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?


Concept/skill: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

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


Concept/skill: 
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

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

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?


Concept/skill: 
Analyze and solve pairs of simultaneous linear equations.

Answer: 
Yes. 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.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_000972_neg_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_pos_0

8
standard_description: Read, write, and compare decimals to thousandths.
problem_activity: Narrative
The purpose of this activity is for students to round to the nearest tenth and hundredth. Students have rounded in earlier grades but this is the first time they round to tenths or hundredths. This is a direct extension of rounding to the nearest ten, hundred, thousand, and other whole number values. Locating the numbers on the number line will recall this earlier work.
The launch introduces the context of a doubloon, a major currency in Portugal and Spain in the seventeenth, eighteenth, and nineteenth centuries. Students round the weight of a doubloon to the nearest tenth and hundredth of a gram. In both cases, older doubloons are still heavier after rounding. Rounded to the nearest gram, however, they are the same. This is important from a practical perspective because it is easier to measure a weight to the nearest gram than it is to the nearest tenth of a gram or hundredth of a gram. It is also important because the numbers 6.9 and even 6.87 are not as complex as 6.867, and having fewer digits helps visualize the value more quickly.
MLR8 Discussion Supports.
During small-group discussion, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “I heard you say . . . .” Original speakers can agree or clarify for their partner.
Advances: Listening, Speaking
Launch
Groups of 2
Display the image.
“This is a doubloon. What do you notice? What do you wonder?” (I notice 1798. I wonder if that’s the year it was made. I wonder if people actually used these as money. I notice there are words in a foreign language. I wonder what they say. I wonder what language it is.)
“Doubloons were a major currency in Portugal and Spain in the seventeenth, eighteenth, and nineteenth centuries. Today, we are going to study the weights of doubloons.”
Activity
“Take a couple of minutes to work on the first problem.”
2 minutes: partner work time
“What does it mean to measure a weight to the nearest tenth of a gram?” (It means you find the closest tenth of a gram to the weight.)
“How did you decide which kind of doubloon was on the scale?” (The older ones are heavier and the tenth they are closest to is 6.9 not 6.8.)
“The weight of the doubloon is rounded to the nearest tenth of a gram.”
“Now complete the rest of the problems.”
6–8 minutes: partner work time
Student Facing
[IMG]
Until 1728, doubloons weighed 6.867 grams.
After 1728, they weighed 6.766 grams.
You have a scale that measures weight to the nearest tenth of a gram.
Was the doubloon on the scale made before or after 1728?
[IMG]
If you had a scale that measured to the nearest gram, explain why you would not be able to tell what year the doubloon was made based on the weight listed on the scale.
Which doubloons weigh more, the ones made before 1728 or the ones made after 1728? Explain or show your reasoning.
Show the weights of the doubloons on the number line.
[IMG]
Use the number lines to find which hundredth of a gram the doubloon weights are each closest to.
[IMG]
[IMG]
Student Response
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Activity Synthesis
Invite students to share their answers for rounding the weights of the doubloons.
“How did you round the weight of the older doubloon to the nearest tenth? What about the nearest hundredth?” (It was closer to 6.9 than to 6.8, so I rounded to 6.9 to the nearest tenth. It was closer to 6.87 than to 6.86 so I rounded to 6.87 to the nearest hundredth.)
“How did the number lines help you round the numbers?” (I could see on the number line which tenth or which hundredth the number was closest to and then knew to round it there.)


aligns: yes