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: Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^(2) × 3^(–5) = 3^(–3) = 1/3^(3) = 1/27.

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.

Use square root and cube root symbols to represent solutions to equations of the form x^(2) = p and x^(3) = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that Ö2 is irrational.

Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
problem_activity: Activity
In previous activities and lessons, students found the exact area of a square in order to find an approximation for the square root of an integer. In this activity, students start with a square root of an integer, and draw a square to verify that a given approximation of the square root is reasonable. This is the first time students have seen or drawn squares that do not have vertices at the intersection of grid lines, so it may take them a few minutes to make sense of the new orientation.
Teacher Notes for IM 6–8 Math Accelerated
After arranging students in groups of 2, display for all to see
\(\sqrt{7}\)
and
\(\sqrt{50}\)
and ask, “What two whole numbers does each square root lie between? Be prepared to explain your reasoning.” After a brief quiet think time, invite students to share their reasoning. For example, students may make a list of perfect squares and then find which two the number was between. Once 2–3 strategies are shared, have students continue with the activity as indicated in the launch.
Launch
Give students access to four-function calculators. Display the diagram for all to see. Ask students what is the same and what is different about this diagram and diagrams they have seen in earlier activities. Arrange students in groups of 2. Give students 2–3 minutes of quiet work time followed by partner and whole-class discussions.
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer for data collection and organizing information.
Supports accessibility for: Language; Organization
Writing, Speaking, Listening: MLR1 Stronger and Clearer Each Time.
After students have had time to identify a point on the number line that is closer to
\(\sqrt{3}\)
, ask them to write a brief explanation of their reasoning. Give students time to meet with 2–3 partners, to share and get feedback on their writing. Display prompts that students can ask each other that will help students strengthen their ideas and clarify their language. For example, “How do you know that
\(\sqrt{3}\)
is between 1.5 and 2?” and “How did you determine the area of the square that you drew?” Students can borrow ideas and language from each partner to refine and clarify their original explanation. This will help students refine both their ideas and their verbal and written output.
Design Principles(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
Diego said that he thinks that
\(\sqrt{3}\approx 2.5\)
.
[IMG]
Use the square to explain why 2.5 is not a very good approximation for
\(\sqrt{3}\)
. Find a point on the number line that is closer to
\(\sqrt{3}\)
. Draw a new square on the axes and use it to explain how you know the point you plotted is a good approximation for
\(\sqrt{3}\)
.
Use the fact that
\(\sqrt{3}\)
is a solution to the equation
\(x^2 = 3\)
to find a decimal approximation of
\(\sqrt{3}\)
whose square is between 2.9 and 3.1.
Student Response
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Student Facing
Are you ready for more?
A farmer has a grassy patch of land enclosed by a fence in the shape of a square with a side length of 4 meters. To make it a suitable home for some animals, the farmer would like to carve out a smaller square to be filled with water, as in the figure.
What should the side length of the smaller square be so that half of the area is grass and half is water?
[IMG]
Student Response
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for free access to Extension Student Response.
Activity Synthesis
Invite one or two students to share their squares. Then tell students, “The square of a point on the number line can be visualized as the area of a literal square. This can help us estimate the value of a square root. Simply squaring the number can as well. Let’s check the squares of some numbers that are potential approximations of
\(\sqrt{3}\)
.”
Then ask students to suggest decimal approximations, and check together as a class by finding their squares. Students should be using each guess to make a better guess next. For example, if they try 1.5, then the square is 2.25, which is too low. This suggests trying bigger. Because we know that 2 is too big (because
\(2^2 = 4\)
, it should be somewhere in between 1.5 and 2.) For example, students might suggest this order:
\(1^2 = 1\)
and
\(2^2=4\)
\(1.5^2 = 2.25\)
\(1.8^2 = 3.24\)
\(1.7^2 = 2.89\)
\(1.72^2 = 2.9584\)
\(1.73^2 = 2.9929\)
So 1.73 is a pretty good approximation of
\(\sqrt{3}\)
.


correct_option_index: 2