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: 
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.

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:
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 and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. 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).

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.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "table-layout:fixed; width:70%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "No solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Exactly 1 solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Infinite number of solutions"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9=4x+7}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y=3y+5}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y={x\\over2}}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${3x+6y-9=0}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${x=y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${1.25x=1.25y}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9.9=6x+8y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9x=2.5y-8.8}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=1.5}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=2.5}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {}]}]}]}]}
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_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_pos_0

8
standard_description: Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
problem_activity: Narrative
The purpose of this activity is for students to round given numbers to the nearest ten and hundred and see that the result can be the same for some numbers. Students think about what it means to round a number that is exactly halfway between two tens or two hundreds and are introduced in the synthesis to the convention that these numbers are rounded up (MP3).
MLR1 Stronger and Clearer Each Time:
Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to “What does 97 round to when we are rounding to the nearest 10? To the nearest hundred? Why does that happen?” Invite listeners to ask questions, to press for details and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.
Advances: Writing, Speaking, Listening
Representation: Internalize Comprehension.
Synthesis: Invite students to identify which details were most useful to solve the problems. Display the sentence frame, “The next time I round numbers to the nearest ten and nearest hundred, I will pay attention to . . . .“
Supports accessibility for: Conceptual Processing
Launch
Groups of 2
“Let’s round some numbers to the nearest ten and the nearest hundred. Remember ‘round to the nearest ten (or hundred)’ is another way of saying find the nearest multiple of 10 (or 100).”
Activity
“Work with your partner to complete the table.”
3–5 minutes: partner work time
Monitor for students who notice 97 and 601 round to the same number whether rounding to the nearest ten or the nearest hundred.
Select previously identified students to share their responses.
“Why did 97 and 601 round to the same number when we rounded to the nearest ten and the nearest hundred?” (The closest multiple of 10 was also a multiple of 100. 100 and 600 are multiples of 100, but they are also multiples of 10.)
“Complete the second problem with your partner. Explain your reasoning.”
3–5 minutes: partner work time
Monitor for the following in student explanations to share in the synthesis:
415 is halfway between 410 and 420.
415 is the same distance from 410 and 420.
415 has two closest multiples of 10: 410 and 420.
750 is halfway between 700 and 800.
750 is the same distance from 700 and 800.
750 has two closest multiples of 100: 700 and 800.
Student Facing
Round each number to the nearest ten and the nearest hundred. Use number lines if you find them helpful.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col", "style": "text-align:center"}, "_value": "number"}, {"_attributes": {"scope": "col", "style": "text-align:center"}, "_value": "nearest ten"}, {"_attributes": {"scope": "col", "style": "text-align:center"}, "_value": "nearest hundred"}]}]}], "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}, "_value": "18"}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "97"}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "312"}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "439"}, {}, {}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "601"}, {}, {}]}]}]}]}
Kiran and Priya are rounding some numbers and are stuck when trying to round 415 and 750.
Kiran said, “415 doesn’t have a nearest multiple of 10, so it can’t be rounded to the nearest ten.”
Priya said, “750 doesn’t have a nearest multiple of 100, so it can’t be rounded to the nearest hundred.”
Do you agree with Kiran and Priya? Explain your reasoning.
Student Response
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Activity Synthesis
Select previously identified students to share their responses.
“Kiran and Priya are correct. There isn’t one closest multiple of 10 to 415 because it’s right in the middle of two multiples of 10. There isn’t a closest multiple of 100 to 750 because it’s right in the middle of two multiples of 100.”
“In cases like this, we round up. For example, since 415 is halfway between 410 and 420, we would round up to 420 to round 415 to the nearest ten.”
“Since 750 is halfway between 700 and 800, we would round up to 800.”
”In both of these situations we could go the other way, but it is helpful if we all do the same thing in these situations.”


aligns: no