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 to multiply a fraction or whole number by a fraction.

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: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

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_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
problem_activity: Activity
The large quantities involved in these questions lend themselves to arithmetic with powers of 10, giving students the opportunity to make use of scientific notation before it is formally introduced. This activity was designed so students could practice modeling skills such as identifying essential features of the problem and gathering the required information (MP4). Students use powers of 10 and the number line as tools to make it easier to calculate and interpret results.
Notice the ways in which students use relevant information to answer the questions. Identify students who can explain why they are calculating with one operation rather than another. Speed is not as important as carefully thinking through each problem.
Teacher Notes for IM 6–8 Math Accelerated
Adjust time to 15 minutes. Give students 10 minutes of work time.
Launch
From the warm-up, students have decided what information they need to solve the problem. Invite students to ask for the information they need. Provide students with only the information they request. Display the information for students to see throughout the activity. If students find they need more information later, provide it to the whole class then.
Here is information students might ask for in order to solve the problems:
The mass of an average classroom meter stick is roughly 0.2 kg.
The length of an average classroom meter stick is 1 meter.
The mass of the Moon is approximately
\(7 \boldcdot 10^{22}\)
kg.
The Moon is roughly
\((3.8) \boldcdot 10^8\)
meters away from Earth.
The distances to various astronomical bodies the students might recognize, in light years, as points of reference for their last answer. (Consider researching other distances in advance or, if desired, encouraging interested students to do so.)
Arrange students in groups of 2–4 so they can discuss how to use the information to solve the problem. Give students 15 minutes of work time.
Student Facing
How many meter sticks does it take to equal the mass of the Moon? Explain or show your reasoning.
Label the number line and plot your answer for the number of meter sticks.
[IMG]
If you took all the meter sticks from the last question and lined them up end to end, will they reach the Moon? Will they reach beyond the Moon? If yes, how many times farther will they reach? Explain your reasoning.
One light year is approximately
\(10^{16}\)
meters. How many light years away would the meter sticks reach? Label the number line and plot your answer.
[IMG]
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?
Here is a problem that will take multiple steps to solve. You may not know all the facts you need to solve the problem. That is okay. Take a guess at reasonable answers to anything you don’t know. Your final answer will be an estimate.
If everyone alive on Earth right now stood very close together, how much area would they take up?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Select previously identified students to share how they organized their relevant information and how they planned to use the information to answer the questions. The important idea students should walk away with is that powers of 10 are a great tool to tackle challenging, real-world problems that involve very large numbers.
It might be illuminating to put 35 million light years into some context. It is over a thousand trillion times as far as the distance to the Moon, or about the size of a supercluster of galaxies. The Sun is less than
\(1.6 \times 10^\text{-5}\)
light year away from Earth.
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: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “I agree/disagree because . . ..”
Supports accessibility for: Language; Social-emotional skills
Speaking: MLR8 Discussion Supports.
Give students additional time to make sure everyone in their group can explain their response to the first question. Then, vary who is called on to represent the ideas of each group. This routine will prepare students for the role of group representative and to support each other to take on that role.
Design Principle(s): Optimize output (for explanation); Cultivate conversation


aligns: no