Decide which of two quantities is greater, by matching for small sets and by place value for larger numbers.
NUM.COMPARE_QUANTITIES“I can tell which number is bigger and explain how I know.”
Compare 4,081 and 4,108 and justify using place value rather than digit count.
Depth is a column, not a second topic. The same topic is taught at these depths in different years; the path table below places each rung.
Match objects one-to-one to decide which group has more.
Compare two- and three-digit numbers using place value and <, >, =.
Compare multi-digit numbers, decimals, and negative numbers on one number line.
| Course | Unit | Depth | Role | Checkpoint |
|---|---|---|---|---|
| Kindergarten | Counting to 10 | intro | introduce | |
| Kindergarten | Teens and beyond | intermediate | review | |
| Grade 1 | Tens and ones | advanced | master | ✓ |
Discrete movable objects arranged, grouped, and recounted to establish cardinality and conservation.
Separates "how many" from "the last word I said".
engine: prop_catalog instances + drag layer
A 10x10 grid of 1–100 where +10 is a row and +1 is a column.
Turns base-ten structure into a map you can walk.
engine: grid with path highlighting
A measured line for locating, comparing, ordering, and jumping between numbers.
One representation carries whole numbers, fractions, integers, and irrationals.
engine: axis scene + tick generator + hop actor
Scene archetype: compare_bars
NUM.CMP.LONGER_IS_MORESays the number with more digits printed wider is larger.
Repair: Align the numerals by place-value column before comparing.
NUM.CMP.SYMBOL_DIRECTIONReverses < and >.
Repair: Read the statement aloud left to right as a sentence and check it is true.
One row per instructional system. Rows marked cluster carry authored treatment for this family of topics; rows marked template are derived from the system’s general pattern and are not topic-specific research. Confidence is recorded on every row.
Counting and cardinality is its own Kindergarten domain, kept separate from operations, and it insists on the cardinality principle explicitly: the last number said is the count, and rearranging does not change it. Comparison of written numerals is expected by the end of the year.
Counting is folded immediately into number bonds, so a child who can count to ten is at once asked to split ten into parts. Numbers to 10 are taken to genuine mastery before 20 is met at all.
Counting is introduced alongside comparison and simple relational problems from the first weeks, so "how many more" appears almost as early as "how many". Ordinal position and sequence direction are treated as distinct ideas requiring their own work.
Levels 7A-6A are literally number sequence: reading numerals, tracing them, and reciting to progressively higher bounds, one worksheet at a time with no objects involved. Counting is treated as a fluency to be automated rather than a concept to be understood.
The first year builds the number sequence through structured objects arranged in fives and tens, so quantity is seen as composed rather than counted. Blocks arranged on the board become the shared reference for the whole class.
Counting is drilled orally and in writing, with daily calendar and skip-counting recitation, then reappears in mixed review for years. Number recognition and formation are taught explicitly as memory work.
Quantity comes before the numeral and both come long before the written symbol: number rods and spindle boxes give a physical length or handful for each quantity, and the child later matches numeral cards to them. The zero spindle box makes "none" a deliberate lesson.
Beast Academy treats counting as the beginning of combinatorics rather than a rote skill: early chapters ask how many ways rather than how many things, and systematic listing is introduced as a technique from the outset.
Reception and Year 1 build the count with rekenreks and ten frames, with heavy emphasis on subitising small quantities so that counting by ones is retired early. Variation is used deliberately: the same six objects are shown in many arrangements.
Kindergarten units use counting collections and "how many do you see" routines, eliciting how a child saw a quantity rather than only what the total was. The synthesis names the grouping strategies children used.