standard_description: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
problem_activity: 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
[IMG]
[IMG]
[IMG]
Calculate the lengths of the 5 unlabeled sides.
Student Response
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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
\(s\)
.
How do the diagonals of
\(EFGH\)
compare to the diagonals of
\(ABCD\)
?
What is the maximum possible length of a diagonal of a rhombus of side length
\(s\)
?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
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.)


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