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 matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
problem_activity: Activity
This activity connects the ideas of roots, rational exponents, graphs of exponential functions, and decimal approximations. Students graph
\(y=2^x\)
for a few integer values of
\(x\)
, approximate a smooth curve passing through those points, and use that curve to estimate the value of
\(2^x\)
for various rational inputs. Students then test these estimates using their understanding of the connection between rational exponents and roots.
Launch
Arrange students in groups of 2–3. Encourage them to compare their answers with their group after each question.
Since the first problem in this activity was designed to be completed without technology, ask students to put away any devices until after they complete it.
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to prepare students for the whole class discussion about the graph of
\(y=2^x\)
. After students complete the table and sketch the graph of
\(y=2^x\)
, invite them to create a visual display of their work. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their tables and graphs. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast the estimates for
\(2^x\)
for various values of
\(x\)
between 0 and 2.
Design Principle(s): Cultivate conversation
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer collaboration. When students share their work with a partner, display sentence frames to support conversation such as: “First, I _____ because. . .”, “How did you get . . .?”, “That could/couldn’t be true because . . . .”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Complete the table as much as you can without using a calculator. (You should be able to fill in three spaces.)
{"table": [{"_attributes": {"align": "center", "border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "50"}, "_value": "\\(x\\)"}]}], "td": [{"_value": "0"}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "14"}, "_value": "\\(\\frac13\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "781"}, "_value": "\\(\\frac23\\)"}]}, {"_value": "1"}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "4325"}, "_value": "\\(\\frac43\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "11218"}, "_value": "\\(\\frac53\\)"}]}, {"_value": "2"}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "1139"}, "_value": "\\(2^x\\)"}], "_value": "(using exponents)"}], "td": [{"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "2323"}, "_value": "\\(2^0\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "2115"}, "_value": "\\(2^{\\frac13}\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "2132"}, "_value": "\\(2^{\\frac23}\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "1930"}, "_value": "\\(2^1\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "25624"}, "_value": "\\(2^{\\frac43}\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "25635"}, "_value": "\\(2^{\\frac53}\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "1131"}, "_value": "\\(2^2\\)"}]}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "1139"}, "_value": "\\(2^x\\)"}], "_value": "(decimal approximation)"}], "td": [{}, {}, {}, {}, {}, {}, {}]}]}]}]}
Plot the points that you filled in.
[IMG]
Connect the points as smoothly as you can.
Use this graph of
\(y=2^x\)
to estimate the value of the other powers in the table, and write your estimates in the table.
Let’s investigate
\(2^{\frac13}\)
:
Write
\(2^{\frac13}\)
using radical notation.
What is
\(\left( 2^{\frac13}\right)^3\)
?
Raise your estimate from the table of
\(2^{\frac13}\)
to the third power. What should it be? How close did you get?
Let’s investigate
\(2^{\frac23}\)
:
Write
\(2^{\frac23}\)
using radical notation.
What is the value of
\(\left( 2^{\frac23}\right)^3\)
?
Raise your estimate from the table of
\(2^{\frac23}\)
to the third power. What should it be? How close did you get?
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?
Answer these questions using the fact that
\((1.26)^3 = 2.000376\)
.
Explain why
\(\sqrt[3]{2}\)
is very close to
\(1.26\)
. Is it larger or smaller than
\(1.26\)
?
Is it possible to write
\(\sqrt[3]{2}\)
exactly with a finite decimal expansion? Explain how you know.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Display several student graphs of
\(y=2^x\)
for all to see and invite students to share how close their estimates for the cubes of
\(2^{\frac13}\)
and
\(2^{\frac23}\)
were. The important takeaway from this activity is that whether we write the number as
\(2^{\frac23}\)
or
\(\sqrt[3]{2^2}\)
, the expression is just a number and its value can be approximated.


aligns: no