B. Number · Number Sense: Numbers to 10 000, Fractions and Decimal TenthsBuilding and Breaking Apart Numbers to 10 000
B1.1 A1About 35 minutes
Learning goals- I can compose (build) a number from different parts.
- I can decompose (break apart) a number in more than one way.
- I can trade 1 thousand for 10 hundreds, 1 hundred for 10 tens, and back.
- I can use money to show big numbers.
Words to know
compose
Put parts together to make a number.
decompose
Break a number into parts.
regroup (trade)
Change one unit for 10 of the next smaller unit, or 10 units for 1 of the next bigger unit.
flexible
Able to think about a number in more than one way.
Remember from Grade 3
In Grade 3 you broke apart numbers to 1000 in many ways. For example, 345 is 3 hundreds 4 tens 5 ones, but it is also 34 tens 5 ones, or 300 + 45. Now we'll do the same thing with numbers up to 10 000.
Let's start with a story
The school is raising $2400 for new library books. The principal gets the money in $100 bills. How many $100 bills is that?
$2400 is 2 thousands and 4 hundreds. Each thousand is 10 hundreds, so 2 thousands = 20 hundreds. 20 + 4 = 24 hundreds.
So the principal has 24 bills of $100. We just decomposed 2400 in a new way: 24 hundreds!
One number, many ways
Here are some ways to decompose 3450:
- 3 thousands, 4 hundreds, 5 tens
- 34 hundreds, 5 tens
- 345 tens
- 3000 + 450
- 2000 + 1450
- 3400 + 50
All of these make 3450. Breaking numbers apart in different ways helps you do mental math and solve problems.
Trading (regrouping)
You can always trade:
- 1 thousand ↔ 10 hundreds
- 1 hundred ↔ 10 tens
- 1 ten ↔ 10 ones
The number stays the same. Only the way you show it changes.
Worked exampleExample 1: Compose a number from mixed-up parts
What number is 4 thousands, 15 hundreds and 3 tens?
- 15 hundreds is too many for the hundreds place. Trade 10 hundreds for 1 thousand.
- Now we have 4 + 1 = 5 thousands and 5 hundreds left over, plus 3 tens.
- 5 thousands, 5 hundreds, 3 tens, 0 ones.
5530. You can check: 4000 + 1500 + 30 = 5530.
Worked exampleExample 2: Decompose in three ways
Show 6200 in three different ways.
- Way 1: 6 thousands and 2 hundreds (6000 + 200).
- Way 2: Trade all the thousands for hundreds: 6 thousands = 60 hundreds. 60 + 2 = 62 hundreds.
- Way 3: Trade one thousand: 5 thousands and 12 hundreds (5000 + 1200).
6000 + 200, 62 hundreds, and 5000 + 1200 all equal 6200.
Worked exampleExample 3: Use money
Show $1375 using $100 bills, $10 bills and $1 coins (loonies) in two ways.
- Way 1: 13 × $100, 7 × $10 and 5 loonies. 1300 + 70 + 5 = 1375.
- Way 2: trade one $100 bill for ten $10 bills: 12 × $100, 17 × $10 and 5 loonies. 1200 + 170 + 5 = 1375.
| Number | Thousands, hundreds, tens, ones | Only hundreds and ones | Only tens and ones |
|---|
| 2600 | 2 thousands, 6 hundreds | 26 hundreds | 260 tens |
| 4085 | 4 thousands, 8 tens, 5 ones | 40 hundreds, 85 ones | 408 tens, 5 ones |
| 7350 | 7 thousands, 3 hundreds, 5 tens | 73 hundreds, 50 ones | 735 tens |
Stick with it
If a question looks hard, try a different way to break the number apart. Good mathematicians are flexible and don't give up after the first try. Ask yourself: "Is there a friendlier way to split this number?"
Just for funRiddle: I am 24 hundreds. I am also 240 tens. I am also 2400 ones. Who am I?
Answer: 2400, the number with the most nicknames in the class!
Hands-on activityNumber building with paper money
You need: paper, scissors (ask a grown-up to help), a marker. Or use play money from a board game.
- Make paper money: 10 bills of $1000, 40 bills of $100, 20 bills of $10 and 10 coins of $1 (small paper circles). Write the value on each.
- Your grown-up names a number, like $3260.
- Build it with the fewest pieces you can. (3 × $1000, 2 × $100, 6 × $10.)
- Now build the same amount a different way, with no $1000 bills at all. (32 × $100 and 6 × $10.)
- Find one more way. Write each way as a number sentence, like 3000 + 200 + 60 = 3260.
Math talk: "Which way used the most pieces? Why would a bank use the fewest pieces?"
Your turn
1How many hundreds are in 4000?
40 hundreds. Each thousand is 10 hundreds, and 4 × 10 = 40.
2What number is 3 thousands and 14 hundreds?
4400. 14 hundreds = 1 thousand and 4 hundreds. 3 + 1 = 4 thousands, 4 hundreds.
5Write 2350 as hundreds and tens only.
6How many $10 bills make $1000?
100 $10 bills. 1000 = 100 tens.
7Show 6070 in two different ways.
Examples: 6 thousands and 7 tens (6000 + 70); 60 hundreds and 7 tens; 5000 + 1070; 607 tens.
More practice: warm-up
1Fill in: 3000 + 400 + 20 = □
2How many hundreds make 1000?
4What number is 20 hundreds?
More practice: on your own
1What number is 2 thousands, 11 hundreds, 5 tens?
3150. 11 hundreds = 1 thousand and 1 hundred, so 3 thousands, 1 hundred, 5 tens.
1300. 4300 − 3000 = 1300.
3How many tens are in 2500?
250 tens. 2500 = 25 hundreds, and each hundred is 10 tens: 25 × 10 = 250.
4Show $2040 with only $100 bills and $10 bills.
20 × $100 and 4 × $10. 2000 + 40 = 2040.
5Fill in: 8 thousands + □ hundreds = 9500
15 hundreds. 9500 − 8000 = 1500, which is 15 hundreds.
Challenge
1A bakery packs cookies in boxes of 100, bags of 10 and singles. It has 3462 cookies. Show two ways to pack all of them.
Example 1: 34 boxes, 6 bags, 2 singles (3400 + 60 + 2). Example 2: 33 boxes, 16 bags, 2 singles (3300 + 160 + 2). Other ways work if the parts add to 3462.
2Kai says 5 thousands, 0 hundreds, 23 tens is the same as 5023. Is he right? Explain.
No. 23 tens = 230, not 23. 5000 + 230 = 5230.
3Which is more: 48 hundreds or 4 thousands, 9 hundreds? Explain.
4 thousands, 9 hundreds (4900) is more. 48 hundreds = 4800. 4900 > 4800.
Quiz
1. How many hundreds are in 3000?
2. What number is 6 thousands and 12 hundreds?
4. Which is not equal to 2700?
5. How many $10 bills are in $350?
Big idea: Composing and decomposing numbers flexibly (B1.1) is what makes mental math and the standard algorithms make sense later. Knowing that 4300 is also 43 hundreds or 3000 + 1300 helps children subtract and estimate.
Common difficulties:
- Not seeing "hidden" hundreds: children often say 3000 has 0 hundreds. That's true for the hundreds digit, but there are 30 hundreds in all. Use money or drawings.
- Too many in one place (6 thousands 12 hundreds): children may write 6120. Practise trading 10 hundreds for 1 thousand.
- Mixing up tens and ones (23 tens = 23).
How to help: Use real or play money; ask "How many $100 bills would make $2400?" Celebrate when your child finds a new way to split a number. Strand A: encourage perseverance and flexible thinking ("Is there another way?").