6th Grade Math

129 core topics + 11 prerequisite topics taught as needed · approximately 31 hours of instruction including spaced review

How the course runs

An adaptive diagnostic (up to 40 questions) places the student on the course's knowledge graph — topics already known are credited, and instruction begins exactly at the learning frontier. Every topic is taught with a worked-example lesson and auto-graded practice; a topic is mastered at 75%+ and then maintained through spaced reviews on an expanding schedule. A cumulative quiz follows every 6 lessons. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.

Core curriculum

Arithmetic Foundations · 7 topics

Adding & Subtracting Whole Numbers [E]	Multi-digit addition and subtraction.
Multiplication [E]	Multiplying whole numbers.
Division [E]	Dividing whole numbers.
Order of Operations [M]	Parentheses first, then multiplication/division, then addition/subtraction.
Negative Numbers: Adding & Subtracting [M]	Working with numbers below zero on the number line.
Negative Numbers: Multiplying & Dividing [M]	Sign rules for products and quotients.
Exponents [M]	Repeated multiplication in shorthand.

Fractions · 7 topics

Equivalent Fractions [M]	Different fractions can name the same amount.
Simplifying Fractions [M]	Reducing a fraction to lowest terms.
Adding Fractions (Like Denominators) [M]	Same-denominator addition.
Adding Fractions (Unlike Denominators) [M]	Rewrite over a common denominator first.
Multiplying Fractions [M]	Multiply straight across.
Dividing Fractions [M]	Multiply by the reciprocal.
Mixed Numbers & Improper Fractions [M]	Converting between forms.

Decimals, Percents & Ratios · 5 topics

Decimal Addition & Subtraction [M]	Line up the decimal points.
Fractions ↔ Decimals [M]	Converting between the two notations.
Percent of a Number [M]	Percent means per hundred.
Percent Increase & Decrease [M]	Applying a percent change to a quantity.
Ratios & Proportions [M]	Two quantities that scale together.

Expressions & Equations · 3 topics

Evaluating Expressions [M]	Substituting a value for a variable.
Combining Like Terms [M]	Adding the coefficients of matching variable parts.
One-Step Equations [M]	Undoing a single operation.

Elementary Math Skills · 10 topics

Place Value & Comparing [E]	Each digit's position multiplies its value by ten.
Rounding & Estimation [E]	Look one digit to the right: 5 or more rounds up.
Multi-Digit Multiplication [E]	Split by place value, multiply each part, add.
Long Division [M]	Divide, multiply, subtract, bring down — repeat.
Time & Elapsed Time [E]	Count up to the next hour, then keep going.
Money Problems [E]	Money is decimals with two places — line up the point.
Units & Measurement [E]	Bigger unit → multiply; smaller unit → divide.
Perimeter & Area of Rectangles [E]	Perimeter walks around; area covers.
Reading Graphs & Tables [E]	Read the values first; the math is the easy part.
Multi-Step Word Problems [M]	One sentence at a time: what do I know, what changes, what's asked?

Ratios, Data & Geometry (Middle School) · 12 topics

GCF & LCM [E]	Greatest common factor and least common multiple.
Prime Factorization [E]	Every whole number is a unique product of primes.
Absolute Value & Distance [E]	Absolute value is distance from zero.
Unit Rates [E]	Per-one comparisons: dollars per item, miles per hour.
Ratio Tables & Equivalent Ratios [E]	Scaling both parts of a ratio keeps it equivalent.
Area: Triangles & Trapezoids [E]	Half of base times height — and its trapezoid cousin.
Area of Composite Figures [E]	Split odd shapes into rectangles and triangles.
Volume: Rectangular Prisms [E]	Length × width × height, fractional edges included.
Surface Area & Nets [E]	Unfold the box: surface area is the area of its net.
Mean, Median & Range [E]	Three ways to summarize a data set with one number.
Reading Data Displays [E]	Pulling answers out of dot plots, tables, and bar graphs.
Probability Basics [E]	Favorable outcomes over total outcomes.

Grade 5: Fraction Operations in Depth · 12 topics

Rewriting with Common Denominators [E]	Before unlike fractions can meet, rename them with the same-size pieces.
Adding Unlike Fractions in Context [M]	Rename to a common denominator, then add the numerators — story or not.
Subtracting Unlike Fractions in Context [M]	How much more, how much left — common denominator first, then subtract tops.
Estimating Sums with Benchmark Fractions [M]	Round each fraction to 0, 1/2, or 1 before computing — the estimate polices the answer.
Fraction × Fraction with Area Models [M]	Columns for one fraction, rows for the other — the overlap is the product.
Fraction of an Amount [M]	“Of” means multiply: divide by the bottom, multiply by the top.
Multiplication as Scaling [M]	Compare the factor to 1 — that alone says whether the product grows or shrinks.
Dividing a Unit Fraction by a Whole Number [M]	Splitting 1/b into n equal shares makes pieces n times smaller: 1/(b × n).
Dividing a Whole by a Unit Fraction [M]	n ÷ 1/b asks how many 1/b-size pieces fit in n — every whole holds b of them.
Mixed Numbers with Unlike Denominators [M]	Handle wholes and parts separately — but give the parts a common denominator first.
Multiplying Mixed Numbers [M]	Convert each mixed number to an improper fraction, then multiply straight across.
Multi-Step Fraction Word Problems [H]	Chain the moves: combine the fractions first, then compare to the whole.

Grade 5: Patterns, Expressions & Data · 12 topics

Nested Grouping Symbols [E]	Work from the innermost parentheses outward.
Comparing Expressions [E]	Parentheses can make the same numbers give a bigger or smaller value.
Writing Expressions from Words [E]	Turn a phrase into symbols — parentheses show what happens first.
Interpreting Expressions [E]	See what an expression says without grinding out the arithmetic.
Rules & Input-Output Tables [E]	Apply a two-step rule forward — or undo it backward.
Generating Number Patterns [E]	Follow the rule step by step to grow the pattern.
Comparing Two Patterns [M]	Line up matching terms and see how the two rules relate.
Ordered Pairs from Patterns [M]	Pair matching terms as (x, y) points ready to plot.
Reading Points in Real-World Graphs [E]	A point (x, y) tells a story: x of one thing, y of another.
Classifying Triangles [E]	Name triangles by their sides or by their angles.
The Quadrilateral Hierarchy [M]	Categories nest: a square is a rectangle is a parallelogram.
Multi-Step Conversion Problems [M]	Convert to one unit first, then finish the problem.

Grade 6: Rational Numbers & the Coordinate Plane · 12 topics

Opposites in Context [E]	Every number has a mirror twin on the other side of zero.
Absolute Value in Context [E]	Absolute value answers 'how far from zero?' — order answers 'which is greater?'
Comparing & Ordering Rational Numbers [E]	Smaller means farther left on the number line — even below zero.
Rational Numbers on the Number Line [E]	Fractions and decimals claim exact spots between the integers.
Plotting in All Four Quadrants [E]	Negative coordinates open up the other three quadrants.
Reflections Across the Axes [E]	Reflecting a point just flips the sign of one coordinate.
Distances on the Coordinate Plane [M]	Same x or same y: subtract and take the absolute value.
Rectangles on the Coordinate Plane [M]	Coordinates give the side lengths; perimeter and area follow.
Dividing Fractions: Word Problems [M]	'How many of this size fit?' is a division by a fraction.
Decimal Operations in Context [E]	Money problems are decimal arithmetic with two places, always.
Dividing Decimals in Context [M]	Slide both decimal points until the divisor is whole, then divide.
Percent of a Quantity: Parts & Wholes [M]	Percent problems run in three directions: find the part, the percent, or the whole.

Grade 5: Volume & Measurement Applications · 12 topics

The Volume Formula V = l × w × h [E]	Multiply the three edge lengths — no more counting cubes one by one.
Base Area × Height: Thinking in Layers [E]	A prism is a stack of identical layers, so volume is base area times height.
Volume Word Problems [M]	Spot the three dimensions hiding in a story, then multiply them.
Finding a Missing Dimension [M]	Divide the volume by the dimensions you know to uncover the one you don't.
Comparing Volumes [E]	Compute each volume, then subtract to see how far apart they are.
Volume of Composite Solids [M]	Split an L- or step-shaped solid into two prisms and add their volumes.
Volume by Subtraction [M]	When a piece is carved away, find the whole, find the hole, and subtract.
Packing Problems: How Many Fit? [M]	Count how many fit along each edge, then multiply the three counts.
Liquid Volume: Liters & Milliliters [E]	Convert between liters and milliliters to solve pouring and filling stories.
Mass Word Problems: Grams & Kilograms [E]	Turn kilograms into grams before adding, subtracting, or scaling masses.
Cubic Centimeters Meet Milliliters [M]	One cubic centimeter holds exactly one milliliter — volume becomes capacity.
Multi-Step Measurement Problems [H]	Chain a volume computation with a unit conversion to finish a real task.

Grade 5: Multi-Step Word Problems · 12 topics

Adding Up the Shopping Cart [M]	Multiply each price by its count, then add the lines like a receipt.
Unit Prices [M]	Divide the total by the count to find the price of just one.
Making Change [M]	Total the purchase first, then subtract it from what was handed over.
Budgets: Spending & Saving [M]	Track what goes out against what you started with — and what still fits.
Two-Step Whole-Number Problems [M]	Do the sentences in order: build the total first, then adjust it.
Equal Shares or Equal Groups? [E]	The same division answers two different questions — know which one you asked.
What the Remainder Means [M]	Round up, drop, or report the leftover — the story decides, not the arithmetic.
Recipe Problems: Add, Then Multiply [M]	Combine the fraction amounts for one batch, then scale up by the number of batches.
Fraction of a Group, Then One More Step [M]	Find the fraction of the group first — then answer what the story actually asks.
Mixing Decimals & Fractions [H]	Friendly numbers let fractions and decimals share one problem — take the fraction first.
Choosing the Operation [M]	Match the story's action to the right operation before touching the numbers.
Multi-Step Problem Marathon [H]	Three or more moves in one story — plan the steps before computing any of them.

Grade 6: Equations & Inequalities in Context · 12 topics

Writing Addition & Subtraction Equations [E]	Turn a join-or-separate story into a one-step equation.
Writing Multiplication & Division Equations [E]	Equal-groups and fair-share stories become px = q or x ÷ p = q.
Solving Add & Subtract Equations in Context [E]	Undo one addition or subtraction to answer the story's question.
Solving Multiply & Divide Equations in Context [E]	Undo one multiplication or division to find the missing amount.
Is It a Solution? [E]	Substitute a value for the variable and see if both sides agree.
Money Equations [M]	One-step equations where the amounts are dollars and cents.
Evaluating Expressions in Context [E]	Substitute a number for the variable and follow the order of operations.
Dependent & Independent Variables [E]	Which quantity drives the relationship, and which one responds?
Using Two-Variable Equations [E]	Feed the independent variable into a rule to get the dependent one.
Writing Inequalities from Constraints [E]	Translate 'at least', 'more than', 'at most', 'fewer than' into symbols.
Graphing Inequalities on a Number Line [E]	Open or closed circle, shaded left or right — read and draw both ways.
Boundary Values of an Inequality [E]	Find the smallest or greatest whole number an inequality allows.

Grade 6: Ratios, Rates & Statistical Thinking · 13 topics

Equivalent Ratios in Context [E]	Scale both parts of a ratio by the same number to keep it equivalent.
Solving Problems with Ratio Tables [M]	Extend a ratio table by scaling a known row up or down.
Unit Rate as a Tool [M]	Divide to a per-one rate, then multiply back up to answer.
Unit Pricing & the Better Buy [M]	Reduce each option to price per unit, then the smaller one wins.
Finding a Part or the Whole from a Ratio [M]	One share of the ratio unlocks every part and the total.
Percent as a Rate per 100 [E]	A percent is just a ratio scaled to a denominator of 100.
Finding a Percent of a Number [M]	Percent of a number = (percent ÷ 100) × the number.
Recognizing a Statistical Question [E]	A statistical question anticipates variety in its answers.
Mean, Median & Mode [M]	Three single-number summaries of the 'center' of a data set.
The Mean as a Fair Share & Balance Point [M]	The mean is what everyone gets when totals are shared out evenly.
Range: Measuring Spread [E]	Range is the distance from the smallest value to the largest.
Interquartile Range [M]	The IQR is the spread of the middle half of the data.
Describing the Shape of a Distribution [E]	Read a distribution's shape: symmetric, skewed, or uniform.

Prerequisite material — taught automatically when the diagnostic finds gaps

Math Kangaroo: Pre-Écolier & Écolier · 2 topics

Counting Pictures & Objects	Careful counting beats fast counting.
Sides & Corners	Shapes hide multiplication; cuts make one more piece.

Grade 5: Decimals, Volume & the Coordinate Plane · 7 topics

Powers of 10 & Patterns	Each power of ten shifts every digit one place.
Multiplying Decimals	Multiply as whole numbers, then place the decimal.
Dividing Decimals	Divide, keeping the decimal point lined up.
Volume by Unit Cubes	Volume counts the unit cubes that fill a solid.
Coordinate Plane (First Quadrant)	(x, y): right then up from the origin.
Converting Measurement Units	Multiply or divide by the conversion factor.
Order of Operations with Grouping	Parentheses first, then multiply/divide, then add/subtract.

Grade 4: Multi-Digit Operations & Place Value · 2 topics

What the Remainder Means	The story, not the arithmetic, decides what to do with a remainder.
Division Stories with Remainders	Divide, then round up, drop, or report the leftover — as the story asks.

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