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: 
Solve quadratic equations in one variable.

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


Answer: 
Yes. This task shows one quadratic equation with 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_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Fluently add and subtract multi-digit whole numbers using the standard algorithm.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
problem_activity: Narrative
The purpose of this activity is for students to make a reasonable estimate for a given product.
In addition to estimating the product, students also decide whether the estimate is too large or too small. In the activity synthesis, students consider about how far their estimate is from the actual product.
I
n the next activity, students will evaluate the expressions using a strategy of their choice.
Students choose between several different possible estimates and justify their choice before they calculate the product (MP3).
Engagement: Provide Access by Recruiting Interest.
Synthesis: Revisit math community norms to prepare students for the whole-class discussion.
Supports accessibility for: Social Emotional Functioning, Language
Launch
Groups of 2
Activity
5–7 minutes: independent work time
2–3 minutes: partner discussion
Monitor for students who:
relate the given expression to each proposed answer by rounding or changing one or both factors.
estimate by rounding the factors.
use benchmark numbers.
use place value reasoning or the properties of operations to explain why their estimate is reasonable.
Student Facing
Which estimate for the product
\(18 \times 149\)
is most reasonable? Explain or show your reasoning.
\(2,\!000\)
\(4,\!000\)
\(3,\!000\)
\(1,\!500\)
Are any of the estimates unreasonable? Explain or show your reasoning.
Do you think the actual product will be more or less than your estimate? Explain or show 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.
Advancing Student Thinking
If students do not choose a reasonable estimate for
\(18 \times 149\)
, ask, “Can you explain how you chose your estimate?”
Activity Synthesis
Invite students to share their estimates for the product and their reasoning.
“Why do you think
\(3,\!000\)
is a good estimate?” (18 is close to 20 and 149 is very close to 150 and I know
\(20 \times 150 = 3,\!000\)
.)
“Is the value of
\(18 \times 149\)
greater than or less than
\(3,\!000\)
? How do you know?” (Less, because 18 is less than 20 and 149 is less than 150.)
“Can you estimate how much less?” (About 300, because I added 2 to 18 to get 20 and
\(2 \times 149\)
is about 300.)
Display:
\(18 \times 149\)
is about
\(2,\!700\)
.
“In the next activity, we’ll check to see how good our estimate is.”


aligns: no