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 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.
{"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: 
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.

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, 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_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
problem_activity: Optional activity
This is a matching activity where each student receives a card showing a triangle and works to form a group of three. Each card has a triangle with the measure of only one of its angles given. Students use what they know about transformations and estimates of angle measures to find partners with triangles congruent to theirs. Each unique triangle’s three interior angles are then displayed for all to see. Students notice that the sum of the measures of the angles in each triangle is 180 degrees.
During this activity, students can use MP7, thinking about applying rigid motions to their triangle to see if it might match up with another student's triangle. Or they may identify that their triangle is acute, right, or obtuse and use this as a criterion when they search for a congruent copy (also MP7).
You will need the Find All Three blackline master for this activity.
Launch
Provide access to geometry toolkits. Distribute one card to each student, making sure that all three cards have been distributed for each triangle. (If the number of students in your class is not a multiple of three, it’s okay for one or two students to take ownership over two cards showing congruent triangles.) Explain that there are two other students who have a triangle congruent to theirs that has been re-oriented in the coordinate plane through combinations of translations, rotations, and reflections. Instruct students to look at the triangle on their card and estimate the measures of the other two angles. With these estimates and their triangle in mind, students look for the two triangles congruent to theirs with one of the missing angles labeled.
Prepare and display a table for all to see with columns angle 1, angle 2, angle 3 and one row for each group of three students. It should look something like this:
{"table": [{"_attributes": {"border": "1", "cellpadding": "1", "cellspacing": "1", "style": "width:500px"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "student groups"}, {"_attributes": {"scope": "col"}, "_value": "angle 1"}, {"_attributes": {"scope": "col"}, "_value": "angle 2"}, {"_attributes": {"scope": "col"}, "_value": "angle 3"}]}]}], "tbody": [{"tr": [{"td": [{"_attributes": {"scope": "row"}}, {}, {}, {}]}, {"td": [{"_attributes": {"scope": "row"}}, {}, {}, {}]}]}]}]}
Once the three partners are together, they complete one row in the posted table for their triangle’s angle measures. Whole-class discussion to follow.
Students might ask if they can use tracing paper to find congruent triangles. Ask how they would use it and listen for understanding of transformations to check for congruence. Respond that this seems to be a good idea.
Representation: Provide Access for Perception.
Display or provide students with a physical copy of the written directions and read them aloud. Check for understanding by inviting students to rephrase directions in their own words. Consider keeping the display of directions visible throughout the activity.
Supports accessibility for: Language; Memory
Student Facing
Your teacher will give you a card with a picture of a triangle.
The measurement of one of the angles is labeled. Mentally estimate the measures of the other two angles.
Find two other students with triangles congruent to yours but with a different angle labeled. Confirm that the triangles are congruent, that each card has a different angle labeled, and that the angle measures make sense.
Enter the three angle measures for your triangle on the table your teacher has posted.
Student Response
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Activity Synthesis
Begin the discussion by asking students:
"What were your thoughts as you set about to find your partners?"
"How did you know that you found a correct partner?"
Expect students to discuss estimates for angle measures and their experience of how different transformations influence the position and appearance of a polygon.
Next look at the table of triangle angles and ask students:
"Is there anything you notice about the combinations of the three angle measures?"
"Is there something in common for each row?"
Guide students to notice that the sum in each row is the same, 180 degrees. Ask whether they think this is always be true for any triangle. Share with students that in the next activity, they will work towards considering whether this result is true for all triangles.


aligns: no