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Brilliant

Algebra Fundamentals

via Brilliant

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Overview

Explore how algebra works and why it matters, and build a strong foundation of skills across many algebra topics including equations, rates, ratios, and sequences.

By the end of this course, you’ll know both traditional algebraic techniques and many unique problem-solving approaches that aren’t typically covered in school. You'll also improve your algebraic intuition and hone your strategic thinking for approaching difficult problems.

Syllabus

  • Introduction: Supercharge your algebraic intuition and problem solving skills!
    • Scale and Lever Logic: Warm up the skills and intuition that algebra requires by balancing scales and measuring weights.
    • Magic Sum Puzzles: How can numbers be arranged so that these three specific sets all sum to 12?
    • Sequences: Find and describe the patterns in these visual sequences.
  • Simplifying Shortcuts: Save time with this clever thinking.
    • Shortcuts 101: Learn some clever techniques to simplify problems, saving yourself time and effort.
    • Guess, Check, and Revise: In this strategic shortcut, use an easy number as a test case and then update to find the true answer.
    • Arithmetic Tricks I: These problems can be quickly solved in your head if you find the trick to each of them.
    • Arithmetic Tricks II: Practice using rearrangement of terms, factoring, distribution, canceling, and any other tricks you know!
    • Difference of Squares: Representing an algebraic identity geometrically can lead to deep insights.
    • The Gauss Trick: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 can be mentally evaluated faster than you might expect!
    • Odd Square Sums: Manipulate these odd arrays of dots to find another useful summation shortcut.
    • Pentagonal and Hexagonal Numbers: Extend the tricks from triangular and square numbers to the next two sets of polygonal numbers.
    • Master of All Shortcuts: These problems require creative combinations of all the shortcuts that you've seen so far.
  • Arithmetic Logic and Magic: Puzzles that are like Sudoku, but trickier!
    • Magic Rectangles Part I: Is it possible to fill these grids so that the sums of the rows and columns are all correctly predicted?
    • Magic Rectangles Part II: What if, instead of specific target sums, you only know that all of the sums need to match?
    • Magic Perimeters: In this exploration, the digits need to be filled in around the perimeter of each shape.
    • Learn Calcdoku: It's like Sudoku except that each specified region must obey a specific arithmetical rule.
    • Beginner Calcdoku: Do this round of puzzles to get warmed up!
    • Thinking About Calcdoku: Go a bit meta to explore some of the mathematics at work 'behind the scenes' of Calcdoku.
    • More Advanced Calcdoku: Now you're prepared for some intense 4x4 Calcdoku challenges!
    • Calcdoku 5x5: These are the most advanced Calcdoku puzzles in this unit — good luck!
  • Balancing Scales: The unification of logic and algebra.
    • Balancing Scales: Explore several types of balance puzzles and learn some strategies for approaching them.
    • Elimination: Simplify systems combining the shapes that balance on one scale with those that balance on another.
    • Substitution: First isolate a shape on one side of a scale, then use this equivalency to make substitutions elsewhere.
    • Fraction-Related Strategies: When fractional shapes or numbers are involved, a few additional steps are required to isolate variables.
    • More than Two Variables: Extend your thinking to cases where there are more, different unknowns in each puzzle.
    • Balancing Chemical Equations: Apply what you've learned balancing scales to the scientific application of balancing chemical equations.
  • Sequences: 2, 5, 10, 17, 26... What comes next?
    • What Comes Next?: Explore, describe, and then predict the patterns in these sequences.
    • Describing Sequences: Practice describing sequences in three different ways: by property, recursively, and explicitly.
    • Arithmetic Sequences: Focus in on this one sequence type and learn a few tricks that take advantage of its steady behavior.
    • Geometric Sequences: Now focus in on these sequences that evolve using recursive multiplication instead of addition.
    • Geometric Applications: Apply what you know about geometric sequences to visualize the formation of fractal figures.
    • Fibonacci and More: Solve challenging problems that employ recursively-described sequences such as the Fibonacci sequence.
  • Rates and Ratios: If 4 cows make 4 gallons of milk in 4 days, how much milk do 8 cows make in 8 days?
    • Applying Rates and Ratios: Explore rates and ratios in some of the real-life situations in which they show up.
    • Proportionality: Hone your skills solving problems that employ proportionality and inverse proportionality.
    • Joint Proportionality: In this exploration, multiple variables can change simultaneously, all contributing to an overall effect.
    • Scaling 3D Shapes: Explore how scaling an object can counter-intuitively affect other properties of that object.
    • Inverse Square Laws: Understand why quantities related by physical and geometric laws have non-linear relationships.
    • Mixing Problems: Extend the rates and ratio strategies in this unit to solve a sequence of challenging mixing puzzles.

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