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
Teachers with a valid work email address can
click here to register or sign in
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 following problem or activity enable students to learn the full intent of the following concept or skill?

shot_info: im_lesson_009431_pos_0

1
standard_description: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
problem_activity: Activity
This task culminates in writing a formula for the area of triangles. By now students are likely to have developed the intuition that the area of a triangle is half of that of a parallelogram with the same base and height. This activity encapsulates that work in an algebraic expression.
Students first find the areas of several triangles given base and height measurements. They then generalize the numerical work to arrive at an expression for finding the area of any triangle (MP8).
If needed, remind students how they reasoned about the area of triangles in the previous lesson (i.e. by composing a parallelogram, enclosing with one or more rectangles, etc.). Encourage them to refer to their previous work and use tracing paper as needed. Students might write
\(b \boldcdot h \div 2\)
or
\(b \boldcdot h \boldcdot \frac12\)
as the expression for the area of any triangle. Any equivalent expression should be celebrated.
At the end of the activity, consider giving students a chance to reason more abstractly and deductively, i.e., to think about why the expression
\(b \boldcdot h \div 2\)
would hold true for all triangles. See Activity Synthesis for prompts and diagrams that support such reasoning.
Launch
Arrange students in groups of 2–3. Explain that they will now find the area of some triangles using what they know about base-height pairs in triangles and the relationships between triangles and parallelograms.
Give students 5–6 minutes to complete the activity and access to geometry toolkits, especially tracing paper. Ask them to find the area at least a couple of triangles independently before discussing with their partner(s).
Student Facing
For each triangle:
Identify a base and a corresponding height, and record their lengths in the table.
Find the area of the triangle and record it in the last column of the table.
[IMG]
|              triangle               | base (units) | height (units) | area (square units) |
|-------------------------------------|--------------|----------------|---------------------|
|      **          A         **       |              |                |                     |
|      **          B         **       |              |                |                     |
|      **          C         **       |              |                |                     |
|      **          D         **       |              |                |                     |
| **          any triangle         ** |    \(b\)     |     \(h\)      |                     |
In the last row, write an expression for the area of any triangle, using
\(b\)
and
\(h\)
.
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 be inclined to write an expression using the variables
\(b\)
and
\(h\)
and instead replace the variables with numbers of their choice. Ask them to reflect on what they did with the numbers for the first four triangles. Then, encourage them to write the same operations but using the letters
\(b\)
and
\(h\)
rather than numbers.
Activity Synthesis
Select a few students to share their expression for finding the area of any triangle. Record each expression for all to see.
To give students a chance to reason logically and deductively about their expression, ask, “Can you explain why this expression is true for
any
triangle?”
Display the following diagrams for all to see. Give students a minute to observe the diagrams. Ask them to choose one that makes sense to them and use that diagram to explain or show in writing that the expression
\(b \boldcdot h \div 2\)
works for finding the area of any triangle. (Consider giving each student an index card or a sheet of paper on which to write their reasoning so that their responses could be collected, if desired.)
[IMG]
When dealing only with the variables
\(b\)
and
\(h\)
and no numbers, students are likely to find Jada's and Lin's diagrams more intuitive to explain. Those choosing to use Elena's diagram are likely to suggest moving one of the extra triangles and joining it with the other to form a non-rectangular parallelogram with an area of
\(b \boldcdot h\)
. Expect students be less comfortable reasoning in abstract terms than in concrete terms. Prepare to support them in piecing together a logical argument using only variables.
If time permits, select students who used different diagrams to share their explanation, starting with the most commonly used diagram (most likely Jada's). Ask other students to support, refine, or disagree with their arguments. It time is limited, consider collecting students' written responses now and discussing them in an upcoming lesson.
Representation: Develop Language and Symbols.
Create a display of important terms and concepts. Invite students to suggest language or diagrams to include that will support their understanding. Include the following terms and maintain the display for reference throughout the unit: opposite vertex, base (of a triangle), height (of a triangle), and formula for finding area.
Supports accessibility for: Memory; Language
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion when students share their expressions for finding the area of any triangle. Provide students with 1–2 minutes of quiet think time to begin to consider why their expression is true for any triangle, before they continue work with a partner to complete their response. Select 1 or 2 pairs of students to share with the class, then call on students to restate their peers' reasoning. This will give more students the chance to use language to interpret and describe expressions for the area of triangles.
Design Principle(s): Support sense-making; Maximize meta-awareness


aligns: yes