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.)


Your response, in json format:
{
  "explanation": "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.


Your response, in json format:
{
  "explanation": "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.
{"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?


Standard options: 
A. Solve linear equations in one variable.
B. Analyze and solve pairs of simultaneous linear equations.


Your response, in json format:
{
  "explanation": "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: Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
problem_activity: Activity
The purpose of this activity is for students to practice using precise language to describe how they estimated volumes of objects. Starting from an object of known volume, students must consider the difference in dimensions between the two objects. The focus here is on strategies to estimate the volume and units of measure used, not on exact answers or calculating volume using a formula (which will be the focus of later lessons). Notice students who:
have clear strategies to estimate volume of contents inside container
have an estimate that is very close to the actual volume
Launch
Arrange students in groups of 2.
Option 1: Bring in real containers and have students estimate how much rice each would hold, one at a time, preferably with one container whose volume is stated so students have a visual reference for their estimates. Also bring plenty of dried rice and measuring tools, such as tablespoons or cups. After collecting students’ estimates, you can demonstrate how much rice each container holds using whichever units of measure the class deems reasonable. Note that 1 tablespoon is 0.5 ounces or around 15 milliliters. 1 cup is 8 ounces or around 240 milliliters. 1 milliliter is the same as 1 cubic centimeter.
Option 2: Display images one at a time for all to see. Give students 1–2 minutes to work with their partner and write down an estimate for the objects of unknown volumes in the picture. Follow with a whole-class discussion.
[IMG]
[IMG]
[IMG]
[IMG]
Representation: Internalize Comprehension.
Provide access to 3-D models of the objects depicted in the images for students to manipulate. Ask students to use the 3-D models to estimate the quantities in the other figures. Encourage students to place objects with known volumes inside or next to others to ensure an accurate estimate.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
Your teacher will show you some containers.
If the pasta box holds 8 cups of rice, how much rice would you need for the other rectangular prisms?
If the pumpkin can holds 15 fluid ounces of rice, how much do the other cylinders hold?
If the small
cone
holds 2 fluid ounces of rice, how much does the large cone hold?
If the golf ball were hollow, it would hold about 0.2 cups of water. If the baseball were hollow, how much would the
sphere
hold?
Student Response
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for free access to Student Response.
Anticipated Misconceptions
Students may think there is a single right answer. Measurements are always approximate. Some of the measurements given by the authors of this activity were calculated using estimates from the photos, and may not be very precise. Measurements listed on the sides of packages are more accurate, but actual contents may vary slightly.
Activity Synthesis
For each set of containers, display the image and select previously identified students to share their strategies for estimating the volume. Once strategies for each set of containers are shared, discuss:
“How do the estimates differ if we measure using water verses rice?” (Measuring with rice leaves a bit of empty space between the grains, while water, being liquid, leaves no empty space, so it’s more accurate.)
“If the containers we used were much larger (like a water tank), would our units of measure change? Why?” (If we were measuring larger volumes, we might want to use a larger unit, like gallons. 4000 ml sounds big, but it’s only a bit more than 1 gallon, which isn’t that much water.)
Conclude the discussion by asking students to compare some other units of measure for volume that they know of. Have students recall what they know about unit conversion between some units of measure. Example:
Fluid ounces, quarts, cups, liters, milliliters
Cubic feet, cubic meters, cubic yards
Note that cubic centimeters are special, because 1 cc = 1 ml
If it comes up, here is the scoop on
ounces
: units called “ounces” are used to measure both volume and weight. It is important to be clear about what quantity you are measuring! To differentiate between them, people refer to the units of measure for volume as “fluid ounces.” For water, 1 fluid ounce is very close to 1 ounce by weight. This is not true for other substances! For example, mercury is much denser than water. 1 fluid ounce of mercury weighs about 13.6 ounces! Motor oil is less dense than water (that’s why it floats), so 1 fluid ounce of oil weighs only about 0.8 ounces. The metric system is not so confusing for quantities that would be measured in ounces, since it’s common to measure mass instead of weight and measure it in grams, whereas volume is measured in milliliters.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each strategy that is shared, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This will provide more students with an opportunity to speak as they make sense of the reasoning of others.
Design Principle(s): Support sense-making


correct_option_index: 0