MATH 170 · Math of AI · What this toolkit makes

What kinds of manipulatives can this produce?

These are the shapes a manipulative can take. Each one is a small machine for a single idea — a thing to drag, a step to press, an answer to commit to before the page will tell you anything. Below each is one page from this course that uses it, so you can see the archetype doing real work rather than posing.

Setting the colour

These pages have exactly one colour decision in them, and it is a hue. Black text, white paper, and one spot colour — the way a two-colour textbook is printed. Pick a hue below and everything on this page, including the thumbnails, changes with it. Nothing else is adjustable, and that is the design: one knob is a decision, five knobs are a project.

The hue for the whole course is set in the Studio — the small editing app the instructor runs on their own laptop — on its dashboard, and it takes effect for every page at the next publish. This page previews the seven and hands you the CSS; it does not change anything beyond itself.

Choosing a colour needs JavaScript. Without it this page stays on the default, Goldenrod, and everything below still reads.

Measured contrast, this palette, both themes.
pairlightdark
Ratios appear here when scripting is available; every number is in design/palettes.json either way.

All seven palettes pass every required pair. The numbers were computed, not estimated.

Why one row says “see note”. The skin prints black text directly on the accent, so the accent has to be light enough to carry that text at 4.5:1 — and anything that light cannot also separate from white paper at 3:1. The two requirements pull opposite ways, and no hue satisfies both. Every accent mark therefore carries a 1px ink hairline: that outline is the boundary the standard asks for, it runs above 8:1 against the accent, and nothing on these pages is ever distinguished by colour alone. (In dark theme the accent separates from the page on its own at better than 9:1, so there the hairline is a habit rather than the load-bearing edge — which is why its dark row reads “advisory”.)

The CSS, if you are editing a file by hand

The Studio applies this for you; this block is for anyone editing a file by hand.

The archetypes

Roughly in the order a student meets them, which is also roughly what they cost to build.

Three bins; two cards resting in them and one card dropping into the third.

1 · Card sorter

Teaches: categories become real when a case moves between them; a one-clause change to the story moves it.

In this course: Learning-Type Sorter week-01-learning-type-sorter.html

The move: drag a card into a bin (or focus it and press 1/2/3).

A table of cells with a curtain half drawn across it; the columns behind it are filled.

2 · Progressive reveal

Teaches: the arithmetic didn't know what the columns meant — so meaning is something we add, and can add wrongly.

In this course: You Are the Machine: Three-Pass Reveal week-01-sandwich-reveal.html

The move: press Reveal, one pass at a time.

Axes with hollow circles and filled squares, and a line with a handle at each end.

3 · Draggable plane

Teaches: an equation and a picture are one object; moving either moves both.

In this course: Draw the Decision Line week-02-decision-line.html

The move: drag a point or a handle (arrow keys move it one step, Shift ten).

Four panels — a graph, a formula, a table and a sentence — threaded by one connector.

4 · Linked representations

Teaches: fluency is translation; the same fact in four notations.

In this course: One Line, Four Ways week-02-line-explorer.html

The move: drag the line, and watch three other panels move.

Three bars of different heights above a slider whose thumb sits about two-thirds along.

5 · Parameter instrument

Teaches: the shape of a dependence — what grows, what saturates, what flips.

In this course: Turning Scores into a Decision week-13-softmax.html

The move: pull a slider and read the number that matters.

A scatter of dots with two centroid crosses and their faint previous positions, above a row of three step buttons.

6 · Algorithm stepper

Teaches: that “the algorithm converged” is a sentence with a reason attached.

In this course: You Are the Algorithm week-06-cluster-by-hand.html

The move: run one half-step yourself and have it marked — the stepper's Step button, handed to the student. (Its ancestor, K-means, Step by Step, pressed Step for you; it was retired on 2026-08-05 and is in the archive.)

A four-node graph beside a four-by-four grid; one edge and its two symmetric cells are marked.

7 · Structure editor

Teaches: a graph and its matrix are the same object seen twice; edit either.

In this course: The Graph IS the Matrix week-11-adjacency.html

The move: click a matrix cell, or drag a node.

A four-row truth table; three rows are filled in and the fourth waits with a caret.

8 · Typed-answer checker

Teaches: commitment first, feedback second — the difference between recognising an answer and having one.

In this course: The Four Cousins of If-Then week-07-truth-tables.html

The move: type T or F and press Enter.

A five-by-five pixel grid with a rough digit drawn in it, an arrow, and a two-by-two result matrix.

9 · Drawing pad + classifier

Teaches: a classifier's decision is arithmetic you can read, not an oracle.

In this course: Trainable Machine week-03-trainable-machine.html

The move: draw on the pad, train, test.

A three-level game tree with one branch traced from the root down to a filled leaf.

10 · Game against the computer

Teaches: that a good move can be derived, and that the derivation is finite.

In this course: Minimax on Tic-Tac-Toe week-08-minimax.html

The move: take your turn; the machine answers and shows why.

Axes carrying six named cursors; one of them is filled with the spot colour.

11 · Shared plane (multi-student)

Teaches: that the class is a dataset, and that unsupervised means nobody told it the groups.

In this course: The Shared Plane classroom/ — the one piece that needs a server

The move: move your own cursor; watch twenty others.

Most pages combine two or three of these. Softmax is a parameter instrument with a ramp; word2vec is a stepper wearing five instruments; the graph sandbox is a structure editor with a scoring loop; the two Week 9 games are “game against the computer” with a graph where the grid was. The archetypes are a vocabulary for asking, not a menu of separate products.

What they have in common

What this deliberately is not

It is not a homework platform. There are no accounts, no scores kept anywhere, no analytics, and no cookie beyond the two keys that remember your theme and your colour. The multi-student piece holds a room in memory and forgets it when the class ends.

The gallery is where the pages themselves live — twenty-nine of them, one idea each.