Each section models a teaching approach: how a concrete prop makes an abstract operation visible. Multiplication and division use a folded paper strip to show equal lengths; mental addition uses an egg tray to show decomposition through 10; the fractions section is a reference card for building automatic recall. Any of these can be recreated in class with food, packaging, or scrap paper.
When working across the number line, fractions, decimals, percentages, ratios, and arithmetic sections, consistently visualize how they represent the exact same thing. Mark a point once and name it four ways: fraction, decimal, percent, and a short ratio description. Students learn that 3/5, 6/10, 0.6, and 60% are the same location, or that 5 as a whole number, 5/1 as a fraction, and 500% are all the same. This way, conversions become an intuitive relabeling exercise rather than a new topic.
Mark a point on the bar and name it four ways: fraction, decimal, percent, and a count out of 100. Drag the red dot to explore — the grid lets you count the squares.
Think of this as the fractions times table. Memorize halves, quarters, fifths, and tenths first; add thirds and eighths next. Two minutes a day builds recall and frees working memory for problem setup.
Each tile is one fact to memorise: a fraction, its decimal, and its percent — three ways of writing the same number. Halves, quarters, fifths, and tenths are the priority; learn those until they are automatic, then add thirds and eighths.
Once students can name a point on the 0 to 1 bar four ways, the line extends past 1. Whole numbers sit at 1, 2, 3, and those same points are also 1/1, 2/1, 3/1 — fractions with 1 in the denominator. Every arithmetic operation now lives on this line as a movement of a number, or as a change to the length of a segment.
Addition slides right. Subtraction slides left, which is the same move as adding a negative number — direction on the line is the only thing that changes. Multiplication unfolds copies of the starting length end to end. Division folds that same length back into equal parts. Dividing by n and multiplying by 1/n land on the same point, because the fold and the unfold are the same fraction run in reverse — which is why the fractions from the earlier table reappear here as operations.
Pick an operation and two numbers, then press Play. The line stays blank until the operation happens, so you see the movement itself — not an answer sitting there from the start.
Use contexts students already feel. If a base recipe uses 100 g of rice and 300 g of water, multiplying the batch by five multiplies both amounts by five → 500 g rice and 1500 g water. Multiply by ten for a large group → 1000 g rice and 3000 g water. The ratio is the same; only the scale changes.
Unit rates are the same picture with one line scaled to 1. If one person needs about 100 g of rice, ten people need 1000 g. To cook for 6 people, match 6 on the People line and read 600 g on the Rice line.
Multicomponent recipes scale the same way. A tuna sandwich might take 60 g tuna, 20 g mayonnaise, 5 g chili, and two slices of bread. For six sandwiches, multiply every ingredient by six. The ratios stay constant while the totals grow.
1 person needs 100 g of rice. Slide to change the number of people — each one gets their own bowl. The table shows that dividing both numbers always gives back 1 : 100.
Base recipe: 100 g rice + 300 g water (1 : 3 ratio). Slide the batch multiplier — you can count the blocks stacking up. Divide both totals by the multiplier and you always get back to 100 : 300.
Students get lost when letters look like just symbols instead of developing an understanding that each letter just represents an unknown but very real quantity. To represent the container approach while retaining the second principle of using concrete and easily grasped real-world examples, we start with donuts as an example. Boxes of donuts naturally come in fixed pack sizes (1 individual donut, 2 or 4 or 6 donut packs, or 12 donuts in a box) and stores sometimes use colored paper bags.
We can give an example where the donut shop always gives out the box of donuts in a colored paper bag, where they use different colors for the different sizes of boxes. So we know that all blue bags hold the same count, all yellow bags hold the same count, but you can't tell from the outside – exactly like how we can know x and y represent some number, that every instance of x represents that number and every instance of y represents that number, and we just don't know what that number is so we use a letter as just a placeholder.
A common area of confusion for students is why the different letters can't be added together when different numbers can, what exactly the difference is and why it has to be that way. The containers frame makes it more intuitive to understand why we can't add or subtract
If someone hands you three blue bags and you open one and see a 2-pack, you know three blue bags mean 2+2+2 = 6 donuts – that's 3×2. But if you have two blue bags and one yellow bag, that doesn't give you the information to add up a total until you also find out how many are in the yellow bag. That's why 2x + x is 3x but 2x + y isn't.
This can then also be used to represent solving algebraic equations. In this example, if you bring two blue bags and a yellow bag to a party, and the host brings out a plate of 10 donuts after unpacking, you can work out how many donuts were in the yellow bag. Blue bags are 2 each, so 2×2 = 4. This means that 4 + yellow bag = 10. That forces the yellow bag to be 6 because 10 − 4 = 6. In symbols: with x=2, 2x + y = 10 implies y = 6.
The expression 2x + 3 reads as two x-bags and three loose donuts. If x is a blue 2-pack, that's 2·2 + 3 = 7. Students see why numbers and letters can sit in the same expression: they all still refer to actual values in real life, it's just whether we can "see" what the number is or if we're just using the placeholder
3(x + 2) can be thought of as three identical party kits. Each kit has one bag and two loose donuts. Total = three bags and six donuts → 3x + 6. This stops the common mistake of calling it x + 5.
x·x builds a tray (x²) that holds x bags, each with x donuts. Then 3x·2x means six such trays → 6x². Exponents feel like counts of trays, not a symbol trick.
Letters are just bags with an unknown number of donuts inside. Same color = same count. Click each step to build from like terms all the way to exponents.
A number is not a single object — it is a bundle of other numbers. 10 is also 1+9, 2+8, 3+7, 4+6, 5+5, and it is also 11−1, 12−2, 13−3. Moving fluently between these forms is what lets people do arithmetic in their head, because 10s are easy to work with and whatever is left over can work on its own.
For mental math, 10s function like fixed-size trays. If two people bring 12 eggs and 9 eggs and only 10-egg trays are available, the eggs regroup themselves into 10s: one tray fills from the 12 with 2 eggs left over, the next tray fills from the 9 and uses one of those leftover eggs to cap itself, and the last egg stays loose. Two full trays plus one loose egg is 21. The move — rearranging numbers across plus and minus signs without changing the total — is the same move algebra asks for later when rearranging an equation across its equals sign.
Pick a target. See every way two smaller numbers combine to make it, and every way a slightly larger number can drop back to it by losing a few. Then watch how 12 + 9 turns into two full trays of 10 plus one egg.