few_shots: Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation. Solving this problem does not require use of the Pythagorean identity.
Answer: B

Example 2:
Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Standard options: 
A. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
B. Solve quadratic equations in one variable.


Thought: This task shows one quadratic equation with one variable, x.
Answer: B

Example 3:
Problem/activity:
Task
Suppose $A$ is an angle with vertex $P$, as pictured below:
[IMG]
Draw a circle with center $P$ and with  radius $r \gt 0$.  Explain why the circle
meets each ray of angle $A$ in a single point.  Label these points $Q$ and $R$ respectively.
Draw circles with centers $Q$ and $R$ respectively and radius $r$.  These circles meet at $P$ and a second point to be labelled $B$.
Show that ray $\overrightarrow{PB}$  bisects angle $P$.


Standard options: 
A. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
B. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.


Thought: This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.
Answer: A

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001894

HS_0

im_task_000870

HS_1

im_task_001160

HS_2
options: Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

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.

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
problem_activity: Activity
In grade 3, students recognized that area is additive. They learned to find the area of a rectilinear figure by decomposing it into non-overlapping rectangles and adding their areas. Here students extend that understanding to non-rectangular shapes. They compose tangram pieces—consisting of triangles and a square—into shapes with certain areas. The square serves as a unit square. Because students have only one square, they need to use these principles in their reasoning:
If two figures can be placed one on top of the other so that they match up exactly, then they have the same area.
If a figure is decomposed and rearranged to compose another figure, then its area is the same as the area of the original figure.
Each question in the task aims to elicit discussions about these two principles. Though they may seem obvious, these principles still need to be stated explicitly (at the end of the lesson), as more-advanced understanding of the area of complex figures depends on them.
The terms
compose
,
decompose
, and
rearrange
will be formalized in an upcoming lesson, but throughout this lesson, look for opportunities to demonstrate their use as students describe their work with the tangram pieces. When students use “make” or “build,” “break,” and “move around,” recast their everyday terms using the more formal ones.
As students work, notice how they compose the pieces to create shapes with certain areas. Look for students whose reasoning illustrates the ideas outlined in the Activity Synthesis.
Demonstrate the use of the word “compose” by repeating students' everyday language use and then recast using the formal terms here.
Digital
Launch
Give each group of 2 students the following set of tangram pieces from the blackline master or from commercially available sets. Note that the tangram pieces used here differ from a standard set in that two additional small triangles are used instead of a parallelogram.
Square: 1
Small triangles: 4
Medium triangle: 1
Large triangles: 2
It is important not to give them more than these pieces.
Give students 2–3 minutes of quiet think time for the first three questions. Ask them to pause afterwards and compare their solutions to their partner's. If they created the same shape for each question, ask them to create a different shape that has the same given area before moving on. Then, ask them to work together to answer the remaining questions.
Classrooms using the digital activities can use physical tangram pieces or an applet with the same shapes to determine the relationships between the areas. Applet is adapted from the work of
Harry Drew
in GeoGebra.
Speaking, Conversing: MLR2 Collect and Display.
Circulate and listen to the ways students describe composing, decomposing, and rearranging the shapes. On a display, write down common phrases you hear students say about each, such as “building,” “breaking apart,” “moving.” Include relevant pictures or drawings. Update the display as needed throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students use mathematical language during their paired and whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
This applet has one square and some small, medium, and large right triangles. The area of the square is 1 square unit.
Click on a shape and drag to move it. Grab the point at the vertex and drag to turn it.
Notice that you can put together two small triangles to make a square. What is the area of the square composed of two small triangles? Be prepared to explain your reasoning.
Use your shapes to create a new shape with an area of 1 square unit that is not a square. Draw your shape on paper and label it with its area.
Use your shapes to create a new shape with an area of 2 square units. Draw your shape and label it with its area.
Use your shapes to create a
different
shape with an area of 2 square units. Draw your shape and label it with its area.
Use your shapes to create a new shape with an area of 4 square units. Draw your shape and label it with its area.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Find a way to use all of your pieces to compose a single large square. What is the area of this large square?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Print
Launch
Give each group of 2 students the following set of tangram pieces from the blackline master or from commercially available sets. Note that the tangram pieces used here differ from a standard set in that two additional small triangles are used instead of a parallelogram.
Square: 1
Small triangles: 4
Medium triangle: 1
Large triangles: 2
It is important not to give them more than these pieces.
Give students 2–3 minutes of quiet think time for the first three questions. Ask them to pause afterwards and compare their solutions to their partner's. If they created the same shape for each question, ask them to create a different shape that has the same given area before moving on. Then, ask them to work together to answer the remaining questions.
Classrooms using the digital activities can use physical tangram pieces or an applet with the same shapes to determine the relationships between the areas. Applet is adapted from the work of
Harry Drew
in GeoGebra.
Speaking, Conversing: MLR2 Collect and Display.
Circulate and listen to the ways students describe composing, decomposing, and rearranging the shapes. On a display, write down common phrases you hear students say about each, such as “building,” “breaking apart,” “moving.” Include relevant pictures or drawings. Update the display as needed throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students use mathematical language during their paired and whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
Your teacher will give you one square and some small, medium, and large right triangles. The area of the square is 1 square unit.
Notice that you can put together two small triangles to make a square. What is the area of the square composed of two small triangles? Be prepared to explain your reasoning.
Use your shapes to create a new shape with an area of 1 square unit that is not a square. Trace your shape.
Use your shapes to create a new shape with an area of 2 square units. Trace your shape.
Use your shapes to create a
different
shape with an area of 2 square units. Trace your shape.
Use your shapes to create a new shape with an area of 4 square units. Trace your shape.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Find a way to use all of your pieces to compose a single large square. What is the area of this large square?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may consider the area to be the number of pieces in the compositions, instead of the number of square units. Remind them of the meaning of area, or prompt them to review the definition of area discussed in the warm-up activity.
Because the 2 large triangles in the tangram set can be arranged to form a square, students may consider that square to be the square unit rather than the smaller square composed of 2 small triangles. Ask students to review the task statement and verify the size of the unit square.
Activity Synthesis
Invite previously identified students (whose work illustrates the ideas shown here) to share. Name these moves explicitly as they come up:
compose
,
decompose
, and
rearrange
.
First question: Two small triangles can be composed into a square that matches up exactly with the given square piece. This means that the two squares—the composite and the unit square—have the same area.
Tell students, “We say that if a region can be placed on top of another region so that they match up exactly, then they have the same
area
.”
Second question: Two small triangles can be rearranged to compose a different figure but the area of that composite is still 1 square unit. These three shapes—each composed of two triangles—have the same area. If we rotate the first figure, it can be placed on top of the second so that they match up exactly. The third one has a different shape than the other two, but because it is made up of the same two triangles, it has the same area.
[IMG]
Emphasize: “If a figure is decomposed and rearranged as a new figure, the
area
of the new figure is the same as the area of the original figure.”
Third and fourth questions: The composite figures could be formed in several ways: with only small triangles, with two triangles and a medium triangle, or with two small triangles and a square.
Last question: A large triangle is needed here.  To find its area, we need to either compose 4 smaller triangles into a large triangle, or to see that the large triangle could be decomposed into 4 smaller triangles, which can then be composed into 2 unit squares.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of these terms. Include the following terms and maintain the display for reference throughout the unit: area, compose, decompose, and rearrange.
Supports accessibility for: Conceptual processing; Language


correct_option_index: 1