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

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

Use properties of rational and irrational numbers.

Reason quantitatively and use units to solve problems.

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

Perform operations on matrices and use matrices in applications.

Represent complex numbers and their operations on the complex plane.

Represent and model with vector quantities.

Use complex numbers in polynomial identities and equations.

Perform arithmetic operations with complex numbers.

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

Perform operations on vectors.

Extend the properties of exponents to rational exponents.

Know that there are numbers that are not rational, and approximate them by rational numbers.
problem_activity: Activity
In this activity, students continue to reason about half-lives and exponential functions to solve problems. Both questions provide possible reasoning for a calculation and then prompt students to construct an argument to justify their position (MP3).
The context continues to be the decay of carbon-14 and students investigate two different situations. In each case, the amount of time that has passed since the carbon-14 began to decay is not a multiple of the half-life. This means that students need to make estimates, either in terms of a fraction of a half-life or a whole number of half-lives.
Launch
In case students are unfamiliar with papyrus, explain that it is a material that was used as a writing surface in ancient times, similar to how paper is used today. Papyrus is made out of the papyrus plant and once it is harvested and turned into a type of paper, the amount of carbon-14 in the papyrus starts decreasing.
Engagement: Develop Effort and Persistence.
Break the class into small discussion groups and then invite a representative from each group to report back to the whole class. While students work, provide tools for creating a visual display for them to record information for sharing. Encourage students to record their group’s thinking by writing out explanations using equations, words, pictures, or other symbolic representations such as timelines, that explain their approaches.
Supports accessibility for: Language; Social-emotional skills; Attention
Student Facing
The half-life of carbon-14 is about 5,730 years.
Pythagoras lived between 600 BCE and 500 BCE. Explain why the age of a papyrus from the time of Pythagoras is about half of a carbon-14 half-life.
Someone claims they have a papyrus scroll written by Pythagoras. Testing shows the scroll has 85% of its original amount of carbon-14 remaining. Explain why the scroll is likely a fake.
Student Response
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Student Facing
Are you ready for more?
A copy of the Gutenberg Bible was made around 1450. Would more or less than 90% of the carbon-14 remain in the paper today? Explain how you know.
Student Response
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Anticipated Misconceptions
As in the previous activity, students may mistakenly think that after about
\(\frac{1}{2}\)
of a half-life, only about
\(\frac{1}{4}\)
of the carbon-14 remains. Ask them if more or less than
\(\frac{1}{2}\)
of the carbon-14 remains after 2,600 years. (Less since this is less than a half-life.)
Activity Synthesis
Focus the discussion on the estimates students made and the properties of exponential functions that they used. Here are some questions for discussion:
“About how old is the papyrus? About how many half-lives of carbon-14 is that?” (2,600 years,
\(\frac{1}{2}\)
half-life)
“By what factor does the amount of carbon-14 decrease in one half-life? What about in half of a half-life?” (
\(\frac{1}{2}\)
and
\(\frac{1}{\sqrt{2}}\)
)
One thing to note to students is that carbon-14 dating, while improving, is not an exact calculation but rather a good approximation for dating items from long ago. Consider telling students that carbon-14 dating is only used for objects that are less than about 50,000 years old. Ask,
“About how many half lives is that?” (9)
“About what fraction of the carbon-14 would be left in a 50,000-year-old artifact?” (
\(\frac{1}{2^9}\)
or
\(\frac{1}{512}\)
)
Speaking, Listening: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they heard to a partner before selecting one or two students to share with the class. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


correct_option_index: 12