few_shots: 
Example 1:
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.

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


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:
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).

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


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

Example 3:
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.

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 $P$.


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, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

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: Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.
problem_activity: Warm-up
This warm-up is the first time students are asked to find
\(A\%\)
of
\(B\)
when
\(B\)
is not 100 or 1.
Students may approach the problem in a few different ways, with or without filling in values on the provided double number line diagram. For example, they may understand from the fundraising context of the problem that \$40 is 100% because it is the amount of the goal. From there, they may simply find half of \$40 for the 50% value and add that value to \$40 to find the 150% value. Other students may use equivalent ratio reasoning to calculate the value at 50% and 150%. As students work, notice the different strategies used and any misconceptions so they can be addressed during discussion.
Launch
Remind students that in the previous lesson, we found percentages of 100 and of 1 using double number lines. Explain that in this lesson we will find percentages of other numbers. Give students 2 minutes of quiet work time, and follow with a whole-class discussion. Encourage students to create a double number line to help them answer the questions if needed.
Student Facing
Each of three friends—Lin, Jada, and Andre—had the goal of raising
$
40. How much money did each person raise? Be prepared to explain your reasoning.
Lin raised 100% of her goal.
Jada raised 50% of her goal.
Andre raised 150% of his goal.
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
Students may be surprised by a percentage greater than 100. If they are puzzled by this, explain that Andre raised more money than the goal.
Activity Synthesis
Consider displaying this double number line for all to see.
[IMG]
Invite a few students to share their solving strategies. One way to highlight the different techniques students use is to invite several students to explain how they calculated the amount of money raised by Andre. We can find 150% of 40 in several ways. For example, we can add the values of 50% of 40 and 100% of 40. We can also reason that since
\(100 \boldcdot (1.5) = 150\)
then
\(40 \boldcdot (1.5) = 60\)
, which means \$60 is 150% of \$40.
If not uncovered in students’ explanations, explain that we are finding percentages of \$40, since this number—not 100 cents or 1 dollar—is the fundraising goal for the three friends. Since 100% of a goal of \$40 is \$40, the 100% and \$40 are lined up on the double number line.
Students who relied on the visual similarity between, for example, \$0.25 is 25% of 1 dollar in the previous lesson find this strategy unworkable here (as \$50 is not 50% of \$40). To encourage students to use their understanding of equivalent ratios to reason about percent problems, ask the class to explain—either when the above misconception arises or as a closing question—why \$50 is not 50% of \$40, but 50% of 100 cents is 50 cents.


aligns: no