few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

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
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Mathematical concept/skill options: 
A. Prove and apply trigonometric identities.
B. Model periodic phenomena with trigonometric functions.
C. Interpret functions that arise in applications in terms of the context.
D. Build a function that models a relationship between two quantities.
E. Define, evaluate, and compare functions.
F. Use functions to model relationships between quantities.
G. Understand the concept of a function and use function notation.
H. Analyze functions using different representations.
I. Extend the domain of trigonometric functions using the unit circle.
J. Interpret expressions for functions in terms of the situation they model.
K. Build new functions from existing functions.
L. Construct and compare linear and exponential models and solve problems.


Thought: This problem applies trigonometric identities, including the angle addition formula for cosine.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Define, evaluate, and compare functions.

Build new functions from existing functions.

Extend the domain of trigonometric functions using the unit circle.

Model periodic phenomena with trigonometric functions.

Interpret functions that arise in applications in terms of the context.

Construct and compare linear and exponential models and solve problems.

Prove and apply trigonometric identities.

Use functions to model relationships between quantities.

Understand the concept of a function and use function notation.

Analyze functions using different representations.

Interpret expressions for functions in terms of the situation they model.

Build a function that models a relationship between two quantities.
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
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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


correct_option_index: 4

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