options: Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
problem_activity: Warm-up
Students calculate a scale factor given the areas of the circular base of a cone and the base of its dilation. This connects the concept of surface area dilation to cross sections, and gives practice with non-integer roots.
Monitor for pairs of students who initially consider an answer of 20.25 but come to a consensus that the scale factor is 4.5.
Launch
Arrange students in groups of 2. Provide access to scientific calculators. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.
Student Facing
The image shows a cone that has a base with area
\(36\pi\)
square centimeters. The cone has been dilated using the top vertex as a center. The area of the dilated cone’s base is
\(729\pi\)
square centimeters.
[IMG]
What was the scale factor of the dilation?
Student Response
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Anticipated Misconceptions
Some students may struggle to find the square root of 20.25. Remind students that their calculators can find square roots, and prompt them to use an estimate to check the reasonableness of the calculator output.
Activity Synthesis
Select a pair of students to explain their reasoning. If the pair considered 20.25 but moved to an answer of 4.5, ask how they knew 20.25 wasn’t correct.
Discuss with students how they can decide if 4.5 is a reasonable value for the square root of 20.25, considering the fact that 4
2
= 16 and 5
2
= 25. If time allows, ask students to calculate the scale factor for the solid’s volume (4.5
3
= 91.125).


correct_option_index: 5

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