options: Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Compute fluently with multi-digit numbers and find common factors and multiples.

Represent complex numbers and their operations on the complex plane.

Perform operations on matrices and use matrices in applications.

Apply and extend previous understandings of numbers to the system of rational numbers.

Perform operations on vectors.

Use properties of rational and irrational numbers.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Perform arithmetic operations with complex numbers.

Extend the properties of exponents to rational exponents.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Reason quantitatively and use units to solve problems.

Use complex numbers in polynomial identities and equations.

Represent and model with vector quantities.
problem_activity: Optional activity
In both subsequent lessons and units, students will work extensively with similar right triangles. This activity gives students opportunities to practice the Pythagorean Theorem, and apply similar triangles in a contextual problem.
Launch
If students struggled to use the Pythagorean Theorem in a previous lesson:
Begin by asking students what they notice and wonder about the image from this activity.
Draw students’ attention to the right triangles, and ask them what must be true about the side lengths.
Provide some possible lengths for sides and have students practice finding an unknown side length in an isosceles right triangle to allow students to practice working with the square root of 2.
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to increase awareness of the language used to talk about similar right triangles. Before revealing the questions in this activity, display the image of the quilt design made of right isosceles triangles. Ask students to write down possible mathematical questions that could be asked about the image. Invite students to compare their questions before revealing the actual questions. Listen for and amplify any questions about the features of the right triangles. For example, “Are all the right triangles similar?”, “What is the scale factor between the small and large right triangle?”, and “What are the side lengths of the right triangles?”
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
Here is a quilt design made of right isosceles triangles. The smallest squares in the center have an area of 1 square unit. Find the dimensions of the triangles.
[IMG]
Are the triangles similar? If so, what are the scale factors?
This quilt is meant for a baby (1 unit = 6 inches). To make a quilt for a queen-size bed, it needs to be 90 inches wide. What dimensions should the center squares of the big quilt have to reach that width?
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
Here is a quilt design made of right triangles. Find as many different size triangles as you can.
[IMG]
Write similarity statements for 2 pairs of triangles.
The smallest triangles have legs 2 units and 3 units long. Write some equivalent ratios or equations that will help you determine the dimensions of triangle
\(LCQ\)
. Use your equivalent ratios or equations to find the dimensions of triangle
\(LCQ\)
.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students who struggle with the number of similar triangles in the image can trace each size triangle separately onto tracing paper and label the corresponding sides using colored pencils.
Activity Synthesis
A key point to emphasize in this discussion is that ratios in which each ratio compares side lengths within each figure are also equivalent across similar figures. That is, for a triangle with side lengths
\(a, b, c\)
, and a similar triangle with corresponding side lengths
\(x,y,z\)
, we have equivalent ratios such as
\(\frac{a}{b} = \frac{x}{y}\)
.
Invite students to share strategies for scaling the figure up from a baby blanket to a quilt for a queen-size bed.


correct_option_index: 11