Mathematics & Reasoning
Objectives
- D05.S1.10.01 Use properties of exponents and roots to simplify expressions and solve problems involving exponential growth and decay.
- D05.S1.10.02 Extend number reasoning to complex numbers and explain why new kinds of numbers arise from real problems.
- D05.S2.10.01 Build and interpret polynomial and rational functions that model real relationships and describe their key features.
- D05.S2.10.02 Model exponential growth and decay in population, finance, or environmental systems and explain the effect of the rate.
- D05.S3.10.01 Use trigonometric ratios and the Pythagorean theorem to solve right-triangle problems in real contexts such as surveying or navigation.
- D05.S3.10.02 Prove and apply theorems about circles and use coordinate geometry to describe and analyze figures.
- D05.S4.10.01 Understand independence and conditional probability and use them to interpret real situations such as screening tests and risk.
- D05.S4.10.02 Use probability rules to compute probabilities of compound events and interpret expected value in everyday decisions.
Essential questions
- How do I use exponents and roots to model growth and decay, and why do new kinds of numbers such as complex numbers arise from real problems?
- How do polynomial, rational, and exponential functions model real relationships, and what do their key features mean in context?
- How do I use trigonometry, circles, and coordinate geometry, and how do probability and expected value read real risk and decisions?
Big ideas
- Exponents, roots, and complex numbers extend number to model growth, decay, and problems that older numbers cannot reach.
- Polynomial, rational, and exponential functions model real relationships, and their key features carry meaning in context.
- Trigonometry, circle theorems, and coordinate geometry describe the world, while independence, conditional probability, and expected value read real risk.
Lessons
- 1 Exponents and the Power Rules 55 minutes
- 2 Roots, Rational Exponents, and the Numbers In Between 55 minutes
- 3 Exponential Growth and Decay: Base and Rate 55 minutes
- 4 Complex Numbers: Why New Numbers Arise 55 minutes
- 5 Polynomial Functions: Shape and Key Features 55 minutes
- 6 Rational Functions: Ratios That Model Rates 55 minutes
- 7 Exponential Functions and Their Features 55 minutes
- 8 Modeling Growth and Decay: Population, Finance, Environment 55 minutes
- 9 Trigonometric Ratios from Similar Triangles 55 minutes
- 10 Surveying and Navigation with Right Triangles 55 minutes
- 11 Circles and Coordinate Geometry 55 minutes
- 12 Probability Rules and Compound Events 55 minutes
- 13 Independence and Conditional Probability: Reading Screening Tests 55 minutes
- 14 Expected Value: Reading Risk and Decisions 55 minutes
Assessment plan
Solved and reasoned exponent-and-root problems with growth and decay; a complex-number extension explanation; polynomial and rational function models with key features; an exponential model of population, finance, or an environmental system; right-triangle problems in surveying or navigation; circle-theorem and coordinate-geometry proofs; and probability work interpreting screening tests and expected value, gathered as portfolio artifacts.
Unit 5 — Mathematics & Reasoning
Overview
This unit grows from Grade 9’s functions and proof into the next rung of modeling, building on linear and exponential functions, quadratics, and coordinate proof. Learners use properties of exponents and roots to simplify expressions and solve problems involving exponential growth and decay, and extend number reasoning to complex numbers, explaining why new kinds of numbers arise from real problems. They build and interpret polynomial and rational functions that model real relationships and describe their key features, and model exponential growth and decay in population, finance, or environmental systems, explaining the effect of the rate. They use trigonometric ratios and the Pythagorean theorem to solve right-triangle problems in real contexts such as surveying or navigation, and prove and apply theorems about circles and use coordinate geometry to describe and analyze figures. They understand independence and conditional probability and use them to interpret real situations such as screening tests and risk, and use probability rules to compute probabilities of compound events and interpret expected value in everyday decisions. Mathematics remains sense-making with real things, drawn from many traditions, for everyone.
Essential questions
- How do I use exponents and roots to model growth and decay, and why do new kinds of numbers such as complex numbers arise from real problems?
- How do polynomial, rational, and exponential functions model real relationships, and what do their key features mean in context?
- How do I use trigonometry, circles, and coordinate geometry, and how do probability and expected value read real risk and decisions?
Lessons
| Lesson | Title | Standards |
|---|---|---|
| L.10.005.01 | Exponents and the Power Rules | D05.S1.10.01 |
| L.10.005.02 | Roots, Rational Exponents, and the Numbers In Between | D05.S1.10.01 |
| L.10.005.03 | Exponential Growth and Decay: Base and Rate | D05.S1.10.01 |
| L.10.005.04 | Complex Numbers: Why New Numbers Arise | D05.S1.10.02 |
| L.10.005.05 | Polynomial Functions: Shape and Key Features | D05.S2.10.01 |
| L.10.005.06 | Rational Functions: Ratios That Model Rates | D05.S2.10.01 |
| L.10.005.07 | Exponential Functions and Their Features | D05.S2.10.02 |
| L.10.005.08 | Modeling Growth and Decay: Population, Finance, Environment | D05.S2.10.02 |
| L.10.005.09 | Trigonometric Ratios from Similar Triangles | D05.S3.10.01 |
| L.10.005.10 | Surveying and Navigation with Right Triangles | D05.S3.10.01 |
| L.10.005.11 | Circles and Coordinate Geometry | D05.S3.10.02 |
| L.10.005.12 | Probability Rules and Compound Events | D05.S4.10.02 |
| L.10.005.13 | Independence and Conditional Probability: Reading Screening Tests | D05.S4.10.01 |
| L.10.005.14 | Expected Value: Reading Risk and Decisions | D05.S4.10.02 |