few_shots: 
Example 1:
Concept/skill: 
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

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?


Answer: 
Yes. This problem solves a real-world problem involving mixing ingredients for cookies. A student doing this problem would need to reason about equivalent ratios of butter amounts to sugar amounts.

Example 2:
Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

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
$$-\frac{1}{4}$$


Answer: 
Yes. This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).

Example 3:
Concept/skill: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

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?


Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_000972_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
problem_activity: Activity
In this activity, students work together to investigate rotation symmetry and communicate their thinking in a visual display. As you monitor, ask groups if they have found all possible angles of rotation that produce symmetry and to explain how they know these angles apply to all shapes of that type.
Launch
Arrange students in the same groups as the previous class. Provide each group with tools for creating a visual display. Assign each group a different shape from Self Reflection and provide enough copies of the shape for everyone in the group.
Give students 5 minutes of work time followed by 5 minutes to put together their visual display. Explain that they will not have enough time to make the visual display perfect, so the purpose is just to get their rough ideas down in an organized way.
Student Facing
Determine all the angles of rotation that create symmetry for the shape your teacher assigns you. Create a visual display about your shape. Include these parts in your display:
the name of your shape
the definition of your shape
drawings of each rotation that creates symmetry
a description in words of each rotation that creates symmetry, including the center, angle, and direction of rotation
one non-example (a description and drawing of a rotation that does
not
result in symmetry)
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?
Finite figures, like the shapes we have looked at in class, cannot have translation symmetry. But with a pattern that continues on forever, it is possible. Patterns like this one that have translation symmetry in only one direction are called
frieze patterns
.
[IMG]
What are the lines of symmetry for this pattern?
What angles of rotation produce symmetry for this pattern?
What translations produce symmetry for this pattern if we imagine it extending horizontally forever?
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
The purpose of this discussion is to identify what types of shapes have rotation symmetry.
Ask groups to display their visual displays in the classroom for all to see in order of the number of angles of rotation that create symmetry for their shape. They will have to communicate with other groups to accomplish this. If possible, display the reflection symmetry visual displays from the previous class for comparison.
Invite students to do a “gallery walk” in which they leave written feedback on sticky notes for the other groups. Here is guidance for the kind of feedback students should aim to give each other:
Was there anything about the organization of the visual display that made the ideas especially clear? Was there anything about the organization that could be improved?
Was there anything about the way the ideas are explained that made the ideas especially clear? Was there anything about the explanations that could be improved?
Are there any angles of rotation creating symmetry that the group missed?
Are there any angles of rotation that don’t create symmetry for
all
shapes of the given type?
After the gallery walk, ask students to share their observations about rotation symmetry. Consider asking:
“Which shapes had the fewest angles of rotation that created symmetry? Which had the most?” (The isosceles trapezoid does not have rotation symmetry because the only rotations that take it to itself are 0 degrees or 360 degrees. For the circle, any angle of rotation creates symmetry.)
“What do you notice about the order of the shapes in terms of reflection symmetry versus the order of the shapes in terms of rotation symmetry?” (The circle has the most lines of symmetry and the most angles of rotation that create symmetry. It seems to be the most symmetric shape overall. The parallelogram that is not a rhombus has 180 degree rotation symmetry, but no lines of symmetry; but the rhombus has both 180 degree rotation symmetry and 2 lines of symmetry.)
If questions about 0 or 360 degree rotations come up, explain that all shapes, no matter how strange, look the same when rotated by 0 degrees or 360 degrees. If these rotations are the only ones that take the shape to itself, then it is
not
considered to have rotation symmetry.
Speaking: MLR 7 Compare and Connect.
Use this routine when students present their categories and their sequence of rotations, translations, and reflections to get from the original figure to the image. Ask students to consider what is the same and what is different about the sequences of transformations. Draw students' attention to the way that transformations are described precisely (i.e., order, numbers, words). These exchanges strengthen students' mathematical language use and reasoning about transformations.
Design Principle(s): Maximize meta-awareness
Engagement: Develop Effort and Persistence.
Break the class into small group discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Language; Social-emotional skills; Attention


aligns: yes