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 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_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_pos_0

8
standard_description: Write, read, and evaluate expressions in which letters stand for numbers.
problem_activity: Activity
The purpose of this task is to give students experience working with exponential expressions and to promote making use of structure (MP7) to compare exponential expressions. To this end, encourage students to rewrite expressions in a different form rather than evaluate them to a single number.
For students who are accustomed to viewing the equal sign as a directive that means “perform an operation,” tasks like these are essential to shifting their understanding of the meaning of the equal sign to one that supports work in algebra, namely, “The expressions on either side have the same value.”
Launch
Before students start working, it may be helpful to demonstrate how someone would figure out whether or not an equation is true without evaluating each expression. For example:
Is
\(4^2=2^3\)
true? Well, let's see. We can rewrite each side like this:
\(\displaystyle 4 \boldcdot 4 = 2 \boldcdot 2 \boldcdot 2\)
Then we can replace one of those
\(2 \boldcdot 2\)
's with a 4, like this:
\(\displaystyle 4 \boldcdot 4 = 4 \boldcdot 2\)
Now we can tell this equation is not true.
These problems can also be worked by directly evaluating expressions, which is fine, as it serves as practice evaluating exponential expressions.
Representation: Internalize Comprehension.
Begin with the demonstration as described in the launch to support connections between new situations involving evaluating exponential expressions and prior understandings. Use color or annotations to highlight what changes and what stays the same at each step.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
Decide whether each equation is true or false, and explain how you know.
\(2^4=2 \boldcdot 4\)
\(3+3+3+3+3=3^5\)
\(5^3=5 \boldcdot 5 \boldcdot 5\)
\(2^3=3^2\)
\(16^1=8^2\)
\(\frac12 \boldcdot  \frac12 \boldcdot  \frac12 \boldcdot  \frac12 = 4 \boldcdot \frac12\)
\(\left( \frac12 \right)^4=\frac{1}{8}\)
\(8^2 = 4^3\)
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Activity Synthesis
Invite students who evaluated the expressions and students who used structure or the meaning of exponents and operations to present their work. Compare and connect the strategies by noting where the use of structure might prove more efficient, where evaluating might be simpler, where thinking about the meaning of exponents and operations can make the true or false determination simpler.
Some guiding questions to highlight the meaning of exponents:
“Can we switch the order with exponents like we can with addition and multiplication—specifically, are
\(a^b\)
and
\(b^a\)
equivalent? How do you know?” (No, you can try different values of
\(a\)
and
\(b\)
or use the meaning of exponents to see that
\(a\)
multiplied
\(b\)
times is not always the same as
\(b\)
multiplied
\(a\)
times.)
“What change can we make to the equation
\(3+3+3+3+3=3^5\)
to make it true?” (Change addition to multiplication on the left, or change the exponent to multiplication on the right.)
“Your friend claims the equation
\(\frac12 \boldcdot  \frac12 \boldcdot  \frac12 \boldcdot  \frac12 = 4 \boldcdot \frac12\)
is true. What do you think they are misunderstanding? How can you convince them it is false?” (Possibly by saying that the left side shows multiplying
\(\frac12\)
4 times, but the right side shows multiplying 4 by
\(\frac12\)
, which means
adding
4 copies of
\(\frac12\)
, not multiplying them. You can show similar examples with other numbers, but the best way to convince them is to talk about what exponents and multiplication mean.)
“Can we show that
\(8^2 = 4^3\)
is true without evaluating both sides? What understanding about the meaning of exponents and operations can help us?” (
\(8^2\)
means
\(8\boldcdot 8\)
or
\(4\boldcdot 2\boldcdot 4\boldcdot 2\)
, which also equals
\(4\boldcdot 4\boldcdot 2\boldcdot 2\)
or
\(4\boldcdot 4\boldcdot 4\)
. Another way to write
\(4\boldcdot 4\boldcdot 4\)
is
\(4^3\)
. We are using the understanding that we can multiply in any order, or the commutative and associative properties of multiplication.)
Speaking, Writing: MLR8 Discussion Supports.
Revoice language and push for clarity in reasoning when students discuss their strategies for determining whether the equations are true or false. Provide a sentence frame such as “The equation is true (or false) because ______.” This will strengthen students’ mathematical language use and reasoning when discussing the meaning of exponents and operations that can make the equivalence of expressions true or false.
Design Principle(s): Maximize meta-awareness


aligns: no