few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


Answer: 
Yes. An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation.

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: 
Construct a tangent line from a point outside a given circle to the circle.

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: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_neg_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
This activity develops students’ understanding of the vertical method of recording partial quotients and their ability to use it to perform division. One of the quotients here involves a remainder, prompting students to interpret it. Students are reminded that the term “remainder” is used to describe “leftovers” when dividing.
MLR2 Collect and Display.
Synthesis: Direct attention to words collected and displayed from the previous lessons. Invite students to borrow language from the display as needed, and update it throughout the lesson.
Advances: Conversing, Reading
Launch
Groups of 2
Activity
3 minutes: independent work time on the first 2 problems.
Pause after problem 2 to discuss students’ responses.
Display the different ways that students decompose 389 to divide it by 7.
“Most other calculations we’ve seen so far end with a 0, but this one ends with a 4. What does the 4 tell us?” (We cannot make a group of 7 with 4 leftover. 389 is not a multiple of 7, and there are leftovers.)
“When we divide and end up with leftovers we call them remainders, because they represent what is remaining after we divide into equal groups.”
Display:
\(389 = 7 \times 55 + 4\)
“How does this equation show that
\(389 \div 7\)
has a remainder?” (It shows that 389 is not a multiple of 7. It also shows that 7 and 55 make a factor pair for 385, and 389 is 4 more than that.)
3 minutes: independent work time on the last 2 problems.
As students work on the last two problems monitor for students who:
start with the largest multiple of 3 and 10 within 702 that they can think of to decompose the dividend (690, 600).
use the fewest steps to find the quotient.
Student Facing
Jada used partial quotients to find out how many groups of 7 are in 389.
Analyze Jada’s steps in the algorithm.
[IMG]
Look at the three numbers above 389. What do they represent?
Look at the three subtractions below 389. What do they represent?
What is another way you can decompose 389 to divide by 7?
Is 389 a multiple of 7? Explain your reasoning.
Use an algorithm that uses partial quotients to find out how many groups of 3 are in 702.
Is 702 a multiple of 3? Explain your reasoning.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Activity Synthesis
Display responses that show variation in how students decomposed 702 in problem 3.
Highlight that there are countless ways of using partial quotients to divide a number, but it may be more efficient to divide larger groups than smaller groups (for example, removing 350 once, as opposed to removing 70 five times. Removing smaller groups is just as valid, however.)
To help students see the relationship between corresponding partial products and partial quotients, consider illustrating them visually. Some examples:
[IMG]
[IMG]
[IMG]


aligns: no