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: Understand solving equations as a process of reasoning and explain the reasoning.

Use polynomial identities to solve problems.

Write expressions in equivalent forms to solve problems.

Understand and apply properties of operations and the relationship between addition and subtraction.

Represent and analyze quantitative relationships between dependent and independent variables.

Interpret the structure of expressions.

Understand the connections between proportional relationships, lines, and linear equations.

Rewrite rational expressions.

Analyze patterns and relationships.

Use the four operations with whole numbers to solve problems.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Gain familiarity with factors and multiples.

Represent and solve problems involving addition and subtraction.

Understand properties of multiplication and the relationship between multiplication and division.

Perform arithmetic operations on polynomials.

Create equations that describe numbers or relationships

Represent and solve problems involving multiplication and division.

Reason about and solve one-variable equations and inequalities.

Write and interpret numerical expressions.

Work with radicals and integer exponents.

Use properties of operations to generate equivalent expressions.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Represent and solve equations and inequalities graphically.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Solve equations and inequalities in one variable.

Generate and analyze patterns.

Solve systems of equations.

Understand the relationship between zeros and factors of polynomials.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Multiply and divide within 100.

Work with addition and subtraction equations.

Work with equal groups of objects to gain foundations for multiplication.

Add and subtract within 20.
problem_activity: Activity
This activity represents the first time students solve a system of equations using algebraic methods. They first match systems of equations to their graphs and then calculate the solutions to each system. The purpose of matching is so students have a way to check that their algebraic solutions are correct, but not to shortcut the algebraic process since the graphs themselves do not include enough detail to accurately guess the coordinates of the solution.
Digital
Launch
Keep students in groups of 2. Give 2–3 minutes of quiet work time for the first problem and then ask students to pause their work. Select 1–2 students per figure to explain how they matched it to one of the systems of equations. For example, a student may identify the system matching Figure A as the only system with an equation that has negative slope. Give students 5–7 minutes of work time with their partner to complete the activity followed by a whole-class discussion. If students finish early and have not already done so on their own, ask them how they could check their solutions and encourage them to do so.
If using the digital activity, implement the lesson as indicated above. The only difference between the print and digital version is the digital lesson has an applet that will simulate the graphs so the students have another way of checking their solutions.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. Invite students to illustrate connections between slopes and
\(y\)
-intercepts of each line to the corresponding parts of each equation using the same color.
Supports accessibility for: Visual-spatial processing
Conversing: MLR8 Discussion Supports.
Use this routine to support small-group discussion as students describe the reasons for their matches. Arrange students in groups of 2. Invite Partner A to begin with this sentence frame: “Figure ____ matches with the system of equations ____, because ____.” Invite the listener, Partner B, to press for additional details referring to specific features of the graphs (e.g. positive slope, negative y-intercept, coordinates of the intersection point, etc). Students should switch roles for each figure. This will help students justify how features of the graph can be used to identify matching equations.
Design Principle(s): Support sense-making; Cultivate conversation
Student Facing
Here are three systems of equations graphed on a coordinate plane:
[IMG]
Match each figure to one of the systems of equations shown here.
\(\begin{cases} y=3x+5\\ y=\text- 2x+20 \end{cases}\)
\(\begin{cases} y=2x-10\\ y=4x-1 \end{cases}\)
\(\begin{cases} y=0.5x+12\\ y=2x+27 \end{cases}\)
Find the solution to each system and then check that your solution is reasonable on the graph.
Notice that the sliders set the values of the coefficient and the constant term in each equation.
Change the sliders to the values of the coefficient and the constant term in the next pair of equations.
Click on the spot where the lines intersect and a labeled point should appear.
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
Keep students in groups of 2. Give 2–3 minutes of quiet work time for the first problem and then ask students to pause their work. Select 1–2 students per figure to explain how they matched it to one of the systems of equations. For example, a student may identify the system matching Figure A as the only system with an equation that has negative slope. Give students 5–7 minutes of work time with their partner to complete the activity followed by a whole-class discussion. If students finish early and have not already done so on their own, ask them how they could check their solutions and encourage them to do so.
If using the digital activity, implement the lesson as indicated above. The only difference between the print and digital version is the digital lesson has an applet that will simulate the graphs so the students have another way of checking their solutions.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. Invite students to illustrate connections between slopes and
\(y\)
-intercepts of each line to the corresponding parts of each equation using the same color.
Supports accessibility for: Visual-spatial processing
Conversing: MLR8 Discussion Supports.
Use this routine to support small-group discussion as students describe the reasons for their matches. Arrange students in groups of 2. Invite Partner A to begin with this sentence frame: “Figure ____ matches with the system of equations ____, because ____.” Invite the listener, Partner B, to press for additional details referring to specific features of the graphs (e.g. positive slope, negative y-intercept, coordinates of the intersection point, etc). Students should switch roles for each figure. This will help students justify how features of the graph can be used to identify matching equations.
Design Principle(s): Support sense-making; Cultivate conversation
Student Facing
Here are three
systems of equations
graphed on a coordinate plane:
[IMG]
Match each figure to one of the systems of equations shown here.
\(\begin{cases} y=3x+5\\ y=\text-2x+20 \end{cases}\)
\(\begin{cases} y=2x-10\\ y=4x-1 \end{cases}\)
\(\begin{cases} y=0.5x+12\\ y=2x+27 \end{cases}\)
Find the solution to each system and check that your solution is reasonable based on the graph.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The goal of this discussion is to deliberately connect the current topic of systems of equations to the previous topic of solving equations with variables on both sides. For each of the following questions, give students 30 seconds of quiet think time and then invite students per question to explain their answer. The final question looks ahead to the following activity.
“Do you need to see the graphs of the equations in a system in order to solve the system?" (No, but the graphs made me feel more confident that my answer was correct.)
“How do you know your solution doesn’t contain any errors?" (I know my solution does not have errors because I substituted my values for
\(x\)
and
\(y\)
into the equations and they made both equations true.)
“How does solving systems of equations compare to solving equations with variables on both sides like we did in earlier lessons?" (They are very similar, only with a system of equations you are finding an
\(x\)
and a
\(y\)
to make both equations true and not just an
\(x\)
to make one equation true.)
“When you solved equations with variables on both sides, some had one solution, some had no solutions, and some had infinite solutions. Do you think systems of equations can have no solutions or infinite solutions?" (Yes. We have seen some graphs of parallel lines where there were no solutions and some graphs of lines that are on top of one another where there are infinite solutions.)


correct_option_index: 29