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: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

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: 
Analyze and solve pairs of simultaneous linear equations.

Answer: 
Yes. 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.

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_neg_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_pos_0

8
standard_description: Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
problem_activity: Activity
Following the concept developed in the warm-up, students will continue to use the idea that outcomes from previous experiments can help inform the likelihood of an outcome when the experiment is repeated. In this activity, students play a game while collecting data. Bags of colored blocks are set up with different probabilities to win and students use random chance to collect points when the color of the block they choose matches the color written on the bag. In the discussion, students will use the data they have collected in three rounds of the game to make a decision about which bag might be most likely to produce a winning block. As an opening activity, this is meant to motivate students into thinking about likelihood of events. As such, it is not important to resolve the questions in the discussion into perfect agreement.
Students are arranged in groups of 4 and given one of 3 different bags labeled with colors. Create enough bags so that each group will have a turn with each of the different color bags after three rounds.
Launch
Arrange students in groups of 4. Explain to students how to play the game:
There will be 3 rounds of the game. Their group will get a different bag of blocks for each round.
They are not allowed to look into the bag or take more than one block out of the bag at a time.
During a round, each person in the group gets 4 turns to take a block out of the bag.
If the block matches the color that is written on the outside of the bag, the person scores (1 point during round 1, 2 points during round 2, and 3 points during round 3). When the block is any other color, they do not earn any points.
They put the block back in the bag, shake the bag to mix up the blocks, and pass the bag to the next person in between each turn.
At the end of a round, they record everyone's points from that round and wait for a new bag of blocks.
Distribute bags of blocks. Each group gets one of these bags:
A bag labeled "Green" that contains 9 green blocks and 3 of another color.
A bag labeled "Blue" that contains 8 blue blocks and 8 of another color.
A bag labeled "Red" that contains 4 red blocks and 10 of another color.
Engagement: Provide Access by Recruiting Interest.
Begin with a small-group or whole-class demonstration of how to play the game.  Check for understanding by inviting students to rephrase directions in their own words.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Your teacher will give your group a bag of colored blocks.
Follow these instructions to play one round of the game:
Everyone in the group records the color written on the bag in the first column of the table.
​Without looking in the bag, one person takes out one of the blocks and shows it to the group.
If they get a block that is the same color as the bag, they earn:
1 point during round 1
2 points during round 2
3 points during round 3
Next, they put the block back into the bag, shake the bag to mix up the blocks, and pass the bag to the next person in the group.
Repeat these steps until everyone in your group has had 4 turns.
At the end of the round, record each person's score in the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "What color bag?"}, {"_attributes": {"scope": "col"}, "_value": "person 1's score"}, {"_attributes": {"scope": "col"}, "_value": "person 2's score"}, {"_attributes": {"scope": "col"}, "_value": "person 3's score"}, {"_attributes": {"scope": "col"}, "_value": "person 4's score"}]}]}], "tbody": [{"tr": [{"th": [{"_value": "round\u00a01"}], "td": [{}, {}, {}, {}, {}]}, {"th": [{"_value": "round\u00a02"}], "td": [{}, {}, {}, {}, {}]}, {"th": [{"_value": "round\u00a03"}], "td": [{}, {}, {}, {}, {}]}]}]}]}
Pause here so your teacher can give you a new bag of blocks for the next round.
Repeat the previous steps to play rounds 2 and 3 of the game.
After you finish playing all 3 rounds, calculate the total score for each person in your group.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Tyler’s class played the block game using purple, orange, and yellow bags of blocks.
During round 1, Tyler’s group picked 4 purple blocks and 12 blocks of other colors.
During round 2, Tyler’s group picked 11 orange blocks and 5 blocks of other colors.
During round 3, Tyler forgot to record how many yellow blocks his group picked.
For a final round, Tyler’s group can pick one block from any of the three bags. Tyler’s group decides that picking from the orange bag would give them the best chance of winning, and that picking from the purple bag would give them the worst chance of winning. What results from the yellow bag could have lead Tyler’s group to this conclusion? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some groups may choose a bag for the bonus round based on the number of points they got in the earlier rounds. Since points are equal in the bonus round, help them see that the blocks picked out is a more important thing to consider than the points for the round.
Activity Synthesis
Tell students that there is a bonus, fourth round. In this round, a block that matches the color on the bag is worth 25 points. Each person will only have one chance to draw a bonus block. In this round, each group will get to choose which bag they would like to draw from. At least 3 people in the group must agree on the bag they will use.
Give students 5 minutes of small-group discussion to agree on which bag the group will use. Tell students that, after the bonus round, they will be asked to explain their reasoning for choosing the bag. It is ok for more than one group to select the same color bag. Identify groups that use the results from the previous rounds to determine which bag will be more likely to draw a winning block.
Allow each group to use the bag they have selected to play the bonus round and total their points.
After all students have played the bonus round, select groups to share their reasoning for choosing which bag to use. Select previously identified students to share their reasoning as well.
Ask students, "Was one bag more likely to give you a winning cube than the others? Explain your reasoning."
Open one of each color bag to reveal the contents. Ask students, "Based on what you see now, does this change your answer? Explain your reasoning."
The green bag should provide the best chance to win points, but it is not essential for the class to come to this understanding at this point.
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to support whole-class discussion as groups share which bag they would choose during the bonus round. Display sentence frames for students to use as they respond to each group: "I agree because ….” or "I disagree because ….” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students.
Design Principle(s): Support sense-making


aligns: no