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

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:
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_neg_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
problem_activity: Activity
The purpose of this task is to give students a good reason to compute a rate “per 1.” Since the task statement doesn't provide all the information needed to answer the question and does not suggest a solution pathway, it is an opportunity for students to make sense of the problem (MP1) and engage in some aspects of mathematical modeling (MP4).
Students learn about the Burj Khalifa, the tallest high-rise in the world, and try to determine how long it would take to wash all of its 24,348 windows given a specific rate of work by a window-washing crew. The fact that the total number of windows is not a multiple of the corresponding value in the given ratio (15 windows in 18 minutes) motivates students to identify the rate for 1 and then scale that number to answer the question.
This activity does not specify which “per 1” rate students should find. Some students are likely to first calculate
\(\frac56\)
windows per minute instead of
\(1\frac15\)
or 1.2 minutes per window. Students are not expected to name the above quantities as “rates per 1” or “unit rates” at this time. As students work, ask them to explain what the quantity they calculated (either windows per minute or minutes per window) means and how they plan to use it. If students are unsure where to start with the problem, ask, “At this rate, how long will it take the crew to finish 90 windows? 300 windows?”
Launch
Ask students to close their workbooks or devices, and ask if they know what the Burj Khalifa is. Allow students who are familiar with the building to share what they know.
Then, display a picture of the building and a map of its location.
[IMG]
Once students see that the Burj Khalifa is the tallest artificial structure in the world and is located on the coast of the Arabian Gulf, surrounded by desert, ask what else they wonder about the building. Pause here to allow students time to think of a question. Keep the photo and map of the Burj Khalifa displayed.
Select students to share their questions, limiting it to one question per student. Record 5–10 questions for all to see. If no students wonder about the number of windows, say that you also have a question about the building and add to the list: “How long does it take to wash all the windows on the Burj Khalifa?”
Explain that your question came to mind after watching an online video of a window washing crew on the Burj Khalifa, using brushes and squeegees to clean the windows while harnessed to ropes.
Tell students they will now try to answer a question about the window washing. Give students 3–5 minutes of quiet think time to do so.
Answering the question requires knowing how many windows the Burj Khalifa has. The amount,
24,348 windows
, is readily available through online searches if students are allowed and able to go online. Otherwise, have the number ready to share, and consider announcing the value to the whole class only after several students have requested the information. Recognizing missing information and the steps needed to acquire it is part of the mathematical modeling process.
Some facts on the Burj Khalifa:
Height: 2,722 ft (829.8 meters)
Tallest artificial structure in the world (as of January 2016)
154 usable floors, 9 maintenance floors, 46 spire levels, 2 below-ground parking lots (Wikipedia)
57 total elevators
It took 6 hours for an individual to climb to the top of the Burj Khalifa in 2011.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, ask students to identify the rate for 1 window or 1 minute. Next, ask students to find how long it would take to wash a portion of the windows, such as 100 or 1000. After that, ask students how long it would take to wash all the windows of the Burj Khalifa.
Supports accessibility for: Organization; Attention
Speaking: MLR7 Compare and Connect.
Use this routine when students present their strategy and representations to find how long it will take to wash all the windows. Ask students to consider what is the same and what is different about each approach. Draw students’ attention to the different quantities they calculated (either windows per minute or minutes per window). These exchanges can strengthen students’ mathematical language use and reasoning to make sense of strategies used to compute a rate “per 1.”
Design Principle(s): Maximize meta-awareness
Student Facing
[IMG]
A window-washing crew can finish 15 windows in 18 minutes.
If this crew was assigned to wash all the windows on the outside of the Burj Khalifa, how long will the crew be washing at this rate?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may get stuck when they try to find the minutes per window because 15 does not divide evenly into 18. Help them reason in terms of equivalent ratios or recall grade 5 methods for determining a quotient that is not a whole number.
Activity Synthesis
Invite a few students to explain their reasoning and solutions. Ask students who used both
\(1\frac15\)
or 1.2 minutes per window and
\(\frac56\)
windows per minute to share. Emphasize that both quantities are valid and have purpose, depending on the question you are answering.
Focus the discussion on strategies for identifying and correcting errors. As rate problems grow more complex, students become more likely to mix up numbers, calculate less-helpful rates per 1, or make arithmetic mistakes that are left unquestioned. If possible, have several students share approaches that did not work, and errors that were made, and ask them to explain how they knew they must have made an error.
If no students offer to share, have an example of a student error ready to display for the class. For example, a student may incorrectly calculate
\(\frac56\)
minutes per window instead of
\(1\frac15\)
minutes per window. Ask students to discuss with a partner what they think the error is and how they would help the student. Select 1–2 students to share their ideas with the class; while they share, make the corrections for all to see.
Lastly, students may point out that the answer, 29,217.6 minutes, is a total time and is not representative of how many normal work days it would take the crew to do the job. Time permitting, pursue this valuable line of thinking, as it shows that students are thinking about the math and the context (MP2).


aligns: no