5th Grade Math

342 core topics + 12 prerequisite topics taught as needed · approximately 100 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. Mixed checks follow every 6 lessons; each unit ends with a 12-item quiz, and course-wide assessments appear at 25%, 50%, 75%, and 100% mastery. 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 · 6 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.
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 · 13 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.
Decimals to Thousandths [E]	Read and write decimals to the thousandths place.
Comparing Decimals to Thousandths [M]	Line up place values; more digits does not mean a bigger number.
Rounding Decimals [M]	Look at the digit just past the place you are rounding to.
Dividing by a Decimal [M]	Shift both numbers so the divisor is a whole number, then divide.
Division with Two-Digit Divisors [M]	Estimate how many times the two-digit divisor fits, then adjust.

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: Liters and Milliliters [M]	Liquid volume converts like mass: 1 liter is 1,000 milliliters.
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.

Grades 4-5 - Fractions & Decimals (Mastery) · 48 topics

Equivalent Fractions in an Array [H]	One row of an array names the same amount two ways: by rows and by dots.
Splitting Every Piece into More Pieces [H]	Cutting every piece into k pieces multiplies the top and bottom by k.
A Chain of Equal Fractions [H]	In a chain of equal fractions the top and bottom grow by the same factor.
The Fraction That Does Not Belong [H]	Two fractions are equal only when one comes from the other by one factor.
Whole Numbers Written as Fractions [H]	Every whole number is a fraction: w wholes hold w times d of the d-ths.
Equivalent Fractions on a Ruler [H]	One mark on a ruler has many names, one for each size of piece.
Comparing Fractions with the Same Top Number [H]	With equal top numbers, the smaller bottom number gives the larger fraction.
Comparing by the Missing Piece [H]	A fraction just short of one whole is larger when its missing piece is smaller.
Comparing by Renaming Just One Fraction [H]	When one bottom number divides the other, rename only that fraction.
Between 0, One Half, and One [H]	Doubling the top number and comparing it with the bottom locates a fraction.
Who Had More? Comparing in a Story [H]	Comparing fractions of the same whole answers who has more in a story.
Comparing Fractions in a Table [H]	Reading a table of fractions means comparing every row, not just two.
Missing Addend with Like Denominators [H]	An unknown part of a fraction sum is found by subtracting the known part.
Adding Three or Four Like Fractions [H]	Same-size pieces add by counting pieces, however many parts there are.
Subtracting from One Whole [H]	Write one whole as d/d before taking a fraction away from it.
Subtracting a Fraction from a Whole Number [H]	Trade one whole for d/d, then subtract the fraction from those pieces.
Sums That Pass One Whole [H]	A sum of d or more d-ths is renamed as wholes plus the leftover pieces.
Adding When One Bottom Number Fits the Other [H]	Rename the larger pieces as the smaller ones, then add the top numbers.
Subtracting When One Bottom Number Fits the Other [H]	Rename first so both fractions count the same size piece, then subtract.
Fifths and Tenths Together [H]	Each fifth is two tenths, which links fifths to decimal notation.
Thirds and Sixths Together [H]	Halves and thirds both rename as sixths, so sixths add all three.
Mixed Numbers on a Number Line [H]	A point past 1 names whole steps plus the parts counted after them.
How Many Wholes Are Hiding? [H]	Dividing the top by the bottom counts the whole wholes inside a fraction.
Mixed Number against Improper Fraction [H]	Write both amounts the same way, then compare their top numbers.
Adding a Fraction onto a Mixed Number [H]	Add the fraction parts first, then carry any whole into the whole number.
How Much More to the Next Whole? [H]	The gap to the next whole is the bottom number minus the fraction's top.
Totaling Mixed Numbers from a Table [H]	Read each amount from the table, then add or subtract the mixed numbers.
From Repeated Addition to Multiplication [H]	Adding a/b n times is the same as n times a/b, so the top is multiplied.
How Many Copies Make One Whole? [H]	Counting copies of a fraction that fill a whole asks how many fit in d/d.
Is the Product More Than One Whole? [H]	Compare n times a with the bottom number to judge the product's size.
Equal Groups of a Fraction [H]	Equal groups of unit-fraction pieces total the number of pieces over d.
Naming the Point on a Fraction Line [H]	The bottom number counts the parts in one whole; the top counts the marks.
Jumping by Fractions on a Number Line [H]	Each jump of one d-th moves the point one mark along the line.
Decimals in Words [H]	The words name the last place, which fixes how many decimal places to write.
Tenths or Hundredths? Watch the Zero [H]	A zero in the tenths place makes the next digit count hundredths, not tenths.
Renaming Tenths as Hundredths [H]	One tenth is ten hundredths, so a tenths top number times ten gives hundredths.
Decimals Greater Than One [H]	The digits before the point are the wholes; the digits after are the parts.
Comparing a Fraction with a Decimal [H]	Write the fraction in hundredths, then compare it with the decimal.
Locating a Decimal Between Two Tenths [H]	Between two tenths the line is cut into hundredths, one per small step.
One Point, Two Names [H]	A point on a tenths line has one fraction name and one decimal name.
Decimals That Make One Whole [H]	Two decimals make one whole when their hundredths add to one hundred.
Reading Tenths from a Bar Graph [H]	Bars counted in tenths become decimals once the count is placed after the point.
How Many Full Containers? [H]	Counting completely filled containers keeps only the whole number, dropping the leftover.
How Many Containers Are Needed? [H]	Holding every last piece needs one extra container for the leftover part.
How Much Is Left Over? [H]	The leftover is the remainder of pieces, written over the same bottom number.
Enough or Not Enough? [H]	Deciding whether an amount suffices compares what you have with what is needed.
Who Used the Most? Reading a Table [H]	Compare every row of amounts, wholes first and then the fraction parts.
Counting On by a Fraction [H]	Counting on by a/d repeatedly adds a pieces, so the nth number is n times a over d.

Grades 4-5 - Measurement, Data & Geometry (Mastery) · 49 topics

Completing a Conversion Table [H]	Each row of a conversion table multiplies by the same fixed number.
Regrouping a Total into Mixed Units [H]	Divide by the conversion number: the quotient is the big units, the remainder the small ones.
Comparing Measurements in Different Units [H]	Rewrite both measurements in the same unit before deciding which is greater.
Ordering Three Measurements [H]	Convert every measurement to one unit, then order the plain numbers.
Inches, Feet, and Yards [H]	One foot is 12 inches and one yard is 3 feet, so multiply to go smaller.
Pounds and Ounces [H]	One pound is 16 ounces, so multiply pounds by 16 to get ounces.
Cups, Pints, Quarts, and Gallons [H]	Each step down the customary capacity ladder doubles, except gallon to quart.
Adding Mixed-Unit Measurements [H]	Change every measurement to the small unit, then add the totals.
Subtracting Mixed-Unit Measurements [H]	Convert both measurements to the small unit, then subtract.
Multiplying a Mixed-Unit Measurement [H]	Convert one item to the small unit, multiply, then finish the story.
Choosing a Sensible Unit [H]	Match the size of the unit to the size of the thing being measured.
Two-Step Conversions [H]	Convert one step at a time, multiplying down the ladder or dividing up it.
Perimeter of a Regular Polygon [H]	A regular polygon has equal sides, so its perimeter is the side length times the number of sides.
Perimeter of an L-Shaped Figure [H]	Add every side of a rectilinear figure; cutting a corner leaves the perimeter unchanged.
Missing Sides of an L-Shaped Figure [H]	Facing sides of an L-shape match, so a missing piece is the whole minus the known piece.
Same Area, Different Perimeter [H]	Rectangles with equal areas can have different perimeters; divide the area by one side to find the other.
Same Perimeter, Different Area [H]	Rectangles with equal perimeters can have different areas; half the perimeter is length plus width.
Squares: Side, Perimeter, and Area [H]	A square has four equal sides, so perimeter is four times a side and area is a side times itself.
Cost of Covering an Area [H]	Find the area first, then multiply by the price for each square unit.
Cost of Fencing a Border [H]	Fencing follows the perimeter, so find the distance around and multiply by the price per meter.
Area of a Border Path [H]	Subtract the inner rectangle's area from the outer rectangle's area to get the border.
Elapsed Time from a Clock Face [H]	Read the start time off the clock, then count minutes forward to the end time.
Finding the Start Time [H]	Count backwards from the end time by the length of the event.
Finding the End Time [H]	Add the length of each stage to the start time, regrouping 60 minutes into an hour.
Reading a Schedule [H]	Subtract a start time from an end time to get the length of each item on a schedule.
Time Spans That Cross Noon [H]	Count up to 12:00 noon, then add the time after noon.
Change with the Fewest Coins [H]	Find the change first, then take the largest coin that fits as many times as possible.
How Many Can You Buy? [H]	Divide the money by the price and drop the remainder, since a part of an item cannot be bought.
Shopping from a Price List [H]	Read each price from the list, multiply for repeats, add the bill, then subtract to find change.
Reading a Line Plot in Halves [H]	Each mark on a line plot stands for one measurement, so counting marks answers how many.
Building a Line Plot from a List [H]	Each measurement in a list becomes one X above its own tick on the line plot.
A Missing Stack on a Line Plot [H]	Subtract the stacks you can see from the total to find the missing stack.
Bar Graphs with a Scale [H]	When one bar unit stands for several items, multiply the bar height by that number.
Comparing Bars on a Graph [H]	Compare two bars by subtracting the shorter value from the taller one.
Totals from a Bar Graph [H]	Add every bar for the total, and subtract from a known total to find a missing bar.
Picture Graphs with a Key [H]	Multiply the number of symbols by the key value to get the real amount.
Comparing Rows of a Picture Graph [H]	Compare rows by their symbol counts, then multiply the difference by the key value.
Reading a Tally Chart [H]	Each group of tally marks crossed through stands for five, so count by fives then add the singles.
Two-Step Questions from a Table [H]	Combine two columns of a table first, then compare or total the results.
Estimating an Angle [H]	Judge an angle against the benchmarks of 90 degrees and 180 degrees.
Quarter Turns and Directions [H]	A quarter turn is 90 degrees, so each 90 degrees clockwise moves one step around the compass.
Angles Around a Point [H]	Angles that fill a complete turn around a point add to 360 degrees.
Angles Made by Clock Hands [H]	The twelve hour marks split a clock into twelve 30 degree steps.
Lines of Symmetry in Regular Shapes [H]	A regular polygon has as many lines of symmetry as it has sides.
Sides of a Quadrilateral from the Perimeter [H]	Opposite sides of a parallelogram are equal, so half the perimeter is one long side plus one short side.
Can These Angles Make a Triangle? [H]	The three angles of any triangle add to exactly 180 degrees.
Naming a Triangle Two Ways [H]	Every triangle has a name from its sides and a second name from its angles.
Multi-Step Measurement Problems [H]	Convert to one unit, work out the part that repeats, then add or subtract.
Leftovers: Round Up or Round Down? [H]	Count only complete groups when filling containers, but add one more group when nothing may be left behind.

Grades 4-5 - Operations & Number Sense (Mastery) · 50 topics

Naming Places to the Millions [H]	Places are counted from the right: ones, tens, hundreds, thousands, and on.
Reading a Place-Value Chart [H]	A place-value chart lists one digit per place, in order from left to right.
Expanded Form of a Large Number [H]	Expanded form adds up what each digit is worth in its own place.
Building a Number from Clues [H]	Each clue fixes one digit; write the digits in place order to build the number.
Comparing What Two Digits Are Worth [H]	The same digit is worth different amounts in different places; subtract to compare.
Reading a Number Line of Large Numbers [H]	Equally spaced marks count by the same amount every jump.
Comparing Numbers to the Millions [H]	Compare place by place from the left; the first place that differs decides.
Rounding to Hundred Thousands and Millions [H]	Check the digit one place to the right, then round the target place up or down.
Numbers That Round to a Given Value [H]	Every rounded value comes from a band of numbers with a top and a bottom.
Rounding One Number Two Ways [H]	Rounding to a larger place can move a number the other direction.
Rounding the Numbers in a Table [H]	Round each value in a data table to the same place before comparing or adding.
Subtracting Across Zeros [H]	A row of zeros needs regrouping to travel several places to the left.
Missing Digits in a Sum [H]	Rebuild a hidden digit by adding one column at a time and tracking carries.
Missing Digits in a Difference [H]	Recover a hidden digit in a subtraction by checking each column, borrows included.
A Missing Number in a Subtraction [H]	Addition undoes subtraction, so it recovers a missing part and checks an answer.
Keeping a Running Total [H]	Add every amount in and subtract every amount out to track a total.
Comparing Two Changes [H]	Find each change by subtracting, then compare the changes themselves.
Partial Products with One Digit [H]	Multiply each place separately, then add the partial products.
Multiplying Multiples of Ten and Hundred [H]	Multiply the front digits, then attach every zero from both factors.
Finding a Missing Factor [H]	Divide the product by the known factor to recover the missing one.
The Area Model for a Two-Digit Product [H]	An area model splits both factors by place and multiplies every pair of parts.
Multiplying by a Multiple of Ten [H]	Multiply by the tens digit, then attach the zero that the ten carries.
Two-Digit Products in Stories [H]	Equal groups of a two-digit size multiply, and other steps follow after.
Comparing Two Products [H]	Estimate or compute each product, then compare the results.
Zeros Inside the Quotient [H]	Write a 0 in the quotient whenever the divisor does not fit into a place.
How Many Digits in the Quotient [H]	Compare the divisor with the leading digits to see where the quotient starts.
Naming the Quotient and the Remainder [H]	A full division answer names how many whole groups and how much is left.
How Many More Would Fill the Last Group [H]	The gap in the last group is the divisor minus the remainder.
Buying Enough Packs [H]	When every item must be covered, the leftover forces one more whole pack.
Dividing in Two Stages [H]	Sharing out twice divides twice; multiplying back undoes both stages.
Both Parts of a Division Answer [H]	One division answers two questions: how many full groups and how many are left.
Between Which Two Numbers? [H]	Rounding the numbers first shows which pair of landmarks the answer falls between.
Which Answer Is Reasonable? [H]	An estimate rules out answers that are ten times too big or too small.
Repairing an Unreasonable Answer [H]	Estimation flags a wrong answer; then redo the work carefully to fix it.
Factor or Multiple? [H]	The smaller number is the factor; the larger one is the multiple.
Counting Multiples in a Range [H]	Skip count through the range, or divide each end and compare the counts.
Divisibility Rules [H]	Quick digit tests tell whether 2, 3, 5 or 10 divides a number evenly.
Which Number Has More Factors [H]	List each number's factor pairs and count them before comparing.
Counting Primes [H]	Test each number for small factors; a prime has exactly two factors.
Patterns with a Two-Step Rule [H]	Apply both parts of the rule, in order, to get from one term to the next.
A Missing Term Inside a Pattern [H]	Find the rule from the terms you can see, then step into the gap.
Working Back to the First Term [H]	Undo the rule once for every step back to reach the first term.
How Many Steps to Reach the Goal [H]	Divide the distance to the goal by the step size, rounding up if any gap remains.
Where Do the Parentheses Go? [H]	Parentheses choose which operation happens first, which changes the value.
How Parentheses Change the Value [H]	The same numbers give different values depending on where parentheses sit.
A Missing Number in an Expression [H]	Undo the operations in reverse order to uncover the missing number.
Solving an Equation with a Letter [H]	A letter stands for one unknown number; undo the operations to find it.
Write an Equation, Then Solve [H]	Turn the story into an equation with a letter before doing any arithmetic.
Two Steps from a Price List [H]	Read the prices from the list, then multiply and add or divide as the story asks.
Two Steps from a Bar Graph [H]	Read the bars first, then add, subtract or compare to answer the question.

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.

4th Grade Math - Deep II · 9 topics

Rounding to a Given Place	Look one place to the right, then round the target digit up or down.
Value of a Digit	A digit's value is the digit times what its place is worth.
Expanded Form to Standard Form	Expanded form spells out each place's value; add them for the number.
Is the Estimate Reasonable?	Round each number first, then add to judge a sensible total.
Rounding to Estimate a Quantity	Real amounts are often reported to the nearest hundred or thousand.
Mixed Numbers to Improper Fractions	Multiply the whole by the denominator, then add the numerator.
Improper Fractions to Mixed Numbers	Divide the top by the bottom: the quotient is the whole, remainder the top.
Missing Part of an Equivalent Fraction	Whatever you multiply the bottom by, multiply the top by the same.
Decimals on a Number Line	Splitting 0 to 1 into 10 parts marks tenths; into 100 parts, hundredths.

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