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.


Your response:
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.


Your response:
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.


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

Use coordinates to prove simple geometric theorems algebraically.

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

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

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

Understand and apply the Pythagorean Theorem.

Apply geometric concepts in modeling situations.

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

Reason with shapes and their attributes.

Apply trigonometry to general triangles.

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

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

Find arc lengths and areas of sectors of circles.

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

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Make geometric constructions.

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

Explain volume formulas and use them to solve problems.

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

Prove theorems involving similarity.

Understand similarity in terms of similarity transformations.

Analyze, compare, create, and compose shapes.

Understand and apply theorems about circles.

Understand congruence in terms of rigid motions.

Define trigonometric ratios and solve problems involving right triangles.

Prove geometric theorems.

Experiment with transformations in the plane.
problem_activity: Activity
The goal of this activity is for students to use the definition of similarity, and to practice applying some of the skills they learned when studying congruent figures to similar figures. Students practice writing a sequence of rigid motions and dilations to take one figure onto another, write accurate similarity statements about the figures, and reason about what must be true about corresponding side lengths.
Launch
Ask students to add this definition to their reference charts as you add it to the class reference chart:
One figure is
similar
to another if there is a sequence of rigid motions and dilations that takes the first figure so that it fits exactly over the second. (Definition)
Translation and dilation takes
\(\triangle ABC\)
onto
\(\triangle FDE\)
so
\(\triangle ABC \sim \triangle FDE\)
[IMG]
Tell students
\(\triangle ABC \sim \triangle FDE\)
is a similarity statement, and we read
\(\sim\)
as “is similar to.”
Arrange students in groups of 2. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their written responses for the sequence of transformations that takes one triangle to the other. Give students time to meet with 2–3 partners to share and get feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “What is the directed line segment of the translation?” and “What is the center and scale factor of the dilation?” Invite students to go back and revise or refine their written explanation based on the feedback from peers. This will help students use precise language to describe the sequence of transformations that takes one triangle to the other.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Representation: Internalize Comprehension.
Activate or supply background knowledge about symbolic notation. Provide students with access to explanations of symbols used in the caption for parallel and perpendicular, and connect to the right angle symbol in the image which shows that two segments are perpendicular.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Are the triangles
similar
?
\(\overline{AB}\parallel\overline{QR},\overline{AB}\perp\overline{AE},\overline{QR}\perp\overline{QT}\)
[IMG]
Write a sequence of transformations (dilation, translation, rotation, reflection) to take one triangle to the other.
Write a similarity statement about the 2 figures, and explain how you know they are similar.
Compare your statement with your partner’s statement. Is there more than one correct way to write a similarity statement? Is there a wrong way to write a similarity statement?
Student Response
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for free access to Student Response.
Anticipated Misconceptions
If students struggle to define sequences of rigid motions and dilations, review an example from the card sort.
Activity Synthesis
Ask students to determine if their sequence of rigid motions and dilations will always work for any pair of triangles that have all pairs of corresponding angles congruent and all pairs of corresponding sides lengths in the same proportion. (Probably not, some figures would require rotations and dilations. But the general method of dilating and using rigid motions will always work.)
For students who dilated first, ask:
How do you choose the dilation to use? (Dilate from any vertex, by the scale factor given by the ratio of a pair of corresponding sides.)
Once you dilate, how do you know you’ll be able to use rigid motions to take one triangle to the other? (The triangles will be congruent, because their corresponding angles will remain congruent, and all their side lengths will be multiplied by the value
\(k\)
that you need to make them the same length as the corresponding side. Since all the corresponding angles and all the corresponding sides will match, you’ll be able to take one triangle to the other.)
For students who used rigid motions first, ask:
How do you know rigid motion will line up parts of the figures? (Because corresponding angles are congruent, we’ll be able to line up both of the rays that meet at the vertex by translating and rotating, because you can always take an angle onto another congruent angle.)
How do you know that dilation will then line up the remaining two points? (The sides lengths will be in the same proportion, because we’re given that information, and the angles all line up, so the triangles are dilations of each other.)


correct_option_index: 20