Base Ten Blocks — Carrying and Borrowing Visualized

Virtual base ten blocks that act out a sum one step at a time: hundreds, tens and ones are put together, ten ones are traded for a ten rod and carried, or a rod is broken into ten ones to borrow. The written column method beside the picture changes at the same moment, and each step has one sentence you can read aloud. Use it when a child asks why you carry the 1, or on a classroom screen.

How to use

1. Choose "Addition", "Subtraction", "Multiplication" or "Division" at the top and type the two numbers. A worked example is already filled in.
2. Press "Next step" to move the blocks one step and read the sentence under the written method aloud. "Previous step" goes back and "Play" runs the steps by itself at the chosen "Speed".
3. Press "New problem" for another one. With "Only problems with carrying" (or borrowing) ticked, every new problem needs regrouping; "2-digit" and "3-digit" set the size.
4. For multiplication pick "Array of groups" or "Area model"; for division pick "Share equally" or "Make groups".
5. In "Make a ten by hand" the child drags small cubes into the "Ten box". When ten are in, they turn into a rod; dragging a rod down to the ones breaks it into ten cubes.
6. "Full screen" shows the board on a classroom TV. The arrow keys, the space bar and F work as shortcuts.

How it works

• Addition and subtraction use whole numbers from 0 to 999. Multiplication takes 1 to 12 times 1 to 12, or a two-digit number times a one-digit number. Division takes 1 to 99 divided by 1 to 9, with a remainder when it does not come out exactly.
• A big square is 100, a rod is 10, a small cube is 1, and a large cube is 1,000. The shapes differ, so colours are not needed to tell them apart.
• In Korean, Japanese and Chinese the array reads a × b as "a, taken b times" (b groups of a); in the other languages as a groups of b. Both give the same product.
• The page remembers the last problem, picture and speed in this browser only.

How to explain carrying in addition to a child

Carrying is a trade. Each place can hold at most nine, because ten of anything make one of the next bigger piece. In 47 + 38 the ones are 7 + 8 = 15, which is too many for the ones place. Ten of the fifteen small cubes are pressed into one rod, the rod moves over to the tens, and 5 ones stay behind. That is exactly what the small 1 above the tens column means: one extra ten.

Then the tens are added together with the carried rod: 4 + 3 + 1 = 8 tens, so the answer is 85. Watching the ten cubes turn into a rod makes it clear that nothing was added or lost; the same amount is just written with fewer pieces.

A common mistake is writing 15 in the ones place, which gives 715. Ask the child how many small cubes are left in the ones column after the trade: the answer is always a single digit.

Borrowing in subtraction explained with blocks

In 52 − 27 there are only 2 ones, so 7 cannot be taken away. One ten rod is broken into ten small cubes, which makes 12 ones and leaves 4 tens. Now 12 − 7 = 5 and 4 − 2 = 2, so 52 − 27 = 25. In the written method the 5 is crossed out and 4 written above it, and the 2 becomes 12.

When the tens place is empty, as in 503 − 7, a hundred is broken into ten tens first, and then one of those tens is broken into ten ones. The written method shows the 0 crossed out, a 10 that is crossed out again, and a 9 above it.

Children often subtract the smaller digit from the larger one in each column (getting 35 for 52 − 27). Pointing to the blocks helps: there really are only 2 small cubes, and you cannot take 7 from them without breaking a rod.

Multiplication arrays and the area model, division by sharing

The array builds a multiplication one group at a time and counts on: 3, 6, 9, 12 for four groups of 3. The area model draws a rectangle and splits it by place value, so 23 × 4 becomes 20 × 4 + 3 × 4 = 80 + 12 = 92; this is the distributive law that written multiplication relies on.

Sharing deals the tens first: 75 ÷ 3 gives each of three plates 2 rods, the leftover rod is traded for ten cubes, and the 15 ones are dealt out, 5 to each plate, so each plate has 25. Making groups instead circles groups of 3 until fewer than 3 are left; what is left over is the remainder.

Questions and answers

Why do you carry the 1 in addition?

Because ten ones are the same as one ten. When a column adds up to 10 or more, ten of its pieces are traded for one piece of the next place, and the small 1 records that extra piece.

What does regrouping mean in math?

Regrouping is trading ten pieces of one place for one piece of the next place, or the other way round. Carrying in addition and borrowing in subtraction are both regrouping.

How do you subtract when the top digit is smaller?

Break one piece of the next place into ten. In 52 − 27, one ten becomes ten ones, so the ones are 12 − 7 = 5 and the tens are 4 − 2 = 2.

What are base ten blocks?

Small cubes worth 1, rods of ten cubes worth 10, flat squares of a hundred cubes worth 100 and large cubes worth 1,000. They let a child see place value as real amounts.

Can I use it on an interactive whiteboard or a classroom TV?

Yes. Press "Full screen" and step with the buttons or the arrow keys. The pieces in "Make a ten by hand" can be dragged with a finger on a touch screen.

How does the area model show 23 × 4?

It draws a rectangle 23 squares wide and 4 tall, splits the width into 20 and 3, and adds the two parts: 80 + 12 = 92.