5th Grade Math

191 core topics + 3 prerequisite topics taught as needed · approximately 45 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 · 4 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.

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 · 2 topics

Decimal Addition & Subtraction [M]	Line up the decimal points.
Fractions ↔ Decimals [M]	Converting between the two notations.

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?

Grade 5: Decimals, Volume & the Coordinate Plane · 8 topics

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

Grade 4: Multi-Digit Operations & Place Value · 12 topics

Ten Times the Place [E]	Each step left in a number makes a digit worth ten times more.
Comparing Numbers to 1,000,000 [E]	Compare big numbers digit by digit from the left.
Big-Number Word Problems [M]	Add and subtract six-digit numbers inside real stories.
Factors & Factor Pairs [E]	Factors come in pairs that multiply to make the number.
Multiples [E]	The multiples of a number are its skip-counting list.
Prime or Composite? [M]	A prime has exactly two factors: 1 and itself.
Times as Many [E]	“3 times as many” means multiply — or divide to go backwards.
More Than vs. Times as Many [M]	“3 more” adds, “3 times as many” multiplies — don't mix them up.
What the Remainder Means [M]	The story, not the arithmetic, decides what to do with a remainder.
Division Stories with Remainders [M]	Divide, then round up, drop, or report the leftover — as the story asks.
Follow the Pattern Rule [E]	Apply a rule again and again — the first term counts as step one.
Find the Pattern's Rule [M]	Compare neighboring terms to uncover the rule, then test it everywhere.

Grade 4: Fraction Sense · 12 topics

Fractions from Equal Parts [E]	The bottom counts the equal parts; the top counts the parts you take.
Seeing Equivalent Fractions [E]	Cutting every part into k smaller pieces changes the name, not the amount.
Decomposing Fractions [E]	Every fraction is a stack of unit-fraction pieces you can split apart.
Comparing Fractions with Benchmarks [M]	Judge each fraction against 1/2 before reaching for common denominators.
Adding Mixed Numbers (Like Denominators) [M]	Add the wholes, add the parts, then trade a full set of parts for a whole.
Subtracting Mixed Numbers [M]	When the top fraction part is too small, break one whole into d parts.
Whole Number × Unit Fraction [E]	n copies of 1/d stack up to n/d — multiplication is repeated addition.
Multiplying a Fraction by a Whole Number [M]	n × a/b is n × a of the 1/b pieces: multiply the top, keep the bottom.
Tenths & Hundredths as Decimals [E]	Decimals are just fractions with denominators 10 and 100 in disguise.
Adding Tenths and Hundredths [M]	Trade every tenth for ten hundredths, then add same-size pieces.
Comparing Decimals to Hundredths [M]	Longer doesn't mean larger — line up the tenths place and compare.
Fraction Word Problems [M]	Pick the picture first — parts left over, laps repeated, or days combined.

Grade 4: Angles, Lines & Measurement · 12 topics

Angles as Turns [E]	An angle measures how far one ray turns away from another.
Adding Angles [E]	A ray through an angle cuts it into parts that add up to the whole.
Finding Missing Angles [M]	Subtract the known parts from 90°, 180°, or 360° to find the rest.
Classifying Triangles [E]	Sort triangles by their angles or by how many sides match.
Classifying Quadrilaterals [M]	Name quadrilaterals by their parallel sides and right angles.
Lines of Symmetry [E]	A line of symmetry folds a shape exactly onto itself.
Perimeter Word Problems [M]	Fence, frame, and rope problems: work the perimeter formula backward too.
Area Word Problems [M]	Tiles, rugs, and L-shaped rooms: multiply, divide back, or split and add.
Kilometers, Meters & Centimeters [E]	Move between km, m, and cm with the anchors 1,000 and 100.
Kilograms & Grams [E]	1 kilogram is 1,000 grams — convert first, then solve the story.
Hours, Minutes & Seconds [E]	Time converts by sixties: hours to minutes to seconds.
Line Plots with Fractions [M]	Read X-mark plots in quarter units, then count, compare, and total.

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 4: Multi-Step Word Problems · 12 topics

Multiply, Then Adjust [E]	First count the groups by multiplying, then add or subtract the rest.
Combine, Then Share [M]	Get one total first, then divide it into equal groups.
Three-Step Problems [M]	Three operations in one story — take them one sentence at a time.
Too Much Information [E]	Cross out the number the question never needed.
Estimate First [E]	Round to friendly numbers before computing to know what answer to expect.
Adding Up the Bill [M]	Total a shopping list: multiply repeated items, then add everything up.
Counting the Change [M]	Total the whole basket first, then subtract it from what was paid.
Elapsed Time Across Hours [M]	Count up to the next hour, jump the whole hours, then finish the minutes.
Schedules with Several Steps [M]	Add up every stage of the schedule before touching the clock.
More Than Meets Times As Many [M]	One story, both kinds of comparison — multiply one step, add the other.
Pick the Right Equation [M]	Translate the story into an equation before any arithmetic happens.
Working Backwards [M]	Start from the ending amount and undo each step in reverse order.

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 4: Fractions & Decimals — Deeper Practice · 13 topics

Equivalent Fractions on a Number Line [E]	The same point on the line has many names — finer ticks, bigger numbers.
Recognizing Equivalent Fractions [E]	Multiply top and bottom by the same number — never add the same number.
Comparing with Common Denominators [M]	Give both fractions the same size pieces, then just count them.
Ordering Three Fractions [M]	Put them over one common denominator and the order reads straight off.
Adding Like-Denominator Fractions in Context [E]	Same-size pieces just pile up — add the numerators, keep the denominator.
Subtracting Like-Denominator Fractions [E]	Take pieces away from same-size pieces — subtract the tops, keep the bottom.
Mixed-Number Addition Word Problems [M]	Add the wholes, add the parts, then trade a full set of parts for a whole.
Mixed-Number Subtraction Word Problems [M]	When the fraction you take is too big, break one whole into d parts.
Fraction × Whole Number Word Problems [M]	n groups of a/b is n×a of the 1/b pieces — multiply the top, keep the bottom.
Decimal Notation for Tenths & Hundredths [E]	A decimal is a fraction with denominator 10 or 100, written by place.
Relating Fractions and Decimals [M]	3/10 and 0.3 are the same amount — two ways to write one number.
Comparing Decimals in Context [M]	Line up the tenths place first — more digits does not mean more value.
Adding Tenths and Hundredths [M]	Trade each tenth for ten hundredths, then add same-size pieces.

Grade 4: Measurement, Data & Geometry — Deeper Practice · 12 topics

Length Problems: Convert, Then Combine [M]	Turn every length into one unit first, then add or subtract.
Mass Problems: Kilograms and Grams [M]	Convert kilograms to grams, then total, take away, or compare.
Liquid Volume: Litres and Millilitres [M]	Liquid volume converts like mass: 1 litre is 1,000 millilitres.
Time Problems: Hours, Minutes, Seconds [M]	Change to one time unit using sixties, then add or subtract.
Elapsed Time Across the Hour [M]	Total the minutes across each o'clock, then adjust for breaks.
Perimeter: Finding a Missing Side [M]	Perimeter is the total edge, so a missing side is the total minus the rest.
Area: Finding a Missing Length [M]	Area is length times width, so divide the area by a known side.
Area of Rectilinear Figures [M]	Split an L or staircase into rectangles, or subtract a cut-out corner.
Line Plots: Fraction Questions [M]	Read a fraction line plot to find spread, totals, and combined lengths.
Adding Angles on a Diagram [M]	Angle measure adds up: parts total the whole, and rays split it.
Classifying Angles and Triangles [M]	Sort angles as acute to reflex, and triangles by angles or sides.
Money: Change and Multi-Step Spending [M]	Total the cost, subtract from what is paid, then spend the change.

Grade 4: Whole-Number Computation Fluency · 13 topics

Adding Within 1,000,000 [E]	Stack six-digit numbers by place value and carry across every column.
Subtracting Within 1,000,000 [E]	Regroup across many columns to subtract six-digit numbers.
Estimating Sums & Differences [E]	Round each number first to get a quick check on a big sum or difference.
Two-Digit by Two-Digit Multiplication [M]	Multiply by the tens and the ones, then add the two partial products.
Four-Digit by One-Digit Multiplication [M]	Multiply each place of a four-digit number by the single digit, carrying up.
Estimating Products [M]	Round the factors to easy numbers to predict roughly how big a product is.
Dividing by a One-Digit Number [M]	Long-divide a three- or four-digit number that comes out even.
Division with Remainders [M]	When the divisor doesn't fit evenly, whatever is left is the remainder.
Estimating Quotients [M]	Swap the dividend for a nearby number the divisor divides evenly.
Understanding Remainders [M]	A remainder is always smaller than the divisor, and rebuilds the dividend.
Order of Operations [M]	Do multiplication and division before addition and subtraction.
Parentheses First [M]	Whatever is inside parentheses gets done before anything else.
Multi-Step Expressions [M]	Combine several operations in one expression, obeying the full order rules.

Grade 4: Fraction & Decimal Mastery · 13 topics

Adding Mixed Numbers with Regrouping [M]	When the fraction parts spill past one whole, carry a whole and keep the rest.
Subtracting Mixed Numbers with Regrouping [M]	When the top fraction is too small, borrow one whole and turn it into d parts.
Combining Mixed Numbers Toward a Goal [M]	Add up what is done, then subtract from the goal to see what is left.
Multiplying Fractions in Word Problems [M]	n groups of a/b is n×a of the 1/b pieces — multiply the top, keep the bottom.
Fractions Against the 1/2 Benchmark [M]	A fraction beats 1/2 exactly when its top is more than half its bottom.
Ordering Fractions with Common Denominators [M]	Rewrite them all over one common denominator, then order the numerators.
Ordering Decimals [M]	Compare place by place from the left — tenths first, then hundredths.
Adding Tenths and Hundredths [M]	Trade every tenth for ten hundredths, then add same-size pieces.
Money as Fractions and Decimals [M]	0.25 = 1/4 = a quarter — one value, three ways to name it.
Coins as Fractions of a Dollar [M]	Count the cents, then name the total as a fraction or a decimal dollar.
Fractions of a Length [M]	Split the whole length into equal parts, then take as many as the top says.
Fractions of Time and Capacity [M]	The same part-of-a-whole idea works on minutes, liters, and hours.
Adding Mixed-Number Measurements [M]	Combine measured amounts by adding wholes, adding parts, and carrying.

Grade 4: Multi-Step Reasoning & Patterns · 13 topics

Four Operations, Bigger Numbers [M]	Chain add, subtract, multiply, and divide through one larger story.
Scale Up, Then Share Out [M]	Build a big total by multiplying, then divide it into equal groups.
Money in Several Steps [M]	Unit prices, totals, and change — thinking in whole cents keeps it exact.
Factor Pairs in Arrays [M]	Every equal-row arrangement of objects matches a factor pair.
Prime or Composite in Context [M]	Whether a group splits into equal teams depends on prime vs. composite.
Making Identical Groups [M]	A common factor of two amounts is a group size that divides both.
Later Terms of Number Patterns [M]	Apply the rule step by step to reach a term far down the pattern.
Growing Shape Patterns [M]	A figure pattern that grows by a fixed amount is a number pattern in disguise.
Features of a Pattern [M]	Spot the always-true property hidden in a pattern's rule.
Interpreting Remainders [M]	Round up, drop it, or report it — the story decides what a remainder means.
Remainders in Multi-Step Problems [M]	Build the total first, then divide and read the remainder the story's way.
Times as Many, Both Directions [M]	Multiply to scale up, divide to scale back, and subtract for the gap.
Mixing 'More Than' and 'Times as Many' [M]	Untangle a story that adds one comparison and multiplies another.

Prerequisite material — taught automatically when the diagnostic finds gaps

Math Kangaroo: Pre-Écolier & Écolier · 3 topics

Counting Pictures & Objects	Careful counting beats fast counting.
Fair Sharing	Division with (and without) leftovers.
Sides & Corners	Shapes hide multiplication; cuts make one more piece.

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