Numbers in words, rounding and regrouping
In this topic you will
- Write two-digit numbers correctly in words.
- Round two-digit numbers to the nearest $10$.
- Regroup two-digit numbers in different ways using place value.
Key Words
- closest 10
- hyphen
- nearest 10
- regroup
- round, rounding
Show Definitions
- closest 10: The multiple of $10$ that a number is nearest to (for example, $47$ is closest to $50$).
- hyphen: A punctuation mark ( $-$ ) used when writing some two-digit numbers in words, such as “twenty-one”.
- nearest 10: The ten (like $20, 30, 40$) that a number rounds to based on which ten it is closer to.
- regroup: To rewrite a number in a different way using place value, such as splitting it into tens and ones (for example, $34 = 30 + 4$).
- round / rounding: Changing a number to a nearby, easier number (like the nearest $10$) while keeping its value close to the original.
Reading number words
Sometimes numbers are written in words. You need to read and understand numbers and number words.
EXERCISES
Exercise $1.9$
$1$. Write each of the numbers represented in words.
a.
b.
c. $93$
d. $31$
👀 Show answer
a. $84$ is eighty-four.
b. $57$ is fifty-seven.
c. $93$ is ninety-three.
d. $31$ is thirty-one.
$2$. Read the number words and write the number.
a. forty-two
b. seventy-six
c. twenty-five
d. sixty-eight
👀 Show answer
a. forty-two = $42$
b. seventy-six = $76$
c. twenty-five = $25$
d. sixty-eight = $68$
$3$. Use these number words to write some $2$-digit numbers in words. How many different numbers can you write?
twenty seventy forty eight nine
👀 Show answer
You can make these $2$-digit numbers:
twenty ($20$), forty ($40$), seventy ($70$), twenty-eight ($28$), twenty-nine ($29$), forty-eight ($48$), forty-nine ($49$), seventy-eight ($78$), seventy-nine ($79$).
That is $9$ different numbers.
Think like a Mathematician
Investigation: Marcus says there is only one 2-digit number word with 12 letters, seventy-seven. Is Marcus correct?
Equipment: Paper and pen (optional: a list of the number words from 10 to 99)
Method:
- Write the number words for the two-digit numbers (10–99).
- For each number word, count the letters ignoring spaces and hyphens (for example, “seventy-seven” becomes “seventyseven”).
- Record any number words that have exactly 12 letters.
- Decide whether “seventy-seven” is the only one, or whether there are others.
Follow-up Questions:
👀 show answer
- 1: “Seventy” has 7 letters and “seven” has 5 letters, so “seventy-seven” has 7 + 5 = 12 letters (ignoring the hyphen).
- 2: Examples include seventy-three (7 + 5 = 12) and seventy-eight (7 + 5 = 12), so there is more than one.
- 3:No. Marcus is not correct because there are other two-digit number words with 12 letters (for example, seventy-three and seventy-eight) when you ignore spaces and hyphens.
EXERCISES
$4$. Round each number to the nearest $10$. Use the number line to help you.
a. $64$
b. $72$
c. $81$
d. $56$
e. $49$
f. $23$
g. $27$
h. $35$
i. $30$
👀 Show answer
a.$64$ rounds to $60$.
b.$72$ rounds to $70$.
c.$81$ rounds to $80$.
d.$56$ rounds to $60$.
e.$49$ rounds to $50$.
f.$23$ rounds to $20$.
g.$27$ rounds to $30$.
h.$35$ rounds to $40$.
i.$30$ rounds to $30$.
$5$. A number rounded to the nearest $10$ is $60$.
What could the number be?
👀 Show answer
Any number from $55$ to $64$ rounds to $60$ (to the nearest $10$).
For example: $58$, $60$, or $63$.
$6$. Find $4$ different ways to regroup $31$. Draw or write your answers.
Compare the ways you regrouped $31$ with a partner.
What did you do the same? What did you do differently?
👀 Show answer
Four regroupings of $31$:
$31 = 3$ tens $+ 1$ one.
$31 = 2$ tens $+ 11$ ones.
$31 = 1$ ten $+ 21$ ones.
$31 = 0$ tens $+ 31$ ones.
Compare: You might both have used tens and ones, but you may have traded different numbers of tens for ones (for example, one person might write $2$ tens and $11$ ones, while another writes $1$ ten and $21$ ones).
Think like a Mathematician
Investigation: Is Arun correct? If he is correct, will Arun’s method always tell you how many different ways there are to regroup a number?
Arun says:$21$ is $2$ tens and $1$ one. $2 + 1 = 3$ so there will only be $3$ ways to regroup $21$ using tens and ones.
Method:
- Write $21$ as tens and ones.
- Regroup by trading $1$ ten for $10$ ones, and record the new tens-and-ones form.
- Keep regrouping until there are no tens left.
- Count how many different tens-and-ones forms you found.
- Decide whether Arun’s method ($2 + 1$) always matches the true number of regroupings.
Follow-up Questions:
👀 show answer
1. The regroupings of $21$ using tens and ones are:
- $2$ tens and $1$ one ($21$)
- $1$ ten and $11$ ones ($10 + 11 = 21$)
- $0$ tens and $21$ ones ($21$)
2. There are $3$ different ways.
3.Yes, Arun is correct for $21$. The number really is $3$.
4.No, Arun’s method does not always work. For any number that is written as $T$ tens and $O$ ones, you can regroup by trading tens for ones, which gives exactly $T + 1$ different tens-and-ones forms (from $T$ tens down to $0$ tens). So the number of ways depends on the number of tens, not on $T + O$. For example, $31$ is $3$ tens and $1$ one: Arun’s method would give $3 + 1 = 4$, but the true number of regroupings is $3 + 1 = 4$ ways because there are $3$ tens (not because $3 + 1$). If the ones digit changes, the number of regroupings stays the same as long as the tens digit stays the same.