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: Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Apply and extend previous understandings of numbers to the system of rational numbers.

Perform arithmetic operations with complex numbers.

Perform operations on vectors.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Use complex numbers in polynomial identities and equations.

Reason quantitatively and use units to solve problems.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Compute fluently with multi-digit numbers and find common factors and multiples.

Use properties of rational and irrational numbers.

Perform operations on matrices and use matrices in applications.

Extend the properties of exponents to rational exponents.

Represent complex numbers and their operations on the complex plane.

Represent and model with vector quantities.
problem_activity: Activity
The purpose of this activity is to see how to tell, from the equation, whether the sum will be positive, negative, or 0, without having to draw a number line diagram every time.
In this activity, students return to the context of height and depth to continue making sense of adding signed numbers. They use the structure of number line diagrams (MP7) to examine a variety of situations, which include starting in the positives, starting in the negatives, moving away from 0, moving towards 0, crossing over 0, and ending exactly on 0. From this variety of situations, students see that the lengths of the arrows in the number line diagrams give important information about the situation. This opens up the discussion of comparing the magnitude of the addends. When the addends have opposite signs, the longer arrow (the number with the larger magnitude) determines the sign of the sum.
\(400 + (\text-150) = 250\)
and
\(\text-200 + 150 = \text-50\)
.
This activity purposefully gives many problems to figure out, but only asks students to draw a diagram for 3 rows. Watch for students who think they need to draw a number line for every problem. Encourage them to look for and generalize from regularity in the problems (MP8) so that they don't need to always draw a diagram.
For students using the digital version of the materials, be sure they do not use the applet to represent and answer every question and bypass good thinking. Encourage students to figure out an answer for each question first, and then using the applet to check their work.
In preparation for dealing with subtraction of signed numbers, some of the problems involve determining how much the temperature changed, given the initial and final temperatures.
Digital
Launch
Explain that a
mountaineer
is someone who climbs mountains, and a
spelunker
is someone who explores caves and caverns. Arrange students in groups of 2. Give students 3 minute of quiet work time followed by partner discussion after the first question. Then have students work on the second question and follow with whole-class discussion.
Representation: Internalize Comprehension.
Represent the same information through different modalities by using physical objects to represent abstract concepts. Create an interactive display that allows students to experience physically moving an object (mountaineer) up and down a number line. Highlight connections between the act of physically moving the object up and down with arrows on a number line diagram.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
Explore the applet and then answer the questions.
A mountaineer is climbing on a cliff. She is 200 feet above the ground. If she climbs up, this will be a positive change. If she climbs down, this will be a negative change.
[IMG]
Complete the table.
|   | starting elevation                                   (feet) | change                                   (feet) | final elevation                                   (feet) |
|---|-------------------------------------------------------------|-------------------------------------------------|----------------------------------------------------------|
| A |                                                             |                                                 |                                                          |
| B |                                                             |                                                 |                                                          |
| C |                                                             |                                                 |                                                          |
| D |                                                             |                                                 |                                                          |
Write an addition equation and draw a number line diagram for B. Include the starting elevation, change, and final elevation in your diagram.
A spelunker is down in a cave next to the cliff. If she climbs down deeper into the cave, this will be a negative change. If she climbs up, whether inside the cave or out of the cave and up the cliff, this will be a positive change.
Complete the table.
|   | starting elevation                                   (feet) | change                                   (feet) | final elevation                                   (feet) |
|---|-------------------------------------------------------------|-------------------------------------------------|----------------------------------------------------------|
| A |                                                             |                                                 |                                                          |
| B |                                                             |                                                 |                                                          |
| C |                                                             |                                                 |                                                          |
| D |                                                             |                                                 |                                                          |
| E |                                                             |                                                 |                                                          |
Write an addition equation and draw a number line diagram for C and D. Include the starting elevation, change, and final elevation in your diagram.
​What does the expression
\(\text-45 + 60\)
tell us about the spelunker? What does the value of the expression tell us?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Explain that a
mountaineer
is someone who climbs mountains, and a
spelunker
is someone who explores caves and caverns. Arrange students in groups of 2. Give students 3 minute of quiet work time followed by partner discussion after the first question. Then have students work on the second question and follow with whole-class discussion.
Representation: Internalize Comprehension.
Represent the same information through different modalities by using physical objects to represent abstract concepts. Create an interactive display that allows students to experience physically moving an object (mountaineer) up and down a number line. Highlight connections between the act of physically moving the object up and down with arrows on a number line diagram.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
A mountaineer is climbing on a cliff. She is 400 feet above the ground. If she climbs up, this will be a positive change. If she climbs down, this will be a negative change.
Complete the table.
|   | starting                                         elevation                                         (feet) | change                                         (feet) | final                                         elevation                                         (feet) |
|---|-----------------------------------------------------------------------------------------------------------|-------------------------------------------------------|--------------------------------------------------------------------------------------------------------|
| A |                                                                                                           |                                                       |                                                                                                        |
| B |                                                                                                           |                                                       |                                                                                                        |
| C |                                                                                                           |                                                       |                                                                                                        |
| D |                                                                                                           |                                                       |                                                                                                        |
[IMG]
Write an addition equation and draw a number line diagram for B. Include the starting elevation, change, and final elevation in your diagram.
[IMG]
​
A spelunker is down in a cave next to the cliff. If she climbs down deeper into the cave, this will be a negative change. If she climbs up, whether inside the cave or out of the cave and up the cliff, this will be a positive change.
Complete the table.
|   | starting elevation                                   (feet) | change                                   (feet) | final elevation                                   (feet) |
|---|-------------------------------------------------------------|-------------------------------------------------|----------------------------------------------------------|
| A |                                                             |                                                 |                                                          |
| B |                                                             |                                                 |                                                          |
| C |                                                             |                                                 |                                                          |
| D |                                                             |                                                 |                                                          |
| E |                                                             |                                                 |                                                          |
Write an addition equation and draw a number line diagram for C and D. Include the starting elevation, change, and final elevation in your diagram.
[IMG]
​
[IMG]
​
What does the expression
\(\text-75 + 100\)
tell us about the spelunker? What does the value of the expression tell us?
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
Some students may struggle with working backwards to fill in the change when given the final elevation. Ask them "What do you do to 400 to get to 50?" for example.
Activity Synthesis
The most important thing for students to get out of this activity is how to tell from the equation whether the sum will be positive, negative, or 0, without having to draw a number line diagram every time.
Write all of the equations where the two addends have the same sign. The ask, “What happens when the two addends have the same sign?”
Write all of the equations where the two addends have the opposite sign. The ask, “What happens when the two addends have opposite signs and . . .
the number with the larger magnitude is positive?”
the number with the larger magnitude is negative?”
Ask, “How can you tell when the sum will be zero?”
Speaking, Listening, Writing: MLR1 Stronger and Clearer Each Time.
To provide students with an opportunity to generalize about the sum of two addends, ask students to draft an initial response to the question “How can you tell from the equation whether the sum will be positive, negative, or 0?”. Ask students to meet with 2–3 partners for feedback. Provide students with prompts for feedback that will help each other strengthen their ideas and clarify their language (e.g., “What if the signs of both numbers are the same/different?” or “How do you know if the sum will be positive/negative?”, etc.). Students can borrow ideas and language from each partner to refine their explanation. This will help students to use mathematical language to generalize about the sums of rational numbers.
Design Principle(s): Optimize output (for generalization)


correct_option_index: 4