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?


Standard options: 
A. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
B. 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.
C. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)


Thought: 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.
Answer: B

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
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Thought: 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).
Answer: B

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.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
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?


Standard options: 
A. Solve linear equations in one variable.
B. Analyze and solve pairs of simultaneous linear equations.


Thought: 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.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
problem_activity: Activity
Up to this point in the unit, students have written and interpreted equations representing exponential change which were meaningful for non-negative input values. In this lesson, they interpret the meaning of a negative value in a context. The independent variable is time,
\(t\)
, so negative values of
\(t\)
refer to times
before
\(t\)
is 0.
For the last question, to find the time when the coral had the given volume, students may:
Calculate the values of
\(y\)
when
\(t\)
is -3, -4, -5, etc. (i.e., extend the table of values, if they previously created one) until they reach 37.5 or a value close to it.
Repeatedly divide by 2 (or multiply by
\(\frac12\)
) the value of
\(y\)
when
\(t\)
is -2 until it approaches or hits 37.5.
Use the equation
\(1,\!200 \boldcdot 2^t = 37.5\)
to find the value of
\(2^t\)
and then reason about
\(t\)
from there.
Look for students using these or other strategies so they can share later.
Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2. Encourage them to think quietly about the questions before discussing with their partner. Creating a table or spreadsheet may help students organize the work in the second question. If needed, encourage students to do so.
Reading, Conversing, Writing: MLR5 Co-Craft Questions.
To help students make sense of the language of exponential situations, start by displaying the task statement “A marine biologist estimates that a structure of coral has a volume of 1,200 cubic centimeters and that its volume doubles each year.” Give students 1–2 minutes to write their own mathematical questions about the situation. Invite students to share their questions with the class, then reveal the activity’s questions. Listen for and amplify any questions referring to the value of
\(t\)
, especially those wondering about time in the past. This will build student understanding and interpretation of the language used to represent negative exponents.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Representation: Internalize Comprehension.
Begin the activity with concrete or familiar contexts. Encourage students to create a timeline next to their tables, selecting a starting point year for zero and then labeling the years correlated with the positive exponents prior to labeling years associated with negative exponents.
Supports accessibility for: Conceptual processing; Memory
Student Facing
A marine biologist estimates that a structure of coral has a volume of 1,200 cubic centimeters and that its volume doubles each year.
[IMG]
Write an equation of the form
\(y=a \boldcdot b^t\)
representing the relationship, where
\(t\)
is time in years since the coral was measured and
\(y\)
is volume of coral in cubic centimeters. (You need to figure out what numbers
\(a\)
and
\(b\)
are in this situation.)
Find the volume of the coral when
\(t\)
is 5, 1, 0, -1, and -2.
What does it mean, in this situation, when
\(t\)
is -2?
In a certain year, the volume of the coral is 37.5 cubic centimeters. Which year is this? Explain your reasoning.
Student Response
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Anticipated Misconceptions
Students may struggle to think of how to start finding the values for
\(a\)
and
\(b\)
. Suggest that they make a table of corresponding values of
\(t\)
and
\(y\)
, and think about starting with
\(t\)
and calculating
\(y\)
.
Activity Synthesis
Review how students found an equation,
\(y = 1,200 \boldcdot 2^t\)
, to represent the volume of coral. Make sure that they recall that the 1,200 represents the 1,200 cubic centimeters of coral when it was first measured and the 2 indicates the doubling of its volume each year.
Select students to share their interpretations of negative values of
\(t\)
and how they found the value of
\(t\)
when given a volume. Ask questions such as:
“In this situation, what does it mean to say that when
\(t\)
is -3,
\(y\)
is
\(1,200 \boldcdot 2^{\text-3}\)
or 150?” (3 years before the coral was measured to be 1,200 cubic centimeters, its volume was 150 cubic centimeters).
“How did you estimate or find when the volume of the coral was 37.5 cubic centimeters?”
It may be helpful to display a graph of
\(y = 1,200 \boldcdot 2^t\)
, such as the one embedded in the digital version of these materials. Display the graph (either with or without the points from the table plotted) and ask students to identify the volume of coral when
\(t\)
is -3 and the time when the volume of coral is 37.5 cubic centimeters.
If using the digital applet, consider initially turning off the function and the point, and hiding the expression list. Ask students to gesture at points on the graph corresponding to questions in the activity. Then, turn on the function and the point for demonstration. Remember that if you click on a plotted point, its coordinates are revealed.


correct_option_index: 0