Adding a one-digit number to a two-digit number by making a ten
There are 47 bees in a hive and 6 more fly in. The 7 ones and the 6 ones make more than nine, so the tens change too. How do you find the total without counting on one bee at a time?
Where you are 6 in Multi-digit addition
When you reach for it
The child already adds a one-digit number to a two-digit number when the ones stay small. In Adding two-digit numbers without regrouping, 43 + 5 is 4 tens and 3 + 5 = 8 ones, so 48. The ones join the ones and the tens are not touched.
Now the ones get too big. There are 47 bees in a hive and 6 more fly in. The ones are 7 and 6, and 7 + 6 = 13. Thirteen ones cannot sit in the ones place, because a ones digit goes only up to 9. Ten of those ones are really a new ten. So the tens change too, and the answer is not “4 tens and 13 ones” but 5 tens and 3 ones, which is 53.
This lesson is about that kind of sum: a two-digit number plus a one-digit number, where the ones make a new ten. The move that handles it is one the child already knows from Making ten and bridging through ten. There, 8 + 5 was found by filling the 8 up to 10 and adding what was left. Here, 47 + 6 is found by filling the 47 up to the next ten, 50, and adding what is left:
The child needs three things first. The bonds to ten, so that 7 is seen at once as needing 3 (the bonds to ten). Splitting a small number into two parts, so that 6 is 3 and 3. And Tens and ones, so that 50 and 3 more is fifty-three with no counting.
The pictures are the same as before: ten frames, bundles of ten sticks or base-ten drawings of rods and loose squares, an open number line, and a hundred chart.
The procedure
The four days build one idea. Day 1 sees the new ten in pictures: ten loose ones become one rod. Day 2 writes the move as numbers and shows why it is the same move as within twenty. Day 3 puts it on the open number line and the hundred chart. Day 4 asks, before adding, whether the ones will make a new ten at all, and runs the move up to 100.
Day 1: ten loose ones become a new rod
Build the first number. Show 47 with bundles of ten sticks: 4 bundles and 7 loose sticks. Then bring 6 more loose sticks for the 6 bees. The loose sticks go with the loose sticks, as in the earlier lesson. But now there are 7 + 6 = 13 loose sticks on the table.
Make the new bundle. Thirteen loose sticks is enough to make a bundle. Take ten of them and put a rubber band around them. That bundle is a ten, just like the other four, so it moves over to join them. Now there are 5 bundles and 3 loose sticks: fifty-three.
The order matters for how easy it is to see. The 7 loose sticks already there need 3 more to be ten. So take 3 of the new sticks, put them with the 7, and band the ten. The other 3 new sticks stay loose. That is filling up to the next ten.
Figure 1 shows the same thing as a base-ten drawing. On the left, before bundling: 4 rods and 13 loose squares. On the right, after: the 7 plain squares and 3 of the shaded ones have become a fifth rod, and 3 shaded squares are left loose. Both pictures show the same squares, so both show the same number. The right one is just easier to read.
Ten frames. Ten frames show the same move. A full frame is a ten. To show 29, use two full frames and a third frame with 9 counters. Adding 4 means placing 4 more counters, filling the boxes in order. The first one fills the third frame, and the other 3 start a fourth frame (Figure 2). Now the frames read 10, 10, 10 and 3: thirty-three. Example 2 does this sum.
Day 2: split the number with a bond
Writing the move. Once the child has bundled sticks a few times, the picture can be said in words: “47 needs 3 to make 50. Six is 3 and 3. So 50 and 3 more is 53.” Written as numbers, the 6 is split into the part that fills the ten and the part that is left:
Two facts the child already has do all the work. The first is a bond of ten: 7 needs 3, so 47 needs 3 to reach 50. Only the ones digit decides how many it needs. The second is a bond of the small number: 6 is 3 and 3.
Why the split is allowed. The 6 bees are the same 6 bees whether they fly in all together or as 3 and then 3. Splitting them changes nothing about how many there are, because 3 and 3 make 6. So 47 + 6 and 47 + 3 + 3 are the same total. The split must be an exact bond of the small number: 6 can be 3 and 3, or 4 and 2, but here we want the part that fills the ten, and 47 needs exactly 3.
The same move, one row up. Put 7 + 6 next to 47 + 6:
The ones do the same thing in both: 7 and 6 make a ten and 3. In 47 + 6 there are also 4 tens that were there all along. They ride along untouched, and the new ten joins them. So the answer has one more ten than the first number, and its ones digit is the same as in 7 + 6. This is why the move works for every sum in this lesson: it is the make-a-ten move within twenty, with some tens carried along.
The small number may come first. In 7 + 64 the one-digit number is written first. The order of adding does not change the total (the commutative property of addition), so think of it as 64 + 7 and fill the 64. Example 4 does this.
Day 3: the open number line and the hundred chart
A stop at the ten. On an open number line, the move is two hops with a stop at the ten. To add 9 to 35, mark 35 and hop to the next ten, 40. That hop is 5. The 9 is 5 and 4, so 4 is left: one more hop from 40 to 44. The line is drawn freehand and need not be to scale, but the stop at 40 is always marked, because that is the point of the move. From a ten, the last hop can be said at once. Example 5 draws it.
A child could also make nine hops of one from 35. The two-hop way says the same thing in two steps the child knows by heart. This kind of drawing grows into Adding on an open number line.
The end of a row is a ten. Each row of a hundred chart ends with a ten: 10, 20, 30 and so on, all in the last column. Adding a one-digit number means stepping right, one square for each one. When the steps reach the end of the row, the next step goes to the start of the next row. That move to a new row is the new ten.
For 58 + 7, step right from 58: 59, 60. That is 2 steps, and 60 ends the row. The 7 is 2 and 5, so 5 steps are left, and they run along the next row: 61, 62, 63, 64, 65 (Figure 4). The chart has done the same split as the bond: 2 to finish the ten, 5 more after it.
Why the tens digit goes up by one. In the earlier lesson a step straight down was ten more. Here the child is not going down a whole row; they finish one row and start the next. The tens digit goes up by one because a new ten has been finished, and the ones start again from 1.
Day 4: will the ones make a new ten?
Look at the ones first. Before adding, look at the ones digit of the two-digit number and the one-digit number. If they add to 9 or less, the ones simply join and the tens stay, as in the earlier lesson. If they add to 10 or more, a new ten is made and the tens digit goes up by one. In 34 + 5 the ones make 4 + 5 = 9, so no new ten: 39. In 34 + 9 the ones make 4 + 9 = 13, a new ten: 43. Example 7 compares three such sums.
Landing on a ten. When the ones add to exactly 10, the whole one-digit number is used to fill the ten and nothing is left. 34 + 6 = 40. The ones digit of the answer is 0.
At most one new ten. A one-digit number is at most 9, and a ones digit is at most 9, so the ones add to at most 9 + 9 = 18. That is one ten and some ones, never two tens. So adding a one-digit number can push the tens digit up by one, never by two.
Up to a hundred. The move works the same way at the top of the chart. 93 + 7 fills 93 up to the next ten, which is 100, and the 7 is used up exactly. Ten tens is a hundred. Example 8 does this one.
Worked examples
Read each problem aloud. Sticks, a ten frame or a hundred chart on the table let the child build or walk every one. Once the child can say “fill to the ten, then the rest” in their head, they are adding by making a ten.
Day 1: Bees fly into the hive
Helmi watches a bee hive. There are 47 bees inside, and then 6 more bees fly in. How many bees are in the hive now?
SolutionThe 6 bees join the 47, so we add: 47 + 6.
47 is 4 tens and 7 ones. The 6 bees are 6 ones. The ones make 7 + 6, which is more than nine, so a new ten will be made. Fill the 7 up to ten: 7 needs 3. Take 3 of the 6, and 3 are left.
Now 47 + 3 = 50, the next ten, and 50 + 3 = 53:
There are 53 bees in the hive. In Figure 1 the 7 plain squares and 3 shaded squares make the new rod, so there are 5 tens and 3 ones.
Check by counting on from 47: forty-eight, forty-nine, fifty, fifty-one, fifty-two, fifty-three. Six counts, ending at 53, and the count passes fifty on the way.
Day 1: Kazoos in the box
The kazoo band keeps its kazoos in a box. There are 29 kazoos in the box, and Eetu brings 4 more. How many kazoos are there now?
SolutionWe add: 29 + 4.
29 is 2 tens and 9 ones. A 9 needs just 1 to be ten, so 29 needs 1 to reach 30. The 4 is 1 and 3. One kazoo fills the ten and 3 are left:
There are 33 kazoos. On ten frames (Figure 2), the first light counter fills the third frame and the other 3 start the fourth, so the frames read 10, 10, 10 and 3.
A number ending in 9 is the easiest to fill: it always needs just 1.
Day 2: The same move, one row up
Find 6 + 8. Then use it to find 46 + 8.
SolutionFor 6 + 8, fill the 6 to ten: it needs 4. The 8 is 4 and 4. So 6 + 4 = 10 and 10 + 4 = 14:
For 46 + 8, the ones digit is 6 again, so 46 needs the same 4 to reach the next ten, 50. The 8 splits the same way, 4 and 4:
The two sums match. 46 is 4 tens and 6 ones, and the ones do exactly what they did in 6 + 8: they make a ten and 4. The 4 tens just come along. So the answer has 4 tens, plus the new ten, plus 4 ones: 5 tens and 4 ones, which is 54.
A child who knows 6 + 8 = 14 can use it here: the ones digit of the answer is the 4 from 14, and the tens digit is one more than the 4 in 46.
Day 2: The small number first
Helmi counts bees coming home to the hive. In the morning 7 bees fly in. In the afternoon 64 more fly in. How many bees flew in that day?
SolutionThe two groups of bees are put together: 7 + 64.
The order of adding does not change the total (the commutative property), so find 64 + 7 instead. It is easier to fill the big number than to add 64 to 7.
64 has 4 ones, which needs 6 to make ten, so 64 needs 6 to reach 70. The 7 is 6 and 1:
So 7 + 64 = 71, and 71 bees flew in that day.
Filling the 7 instead also works: 7 needs 3, so take 3 from 64, leaving 61, and 10 + 61 = 71. The total agrees, but it asks the child to add a ten to a two-digit number as an extra step. Filling the bigger number keeps the work small.
Day 3: Marching on the open number line
Ruut leads the kazoo band down the hall. The band marches 35 steps, stops to play a tune, then marches 9 more steps. How many steps did the band march? Show it on an open number line.
SolutionWe add: 35 + 9.
Mark 35 on an open number line. The next ten is 40, and 35 needs 5 to get there, so make a hop of 5 to 40. The 9 is 5 and 4, so 4 is left: hop 4 more, from 40 to 44.
The band marched 44 steps. The stop at 40 splits the 9 into the hop that finishes the ten and the hop after it, the same split a bond of 9 gives.
Day 3: To the next row of the hundred chart
Use a hundred chart to find 58 + 7.
SolutionAdding 7 is 7 steps to the right, one for each one. Start at 58. Step right: 59, 60. That is 2 steps, and 60 is at the end of the row, so the next step goes to the start of the next row. There are 7 - 2 = 5 steps left: 61, 62, 63, 64, 65.
So 58 + 7 = 65. The chart split the 7 into 2 and 5 without being asked: 2 steps finish the row, which is the ten, and 5 steps run along the new row. Moving down to the new row is the tens digit going from 5 to 6.
Day 4: Does it make a new ten?
Find 34 + 5, 34 + 6 and 34 + 9. Which of them make a new ten?
SolutionAll three start from 34, which is 3 tens and 4 ones, and 34 needs 6 to reach 40.
In 34 + 5 the ones are 4 + 5 = 9. That is less than ten, so no new ten is made. The ones join and the 3 tens stay:
In 34 + 6 the ones are 4 + 6 = 10, exactly a ten. All 6 are used to fill the ten and nothing is left:
In 34 + 9 the ones are 4 + 9 = 13. Fill to 40 with 6 of the 9, and 3 are left:
34 + 6 and 34 + 9 make a new ten; 34 + 5 does not. On the hundred chart, 34 + 5 stops at 39, still in the row of the thirties, while 34 + 6 lands at the end of that row and 34 + 9 goes on into the next.
Day 4: Up to a hundred
There are 93 bees in the hive, and 7 more fly in. How many bees are in the hive now?
SolutionWe add: 93 + 7.
93 is 9 tens and 3 ones, and 3 needs 7 to make ten. So 93 needs exactly 7 to reach the next ten. All 7 bees fill that ten, and none are left:
There are 100 bees in the hive. With sticks, the 3 loose sticks and 7 new ones make a tenth bundle, and ten bundles of ten is one hundred. On the hundred chart, 7 steps right from 93 land on the last square, 100.
Boundaries
- The ones must make a new ten. The move is for sums whose ones add to 10 or more, as in 7 + 6 = 13. When the ones add to 9 or less, as in 43 + 5, there is no ten to fill; the ones simply join, as in Adding two-digit numbers without regrouping.
- The split must be an exact bond. The two parts must add back to the number being split: 6 is 3 and 3, never 3 and 4. The first part is set by the ones digit of the two-digit number, because that is what decides how far away the next ten is.
- One number has one digit. This lesson adds a one-digit number to a two-digit number, so at most one new ten is made, because the ones add to at most 9 + 9 = 18. Adding two numbers that both have tens and ones, with a new ten, is Regrouping in addition.
- The totals stay at 100 or less. 93 + 7 = 100 is the top of the hundred chart. A sum such as 96 + 7 fills to 100 and goes past it, to 103, which this lesson does not reach.
- Order does not matter. 7 + 64 = 64 + 7, because the order of adding two numbers never changes the total. It is usually easier to fill the two-digit number.
- This is adding, not taking away. Taking away across a ten, as in 53 - 6, crosses the ten the other way and belongs to subtracting bigger numbers.
Where this goes next
Filling up to the next ten is the first time the ones of a sum spill into the tens, and every way of adding larger numbers has to deal with that spill. When both numbers have tens and ones, the same “ten ones make a ten” becomes Regrouping in addition, and writing that new ten above the tens column is the carry in The standard addition algorithm. The two hops with a stop at the ten grow into longer walks on Adding on an open number line, and adding each place on its own, then combining, is The partial sums method. Using the next ten as a landmark stays one of the quickest mental strategies for adding in your head.
Exercises
Day 1: ten loose ones become a new rod
- 28 + 5 = \square
- 56 + 7 = \square
- 19 + 3 = \square
- Picture A shows a sum. The plain pieces are the first number and the shaded squares are being added. How many shaded squares finish a new rod? Write the sum and find the total.
- There are 54 bees in the hive, and 8 more fly in. How many bees are in the hive now?
Day 2: split the number with a bond
- Split the 5 to finish the ten: 27 + 5 = 27 + 3 + \square. What goes in the box, and what is 27 + 5?
- 68 + 4 = \square
- Find 5 + 9. Then use it to find 45 + 9.
- 6 + 77 = \square
- The kazoo band has 49 kazoos. Ruut brings 7 more. How many kazoos does the band have now?
Day 3: the open number line and the hundred chart
- Show 26 + 7 on an open number line with a stop at the ten. Which numbers do you land on?
- Show 89 + 5 on an open number line with a stop at the ten. Which numbers do you land on?
- On a hundred chart, start at 76 and move 6 squares on. Where do you land? Write the sum you found.
- Use a hundred chart to find 43 + 9. How many steps finish the row?
- 57 + \square = 60
Day 4: will the ones make a new ten?
- Find 62 + 7, 62 + 8 and 62 + 9. Which of them make a new ten?
- 92 + 8 = \square
- 84 + 9 = \square
- Eetu sees 65 bees in the hive, and 8 more fly in. How many bees are in the hive now?
- 39 + \square = 45
Explain
- To find 36 + 7, why do you split the 7 into 4 and 3, and not into 5 and 2?
- You know 8 + 6 = 14. Use it to find 78 + 6, and say why it works.
- When you add a one-digit number to a two-digit number and the ones make a new ten, does the tens digit always go up by exactly one? Say why, and give an example.
- On a hundred chart, why does 67 + 8 move you to the next row, but 61 + 8 does not? Find both sums.
Answers
- 33. 28 needs 2 to reach 30; 5 is 2 and 3, so 28 + 5 = 30 + 3 = 33.
- 63. 56 needs 4 to reach 60; 7 is 4 and 3, so 56 + 7 = 60 + 3 = 63.
- 22. 19 needs 1 to reach 20; 3 is 1 and 2, so 19 + 3 = 20 + 2 = 22.
- 4 shaded squares finish a new rod, and 66 + 7 = 73. The plain pieces are 6 rods and 6 squares, which is 66. The 6 plain squares need 4 more to be ten, so 4 shaded squares make the seventh rod and 3 stay loose: 7 tens and 3 ones.
- 62 bees. 54 needs 6 to reach 60; 8 is 6 and 2, so 54 + 8 = 60 + 2 = 62.
- 2, and 27 + 5 = 32. The 5 is 3 and 2. 27 + 3 = 30, and 30 + 2 = 32.
- 72. 68 needs 2 to reach 70; 4 is 2 and 2, so 68 + 4 = 70 + 2 = 72.
- 5 + 9 = 14 and 45 + 9 = 54. 5 needs 5 to be ten, and 9 is 5 and 4, so 5 + 9 = 10 + 4 = 14. 45 needs the same 5 to reach 50, so 45 + 9 = 50 + 4 = 54: one more ten than 45, and the same ones digit as 14.
- 83. Think of 77 + 6. 77 needs 3 to reach 80; 6 is 3 and 3, so 6 + 77 = 80 + 3 = 83.
- 56 kazoos. 49 needs 1 to reach 50; 7 is 1 and 6, so 49 + 7 = 50 + 6 = 56.
- 30, then 33. Hop 4 from 26 to 30, then 3 more to 33, because 7 is 4 and 3: 26 + 7 = 33.
- 90, then 94. Hop 1 from 89 to 90, then 4 more to 94, because 5 is 1 and 4: 89 + 5 = 94.
- 82, so 76 + 6 = 82. Four steps, 77, 78, 79, 80, finish the row; two more, 81, 82, start the next row.
- 52, and 7 steps finish the row. From 43, the steps 44 to 50 are 7 steps, ending the row at 50. The 9 is 7 and 2, so 2 more steps, 51, 52: 43 + 9 = 52.
- 3. 57 has 7 ones, which needs 3 to make ten, so 57 + 3 = 60.
- 62 + 7 = 69, 62 + 8 = 70, 62 + 9 = 71. The ones make 2 + 7 = 9, 2 + 8 = 10 and 2 + 9 = 11, so 62 + 8 and 62 + 9 make a new ten and 62 + 7 does not.
- 100. 92 needs exactly 8 to reach the next ten, which is 100, so nothing is left: 92 + 8 = 100.
- 93. 84 needs 6 to reach 90; 9 is 6 and 3, so 84 + 9 = 90 + 3 = 93.
- 73 bees. 65 needs 5 to reach 70; 8 is 5 and 3, so 65 + 8 = 70 + 3 = 73.
- 6. From 39, 1 reaches 40 and 5 more reaches 45, and 1 and 5 make 6: 39 + 6 = 45.
- 36 has 6 ones, and 6 needs 4 to make ten, so 36 needs 4 to reach the next ten, 40. The 7 is split so its first part fills that ten exactly: 4, leaving 3. Then 36 + 7 = 40 + 3 = 43. With 5 and 2, 36 + 5 = 41 goes past 40 and the ten is not a clean stop. The sum is still 43, but the step is no longer easy.
- 84. 78 is 7 tens and 8 ones. The 8 ones and the 6 make 8 + 6 = 14, which is a ten and 4 ones. The 7 tens ride along, and the new ten joins them to make 8 tens. So 78 + 6 = 84: 8 tens and 4 ones.
- Yes. The ones digit is at most 9 and the one-digit number is at most 9, so the ones add to at most 9 + 9 = 18. That is one ten and some ones, never two tens. So exactly one new ten joins the tens. For example, 85 + 7: the ones make 5 + 7 = 12, one ten and 2, so 85 + 7 = 92 and the tens digit goes from 8 to 9.
- 67 + 8 = 75 and 61 + 8 = 69. A row ends at the next ten, 70. From 67 that is 3 steps away, and 8 is more than 3, so the steps run off the end of the row into the next one. From 61 the end of the row is 9 steps away, and 8 steps stop at 69, before the end. The ones make 7 + 8 = 15 in one sum and 1 + 8 = 9 in the other.