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 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.

Use place value understanding and properties of operations to perform multi-digit arithmetic. (A range of algorithms may be used.)

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

Understand the place value system.

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

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.

Extend the counting sequence.

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

Understand place value.
problem_activity: Narrative
The purpose of this activity is for students to solve multi-step distance problems using centimeters, meters, and kilometers. This gives students an opportunity to think about which units are most helpful for communicating a distance (MP6). When the distance is short, like the length of a single footstep, centimeters or meters both work well. For a longer distance, like the distance a person walks in a day, it is reasonable to use meters or kilometers, but the number of centimeters is very large and more difficult to visualize.
To add movement or make this activity interactive, consider providing groups of 2 or 4 with a centimeter ruler or meter stick to measure their or a classmate’s step before working on the task. While students could use the measurements of their own steps to complete the table, the arithmetic may be more complex and, as a result, it may be harder to observe patterns.
Engagement: Provide Access by Recruiting Interest.
Optimize meaning and value. Invite students to share the places they could walk in the classroom, at home, in the community, and so on, that would be equivalent to the distances presented in the chart.
Supports accessibility for: Conceptual Processing; Visual-Spatial Processing
Required Materials
Materials to Gather
Metersticks
Launch
Groups of 2 or 4
“If someone wants to measure the length of one of their footsteps, which unit do you think they should use? Millimeters, centimeters, meters, or kilometers?”
Highlight that millimeters are too small and kilometers are too large. Both centimeters and meters make sense and the first part of this activity will use those two units of measure.
“There are 100 centimeters in a meter. About how many steps are in 1 meter?” (2 or 3)
“There are 1,000 meters in a kilometer. About how many steps are in a kilometer?” (2,000 or 3,000)
Give students access to a meter stick.
Activity
5 minutes: independent work time
5 minutes: small-group work time
Monitor for students who use the following strategies when determining how many kilometers Lin walks during the day:
multiply 50 centimeters by 8,500 steps to determine the distance in centimeters that Lin walked and divide 425,000 by 100,000
multiply 0.5 meters by 8,500 steps to determine the distance in meters that Lin walked and then divide 4,250 by 1,000
Student Facing
Lin has a watch that counts the number of steps she takes during the day and displays those steps in centimeters, meters, or kilometers.
Here is a list of activities Lin did on Monday. Next to each activity, write whether it would make sense to display the distance in cm, m, or km.
walked to her friend’s desk
walked to the front of the classroom
walked from her classroom to the bus
ran twice around the playground
The table shows the amount of steps Lin’s watch displayed for each activity. If each of her steps is 50 centimeters, how many centimeters and meters did Lin walk for each activity?
|               activity               | number of steps | distance (cm) | distance (m) |
|--------------------------------------|-----------------|---------------|--------------|
|     walked to her friend’s desk      |        5        |               |              |
| walked to the front of the classroom |       12        |               |              |
| walked from her classroom to the bus |       250       |               |              |
|   ran twice around the playground    |      1,000      |               |              |
At the end of the day, Lin’s watch displayed 8,500 steps. Would it make sense for her watch to record the distance in centimeters, meters, or kilometers? Why?
How many kilometers did Lin walk that day?
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Student Response
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Activity Synthesis
“How did you determine how many kilometers Lin walked during the day?”
Ask previously selected students to share their solutions.
Display student work or write these equations for all to see:
\(8,\!500 \times 50= 425,\!000\)
\(8,\!500 \times 0.5 = 4,\!250\)
“How does each of these equations represent the situation?” (
\(8,\!500 \times 50= 425,\!000\)
represents 8,500 steps that are each 50 centimeters long so that would be a total of 425,000 centimeters.
\(8,\!500 \times 0.5 = 4,\!250\)
represents 8,500 steps that are each 0.5 meter long so that would be a total of 4,250 meters.)
“What is the same about these equations?” (They have the same number of steps. They are both multiplication equations.)
“What is different about these equations?” (One multiplies the number of steps by 50 centimeters and one multiplies the number of steps by 0.5 meter. The products are different. 4,250 is 100 times smaller than 425,000 because it represents meters instead of centimeters.)
Display:
\(425,\!000 \div 100,\!000= 4.25\)
\(4,\!250 \div 1,\!000 = 4.25\)
“How does each of these equations represent the situation?” (They both represent the number of kilometers that Lin walked. 425,000 is the distance she walked in centimeters so if you divide it by 100,000, you get the number of kilometers she walked. 4,250 is the distance in meters that she walked so we only have to divide by 1,000 to figure out how many kilometers she walked.)
Display: 4.25 km, 4,250 m, 425,000 cm
“Which of these do you think best communicates how far Lin walked this day?” (4.25 kilometers because I can picture how long a kilometer is.)


correct_option_index: 3