Grade 3 · secular · Common Core-aligned
This is the year multiplication and division actually click - not as a set of memorized facts, but as pictures your child can draw and reason from. Everything else in the year - measuring area, comparing fractions, reading a graph - turns out to be that same equal-groups picture wearing a different hat. By June, your child should be able to look at a rectangle, a clock, or a bar graph and say "oh, this is just groups of things again."
The whole year, in plain English. Tap any unit to see every skill inside, nothing is hidden.
This unit gives a name and a picture to something your child already half-knows: counting things in equal groups. They'll learn two pictures - equal groups and arrays (rows and columns of dots or tiles) - and the words that go with them: factor, product, row, column. The array picture is the star here because rotating it shows, without any rule to memorize, that 4 groups of 6 and 6 groups of 4 come out the same.
Given a picture of equal groups (e.g., 4 baskets of 3 apples), the student writes the total as repeated addition and as a multiplication equation.
The student builds an array with a stated number of rows and columns and states the total.
The student names the factors and product in a given multiplication equation and in a matching array.
Given an array, the student rotates it 90 degrees (physically or by redrawing) and explains why the total does not change.
The student compares an equal-groups picture and an array for the same fact and explains what stays the same and what looks different.
Given a word problem describing equal groups, the student selects and draws the matching picture and writes the equation.
The student identifies a case where repeated addition describes a situation but an array does not fit naturally (e.g., 3 groups of a half-sandwich), and explains why.
Given a multiplication fact never shown in class, the student draws an array, rotates it, and explains using the picture why the two factor orders give the same product.
Division isn't a new topic here - it's the same array from Unit 1, looked at from a different angle. Every array hiding a multiplication fact is also hiding two division facts, and your child learns to see all four before drilling any one of them. The unit moves from splitting real arrays, to naming fact families, to solving for missing factors, to fast recall, ending with two-step word problems.
Given an array of up to 10 rows and 10 columns, state the two multiplication facts and two division facts it represents.
Identify the divisor, dividend, and quotient in a division fact when the fact and its array are both shown.
Divide a set of up to 100 objects into equal rows, using an array drawing, to find the number in each row.
Recall the quotient for any division fact with a divisor and dividend within 10x10, without counting or drawing.
Find the missing factor in an equation such as 6 x ? = 42 or ? x 7 = 56, presented with the product in either position.
Explain why a set of objects cannot always be split into equal whole-number groups, using a specific non-even example.
Solve a two-step word problem requiring one multiplication step and one related division step, drawn from an unfamiliar context.
Identify the pattern of increasing products in a multiplication table (e.g. the 9s pattern) and use it to predict an unfamiliar fact.
Compare two array drawings that represent the same fact family rotated 90 degrees and judge whether they show the same facts.
Numbers get bigger - up to 10,000 - and rounding gets taught as a real choice ('which friendly number is this closer to on a line?') instead of a digit-underlining trick. Your child builds a place value chart through the thousands, places numbers on number lines with no marked ticks except two endpoints, and rounds to the nearest 10 and 100 using that line. They also multiply single digits by multiples of 10 using the array knowledge from Units 1-2.
Given a 4-digit number, write how many thousands, hundreds, tens, and ones it has using a place value chart.
Write a 4-digit number in expanded form as the sum of thousands, hundreds, tens, and ones.
Place a given whole number as a point on an open number line marked only with two benchmark endpoints.
Round a given whole number to the nearest 10 by identifying which benchmark on a number line it is closer to.
Round a given whole number to the nearest 100 using the same number-line reasoning applied to the nearest 10.
Explain why a number can round differently depending on whether the purpose calls for nearest 10 or nearest 100.
Multiply a one-digit number by a multiple of 10 (10-90) using an array or equal-groups picture from Units 1-2.
Estimate a sum or difference by rounding both numbers first, then compare the estimate to a computed exact answer to judge reasonableness.
Decide, for an unfamiliar word problem with no rounding cue given, whether an exact answer or an estimate best fits the situation, and justify the choice.
Same rectangle, two different questions: how much space is inside it (area), and how far around the outside (perimeter). Your child tiles rectangles with unit squares to build area as an honest count, then connects that count to the array's rows-times-columns from Units 1-2, now relabeled length-times-width. Perimeter goes on the very same rectangles on purpose, so the two ideas get compared side by side instead of learned as unrelated facts.
Find the area of a rectangle by tiling it with unit squares and counting them.
Explain why the number of unit squares in a tiled rectangle equals rows times columns.
Compute the area of a rectangle using the formula length times width, given whole-number side lengths.
Distinguish area problems from perimeter problems by identifying which unit label (square units vs. linear units) and which count (interior squares vs. distance around) each requires.
Find an unknown side length of a rectangle given its perimeter and one known side.
Build rectangles from a fixed number of unit tiles and record their area and perimeter accurately.
Predict, for a new fixed number of tiles not yet built, whether area or perimeter will stay the same across different rectangles.
Determine the rectangle dimensions that would use a given number of square tiles most efficiently for a real fencing or flooring scenario not previously discussed in class.
Round three-digit rectangle side lengths to the nearest ten to estimate area before computing the exact product.
Fractions arrive as pictures, not symbols, first. Your child folds and divides shapes and number lines into equal parts, names unit fractions like 1/4, and places fractions as points or hops on a line rather than as two numbers stacked on top of each other. A big chunk of this unit is spent on unequal partitioning, because that's the single most common way this goes wrong.
Partition a rectangle or circle into b equal-sized parts, b up to 8, using folding or drawn lines.
Shade a of b equal parts on a fraction bar to represent a given fraction a/b.
Identify which of several partitioned shapes shows equal parts versus unequal parts, given shapes cut to look plausible but unequal.
Explain why a fraction label is meaningless if the parts of the whole are not equal in size.
Partition a number line segment from 0 to 1 into b equal lengths and label each tick with the correct unit fraction.
Locate and label a given fraction a/b as the point reached after a hops of length 1/b from 0.
Compare a fraction bar model and a number line model of the same fraction and explain that both represent the same distance from a whole.
Determine whether 1/2 of two different-sized wholes (a long strip and a short strip, both cut fairly) names the same actual size of piece.
Given an unfamiliar real object (a garden bed, a ribbon length) never used in class examples, partition it into equal fourths and mark the 1/4 point on a corresponding number line.
Now that your child can build and place fractions, this unit compares them - using bars and number lines, never by comparing the numerators or denominators alone. Same-denominator and same-numerator comparisons come first because the picture gives an obvious answer; comparing against the benchmark 1/2 comes next for trickier pairs; whole numbers as fractions (3 = 3/1) closes the unit out.
Given two fraction bars split into the same number of equal parts, state which shaded amount is greater by counting shaded parts.
Given two fraction bars with the same numerator but different denominators, explain why the fraction with fewer parts is greater.
Locate two given fractions as points on the same number line and state which point is farther from zero.
Identify two fractions shown on stacked, equal-length fraction bars as equivalent when their shaded lengths match exactly.
Compare two fractions that share neither numerator nor denominator by reasoning about each fraction's distance from the benchmark 1/2.
Explain why two fractions with different denominators cannot be compared by comparing the denominators alone.
Represent a whole number such as 3 as a fraction (3/1) and place it correctly on a number line alongside fractions less than one.
Given a new pair of fractions and two blank equal-length strips with no marks, choose and draw a model that shows which fraction is greater, then justify the choice.
A clock face gets introduced as a number line bent into a circle - the same skip-counting-by-5s your child already has gets pointed at a new, round context. The unit builds clock reading first, then elapsed time using a timeline drawing, adding one complication at a time: within an hour, crossing an hour, comparing two gaps, crossing noon or midnight. Unlike most of this course, clock-reading itself is taught fairly directly - it's a genuinely new structure, not something to rediscover.
Given an analog clock with only hour marks labeled, skip-count by 5s aloud or in writing to name the exact minute the minute hand points to.
Read any analog clock face and state the time to the nearest minute, including times where the minute hand sits between labeled 5-minute marks.
Explain why the clock face can be thought of as a number line bent into a circle, using the wrap-around at 12 as evidence.
Draw a timeline with labeled hops of 5 minutes to represent the elapsed time between a given start time and end time that do not cross an hour boundary.
Solve a word problem requiring addition or subtraction of a time interval that crosses one hour boundary, using a drawn timeline.
Compare two different start-and-end time pairs and determine whether they represent the same elapsed time.
Solve a multi-step elapsed-time word problem set in an unfamiliar real-world context (e.g. a multi-stop bus schedule) that crosses both an hour boundary and the a.m./p.m. divide, with no explicit cue that a timeline should be used.
Recall the skip-count sequence by 5s to 60 from memory within 30 seconds.
Classify a set of mixed elapsed-time word problems by which structural feature makes them hard (crosses an hour, crosses noon/midnight, or neither) before solving any of them.
This closing unit reveals that a graph's scale - one symbol equals 5, one gridline equals 5 - is just an equal group from Unit 1 wearing a costume. Your child reads scaled picture graphs, then scaled bar graphs, then builds their own bar graph from a data table, and uses rounding from Unit 3 to sanity-check totals they compute from a graph.
Find the total number of items shown by a set of symbols on a scaled picture graph, given a key stating one symbol equals a fixed number.
Determine the total represented by a bar on a scaled bar graph by multiplying the bar's grid length by the graph's scale value.
Solve a two-step 'how many more/fewer' comparison problem using data from a scaled bar or picture graph.
Explain why a graph's scale must be read before its bars or symbols can be compared to another graph of different data.
Construct a scaled bar graph from a given data table, choosing and labeling an appropriate scale.
Compare two bar graphs of the identical data set drawn at different scales and judge whether either graph misrepresents the data.
Round a graph-derived total to the nearest ten to judge whether a computed answer is reasonable.
Recall the meaning of graph vocabulary (scale, key, axis) introduced in this unit.
From the parent guide
This is the year multiplication and division actually click - not as a set of memorized facts, but as pictures your child can draw and reason from. Everything else in the year - measuring area, comparing fractions, reading a graph - turns out to be that same equal-groups picture wearing a different hat. By June, your child should be able to look at a rectangle, a clock, or a bar graph and say "oh, this is just groups of things again."
Unit 1 · what to expect
This unit gives a name and a picture to something your child already half-knows: counting things in equal groups. They'll learn two pictures - equal groups and arrays (rows and columns of dots or tiles) - and the words that go with them: factor, product, row, column. The array picture is the star here because rotating it shows, without any rule to memorize, that 4 groups of 6 and 6 groups of 4 come out the same.
The full guide covers all 8 units: where kids get stuck, what to say, and how to tell it's working. Included with the course.
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