few_shots: 
Example:
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_001392_neg_0

8
standard_description: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
problem_activity: Narrative
The purpose of this activity is for students to relate multiplying a non-unit fraction by a whole number to multiplying a unit fraction by the same whole number. After finding the value of
\(\frac{1}{5} \times 3\)
in a way that makes sense to them, they then consider the value of the products
\(\frac{2}{5} \times 3\)
and
\(\frac{3}{5} \times 3\)
. In the synthesis students address how they can use the value of
\(\frac{1}{5} \times 3\)
to find the value other expressions.
Launch
Groups of 2
Activity
8 minutes: independent work time
Monitor for students who:
draw a diagram
use division to solve
recognize a relationship between
\(\frac{1}{5}\times3\)
,
\(\frac{2}{5}\times3\)
, and
\(\frac{3}{5}\times3\)
.
Student Facing
Find the value of each expression. Explain or show your reasoning. Draw a diagram if it is helpful.
\(\frac{1}{5} \times 3\)
\(\frac{2}{5} \times 3\)
\(\frac{3}{5} \times 3\)
Student Response
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Activity Synthesis
Ask previously selected students to share their solutions.
Display:
\(\frac{1}{5} \times 3\)
,
\(\frac{2}{5} \times 3\)
,
\(\frac{3}{5} \times 3\)
“How are the expressions the same?” (They all have a 3. They all have some fifths and there is a product.)
“How are the expressions different?” (The number of fifths is different. There is 1 and then 2 and then 3.)
“How can you use the value of
\(\frac{1}{5} \times 3\)
to help find the value of
\(\frac{2}{5} \times 3\)
?” (I can just double the result because it’s
\(\frac{2}{5}\)
instead of
\(\frac{1}{5}\)
.)
“What about
\(\frac{3}{5} \times 3\)
?” (That’s just another
\(\frac{1}{5} \times 3\)
.)
Display diagram from student solution or a student generated diagram like it.
“How does the diagram show
\(\frac{1}{5} \times 3\)
?” (There is 3 total and
\(\frac{1}{5}\)
of it is shaded.)
Display:
\(\frac{2}{5} \times 3\)
.
“How could you adapt the diagram to show
\(\frac{2}{5} \times 3\)
?” (I could fill in 2 of the fifths in each whole instead of 1.)
“In the next activity we will study a diagram for
\(\frac{2}{5} \times 3\)
more.”


aligns: yes