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: 
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_pos_0

7

fl_problem_001392_pos_0

8
standard_description: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))^(3) = 5(^(1/3))^(3) to hold, so (5^(1/3))^(3) must equal 5.
problem_activity: Activity
This activity requires students to make sense of negative powers of 10 as repeated multiplication by
\(\frac{1}{10}\)
in order to distinguish between equivalent exponential expressions. If students have time, instruct them to write the other expressions in each table as a power of 10 with a single exponent as well. Look for students who have productive debate regarding the interpretation of, for example,
\((10^2)^{\text-3}\)
versus
\((10^{\text-2})^3\)
and ask them to share their reasoning later (MP3).
Launch
Arrange students in groups of 2. Give students 15 minutes of partner work time followed by a whole-class discussion. Ask students to explain their reasoning to their partner as they work. If there is disagreement, tell students to work to reach an agreement.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time.
Supports accessibility for: Organization; Attention
Student Facing
Match each exponential expression with an equivalent multiplication expression:
\(\left(10^2\right)^3\)
\(\left(10^2\right)^{\text -3}\)
\(\left(10^{\text -2}\right)^3\)
\(\left(10^{\text -2}\right)^{\text-3}\)
{"table": [{"_attributes": {"align": "center", "border": "1"}, "tbody": [{"tr": [{"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "43590"}, "_value": "\\(\\frac{1}{(10 \\boldcdot 10)} \\boldcdot \\frac{1}{(10 \\boldcdot 10)} \\boldcdot \\frac{1}{(10 \\boldcdot 10)}\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "43636"}, "_value": "\\(\\left(\\frac{1}{10} \\boldcdot \\frac{1}{10}\\right)\\left(\\frac{1}{10} \\boldcdot \\frac{1}{10}\\right)\\left(\\frac{1}{10} \\boldcdot \\frac{1}{10}\\right)\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "43680"}, "_value": "\\(\\frac{1}{ \\frac{1}{10} \\boldcdot \\frac{1}{10} }\\boldcdot \\frac{1}{ \\frac{1}{10} \\boldcdot \\frac{1}{10} } \\boldcdot \\frac{1}{ \\frac{1}{10} \\boldcdot \\frac{1}{10} }\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "43719"}, "_value": "\\((10 \\boldcdot 10)(10 \\boldcdot 10)(10 \\boldcdot 10)\\)"}]}]}]}]}]}
Write
\((10^2)^{\text-3}\)
as a power of 10 with a single exponent. Be prepared to explain your reasoning.
Match each exponential expression with an equivalent multiplication expression:
\(\frac{10^2}{10^5}\)
\(\frac{10^2}{10^{\text -5}}\)
\(\frac{10^{\text -2}}{10^5}\)
\(\frac{10^{\text -2}}{10^{\text -5}}\)
{"table": [{"_attributes": {"align": "center", "border": "1"}, "tbody": [{"tr": [{"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "43945"}, "_value": "\\(\\frac{ \\frac{1}{10} \\boldcdot \\frac{1}{10} }{ \\frac{1}{10} \\boldcdot \\frac{1}{10} \\boldcdot \\frac{1}{10}\\boldcdot \\frac{1}{10}\\boldcdot \\frac{1}{10} }\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "43989"}, "_value": "\\(\\frac{10 \\boldcdot 10}{10 \\boldcdot 10 \\boldcdot 10 \\boldcdot 10 \\boldcdot 10}\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "44030"}, "_value": "\\(\\frac{ \\frac{1}{10} \\boldcdot \\frac{1}{10} }{ 10 \\boldcdot 10\\boldcdot 10\\boldcdot 10\\boldcdot 10 }\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "44069"}, "_value": "\\(\\frac{ 10 \\boldcdot 10 }{ \\frac{1}{10} \\boldcdot \\frac{1}{10} \\boldcdot \\frac{1}{10}\\boldcdot \\frac{1}{10}\\boldcdot \\frac{1}{10}}\\)"}]}]}]}]}]}
Write
\(\frac{10^{\text -2}}{10^{\text -5}}\)
as a power of 10 with a single exponent. Be prepared to explain your reasoning.
Match each exponential expression with an equivalent multiplication expression:
\(10^4 \boldcdot 10^3\)
\(10^4 \boldcdot 10^{\text -3}\)
\(10^{\text -4} \boldcdot 10^3\)
\(10^{\text -4} \boldcdot 10^{\text -3}\)
{"table": [{"_attributes": {"align": "center", "border": "1"}, "tbody": [{"tr": [{"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "44231"}, "_value": "\\((10 \\boldcdot 10 \\boldcdot 10 \\boldcdot 10) \\boldcdot ( \\frac{1}{10} \\boldcdot \\frac{1}{10}\\boldcdot \\frac{1}{10})\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "44275"}, "_value": "\\(\\left(\\frac{1}{10} \\boldcdot \\frac{1}{10} \\boldcdot \\frac{1}{10} \\boldcdot \\frac{1}{10}\\right) \\boldcdot \\left( \\frac{1}{10} \\boldcdot \\frac{1}{10} \\boldcdot \\frac{1}{10}\\right)\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "44316"}, "_value": "\\(\\left(\\frac{1}{10}\\boldcdot \\frac{1}{10} \\boldcdot \\frac{1}{10} \\boldcdot \\frac{1}{10}\\right) \\boldcdot \\left(10 \\boldcdot 10 \\boldcdot 10\\right)\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "44356"}, "_value": "\\((10 \\boldcdot 10 \\boldcdot 10 \\boldcdot 10) \\boldcdot (10 \\boldcdot 10 \\boldcdot 10)\\)"}]}]}]}]}]}
Write
\(10^{\text-4} \boldcdot 10^3\)
as a power of 10 with a single exponent. Be prepared to explain your reasoning.
Student Response
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click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Priya, Jada, Han, and Diego stand in a circle and take turns playing a game.
Priya says, SAFE. Jada, standing to Priya's left, says, OUT and leaves the circle. Han is next: he says, SAFE. Then Diego says, OUT and leaves the circle. At this point, only Priya and Han are left. They continue to alternate. Priya says, SAFE. Han says, OUT and leaves the circle. Priya is the only person left, so she is the winner.
Priya says, “I knew I’d be the only one left, since I went first.”
Record this game on paper a few times with different numbers of players. Does the person who starts always win?
Try to find as many numbers as you can where the person who starts always wins. What patterns do you notice?
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 students may struggle to distinguish between
\(\left(10^{\text-2}\right)^3\)
and
\(\left(10^2\right)^{\text-3}\)
. Breaking down each expression into parts that emphasize repeated multiplication will help to illustrate the difference. For the first expression,
\(10^{\text-2} = \frac{1}{10} \boldcdot \frac{1}{10}\)
. The outer exponent of 3 means that
\(\frac{1}{10} \boldcdot \frac{1}{10}\)
is multiplied repeatedly 3 times. Similarly for the second expression,
\(10^2 = 10 \boldcdot 10\)
and the outer exponent of
\(\text- 3\)
means that the
reciprocal
of
\(10 \boldcdot 10\)
is multiplied repeatedly 3 times. In the end, both expressions are equal to
\(10^{\text-6}\)
.
Activity Synthesis
Select students who had disagreements during the activity to share what they disagreed about and how they came to an agreement. Consider asking:
“How is
\((10^{\text-2})^3\)
different from
\((10^2)^{\text-3}\)
? How are they the same?”
“Do you agree or disagree? Why?”
“Could you restate __'s reasoning in a different way?”
It is important for students to understand that the exponent rules work even with negative exponents. To see why, the whole class discussion must make a clear connection between the exponent rules and the process of multiplying repeated factors that are 10 and
\(\frac{1}{10}\)
. Contrast the expanded version of
\(\left(10^{\text-2}\right)^3\)
and
\(\left(10^2\right)^{\text-3}\)
. For
\((10^{\text-2})^3\)
, there are 3 factors that are
\(10^{\text-2}\)
, where
\(10^{\text-2}\)
is two factors that are
\(\frac{1}{10}\)
, so
\(\displaystyle \left(10^{\text-2}\right)^3 = \left( \frac{1}{10} \boldcdot \frac{1}{10} \right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right) = \frac{1}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10} = 10^{\text-6}.\)
For
\((10^2)^{\text-3}\)
, there are 3 factors that are
\(\frac{1}{10^2}\)
, so
\(\displaystyle (10^2)^{\text-3} = \left( \frac{1}{10 \boldcdot 10} \right)\left(\frac{1}{10 \boldcdot 10}\right)\left(\frac{1}{10 \boldcdot 10}\right) = \frac{1}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10} = 10^{\text-6}.\)
Both
\(\left(10^{\text-2}\right)^3\)
and
\(\left(10^2\right)^{\text-3}\)
are equal to
\(10^{\text-6}\)
in the same way that
\(\text-2 \boldcdot 3 = 2 \boldcdot \text-3 = \text-6\)
.
Listening, Conversing: MLR8 Discussion Supports.
As students share the disagreements they had with their partner, press for details in students’ statements by requesting that students elaborate on an idea or give an example of their process of reconciliation. For example, ask students, “How did you explain your reasoning to your partner in a way that convinced them?” In this discussion, it may be useful to revoice student ideas to model mathematical language use by restating a statement in order to clarify, apply appropriate language, or involve more students.
Design Principle(s): Maximize meta-awareness


aligns: no