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: 
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_pos_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_pos_0

8
standard_description: Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.
problem_activity: Narrative
The purpose of this activity is for students to solve problems about a locker game by using the ideas of factors and multiples and by observing patterns in the numbers. The situation offers many possible explorations, but the questions around which lockers are touched and by how many people are designed to elicit understandings about multiplication.
Discussion about which lockers will be open or closed or by whom will likely move students away from thinking about factors and multiples and should not be the focus. The final question challenges students who are ready for more to look for additional patterns and generalize their observations. It is not expected that all students complete it. Students who are ready may simulate opening and closing the lockers in different ways, such as:
using actual lockers, if available
using a printed set of squares labeled 1 through 20
using two color counters and 10-frames, if available
using the head and tail sides of coins
MLR7 Compare and Connect
. Synthesis: Invite groups to prepare a visual display that shows the strategy they used to determine which lockers will be open. Encourage students to include details that will help others interpret their thinking. For example, specific language, using different colors, shading, arrows, labels, notes, diagrams or drawings. Give students time to investigate each other’s work. During the whole-class discussion, ask students, “Which students open or close locker 10? What about locker 20? What patterns do you notice about the number of factors for a locker number and whether it is open or closed?”
Advances: Representing, Conversing
Representation: Internalize Comprehension
. Support students in transferring their knowledge to a new context. Begin by asking, “Does this problem remind anyone of the concepts we have been studying?” Invite students to reference the vocabulary display, and continue to refer them to it throughout the lesson.
Supports accessibility for: Conceptual Processing, Language, Attention
Launch
Groups of 3–4
“Let’s solve some problems about a game you read about earlier, where students take turns opening and closing lockers.”
“Silently read and think about each question.”
1 minute: quiet think time
“Work in your group to answer each question. Consider the representations you created or saw earlier to help you think about the problems.”
Activity
15–20 minutes: group work time
Monitor for students who:
represent each contact with the lockers in a systematic way
represent the open and closed lockers in various ways
reason about factors and multiples and use the vocabulary in their explanations
Encourage students who use a pattern to answer the final question to share their explanation. Invite them to consider a situation with more lockers such as 50 or 100.
Student Facing
The 20 students in Tyler’s fourth-grade class are playing a game in a hallway with 20 lockers in a row.
Your goal is to find out which lockers will be touched as all 20 students take their turn touching lockers.
[IMG]
[IMG]
Which locker numbers does the 3rd student touch?
Which locker numbers does the 5th student touch?
How many students touch locker 17? Explain or show how you know.
Which lockers are only touched by 2 students? Explain or show how you know.
Which lockers are touched by only 3 students? Explain or show how you know.
Which lockers are touched the most? Explain or show how you know.
If you have time: Which lockers are still open at the end of the game? Explain or show how you know.
Student Response
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Activity Synthesis
“Take a few minutes to share your answers with students in a different group.”
Invite students to share how they kept track of the number of touches for each locker.
See lesson synthesis.


aligns: yes