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 following problem or activity 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: Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
- The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
- The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
- 10 can be thought of as a bundle of ten ones — called a “ten.”
problem_activity: Narrative
The purpose of this activity is for students to write three-digit numbers as the sum of the value of each digit,
expanded form
. Students connect the order and values of the addends in expanded form to the order and value of each place in a three-digit number. Use expanded form and its definition interchangeably throughout the activity so that students feel comfortable with the new vocabulary. When students represent numbers as sums by place value, they interpret the three-digit numbers in terms of its digits and the operation of addition (MP7).
MLR8 Discussion Supports.
Provide all students with an opportunity to produce mathematical language by inviting students to chorally repeat “expanded form” in unison 1–2 times.
Advances: Speaking
Required Materials
Materials to Gather
Base-ten blocks
Launch
Groups of 2
Display Andre, Tyler, and Mai’s situation and the image of their blocks.
“What would the expression look like?”
1 minute: independent work time
1 minutes: partner discussion
Share responses.
Display 357 and
\(300 + 50 + 7\)
.
“We can represent the value of the blocks by writing a three-digit number.”
“A number can also be represented as a sum of the value of each of its digits. This is called
expanded form
.”
“Like a three-digit number, expanded form shows the sum starting with the place that has the greatest value on the left to the place with the least value on the right.”
As needed, discuss reasons why any expressions generated in the launch would or would not be examples of expanded form.
Activity
“Now you will practice writing the value of the base-ten diagrams in expanded form and as a three-digit number.”
7 minutes: partner work time
Student Facing
Andre has 3 hundreds. Tyler has 5 tens. Mai has 7 ones. They want to represent the amount they have using an equation.
[IMG]
Write an expression to represent the sum of their values.
__________ + __________ + __________
Write the total value as a three-digit number:
_______________
Write each number as the sum of hundreds, tens, and ones, and as a three-digit number.
[IMG]
Expanded form: _________________________
Three-digit number: _________________________
[IMG]
Expanded form: _________________________
Three-digit number: _________________________
[IMG]
Expanded form: _________________________
Three-digit number: _________________________
[IMG]
Expanded form: _________________________
Three-digit number: _________________________
Student Response
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for free access to Student Response.
Activity Synthesis
“What is the value of
\(40 + 100 + 3\)
? Explain how you know.” (143. I saw there was 1 hundred, 4 tens, and 3 ones. I just rearranged it in my head like expanded form.)
“We know we can rearrange the addends to add in any order when we find the value of a sum. When a number is written in expanded form the values are written in place value order.”
“How would we write 143 in expanded form?” (
\(100 + 40 + 3\)
)


aligns: no