few_shots: 
Example 1:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Concept/skill: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

Example 3:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Concept/skill: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

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_000972_neg_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.
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