options: Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
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
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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
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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


correct_option_index: 0

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