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?


Mathematical concept/skill options: 
A. Analyze proportional relationships and use them to solve real-world and mathematical problems.
B. Understand ratio concepts and use ratio reasoning to solve problems.


Thought: This problem focuses on the ratio between two ingredients. Students do not have to analyze proportional relationships in-depth, which is reserved for a higher grade level, but rather use ratios to find missing values.
Answer: B

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:
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?


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


Thought: This set of problems involves solving linear equations and pairs of linear equations.
Answer: L

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: Use functions to model relationships between quantities.

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

Construct and compare linear and exponential models and solve problems.

Analyze functions using different representations.

Define, evaluate, and compare functions.

Extend the domain of trigonometric functions using the unit circle.

Build new functions from existing functions.

Model periodic phenomena with trigonometric functions.

Build a function that models a relationship between two quantities.

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

Understand the concept of a function and use function notation.

Prove and apply trigonometric identities.
problem_activity: Activity
In this activity, students practice constructing an exponential function and evaluating it at a fractional input value as they translate between decades and years. They also think about how to express a percent increase over a fractional interval of input—specifically, how to express the annual growth rate of a population given the growth rate for one decade. They analyze and articulate why the growth rate for one year is
not
one tenth of that of one decade (MP3).
Launch
Tell students that, as of 2017, Nigeria is the seventh-most populous country in the world. It has grown at a fairly steady (and relatively high) rate since 1980.
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to orient students to the language of exponential functions in context. Ask students to keep their books or devices closed, and display only the image of the map and the problem statement, without revealing the questions that follow. Arrange students in groups of 2, and invite them to create a list of mathematical questions that could be asked about the situation.  Listen for and amplify questions about growth and decay factors, percent increase or decrease, fractional intervals, and comparisons of growth rates. Invite 2–3 groups to share their questions with the class.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
[IMG]
In 1990, Nigeria had a population of about 95.3 million. By 2000, there were about 122.4 million people, an increase of about 28.4%. During that decade, the population can be reasonably modeled by an exponential function.
Express the population of Nigeria
\(f(d)\)
, in millions of people,
\(d\)
decades since 1990.
Write an expression to represent the population of Nigeria in 1996.
A student said, “The population of Nigeria grew at a rate of 2.84% every year.”
Explain or show why the student’s statement is incorrect.
Find the correct annual growth rate. Explain or show your reasoning.
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
If students struggle to identify the input to their function where
\(d\)
is not an integer, remind them that there are 10 years in a decade. Then ask them how many decades is 6 years? What about one year?
Activity Synthesis
Focus the discussion on why the annual growth rate is not
\(\frac1{10}\)
of 28.4%. Consider asking students:
“Why might someone think the annual population growth rate is 2.84%?” (Because 28.4 divided by 10 is 2.84, and they assume that population was growing linearly, or by the same amount each year, rather than by the same factor each year.)
“How can we determine the growth factor for one year?” (We know that when the annual growth factor, say
\(b\)
, is raised to an exponent of 10, it must equal 1.284, or
\(b^{10} = 1.284\)
, so
\(b\)
must be
\(1.284^\frac{1}{10}\)
or
\(\sqrt[10]{1.284}\)
because raising that number to an exponent of 10 gives 1.284.)
“How do we express the factor as a growth rate? (
\(1.284^\frac{1}{10} \approx 1.0258\)
, so the growth rate is 2.58%.)
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students share their reasoning about the last question, scribe their thinking on a visible display. To illustrate why the growth factor for one year is not
\(\frac{1}{10}\)
of that of a decade, it may be helpful to return to a diagram that is configured like a double number line (especially if used in the last lesson). Use the double number line of years and decades to annotate and show the growth factor. When students explain that the growth factor for one decade is 1.284, draw an arrow across the decade line and label it
\(\boldcdot 1.284\)
. Then draw small arrows for the years labeled with
\(\boldcdot b\)
for each segment. Label the decade line with the corresponding equation. Encourage students to refer to the diagram when explaining why
\(b\)
is not one-tenth the growth rate of a decade. When students first explain the use of the fractional exponent to calculate the yearly population growth, label the line representing years with the corresponding equation.
Supports accessibility for: Visual-spatial processing; Conceptual processing


correct_option_index: 2