Compute percent change relative to the original amount, and use multipliers for successive changes.
RATIO.PERCENT_CHANGE“I can find a percent increase or decrease and use a single multiplier for it.”
A $80 coat is marked up 25% then discounted 25%. Find the final price and explain why it is not $80.
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.
Find a percent of an amount and add or subtract it.
Use a single multiplier (x1.25, x0.75) and find the original from the new amount.
Chain successive percent changes and compare with a single equivalent change.
| Course | Unit | Depth | Role | Checkpoint |
|---|---|---|---|---|
| Grade 7 | Percent applications | advanced | review | ✓ |
| Grade 7 | Percent applications | intermediate | review | |
| Grade 7 | Percent applications | intro | introduce |
Rectangular bars segmented to show parts inside a whole, with the unknown drawn as a labelled gap.
Word problems become a picture you can read the equation off.
engine: segmented bar actor with labelled brace
Two parallel scales locked together so equivalent ratios line up vertically.
Makes "per one" visible and kills cross-multiply-first habits.
engine: two coupled axes with linked tick mapping
A goal bar filling over time with deposits, withdrawals, and interest visibly compounding.
Compound growth watched rather than computed once.
engine: existing piggy-bank scene archetype
Scene archetype: bar_model
RATIO.PCT.BASE_CONFUSIONComputes percent change relative to the new amount.
Repair: Always mark the original as 100% on a percent bar first.
RATIO.PCT.REVERSEFinds the original by subtracting the same percent.
Repair: Divide by the multiplier rather than reversing the addition.
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.
Common Core places percent increase, decrease, markup, and discount at a specific grade and expects it to be built on a named prerequisite from the year before rather than re-taught from scratch. The standard is usually phrased as understanding plus application, so the expectation is both a correct procedure and an explanation of why it works, developed through a documented progression of representations.
Singapore introduces percent increase, decrease, markup, and discount concretely with objects the learner can move, then moves to a drawn model — usually a bar or a number bond — that keeps the structure visible, and only then to symbols. The drawn model is retained as the tool for word problems rather than discarded once the algorithm appears, which is why the same bar picture reappears years later for ratio and algebra.
The Russian tradition treats percent increase, decrease, markup, and discount as structure to be analysed rather than a procedure to be executed. A typical lesson opens with a problem whose condition is schematised — a labelled segment drawing or a quantity table — so the relationships are visible before arithmetic begins, and closes with variations that break any pattern-matching the learner may have adopted. Non-routine and multi-step versions appear early rather than as extension.
Kumon reaches percent increase, decrease, markup, and discount as a numbered worksheet level rather than a grade, and teaches it by graded repetition: the first pages of the level are barely harder than the last pages of the level before, so the method is inferred rather than explained. There is no manipulative and usually no context — the learner meets the bare computation, repeated until it is both accurate and fast, and repeats the level if the standard completion time is not met.
A Japanese lesson on percent increase, decrease, markup, and discount usually opens with one problem the class has not been shown how to do, worked independently for a stretch, after which several student approaches are put on the board side by side and ordered from the concrete to the general. The teaching happens in that comparison rather than before it, and the board keeps the whole argument visible so the class can see why the efficient method is efficient.
The classical American approach introduces percent increase, decrease, markup, and discount as a small increment inside a lesson that also reviews a dozen earlier ideas: the teacher shows a worked example, the learner imitates it, and the topic then reappears in mixed practice for months afterwards. Retention comes from distributed review rather than from a single deep unit, and fact fluency is drilled to automaticity on a timer before the topic is used elsewhere.
Montessori presents percent increase, decrease, markup, and discount first as a material the child manipulates, chosen so that the structure of the mathematics is physically present in the object rather than explained about it. The child works with that material until the answer becomes predictable, then moves to a deliberately more abstract material representing the same idea, and only reaches written notation when the material has become redundant.
AoPS approaches percent increase, decrease, markup, and discount by handing the learner a problem that the standard method would solve easily but that they have not yet been given the standard method for, and letting the method be reconstructed from the attempt. The treatment goes materially deeper than a grade-level curriculum, favours a structural argument over a computation, and follows up with non-routine variations chosen to defeat pattern-matching.
The English mastery approach teaches percent increase, decrease, markup, and discount to the whole class in small steps, using a carefully varied sequence of examples in which one feature changes at a time so the learner can see what the idea does and does not depend on. Concrete and pictorial representations — usually a bar model or a part-whole diagram — sit alongside the symbols rather than before them, and pupils who finish early get deeper problems on the same topic rather than the next one.
IM builds percent increase, decrease, markup, and discount out of a task students can start with what they already have, then names the mathematics in a whole-class synthesis once several approaches are on the table. Representations are introduced in a planned order within the unit — commonly a diagram, then a table, then the symbolic form — and a short cool-down at the end of the lesson checks whether the idea landed before the sequence moves on.