International Baccalaureate · PYP 4

Addition

⌚ ~74 min ▢ 7 blocks ✓ 38 exercises ➣ Prerequisites : Numbers to 999 999, Addition, Comparing, Ordering and Rounding, Odd and Even Numbers

I Calculating in Your Head with Large Numbers

Discover The Clever Path

At the market, Hugo must pay $47 + 9$ dollars. No paper, no column addition: it would be too slow. He thinks: $47 + 10 = 57$, and $9$ is one less than $10$, so $57 - 1 = 56$.
Mental calculation is the art of finding the clever path: using whole tens, doubles and simple decompositions instead of writing the calculation down. The gestures learned before now work on big numbers.

Method — Adding or Subtracting a Near-Ten

A near-ten is a number close to a whole ten: $8$, $9$, $18$, $19$, $28$, $29$, $38$, $39$. We go through the whole ten, then we correct.

Adding 9, 19, 29, 39: add $10$, $20$, $30$, $40$, then take $1$ away. $456 + 29 = 486 - 1 = 485$.
Adding 8, 18, 28, 38: add $10$, $20$, $30$, $40$, then take $2$ away. $456 + 38 = 496 - 2 = 494$.
Subtracting 9, 19, 29, 39: take $10$, $20$, $30$, $40$ away, then add $1$ back. $456 - 29 = 426 + 1 = 427$.
Subtracting 8, 18, 28, 38: take $10$, $20$, $30$, $40$ away, then add $2$ back. $456 - 38 = 416 + 2 = 418$.

This is a choice: it suits numbers close to a ten. For other numbers, we choose another way: split the number ($456 + 23 = 456 + 20 + 3$), or set the operation out in columns (next section).

Example

Compute mentally: (a) $24\,536 + 29$, (b) $18\,342 - 19$, (c) $39\,985 + 28$, (d) $70\,312 - 39$.

Answer

Only the near-ten is replaced by its whole ten; the large number is not changed. The gesture is the same as with small numbers.

(a) $29$ is $30$ minus $1$: $24\,536 + 30 = 24\,566$, then $24\,566 - 1 = \mathbf{24\,565}$.
(b) $19$ is $20$ minus $1$: $18\,342 - 20 = 18\,322$, then $18\,322 + 1 = \mathbf{18\,323}$ (one too many was taken away, it is given back).
(c) $28$ is $30$ minus $2$: $39\,985 + 30 = 40\,015$, then $40\,015 - 2 = \mathbf{40\,013}$. Adding $30$ goes past $40\,000$: every digit changes, except the ones digit.
(d) $39$ is $40$ minus $1$: $70\,312 - 40 = 70\,272$, then $70\,272 + 1 = \mathbf{70\,273}$.

Skills to practice

Adding and Subtracting Near-Tens

II Adding in Columns up to 100 000

Discover

On Saturday the stadium welcomed $4\,738$ spectators, on Sunday $2\,589$. The manager wants the total for the weekend. There are far too many people to count with cubes, and the numbers are bigger than $1\,000$. Say how to find the total.

Answer

The column addition works exactly as with three digits: one more column on the left, the thousands. Ones: $8 + 9 = 17$, write $7$, carry $1$ ten. Tens: $1 + 3 + 8 = 12$, write $2$, carry $1$ hundred. Hundreds: $1 + 7 + 5 = 13$, write $3$, carry $1$ thousand. Thousands: $1 + 4 + 2 = 7$.

$4\,738 + 2\,589 = 7\,327$: the stadium welcomed $7\,327$ spectators.

Example

Compute $68\,457 + 47\,896$ in columns.

Answer

Five-digit numbers are added like smaller ones: ones under ones, from right to left, with a carry to the next column whenever a column makes $10$ or more.

Ones: $7 + 6 = 13$. Write $3$ and carry $1$ ten.
Tens: $1 + 5 + 9 = 15$. Write $5$ and carry $1$ hundred.
Hundreds: $1 + 4 + 8 = 13$. Write $3$ and carry $1$ thousand.
Thousands: $1 + 8 + 7 = 16$. Write $6$ and carry $1$ ten thousand.
Ten thousands: $1 + 6 + 4 = 11$. Write $1$ and carry $1$ hundred thousand. There is no hundred thousands column yet: the carry opens a new column on the left.

So $68\,457 + 47\,896 = 116\,353$. The result has six digits: two numbers smaller than $100\,000$ can have a sum larger than $100\,000$.

Skills to practice

Solving Real-World Problems
Adding Numbers up to 100 000

III Estimating a Sum or a Difference

Example — Estimating a Sum by Rounding

A library lends $386$ books in March and $213$ books in April. Estimate how many books it lends in the two months, by rounding each number to the nearest hundred.

Answer

The tens digit of $386$ is $8$, so $386$ rounded to the nearest hundred is $400$. The tens digit of $213$ is $1$, so $213$ rounded to the nearest hundred is $200$. $$\begin{aligned}[t] 386 + 213 &\approx 400 + 200 && \text{(each number rounded to the nearest hundred)}\\ &= 600 \end{aligned}$$ The sign $\approx$ is read “is about”: $386 + 213 \approx 600$ is read “$386 + 213$ is about $600$”. The library lends about $600$ books in the two months.
The estimate is quick, and it is close: the exact sum is $386 + 213 = 599$.

Example — Estimating a Difference

A school library owns $6\,482$ books. $2\,715$ of them are lent out. Estimate how many books are left on the shelves, by rounding each number to the nearest hundred.

Answer

The tens digit of $6\,482$ is $8$, so $6\,482$ rounded to the nearest hundred is $6\,500$. The tens digit of $2\,715$ is $1$, so $2\,715$ rounded to the nearest hundred is $2\,700$. $$\begin{aligned}[t] 6\,482 - 2\,715 &\approx 6\,500 - 2\,700 && \text{(each number rounded to the nearest hundred)}\\ &= 3\,800 \end{aligned}$$ The sign $\approx$ is read “is about”: $6\,482 - 2\,715 \approx 3\,800$ is read “$6\,482 - 2\,715$ is about $3\,800$”. About $3\,800$ books are left on the shelves.
The estimate is quick, and it is close: the exact difference is $6\,482 - 2\,715 = 3\,767$.

Skills to practice

Estimating a Sum or a Difference
Estimating Sums and Differences by Rounding

IV Odd and Even Numbers

Definition — Even and Odd Numbers

An even number is a number of whole items that can be split into two equal groups: every item has a partner and nothing is left over.
Examples: $0, 2, 4, 6, 8, 10, 12, \dots$
$0$ is even too: with $0$ objects, the two groups are empty and nothing is left over.
An odd number is a number of whole items that cannot be split into two equal groups: one item is always left without a partner.
Examples: $1, 3, 5, 7, 9, 11, 13, \dots$

Skills to practice

Investigating Odd and Even Sums and Products

V Going further

Skills to practice

Reasoning about Sums
Quiz