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: Explain volume formulas and use them to solve problems.

Prove geometric theorems.

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

Understand and apply theorems about circles.

Understand and apply the Pythagorean Theorem.

Find arc lengths and areas of sectors of circles.

Understand congruence in terms of rigid motions.

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

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

Experiment with transformations in the plane.

Apply trigonometry to general triangles.

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

Make geometric constructions.

Prove theorems involving similarity.

Define trigonometric ratios and solve problems involving right triangles.

Analyze, compare, create, and compose shapes.

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

Use coordinates to prove simple geometric theorems algebraically.

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

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

Reason with shapes and their attributes.

Apply geometric concepts in modeling situations.

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

Understand similarity in terms of similarity transformations.

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

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

Visualize relationships between two-dimensional and three-dimensional objects.
problem_activity: Activity
Students have seen images showing a sequence of transformations in the first lesson of this unit, however they have not heard the term sequence of transformations. They have also not been asked to describe the moves in the sequence using precise language. The launch of this activity introduces this term and gives students an opportunity to describe the sequence of more than one transformation.
For the second problem, encourage students to find a different sequence of transformations than the one shown in the image. Each time a reflection is mentioned, ask students where the line of reflection is located and when a rotation is mentioned, ask for the center of the rotation and the number of degrees. Monitor for students who apply different transformations (or apply transformations in a different order).
Digital
Launch
Arrange students in groups of 2, and provide access to their geometry toolkits. Display the image for all to see. Ask students if they can imagine a single translation, rotation, or reflection that would take one bird to another? After a minute, verify that this is not possible.
Ask students to describe how we could use translations, rotations, and reflections to take one bird to another. Collect a few different responses. (One way would be to take the bird on the left, translate it up, and then reflect it over a vertical line.) Tell students when we do one or more transformations in a row to take one figure to another, it is called a
sequence of transformations
.
[IMG]
If using the digital activity, you may want to review the transformation tools in the applet. (The instructions are repeated in the activity for students' reference.)
Give students 2 minutes of quiet work time to engage in the task followed by 3 minutes to discuss their responses with a partner and complete any unfinished questions. Follow with a whole-class discussion.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as “To take Figure
\(A\)
to Figure
\(B\)
, I _____ because…”, “I noticed _____ so I...”, “Why did you...?”, or “I agree/disagree because….”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Here are some figures on an isometric grid. Explore the transformation tools in the tool bar. (Directions are below the applet if you need them.)
Name a transformation that takes Figure A to Figure B. Name a transformation that takes Figure B to Figure C.
What is one
sequence of transformations
that takes Figure A to Figure C? Explain how you know.
Translate
Select the Vector tool.
[IMG]
Click on the original point and then the new point. You should see a vector.
Select the Translate by Vector tool.
[IMG]
Click on the figure to translate, and then click on the vector.
Rotate
Select the Rotate around Point tool.
[IMG]
Click on the figure to rotate, and then click on the center point.
A dialog box will open. Type the angle by which to rotate and select the direction of rotation.
Reflect
Select the Reflect about Line tool.
[IMG]
Click on the figure to reflect, and then click on the line of reflection.
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
Arrange students in groups of 2, and provide access to their geometry toolkits. Display the image for all to see. Ask students if they can imagine a single translation, rotation, or reflection that would take one bird to another? After a minute, verify that this is not possible.
Ask students to describe how we could use translations, rotations, and reflections to take one bird to another. Collect a few different responses. (One way would be to take the bird on the left, translate it up, and then reflect it over a vertical line.) Tell students when we do one or more transformations in a row to take one figure to another, it is called a
sequence of transformations
.
[IMG]
If using the digital activity, you may want to review the transformation tools in the applet. (The instructions are repeated in the activity for students' reference.)
Give students 2 minutes of quiet work time to engage in the task followed by 3 minutes to discuss their responses with a partner and complete any unfinished questions. Follow with a whole-class discussion.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as “To take Figure
\(A\)
to Figure
\(B\)
, I _____ because…”, “I noticed _____ so I...”, “Why did you...?”, or “I agree/disagree because….”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Here are some figures on an isometric grid.
[IMG]
Name a transformation that takes Figure
\(A\)
to Figure
\(B\)
. Name a transformation that takes Figure
\(B\)
to Figure
\(C\)
.
What is one
sequence of transformations
that takes Figure
\(A\)
to Figure
\(C\)
? Explain how you know.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Experiment with some other ways to take Figure
\(A\)
to Figure
\(C\)
. For example, can you do it with. . .
No rotations?
No reflections?
No translations?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Select students with different correct responses to show their solutions. Be sure to highlight at least one rotation. If no students mention that, demonstrate a way to take A to C that involves a rotation. Whether or not students use the geogebra applet, it may be helpful to display the applet to facilitate discussion:
ggbm.at/jqvTEgsj
Emphasize that there are many ways to describe the translation that takes figure A to figure B. All one needs is to identify a pair of corresponding points and name them in the correct order (and to use the word “translate”).
For students who used a reflection to take B to C, emphasize that reflections are determined by lines and we should name the line when we want to communicate about it.
After a student or the teacher uses a rotation, emphasize that a rotation is defined by a center point and an angle (with a direction). The center point needs to be named and the angle measure or an angle with the correct measure needs to be named as well (as does the direction). Reinforce to students that when we do more than one transformation in a row, we call this a sequence of transformations.
Speaking: Math Language Routine 7 Compare and Connect.
This is the first time Math Language Routine 7 is suggested as a support in this course. In this routine, students are given a problem that can be approached using multiple strategies or representations and are asked to prepare a visual display of their method. Students then engage in investigating the strategies (by means of a teacher-led gallery walk, partner exchange, group presentation, etc.), comparing approaches, and identifying correspondences between different representations. A typical discussion prompt is: “What is the same and what is different?” regarding their own strategy and that of the others. The purpose of this routine is to allow students to make sense of mathematical strategies by identifying, comparing, contrasting, and connecting other approaches to their own, and to develop students’ awareness of the language used through constructive conversations.
Design Principle(s): Maximize meta-awareness: Support sense-making
How It Happens:
Use this routine to compare and contrast different strategies for transforming Figure A to Figure C. Invite students to create a visual display showing how they made sense of the problem and why their solution makes sense for transforming Figure A to Figure C.
Students should include these features on their display:
a sketch of the figures (not necessary if using the applet)
a sketch of the figures after each transformation (not necessary if using the applet)
a written sequence of transformations with an explanation
Before selecting students to show their solutions to the class, first give students an opportunity to do this in a group of 3–4. Ask students to exchange and investigate each other’s work. Allow 1 minute for each display and signal when it is time to switch.
While investigating each other’s work, ask students to consider what is the same and what is different about each approach. Next, give each student the opportunity to add detail to their own display for 1–2 minutes.
As groups are presenting, circulate the room and select 2–3 students to share their sequence of transformations taking Figure A to Figure C. Be sure to select a variety of approaches, including one that involves a rotation.
Draw students’ attention to the different ways the figures were transformed (e.g., rotations, reflections, and translations) and how the sequence of transformation is expressed in their explanation. Also, use the bullet points in the Activity Synthesis to emphasize specific features of translations, reflections, and rotations.
After the selected students have finished sharing with the whole class, lead a discussion comparing, contrasting, and connecting the different approaches.
Consider using these prompts to amplify student language while comparing and contrasting the different approaches: “Why did the approaches lead to the same outcome?”, “What worked well in __’s approach? What did not work well?”, and “What would make __’s strategy more complete or easy to understand?”
Consider using these prompts to amplify student language while connecting the different approaches: “What role does a translation play in each approach?” “Is it possible to use the all three types of transformations?”, and “What transformation do you see present in all the strategies?”
Close the discussion by inviting 3 students to revoice the strategies used in the presentations, and then transition back to the Lesson Synthesis and Cool-Down.


correct_option_index: 11