Interactive course — Foundational Math

Algebra Fundamentals

Supercharge your algebraic intuition and problem solving skills!



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 more generally have improved algebraic intuition, honed for the strategic thinking that you need when approaching difficult problems.

Topics covered

  • Arithmetic Sequences
  • Algebra Shortcuts
  • Balanced Scale Puzzles
  • Calcdoku Puzzles
  • Difference of Squares
  • Fibonacci Numbers
  • Geometric Sequences
  • Inverse Square Laws
  • Joint Proportionality
  • Linear Equations
  • Magic Squares
  • Solving Systems of Equations

Interactive quizzes


Concepts and exercises


Course map

Prerequisites and Next Steps

  1. 1


    Supercharge your algebraic intuition and problem solving skills!

    1. Scale and Lever Logic

      Warm up the skills and intuition that algebra requires by balancing scales and measuring weights.

    2. Magic Sum Puzzles

      How can numbers be arranged so that these three specific sets all sum to 12?

    3. Sequences

      Find and describe the patterns in these visual sequences.

  2. 2

    Simplifying Shortcuts

    Save time with this clever thinking.

    1. Shortcuts 101

      Learn some clever techniques to simplify problems, saving yourself time and effort.

    2. Guess, Check, and Revise

      In this strategic shortcut, use an easy number as a test case and then update to find the true answer.

    3. Arithmetic Tricks I

      These problems can be quickly solved in your head if you find the trick to each of them.

    4. Arithmetic Tricks II

      Practice using rearrangement of terms, factoring, distribution, canceling, and any other tricks you know!

    5. Difference of Squares

      Representing an algebraic identity geometrically can lead to deep insights.

    6. The Gauss Trick

      1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 can be mentally evaluated faster than you might expect!

    7. Odd Square Sums

      Manipulate these odd arrays of dots to find another useful summation shortcut.

    8. Pentagonal and Hexagonal Numbers

      Extend the tricks from triangular and square numbers to the next two sets of polygonal numbers.

    9. Master of All Shortcuts

      These problems require creative combinations of all the shortcuts that you've seen so far.

  3. 3

    Arithmetic Logic and Magic

    Puzzles that are like Sudoku, but trickier!

    1. Magic Rectangles Part I

      Is it possible to fill these grids so that the sums of the rows and columns are all correctly predicted?

    2. Magic Rectangles Part II

      What if, instead of specific target sums, you only know that all of the sums need to match?

    3. Magic Perimeters

      In this exploration, the digits need to be filled in around the perimeter of each shape.

    4. Learn Calcdoku

      It's like Sudoku except that each specified region must obey a specific arithmetical rule.

    5. Beginner Calcdoku

      Do this round of puzzles to get warmed up!

    6. Thinking About Calcdoku

      Go a bit meta to explore some of the mathematics at work 'behind the scenes' of Calcdoku.

    7. More Advanced Calcdoku

      Now you're prepared for some intense 4x4 Calcdoku challenges!

    8. Calcdoku 5x5

      These are the most advanced Calcdoku puzzles in this unit — good luck!

  4. 4

    Balancing Scales

    The unification of logic and algebra.

    1. Balancing Scales

      Explore several types of balance puzzles and learn some strategies for approaching them.

    2. Elimination

      Simplify systems combining the shapes that balance on one scale with those that balance on another.

    3. Substitution

      First isolate a shape on one side of a scale, then use this equivalency to make substitutions elsewhere.

    4. Fraction-Related Strategies

      When fractional shapes or numbers are involved, a few additional steps are required to isolate variables.

    5. More than Two Variables

      Extend your thinking to cases where there are more, different unknowns in each puzzle.

    6. Balancing Chemical Equations

      Apply what you've learned balancing scales to the scientific application of balancing chemical equations.

  5. 5


    2, 5, 10, 17, 26... What comes next?

    1. What Comes Next?

      Explore, describe, and then predict the patterns in these sequences.

    2. Describing Sequences

      Practice describing sequences in three different ways: by property, recursively, and explicitly.

    3. Arithmetic Sequences

      Focus in on this one sequence type and learn a few tricks that take advantage of its steady behavior.

    4. Geometric Sequences

      Now focus in on these sequences that evolve using recursive multiplication instead of addition.

    5. Geometric Applications

      Apply what you know about geometric sequences visualize the formation of fractal figures.

    6. Fibonacci and More

      Solve challenging problems that employ recursively-described sequences such as the Fibonacci sequence.

  6. 6

    Rates and Ratios

    If 4 cows make 4 gallons of milk in 4 days, how much milk do 8 cows make in 8 days?

    1. Applying Rates and Ratios

      Explore rates and ratios in some of the real-life situations in which they show up.

    2. Proportionality

      Hone your skills solving problems that employ proportionality and inverse proportionality.

    3. Joint Proportionality

      In this exploration, multiple variables can change simultaneously, all contributing to an overall effect.

    4. Scaling 3D Shapes

      Explore how scaling an object can counter-intuitively affect other properties of that object.

    5. Inverse Square Laws

      Understand why quantities related by physical and geometric laws have non-linear relationships.

    6. Mixing Problems

      Extend the rates and ratio strategies in this unit to solve a sequence of challenging mixing puzzles.