4th Grade Math

298 core topics + 3 prerequisite topics taught as needed · approximately 88 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 · 4 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.
Mixed Numbers & Improper Fractions [M]	Converting between forms.

Decimals, Percents & Ratios · 1 topics

Decimal Addition & Subtraction [M]

Line up the decimal points.

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 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 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 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.

4th Grade Math - Deep II · 20 topics

Rounding to a Given Place [M]	Look one place to the right, then round the target digit up or down.
Value of a Digit [M]	A digit's value is the digit times what its place is worth.
Expanded Form to Standard Form [M]	Expanded form spells out each place's value; add them for the number.
Reading Number Names [M]	Translate the words for each place group into digits.
Is the Estimate Reasonable? [M]	Round each number first, then add to judge a sensible total.
Rounding to Estimate a Quantity [M]	Real amounts are often reported to the nearest hundred or thousand.
Mixed Numbers to Improper Fractions [M]	Multiply the whole by the denominator, then add the numerator.
Improper Fractions to Mixed Numbers [M]	Divide the top by the bottom: the quotient is the whole, remainder the top.
Missing Part of an Equivalent Fraction [M]	Whatever you multiply the bottom by, multiply the top by the same.
Simplest Form [M]	Divide the top and bottom by their greatest common factor.
Fraction of a Set [M]	Split the set into equal groups, then take as many as the numerator says.
Least Common Multiple [M]	The LCM is the smallest number both counts land on when you skip-count.
Greatest Common Factor [M]	The GCF is the largest number that divides both amounts evenly.
Input-Output Tables [M]	Apply the rule's steps in order to turn an input into its output.
Finding the Rule of a Table [M]	Test a candidate rule against every row of the table.
Parallel and Perpendicular Lines [M]	Parallel lines never meet; perpendicular lines cross at a square corner.
Points, Lines, Rays, and Segments [M]	A point is a spot; lines, rays, and segments differ by their endpoints.
An Angle as a Fraction of a Turn [M]	A full turn is 360°, so a fractional turn is that fraction of 360.
Comparing Area and Perimeter [M]	Area multiplies the sides; perimeter adds all four - they can differ.
Decimals on a Number Line [M]	Splitting 0 to 1 into 10 parts marks tenths; into 100 parts, hundredths.

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.

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