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: Read, write, and compare decimals to thousandths.

Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

Use place value understanding to round decimals to any place.
problem_activity: Narrative
The purpose of this activity is to provide further practice relating the different forms of decimals. This includes expanded form, word form, and decimal form. Using the balance and weight of gold nuggets as a context, students go back and forth between different ways of representing these weights (MP2). The synthesis highlights the meaning of the digits in a decimal and how that relates to the expanded form of the decimal.
Launch
Display the image.
“What decimal can I write for the weight of the gold nugget?” (0.124)
Write equation:
\(0.124 = \left(1 \times 0.1\right) + \left(2 \times 0.01\right) + \left(4 \times 0.001\right)\)
“How does the balance show the expanded form of 0.124?” (There is 1 tenth of an ounce weight, 2 hundredth of an ounce weights, and 4 thousandth of an ounce weights.)
Activity
8–10 minutes: independent work time
Monitor for students who see, in the first problem, that the digits in the decimal for the gold weights are the same as the number of weights for that decimal place value.
Student Facing
[IMG]
Weights are used to balance some gold nuggets. Write the weight of each gold nugget in expanded form.
three 0.1 ounce weights, five 0.01 ounce weights, and eight 0.001 ounce weights
six 0.1 ounce weights and two 0.001 ounce weights
two 0.01 ounce weights and six 0.1 ounce weights
Here are the weights of some gold nuggets in word form. Write the weights in expanded form.
two hundred eighty three thousandths of an ounce
four hundred nine thousandths of an ounce
A gold nugget weighs 0.527 ounces.
What is the value of each of the digits in the decimal 0.527?
How does the expanded form of 0.527 show the value of each digit in the decimal?
Student Response
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Activity Synthesis
Invite students to share the expanded form of the decimal 0.527.
Display the expression:
\(\left(5 \times 0.1 \right) +\left(2 \times 0.01\right) +\left( 7 \times 0.001\right)\)
“What is the value of the 5 in 0.527?” (5 tenths)
“How does the expanded form show this?” (It shows the 5 is
\(5 \times 0.1\)
or 5 tenths.)
“What is the value of the 7 in 0.527?” (7 thousandths)
“How does the expanded form show this?” (It shows the 7 is
\(7 \times 0.001\)
or 7 thousandths.)
“How is going from word form to expanded form different than going from decimal form to expanded form?” (The decimal form shows the place values. With the word form, everything is given in terms of thousandths, so I need to figure out what the individual place values of the number are.)


correct_option_index: 0

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