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: 
Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Students do not need to learn formal names such as “right rectangular prism.”)

Answer: 
Yes. This activity asks students to compose three-dimensional shapes, using geoblocks, to create composite shapes.

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_pos_0

1
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
The purpose of this activity is to bring together identities and the work students did earlier with rational equations. For each question, students first rewrite the fraction
\(\frac{2}{15}\)
using the given formula and then prove that the formula is an identity.
The whole-class discussion at the end of the activity should focus on how students showed that the formula is an identity. Monitor for groups who use different methods to share during the discussion. For example, they may identify different common denominators when adding together the fractions in the expression on the right side of the formula.
Launch
Arrange students in groups of 2. Tell students to complete each problem individually, and then compare their work with their partner. Partners should discuss any similarities and differences in their methods, and decide which method they prefer.
If students need some practice adding fractions with unlike denominators, begin the activity by displaying the equation
\(\frac78 = \frac12 + \frac13 + \frac{1}{24}\)
for all to see and ask students how they can figure out without using a calculator that the equation is true. After some quiet work time, select 1–2 students to share their thinking about why the equation is true. The purpose of looking at this equation before the main activity is to make sure students recall that adding fractions requires a common denominator.
Speaking: MLR8 Discussion Supports.
Use this routine to support students in producing statements justifying their identity formulas. Provide sentence frames for students to use when comparing their methods and discussing which method they prefer, such as “I prefer _____ because _____ .”, or “We both have _____ which represents _____ .”
Design Principle(s): Support sense-making
Student Facing
[IMG]
In Ancient Egypt, all non-unit fractions were represented as a sum of distinct unit fractions. For example,
\(\frac49\)
would have been written as
\(\frac13 + \frac19\)
(and not as
\(\frac19+\frac19+\frac19+\frac19\)
or any other form with the same unit fraction used more than once). Let’s look at some different ways we can rewrite
\(\frac{2}{15}\)
as the sum of distinct unit fractions.
Use the formula
\(\frac{2}{d}=\frac{1}{d}+\frac{1}{2d}+\frac{1}{3d}+\frac{1}{6d}\)
to rewrite the fraction
\(\frac{2}{15}\)
, then show that this formula is an identity.
Another way to rewrite fractions of the form
\(\frac{2}{d}\)
is given by the identity
\(\frac{2}{d} = \frac{1}{d} +\frac{1}{d+1} + \frac{1}{d(d+1)}\)
. Use it to re-write the fraction
\(\frac{2}{15}\)
, then show that it is an identity.
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
For fractions of the form
\(\frac{2}{pq}\)
, that is, fractions with a denominator that is the product of two positive integers, the following formula can also be used:
\(\frac{2}{pq}=\frac{1}{pr} + \frac{1}{qr}\)
, where
\(r = \frac{p+q}{2}\)
. Use it to re-write the fraction
\(\frac{2}{45}\)
, then show that it is an identity.
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Some students may not be sure how to find a common denominator when the denominators all involve variables. Remind them that a common denominator is a multiple of all of the denominators. If needed, students can guess possible common denominators, and then test them by checking whether they're divisible by each of the denominators.
Activity Synthesis
The purpose of this discussion is for students to share how they proved the given formulas are identities. Ask 2–4 previously identified groups who used different methods to share how they each showed that the formula is an identity, recording their strategies for all to see.


aligns: yes