A number can look different without changing its value. Ten separate ones and one bundle of ten both stand for ten. Regrouping uses this idea: exchange a group for an equal-value group, then read or calculate with the new arrangement.
Use this lesson when you can count groups of ten and read three-digit numbers. Work with an adult or teacher if you need help reading the instructions or making paper cards. You may point, move cards, draw or explain aloud. You do not have to complete every activity in one sitting.
Prepare a hundreds–tens–ones mat
Draw three columns and label them Hundreds, Tens and Ones. Use large paper cards marked 100, 10 and 1. A card marked 100 stands for one hundred; it does not have to be physically one hundred times larger than a ones card. Keep cards with the same value in the same column.
Ask an adult to help with cutting if needed. Avoid small loose counters where swallowing or handling them could be a concern. An equally good alternative is to draw squares for hundreds, rods for tens and dots for ones, following the key on the models.
The supply is not part of the number on your mat. When you exchange ten ones, move those ten cards back to the supply and take one tens card. Do both actions together. Otherwise, you might accidentally add ten to the quantity.
Worked case 1: build and read 268
Place two hundreds cards, six tens cards and eight ones cards on the mat. Read the quantity as two hundred and sixty-eight. Its expanded form is 200 + 60 + 8 = 268.
| Digit | Place | Value contributed |
|---|---|---|
| 2 | Hundreds | 200 |
| 6 | Tens | 60 |
| 8 | Ones | 8 |
A digit is a written symbol such as 6. Its place tells us which group it counts. Its value here is sixty because it counts six tens. Moving the digit to a different place would usually change the number.
Try exchanging one tens card for ten ones cards. You now have two hundreds, five tens and eighteen ones. Count their value: 200 + 50 + 18 = 268. This is another model of the same number. Do not join the counts into “2518”: those counts describe groups of different sizes, not ordinary digit positions.
Worked case 2: add 268 and 157
Build 268 and 157 separately, then join their cards in the matching columns. There are three hundreds, eleven tens and fifteen ones. This arrangement is allowed in a model, even though each place in the usual written number has just one digit.
| Stage | Hundreds | Tens | Ones | Value |
|---|---|---|---|---|
| First number | 2 | 6 | 8 | 268 |
| Second number | 1 | 5 | 7 | 157 |
| Joined cards | 3 | 11 | 15 | 300 + 110 + 15 = 425 |
| Trade ten ones | 3 | 12 | 5 | 300 + 120 + 5 = 425 |
| Trade ten tens | 4 | 2 | 5 | 400 + 20 + 5 = 425 |
- Combine the ones: 8 + 7 = 15 ones. Exchange ten of them for one ten. Five ones remain.
- Combine the tens, including the exchanged ten: 6 + 5 + 1 = 12 tens. Exchange ten tens for one hundred. Two tens remain.
- Combine the hundreds, including the exchanged hundred: 2 + 1 + 1 = 4 hundreds.
So 268 + 157 = 425. If you write the calculation in columns, align hundreds under hundreds, tens under tens and ones under ones. The small 1 carried from the ones column means one ten. The small 1 carried from the tens column means one hundred. Say the unit with the number so that the marks have meaning.
Check another way: 268 + 100 = 368; add 50 to make 418; add 7 to make 425. The partitioned addition agrees with the cards.
Worked case 3: subtract across a zero, 302 − 146
Build three hundreds and two ones. Leave the tens column empty but keep its label. The zero in 302 records that there are no tens in this usual grouping; it keeps the hundreds and ones in their correct places.
You need to remove six ones but currently have only two loose ones. You cannot exchange a ten directly because the tens column is empty. Make two linked exchanges before removing anything.
- Exchange one hundred for ten tens. The mat becomes two hundreds, ten tens and two ones: 200 + 100 + 2 = 302.
- Exchange one of those ten tens for ten ones. The mat becomes two hundreds, nine tens and twelve ones: 200 + 90 + 12 = 302.
- Remove six ones from twelve, leaving six ones. Remove four tens from nine, leaving five tens. Remove one hundred from two, leaving one hundred.
The answer is 156. The model before removal is 2 hundreds, 9 tens and 12 ones. If you annotate a written calculation, those are the revised amounts above the columns. Do not leave ten tens after also exchanging one of them; that would count the exchanged ten twice.
Check by putting back what was removed: 156 + 146 = 302. A rough check also helps: taking about 150 from about 300 should leave about 150, so 156 is a sensible size.
Find the incorrect trade
A learner starts with three hundreds, two tens and seven ones. They replace two tens with one hundred and claim the mat still represents 327. The new mat has four hundreds, zero tens and seven ones.
Check the values rather than counting the number of cards. Two tens are worth 20, but one hundred is worth 100. The replacement adds 80 without permission: the new value is 407. A correct tens-to-hundreds exchange requires ten tens. This learner does not have ten tens available.
Ask, “What value left the mat? What value entered?” A correct exchange gives the same answer to both questions. The number of pieces can change; the total value must stay the same.
Three practice tasks with complete solutions
Practice 1: show a different grouping
Build 451. Exchange one hundred for tens. Write the new counts and check their value.
Solution: 451 begins as four hundreds, five tens and one one. Exchanging one hundred gives three hundreds, fifteen tens and one one. The value is 300 + 150 + 1 = 451. The hundreds count falls by one while the tens count rises by ten.
Practice 2: add with two exchanges
Use a mat or drawing to calculate 346 + 278. Explain each exchange.
Solution: joining gives five hundreds, eleven tens and fourteen ones. Exchange ten ones to get five hundreds, twelve tens and four ones. Exchange ten tens to get six hundreds, two tens and four ones. Therefore 346 + 278 = 624. Check by adding 200, then 70, then 8 to 346: 546, 616, 624.
Practice 3: subtract with an empty tens column
Calculate 504 − 237. Say the regrouped amounts before removing cards.
Solution: exchange one of five hundreds to get four hundreds, ten tens and four ones. Exchange one ten to get four hundreds, nine tens and fourteen ones. Remove two hundreds, three tens and seven ones. Two hundreds, six tens and seven ones remain, so the answer is 267. Check: 267 + 237 = 504.
Talk through your own model
Cover the worked answer and describe one exchange using this sentence: “I replaced ___ with ___; both are worth ___.” Then ask a partner to rebuild your model from your words. If their model differs, compare the values before deciding whether either model is wrong.
If three columns feel too busy, practise the exchange with ten ones first. Return to three-digit quantities when you can explain why the exchange preserves ten. An adult can move the cards while you direct the changes; controlling small pieces is not the skill being checked.
Questions about regrouping
Does regrouping change the answer?
A correct exchange changes how a quantity is grouped, not its value. An addition or subtraction changes the quantity when amounts are joined or removed.
Why do we start with the ones?
Starting with ones lets each exchange feed into the next column. Other correct methods exist, but this order makes the column method easier to follow.
Can a model have more than nine ones?
Yes. Twelve ones is a valid model quantity. To write the usual place-value form, exchange ten of them for one ten and leave two ones.
Why is the zero in 302 important?
It records an empty tens place. Without it, writing 32 would mean three tens and two ones rather than three hundreds and two ones.
Is a hundreds card worth more because it is bigger?
No. In this card model its label defines its value. The different shapes help you recognise the units; they are not area measurements.
Must I use cut-out cards?
No. Draw the symbols, point to the supplied models or direct an adult to move cards. Keep the hundreds, tens and ones clearly labelled.
How can I tell whether a trade is correct?
Compare the total value before and after. Ten ones must exchange for one ten, and ten tens must exchange for one hundred.
For an adult supporting the activity
Ask for the value of a group, not only the number of pieces. Clear mathematical language and connected concrete or drawn models are supported in the Institute of Education Sciences mathematics practice guide. Use the learner’s explanation to decide whether to stay with cards or move to written columns.
WhatsApp Us Now