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: 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.)

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

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.
problem_activity: Activity
The purpose of this activity is for students to continue interpreting signed numbers in context and to begin to compare their relative location. A vertical number line shows the heights above sea level or depths below sea level of various animals. The number line is labeled in 5 meter increments, so students have to interpolate the height or depth for some of the animals. Next, they are given the height or depth of other animals that are not pictured and asked to compare these to the animals shown.
As students work, monitor for whether they are expressing relative distances in words, for example “3 meters below,” or if they are expressing the same idea with notation, as in -3 meters. Both are acceptable; these ideas are connected in the discussion that follows (MP2). Also monitor for students who notice that there are two possible answers for the last question.
Launch
Display the image for all to see. Tell students to measure the height or depth of each animal's eyes, to the nearest meter. Remind students that we choose sea level to be our zero level, in the same way that we chose a zero level for temperature.
Representation: Internalize Comprehension.
Represent the same information with additional structure. If students are unsure where to begin, suggest that they extend a straight horizontal line at each depth to determine the height or depth of each animal.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Conversing, Representing: MLR2 Collect and Display.
Use this routine to capture the language students use during discussion with a partner. Collect the language of opposites: “above” or “below.” For example, “The albatross is 3 meters above the penguin or the penguin is 3 meters below the albatross.” Then, identify students who use negative numbers to describe these differences to share their reasoning during the whole-class discussion. Ensure students connect this language to, “The difference in height is +3 (to represent above) or -2.5 (to represent below).”
Design Principle(s): Maximize meta-awareness
Student Facing
Here is a picture of some sea animals. The number line on the left shows the vertical position of each animal above or below sea level, in meters.
[IMG]
How far above or below sea level is each animal? Measure to their eye level.
A mobula ray is 3 meters above the surface of the ocean. How does its vertical position compare to the height or depth of:
The jumping dolphin?
The flying seagull?
The octopus?
An albatross is 5 meters above the surface of the ocean. How does its vertical position compare to the height or depth of:
The jumping dolphin?
The flying seagull?
The octopus?
A clownfish is 2 meters below the surface of the ocean. How does its vertical position compare to the height or depth of:
The jumping dolphin?
The flying seagull?
The octopus?
The vertical distance of a new dolphin from the dolphin in the picture is 3 meters. What is its distance from the surface of the ocean?
Student Response
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Student Facing
Are you ready for more?
The north pole is in the middle of the ocean. A person at sea level at the north pole would be 3,949 miles from the center of Earth. The sea floor below the north pole is at an elevation of approximately -2.7 miles. The elevation of the south pole is about 1.7 miles. How far is a person standing on the south pole from a submarine at the sea floor below the north pole?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students measure to the top or bottom of the animal, remind them that we are using the eyes of the animal to measure their height or depth.
Some students may struggle to visualize where the albatross, seagull, and clownfish are on the graph. Consider having them draw or place a marker where the new animal is located while comparing it to the other animals in the picture.
Activity Synthesis
The main point for students to get out of this activity is that we can represent distance above and below sea level using signed numbers. The depths of the shark, fish, and octopus can be expressed as approximately -3 m, -6 m, and -7.5 m respectively, because they are below sea level.
Signed numbers can also be used to represent the relative vertical position of different pairs of animals. Have selected students share their responses and reasoning for how the heights of the albatross, seabird, and clownfish compare to the dolphin, seagull, and octopus. Record and display their verbal descriptions using signed numbers. For example, if a student says the albatross is 7 meters below the seagull, write "-7".
Finally, ask whether students noticed the ambiguity in the last question (about the height of the new dolphin). Ask such a student to explain why there are two possible answers to the last question.


correct_option_index: 2