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: Find arc lengths and areas of sectors of circles.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Explain volume formulas and use them to solve problems.

Define trigonometric ratios and solve problems involving right triangles.

Understand congruence in terms of rigid motions.

Understand and apply theorems about circles.

Translate between the geometric description and the equation for a conic section.

Analyze, compare, create, and compose shapes.

Visualize relationships between two-dimensional and three-dimensional objects.

Apply geometric concepts in modeling situations.

Prove geometric theorems.

Understand and apply the Pythagorean Theorem.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Use coordinates to prove simple geometric theorems algebraically.

Apply trigonometry to general triangles.

Experiment with transformations in the plane.

Solve real-world and mathematical problems involving area, surface area, and volume.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Classify two-dimensional figures into categories based on their properties.

Understand similarity in terms of similarity transformations.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Make geometric constructions.

Prove theorems involving similarity.

Reason with shapes and their attributes.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
problem_activity: Activity
This activity informally introduces reflections, which appear in addition to some translations and rotations (that were introduced informally in the previous lesson). Students are given a 6-frame cartoon showing the change in position of a polygon. As in the previous lesson, they describe the moves, but this time there are reflections, which may seem impossible as physical moves unless you allow the shape to leave the plane. Students identify the new moves and try to describe them.
After the end of this activity, the three basic moves have been introduced and the next activity will introduce their names (translations, rotations, and reflections).
Launch
Keep students in the same groups, and maintain access to geometry toolkits. Give students 5 minutes of quiet work time, and then invite them to share their responses with their group. Follow with a whole-class discussion. Tell students that they will be describing moves as they did in the previous lesson, but this time there is a new move to look out for. Recall the words the class used to describe slides and turns.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. During the launch, take time to review terms that students will need to access for this activity. Invite students to suggest language or diagrams to include that will support their understanding of moving, sliding, and turning.
Supports accessibility for: Conceptual processing; Language
Student Facing
Here is another set of dance moves.
[IMG]
Describe each move or say if it is a new move.
Frame 1 to Frame 2.
Frame 2 to Frame 3.
Frame 3 to Frame 4.
Frame 4 to Frame 5.
Frame 5 to Frame 6.
How would you describe the new move?
Student Response
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Anticipated Misconceptions
Students may see a reflection as a translation especially since the figures are not on the same frame. Ask these students to trace Frame 2 on tracing paper. Is there any way to turn it into Frame 3 by sliding it? What do they have to do to turn it into Frame 3? (They have to flip the tracing paper over, so, this is a new kind of move.)
In describing reflections, students may confuse the terms horizontal and vertical. Consider posting the terms horizontal and vertical with examples in your room.
Activity Synthesis
The purpose of this discussion is an initial understanding that there is a third type of move that is fundamentally different from the moves encountered in the previous lesson, because it reverses directions. Some possible discussion questions to help them identify these are:
“How is the motion from panel 2 to panel 3 different than the ones we discussed yesterday?”
“Is there anywhere else that happens in this cartoon?”
“What features of the image help us to see that this move is happening?”
To help answer these questions, tell students to pay attention to the direction that the “beak” of the polygon is pointing, left or right. Draw a dotted vertical line in the middle of Frame 2, and say, “Here is a mirror. The polygon in Frame 3 is what the polygon in Frame 2 sees when it looks in the mirror.”
[IMG]
Demonstrate using tracing paper or transparencies to show they are mirror images. Then ask students if there are any other mirror lines in other frames. For the second reflection, from Frame 5 to Frame 6, point out that the mirror line is now a horizontal line: in Frame 5 the beak is pointing down, and in Frame 6 the beak is pointing up, with the head on the right of the body in both cases. Contrast this with a rotation through 180º, which would put the head on the left of the body. Demonstrate with tracing paper or transparencies.


correct_option_index: 17