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Explore graphs of equations, exponents, counting problems, and more, emphasizing intuition and understanding over just finding an answer.
This course will deepen your knowledge of basic algebra and introduce you to some surprisingly useful applications of this powerful mathematical tool.
Some prior experience with algebra is assumed, but you're in good shape to start this course if you can plot points and linear equations on a coordinate plane and use a variable to describe the relationship between the side length of a square and its area.
By the end of this course, you’ll have extended your problem solving skills and be more aware of what common misconceptions can happen in algebra problems. You'll have enough background to go on to our Algebra II course.
Overview
Syllabus
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- Introduction: Strengthen your algebra skills by exploring factorials, exponents, and the unknown.
- Exponent Shortcuts: Learn to solve exponent problems quickly and efficiently.
- Line Exploration: Play with lines and build an understanding of the equations that define them.
- When 1 = 2: What kind of algebraic errors lead to obviously wrong conclusions?
- Equations and Unknowns: Sometimes x is unknown, and sometimes it's unknowable.
- Combination Locks: How much information is needed to pin down the exact combination?
- Equations and Variables: How does the number of variables affect what can be known about the equations?
- Scaling and Adding Equations: Practice solving multi-equation problems.
- Graphing Equations: Explore the connections between lines and the equations that generate them.
- Different Ways of Writing a Line: How many different ways can you write the equation of one line (and why might you want to)?
- Equation Dependencies: What happens when you have multiple equations but some of them convey the same relationships?
- Array Puzzles: Solve these puzzles applying what you've learned about equations and variables.
- Systems of Equations Problem Solving: Stretch your understanding of equation systems to the next level with these challenging problems.
- Manipulating Exponents: A powerful language for the very large and the very small.
- Orders of Magnitude: Learn how to use scientific notation to more skillfully talk about very large and very small measures.
- Same Bases 1: Derive the rules for combining exponential terms by thinking about exponents as repeated multiplication.
- Same Bases 2: Learn to spot patterns that will help you simplify an exponential expression.
- Negative Exponents: Learn how to interpret exponents that have a negative value.
- Fractional Exponents: Extend multiplication patterns even further to make sense of fractional exponents.
- Exponent Intuition: Apply problem solving strategies to further develop your intuition about exponents.
- Like Powers: Simplify expressions when the exponents are the same.
- Powers and Towers: Learn when and why towers of exponents can produce absurdly large values.
- Exponent Simplification Challenges: Try your hand at these problems to put all of your new exponent intuition and skills to the test.
- Algebra in Motion: Mostly, we just want to see the collisions!
- Distance, Rate, and Time: Explore the algebra that describes motion and change.
- Stunt Driving and Collisions: Use these dangerous motion problems to practice and strengthen your skills!
- The Harmonic Average: Learn how to calculate the average of two speeds.
- Boats: Speed becomes more complicated when what you're travelling in (a river) is also moving.
- Velocity and Acceleration: When direction matters, the concept of speed is inadequate. Add direction to speed, and you get velocity!
- Bullets: Use velocity and acceleration to solve these problems about firing and stopping bullets.
- Factorials: Bang! Now it's huge!
- Counting and Factorials: Learn how to efficiently count cases when there are several sequential choices.
- Factorials: Explore and practice techniques for efficiently manipulating factorials.
- Operations and Simplification: The large values produced by factorials become easier to work with when simplified or reduced.
- Permutations: Permutations count the number of ways things can be put in order.
- Combinations: Combinations count the number of ways things can be selected, regardless of order.
- Combinations II: Combine some of the techniques you've learned together to solve these combination problems.
- Common Misconceptions: You'd be surprised how many contest these algebraic facts.
- The Basics: Identify and repair the mistakes in thinking that lead to these common errors.
- Ordering and Reordering: The order in which you evaluate the parts of a calculation can have a huge impact on the outcome.
- Two Kinds of Cancel: Understanding the ideas behind "cancelling" will keep you from making some common errors.
- The Distributive Property: Test the limits of your understanding of addition, multiplication, and how they interact!
- Fraction Follies: Carefully make sense of fraction setups that frequently cause calculation mistakes.
- Powers and Square Roots: Improve your understanding of exponents and roots by simplifying these exponential expressions.
- Find the Mistakes!: Figure out where each of these problem solvers slipped up in their calculations.
