Subtraction with Tens and Ones Using Base Ten Blocks

Every two digit number is made up of tens and ones. For example, 42 means 4 tens and 2 ones, because \( 4 \times 10 + 2 = 42 \). When we subtract two digit numbers, it helps to think about tens and ones separately instead of treating the whole number as one big unknown chunk. This is exactly the idea behind place value, and it is the foundation for every subtraction method used with larger numbers, including subtracting with digits up to 1000.

Working with tens and ones turns a subtraction problem like \( 42 - 15 \) into two smaller, more manageable steps: subtract the ones, then subtract the tens. The tricky part happens when there are not enough ones to subtract from, which is where regrouping (sometimes called borrowing) comes in.

Base ten blocks give a hands on way to see tens and ones. A long rod represents one ten, and a small cube represents one one. To show the number 42, you would use 4 rods and 2 cubes.

When the ones digit on top is greater than or equal to the ones digit being subtracted, no regrouping is needed. Just subtract ones from ones and tens from tens.

Example: \( 58 - 23 \)

• Ones: \( 8 - 3 = 5 \)
• Tens: \( 5 - 2 = 3 \)
• Answer: \( 58 - 23 = 35 \)

With base ten blocks, you would start with 5 rods and 8 cubes, remove 2 rods and 3 cubes, and count what is left: 3 rods and 5 cubes, which is 35.

Sometimes the ones digit on top is smaller than the ones digit being subtracted. In that case, you cannot take away that many ones directly, so you regroup one ten from the tens column into 10 ones.

Example: \( 42 - 15 \)

Look at the ones column first: \( 2 - 5 \) cannot be done with a smaller number of ones on top. So take 1 ten from the 4 tens, leaving 3 tens, and turn it into 10 ones. Now the ones column has \( 10 + 2 = 12 \) ones.

• Ones: \( 12 - 5 = 7 \)
• Tens: \( 3 - 1 = 2 \)
• Answer: \( 42 - 15 = 27 \)

Once the blocks are regrouped, you can physically remove 1 ten rod and 5 ones cubes, leaving 2 rods and 7 cubes, which is 27. This matches a lesson you will use again when working on subtraction with regrouping up to 1000, where the same idea is extended to hundreds, tens, and ones together.

Base ten blocks turn an abstract rule ("borrow from the tens column") into something you can see and touch. Instead of memorizing a procedure, you can watch a ten physically break apart into ten ones, which is exactly what happens on paper when you cross out a tens digit and write a smaller number above it. Once this idea feels natural with blocks, it becomes much easier to subtract using written column methods, mental math, or the strategies covered in subtraction using mental strategies.

• Write or picture the number using tens rods and ones cubes.
• Compare the ones digits. If the top ones digit is smaller, regroup one ten into ten ones.
• Subtract the ones column.
• Subtract the tens column.
• Combine the remaining tens and ones for the final answer.

You can check your work by adding your answer back to the number you subtracted. For \( 42 - 15 = 27 \), check with \( 27 + 15 = 42 \). If both sides match, the subtraction is correct.

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