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?


Topic options: 
A. Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
B. Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
C. Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
D. Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
E. Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.
F. Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.
G. Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.
H. Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.
I. Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
J. Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.
K. Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.


Thought: This problem asks students to reason about ratios of butter amounts to sugar amounts.
Answer: D

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


Topic options: 
A. Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.
B. Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
C. Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
D. Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
E. Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
F. Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.
G. Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
H. Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.
I. Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
J. Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.
K. Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.


Thought: This problem teaches students different ways of writing numbers that express the same value. This helps students better understand the complex number system, which includes rational numbers.
Answer: I

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?


Topic options: 
A. Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.
B. Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
C. Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.
D. Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.
E. Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
F. Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.
G. Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
H. Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.
I. Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
J. Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
K. Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.


Thought: This set of problems asks students to reason with equations, specifically linear equations, such as the one shown in the table in Problem 1.
Answer: F

Now, assign the following problem or activity to one or more relevant topics it teaches.
shot_info: fl_problem_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
problem_activity: In Class Launch
Use after Unit 4, Lesson 9
Display an image of the flag of Nepal, such as this one, for all to see.
[IMG]
Tell students that this is the flag of Nepal, which is a country in South Asia that borders India and China. Ask students “What do you notice? What do you wonder?”
Students may notice:
It looks like it’s made of triangles.
I could break it up into a rectangle and three triangles.
The bottom left corner looks like a right angle.
Students may wonder:
What shapes is the flag made of?
Why does it look like this?
How could we break up the white symbols into simpler shapes?
If it does not come up during the discussion, ask students how they might break up the shape of the flag into simpler shapes so that they could, for example, find its area.
Then tell students that in this task they will see how to construct this flag, and investigate its shape. They will also investigate the shapes involved in a flag of their choosing.
Teacher Instructions
Students will not have seen how to trisect a line segment, which they will need to do in order to construct
\(AC\)
. They can measure first to see where the points that trisect the line should go. Instructions for trisecting a line segment can be found online. There are many ways of doing it, and some are easier than others, so if students find the instructions themselves they should be encouraged to compare several methods before deciding which one they prefer. If students try different methods, it could be beneficial to invite selected students to show the class what they did. The other steps for constructing the flag should be familiar. Having access to geometry software can streamline the construction process.
If students have trouble getting started with finding the angles, they can try giving
\(AB\)
a specific value and calculating the other side lengths.
Once students have completed the task, invite them to share the flags they chose and show how they adapted them to fit on the triangular banner.
Student-Facing Statement
The constitution of Nepal includes instructions for constructing the flag. Here are the instructions:
Step 1: Draw a line
\(AB\)
of the required length.
\(AB\)
will be the bottom edge of the flag.
Step 2: Draw a line
\(AC\)
perpendicular to
\(AB\)
so that the length of
\(AC\)
is
\(\frac43\)
the length of
\(AB\)
.
Step 3: Mark the point
\(D\)
on the line
\(AC\)
so that
\(AD\)
is the same length as
\(AB\)
. Then join the points
\(D\)
and
\(B\)
with a straight line.
Step 4: Mark the point
\(E\)
on the line
\(BD\)
so that
\(BE\)
is the same length as
\(AB\)
.
Step 5: Draw a line
\(FG\)
that goes through
\(E\)
, is parallel to
\(AB\)
, and is the same length as
\(AB\)
.
\(F\)
should be on the line
\(AC\)
.
Step 6: Join the points
\(C\)
and
\(G\)
with a straight line.
Construct the flag according to the instructions. You can make it any size you want.
Find the measure of each of the angles in the flag.
Imagine that there will be a parade featuring the flag of Nepal. For the parade there will be small flags and large flags. Decide what the sizes of each type of flag should be.
How much material would you need to make each type of flag?
If you made the border of the flags by sewing a ribbon along the edge, how long would the ribbon have to be for each type of flag?
Imagine that there will be a parade featuring a flag that is meaningful to you. (Choose a flag or design your own.) At the parade there will be small flags, large flags, and triangular banners like this:
[IMG]
How will you adapt your flag to fit on the triangular shape?
How much material of each color will you need for each kind of flag?
Teacher Instructions
Students who want an extra challenge can find the areas and perimeters of both the small and large flag in the second question, and then compare the areas to see the effect of the scale factor. The relationship between scaling lengths and scaling areas will be a topic in the next unit, so students do not need to have a full understanding of it at this time, but this is a good opportunity to preview the concept.
Another opportunity for an extra challenge is to try describing the exact ratios between the side lengths using radicals. An especially interesting thing students may notice is that the ratio between
\(AF\)
and
\(AD\)
is the same as the ratio between
\(AD\)
and
\(BD\)
,
\(\tfrac{1}{\sqrt2}\)
.
Having access to geometry software can help students visualize the shapes they are working with.
Once students have completed the task, invite them to share the flags they chose and show how they adapted them to fit on the triangular banner.
Student-Facing Statement
The constitution of Nepal includes instructions for constructing the flag. Here are the instructions:
Step 1: Draw a line
\(AB\)
of the required length.
\(AB\)
will be the bottom edge of the flag.
Step 2: Draw a line
\(AC\)
perpendicular to
\(AB\)
so that the length of
\(AC\)
is
\(\frac43\)
the length of
\(AB\)
.
Step 3: Mark the point
\(D\)
on the line
\(AC\)
so that
\(AD\)
is the same length as
\(AB\)
. Then join the points
\(D\)
and
\(B\)
with a straight line.
Step 4: Mark the point
\(E\)
on the line
\(BD\)
so that
\(BE\)
is the same length as
\(AB\)
.
Step 5: Draw a line
\(FG\)
that goes through
\(E\)
, is parallel to
\(AB\)
, and is the same length as
\(AB\)
.
\(F\)
should be on the line
\(AC\)
.
Step 6: Join the points
\(C\)
and
\(G\)
with a straight line.
Clare is constructing the flag and she starts with
\(AB\)
of length 6 inches. Here is her finished flag shape. Use the instructions to fill in as many lengths and angles as you can.
[IMG]
Imagine that there will be a parade featuring the flag of Nepal. For the parade there will be small flags and large flags. For one of those types, decide how big it should be.
How much material would you need to make the flag?
If you made the border of the flag by sewing a ribbon along the edge, how long would the ribbon have to be?
Imagine that there will be a parade featuring a flag that is meaningful to you. (Choose a flag or design your own.) At the parade there will be rectangular flags and triangular banners like this:
[IMG]
How will you adapt your flag to fit on the triangular shape?
How much material of each color will you need for each kind of flag?


correct_option_index: 3

9