standard_description: Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
problem_activity: Activity
This activity gives students an opportunity to put into practice some things they already know about finding percent rates. Additionally, the idea of a fraction of a percent appears for the first time. Encourage students to use any representation they would like to calculate the percentage of appetizers, entrees and desserts. Monitor for students who used various representations and ask them to share during the discussion. The main focus should be on the fractional percentages they encounter in this problem for the first time.
Digital
Launch
Tell students they will be finding some more percentages. Encourage them to use any representation they understand, for example, a double number line or a table. Students in groups of 2. Give students 1–2 minutes of quiet work time, followed by partner then whole-class discussion.
If using the digital activity, students will use an applet to find and check percentages.
Representation: Internalize Comprehension
. Activate or supply background knowledge about diagrams that can be used to represent percent rates such as tables and double number line diagrams.
Supports accessibility for: Memory; Conceptual processing
Writing and Listening: MLR5 Co-Craft Questions.
Before students begin work, display the waiter's situation without revealing the questions. Ask students to write down possible mathematical questions that might be asked about the situation. Invite pairs to compare their questions, and then ask for a few to be shared in a whole-class discussion. Reveal the actual questions about the waiter's situation that students will answer. This will help students make sense of the problem before attempting to solve it.
Design Principle(s): Optimize output for explanation
Student Facing
During one waiter’s shift, he delivered appetizers, entrées, and desserts. What percentage of the dishes were desserts? appetizers? entrées? What do your percentages add up to?
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
Tell students they will be finding some more percentages. Encourage them to use any representation they understand, for example, a double number line or a table. Students in groups of 2. Give students 1–2 minutes of quiet work time, followed by partner then whole-class discussion.
If using the digital activity, students will use an applet to find and check percentages.
Representation: Internalize Comprehension
. Activate or supply background knowledge about diagrams that can be used to represent percent rates such as tables and double number line diagrams.
Supports accessibility for: Memory; Conceptual processing
Writing and Listening: MLR5 Co-Craft Questions.
Before students begin work, display the waiter's situation without revealing the questions. Ask students to write down possible mathematical questions that might be asked about the situation. Invite pairs to compare their questions, and then ask for a few to be shared in a whole-class discussion. Reveal the actual questions about the waiter's situation that students will answer. This will help students make sense of the problem before attempting to solve it.
Design Principle(s): Optimize output for explanation
Student Facing
During one waiter’s shift, he delivered 13 appetizers, 17 entrées, and 10 desserts.
What percentage of the dishes he delivered were:
desserts?
appetizers?
entrées?
What do your percentages add up to?
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
If students round to the nearest percentage, they will get that 33% of the dishes were appetizers and 43% of the dishes were entrées. Along with the 25% desserts, their percentages will sum to 101%. Point out that all of the dishes taken as percentages should sum to 100% and encourage them to be critical of their method and try to figure out where the extra 1% came from.
Activity Synthesis
Select students to share the percentages they calculated for each type of dish the waiter delivered. Depending on the outcome of the warm-up, it may be appropriate once again to display different representations of percentages as rates per 100.
Double number line:
[IMG]
Table:
| number of dishes | percentage |
|------------------|------------|
|        10        |     25     |
|        1         |    2.5     |
|        13        |    32.5    |
|        17        |    42.5    |
Equation: Students may have previously learned to represent relationships like this using an equation in a form
\(y=kx\)
. For example, to find what percent 13 is of 40, they might write
\(13=k \boldcdot 40\)
, and find that
\(k\)
is 0.325 by evaluating
\(13 \div 40\)
. 0.325 is the rate per 1, so 32.5 is the rate per 100.
Students may have never seen a percentage that was not a whole number. Spend a few minutes making sense of this. Ask students:
What do you notice that is different about these percentages from the ones you have looked at before? (Some of these percentages are not whole numbers.)
What do the percentages add up to? (Exactly 100.)
What does 32.5% of 40 mean? (It’s halfway between 32% and 33% of 40.)


aligns: no