These are math exercises from kindergarten to about second grade. They're 5-10 minutes each. Most are in 3D. They're free, no sign-ins, and everything starts unlocked. Starting one is 2 buttons from the menu. It's not a huge download. But there's nothing extra besides the exercises - no progress report or statistics, no support for multiple children, no videos. But otherwise they work - each is finished, they score themselves, have some polish, and mostly flow into each other, skills-wise.

The inspiration was an educational project I was a coder for. It was intended as support for rural areas which might not have a math teacher, where an App was the next best thing. At the time, I figured existing educational Apps already had that covered. As I looked, I saw that wasn't so. The new stuff was impressive - running on every device, web-page dashboards, automatic setting of deadlines, and much better graphics. But the actual exercises were untouched. Back in the day, we took paper classroom worksheets, and computerized them. And that's what these were still doing.

The project I was on really impressed me. But it fell apart before anything was ever published. This isn't that original project, but it captures the spirit of making useful math exercises, taking advantage of a computer.

**The menu and sequence:** The main menu page has 16 categories,
each leading to a sub-menu with up to 6 exercises. 61 total. You can freely navigate and select any exercise.
Inside, double-tapping the upper-right circle jumps up to the sub-menu.
But the exercises also play straight through - finish one and it goes into the next. It actually gives you a choice
to go on or to play this one some more, via pop-up buttons Again and Next.

**Progress:** Each exercises has a rough set of categories it goes through. Sometimes
that's an increase in difficulty, or a "flip", or merely a change in the format. For example:
match words to numbers, then use larger numbers, then flip to matching numbers to words. Problems
are chosen randomly from within the category. For fun, the last few problems are from random categories -
sort of a review. The "Again" button is a bit misleading. It keeps going at the current difficulty (I felt that
"continue" made it seem there was more to see, while "Again" more clearly communicated that you'd
see everything it could do).

You never get a problem wrong - it has you keep trying until you solve it, sometimes with extra help. But the computer counts mistakes to decide how quickly to move to the next category. It tries to be subtle. This means there's no set number of problems in each exercise.

Each exercise has a progress bar, but it's only shown in the menu. This is what determines the sorts of problems you'll get - when the progress bar is about 1/2-full, you're only getting problems from the hardest setting. Likewise, if you test an exercise for a while and the progress bar is nearly empty, you've only been seeing the easiest problems (you probably made lots of mistakes, convincing it to hold you back).

Filling a progress bar generally requires playing long enough to use the "Again" button twice (which will be a dozen problems from the hardest setting). Filling it does nothing, and won't prevent you from playing more. The progress bars on the main "category" menu are an average of all exercises in its submenu. They don't affect anything, but are nice to look at.

To replay an exercise from the start, in the Options menu, tap the "reset all progress" button. There's no way to reset only one exercise - only all of them at once.

**Cheating:** This is an option to browse through the problems in one exercise, seeing how the
format changes. It's crude - all it does is pretend you solved this problem, and skips directly to the next.
It won't show the solution. It this problem has 3 steps, it skips past, giving no indication. But combined
with playing though a few problems (or partly through, then using this trick to skip ahead), it gives a decent tour.
When you see Again/Next, you've seen every style of problem (but you can select "Again" and keep browsing).

To use it, put a finger on each corner of the screen - 4 fingers total - and tap with any one of them. 3 down, any 1 tapping. I use the thumb and forefinger of both hands. The area counting as a corner is pretty large, but it's very sensitive to finger-sliding, which forces you to try again. Once you have it, the 4th finger can tap, tap, tap to quickly skip through. I may have forgotten to add this to some exercises.

Cheating doesn't affect the progress bar, which means you can jump out and back and will be playing the exercise through from the beginning.

**Number Tiles:** At first these need to be dragged into the answer slots. But there's a
tap-to-select option: tapping any empty answer slot turns it on - you should see a question mark, ?, in
that empty slot. With this option off, tapping a number tile merely speaks that number. That seems like a good deal
for younger kids, especially since dragging a number into a slot feels more natural.

Tapping the "?" switches back into "tap to speak, drag to place" mode.

**Tumble Tiles:** These are labelled tiles which roll and spin like 3D blocks. They may be
upside-down (I try to make 6's and 9's distinguishable), or face down (tapping will usually
flip them over). They may need to be carefully pulled over and around obstacles.

The thing I like about them is they feel real. You can arrange them, sort them, move wrong ones into a corner, and place them near candidate slots. Moving them is more of a pain, but it feels like processing the problem in a way that tap-to-place doesn't. In fact, part of the research from the original project was deliberately engaging the spacial senses in this way.

Some exercises have a shortcut - a tile will snap into a space as long as your finger is over it. You don't need to wait for the actual tile to be pulled over the space. This is set per-exercise. There's no way to turn it on if an exercise doesn't allow it. Again, the theory is to simulate a complete movement of an actual tile, which is good for younger children.

**The error cow:** After a few mistakes, a cow may fall from the sky and jump around.
It will go away on its own. Tapping the cow shoves it in the opposite direction, and also hastens its departure.
The Options menu has a button to disable it.

My idea was that if you make a few mistakes in a row, you might get a little frustrated. I always felt "You can do it. Keep Trying" was something that impressed parents, but wasn't all that helpful to children. An angry cow mooing at you, chasing the pieces around just seems to work better.

These exercises are pre-numeracy, or "familiarity". Various ways of counting, seeing the digits, and words "one", "two" ... . These are about nursery school level, I think.

Dragging blocks to make lines length 1 through 10. There's no counting. This is just a typical exercise to get used to how numbers represent a quantity. It shuffles the order slightly - you might get 4, jump to 6, then back to 5. It's surprisingly suspenseful.

At the end the blocks stack themselves and let you knock them down. For fun.A version of Pink Tower from Montessori. The standard has 10 blocks: 1x1x1 up to 10x10x10. For this, I could only fit up to 6x6x6, but you get as many of each as you like (tap the dispensers at the bottom). There's no goal. It starts with random sample towers, but they don't mean anything.

The purpose of Pink Tower, and this, is to get a feel for numbers based on the sizes. A 3x3x3 fits onto a 4x4x4 with a size-1 ledge around two sides, and so on. The blocks will snap into place if they are mostly-aligned. The smaller ones leave footprints. For fun, a red thing walks around, attempting to climb them.

Digit recognition. This is similar to tracing digits. You get a small window into part of a digit, which you can slide around to see the rest. It turns out that the curves in 2's, 3's, 5's, 6's, 8's and 9's are all the same, and can fool you.

The colors are random. After a while it switches fonts (Tahoma and Arial). The window gets smaller. Eventually numbers tilt just a little (we can't tilt them much, since 6's and upside-down 9's are identical). The range starts at 1-5, then to 1-9, then finally adds 0. They may not know 0 is a number yet, but it works here as a shape.

Left on it's own, the window slides around slowly. It's somewhat fun to play using only the auto-slide.

Recognizing the words ("one", "two" ... ). It works as a matching game. Drag the words next to the matching digits. Later it flips to where you're dragging digits to the words. Then it uses incorrect spellings - "one", "won", and "wun". It eventually adds tiles you don't need.

To make it interesting, they come in groups of 3-5. Sometimes all at the start, sometimes one-at-a-time.

These exercises are for learning what numbers mean.
It's hard for us to remember not knowing them. At first we learn to write 1,2,3 ... on things; but that's not any different then writing "a", "b", "c".
We have to learn that 1,2,3 is *counting*. Next, we should know you can count in any order, since only
the total matters. Finally, we
should know you don't have to count them if you have some other way - for
example, recognizing the pattern.

One end goal is knowing you can ask "how many?" about any group and the answer is a definite number. Another is being able to think of "6" as a general amount, without needing it to be 6 of anything.

Counting items in various lines. At first they're in a row, then with gaps, then with varied shapes and colors. They come in groups of three to encourage comparing numbers. For fun, it alternates all 3 at once, or 1-at-a-time. If you get tired of dragging the tiles, tap a blank space for tap-to-place mode.

Counting jumbled items with different colors and sizes. For fun they come in pairs, and the areas change between various-sized rings and rectangles.

This is a standard counting exercise. Unlike a line, we have to actively put these jumbled objects in order as we count.

Another matching exercise, this time with patterns of dots, from 1 to 10 - known as subatizing. The dots come in 5x2, or 4x3, or domino-style. The system gives a few of each before moving to the next. As before, it also flips between dragging the dots to the numbers, or the numbers to the dots.

This one is the opposite of counting. It shows you a number and asks you to build it with that many blocks. Tapping the "crystal" pops out extra blocks. Tap the number when you've placed the correct number of blocks.

Each line starts with some blocks already in it. Sometimes more than you need. This makes it a lot more interesting. It's also surprisingly fun to use extra blocks from one line to fill out the next.

Educationally, this helps us see numbers as having their own meaning. When we count blocks, the number is a description of them. But here, the number is primary and we're using the blocks to describe it (that's the theory, anyway).

Counting items arranged randomly in a grid, up to 20. Tapping them toggles a little dot, but isn't required. It starts at most 10, working it's way up to 20. The color patterns, cube sizes and patterns of squares, are random. Sometimes you get some pretty ones.

This is experimental. As an adult, it's easier to spot clusters of certain sizes, and add. Kids might not be able to do that, so this would be just more boring counting.

Matching digits 11 to 20 with the words. This is the same as the 1-10 version, except no alternate spellings. As usual, you drag words to numbers, followed by numbers to words, then random; and it adds unused tiles, later.

Matching numbers to dot-patterns again, but now up to 20. Tiles are 5x4, then 6x4 (but at most 20 dots). There are no domino patterns for numbers past 12. Later it adds words.

It runs through every combination of matching: dots to digits, words to dots, dots to different dots ... . Ending with random. This is the ultimate version of every previous matching.

Counting on a standard 10-wide number grid, up to 20. The sequences are simple: 1 to 10, 10 down to 1, evens (skip counting), odds, 1 to 20, 20 to 1, and a few more.

The answer is multiple choice. I tried to guess common mistakes, such as 5+1 is 51.

This is for the kindergarten Common Core "understand 11-19 are ten plus a single digit". As usual you have to fill each area with how many it says. But the areas are now always ten tall, and length 10 blocks are added. Those are required - the only way to make 12 is using a single ten-block and 2 singles.

They're shown in groups of three, in sequences. 2, 12, 22 emphasizes the change in tens' place. 9, 10, 11 emphasizes 9 one's, but only a single ten.

There's no way to dispose of cubes - if you pop out too many 10's you have to shove them out of the way.

These are repeats of previous exercises, but using zero.

Another matching; the word "zero" with the digit 0. It later adds dot-tiles, where the 0-dot tile has nothing on it.

Counting cubes in lines, using 0-6. This works the same as the previous version using 1-10. except one of them is always 0.

The "see a digit, make a line with that many blocks" exercise, but this time using 0, and a range only up to 6. As before, lines start with a few blocks. This allows us to "make" zero blocks by dragging everything out.

Later on, this exercise adds one new thing -- the numbers are spelled out. You're asked to create "one", "zero" and "four".

This is an experimental combination of sorting and counting. You have to count the objects according to a rule. Rules can be: in the square, out of the square, only cubes or chips, only small or large items, only stretched-out ones, or only a certain color. Tapping the rule reads it out loud (the volume levels are bad, but it should be understandable).

As usual, it ramps up the difficulty. Sometimes it uses the square as a distraction, with some items in and out, when we don't need it.

This is two exercises in one to try to show even/odd. One goal is merely seeing those words. The first part is from the picture. We get some cubes and have to arrange them into pairs, possibly with an extra, then select even or odd.

The second exercise is based on "a number is even if we can split it in half". You have to tap a rod to get an even or as close-to-even split, place the new half under the first (so the lengths are easily compared) and select even or odd. The rod splits where it's tapped. For un-even splts, dragging them end-to-end re-assembles the rod for another try.

The payoff is seeing your two halves are the same length, or there's an off-by-one and it's impossible. Then you click even or odd.

This is a measuring game. You get 2 same-sized blocks to help measure a rod. Sometimes both are free to move. Other times, one is anchored along the rod. You can guess the answer whenever you want.

The idea is, at first, you can leap-frog the 2 cubes along the rod, counting 1,2,3 ... . Eventually, you might figure out to drag a cube to the middle, eye-balling the length of each side and adding. The fixed-in-place cubes are to force you to sometimes do that.

The length of the cubes change. The maximum length of the rod gets larger over time. The rod sometimes has a small tilt. The cubes become sticky when left just below the rod, but won't try to snap into a correct spot.

Measuring the same thing using different units. You get one rod and up to 3 differently-scaled rulers. You have to measure it with them all. It starts with only 1 ruler, then increases to 2 then 3 rulers.

Measuring is harder than counting, since it depends on the units. I thought the multiple-ruler was a nice way to show that.

Sorting, based on a variety of rules. They're the same as before: color, size, chip vs. cube, regular vs. stretched. Some have you sort into 3 categories (color and size). Others into only 2 groups (shape, stretched vs. normal).

Sorting is done by dragging into areas. You have to pull them into the edge and wait for them to pop in or out. To make it easier, you're not allowed to drag something into the wrong area (doing so counts as a mistake, for progress).

Three mini-games involving equal/not-equal, and less/equal/more. Minigame #1 is deciding if two counts are the same or different. For fun it randomly mixes words "same," "different," "equal" and "not equal." In minigame #2 the items are always unequal and you have to label both with "greater", "fewer", "less" or "more" (the possible words are still chosen randomly). It later switches from blocks to dots on tiles.

Mini-game #3 combines the first two. The two counts can be the same or be different. First you choose same/different. Then you have to choose from between <, = and >.

.Counting up to 40. By 1's, then skip-counting, but not too hard. Everything is in proper multiples - 5, 10, 15 but never 12, 17, 22. As before, the answers are multiple-choice.

Skip counting between 1 and 100, including off-center multiples (ex: 12, 17, 22).

The in-between numbers are sometimes filled in (like the picture) and sometimes not. I feel like seeing it both ways is more helpful.

Skip-counting where you just tap the correct spot. This is easier, but it's a faster drill for seeing the patterns. The rows sometimes start at 0 (so, 0-9 instead of 1-10).

Bunnies hop across. They have no educational value (if you have disabled the wrong-answer-cow, the bunnies will also stay home).

This is a game where you have to build a larger number from smaller ones, and some of the options are dead ends. In the picture we're making 8 using 5,3,3,3,2,2,2 and 1 (the light blue block lying on it's side). It's easy, and there are several ways to fill the remaining 3 spaces. But a later problem might give us only 2, 4, 4 and 5. Placing the 2 or 5 leaves us stuck. The "hard" problems are actually from a pre-made list of every set of sub-blocks with only a single solution

The 3+2 in the upper-left does nothing. It's displays your work as an equation, as practice for when we see them later. That's one of the reasons for this game

This exercise has you match an equation to a picture, and vice-versa. In the picture, we've identified ppRRRR as representing 2+4, and we're working on identifying BbYYYY, which represents 1+1+4. Notice how the order matters for this exercise. The 1+4+1 tile in the upper right is incorrect, since it represents a picture like XOOOOY.

It starts with only 2 addends, going to 3 at higher difficulties. In the exercises where you have to make the picture based on the written sum, the picture only accepts correct length tiles, in the correct order. It bounces out any others.

These are exercises using 2 bars on a number-line to represent an equation with 2 numbers. Plus and minus (which looks fine here - the second bar goes backwards).

The range is 0 to 12, for no special reason (it fit on the screen and seemed large enough). The bars slide using the ball on the end. Often one bar is locked. If you can move both, the ball will be on the end of the "active" bar. Tapping it switches it to the other bar.

Introduction to number lines, in two parts. Part #1 is simply figuring out how it works by moving the bar to each number. It asks for them in a scrambled order - slide it there using the knob, then tap the tile. Numbers are displayed as you get them. Filling it the final few is almost fun.

Part #2 is shown in the picture. It choses two numbers and asks how far they are apart. For optional help, you can measure the distance by sliding the second bar (the system does that automatically if you make a mistake).

At first, you only measure "up" - the second number is always larger than the first. Later if switches that order. But the answers are always positive numbers.

This exercise asks you to explicitly label a number-line with the equation it represents. In the picture, the line represents 4+3. You may recognize this as a repeat of the same exercise using blocks. As before, the order must be the same. 4+3 and 3+4 count as different equations.

In part #2 we need to adjust both bars in the number-line to match the equation it shows us (we need to move both bars - this is where we need to tap the ball to switch).

This is the same as the previous, except using subtraction (the second yellow bar runs backwards.) Later, it alternates addition and subtraction.

Solving equations where a number in the middle is unknown, for example "3+__=7". You choose the number by moving the slider; tap the tile when done. I think this only really makes sense when you watch it work:

In the picture, we were given "9-??=1". The blue bar is locked at 9. The yellow ball is also on the 9 (it has length 0). So far we've slid it 4 down. The tile above automatically changed 1,2,3,4 as we did that. If we tapped the "4" tile now, the system would yell at us and attempt to show us how the 5 doesn't match the 1 we're looking for.

Either number can be missing; combined with plus or minus, that gives four types of problems. It uses words "add" and "subtract" at first, then switches to "+" and "-". Values are limited so the total can never be negative.

Sorting regular N-gons (triangles, squares ... ) by number of sides. As before, you have to pull them a little to get them to pop in or out of an area, and they can't be pulled into the wrong one.

It works it's way up to at most 10 sides, and "closer together" side counts.

Another matching exercise to show the Latin number prefixes for shapes - "tri", "pente", "octo" ... . As usual, it has you match the words to the numbers, then the reverse, then misspellings, then random.

This is more matching using the full name for the regular N-gons: triangle, square, pentagon up to decagon.

The types of triangles and 4-sided shapes: isosceles, scalene parallelogram and so on. Each one that can has two sample shapes (only 1 square, but 2 types of rectangles). It's obviously another matching game.

A fun exercise building larger shapes out of triangles.

There are 3 triangle types: equilateral, right with same-length sides, and right with different-length sides. The 9 pictures to make are: square from 2 tris, rectangle from 2 tris, bigger triangle from 2 tris, square from 4 tris, big right triangle from 4 tris (same sides and different sides), large equilateral triangle from 4 small ones, hexagon from 4 equilateral tris, and a funny pentagon.

For fun, it has you sometimes make several of those shapes at once. It also rotates them, to pre-set angles.

Roman numerals, from I, II up to XX. It also has wrong "spellings", for example VX, VVV and XY for 15.

A clock with roman numerals. The only action is placing the correct numbers in all 12 places, in the order it tells you.

The clock moves ahead a few hours each time. As it does, a sun (with glaring sunlight,) and moon (which brings darkness) spin based on the time of day (a 24-hour cycle, even though you place only 1 through 12). Watching it spin through a few days is kind of fun.

Flash cards in groups of 4 to 6 with a theme. The themes make more sense once you see them. They are: 10+(random 1-9), the reverse of that; all same sum (3+8, 6+5, 10+1); constant plus interval (7+2, 7+4, 7+6, 7+8, 7+10), the reverse of that, both numbers get +1 (3+6, 4+7, 5+8, 6+9); and commuting (random pairs: 3+6, 6+3, 1+14, 14+1, 12+6, 6+12.)

Once you notice that each set follows a pattern, these are a lot more fun to answer. It also has hopping bunnies, for no educational reason.

This exercise is about adding several numbers by grouping them. If we had 6+3+4+7 we'd re-arrange to make 6+4 and 7+3. Tiles are automatically combined by dragging one over another. You're done when you have one tile left, with the final sum.

The plus-signs are only a hint. They vanish as soon as you touch anything. The tiles can also be freely moved anywhere.

There are 3 versions: combining digits to make 10's; then making any multiples of 10, for example 17 and 13; then making 5's and 15's.

This exercise is about splitting a number so you can land on 10. For example 8+6 can be done as "8+2 to get 10, then 4 more" - we split the 6. It uses a new "splitter tool": tap a tile and blocks pop up below. Slide/tap those to select the split, then tap the original tile (now grey) to do it. In the picture we've tapped the 5 and selected 2+3 as the split.

At first you merely have to combine three digits to make 10. It might give you 4,2,3 and 4; you have to figure out which three add to 10, then add the remaining one to that. After that you get simple splitting problems like 8+6. Then combinations where you make some 10's and split to make another (ex: 6+4 + 7+5).

The final part makes you combine pairs to get the "mystery number". If the mystery number is 26, you get three pairs which all add to 26 (they're mixed-up, so it's more fun than it seems).

This exercise is subtraction flash cards in groups with a theme. Numbers go up to 20. There are 5 patterns: constant second number (7-3, 8-3, 9-3), constant first number (8-2, 7-2, 6-2), same answer (7-1, 8-2, 9-3), doubles (6-3, 12-6, 16-8) and tens with commute: (in pairs: 10-2, 10-8 or 10-4, 10-6).

Base-10 is another thing that's so obvious to us as adults that we forget we needed to learn it. These exercises are about the basics of it: the names of the places, knowing higher places count for a lot more, that 942 is like three digits counting different things, and how 0 is important as a place-holder.

This exercise is building a large number out of cubes. We get a 10's rod, a new 100's square and a giant 1000's cube. It automatically stacks them nicely. You're done once you place cubes making the number. It only uses up to 100's at first. It also tries to use a lot of 0's, for example 31 vs. 301 so you can see how have to think what the 3 represents.

The reverse of the previous exercise -- it shows you a stack of cubes and you have to choose the number it makes. Since you drag in one digit at a time, this is about knowing what to count. If you're counting the 4 in 245, you have to know it's the long skinny ones.

Tapping also shows and says the "ones", "tens", "hundreds" and "thousands" place names.

The next three are sorting games using place value. In 2000, 406, and 98, the smaller numbers have larger digits, so maybe they trick you. Three exercises on this is too many, but they all seemed fun.

Sort a list by swapping adjacent pairs. To swap, either tap the < between them, or attempt to drag one towards the other. When done, tap the ?? to check. It goes up to 5 items, and uses > later on (you have to order them high to low in that case).

You can make it into a game, trying to get the least swaps. Because it's possible to accidentally swap values where one gets closer to where it should be, but the other gets further away.

This exercise has you divide numbers into more or less than a target number. It starts with 2-digit numbers, later moving to 3. As a hint for the 2-digit numbers, tapping any tile animates a small 10x10 grid showing that many tiny cubes.

It's straight-forward, but it also feels like a little game. One of the patterns is mixed-up digits, such as 376, 367, 673. Those are surprisingly fun.

This exercise is sorting a list of numbers by choosing them in order. At first you choose them low to high, shown with a < symbol between. Then it switches to descending, using >. Later it flips to a right-to-left fill. It's a little fun since there are 4 combination where you have to figure out whether you pick the smallest or largest.

The numbers to sort come in several patterns: rearranged 2 digits (704, 407, 740 ... ), rearranged same digit (404, 440, 44, 400 ... ), three digits scrambled (456, 645, 564 ... ), and the old "fool you with a larger digit in a less significant place" (555, 399, 721).

Learning the words for numbers up to 99,999. For example: "fifty thousand five hundred seventeen". It flips back and forth between digits first and words first, and word tiles have some intentional misspellings for the 10's ("fivety").

For the version where you guess the words, incorrect word tiles are "bounced out". But correct tiles may are allowed to be dragged into the wrong place, and may need to be sorted. To avoid possible confusion, words never repeat (330 is fine since one is a "three", and the other a "thirty", but 303 is no good).

Flash cards adding 2 numbers with no carries. Hopefully we can get used to adding tens add to tens, and ones to ones.

There are 6 patterns of sums: A) Only a tens place: 40+30, 20+20. B) Merely adding zero's: 30+6, 50+2. C) 2-digit plus 1-digit: 21+6, 94+2, D) Adding only the tens place: 34+20, 70+19, E) pairs adding a value to 1's, then to 10's: 42+3 / 42+30, and F) pairs with a digit flip: 23+16 / 23 + 61.

The tiles slide from 1 to 99. The area just below works as a fast slide. As a hint, it will sometimes slide itself to roughly center the answer.

The is a made-up game where the real purpose is to cause you to sometimes pause and "make change". The same idea as being the banker in Monopoly when you need to ask someone to trade you back their 5's for 20's. It's practice for borrow and carry.

The game does only one thing over and over. It asks you to pay yourself a specific small amount from the bank. Since the bank only has limited chips in limited values, you'll occasionally have to trade your small-value chips back to the bank for higher-value ones. The goal is to reach a certain total. Later it gets fancy and the goal may be to have three chips worth 5. On higher difficulty the chip values sometimes change, from 1,3,5 and 10 to 1,2,4 and 8. Later on it decreases how many chips there are, forcing you to make more frequent swaps as they run out.

Trades are made by tapping the Exchange button. A box comes up, tap to load chips onto each side. When the side have an equal value, tapping Exchange again makes the trade (the box spins 180 degrees, then dumps each half).

For such a seemingly pointless game, it's fun. I sometimes enjoy changing up to the largest chips, as soon as possible. Educationally, it's to understand carry and borrow. Carry is really just "oh, I can't have thirteen 1's - I have to trade ten of then for a 10".

These three exercises are standard multi-digit addition, with carry. They walk you through the process. They start with the sum written across, like 364+283. You have to drag them into a column, lined-up properly. Either number can be on top. Then you have to tap the ones' place, to show you understand it goes right-to-left. Then you add in the usual way. The carry is a toggle - tap it to change.

Each of the three exercises uses harder numbers. We start with a pair of 2-digit numbers. Later we get carry-past-the-end; up to 4 digits; different lengths (324+98); a third addend; and carries of 2. It also has some fun ones, like sums with repeating digits, or round like 6000.

Tapping the current column gives an adding hint - the 10x4 mini-grid. It fills as you tap each digits. In the first picture it's showing the first column as 9+2=11. You don't to bring it up, it does nothing, and it will vanish on its own. It's just a reminder.

These are warm-ups for multiplication, playing around with the idea of multiples. In Skip Counting were got a taste of how 4 can "make" 8, 12 and 16, which makes them special numbers for 4. These exercises try to reinforce that.

This is a game for playing with multiples. It gives you a large number and some small ones. You have to pick the number that divide it. But it's more fun than that. A sample round of the game might: ask you to make 35 with options 4, 6 and 7. Tapping the 6 button makes a grid 6 wide. Sliding the arrow up runs through 6, 12, 18 ... 30, 36. You see there's no way to get 35 using 6's. Clicking the 7 button turns the grid 7 wide, and resets it. Sliding the arrow now makes 7, 14, 21, 28, 35. If you wait for the small delay while the cubes fill the rows, you "win" when they fill it to 35.

The target numbers go up to 60. The divisor numbers range from 3 to 13, chosen so only one will work (which is fun: we know 5 divides 35, but that wasn't an option this time).

This uses a skip-counting format on a basic number grid to play with identifying multiples. Like skip-counting, a small number is highlit. But then it selects only certain multiples, in a random order, to match from the tiles below. In the picture the divisor is 4 and we have to place 8, 12, 28 and 40, with a extra 10 tile as a distraction.

The maximum number is 60. Sots are filled in either backwards, randomly, or all-at-once. It works out to be lots more interesting than skip-counting.

Introduction to common multiples. At first you pick multiples of one number (the one written on the area of the hollow square). But you eventually get 2 numbers, then a "both" category. If a number belongs in "both" you're allowed to drag it into either, but it won't turn green, as a hint it needs to be moved.

It's split over two exercises. The first starts with a single number, then quickly moves to two, without a "both". In other words, if the divisors are 4 and 6, the tiles will never include 24 or 48. The second uses larger numbers, then the Both category, and later adds divisors like 4 and 8, or 3 and 12, where one divides the other.

There's no visual help - it's assumed you understand the "numbers you can skip-count from N" idea. Mistakes - a tile in the wrong spot - give a hint by bracketing the number with the surrounding multiples.

This is to show how multiplication is addition. It only shows the symbolic part: how 3x5 is a shortcut for 3+3+3+3+3, and vice-versa. We don't give the answers, in this one..

It starts showing a multiplication problem such as "3, 5 times" and asks you to drag the blue triangle to create the 3+3+3+3+3 representation. Tapping the greyish background verifies your answer. It has two more phrasings: "3 times 5" and "3x5", mixing to show how all three mean the same thing.

After "3+3+3+3" it moves into tiles with dots. To answer "3x5" you need to drag in 5 tiles (which each have 3 dots). Distraction tiles have the wrong number of dots. Then if flips around: you're shown "3+3+3+3+3" and need to fill in __ x __. Either order (3x5 or 5x3) is correct. Likewise you're shown five tiles with 3 dots each, with the same __ x __ as your answer.

Midway, you're shown squares: 4x4, 7x7, 3x3. Before this we avoided them. The last part is a random selection of any three of the four problem types. Numbers for all problems range from 2-9 (1st number) and 2-8 (the second number. The limit of 8 is because that's how many tiles fit).

The second part of showing how multiplication is really addition. The 2 new things this shows are: how the addends must all be the same: 3+3+2+2+3 doesn't represent anything times anything. And the 2-ways-to-see-it problem: 3+3+3+3 is equally valid as 3x4 ("3, 4 times") or as 4x3 ("4 copies of 3").

The questions are either sums (3+3+3+3) or a row of dot-tiles. Often 1 or 2 of them are ill-formed - they aren't all the same number, so don't have an answer. The answers are the usual group of draggable tiles with 3x5 and so on, with "not" tiles for the invalid ones. The order of numbers on a tile is random: 3x5 or 5x3. You may see 3+3+3+3+3, not the a tile with the 3&5 order you prefer, and have to use the tile with the flipped order.

The numbers range from 2-8. For fun there are 5 arrangements of the 3-5 questions in each round. But it never gets any more complicated past the 4th problem.

This exercise is based on how you don't need to memorize 7 times 9, since it's obviously 7 less than 70: we can check a sum by skip-counting from one we know.

It chooses one main equation for the center and puts the four +/-1 equations in a cross around it. In the picture we picked 10x6. The sides are 9x and 11x, the top/bottom are x5 and x7. If has you fill them in across the arms, making a 3-number skip-count.

Five variants: #1) 10 times X in center, #2) 5 times X in center, #3) any numbers (from 3 to 12). Variant #4 changes the arms into "times 2" and "divided by 2". If the center is 4x6, the left could be 2x6 and the right 8x6. #5) Uses a square, such as 5x5. Across are the next higher and lower squares, keeping up/down the same as normal (4x5, 6x5).

I don't think it works as intended. When I tried it, it seemed just like five similar flashcards. I never found it simpler to look at the center answer and skip-count up and down to get the arms. But it's not bad the way it is - it's a slightly more interesting version of flashcards.

These are multiplication problems with the answers as close as possible together. If we start with 3x7=21, next is 2x11=22, 6x4=24 and possibly even a repeat of 24 using 8x3. It goes up to 80 with problems as big as 3x26. It sometimes runs right-to-left, high-to-low.

It works out to flashcards with a very big assist - the answer is always within the 3 or 4 first numbers. That's why it seems fair to have larger values. Say we're asked 17x4 and the previous answer was 66. We only need to figure out that it ends in an 8.

Fix slight hiss/hollow-ringing on audio. Redo funny levels. Add more "that was correct" sounds.

The number-line has "how far from 16 to 9" problems, which is informal intro to subtraction. How, and if, to turn that into the standard subtraction trick where 56-27 involves a number-line and 27(+3+10+10+6)=56

Fractions? Negative numbers. Subtraction with borrow?

No information is collected by this App. No names, ages or locations; there's nothing to buy so there's no financial data.

A better App would at least collect scrubbed usage statistics (for example, which exercises no one ever spends more than 15 seconds on.) This doesn't even do that.

Every exercise keeps going until you get it right. If you're taking a test, "you got it wrong, now moving on" is fine. But for learning I think trying until you get it is better. The questions are still scored, secretly. Make enough mistakes and the system stops giving you harder problems, and may even make them easier.

A drawback with the "try until correct" method is it encourages trial&error. Worst case, students learn to quickly tap every answer. The trick is to give a longish delay. Even 2 seconds could be enough to encourage them to use this time to think about the next guess.

**Randomness:** most exercises choose the problems at random,
but that means a lot of things. Most limit the selection in a few ways to
make sure it's good. Some use them all once, in a random order. Others
can repeat a problem, just not the same one twice in a row. Some use
weighting to prefer certain types, but are still random.

The random colors cheat. It first randomly chooses a category, such as "2 colors" or "several similar colors". Then it picks the colors at random.

**Q:** Why the cutesy name in an otherwise non-cutesy App?

**A:** Because it took me 2 weeks to name my cat. I finally settled on "Abby", which is not a good cat name. For this I was thinking "Early Numeracy,"
or "Math Game_056." It's hard thinking of a name you don't hate.

**Q:** Clearly the first two "correct" sounds are a small bell being dropped on a hammer. But what is the
last one?

**A:** The third "correct" sound is conceptual: a glass of apple juice being poured and drunk. It seemed more fun than the usual cheering or trumpets, and apple juice is delicious.

**Q:** Why do the number sounds sometimes mess up?

**A:** There are two sets of 1-9 sounds: normal, for taps, and fast for when the computer does a
count-off. To mix it up, there's a third set of slow numbers They rarely, randomly play instead of the
normal ones.

Q: What's going on with the sound levels and voicing?

A: What I want to know is why sounds are so loud on every other App. There are 16 levels, but 3 is the loudest you ever need. But beyond that, avoiding hiss and hollowness is surprisingly tough. Pronouncing a word so is sounds "normal" is also hard: every syllable can pontetially be too fast, too slow, rising and falling in a funny way. Try to say "several" in the absolutely correct way. It's so hard!

**Q:** Where can I find a real rug with that pattern?

**A:** If you're serious, I'm sorry - it's not based on any specific rug. But if you're mocking it, let me tell you that
is a great rug. It goes in the dining room and is a beautiful non-ironic red plush. The gold is patterned like trim or curtain
rods with knobs, or filigree. It classes up your meals. Then it can be streets, or the fat parts can be cities for when you
play under the table with your hot-wheel cars. It's a great rug and you know it.

**Q:** Are there details on how the cow works?

**A:** Yes. The cow randomly appears after any mistake (the chance increases slightly after each mistake and is lower if you've recently been cow'ed.) Where it falls is random. The little jumps it makes are also random (both the timing and the force, which is why it sometimes seems to stampede.) When you move a block or number tile, the cow will face it, enabling you to lead it around.

It will leave after a minute or so, or randomly after being tapped. Tapping the cow also shoos it away from your finger.
There can only be one cow at a time. There are 4 moo sounds (none of which are from a real cow,) played semi-randomly.

The cow is made with simple game-physics joints. The head-bob, tail wag and leg flex are the result of tweaking settings, and involve no coding. Even the way it turns itself feet-down is code-free - it just naturally does that when air-borne since the feet are heavier than the head. If you've used Unity3D, it's all just character joints and settings, especially the Spring numbers. They are tricky, and in my first try it was un-cow-like junk, but it can be done.

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