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: 
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: 
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_neg_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.
problem_activity: Activity
The goal of this activity is for students to recognize rewriting a quadratic expression from standard form to vertex form essentially entails completing the square.
Students had quite a bit of experience completing the square, mostly in the context of solving equations, in which they know to add or subtract the same number from both sides of an equation to keep the equation true. Here students are dealing with expressions and need to be careful to keep each expression equivalent to the original. If they add a number to the expression, they need to remember to subtract the same number to keep the value of the expression unchanged.
Launch
Arrange students in groups of 2. Give students quiet time to think about the first two questions and then time to share their thinking with a partner.
Pause for a class discussion after the second question and invite one or more students to demonstrate their strategy. Highlight approaches that involve completing the square. If no students took that approach, display and discuss the following:
\(\displaystyle \begin {align} &x^2 - 6x + 2\\ &x^2 - 6x + 9 - 9 + 2 \\&(x - 3)^2 - 7 \end {align}\)
Then, ask students to proceed with the rest of the activity. Provide access to devices that can run Desmos or other graphing technology.
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students share their reasoning about how to convert standard form back into vertex form, scribe their thinking on a visible display. Use annotations and labels to show the structure of the steps using the form included in the launch. For example, labeling the first step as “standard form,” labeling +9 as “add nine to complete the square,” and drawing an arrow to the -9 and labeling it as “keeping it equivalent.” Continue to add in any key phrases students contributed that illustrate the steps.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
Here are two expressions in vertex form. Rewrite each expression in standard form. Show your reasoning.
\((x+5)^2+1\)
\((x-3)^2-7\)
Think about the steps you took, and about reversing them. Try converting one or both of the expressions in standard form back into vertex form. Explain how you go about converting the expressions.
Test your strategy by rewriting
\(x^2+10x+9\)
in vertex form.
Let’s check the expression you rewrote in vertex form.
Use graphing technology to graph both
\(x^2+10x+9\)
and your new expression. Does it appear that they define the same function?
If you convert your expression in vertex form back into standard form, do you get
\(x^2+10x+9\)
?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students may wonder why they need to both add and subtract a number from the standard form expression in order to complete the square. In previous lessons, students completed the square while solving an equation, often adding the same number to each side of the equal sign to maintain equality. Here, there is no equal sign. Emphasize that each move must generate an equivalent expression. Show a few expressions such as 5,
\(5+0\)
,
\(5-2+2\)
,
\(5+7-7\)
,
\(5+8\)
, and
\(5+8-3\)
. Ask them which ones are equal to 5. Next, ask them to write 2 more expressions that include the number 5 and also equal 5. Encourage students to notice that the sum of the numbers excluding 5 must be 0.
Activity Synthesis
Invite students to share their graph and response to the last question. Emphasize that while graphing is a quick way to check whether two expressions define the same function, it is not always reliable. If we make an algebraic error, the graph would almost certainly show it. But having two graphs that
appear to be
identical does not
prove
that two expressions are indeed equivalent. We can only be sure that two expressions define the same function by showing equivalence algebraically. For example, when students convert the expression in vertex form back to standard form, does it produce the original expression?
Make sure students understand that whatever operation is performed on an expression to complete the square, it should not change the value of the expression. Adding opposites (for example, 9 and -9, or -25 and 25), or adding and subtracting the same number, has the effect of adding 0, which keeps the original and the transformed expressions equivalent.


aligns: no