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: 
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.

Answer: 
Yes. 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.

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 to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

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 problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_000972_pos_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
problem_activity: Activity
In this activity students continue the work of generating equivalent expressions as they decide where to place a set of parentheses and explore how that placement affects the expressions.
Launch
Arrange students in groups of 2. Tell students to first complete both questions independently. Then, trade one of their expressions with their partner. The partner’s job is to decide whether the new expression is equivalent to the original or not, and explain how they know.
Student Facing
Diego was taking a math quiz. There was a question on the quiz that had the expression
\(8x - 9 - 12x + 5\)
. Diego’s teacher told the class there was a typo and the expression was supposed to have one set of parentheses in it.
Where could you put parentheses in
\(8x - 9 - 12x + 5\)
to make a new expression that is still equivalent to the original expression? How do you know that your new expression is equivalent?
Where could you put parentheses in
\(8x - 9 - 12x + 5\)
to make a new expression that is not equivalent to the original expression? List as many different answers as you can.
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
Students may not realize that they can break up a term and place a parentheses, for example, between the 8 and
\(x\)
in the term
\(8x\)
. Clarify that they may place the parentheses anywhere in the expression.
Activity Synthesis
Questions for discussion:
“How can you write the original expression in different ways with fewer terms?” (
\(\text-4x-4\)
,
\(\text-4(x+1)\)
,
\(4(-x-1)\)
)
“Where did you place the parentheses to create an equivalent expression?”
“Share other ways you placed parentheses and the resulting expression with fewer terms.”
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary.  Include the following terms and maintain the display for reference throughout the unit: parenthesis. The display should include examples of how to create equivalent expressions using parentheses by breaking up the components. Be sure to emphasize how the placement of the parentheses affects the expression.
Supports accessibility for: Memory; Language
Speaking, Writing: MLR8 Discussion Supports.
When students interpret the new expression they receive, ask students to press their partner for more details in their explanations by challenging an idea, or asking for an example. Listen for, and amplify language students use to describe how the parentheses influence the resulting expression with fewer terms. This will help call students' attention to the types of details and language to look for to determine if an expression is equivalent to another or not.
Design Principle(s): Maximize meta-awareness; Support sense-making


aligns: no