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?


Standard options: 
A. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
B. 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.
C. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)


Thought: 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.
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
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Thought: 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).
Answer: B

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?


Standard options: 
A. Solve linear equations in one variable.
B. Analyze and solve pairs of simultaneous linear equations.


Thought: The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Verify experimentally the properties of rotations, reflections, and translations:
- Lines are taken to lines, and line segments to line segments of the same length.
- Angles are taken to angles of the same measure.
- Parallel lines are taken to parallel lines.

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
problem_activity: Optional activity
The goal of this task is to introduce the notion of a tessellation and then carefully examine how to create a tessellation with each pattern block shape (square, rhombus, equilateral triangle, isosceles trapezoid). For hexagons, there is only one way to fit them together because there will be gaps unless three hexagons meet at each vertex. The other shapes offer much more flexibility, and students have an opportunity to use their artistic creativity.
Students look for and make use of structure (MP7), both when they try to put copies of the shape together to build a tessellation and when they examine whether or not it is possible to construct a different tessellation.
Digital
Launch
Introduce the definition of a
tessellation
of the plane by polygons: a tessellation covers the plane with copies of the shape with no gaps and no overlaps.
Show the students the image from the previous notice and wonder activity, and ask them if these are tessellations. (Yes.)
Demonstrate how to use tracing paper to create a tessellation.
Arrange students in groups of 2. Each pair uses the same shape for their tessellations.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Include the following term and maintain the display for reference throughout the lesson: tessellation. Consider providing step-by-step directions for students to use tracing paper to create a tessellation. Invite students to suggest language or diagrams to include that will support their understanding.
Supports accessibility for: Memory; Language
Speaking: MLR8 Discussion Supports.
Amplify students' uses of mathematical language to communicate about tessellations; edge to edge, vertex, symmetrical, plane, rigid motion, etc. Encourage students to use these words, revoicing students’ ideas as necessary. Ask students to chorally repeat the phrases that include these words in context.
Design Principle(s): Support sense-making; Optimize output (for explanation)
Student Facing
With your partner, choose one of the six shapes in the toolbar that you will both use.
Select the shape tool by clicking on it. Create copies of your shape by clicking in the work space.
When you have enough to work with, click on the Move tool (the arrow) to drag or turn them.
If you have trouble aligning the shapes, right click to turn on the grid.
When you are finished making your tessellation, you can hide the points.
Compare your tessellation to your partner’s. How are they similar? How are they different?
If possible, make a third tessellation of the plane with your shape (different from the ones you and your partner already created). If not possible, explain why it is not possible.
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
Introduce the definition of a
tessellation
of the plane by polygons: a tessellation covers the plane with copies of the shape with no gaps and no overlaps.
Show the students the image from the previous notice and wonder activity, and ask them if these are tessellations. (Yes.)
Demonstrate how to use tracing paper to create a tessellation.
Arrange students in groups of 2. Each pair uses the same shape for their tessellations.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Include the following term and maintain the display for reference throughout the lesson: tessellation. Consider providing step-by-step directions for students to use tracing paper to create a tessellation. Invite students to suggest language or diagrams to include that will support their understanding.
Supports accessibility for: Memory; Language
Speaking: MLR8 Discussion Supports.
Amplify students' uses of mathematical language to communicate about tessellations; edge to edge, vertex, symmetrical, plane, rigid motion, etc. Encourage students to use these words, revoicing students’ ideas as necessary. Ask students to chorally repeat the phrases that include these words in context.
Design Principle(s): Support sense-making; Optimize output (for explanation)
Student Facing
[IMG]
Pick one of the shapes. Create a
tessellation
by tracing copies of your shape. Make sure to use the same shape as your partner.
Compare your tessellation to your partner’s. How are they similar? How are they different?
If possible, make a third tessellation of the plane with your shape (different from the ones you and your partner already created). If not possible, explain why it is not possible.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Tessellations do not need to be edge to edge. That is, pieces do not need to fit together with edges of the same lengths matching exactly.
Tessellations do not need to have symmetric, repeating patterns, though sometimes the shape forces it (as with the regular hexagons).
Activity Synthesis
Important questions to address include:
“Were you able to make different tessellations with your shape?” (It depends on the shape.)
“If not, why not?” (Three hexagons have to come together at each vertex: once the first hexagon of the pattern is placed, everything else has no flexibility.)
“If so how?” (Triangles can be built into hexagons, or they can make rows that can be translated and stacked on top of one another. Parallelograms can also be made into hexagons or rows. Trapezoids can be made into hexagons or rows. Rhombuses can be made into rows and translated. Squares can also be made into rows and translated.)
“What does it look like to
not
define a tessellation?” (Two octagons can be put together sharing a vertex, but there is a gap that is not large enough for a third octagon.)
Make sure, as students share their ideas for tessellations, to use the language of rigid motions to describe the tessellations. For example, if a student has built a tessellation with parallelograms, choose two parallelograms and ask:
“How can I use rigid motions to move one of these parallelograms into the position of the other?”
“Are there other ways I could do this?”


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