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.
|                                                          | **           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?


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 following problem or activity 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: Add and subtract vectors.
problem_activity: Activity
The purpose of this activity is for students to use rational approximations of irrational numbers to place both rational and irrational numbers on a number line, and to reinforce the definition of a cube root as a solution to the equation of the form
\(x^3=a\)
. This is also the first time that students have thought about negative cube roots.
Launch
No access to calculators. Students in groups of 2. Two minutes of quiet work time, followed by partner then whole-class discussion.
Action and Expression: Develop Expression and Communication.
Maintain a display of important terms and vocabulary. During the launch take time to review the following term with a number line and sample calculations from previous activities that students will need to access for this activity: cube root.
Supports accessibility for: Memory; Language
Student Facing
The numbers
\(x\)
,
\(y\)
, and
\(z\)
are positive, and:
\(\displaystyle x^3= 5\)
\(\displaystyle y^3= 27\)
\(\displaystyle z^3= 700\)
[IMG]
Plot
\(x\)
,
\(y\)
, and
\(z\)
on the number line. Be prepared to share your reasoning with the class.
Plot
\(\text- \sqrt[3]{2}\)
on the number line.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Diego knows that
\(8^2=64\)
and that
\(4^3=64\)
. He says that this means the following are all true:
\(\sqrt{64}=8\)
\(\sqrt[3]{64}=4\)
\(\sqrt{\text -64}=\text-8\)
\(\sqrt[3]{\text -64}=\text -4\)
Is he correct? Explain how you know.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Display the number line from the activity for all to see. Select groups to share how they chose to place values onto the number line. Place the values on the displayed number line as groups share, and after each placement poll the class to ask if students used the same reasoning or different reasoning. If any students used different reasoning, invite them to share with the class.
Conclude the discussion by asking students to share how they placed -
\(\sqrt[3]{2}\)
on the number line.
Representing, Conversing, Listening: MLR7 Compare and Connect.
As students prepare their work before the discussion, look for approaches that favor the algebraic definition, and approaches that use the geometric interpretation. Call students’ attention to the different ways students describe how they operated on numbers cubed to determine the approximate cube roots, and to the different ways these operations are made visible in each representation (e.g., using the fact that
\(\sqrt[3]{a}\)
is a solution to the equation
\(x^3 = a\)
, and finding rational approximations of the irrational numbers to plot the cube roots on the number line). Emphasize language used to make sense of strategies reinforcing the definition of a cube root as a solution to the equation of the form
\(x^3 = a\)
.
Design Principle(s): Maximize meta-awareness; Support sense-making


aligns: no