A bus has children on it and more get on. A parade float has paper flowers and more are added. Neither total needs counting one by one. What do you add to what, and why?

Grade 1 · 1.NBT.C.4 · 2.NBT.B.5

Adding two-digit numbers without regrouping

A bus has 43 children on it and 5 more get on. A parade float has 27 paper flowers and 40 more are added. Neither total needs counting one by one. What do you add to what, and why?

You need

Adding and subtracting within 20Grades K to 1 Tens and onesGrade 1

Unlocks

Adding a one-digit number to a two-digit number by making a tenGrade 1 The partial sums methodGrade 2

Where you are 6 in Multi-digit addition

Number9

Arithmetic5

Addition and subtraction12

Multi-digit addition6

Adding two-digit numbers without regrouping

Adding on an open number line Regrouping in addition Adding a one-digit number to a two-digit number by making a ten The partial sums method The standard addition algorithm

When you reach for it

A child who knows the facts within twenty can find 3 + 5 quickly. Now the numbers get bigger: 43 children are on a school bus and 5 more get on. Counting on five from 43 works, but the same child also meets 27 + 40, and counting on forty times is slow and easy to get wrong. Both sums have a short way, and it is the same way for both.

The short way rests on Tens and ones: a two-digit number is some tens and some ones. 43 is 4 tens and 3 ones. 27 is 2 tens and 7 ones. When we add, we put the tens with the tens and the ones with the ones.

In 43 + 5, the 5 is 5 ones. It joins the 3 ones, making 8 ones. The 4 tens stay as they were. So 43 + 5 = 48.
In 27 + 40, the 40 is 4 tens. It joins the 2 tens, making 6 tens. The 7 ones stay as they were. So 27 + 40 = 67.

That is the whole idea, and this lesson spends four days making it solid. It covers two kinds of sum:

a two-digit number and a one-digit number, such as 43 + 5, where the ones add to the ones;
a two-digit number and a whole number of tens (10, 20, 30 and so on), such as 27 + 40, where the tens add to the tens.

In every sum here the ones stay at 9 or fewer, so no new ten is made. A sum such as 47 + 6, where the ones run past nine, is a different move, named under Boundaries.

The pictures are the ones the child already knows: bundles of ten sticks (or linking cubes snapped into trains of ten) with some loose ones, base-ten drawings of the same thing, a hundred chart, and an open number line.

The procedure

The four days build one idea. Day 1 adds ones to ones. Day 2 adds tens to tens. Day 3 shows both moves on the hundred chart and the open number line. Day 4 puts them side by side and asks, each time, which part of the number the new amount joins.

Day 1: ones join ones

Build the first number. Ask the child to show 43 with bundles of ten sticks: 4 bundles and 3 loose sticks. A bundle is ten sticks held by a rubber band, so it can be counted as one ten without opening it. This is the picture from Tens and ones, and it is worth building with real sticks before drawing anything.

Bring the new ones. Adding 5 means bringing 5 loose sticks. Where do they go? They are loose, so they go with the loose sticks. Now there are 3 + 5 = 8 loose sticks. Nothing happened to the bundles: there are still 4 of them. So the total is 4 tens and 8 ones, which is forty-eight.

43 + 5 = 48

The fact the child used, 3 + 5 = 8, is one they already know. That is the point of this day: a sum with a two-digit number in it is often just a small fact they know, done in the ones place, with the tens carried along unchanged.

Draw it. Once the sticks are familiar, a drawing does the same job. A tall thin rod of ten small squares stands for a bundle, and one small square for a loose stick. Figure 1 shows 43 + 5 this way, with the added ones shaded.

Figure 143 + 5. The 5 new ones (shaded) join the 3 ones. The 4 tens are not touched.

Read the digits. In 43 the 4 counts tens and the 3 counts ones. Adding 5 ones changes the ones digit from 3 to 8 and leaves the tens digit at 4. The child can see this in the numbers themselves: 43, 48, same first digit.

The small number may come first. In 4 + 61 the one-digit number is written first. The order does not change the total, because the same sticks are being put together either way (the commutative property of addition). So 4 + 61 can be found as 61 + 4: the 4 ones join the 1 one. Example 2 does this.

Day 2: tens join tens

A whole number of tens. The numbers 10, 20, 30, \ldots, 90 are tens with no loose ones: 40 is 4 bundles and nothing else. The child met counting by tens in Counting by tens, and adding a single ten in Ten more and ten less. Today the new amount is several tens at once.

Bring the new tens. Show 27 as 2 bundles and 7 loose sticks. Adding 40 means bringing 4 more bundles. Bundles go with bundles. Now there are 2 + 4 = 6 bundles, and the 7 loose sticks were never touched. So the total is 6 tens and 7 ones:

27 + 40 = 67

Figure 227 + 40. The 4 new tens (shaded) join the 2 tens. The 7 ones are not touched.

Once again the child used a small fact, 2 + 4 = 6, but this time it counts bundles, so it means 20 + 40 = 60. The words help: “two tens and four tens is six tens”, just as “two apples and four apples is six apples”. Ten is the thing being counted.

Why a ten only joins the tens. The new amount, 40, has 4 tens and 0 ones. It has no loose sticks to give, so the loose sticks of 27 have nothing to join them, and the ones digit stays 7. All the change happens in the tens digit, which goes from 2 to 6. Putting a ten with the ones would make no sense: a bundle is ten sticks, and it is counted with the other bundles.

The tens number may come first. 50 + 18 is the same as 18 + 50, by the same order-does-not-matter reason as on Day 1. A teen number such as 18 is 1 ten and 8 ones, so it works exactly like any other two-digit number: 1 ten and 5 tens make 6 tens, and the 8 ones stay. Example 4 does this.

Day 3: the hundred chart and the open number line

On the hundred chart, ones go right. Each row of a hundred chart holds ten numbers in order, so one step to the right is one more. Adding 3 to 62 is three steps right: 63, 64, 65. The tens digit stays 6 the whole way along the row.

Tens go down. One step straight down skips a whole row of ten, so it is ten more (the move from Ten more and ten less). Adding 30 is three steps down. From 62 that is 72, 82, 92. Every square in that column ends in 2: the ones digit never changes as you go down, and the tens digit goes up by one for each row.

Figure 3From 62, three rows down is 92. Each row down is one more ten.

So the chart turns the two moves into two directions. Ones move along a row; tens move down a column. A child who walks 62 + 30 on the chart and then writes 6 tens and 3 tens as 9 tens has checked the same answer two ways.

A missing number on the chart. The chart also answers questions with a box. For 25 + \square = 75, put a finger on 25 and find 75 below it: it is five rows down. Five rows is five tens, so \square = 50. The ones digit was 5 both times, which is a sign that only tens were added.

The open number line. An open number line is a line with no scale marked on it. The child marks only the numbers they use and draws a hop for each step. To add 30 to 36, start at 36 and make three hops of ten: 46, 56, 66 (Figure 4, Example 6). A child who is sure of the tens can make one long hop labeled +30 instead. To add 3 to 36, make three small hops of one, or one hop of 3, to 39. The line is drawn freehand, so the hops need not be to scale; a long hop for tens and a short hop for ones is enough. Adding on a line like this grows into Adding on an open number line.

Day 4: which part does it join?

By now the child has two moves. Day 4 is about choosing between them, and that choice is the heart of place value in addition.

Ask what the new amount is made of. Before adding, say what the number being added is: some ones, or some tens. 5 is five ones. 50 is five tens. They sound alike and use the same digit, but they join different parts:

24 + 5: five ones join four ones. The total is 2 tens and 9 ones, 29.
24 + 50: five tens join two tens. The total is 7 tens and 4 ones, 74.

Example 7 looks at this pair closely.

A check with tens and ones. After adding, look at the digits of the answer. When ones were added, the tens digit should be the same as before. When tens were added, the ones digit should be the same as before. If the wrong digit changed, the new amount joined the wrong part.

A look ahead: both tens and ones. Later the child will add two numbers that both have tens and ones, such as 34 + 25. The same rule does all the work: tens with tens, ones with ones. Example 8 shows one, marked as a look ahead; the full lesson is The partial sums method.

Worked examples

Read each problem aloud. Bundles of sticks or a hundred chart on the table let the child build or walk every one. Once they can say the answer by thinking “tens with tens, ones with ones”, they are adding by place value in their head.

Example 1

Day 1: Five more children get on the bus

Urho rides the school bus. When it stops at his corner, 43 children are already on the bus. Urho and 4 friends get on, so 5 more children are on board. How many children are on the bus now?

Solution

The bus started with 43 children and 5 more got on, so we add: 43 + 5.

43 is 4 tens and 3 ones. The 5 children who got on are 5 ones. Ones join ones: 3 + 5 = 8. The 4 tens are not touched. So there are 4 tens and 8 ones:

43 + 5 = 48

There are 48 children on the bus. Figure 1 shows the same sum with rods and loose squares.

Check by counting on from 43: forty-four, forty-five, forty-six, forty-seven, forty-eight. Five counts, and the count ends at 48.

Example 2

Day 1: Balloons on the float

Senni ties 4 balloons to the front of a parade float and 61 balloons along its sides. How many balloons are on the float?

Solution

All the balloons are put together, so we add: 4 + 61.

The small number comes first here. The order of adding does not change the total (the commutative property), so 4 + 61 is the same as 61 + 4. That is easier to think about: 61 is 6 tens and 1 one. The 4 balloons are 4 ones, and ones join ones: 1 + 4 = 5. The 6 tens stay.

4 + 61 = 65

There are 65 balloons on the float. Counting on from the bigger number, sixty-one, then sixty-two, sixty-three, sixty-four, sixty-five, agrees. Counting on 61 times from 4 would also reach 65, but it would take a long time and a slip would be easy.

Example 3

Day 2: Four bundles of flowers

Nooa has made 27 paper flowers for the parade float. Senni brings 4 bundles of 10 flowers each. How many flowers are there now?

Solution

Four bundles of ten is 40 flowers, so we add: 27 + 40.

27 is 2 tens and 7 ones. The 40 new flowers are 4 tens and no ones. Tens join tens: 2 tens and 4 tens make 6 tens. The 7 ones stay as they were. So there are 6 tens and 7 ones:

27 + 40 = 67

There are 67 flowers. Figure 2 shows the 4 shaded rods joining the 2 plain ones.

Check by counting on by tens from 27: thirty-seven, forty-seven, fifty-seven, sixty-seven. Four counts of ten, one for each bundle, and the ones digit is 7 every time.

Example 4

Day 2: Two loops of the bus

On its first loop the bus brings 18 children to school. On its second loop it brings 50 more. How many children did the bus bring in all?

Solution

The two groups are put together: 18 + 50.

18 is 1 ten and 8 ones. 50 is 5 tens and no ones. Tens join tens: 1 ten and 5 tens make 6 tens. The 8 ones have nothing to join them, so they stay 8:

18 + 50 = 68

The bus brought 68 children. The tens number could just as well have come first: 50 + 18 = 68 too, because the order of adding does not change the total.

A teen number can look different from other two-digit numbers, because “eighteen” does not say “one ten” out loud. But it is 1 ten and 8 ones like any other, and it adds the same way.

Example 5

Day 3: Down the hundred chart

Use the hundred chart to find 62 + 30. Then use it to find 62 + 3.

Solution

Adding 30 adds 3 tens. Each step straight down the chart is one more ten, so start at 62 and go down three rows: 72, 82, 92 (Figure 3).

62 + 30 = 92

The tens way agrees: 6 tens and 3 tens make 9 tens, and the 2 ones stay, so 92.

Adding 3 adds 3 ones. Each step to the right along a row is one more, so start at 62 and go three squares right: 63, 64, 65.

62 + 3 = 65

Two sums with the same digit, two different directions. Going down, the ones digit stays 2 and the tens digit climbs. Going right, the tens digit stays 6 and the ones digit climbs.

Example 6

Day 3: The float on an open number line

The parade float is pulled 36 meters along the street, stops, then is pulled 30 meters more. Show on an open number line how far it has gone.

Solution

Both distances are along the same street, so we add: 36 + 30. On an open number line, mark 36. The 30 is 3 tens, so make three hops of ten. Each hop adds one ten and leaves the 6 ones:

36 \to 46 \to 56 \to 66

Figure 436 + 30 on an open number line, as three hops of ten.

The float has gone 66 meters, so 36 + 30 = 66.

A child sure of the tens can make one long hop of 30 straight from 36 to 66. If the float had been pulled only 3 meters more, the hop would be short, from 36 to 39, because 3 is three ones and only the ones change: 36 + 3 = 39.

Example 7

Day 4: Five ones or five tens

Find 24 + 5 and 24 + 50. Both sums add a 5. Why are the answers so far apart?

Solution

24 is 2 tens and 4 ones.

In 24 + 5, the 5 is 5 ones. Ones join ones: 4 + 5 = 9. The tens stay at 2:

24 + 5 = 29

In 24 + 50, the 50 is 5 tens. Tens join tens: 2 + 5 = 7 tens. The ones stay at 4:

24 + 50 = 74

The same fact, 5 more, is used both times, but it counts different things. Five ones is five loose sticks; five tens is five bundles, which is fifty sticks. Fifty is 45 more than five, since 50 - 5 = 45, so 24 + 50 is 45 more than 24 + 5.

Look at the digits of each answer. In 29 the tens digit is still 2, so only ones were added. In 74 the ones digit is still 4, so only tens were added. That is the check from Day 4, and it catches a 5 that has been put in the wrong place.

Example 8

Day 4, looking ahead: Children around the float

This one looks ahead to a later lesson. 34 children walk in front of the parade float and 25 children walk behind it. How many children walk with the float?

Solution

All the children are counted together: 34 + 25. Now both numbers have tens and ones, but the rule is the same.

34 is 3 tens and 4 ones; 25 is 2 tens and 5 ones. Tens join tens: 3 + 2 = 5 tens, which is 50. Ones join ones: 4 + 5 = 9. So there are 5 tens and 9 ones:

34 + 25 = 59

There are 59 children. This is the two moves of this lesson made one after the other: first add the 2 tens, 34 + 20 = 54, then add the 5 ones, 54 + 5 = 59. On the hundred chart, that is two rows down and five squares right.

Boundaries

The ones must stay at nine or fewer. Every sum here has ones that add to at most 9, as in 3 + 5 = 8. In 47 + 6 the ones add to 7 + 6 = 13, which is a new ten and 3 ones, so the tens digit changes too and the answer is 53. That is the job of Adding a one-digit number to a two-digit number by making a ten.
The total stays at 100 or below. 62 + 30 makes 9 tens. 62 + 40 would make 10 tens and 2 ones, which is 102, a number with three digits and past the end of the hundred chart. Ten tens with no ones, as in 60 + 40 = 100, is the largest total in this lesson.
The number added is ones only or tens only. A one-digit number adds to the ones; a whole number of tens adds to the tens. A number with both, as in 34 + 25, is the look ahead in Example 8.
A ten never joins the ones. 40 has 0 ones, so adding it leaves the ones digit alone. In 4 + 40 the 4 tens and the 4 ones are different things and stay apart: the total is 4 tens and 4 ones, 44.
Order does not matter. 4 + 61 = 61 + 4 and 50 + 18 = 18 + 50 because the order of adding two numbers never changes the total.
Down the chart is ten more only inside the chart. On a hundred chart the square below is ten more for every number from 1 to 90. Below the last row there are no more squares.

Where this goes next

Adding tens to tens and ones to ones is the first time a child adds by place value, and every written way of adding larger numbers rests on it. The case where the ones run past nine, named under Boundaries, is the next step. Taking tens away works the same way backwards, as in Subtracting multiples of ten. Adding a number with both tens and ones, as in Example 8, is The partial sums method, and the hops of Example 6 grow into Adding on an open number line. When the ones make a new ten in bigger sums, the move is Regrouping in addition, and adding column by column with that regrouping built in is The standard addition algorithm.

Exercises

Day 1: ones join ones

52 + 4 = \square
71 + 8 = \square
3 + 85 = \square
30 + 6 = \square
There are 32 children on the school bus. At the next stop 6 more get on. How many children are on the bus now?

Figure 5Base-ten pictures for Exercises 6 and 11. The shaded pieces are the ones being added.

Picture A shows a sum. The plain pieces are the first number and the shaded pieces are being added. Write the sum and find the total.

Day 2: tens join tens

45 + 30 = \square
16 + 70 = \square
20 + 59 = \square
40 + 50 = \square
Picture B shows a sum. The plain pieces are the first number and the shaded pieces are being added. Write the sum and find the total.
Nooa has 33 ribbons for the parade float. Senni brings 6 bags with 10 ribbons in each. How many ribbons are there now?

Day 3: the hundred chart and the open number line

On a hundred chart, start at 47 and move straight down 4 rows. Where do you land? Write the sum you found.
On a hundred chart, start at 21 and move 6 squares to the right. Where do you land? Write the sum you found.
Show 58 + 30 on an open number line with hops of ten. Which numbers do you land on?
53 + \square = 93
64 + \square = 69

Day 4: which part does it join?

Find 26 + 3 and 26 + 30.
Find 8 + 41 and 80 + 11.
The bus passes 46 houses on its route before it reaches the school, and 40 more houses after it. How many houses does it pass in all?
\square + 6 = 76

Explain

When you add 30 to 64, why does the 4 stay the same?
Is 52 + 3 the same as 52 + 30? Find both and say why they are different.
Which of these sums make a new ten from the ones: 43 + 6, 47 + 6, 43 + 50? Find all three.
On a hundred chart, why does adding 30 move you straight down three rows?

Answers

56. 52 is 5 tens and 2 ones; ones join ones, 2 + 4 = 6, so 52 + 4 = 56.
79. 1 + 8 = 9 ones and the 7 tens stay: 71 + 8 = 79.
88. Swap the order: 3 + 85 is the same as 85 + 3. Ones join ones, 5 + 3 = 8, and the 8 tens stay: 3 + 85 = 88.
36. 30 is 3 tens and no ones; the 6 ones join the 0 ones: 30 + 6 = 36.
38 children. 32 is 3 tens and 2 ones; 2 + 6 = 8 ones: 32 + 6 = 38.
25 + 3 = 28. The plain pieces are 2 rods and 5 squares, which is 25; the 3 shaded squares are 3 ones. Ones join ones, 5 + 3 = 8, and the 2 tens stay.
75. 4 tens and 3 tens make 7 tens; the 5 ones stay: 45 + 30 = 75.
86. 16 is 1 ten and 6 ones; 1 ten and 7 tens make 8 tens: 16 + 70 = 86.
79. 2 tens and 5 tens make 7 tens; the 9 ones stay: 20 + 59 = 79.
90. 4 tens and 5 tens make 9 tens, with no ones at all: 40 + 50 = 90.
14 + 50 = 64. The plain pieces are 1 rod and 4 squares, which is 14; the 5 shaded rods are 50. Tens join tens, 1 + 5 = 6 tens, and the 4 ones stay.
93 ribbons. Six bags of ten is 60. 3 tens and 6 tens make 9 tens, and the 3 ones stay: 33 + 60 = 93.
87, so 47 + 40 = 87. Each row down is ten more: 57, 67, 77, 87.
27, so 21 + 6 = 27. Each square right is one more: 22, 23, 24, 25, 26, 27.
68, 78 and 88. Three hops of ten from 58, each keeping the 8 ones: 58 + 30 = 88.
40. The ones digit is 3 in both numbers, so only tens were added. 5 tens grew to 9 tens, which is 4 more tens: 53 + 40 = 93.
5. The tens digit is 6 in both numbers, so only ones were added. 4 ones grew to 9, which is 5 more: 64 + 5 = 69.
29 and 56. In 26 + 3 the ones join: 6 + 3 = 9, so 26 + 3 = 29. In 26 + 30 the tens join: 2 + 3 = 5 tens, so 26 + 30 = 56.
49 and 91. 8 + 41 is 41 + 8: 1 + 8 = 9 ones, so 8 + 41 = 49. 80 + 11: 8 tens and 1 ten make 9 tens, and the 1 one stays, so 80 + 11 = 91.
86 houses. 4 tens and 4 tens make 8 tens; the 6 ones stay: 46 + 40 = 86.
70. The ones digit is 6 on both sides, and the 6 being added accounts for all of it, so the box has no ones. The 7 tens of 76 all come from the box: 70 + 6 = 76.
64 is 6 tens and 4 ones. 30 is 3 tens and no ones. The 30 brings only tens, which join the 6 tens to make 9 tens. It brings no ones, so the 4 ones are not touched: 64 + 30 = 94.
No. 52 + 3 = 55 and 52 + 30 = 82. The 3 in 52 + 3 is 3 ones, so it joins the 2 ones. The 30 in 52 + 30 is 3 tens, so it joins the 5 tens. Three tens is much more than three ones.
Only 47 + 6. Its ones add to 7 + 6 = 13, which is a new ten and 3 ones, so 47 + 6 = 53. In 43 + 6 the ones add to 3 + 6 = 9, so 43 + 6 = 49. In 43 + 50 only tens are added, so 43 + 50 = 93.
Each row of the chart holds ten numbers. A step straight down skips the rest of the row and the start of the next one, ten numbers in all, so each step down is ten more. Adding 30 is adding three tens, so it is three steps down, and the ones digit stays the same the whole way.