Unit 8, Lesson 2
Side Lengths and Areas
Accelerated 7
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In this Warm-up, students use what they know about finding the area of a square given its side length to think about the converse: finding the side length of a square given its area.
First, students work with squares whose side lengths and areas can be found by counting length or area units. Next, students find the area of a “tilted” square using strategies from a previous lesson. They reason that if two squares have the same area, their side lengths must also be the same, and verify this using tracing paper.
Arrange students in groups of 2. Provide access to geometry toolkits, including tracing paper.
Give students 2–3 minutes of quiet work time followed by a whole-class discussion.
What is the area of Square B? What is its side length? (Use tracing paper to check your answer to this.)
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The key takeaway from this activity is that if two squares have the same area, they must also have the same side length. This can be reinforced by measuring with tracing paper. Invite 1–3 students to share how they determined that Square A and Square B have the same area.
MLR2: Collect and Display
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The purpose of this activity is for students to estimate the side length of a square using a geometric construction that relates the side length of the square to a point on the number line. Students then verify their estimate using techniques from an earlier lesson.
Once students connect the side length to a point on the number line, they learn that this number has a name and a special notation to denote it: square root and the square root symbol. Students will have many opportunities to deepen their understanding of square roots and practice using square root notation in later activities and lessons.
Arrange students in groups of 2. Give them 1–2 minutes of quiet work time for the first question followed by a brief partner discussion. Have students compare estimates and explain their reasoning to their partner before continuing with the rest of the activity. Follow with a whole-class discussion.
Use the circle to estimate the area of the square shown here. Explain your reasoning.
A coordinate plane with the origin labeled “O.” x-axis, scale -6 to 6 , by 1's. y-axis,-6 through 7, by 1's. Circle and square graphed. circle’s circumference passes through 2 of the squares vertices. The circle’s center is the origin and it’s circumference is indicated by a dashed line that passes through the following approximate points on the axes: Negative 5 point 3 comma 0, 0 comma 5 point 3, 5 point 3 comma 0, and 0 comma negative 5 point 3. The square is tilted so that all its sides are diagonal to the coordinate grid. It has vertices at: 0 comma 0, negative 2 comma 5, 3 comma 7, and 5 comma 2. The circumference of the circle passes through the square’s vertices at negative 2 comma 5 and 5 comma 2 so that the sides of the square, extending from the origin to those 2 vertices, are within the circle. @Kia Johnson I didn't want to say that the sides of the square were the radius felt like taking away some of the cognitive demand) but felt a little wordy. REPLY 11:44 (Fixed some language, now that I am writing for same image on grid): A coordinate plane with the origin labeled “O.” The x-axis has the numbers negative 6 through 6 indicated with tick marks. The y-axis has the numbers negative 6 through 7 indicated with tick marks. A square and a circle are drawn on the plane so that the circle’s circumference passes through 2 of the squares vertices. The circle’s center is the origin and it’s circumference is indicated by a dashed line that passes through the following approximate points on the axes: Negative 5 point 3 comma 0, 0 comma 5 point 3, 5 point 3 comma 0, and 0 comma negative 5 point 3. The square is tilted so that all its sides are diagonal to the coordinate grid. It has vertices at: 0 comma 0, negative 2 comma 5, 3 comma 7, and 5 comma 2. The circumference of the circle passes through the square’s vertices at negative 2 comma 5 and 5 comma 2 so that the sides of the square, extending from the origin to those 2 vertices, are within the circle.
Use the grid to check your answer to the first problem.
A coordinate plane with the origin labeled “O.” x-axis, scale -6 to 6 , by 1's. y-axis,-6 through 7, by 1's. Circle and square graphed. the circle’s circumference passes through 2 of the squares vertices. The circle’s center is the origin and it’s circumference is indicated by a dashed line that passes through the following approximate points on the axes: Negative 5 point 3 comma 0, 0 comma 5 point 3, 5 point 3 comma 0, and 0 comma negative 5 point 3. The square is tilted so that all its sides are diagonal to the coordinate grid. It has vertices at: 0 comma 0, negative 2 comma 5, 3 comma 7, and 5 comma 2. The circumference of the circle passes through the square’s vertices at negative 2 comma 5 and 5 comma 2 so that the sides of the square, extending from the origin to those 2 vertices, are within the circle.
The purpose of this discussion is to introduce students to square roots and the square root symbol in the context of area and side length of squares.
Display the image from the second problem and ask students, “What do you think the actual side length of the square is? That is, what is the number that when squared is equal to 29?” As students share, take each guess and square it, noticing out loud how some come very close to 29. For example, point out that
Tell students that some squares like this one have areas that are whole numbers, but their side lengths are not whole numbers. Explain that there is a point on the number line (the
Tell students that the exact side length of a square with area 29 square units is called “the square root of 29,” which can be written as “
The purpose of this discussion is to check that students understand the definition of square roots as they relate to side lengths of squares. Here are some questions for discussion. After each question, consider displaying a square for all to see and adding labels for the area and side length. An example is provided.
“What does it mean when we write
“If
The area of square
Since the area is between
When we want to talk about the exact side length, we can use the square root symbol. We say “the square root of 73,” which is written as
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Find the areas of Squares D, E, and F.
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