Adding up to ten

You add every day, even before you know the word. Two sweets in one hand and three in the other: how many sweets alto­gether? Four chil­dren at the table and one more comes to sit: how many chil­dren now? Each time you are putting two groups together and count­ing the whole lot.

This les­son is the first step in all the adding you will ever do. The num­bers are small, so you can use your fin­gers, beads, but­tons or a num­ber line to check every answer. Take it slowly. Say each sum out loud. The facts you learn here will be used again in every les­son that fol­lows.

Adding means putting things together and count­ing how many there are alto­gether. In this les­son every answer stays at ten or below, so you can always check it on your fin­gers.

The num­ber lad­der

Before adding any­thing, say the num­bers in order. Going up is called ascend­ing order:

$0, \; 1, \; 2, \; 3, \; 4, \; 5, \; 6, \; 7, \; 8, \; 9, \; 10$

Going down is called descend­ing order:

$10, \; 9, \; 8, \; 7, \; 6, \; 5, \; 4, \; 3, \; 2, \; 1, \; 0$

Think of it as a lad­der of num­bers. Adding is climb­ing up it, and tak­ing away is climb­ing down. To work out $4 + 3$, stand on 4 and take three steps up: 5, 6, 7. So $4 + 3 = 7$.

The pic­ture shows the three steps. Count the hops, not the marks: you start on 4, so the first hop takes you to 5. Here are two more to try on the line.

Exam­ple: 2 + 5

Stand on 2. Take five steps up: 3, 4, 5, 6, 7. You land on 7.

$$2 + 5 = 7$$

Exam­ple: 5 + 2, the quicker way

Now stand on the big­ger num­ber, 5, and take only two steps: 6, 7. You land on 7 again, and it took fewer steps.

$$5 + 2 = 7$$

Adding zero

Zero means noth­ing at all. If you have four sweets and some­body gives you noth­ing, you still have four sweets.

Remem­ber. When you add zero to a num­ber, the answer is that same num­ber.

On the lad­der, adding zero means tak­ing no steps at all. You stay exactly where you are.

Adding one

Adding one is tak­ing a sin­gle step up the lad­der, so the answer is sim­ply the next num­ber you say when you count.

Remem­ber. Adding one gives the num­ber that comes next in count­ing.

Exam­ple: apples in a bas­ket

A bas­ket has 6 apples. Amma puts in 1 more apple. Count on from 6 by one: 7. There are 7 apples in the bas­ket.

$$6 + 1 = 7$$

Adding two

Adding two is two steps up. Stand on 6, step to 7, step to 8 — so $6 + 2 = 8$.

Adding two also skips a num­ber. From 3, adding two jumps over 4 and lands on 5. From 6 it jumps over 7 and lands on 8. That is a quick way to check your­self.

Adding three

Three steps up the lad­der this time.

Exam­ple: beads on a string

Riya has 5 red beads on her string. She adds 3 blue beads. Point at the red ones and say "five", then touch each blue bead and count on: 6, 7, 8.

$$5 + 3 = 8$$

Dou­bles

A dou­ble is a num­ber added to itself. Dou­bles are worth learn­ing by heart, because once you know them a great many other sums become easy.

You can see dou­bles on your own body. Two hands, each with 5 fin­gers, make $5 + 5 = 10$. A spi­der has 4 legs on each side, so $4 + 4 = 8$ legs alto­gether.

Dou­bles also help with the sums next door to them. If you know $4 + 4 = 8$, then $4 + 5$ is just one more, so it is $9$.

A sum that is one more than a dou­ble is called a near dou­ble. Here is the think­ing writ­ten out for $3 + 4$:

$$\begin{aligned}3 + 4 &= 3 + 3 + 1 \\ &= 6 + 1 \\ &= 7\end{aligned}$$

The pairs that make ten

These pairs are called the tens com­ple­ments, and they mat­ter more than any­thing else in this les­son. Each num­ber below has one part­ner that fin­ishes the jour­ney to ten.

Look at each row of the pic­ture. The blue coun­ters and the orange coun­ters always fill all ten places. If you know how many are blue, the empty places tell you how many more you need to reach ten. Hold up ten fin­gers, fold down 7, and the 3 still stand­ing are the part­ner of 7.

Notice the third col­umn. It does not mat­ter which way round you add: $9 + 1$ and $1 + 9$ both give ten. That is true of every addi­tion, and it means you may always start with the big­ger num­ber and count on from there.

Remem­ber. Learn the five pairs — 9 and 1, 8 and 2, 7 and 3, 6 and 4, 5 and 5. Every later les­son leans on them.

Your turn

Fill in the miss­ing num­ber

1. $5 + \underline{\phantom{0}} = 9$
2. $9 + \underline{\phantom{0}} = 9$
3. $0 + \underline{\phantom{0}} = 3$
4. $7 + \underline{\phantom{0}} = 8$
5. $4 + \underline{\phantom{0}} = 7$
6. $\underline{\phantom{0}} + 2 = 4$
7. $\underline{\phantom{0}} + 4 = 5$
8. $\underline{\phantom{0}} + 6 = 9$
9. $\underline{\phantom{0}} + 0 = 8$
10. $\underline{\phantom{0}} + 7 = 8$

Use dou­bling to help you

1. $4 + 4$
2. $4 + 5$
3. $3 + 3$
4. $3 + 4$
5. $2 + 2$
6. $2 + 3$

Com­mon mis­takes

• Count­ing the start­ing num­ber as a step. For $4 + 3$ some chil­dren say "4, 5, 6" and answer 6. Start on 4 and say only the new num­bers: 5, 6, 7.
• Think­ing adding zero makes zero. $7 + 0$ is 7, not 0. Noth­ing was added, so noth­ing changed.
• Count­ing on from the smaller num­ber. It is not wrong, but it is slow and easy to lose count. For $2 + 6$, start on 6 and count two more.
• Mix­ing up the part­ners of ten. 6 goes with 4, not with 3. Check with ten fin­gers or a ten-frame.
• For­get­ting the miss­ing num­ber can be at the front. In $\underline{\phantom{0}} + 2 = 4$ you need the num­ber that 2 is added to, which is 2.

Key terms

Add

Put two groups together and find how many there are alto­gether. The sign is $+$.

Ascend­ing order

Num­bers writ­ten going up: 0, 1, 2, 3 and so on.

Descend­ing order

Num­bers writ­ten going down: 10, 9, 8, 7 and so on.

Count on

Start on one num­ber and say the next num­bers, one for each thing you add.

Dou­ble

A num­ber added to itself, such as $3 + 3 = 6$.

Near dou­ble

A sum one more than a dou­ble, such as $3 + 4 = 7$.

Tens com­ple­ments

The pairs of num­bers that add up to ten: 9 and 1, 8 and 2, 7 and 3, 6 and 4, 5 and 5.

Answers

Fill in the miss­ing num­ber

1. $5 + 4 = 9$, so the miss­ing num­ber is 4.
2. $9 + 0 = 9$, so the miss­ing num­ber is 0.
3. $0 + 3 = 3$, so the miss­ing num­ber is 3.
4. $7 + 1 = 8$, so the miss­ing num­ber is 1.
5. $4 + 3 = 7$, so the miss­ing num­ber is 3.
6. $2 + 2 = 4$, so the miss­ing num­ber is 2.
7. $1 + 4 = 5$, so the miss­ing num­ber is 1.
8. $3 + 6 = 9$, so the miss­ing num­ber is 3.
9. $8 + 0 = 8$, so the miss­ing num­ber is 8.
10. $1 + 7 = 8$, so the miss­ing num­ber is 1.

Use dou­bling to help you

1. $4 + 4 = 8$
2. $4 + 5 = 9$, one more than $4 + 4$.
3. $3 + 3 = 6$
4. $3 + 4 = 7$, one more than $3 + 3$.
5. $2 + 2 = 4$
6. $2 + 3 = 5$, one more than $2 + 2$.