Adding with Regrouping Using Base Ten Blocks

Base ten blocks are a hands-on way to picture place value. A small square stands for one unit, called a "one." A long rectangle made of ten of those units stacked together stands for a "ten." When ten of those tens are put together, they form a "hundred" flat. Because each shape is worth exactly ten of the shape before it, base ten blocks let students see, not just calculate, what happens when a column of numbers adds up to ten or more.

This trade, ten small blocks for one bigger block of the next size, is exactly what "regrouping" (sometimes called "carrying") means when adding numbers on paper. Base ten blocks turn that abstract rule into something students can build and see.

When adding multi-digit numbers, each place value column (ones, tens, hundreds, and so on) can only hold the digits 0 through 9. If a column's blocks add up to ten or more, there are too many blocks to leave in that column, so ten of them get traded for one block in the next column to the left. This keeps every column showing a single digit while still representing the correct total. If you have not yet compared this to place value without regrouping, it helps to first look at adding multi-digit numbers so the column setup feels familiar before blocks are added on top.

Start by building each number with blocks: \(27\) is 2 tens rods and 7 ones squares, and \(15\) is 1 tens rod and 5 ones squares.

Line up the ones squares from both numbers and count them together: \(7 + 5 = 12\). Since 12 is more than a single column can hold, trade 10 of those ones squares for 1 new tens rod, leaving 2 ones behind. Now add the tens column, including the new rod: \(2 + 1 + 1 = 4\). Reading the blocks left to right gives 4 tens and 2 ones, or \(42\), matching \(27 + 15 = 42\).

Bigger sums sometimes require regrouping more than once. Try \(68 + 57\). Add the ones column first: \(8 + 7 = 15\). Trade 10 ones squares for 1 tens rod, leaving 5 ones and carrying 1 extra ten into the tens column. Now add the tens: \(6 + 5 + 1 = 12\) tens. Since that is also ten or more, trade 10 tens rods for 1 hundred flat, leaving 2 tens and carrying 1 hundred. The final blocks show 1 hundred flat, 2 tens rods, and 5 ones squares, which is \(125\), so \(68 + 57 = 125\). This is the same trading idea from the first example, just applied twice, once between ones and tens, and again between tens and hundreds.

• Always add the ones column first, then tens, then hundreds, working from right to left.
• Every time a column reaches 10 or more blocks, trade exactly 10 of them for 1 block in the next column.
• Sketch or use physical blocks side by side for each addend before combining them, so the "before" and "after" totals can be compared.
• Once the block model feels comfortable, connect each trade to the small carried "1" used in the addition strategies written method.
• Practicing the reverse process helps too: comparing this to subtracting with regrouping shows how trading works in both directions.

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• Math 4