Lesson 2

Half a Square

2.1: Diagonals of Rectangles (5 minutes)

Warm-up

Later in this lesson, students will explore isosceles right triangles and reason about them in the context of diagonals of squares. In this warm-up, students practice finding the hypotenuse of a right triangle with the Pythagorean Theorem in the context of finding the diagonal of a rectangle.

In the synthesis of a later activity, students will return to these examples to explore why the ratio of the length of the diagonal to the length of the side of a square is constant and explain why that is not the case for all rectangles, though it is the case for any family of similar rectangles (including squares).

Monitor for students who:

  • leave their answers in radical form
  • estimate the answer
  • use a calculator to get an answer in decimal form

Student Facing

A four sided figure A B C D.
A four sided figure K L M J.

 Calculate the values of \(x\) and \(y\).

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students are struggling, ask them what shapes they see. (Rectangles and right triangles.) If students are still stuck, ask them how they would find \(x\) or \(y\) if there were only a right triangle. (Using the Pythagorean Theorem.)

Activity Synthesis

Focus discussion on an appropriate level of precision for the length of \(x\). Invite students to share their answers in different forms. If no student estimated, ask them approximately how big \(x\) is based on finding that \(x^2 = 65 \text{ or } x=\sqrt{65}\). (I know \(x\) is a little greater than 8, because \(8^2=64\).) Compare that with the results students got if they used a calculator to find a decimal approximation of \(x\). Note that recording all the digits displayed on the calculator isn't helpful. We will use the convention of rounding side lengths to the nearest tenth, but sometimes knowing the exact answer of \(\sqrt{65}\) is helpful.

2.2: Decomposing Squares (15 minutes)

Optional activity

In this activity, students compute the diagonals of some squares and measure the diagonals of others. From the data they collect, they reason that for a square with side length \(s\), the length of the diagonal is about \(1.4s\). They also calculate the diagonal of a unit square exactly, using the Pythagorean Theorem, allowing them to connect the decimal approximation \(1.4\) to the square root of 2.

Students will need a set of squares to measure. They could use various sizes of origami paper, sticky notes, or other convenient objects. If squares are not available, there is a blackline master provided with squares students can use instead.

The goal of this activity is not to simplify radicals or make explicit the connection that, for example, \(\sqrt{50} \) can be rewritten as \( \sqrt{25 \boldcdot 2}= 5\sqrt2\)

Monitor for students who:

  • use a calculator to determine that the diagonal of a unit square is about 1.4
  • leave the diagonal of the unit square as exactly \(\sqrt2\)
  • make a table to record what they learned about squares and their diagonals

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Arrange students in groups of 2. Distribute squares of several different sizes to each group (either the blackline master or other convenient squares).

Action and Expression: Internalize Executive Functions. Provide students with a three-column table to organize their measurements of the side length, diagonal length and ratio of side to diagonal length. The table will also provide a visual support for students to identify a pattern.
Supports accessibility for: Language; Organization

Student Facing

  1. Draw a square with side lengths of 1 cm. Estimate the length of the diagonal. Then calculate the length of the diagonal.
  2. Measure the side length and diagonal length of several squares, in centimeters. Compute the ratio of side to diagonal length for each.
  3. Make a conjecture.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students are struggling to organize their thinking, suggest that they make a table. Help students brainstorm what would be good columns to organize their measurements and calculations, such as “side length,” “diagonal length,” and “ratio of diagonal to side.”

Activity Synthesis

Ask students what patterns they noticed and what conjectures they made. Invite students who organized their thinking using a table to display their work for all to see. If no students made a table, create one as a class, displayed for all to see. Include students who approximated the diagonal length of the unit square using a calculator, and students who left it as \(\sqrt2\). Leave the table visible for all to see during the next activity as well.

If this conjecture is not mentioned by students, point it out in the table and then ask students to explain:

  • The diagonal length of a square seems to be the side length multiplied by 1.4 each time. (All squares are similar to the unit square by a scale factor of the side length, so it makes sense that all the diagonal lengths are multiples of the diagonal length of the unit square.)

Ask students if they agree that both of these things are true:

  • The diagonal of a square with side length \(s\) is about 1.4 times \(s\).
  • The diagonal of a square with side length \(s\) is exactly \(s\sqrt2\).
Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. Encourage students to consider what details are important to share, and to think about how they will explain their thinking using mathematical language. As students describe the patterns they noticed and explain their conjectures, annotate the display to illustrate connections between the ratios of side to diagonal lengths for squares, and the decimal approximation 1.4 to the square root of 2.
Design Principle(s): Support sense-making; Maximize meta-awareness

2.3: Generalize Half Squares (15 minutes)

Optional activity

In the previous activity, students generalized that the diagonals of squares are related to the side length of the square by a factor of about 1.4, or exactly \(\sqrt2\).

In this activity, students apply their generalization about the diagonals of squares to isosceles right triangles. To find the lengths of the unlabeled sides in the second figure, students will need to generalize that a right triangle with one 45 degree angle is isosceles (because it’s half a square, or because both base angles are congruent, or because it’s similar to the isosceles triangle in the first figure).

To find the unknown values in the third figure, students will have to use the ratio of diagonal length to side length to find unknown side lengths. Students may use the approximate ratio \(1.4:1\) (or \(1:0.7\)) to find the side lengths, or they might use the exact ratio \(\sqrt2:1 \text{ or } 1:\frac{1}{\sqrt2}\). Students are not expected or encouraged to rewrite answers of \(\frac{24}{\sqrt2}\) in another form.

Launch

Action and Expression: Internalize Executive Functions. Provide students with a three-column table to organize their work. Use the column headings: the side adjacent to the 45 degree angle, the side opposite to the 45 degree angle, and the hypotenuse. The table will provide a visual support for students to identify the type of triangle and pattern.
Supports accessibility for: Language; Organization

Student Facing

Triangle A B C. Angle B is marked as a right angle. Angle A is labeled 45 degrees. Side A B is labeled 3. Side B C is labeled 3.
Triangle P Q R. Angle Q is marked as a right angle. Angle P is labeled 45 degrees. Side Q P is labeled 14.
Triangle X Y Z. Angle Y is marked as a right angle. Angle X is labeled 45 degrees. Side X Z is labeled 24.

Calculate the lengths of the 5 unlabeled sides.  

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Square \(ABCD\) has a diagonal length of \(x\) and side length of \(s\). Rhombus \(EFGH\) has side length \(s\).

  1. How do the diagonals of \(EFGH\) compare to the diagonals of \(ABCD\)?
  2. What is the maximum possible length of a diagonal of a rhombus of side length \(s\)?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students are struggling, encourage them to analyze the three triangles, look for patterns, and identify the triangles as isosceles right triangles. Students can then use the patterns from the previous activity. 

Activity Synthesis

Make sure all students understand that the three triangles are isosceles right triangles, and represent half of a square, and so are prepared to connect their reasoning from earlier activities to this activity.

Display this list of solutions for triangle \(PQR\):

  • \(QR=14, RP=\sqrt{392}\)
  • \(QR=14, RP=14\sqrt{2}\)
  • \(QR=14, RP = 19.8\)
  • \(QR=14, RP = 1.4 \boldcdot 14 \) or 19.6

Invite students to explain the reasoning behind each. Encourage students to think about which answers are most accurate, and which are most efficient. (The first two methods are equally accurate, but the first probably took more calculation, while the second could be found using scale factors quite easily. The third answer was obtained by using a calculator to evaluate either of the first two answers, which actually loses both efficiency and accuracy. The fourth answer is very efficient, as it was found using scale factors, but is a bit less accurate. For some applications, the fourth answer may be best, and it’s great for estimating.)

Lesson Synthesis

Lesson Synthesis

Create a class display of half a square. This display should be posted in the classroom for the remaining lessons within this unit. 

A 45-45-90 triangle.

Ask students what they notice and what they wonder about the image. (It’s isosceles. It’s half a square. It’s the same triangles we’ve been looking at all class. The sides are \(s, s, \text{ and } s\sqrt2\). Why is it important? Do we have to memorize it? Why does it have \(\sqrt2\) instead of 1.4?)

Explain to students that they don’t need to memorize this information since they have other strategies to calculate missing side lengths and hypotenuses in isosceles right triangles. 

If applicable, tell students that this type of image might be provided as a reference on standardized tests. Ask students how they might decide what types of problems to use it on, and how they would solve those problems. Here are three sample problems to discuss:

  1. The distance between bases on a baseball field is 90 feet. If the 2nd base player stands on second base and throws the ball to the catcher standing at home plate, how far is the throw? (If students are not familiar with the context of baseball, provide a diagram.)
  2. The hypotenuse of an isosceles right triangle is 10 cm. How long is the side?
  3. A rectangle has sides of 3 cm and 4 cm. How long is the diagonal of the rectangle?

Students may say that the throw divides the square in half so it matches the image. An isosceles right triangle also matches the image. Half of this rectangle isn’t similar to half a square, so they would need a different method such as the Pythagorean Theorem. Note that they could use the Pythagorean Theorem to solve all three problems, since that is where the labels on the image come from.

2.4: Cool-down - Another Half Square (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Drawing the diagonal of a square decomposes the square into 2 congruent triangles. They are right isosceles triangles with acute angles of 45 degrees. These congruent angles make all right isosceles triangles similar by the Angle-Angle Triangle Similarity Theorem.

Consider an isosceles right triangle with legs 1 unit long where \(c\) is the length of the hypotenuse. By the Pythagorean Theorem, we can say \(1^2+1^2=c^2\) so \(c=\sqrt2\). The hypotenuse of an isosceles right triangle with legs 1 unit long is \(\sqrt2\) units long.

Now, consider an isosceles right triangle with legs \(x\) units long. By the Angle-Angle Triangle Similarity Theorem, the triangle is similar to the isosceles right triangle with side lengths of 1, 1, and \(\sqrt2\) units. A scale factor of \(x\) takes the triangle with leg length of 1 to the triangle with leg length of \(x\). Therefore, the hypotenuse of the isosceles right triangle with legs \(x\) units long is \(x\sqrt2\) units long. 

Triangle. 
Triangle A B C. Angle A is marked as a right angle. Angle B is labeled 45 degrees. Angle C is labeled 45 degrees. Side A B is labeled 6.
Triangle D E F. Angle F is marked as a right angle. Angle D is labeled 45 degrees. Angle E is labeled 45 degrees. Side D E is labeled 12.

In triangle \(ABC, x=6\) so \(AC\) is 6 units long and \(BC\) is \(6\sqrt2\) units long. 

In triangle \(DEF, x\sqrt2=12\) so \(x=\frac{12}{\sqrt2}\), which means both \(EF\) and \(DF\) are \(\frac{12}{\sqrt2}\) units long.