they correct? If so, how much smaller was the new screen compared to the old screen?
Invite students to share their ideas and progress with the class. If appropriate, students may benefit from an opportunity to clearly present their solution in writing.
The debrief and presentation of student work provides opportunities to summarize takeaways from this lesson. Aside from opportunities to point out how the Pythagorean Theorem can help us tackle difficult problems, this lesson makes explicit some aspects of mathematical modeling. Highlight instances where students had to figure out what additional information they would need to make progress, or restate a question in mathematical terms.