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: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
problem_activity: Activity
This activity encourages students to apply strategies for finding the area of polygons to finding the
surface area
of rectangular prisms. Students use 12 cubes to build a prism, think about its surface area, and use isometric dot paper to draw their prism.
As students build their prisms, notice those with different designs and those with the same design but different approaches to finding surface area (e.g. by counting individual square, by multiplying the edge lengths of rectangular faces, etc.).
Digital
Launch
Read the first two lines of the task statement together. Remind students that we refer to the flat surfaces of a three-dimensional figure as
faces
.
Give students a minute to think about how we know the surface area of the shown prism is 32 square units. Ask 1–2 students to explain their reasoning to the class. Use students’ explanations to highlight the meaning of surface area, i.e., that the area of all the faces need to be accounted for, including those we cannot see when looking at a two-dimensional drawing.
Tell students they will use 12 cubes to build a different prism, draw it, and find its surface area. Consider doing a quick demonstration on how to draw a simple prism on isometric dot paper. (Start with one cube and then add a cube in each dimension.) Tell students that in this activity, we call each face of a single cube, "1 square unit."
Give each student 12 cubes to build a prism and 6–8 minutes of quiet work time. If students are using snap cubes, say that we will pretend all of the faces are completely smooth, so they do not need to worry about the “innies and outies” of the snap cubes.
As students work, consider arranging two students with contrasting designs or strategies as partners. Ask partners to share their answers, explanations, and drawings. Stress that each partner should focus their explanation on how they went about finding surface area. The listener should think about whether the explanation makes sense or if anything is amiss in the reasoning.
For students in digital classrooms, an applet can be used to build and draw prisms. Physical cubes are still recommended and preferred for the building of the figures, however.
Action and Expression: Develop Expression and Communication.
Support multiple forms of communication. Some students may benefit from an explicit demonstration and additional practice to learn how to draw cubes using isometric dot paper. Invite students who are unable to represent their figures using isometric dot paper to explain their reasoning orally, using virtual or concrete manipulatives.
Supports accessibility for: Visual-spatial processing; Conceptual processing; Fine-motor skills
Student Facing
Here is a rectangular prism built from 12 cubes:
[IMG]
It has six
faces
, but you can only see three of them in the sketch. It has a
surface area
of 32 square units.
The applet has 12 blocks, too. They are all in the same spot on the screen, like a hidden stack of blocks. You will always know where the stack is because it sits on a gray square. To use a block, drag the red point to move it. Click on the red points to change from left/right movement to up/down movement.
[IMG]
Use all 12 cubes to build a different rectangular prism (with different edge lengths than shown in the prism here). You can turn the view to see all of the faces of your figure.
How many faces does your figure have?
What is the surface area of your figure in square units?
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
Read the first two lines of the task statement together. Remind students that we refer to the flat surfaces of a three-dimensional figure as
faces
.
Give students a minute to think about how we know the surface area of the shown prism is 32 square units. Ask 1–2 students to explain their reasoning to the class. Use students’ explanations to highlight the meaning of surface area, i.e., that the area of all the faces need to be accounted for, including those we cannot see when looking at a two-dimensional drawing.
Tell students they will use 12 cubes to build a different prism, draw it, and find its surface area. Consider doing a quick demonstration on how to draw a simple prism on isometric dot paper. (Start with one cube and then add a cube in each dimension.) Tell students that in this activity, we call each face of a single cube, "1 square unit."
Give each student 12 cubes to build a prism and 6–8 minutes of quiet work time. If students are using snap cubes, say that we will pretend all of the faces are completely smooth, so they do not need to worry about the “innies and outies” of the snap cubes.
As students work, consider arranging two students with contrasting designs or strategies as partners. Ask partners to share their answers, explanations, and drawings. Stress that each partner should focus their explanation on how they went about finding surface area. The listener should think about whether the explanation makes sense or if anything is amiss in the reasoning.
For students in digital classrooms, an applet can be used to build and draw prisms. Physical cubes are still recommended and preferred for the building of the figures, however.
Action and Expression: Develop Expression and Communication.
Support multiple forms of communication. Some students may benefit from an explicit demonstration and additional practice to learn how to draw cubes using isometric dot paper. Invite students who are unable to represent their figures using isometric dot paper to explain their reasoning orally, using virtual or concrete manipulatives.
Supports accessibility for: Visual-spatial processing; Conceptual processing; Fine-motor skills
Student Facing
Here is a sketch of a rectangular prism built from 12 cubes:
[IMG]
It has six
faces
, but you can only see three of them in the sketch. It has a
surface area
of 32 square units.
Your teacher will give you 12 snap cubes. Use all of your snap cubes to build a different rectangular prism (with different edge lengths than the prism shown here).
How many faces does your figure have?
What is the surface area of your figure in square units?
Draw your figure on isometric dot paper. Color each face a different color.
[IMG]
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
Students may count the faces of the individual snap cubes rather than faces of the completed prism. Help them understand that the faces are the visible ones on the outside of the figure.
Activity Synthesis
After partner discussions, select a student to highlight for the class the strategy (or strategies) for finding surface area methodically. Point out that in this activity each face of their prism is a rectangle, and that we can find the area of each rectangle by multiplying its side lengths and then add the areas of all the faces.
Explain that later, when we encounter non-rectangular prisms, we can likewise reason about the area of each face the way we reasoned about the area of a polygon.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. Call on students to use mathematical language (e.g., cubes, faces, surface area, square units, etc.), to restate and/or revoice the strategy (or strategies) presented. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. This will provide more students with an opportunity to produce language that describes strategies for finding surface area.
Design Principle(s): Support sense-making; Maximize meta-awareness


correct_option_index: 1