Curriculum / Math / 11th Grade / Unit 5: Exponential Modeling and Logarithms / Lesson 6
Math
Unit 5
11th Grade
Lesson 6 of 16
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Find the sum of an infinite geometric series.
The core standards covered in this lesson
A.SSE.B.4 — Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
Suggestions for teachers to help them teach this lesson
This video, "ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12" isn't strictly relevant to the objective, but it is a fun example of how infinite series can be manipulated.
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
To find $${\sum_{n=1}^{100}8\left ( 1\over2 \right )^{n-1}}$$ what would the expression look like? Which term is insignificant?
How could you find the sum below?
$${\sum_{n=1}^{\infty}8\left ( 1\over2 \right )^{n-1}}$$
Consider the following definition:
Fill out the following table for $${h(n)}$$, keeping your answers accurate to two decimal places:
Day 1: Flip the Script is made available by Park City Mathematics Institute under the CC BY-NC-SA 4.0 license. © 2001 - 2018 Park City Mathematics Institute. Accessed Feb. 22, 2018, 4 p.m..
Nicole has four sheets of paper and wants to share them with Mary, Elizabeth, and Mandy. She starts by handing everyone a full sheet, keeping one. Then, Nicole realizes she doesn’t want a full sheet. She breaks her sheet into four equal pieces and shares them, keeping one. Then Nicole realizes she doesn’t want a full $${{{{1\over4}}}}$$ sheet. She breaks her $${{{{1\over4}}}}$$ sheet into four equal pieces and shares them, keeping one. This keeps happening. Write and evaluate an infinite sum based on all the pieces of paper Mandy ends up with.
Day 3: Flipping Out Loud is made available by Park City Mathematics Institute under the CC BY-NC-SA 4.0 license. © 2001 - 2018 Park City Mathematics Institute. Accessed Feb. 22, 2018, 4:13 p.m..
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
Evaluate:
$${\sum_{n=1}^{\infty}8\left ( 1\over3 \right )^n}$$
Lesson 5
Lesson 7
Topic A: Modeling with and Interpreting Exponential Functions
Identify, model, and analyze geometric sequences.
F.IF.A.3 F.IF.B.5 F.LE.A.2
Analyze and construct exponential functions that model contexts.
F.IF.B.4 F.IF.C.8.B F.LE.A.2
Write and change the form of exponential functions that model compounding interest.
F.BF.A.1.A F.LE.B.5
Define and use $$e$$ in continuous compounding situations.
A.SSE.B.3.C F.BF.A.1.A
Describe the derivation of the formula for the sum of a finite geometric series and use it to solve problems.
A.SSE.B.4
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Topic B: Definition and Meaning of Logarithms
Describe and evaluate simple numeric logarithms (Part I).
F.LE.A.4
Describe and evaluate simple numeric logarithms (Part II).
Describe logarithms as the inverse of exponential functions and graph logarithmic functions.
F.BF.B.3 F.BF.B.4.B F.BF.B.4.C F.BF.B.5 F.IF.C.7.E
Evaluate common and natural logs using tables, graphs, and calculators.
F.BF.B.4.C F.LE.A.4
Understand and apply the change of base property to evaluate logarithms.
Develop and use the product and quotient properties of logarithms to write equivalent expressions.
Develop and use the power property of logarithms to write equivalent expressions.
F.BF.B.4.B F.LE.A.4
Solve equations with logarithms.
Use logarithms to solve exponential modeling problems (Part I).
A.SSE.A.1.B F.LE.A.4
Use logarithms to solve exponential modeling problems (Part II).
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