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: 
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 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_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
problem_activity: Optional activity
This is the second of two optional activities in which students explore pH ratings as an application of logarithmic functions. In the first activity, students noticed a pattern in the concentration of positive hydrogen ions to pH ratings and described it informally. In this activity, they represent the relationship with an equation, test it, and use it to solve problems.
Monitor for how students approach the third question. Some students may pay attention only to the exponent -11 in
\(5.6 \times 10^{\text-11}\)
and disregard the 5.6. Identify students who recognize that the pH scale is the base 10 logarithm of whatever number or expression represents the hydrogen ion concentration, not just the part expressed in the power of 10.
Launch
Ask students to look at the table in this activity and the one in the previous activity. Briefly discuss how they are alike and different, and whether the table still shows the same relationship between hydrogen ion concentration and pH, as in the previous task. Make sure students see that the relationship hasn’t changed and that the hydrogen ion concentrations can be written as powers of 10.
Representation: Internalize Comprehension.
Use color coding and annotations to highlight connections between representations in a problem. For example, invite students to write the scientific notation form of the number next to the number itself on the table. Students may choose to color code or draw arrows to brainstorm their interpretation of the connection between this column and the pH output. For example, students may draw an arrow from the exponent to the pH and write “opposite.”
Supports accessibility for: Visual-spatial processing
Student Facing
This table shows the relationship between hydrogen ion concentrations and pH ratings (acidity) for different substances.
|   substance    | hydrogen ion concentration                   (moles per liter) | pH |
|----------------|----------------------------------------------------------------|----|
| mild detergent |                          0.0000000001                          | 10 |
|   toothpaste   |                          0.000000001                           | 9  |
|  baking soda   |                           0.00000001                           | 8  |
|     blood      |                           0.0000001                            | 7  |
|      milk      |                            0.000001                            | 6  |
|     banana     |                            0.00001                             | 5  |
|     tomato     |                             0.0001                             | 4  |
|     apple      |                             0.001                              | 3  |
|     lemon      |                              0.01                              | 2  |
Write an equation to represent the pH rating,
\(p\)
, in terms of the hydrogen ion concentration
\(h\)
, in moles per liter.
Test your equation by using the hydrogen ion concentration of a substance from the table as the input. Does it produce the right pH rating as the output? If not, revise your equation and test it again.
Magnesium hydroxide (also called “milk of magnesia”) is a medication used to treat stomach indigestion. It has a hydrogen concentration
\(5.6 \times 10^{\text-11}\)
mole per liter. Estimate a pH rating for magnesium hydroxide. Explain or show your reasoning.
As shown in the table, apple has a pH of 3 and milk has a pH of 6. How many times more acidic is the apple than milk?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The graph shows points representing the hydrogen ion concentration, in moles per liter, and pH ratings of the different substances you saw earlier.
[IMG]
Which point represents baking soda? Which represents banana? How can you tell?
Vinegar has a pH of 2.4. Where on the graph would a point that represents vinegar be plotted?
Why do you think the graph appears the way it does, with a group of points stacked up along the vertical axis?
How is it like and unlike other graphs of logarithmic functions you have seen so far?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may struggle writing an equation that relates the hydrogen ion concentration to the pH. Refer them back to the previous activity where they noticed a relationship by writing the hydrogen ion concentration using an exponential expression rather than a decimal.
Activity Synthesis
Invite students to share how they constructed the equation in the first question. Ensure that they see pH rating as a negative base 10 logarithm function.
Focus the discussion on the third question. Help students see that the pH scale of milk of magnesia is
\(\text- \log 5.6 \times 10^{\text-11}\)
. This log value can be computed with a calculator, but it can also be estimated. For example, we know that
\(\text- \log 10^{\text-11}\)
is 11 and
\(\text- \log 10^{\text-10}\)
is 10, and that a hydrogen concentration in the amount of the latter is 10 times that of the former. Because the hydrogen ion concentration of milk of magnesia is 5.6 times, not quite 10 times,
\(10^{\text-11}\)
, its pH is between 10 and 11.
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. After each student shares their response to the first question, provide the class with the following sentence frames to help them respond: "I agree because . . .” or "I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. For example, a statement such as, “The equation is a log” can be restated as a question such as, “Does the equation showing the relationship between hydrogen ion concentrations and pH ratings represent a logarithmic function?” This will help students solidify their understanding that pH rating is a negative base 10 logarithm function.
Design Principle(s): Support sense-making


aligns: no