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.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "table-layout:fixed; width:70%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "No solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Exactly 1 solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Infinite number of solutions"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9=4x+7}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y=3y+5}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y={x\\over2}}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${3x+6y-9=0}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${x=y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${1.25x=1.25y}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9.9=6x+8y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9x=2.5y-8.8}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=1.5}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=2.5}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {}]}]}]}]}
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: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
problem_activity: Activity
Students have seen quadratic expressions in both standard form and factored form since the beginning of the unit. In this activity, they learn to distinguish the expressions by their forms and to refer to each form by its formal name. Refining their language about the different forms prepares students to be more precise in their thinking about the graphs of quadratic functions in later lessons (MP6).
Launch
Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.
Writing, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their writing, by providing them with multiple opportunities to clarify their explanations through conversation. Give students time to meet with 1–2 partners to share their response to the final question, “Why do you think that form is called factored form?” Provide listeners with prompts for feedback that will help their partner add detail to strengthen and clarify their ideas. For example, students can ask their partner, “Can you say more about what each expression means?” or “I understand _____, but can you clarify . . .” Next, provide students with 2–3 minutes to revise their initial draft based on feedback from their peers.
Design Principle(s): Optimize output (for explanation)
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer that can be used to brainstorm the activity. Include one column for standard form and one column for factored form. Include a space at the bottom of the organizer for student-generated notes or reminders about standard form and factored form.
Supports accessibility for: Language; Organization
Student Facing
The quadratic expression
\(x^2 + 4x + 3\)
is written in
standard form
.
Here are some other quadratic expressions. The expressions on the left are written in standard form and the expressions on the right are not.
Written in standard form:
\(x^2 – 1\)
\( x^2 + 9x\)
\(\frac12 x^2\)
\(4x^2 – 2x + 5\)
\(\text-3x^2 – x + 6\)
\(1 - x^2\)
Not written in standard form:
\((2x + 3)x\)
\((x+1)(x-1)\)
\(3(x-2)^2 +1\)
\( \text-4(x^2 + x) +7\)
\( (x+8)(\text{-}x+5)\)
What are some characteristics of expressions in standard form?
\((x+1)(x-1)\)
and
\((2x + 3)x\)
in the right column are quadratic expressions written in
factored form
. Why do you think that form is called factored form?
Student Response
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Student Facing
Are you ready for more?
What quadratic expression can be described as being both standard form and factored form? Explain how you know.
Student Response
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for free access to Extension Student Response.
Activity Synthesis
Display the two columns of expressions for all to see. Solicit students’ ideas on the features of each form. Record their responses for all to see. Invite other students to express agreement or disagreement or to clarify their fellow students’ responses.
Define a quadratic expression in
standard form
explicitly as
\(ax^2 + bx + c\)
. Explain that we refer to
\(a\)
as the coefficient of the squared term
\(x^2\)
,
\(b\)
as the coefficient of the linear term
\(x\)
, and
\(c\)
as the constant term.
Ask students:
“How would you write
\((2x + 3)x\)
in standard form?” (
\(2x^2 + 3x\)
)
“The expression
\(2x^2 + 3x\)
only has two terms. Is it still in standard form?” (Yes, there is no constant term, which means
\(c\)
is 0.)
“How would you write
\(\text {-4}(x^2 + \frac14x) +7\)
in standard form?” (
\(\text {-4}x^2 - x + 7\)
)
“What are the values of the coefficients
\(a\)
and
\(b\)
?” (-4 and -1) “What about the constant term?” (7)
Then, clarify that a quadratic expression in
factored form
is a product of two factors that are each a linear expression. For example,
\((x+1)(x-1)\)
,
\((2x + 3)x\)
, and
\(x(4x)\)
all have two linear expressions for their factors. An expression with two factors that are linear expressions and a third factor that is a constant, for example:
\(2(x+2)(x-1)\)
, is also in factored form.
Tell students that each form gives us interesting or useful information about the function that the expression represents, and they will learn more about this in future lessons.


aligns: no