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Arrange students in groups of 2. Provide access to geometry toolkits, including tracing paper. Since the goal of this activity is for students to estimate square roots and not find exact values, do not provide access to calculators. Remind students about the meaning and use of square root notation.
Find the area of each square and estimate the side lengths using your geometry toolkit. Then write the exact length for the sides of each square.
The purpose of this discussion is for students to connect the estimate made with tracing paper to the exact length of the side of a square written in square root notation. This helps students to see that a square root is still a number, and it can be approximated by a value that is easier to see.
For each square, invite several students to share their estimate for the side length and display the results for all to see. Here are some questions for discussion:
“What do you notice about the estimates?” (The estimates are all close to each other. Some have more decimal places than others.)
“How can we check the accuracy of our estimates?” (We can take our estimate and square it to see how close we are to the area of the square.)
The purpose of this activity is for students to further their understanding of square roots as numbers by completing a table and creating a graph of side length-area pairs. Students will use the graph to estimate both a square’s side length when the value is not an integer and the value of a square root not linked to a drawn square (MP2).
Provide access to calculators. Ask students to complete the table and mark their graphs independently, then check with a partner. Follow with a whole-class discussion.
| side length, | 0.5 | 1.5 | 2.5 | 3.5 | ||||
|---|---|---|---|---|---|---|---|---|
| area, | 1 | 4 | 9 | 16 |
| side length, | 4.5 | 5.5 | 6.5 | 7.5 | ||||
|---|---|---|---|---|---|---|---|---|
| area, | 25 | 36 | 49 | 64 |
Plot the points, , on the coordinate plane shown here.
Use the graph to approximate .
The purpose of this discussion is to make sure students understand how they can use the table and graph to estimate a square’s side length when it is not an integer. This thinking helps students relate numbers expressed using square root notation, such as , to numbers they are more familiar with, such as 6.5 and 7.
Invite 1–3 students to share their tables and graphs and display them for all to see. Here are some questions for discussion:
“What relationship does the graph display?” (the relationship between the side length and area of a square)
“How could we use the table to estimate ?” (An area of 45 square units is between the values of 42.25 and 49 in the table. These areas correspond to side lengths of 6.5 and 7 units, so has to be between 6.5 and 7.)
“How did you use the graph to approximate ?” (I saw that the point on the graph with a -value of 45 had an -value that was a little bigger than 6.5.)
The purpose of this activity is to reinforce the difference between the exact and an approximate side length of a square (MP6) and to practice using different strategies (tracing paper, table, graph) to find them.
In this activity, students are asked to determine which square has the larger area. While the squares appear different, they are actually congruent, both with an area of 50 square units.
Arrange students in groups of 2. Provide access to geometry toolkits. Do not provide access to calculators since the goal of the activity is to connect the approximate value with the exact value of a square root.
Display the image of the two squares for all to see and survey the class on which square they think has the larger area. Record the results near the displayed image.
Give students 2–3 minutes of quiet work time, with one student from each pair finding the area of Square A and the other finding the area of Square B. Pause for a partner discussion, followed by a whole-class discussion.
The purpose of this discussion is to highlight how gives us a way to answer exactly, while 7.1 only gets us close.
Here are some questions for discussion:
Tell students that this activity shows us that a square with an area of 50 square units has an exact side length of units, which is approximately 7.1 units. This is true for any square with an area of 50 square units no matter the orientation of the square. We can also say that because and that .
because .
because .
The value of must be between 3 units and 4 units because it is between the values of and .
There are 3 squares on a square grid, arranged in order of area, from smallest, on the left, to largest, on the right. The left most square is aligned to the grid and has side lengths of 3 with an area of 9. The middle square is tilted on the grid so that its sides are diagonal to the grid. The square is labeled with a side length of square root of 10 and an area of 10. The right most square is aligned to the grid and has side lengths of 4 with an area of 16.