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

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: 
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:
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_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
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?


aligns: yes