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: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
problem_activity: Activity
Students extend their understanding of exponents to include negative exponents and explain patterns in the placements of the decimal point when a decimal is multiplied by 10 or
\(\frac{1}{10}\)
. Students use repeated reasoning to recognize that negative powers of 10 represent repeated multiplication by
\(\frac{1}{10}\)
and generalize to the rule
\(10^{\text- n} = \frac{1}{10^n}\)
(MP8).
A table is used to show different representations of decimals, fractions, and exponents. The table is horizontal to mimic the structure of decimals. As students work, notice the strategies they use to go between the different representations of the given powers of 10. Ask students who use contrasting strategies to share later.
Launch
Tell students to complete the table one row at a time to see the patterns most clearly. Ask a student to read the first question aloud. Select a student to explain the idea of a “multiplier” in this context. Give students 5–7 minutes to work. Follow with a whole-class discussion.
Student Facing
Complete the table to explore what negative exponents mean.
[IMG]
As you move toward the left, each number is being multiplied by 10. What is the multiplier as you move right?
How does a multiplier of 10 affect the placement of the decimal in the product?  How does the other multiplier affect the placement of the decimal in the product?
Use the patterns you found in the table to write
\(10^{\text -7}\)
as a fraction.
Use the patterns you found in the table to write
\(10^{\text -5}\)
as a decimal.
Write
\(\frac{1}{100,000,000}\)
using a single exponent.
Use the patterns in the table to write
\(10^{\text -n}\)
as a fraction.
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
Some students may think that, for example,
\(\frac{1}{1,000,000} = 10^{\text-7}\)
because the number 1,000,000 has 7 digits. Ask these students if it is true that
\(\frac{1}{10} = 10^{\text-2}\)
.
Activity Synthesis
One important idea is that multiplying by 10 increases the exponent, thus multiplying by
\(\frac{1}{10}\)
decreases the exponent. So negative exponents can be thought of as repeated multiplication by
\(\frac{1}{10}\)
, whereas positive exponents can be thought of as repeated multiplication by
\(10\)
. Another key point is the effect that multiplying by 10 or
\(\frac{1}{10}\)
has on the placement of the decimal.
Ask students to share how they converted between fractions, decimals, and exponents. Record their reasoning for all to see. Here are some possible questions to consider for whole-class discussion:
“Do you agree or disagree? Why?”
“Did anyone think of this a different way?”
“In your own words, what does
\(10^{\text- 7}\)
mean? How is it different from
\(10^7\)
?”
Introduce the visual display for
\(10^{\text- n} = \frac{1}{10^n}\)
and display it for all to see throughout the unit. For an example that illustrates the rule, consider displaying
\(10^{\text- 3} = \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} = \frac{1}{10^3}\)
.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections. For example, annotate the patterns students notice to help synthesize how negative exponents can be thought of as repeated multiplication by
\(\frac{1}{10}\)
, whereas positive exponents can be thought of as repeated multiplication by
\(10\)
.
Supports accessibility for: Visual-spatial processing
Conversing, Representing, Writing: MLR2 Collect and Display.
During the whole-class discussion, capture the vocabulary and phrases students use to describe their strategies for converting between fractions, decimals, and exponents. Listen for students who make connections between repeated multiplication and the placement of the decimal point. Record this language onto a visual display that can be referenced in future discussions. This routine provides feedback to students that supports sense-making and increases meta-awareness of mathematical language.
Design Principle(s): Support sense-making; Maximize meta-awareness


aligns: no