few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

Example 2:
Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Concept/skill: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
Problem/activity:
Task
Suppose $A$ is an angle with vertex $P$, as pictured below:
[IMG]
Draw a circle with center $P$ and with  radius $r \gt 0$.  Explain why the circle
meets each ray of angle $A$ in a single point.  Label these points $Q$ and $R$ respectively.
Draw circles with centers $Q$ and $R$ respectively and radius $r$.  These circles meet at $P$ and a second point to be labelled $B$.
Show that ray $\overrightarrow{PB}$  bisects angle $P$.


Concept/skill: 
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Answer: 
Yes. This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_neg_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
problem_activity: Activity
In this activity, students begin to build their own table of trigonometric ratios (ratios of side lengths in right triangles, by angle measure of one acute angle). For now, call this the right triangle table so students remember these ratios only apply to right triangles. Building these ratios from the similar triangles they are based on helps students build connections between similar triangles and trigonometry.
Measuring several examples of triangles with the same angle measurements is essential in both the digital and paper versions of this activity. In the paper version, it provides more data points to get a more accurate ratio for the given angle. In both versions, it helps students understand that all right triangles with their angle as one of the angles will be similar and thus have the same ratios.
In the next activity and the cool-down of this lesson, students will look for patterns in the right triangle table, and also apply the ratios in the right triangle table to find unknown angle measures given a ratio of side lengths. Do not use the word trigonometry or any of the names of the ratios yet.
Digital
Launch
Provide access to devices that can run GeoGebra. Students will be working with the applet at
ggbm.at/bfxeb6cr
.
Arrange students in groups of 2-4. Invite students to create a spreadsheet to organize their data and perform calculations. Students can access a GeoGebra spreadsheet in the math tool kit or at
geogebra.org/spreadsheet
. For students needing extra support, consider providing this spreadsheet for them to use:
ggbm.at/jvvqrngb
.
Assign each group a pair of complementary angles. If students did the optional lesson choose to skip 30 and 60 or to give them to a group that needs extra practice.
Explain to students that the computer measurement is precise to at least the ten-thousandths place, but the thousandths place is sufficient for our needs.
Representation: Internalize Comprehension.
Use color coding and annotations to highlight connections between representations in a problem. For example color code the side adjacent to angle
\(A\)
, the side opposite angle
\(A\)
, and the hypotenuse.
Supports accessibility for: Visual-spatial processing.
Student Facing
Your teacher will give you some angles.
Use the applet to build 4 different right triangles for each of your angles.
Record the side lengths of each of the triangles.
Compute these 3 quotients for the acute angles in each triangle:
The length of the leg adjacent to your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the leg adjacent to your angle
Find the mean of each type of quotient.
What do you notice? What do you wonder?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 3-4. Give each group a copy of the blackline master to complete.
Assign each group a pair of complementary angles. If students did the optional lesson choose to skip 30 and 60 or to give them to a group that needs extra practice.
Discuss rounding with students before they begin. Ask students whether it makes sense to measure the short side of a triangle as about 5.2 cm and the hypotenuse as pretty close to 7.1 cm and then report that the ratio of the short side to the hypotenuse is exactly 0.73239437. (No, that's too many digits to report.)
Explain to students that the rule of thumb used by scientists is that if your measurements have two digits then your calculated answer should have two digits as well. For example, if your tools measure to the tenths place, say 5.2 cm, so that your measurements have one digit you’re confident in - the 5 in the one’s place, and one digit you may have had to estimate a bit - the 2 in the tenths place, then your answer should be rounded to two digits - 0.73. In the case where the number starts with a zero we count digits starting at the first non-zero digit, so in this case we will round to the hundredths place.
As students finish their measurements, have them find the mean of each column and record that on the class Right Triangle Table.
Check students’ mean data before they enter it to ensure that it is close to the actual values:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "angle"}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "4258"}, "_value": "\\(\\div\\)"}], "_values": ["adjacent leg", "hypotenuse"]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "4258"}, "_value": "\\(\\div\\)"}], "_values": ["opposite leg", "hypotenuse"]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "4258"}, "_value": "\\(\\div\\)"}], "_values": ["opposite leg", "adjacent leg"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "10 degrees"}, {"_value": "0.985"}, {"_value": "0.174"}, {"_value": "0.176"}]}, {"td": [{"_value": "20 degrees"}, {"_value": "0.940"}, {"_value": "0.342"}, {"_value": "0.364"}]}, {"td": [{"_value": "30 degrees"}, {"_value": "0.866"}, {"_value": "0.500"}, {"_value": "0.577"}]}, {"td": [{"_value": "40 degrees"}, {"_value": "0.766"}, {"_value": "0.643"}, {"_value": "0.839"}]}, {"td": [{"_value": "50 degrees"}, {"_value": "0.643"}, {"_value": "0.766"}, {"_value": "1.192"}]}, {"td": [{"_value": "60 degrees"}, {"_value": "0.500"}, {"_value": "0.866"}, {"_value": "1.732"}]}, {"td": [{"_value": "70 degrees"}, {"_value": "0.342"}, {"_value": "0.940"}, {"_value": "2.747"}]}, {"td": [{"_value": "80 degrees"}, {"_value": "0.174"}, {"_value": "0.985"}, {"_value": "5.671"}]}]}]}]}
This table is needed for the next activity, so it should be recorded to display at that time.
Representation: Internalize Comprehension.
Use color coding and annotations to highlight connections between representations in a problem. For example color code the side adjacent to angle
\(A\)
, the side opposite angle
\(A\)
, and the hypotenuse.
Supports accessibility for: Visual-spatial processing.
Student Facing
Your teacher will give you some angles.
Draw several right triangles using the angles you receive.
Precisely measure the side lengths of the triangles.
Complete the tables by computing 3 quotients for the acute angles in each triangle:
The length of the leg adjacent to your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the leg adjacent to your angle
Find the mean of each column in your table.
What do you notice about your table? What do you wonder about your table?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students might struggle to identify opposite and adjacent legs. Prompt students to highlight the angle in question to accurately identify sides. If you point to “your angle” then the leg you are touching is adjacent and the leg you are not touching is opposite. The hypotenuse is always the longest side.
[IMG]
Activity Synthesis
Invite groups to share what they noticed and wondered about their data. (Each ratio is about the same for the angle. The ratio of the opposite side to the hypotenuse for angle
\(A\)
is the same as the ratio of the adjacent side to the hypotenuse for angle
\(B\)
.)
Speaking: MLR8 Discussion Supports.
Give students additional time to make sure that everyone in their group can explain what they noticed and wondered about the relationships between the quantities represented in the table. Invite groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking, and will improve the quality of explanations shared during the whole-class discussion.
Design Principle(s): Support sense-making; Cultivate conversation


aligns: no