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 numbers to the system of rational numbers.

Represent and model with vector quantities.

Use complex numbers in polynomial identities and equations.

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

Perform operations on vectors.

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

Perform arithmetic operations with complex numbers.

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

Represent complex numbers and their operations on the complex plane.

Use properties of rational and irrational numbers.

Perform operations on matrices and use matrices in applications.

Extend the properties of exponents to rational exponents.

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

Reason quantitatively and use units to solve problems.
problem_activity: Activity
In this task, students explore division situations (in the context of baking cookies) where the number of groups and a total amount are given, but the size of 1 group is unknown. They write multiplication equations in which the missing factor answers the question “how much in each group?” instead of “how many groups?”
Students continue to use equations, diagrams, and the connection between multiplication and division in their reasoning. No grid is provided here, but allow students to use graph paper to support their diagram drawing if desired or needed.
As students work, monitor how they start their diagrams and which quantity they represent first. If they are not quite sure how to show a particular quantity, ask them to refer to earlier diagrams and notice how the number of groups, the size of each group, and the total amount were represented on a single diagram. Select students whose diagrams would be instructive to others to share later.
Launch
Arrange students in groups of 3–4. Give groups 5–6 minutes of quiet work time and 2 minutes to share their responses in their group. Provide access to geometry toolkits (especially graph paper and colored pencils).
Student Facing
To make 5 batches of cookies, 10 cups of flour are required. Consider the question: How many cups of flour does each batch require?
We can write equations and draw a diagram to represent this situation.
\(\displaystyle 5\, \boldcdot {?} = 10\)
\(\displaystyle 10 \div 5 = {?}\)
[IMG]
This helps us see that each batch requires 2 cups of flour.
For each question, write a multiplication equation and a division equation, draw a diagram, and find the answer.
To make 4 batches of cupcakes, it takes 6 cups of flour. How many cups of flour are needed for 1 batch?
To make
\(\frac12\)
batch of rolls, it takes
\(\frac54\)
cups of flour. How many cups of flour are needed for 1 batch?
Two cups of flour make
\(\frac23\)
batch of bread. How many cups of flour make 1 batch?
Student Response
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Anticipated Misconceptions
When writing multiplication equations, some students might simply use the smaller number as a factor and the larger number as the product. Or, when writing division equations, they might simply divide the larger number by the smaller number without attending to what the numbers mean. Encourage them to think about what the numbers represent. To help them see the connection between verbal statements and mathematical equations, support their reasoning with simple examples, e.g., “To make 2 batches of cookies, we need 6 cups of flour. How many cups are needed for 1 batch?” This can be interpreted as “2 groups of what equals 6?” and then
\(2 \boldcdot {?} = 6\)
.
Activity Synthesis
Select previously identified students to share their diagrams and reasoning, or display the ones here for all to see. If other students used alternative approaches, invite a couple of them to share.
[IMG]
[IMG]
To highlight some key ideas in this lesson and in the unit, discuss:
“How are these diagrams like those in previous lessons? How are they different?” (Alike: We can use them to show a total amount, the number of groups, the size of 1 group. Different: We don't know the value of 1 group here. In past diagrams, we were given the size of 1 group, but needed to find the number of groups.)
“How are these equations and the ones in previous lessons alike and different?” (Alike: The multiplication equations have one unknown factor. The division equations have the same setup. Different: The unknown factor here represents a different quantity than in past lessons.)
“How are the diagrams for the second and third questions different from the one in the first question?” (In the first one, the given amount is greater than the size of 1 batch. In the other two, the given amount is less than the size of 1 batch or is a fraction of a batch.)​
Representation: Internalize Comprehension.
Use color and annotations to illustrate connections between representations. As students share their diagrams and reasoning, use color and annotations to scribe their thinking on a display of each problem so that it is visible for all students.
Supports accessibility for: Visual-spatial processing; Conceptual processing


correct_option_index: 3