Virtual base ten blocks

What base ten blocks show

Base ten blocks are a physical picture of our number system. A small cube is one. A rod is ten cubes joined in a row. A flat is ten rods side by side, which makes one hundred. Building 245 with 2 flats, 4 rods and 5 cubes shows a child what the 2, the 4 and the 5 in that number are actually worth.

On this page you add and remove each kind of block and the total updates straight away, written out as hundreds, tens and ones. You can put down more than nine cubes or rods, so a child can build 15 ones and see that it is the same amount as 1 ten and 5 ones. Bundle and Break buttons appear whenever a swap is possible.

Why place value needs a model

Place value is the idea that a digit's value depends on where it sits. That idea is abstract, and many later errors trace back to it: lining up columns wrongly, or writing 3005 for three hundred five. Blocks make the size difference impossible to miss, because a flat is plainly bigger than a cube. Common Core standard 2.NBT.A.1 describes this directly, asking second graders to understand that the three digits of a three-digit number stand for hundreds, tens and ones.

Blocks also explain regrouping. "Carrying" in addition is swapping 10 ones for 1 ten. "Borrowing" in subtraction is breaking 1 ten into 10 ones. When a child does that swap themselves by removing a rod and adding 10 cubes, the written method stops being a trick to memorize and becomes a record of something they did.

Ideas by grade

• Grade 1: build numbers from 11 to 19 as one rod and some cubes. Then build 20, 30 and 40 with rods only.
• Grade 2: build three-digit numbers, including ones with a zero in the middle, like 408. Compare two numbers by comparing flats first, then rods.
• Grade 3: model addition and subtraction with regrouping. Build both numbers, combine them, and swap whenever you reach ten of one kind.
• Grade 4: use the blocks to round to the nearest ten or hundred. Build 999 and talk through what one more cube would set off: ten ones, then ten tens, then ten hundreds, which is a thousand.

A 10-minute regrouping activity

This routine uses 47 + 28, but any two-digit pair whose ones add to more than 9 works.

• Build 47 with 4 rods and 7 cubes. Ask the child to read the total aloud.
• Add 28 by putting down 2 more rods and 8 more cubes.
• Count the cubes. There are 15, which is more than 9, so remove 10 cubes and add 1 rod. The total stays the same.
• Read the new model: 7 rods and 5 cubes, which is 75.
• Now write 47 + 28 in columns and point to the small 1 carried into the tens. Ask the child which block it stands for.

For subtraction, reverse the idea. Build 52 and try to take away 27. There are not enough cubes to take 7, so remove one rod and add 10 cubes first, then take away.

Moving on from blocks

Blocks are a bridge, not a destination. Once a child can describe a swap without doing it, move to written methods. Eurekami lessons draw base-ten blocks from the numbers in each question, and the AI tutor checks each line of a child's working and points to the first line that goes wrong, so a regrouping slip gets caught where it happens.