few_shots: 
Example:
Problem/activity:
Narrative
The purpose of this activity is for students to create composite shapes and compose new shapes from composite shapes. For example, students may find that a rectangular prism can be made from four triangular prisms or from eight smaller cubes. Students then discover that they can combine two of the composed rectangular prisms to make a new rectangular prism.
Required Materials
Materials to Gather
Geoblocks
Launch
Groups of 4
Give each group geoblocks.
Display a large cube.
“This cube can be built from other shapes. See how many ways you can build a cube using other shapes.”
3 minutes: partner work time
Share responses.
Activity
“Choose a different shape. Try to find smaller shapes you can use to build that shape.”
5 minutes: partner work time
Student Response
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for free access to Student Response.
Activity Synthesis
Display a cube and a composite cube—for example, a cube composed of four small cubes and one rectangular prism.
Demonstrate putting the shapes together.
“What new shape did I make with the cube and the cube I made?”
“Put your shapes together to make a rectangular prism.”
Display a rectangular prism and two triangular prisms.
“Find these shapes. Put them together. What new shape can you make?” (a cube, a rectangular block)


Concept/skill: 
Recognize area as an attribute of plane figures and understand concepts of area measurement.

Answer: 
No. This activity does not teach students area as an attribute of plane figures. Instead, the activity focuses on solid, three-dimensional figures.

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

shot_info: im_lesson_009431_neg_0

1
standard_description: Use the structure of an expression to identify ways to rewrite it. For example, see x^(4) – y^(4 )as (x^(2))^(2 )– (y^(2))^(2), thus recognizing it as a difference of squares that can be factored as (x^(2 )– y^(2))(x^(2 )+ y^(2)).
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
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
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: yes