few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


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

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


Concept/skill: 
Solve quadratic equations in one variable.

Answer: 
Yes. This task shows one quadratic equation with 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: 
Construct a tangent line from a point outside a given circle to the circle.

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, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_neg_0

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standard_description: Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. (Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.) For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
problem_activity: Activity
This activity is an opportunity for students to practice rewriting expressions using the distributive property. It is a step up from the same type of work in grade 6 because arithmetic with signed numbers is required.
The row with
\(6a-2b\)
is designed to allow students to figure out how to factor by reasoning based on structure they already understand, instead of learning how to factor based on a procedure that a teacher demonstrates first. The rows with
\(k(4 - 17)\)
and
\(10a-13a\)
are designed to prepare students for combining like terms in the next lesson.
Launch
Draw students’ attention to the organizers that appear above the table, and tell them that these correspond to the first three rows in the table. Let them know that they are encouraged to draw more organizers like this for other rows as needed.
Arrange students in groups of 2. Instruct them to take turns writing an equivalent expression for each row. One partner writes the equivalent expression and explains their reasoning while the other listens. If the partner disagrees, they work to resolve the discrepancy before moving to the next row.
Action and Expression: Internalize Executive Functions.
Provide students with blank diagrams to organize their work for the factored and expanded expressions. Include example diagrams labeled “expanded” and “factored.”
Supports accessibility for: Language; Organization
Student Facing
In each row, write the equivalent expression. If you get stuck, use a diagram to organize your work. The first row is provided as an example. Diagrams are provided for the first three rows.
[IMG]
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "factored"}, {"_attributes": {"scope": "col"}, "_value": "expanded"}]}]}], "tbody": [{"tr": [{"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42054"}, "_value": "\\(\\text- 3(5 - 2y)\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42097"}, "_value": "\\(\\text- 15 + 6y\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42137"}, "_value": "\\(5(a-6)\\)"}]}, {}]}, {"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "4097"}, "_value": "\\(6a-2b\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42180"}, "_value": "\\(\\text- 4(2w-5z)\\)"}]}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42214"}, "_value": "\\(\\text- (2x-3y)\\)"}]}, {}]}, {"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42247"}, "_value": "\\(20x-10y+15z\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42283"}, "_value": "\\(k(4-17)\\)"}]}, {}]}, {"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "4113"}, "_value": "\\(10a-13a\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42333"}, "_value": "\\(\\text- 2x(3y-z)\\)"}]}, {}]}, {"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42375"}, "_value": "\\(ab-bc-3bd\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42413"}, "_value": "\\(\\text- x(3y-z+4w)\\)"}]}, {}]}]}]}]}
Student Response
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Student Facing
Are you ready for more?
Expand to create an equivalent expression that uses the fewest number of terms:
\(\left(\left(\left(\left(x\strut+1\right)\frac12\right)+1\right)\frac12\right)+1\)
. If we wrote a new expression following the same pattern so that there were 20 sets of parentheses, how could it be expanded into an equivalent expression that uses the fewest number of terms?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students are unsure how to proceed, remind them of tools and understandings they have seen recently that would be helpful. For example, “Draw an organizer and think about how the organizer represents terms in the expression.” Also, “Rewrite subtraction as adding the opposite.”
Activity Synthesis
Much of the discussion will take place in small groups. Display the correct equivalent expressions and work to resolve any discrepancies. Expanding the term
\(\text-(2x-3y)\)
may require particular care. One way to interpret it is to rewrite as
\(\text-1 \boldcdot (2x-3y).\)
If any confusion about handling subtraction arises, encourage students to employ the strategy of rewriting subtraction as adding the opposite.
To wrap up the activity, ask:
“Which rows did you and your partner disagree about? How did you resolve the disagreement?”
“Which rows are you the most unsure about?”
“Describe a process or procedure for taking a factored expression and writing its corresponding expanded expression.”
“Describe a process or procedure for taking an expanded expression and writing its corresponding factored expression.”
Writing: MLR3 Clarify, Critique, Correct.
Present an incorrect response for one of the expanded expressions in the table. For example, “An equivalent expression for
\(6a − 2b\)
is
\(4(a−b)\)
because
\(6−2\)
is 4 and
\((a−b)\)
is left on its own.” Prompt students to clarify any of the language and reasoning in the incorrect response and then to identify the error(s). Invite students to work with a partner to write a correct response. This helps students evaluate and improve on the written mathematical arguments of others.
Design Principle(s): Maximize meta-awareness


aligns: no