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: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
problem_activity: Activity
This activity has two parts and serves two purposes. The first two problems aim to help students see the limits of using base-ten diagrams to add and subtract numbers and to think about choosing more efficient methods. The latter three questions prompt students to reason about addition and subtraction of numbers with more decimal places in the context of situations. Students need to determine the appropriate operations for the given situations, perform the calculations, and then relate their answers back to the contexts (MP2).
As students work, check that they remember to line up the decimal points, write zeros when needed, and label their answers with units of measurement (for the last three questions). Identify one student to present each question. For the last question, there are two likely methods for solving (as outlined in the solutions). If both methods arise, they should both be presented.
Launch
Ask students to read the first 2 problems and think about which method of reasoning might work best. Give students 5 minutes of quiet time to work on these problems. Observe which methods students choose and check their work for accuracy.
Afterwards, take a few minutes to discuss their answers and methods. Ask one or more of the following questions before moving onto the second half of the activity:
“Would base-ten diagrams work well for
\(318.8 - 94.63\)
? Why or why not?” (No, it is time consuming to draw and keep track of so many pieces. Yes, I can draw 3 large squares for 300, 1 rectangle for 10, 8 small squares for 8, etc., remove pieces in the amount of 94.63, and count what is left.)
“What challenges did you face in computing
\(0.02 - 0.0116\)
using a diagram?” (I needed to take away 116 ten-thousandths from 2 hundredths, so I had to exchange hundredths with ten-thousandths, which meant drawing or showing many smaller pieces.) “What about using vertical calculations?” (The first number is two digits shorter than the second, which meant some extra steps.)
“When might base-ten diagrams be a good choice for finding a difference (or sum)?” (When the numbers are simple or short so the diagrams do not take too long to draw.)
“When might an algorithm be a good choice for finding a difference (or sum)?” (For a difference like
\(6.739258 - 0.8536672\)
, it will be time consuming to draw pictures and label each place value. The algorithm takes some time as well but is more efficient than drawing pictures.)
After this discussion, have students work independently on the remaining problems. During this time, monitor and select individuals to share their work during the whole-class discussion.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check to make sure students have proper decimal point and place value unit alignment prior to calculating the value of each expression. Some students may benefit from access to graph paper to aide in proper alignment of decimals.
Supports accessibility for: Organization; Visual-spatial processing
Reading: MLR6 Three Reads.
Use this routine to support reading comprehension of the problem about Lin’s grandmother. After a shared reading of the question, ask students “what is this situation about?” (Lin’s grandmother ordered needles; the needles were different lengths). Use this read to clarify any unfamiliar language such as pharmacist or administer. After the second read, students list any quantities that can be counted or measured, without focusing on specific values (length of each type of needle, in inches). After the third read, invite students to discuss possible strategies to solve the problem. If needed, repeat this support with the remaining word problems. This helps students connect the language in the word problem and the reasoning needed to solve the problem.
Design Principle(s): Support sense-making
Student Facing
Find the value of each expression. Show your reasoning.
\(11.3 - 9.5\)
\(318.8 - 94.63\)
\(0.02 - 0.0116\)
Discuss with a partner:
Which method or methods did you use in the previous question? Why?
In what ways were your methods effective? Was there an expression for which your methods did not work as well as expected?
Lin’s grandmother ordered needles that were 0.3125 inches long to administer her medication, but the pharmacist sent her needles that were 0.6875 inches long. How much longer were these needles than the ones she ordered? Show your reasoning.
There is 0.162 liter of water in a 1-liter bottle. How much more water should be put in the bottle so it contains exactly 1 liter? Show your reasoning.
One micrometer is 1 millionth of a meter. A red blood cell is about 7.5 micrometers in diameter. A coarse grain of sand is about 70 micrometers in diameter. Find the difference between the two diameters in
meters
. Show your reasoning.
Student Response
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Anticipated Misconceptions
Students may benefit from using graph paper for calculations; they can place one digit in each box for proper decimal point placement and place value unit alignment. For the last question, students might confuse the amount of zeros in 1 millionth. It might be helpful to write 0.000001 on the board as reference.
Activity Synthesis
Discussions on the first 2 questions are outlined in the Launch section. Focus the discussion here on the last 3 questions, i.e., on how students interpreted the problems, determined which operation to perform, and found the sum or difference (including how they handled any regroupings). Select a few previously identified students to share their responses and reasoning.
If some students chose base-ten diagrams for the calculation, contrast its efficiency with that of numerical calculations. The numbers in these problems have enough decimal places that using base-ten representations (physical blocks, drawings, or digital representations) would be cumbersome, making a numerical calculation appealing.


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