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: Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.
problem_activity: Narrative
The purpose of this activity is for students to round given numbers to the nearest ten and hundred and see that the result can be the same for some numbers. Students think about what it means to round a number that is exactly halfway between two tens or two hundreds and are introduced in the synthesis to the convention that these numbers are rounded up (MP3).
MLR1 Stronger and Clearer Each Time:
Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to “What does 97 round to when we are rounding to the nearest 10? To the nearest hundred? Why does that happen?” Invite listeners to ask questions, to press for details and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.
Advances: Writing, Speaking, Listening
Representation: Internalize Comprehension.
Synthesis: Invite students to identify which details were most useful to solve the problems. Display the sentence frame, “The next time I round numbers to the nearest ten and nearest hundred, I will pay attention to . . . .“
Supports accessibility for: Conceptual Processing
Launch
Groups of 2
“Let’s round some numbers to the nearest ten and the nearest hundred. Remember ‘round to the nearest ten (or hundred)’ is another way of saying find the nearest multiple of 10 (or 100).”
Activity
“Work with your partner to complete the table.”
3–5 minutes: partner work time
Monitor for students who notice 97 and 601 round to the same number whether rounding to the nearest ten or the nearest hundred.
Select previously identified students to share their responses.
“Why did 97 and 601 round to the same number when we rounded to the nearest ten and the nearest hundred?” (The closest multiple of 10 was also a multiple of 100. 100 and 600 are multiples of 100, but they are also multiples of 10.)
“Complete the second problem with your partner. Explain your reasoning.”
3–5 minutes: partner work time
Monitor for the following in student explanations to share in the synthesis:
415 is halfway between 410 and 420.
415 is the same distance from 410 and 420.
415 has two closest multiples of 10: 410 and 420.
750 is halfway between 700 and 800.
750 is the same distance from 700 and 800.
750 has two closest multiples of 100: 700 and 800.
Student Facing
Round each number to the nearest ten and the nearest hundred. Use number lines if you find them helpful.
| number | nearest ten | nearest hundred |
|--------|-------------|-----------------|
|   18   |             |                 |
|   97   |             |                 |
|  312   |             |                 |
|  439   |             |                 |
|  601   |             |                 |
Kiran and Priya are rounding some numbers and are stuck when trying to round 415 and 750.
Kiran said, “415 doesn’t have a nearest multiple of 10, so it can’t be rounded to the nearest ten.”
Priya said, “750 doesn’t have a nearest multiple of 100, so it can’t be rounded to the nearest hundred.”
Do you agree with Kiran and Priya? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Select previously identified students to share their responses.
“Kiran and Priya are correct. There isn’t one closest multiple of 10 to 415 because it’s right in the middle of two multiples of 10. There isn’t a closest multiple of 100 to 750 because it’s right in the middle of two multiples of 100.”
“In cases like this, we round up. For example, since 415 is halfway between 410 and 420, we would round up to 420 to round 415 to the nearest ten.”
“Since 750 is halfway between 700 and 800, we would round up to 800.”
”In both of these situations we could go the other way, but it is helpful if we all do the same thing in these situations.”


aligns: no