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: 
Analyze and solve pairs of simultaneous linear equations.

Answer: 
Yes. 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.

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_pos_0

8
standard_description: Extend polynomial identities to the complex numbers. For example, rewrite x^(2) + 4 as (x + 2i)(x – 2i).
problem_activity: Activity
Earlier, students noticed that when an expression
\(kx+m\)
is squared and written in standard form
\(ax^2+bx+c\)
, the value of
\(b\)
is
\(2km\)
and the value of
\(c\)
is
\(m^2\)
. In this activity, they use these observations to complete expressions to make them perfect squares and then write them as squared factors. Then, they apply this skill to solve quadratic equations.
Launch
To reiterate the connections between
\((kx + m)^2\)
and its equivalent expression of the form
\(ax^2+bx+c\)
, display these two expressions for all to see and ask students:
“If the two expressions are perfect squares and are equivalent, how is
\(a\)
related to
\(k\)
?” (
\(a\)
is
\(k^2\)
.)
“How is
\(c\)
related to
\(m\)
?” (
\(c\)
is
\(m^2\)
.)
“How is
\(b\)
related to
\(k\)
and
\(m\)
?” (
\(b\)
is
\(2km\)
.)
Tell students they will use these insights to complete some squares. Consider keeping students in groups of 2.
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students share their reasoning about the connections between
\((kx+m)^2\)
and
\(ax^2+ bx+c\)
, scribe their thinking on a visible display. Invite students to suggest language or diagrams to include that will support their understanding. Keep the display visible as students continue to work.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
Find the value of
\(c\)
to make each expression in the left column a perfect square in standard form. Then, write an equivalent expression in the form of squared factors. In the last row, write your own pair of equivalent expressions.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "standard form", "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "26145"}, "_value": "\\((ax^2+bx+c)\\)"}]}, {"_attributes": {"scope": "col"}, "_value": "squared factors", "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "2851"}, "_value": "\\((kx+m)^2\\)"}]}]}]}], "tbody": [{"tr": [{"td": [{"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "26166"}, "_value": "\\(100x^2+80x+c\\)"}]}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "26189"}, "_value": "\\(36x^2-60x+c\\)"}]}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "26219"}, "_value": "\\(25x^2+40x+c\\)"}]}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "26251"}, "_value": "\\(0.25x^2-14x+c\\)"}]}, {}]}, {"td": [{}, {}]}]}]}]}
Solve each equation by completing the square:
\(25x^2 + 40x = \text-12\)
\(36x^2 - 60x + 10 = \text-6\)
Student Response
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Anticipated Misconceptions
If students struggle to follow the generalized relationships between
\((kx + m)^2\)
and its equivalent expression of the form
\(ax^2+bx+c\)
(as discussed in the launch), consider revisiting a concrete example from an earlier activity—for example,
\((5x-2)^2\)
. Ask students to relate each term in the squared factor to the terms of its expression in standard form,
\(25x^2 - 20x + 4\)
.
If further scaffolding is helpful, consider using an example with an additional variable (for instance,
\((3x+n)^2\)
and its counterpart in standard form,
\(9x^2 + 2(3)(n)x + n^2\)
) before generalizing the equivalence of the two forms in entirely abstract terms.
Activity Synthesis
Ask students to write the standard-form expression they invented on a piece of scrap paper, except without the constant term. (From the blank row of the table.) Invite them to switch papers with a partner, and complete the square and write each other’s expressions in factored form. If they get stuck, encourage them to talk with their partner to work together and fix any mistakes.
If time permits, invite students to share their responses and strategies. Discuss questions such as:
“How did you know what value to use for
\(c\)
in, say,
\(100x^2 + 80x + c\)
?” (
\(k^2=100\)
, so
\(k\)
is either 10 or -10. If
\(k\)
is 10, then
\(2(10)m=80\)
and
\(m=4\)
. We know that
\(c=m^2\)
, so
\(c\)
is
\(4^2\)
or 16. If
\(k\)
is -10,then
\(2(\text-10) m=80\)
and
\(m= \text-4\)
. Squaring -4 still gives 16.)
“How did you know whether
\(c\)
would be positive or negative?” (
\(c\)
is some number squared, so it will always be positive.)
“In the equation
\(36x^2 - 60x + 10 = \text-6\)
, the expression on the left already has 10 as a constant term but it is not a perfect square. How do we make it a perfect square?” (25 would make it a perfect square, so we can either add 15 to each side of the equation, or first subtract 10 from each side and then add 25.)
As students explain their solution methods on the second question, record and display their reasoning for all to see, or display a worked solution such as:
\(\begin{align} 36x^2 - 60x + 10 &= \text-6\\ 36x^2 - 60x + 10 + 15 &= \text-6 + 15\\ 36x^2 - 60x + 25 &= 9\\ (6x-5)^2 &=9 \end{align}\)
\( \)
\(\begin {align} 6x-5 =3 \quad &\text{or} \quad 6x-5 =\text-3\\ 6x = 8 \quad &\text{or} \quad 6x = 2\\ x = \frac86 =\frac43 \quad & \text{or} \quad x = \frac13 \end{align}\)


aligns: no