Expressions

What this is. Algebra written out the way it is on paper, one step per row. The lowest row is the one you work on: wherever a move can be made there, the operator glows. Pointing at a glowing control (or, on a touchscreen, tapping it once) reads the move out in words in the blank row below, before you make it; clicking it (or tapping it again) makes a copy of the row that slides down into the new row, and then the move plays there, so the step before stays just above for comparison. Beside each row is what happened, in words: the first row reads the expression, and every later row says the move that made it. Tap a glowing control and the terms merge, swap, or distribute. Where nothing can be done, nothing glows, so x + 2 is dim (you can't add 2 to an unknown) while in 5(x + 2) only the times sign glows (you can distribute the 5). A faint ⇄ under an operator swaps its neighbors, which is how terms get next to each other so they can combine. A back arrow in the top left appears when there is a step to undo (the lowest row plays its move backwards, then folds up into the row above), and a forward arrow beside it when there are undone steps to replay. When the expression can be simplified no further it turns green, the row below it shows ∎ (the end-of-proof square), and a Next button appears in the bottom right.

The gray ☰ in the corner of the app opens the options: full screen (useful on a small screen), and five submenus. Language chooses which of seven languages the words are written in; tick several, and every row is written in each of them, each language in its own color, so students who read different languages can work from the same screen. Concepts has the skills to practice. Problems has the seed, a box to type your own problem (in LaTeX, starting from the current one), and new problem / start over. Affordances sets which moves are offered, whether the solution is marked, and whether hidden times signs are drawn (faintly), so that 5x visibly is 5 times x. Links copies a link to this exact problem and shows a QR code, so it can be moved to a phone. Choices are remembered in this browser and put in the address. The app's version number is at the foot of the menu: mention it when sending feedback.

On a keyboard, ↑ or ← (or Backspace, or z) undoes the last move, and ↓ or → replays the steps you just undid, in the same order. Press as fast as you like: the keys are queued. Pressing the other arrow while a move is still playing turns it around from where it is, so a student can go back and forth over one change to watch exactly what it does. Pointing at a glowing control tints what it will act on, and says the operation in words in the next row. A division is drawn as a fraction, and the bar is the control. When the same number is a factor above and below the bar, the pair glows: tap it to pull them out as 5/5, which the bar then turns into 1 ("same over same is 1"; nothing is crossed out). When two numbers share a factor without being equal (6 over 9), they glow too: tap one to break it into primes (2×3), and now the 3s match. A sum over (or under) the bar whose terms share a number with the other side, like (2x+2)/2, is not finished: the shared numbers glow, and tapping pulls the 2 out front, 2(x+1)/2, where it cancels (the reverse of distributing). Two fractions side by side can be joined into one (tops times tops, bottoms times bottoms; the times sign splits, one copy up and one down, as the two bars merge), and a fraction times something else, like (2/3)·x, pulls that factor up into the numerator. A long expression shrinks to fit the screen (all the rows together, like lines on one page), and the rows scroll when there are more than fit.

Every legal move is available, including unhelpful ones. That is on purpose: choosing among moves, and backing out of a poor one, is part of the skill.

Setting it up. The app is configured by URL. Add any of these to this page's address:

Parameter	Meaning
`skills=add_sub,order,like_terms,reorder,distribute`	Which skills to practice (`add_sub`, `mul_div`, `order`, `num_distribute`, `like_terms`, `reorder`, `unlike_terms`, `cancel`, `reduce`, `distribute`)
`prereqs=1`	Also practice everything each chosen skill builds on
`seed=period3`	A number or any word: it names one sequence of problems, so the same link gives everyone the same problems. Leave it out (or blank) for a fresh random problem on every load, and different next problems for each person
`n=3`	Which problem in the seed's sequence (with `seed`)
`problem=3735928559-4`	Written by the app itself: the problem on screen when there is no seed (its random seed and position), so the address always leads straight back to it. To report a problem with a problem, send the address (the version is at the foot of the ☰ menu). To give every student a fresh problem, share the page's plain address, without this
`lang=en,es`	The languages the words are written in, one or several: `en`, `es`, `bn` (Bengali), `fa` (Persian), `ar` (Arabic), `sw` (Swahili), `prs` (Dari). English by default
`expr=5(x+2)`	Use your own expression instead. Any single letter can be the unknown (`3t + 2`, `2x + 3y`); unlike letters never combine
`moves=solving`	Only offer productive moves: any move toward a solution, not only the shortest, but never one that could send you round in a circle (default: all legal moves)
`times=1`	Draw the hidden times signs faintly (`3×x`) so implicit multiplication visibly is multiplication
`mark=0`	Don't turn green at the solution: students decide when they're done
`first=Ada&last=Lovelace`	Save a student's name in this browser, so their activity is logged under it

Activity is logged locally in the browser (no server involved) down to each move and undo.