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: Generalize place value understanding for multi-digit whole numbers.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

Use place value understanding and properties of operations to perform multi-digit arithmetic.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

Understand place value.

Extend the counting sequence.

Work with numbers 11–19 to gain foundations for place value.

Use place value understanding and properties of operations to add and subtract.

Perform operations with multi-digit whole numbers and with decimals to hundredths.

Understand the place value system.

Use place value understanding and properties of operations to perform multi-digit arithmetic. (A range of algorithms may be used.)
problem_activity: Narrative
The purpose of this activity is for students to round to the nearest tenth and hundredth. Students have rounded in earlier grades but this is the first time they round to tenths or hundredths. This is a direct extension of rounding to the nearest ten, hundred, thousand, and other whole number values. Locating the numbers on the number line will recall this earlier work.
The launch introduces the context of a doubloon, a major currency in Portugal and Spain in the seventeenth, eighteenth, and nineteenth centuries. Students round the weight of a doubloon to the nearest tenth and hundredth of a gram. In both cases, older doubloons are still heavier after rounding. Rounded to the nearest gram, however, they are the same. This is important from a practical perspective because it is easier to measure a weight to the nearest gram than it is to the nearest tenth of a gram or hundredth of a gram. It is also important because the numbers 6.9 and even 6.87 are not as complex as 6.867, and having fewer digits helps visualize the value more quickly.
MLR8 Discussion Supports.
During small-group discussion, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “I heard you say . . . .” Original speakers can agree or clarify for their partner.
Advances: Listening, Speaking
Launch
Groups of 2
Display the image.
“This is a doubloon. What do you notice? What do you wonder?” (I notice 1798. I wonder if that’s the year it was made. I wonder if people actually used these as money. I notice there are words in a foreign language. I wonder what they say. I wonder what language it is.)
“Doubloons were a major currency in Portugal and Spain in the seventeenth, eighteenth, and nineteenth centuries. Today, we are going to study the weights of doubloons.”
Activity
“Take a couple of minutes to work on the first problem.”
2 minutes: partner work time
“What does it mean to measure a weight to the nearest tenth of a gram?” (It means you find the closest tenth of a gram to the weight.)
“How did you decide which kind of doubloon was on the scale?” (The older ones are heavier and the tenth they are closest to is 6.9 not 6.8.)
“The weight of the doubloon is rounded to the nearest tenth of a gram.”
“Now complete the rest of the problems.”
6–8 minutes: partner work time
Student Facing
[IMG]
Until 1728, doubloons weighed 6.867 grams.
After 1728, they weighed 6.766 grams.
You have a scale that measures weight to the nearest tenth of a gram.
Was the doubloon on the scale made before or after 1728?
[IMG]
If you had a scale that measured to the nearest gram, explain why you would not be able to tell what year the doubloon was made based on the weight listed on the scale.
Which doubloons weigh more, the ones made before 1728 or the ones made after 1728? Explain or show your reasoning.
Show the weights of the doubloons on the number line.
[IMG]
Use the number lines to find which hundredth of a gram the doubloon weights are each closest to.
[IMG]
[IMG]
Student Response
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Activity Synthesis
Invite students to share their answers for rounding the weights of the doubloons.
“How did you round the weight of the older doubloon to the nearest tenth? What about the nearest hundredth?” (It was closer to 6.9 than to 6.8, so I rounded to 6.9 to the nearest tenth. It was closer to 6.87 than to 6.86 so I rounded to 6.87 to the nearest hundredth.)
“How did the number lines help you round the numbers?” (I could see on the number line which tenth or which hundredth the number was closest to and then knew to round it there.)


correct_option_index: 7