standard_description: Construct a tangent line from a point outside a given circle to the circle.
problem_activity: Activity
This activity builds on the previous one as students plot a different relationship, comparing the length of each square diagonal to its area. Students use the length measurements from the previous activity and use them to calculate the area of the squares. Unlike in the previous activity, this time both the table and the graph show that the relationship between diagonal length and area is
not
close to being proportional. As a result, the measurement error does not play a role in this activity. Rather, the focus is on identifying that a relationship is not proportional: this is an example of MP7 because students will identify that there is no constant of proportionality in the table (not even allowing for measurement error) and will correspondingly notice that the plotted points are not close to lying on a line.
Launch
Keep students in the same groups. Tell them to calculate the area of the same 3 squares they measured in the previous activity. Give students 4 minutes of group work time. Display this blank grid and have students plots points for their measurements.
[IMG]
Give students 2–3 more minutes of group work time followed by whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge about calculating the area of a square. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
In the table, record the length of the diagonal for each of your assigned squares from the previous activity. Next, calculate the area of each of your squares.
|          | diagonal (cm) | area (cm                       2                      ) |
|----------|---------------|---------------------------------------------------------|
| square A |               |                                                         |
| square B |               |                                                         |
| square C |               |                                                         |
| square D |               |                                                         |
| square E |               |                                                         |
| square F |               |                                                         |
| square G |               |                                                         |
| square H |               |                                                         |
| square I |               |                                                         |
Pause here so your teacher can review your work. Be prepared to share your values with the class.
Examine the class graph of these values. What do you notice?
How is the relationship between the diagonal and area of a square the same as the relationship between the diagonal and perimeter of a square from the previous activity? How is it different?
Student Response
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Student Facing
Are you ready for more?
Here is a rough map of a neighborhood.
[IMG]
There are 4 mail routes during the week.
On Monday, the mail truck follows the route A-B-E-F-G-H-A, which is 14 miles long.
On Tuesday, the mail truck follows the route B-C-D-E-F-G-B, which is 22 miles long.
On Wednesday, the truck follows the route A-B-C-D-E-F-G-H-A, which is 24 miles long.
On Thursday, the mail truck follows the route B-E-F-G-B.
How long is the route on Thursdays?
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Some students may struggle to calculate the area of the squares, or may use the length of the diagonal as if it were the side length. Prompt them with questions like “How can you calculate the area of a rectangle?” “What is the length and width of your square?”
Some students may measure the side length of each square again, instead of dividing the perimeter from the previous activity by 4. This strategy is allowable, although you can also prompt them to consider if they no longer had access to a ruler, is there a way they could use the information they already recorded to find this measurement.
Some students may struggle to organize the information. Prompt them to add a column to the table in the previous activity to record the side length of the squares.
Activity Synthesis
The goal of this discussion is for students to recognize that the relationship between the length of the diagonal and the area of squares of different sizes is not a proportional relationship.
Invite students to share their observations about the graph. Important points include:
As the length of the diagonal increases, the area of the square increases.
The points do not look like they lie on a line and definitely do not lie on a line that goes through
\((0,0)\)
.
The area of the square looks to grow at a faster rate for larger diagonals than for the short diagonals.
To reinforce that the relationship is not proportional, instruct students to calculate the quotient of the area divided by the length of the diagonal for their square. Consider displaying a table of their measurements and adding a third column to record their quotients. Students should realize that the differences between these quotients are too large to be accounted for by just measurement error.
| diagonal (cm) | area (cm                   2                  ) | \(\text{area} \div \text{diagonal}\) |
|---------------|-------------------------------------------------|--------------------------------------|
|      8.2      |                      33.6                       |                 4.1                  |
|      4.2      |                        9                        |                 2.1                  |
|      6.8      |                       23                        |                 3.4                  |
|      2.2      |                       2.4                       |                 1.1                  |
|     12.6      |                      77.4                       |                 6.1                  |
If time permits, consider using this picture to reinforce that length scales linearly while area does not:
[IMG]
“If the side length of the small square is
\(s\)
, what is the side length of the large square?” (
\(2s\)
)
“If the diagonal of the small square is
\(d\)
, what is the diagonal of the large square?” (
\(2d\)
)
“If the area of the small square is
\(A\)
, what is the area of the large square?” (
\(4A\)
)
“Is the relationship between the diagonal and the area proportional?” (No, when the diagonal doubles, the area quadruples; if the relationship were proportional, both the diagonal and the area would double.)
Speaking, Listening: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they heard to a partner before selecting one or two students to share with the class. Ask the original speaker(s) if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


aligns: no