Comparing and ordering whole numbers
Worku has 736 reading points and Tuya has 763. The two numbers use the same three digits. Which one is more, and how can you tell without counting every point?
Where you are 15 in Place value
When you reach for it
Worku, Tuya and Kawika are in a summer reading club. Every book read earns points, and the club keeps a running total for each reader. At recess the same three friends count their jumps with a jump rope and add them up over a week. Both games keep asking the same two questions. Who has more? And how do the totals line up, from least to greatest?
Counting on from the smaller number answers the first question, but it is slow when the numbers are 489 and 512. Place value is faster. A three-digit number is some hundreds, some tens and some ones (Reading and writing numbers to 1,000), and each column of the The place value chart counts a block ten times bigger than the column on its right. That makes the hundreds digit the one to look at first. Most of the time it settles the question alone; when it does not, the tens do, and only rarely the ones.
Two-digit numbers were compared tens first in Comparing numbers and the symbols >, < and =, and the signs >, < and = were introduced there. This lesson uses those signs for numbers up to 1{,}000 and does not reteach them.
The lesson runs over three days.
- Day 1, column by column. Comparing two three-digit numbers: hundreds first, then tens, then ones, and why that order is right.
- Day 2, numbers that look different. A two-digit number against a three-digit one, and a pile of blocks that has not been traded yet.
- Day 3, putting numbers in order. Ordering three or four numbers, and building the largest and smallest number from digit cards.
Base ten blocks, a three-column chart on paper, digit cards and a number line from 0 to 1{,}000 marked at every hundred are all the lesson needs.
The procedure
Day 1: hundreds first, then tens, then ones
Build both numbers with blocks and look at the flats first. If one number has more flats, it is the bigger number, whatever its rods and small squares say. The reason is size. A three-digit number has at most 9 rods and 9 small squares, which are worth at most
and 99 is less than one flat. So the other number’s extra rods and small squares can never make up even one missing flat.
If the flats match, they cancel out: put them aside, since they add the same amount to both numbers. Now compare the rods. One more rod is 10, and the small squares can add at most 9, so again the rods win over anything the small squares do. If the rods match too, the small squares decide. If all three counts match, the numbers are equal.
The chart says the same thing without the blocks. Write the two numbers in two rows, one digit per cell, ones under ones. Read the columns from the left. The first column where the digits differ decides: the row with the bigger digit there is the bigger number. Columns further right do not matter once a difference has been found.
On a number line the bigger number is the one further to the right, because the count moves right as numbers grow. Comparing with place value finds the same answer without drawing the line.
The result is written with a sign, as in Comparing numbers and the symbols >, < and =. The wide open side faces the bigger number, and every comparison can be read both ways: a > b and b < a say the same fact.
Day 2: line up from the right, and trade first
A two-digit number has no hundreds. In the chart its hundreds cell is empty, which means 0 flats. So two numbers are always lined up from the right end, ones under ones, before any digits are compared. Pairing the first digit of 83 with the first digit of 308 would match tens against hundreds, which is not a comparison of like with like.
Lined up that way, the hundreds rule handles every case. A three-digit number has at least one flat, so it is at least 100. A two-digit number has no flats, so it is at most 99. Every three-digit number is greater than every two-digit number, and every two-digit number is greater than every one-digit number. Counting digits is a fast first check, as long as no 0 is written at the left end.
The column rule needs one digit in each column. A pile of 4 flats and 13 rods does not fit the chart until the ten extra rods are traded for a flat (Base ten blocks and trading ten for one). The same goes for a number given as 6 hundreds and 11 ones, or in words: rewrite it as digits, one per column, and then compare.
Day 3: ordering
To put several numbers in order is to compare them two at a time. A tidy way to do it: sort them first by how many digits they have, then sort each group by its hundreds digit, then by tens, then by ones. The order from least to greatest is written as a chain,
which says a < b and b < c. Less than passes along the chain: if a is to the left of b on the number line and b is to the left of c, then a is to the left of c. From greatest to least, the chain uses > and runs the other way.
Building the biggest number from digit cards turns the column rule around: the biggest card goes where it counts most, in the hundreds column (The place value chart plays this game with three cards). A 0 card adds one more condition. A three-digit number cannot start with 0: the cards 0, 5, 7 laid as 057 make 57, a two-digit number.
Worked examples
Examples 1 to 3 go with Day 1, Examples 4 and 5 with Day 2, and Examples 6 and 7 with Day 3.
Day 1: Worku’s and Tuya’s points
At the end of June, Worku has 736 reading points and Tuya has 763. Who has more? Write the comparison both ways.
SolutionPut both numbers in the chart (Figure 1) and read the columns from the left.
Hundreds: both have 7. In blocks, each has 7 flats, so the flats add the same amount to both and cannot decide.
Tens: Worku has 3, Tuya has 6. This is the first column that differs, so it decides. Tuya has 3 more rods, which is 30 more. Worku’s extra small squares, 6 against 3, are only 3 more, and even 9 small squares could not make up one rod. Tuya has more points.
The two numbers use the same digits, 7, 3 and 6. What makes them different is which column each digit sits in: the 6 is worth 60 in Tuya’s number and 6 in Worku’s.
Day 1: A whole week of jumps
Over one week of recess, Kawika jumped rope 489 times and Worku jumped 512 times. Who jumped more?
SolutionLook at the hundreds first: 4 for Kawika, 5 for Worku (Figure 2).
Worku has one more flat. Kawika’s rods and small squares, 8 rods and 9 small squares, are worth 89, which is less than a flat. So Kawika’s total is less than 5 hundreds: counting on from 489, it takes 11 more jumps to reach 500. Worku’s total is 500 and then some more. Worku jumped more.
Kawika’s tens and ones digits, 8 and 9, are bigger than Worku’s, 1 and 2. They never come into it, because the hundreds already differ. The tens are looked at only when the hundreds match.
Day 1: Three signs
Write >, < or = in each box.
- 645 \;\square\; 641
- 270 \;\square\; 207
- 5 hundreds 3 tens 8 ones \;\square\; 538
In the first pair the hundreds match (6 and 6) and the tens match (4 and 4). The ones are the first column that differs: 5 ones against 1 one. So 645 > 641.
In the second pair the hundreds match (2 and 2). The tens differ: 7 tens against 0 tens. That decides it, and the ones never matter: 270 > 207. The 0 in 207 says it has no rods at all, while 270 has seven of them.
In the third pair, write the left side as digits first: 5 hundreds, 3 tens and 8 ones is 538. Now every column matches, so the two are the same number written two ways.
Day 2: Eighty-three or three hundred eight?
In the first week of the club, Tuya earned 83 points and Kawika earned 308. Both numbers have an 8 and a 3. Who earned more?
SolutionLine the numbers up from the right end, ones under ones (Figure 3).
Tuya’s 83 has 3 ones and 8 tens, and its hundreds cell is empty: 0 hundreds. Kawika’s 308 has 8 ones, 0 tens and 3 hundreds. The hundreds column comes first, and 3 hundreds is more than 0 hundreds. Kawika earned more.
The 8 in Tuya’s number is worth 80; the 3 in Kawika’s is worth 300. Any number with a flat is at least 100, and 83 has no flat, so there was no contest. Counting the digits gives the same answer at a glance: two digits against three.
Day 2: A pile that is not traded yet
Tuya keeps her week of jumps with base ten blocks. At the end of the week she has 4 flats, 13 rods and 6 small squares. Worku’s week is 527 jumps. Who jumped more?
SolutionThe column rule needs one digit in each column, and 13 rods is more than one digit. Trade first: ten of the rods make one more flat (Base ten blocks and trading ten for one). Tuya now has 5 flats, 3 rods and 6 small squares, which is 536. The trade changed how the blocks are grouped, not how many jumps they stand for:
Now compare 536 with 527. The hundreds match, 5 and 5. The tens differ: 3 against 2. So 536 > 527, and Tuya jumped more.
Comparing before the trade would have pitted 4 flats against Worku’s 5 hundreds and given the wrong winner. The ten rods hiding in Tuya’s pile are a whole flat.
Day 3: Four weeks in order
The club’s totals for its first four weeks were 308, 380, 83 and 830 points. Put them in order from least to greatest, and then from greatest to least.
SolutionSort by the number of digits first. Only 83 has two digits, so it is the least of the four; Example 4 already showed 83 < 308.
The other three all have three digits. Compare their hundreds: 308 and 380 have 3 hundreds, and 830 has 8. So 830 is the greatest.
That leaves 308 against 380. The hundreds match, so look at the tens: 0 tens against 8 tens. So 308 < 380. From least to greatest:
From greatest to least, the same chain runs the other way:
On the number line (Figure 4) the four dots sit in exactly this order from left to right.
All four numbers are built from an 8, a 3 and sometimes a 0. Where the 8 sits decides most of the order: worth 8 in 308, 80 in 83 and 380, and 800 in 830.
Day 3: Four cards, three places
Worku, Tuya and Kawika play a card game with the cards 0, 6, 4 and 9. A player picks three of the four cards and lays them down to make a three-digit number. What is the largest number a player can make? What is the smallest three-digit number?
SolutionFor the largest, the hundreds column counts the most, so it gets the biggest card, 9. Any number starting with 9 beats any number starting with a smaller digit, by the Day 1 rule. Of the cards left, 6 is biggest, so it goes in the tens column. The ones column gets the bigger of 4 and 0, which is 4. The 0 is the card left out. The largest number is 964.
For the smallest, the hundreds column should get the smallest card, but the smallest card is 0, and a number that starts with 0 is not a three-digit number: 069 is just 69. So the hundreds column gets the smallest card that is not 0, which is 4. Now the 0 can be used, and the tens column is the place for it, since the tens count more than the ones. The ones get the smallest card left, 6. The smallest three-digit number is 406.
Check the two answers against each other with the column rule: 9 hundreds against 4 hundreds, so 964 > 406, as it must be.
Boundaries
- The column rule works only when each column holds one digit, 0 to 9. A count of ten or more in one column, such as 13 rods, has to be traded before the digits are compared.
- Numbers must be lined up from the right end, ones under ones. A 0 written at the left end changes nothing (083 is 83), so counting digits is a fair first check only when no number starts with 0.
- Two numbers are equal only when every column matches. Using the same digits is not enough: 736 and 763 use the same three digits and are different numbers.
- A comparison says which number is more, not how much more. Finding that Tuya has 27 more points than Worku in Example 1 is a subtraction, 763 - 736 = 27.
- The lesson stays with whole numbers up to 1{,}000. The number 1{,}000 has four digits and is greater than every three-digit number, for the same reason a three-digit number beats a two-digit one.
- For numbers with a decimal point, more digits does not mean bigger, and lining up from the right end no longer works; Comparing and ordering decimals lines them up at the point instead.
Where this goes next
Bigger numbers are compared the same way, one more column at a time. Numbers to one million adds thousands, ten thousands and hundred thousands, and the leftmost column where two numbers differ still decides, because each column is still worth ten of the one to its right. Rounding uses the same comparisons to say which hundred or ten a number is nearest: 489 is nearer 500 than 400. Comparing and ordering decimals carries the rule to the right of the ones column, with the decimal point as the anchor. Putting a list of numbers in order is also the first step in finding the middle value of a set of data, in Median and mode.
Exercises
Day 1: column by column
- Write >, < or =: 452 \;\square\; 425.
- Write >, < or =: 618 \;\square\; 681.
- Write >, < or =: 799 \;\square\; 801.
- Write >, < or =: 364 \;\square\; 369.
- Write >, < or =: 760 \;\square\; 706.
- Worku has 391 reading points and Kawika has 319. Who has more? Write the comparison both ways.
- Write >, < or =: 7 hundreds 0 tens 5 ones \;\square\; 705.
- Compare 247 and 271. Which column decides, and what is the comparison?
Day 2: numbers that look different
- Write >, < or =: 96 \;\square\; 196.
- Write >, < or =: 405 \;\square\; 54.
- Write >, < or =: 990 \;\square\; 99.
- Tuya’s week of jumps is 3 flats, 15 rods and 2 small squares. Kawika’s week is 445 jumps. Who jumped more?
- Write >, < or =: 6 hundreds 11 ones \;\square\; 611.
- Which digits can go in the box to make 5\square 3 > 587 true?
- Write >, < or =: 2 hundreds 9 tens \;\square\; 3 hundreds.
Day 3: putting numbers in order
- Put in order from least to greatest: 517, 157, 571, 175.
- Kawika’s jump counts at four recesses were 64, 406, 460 and 46. Put them in order from greatest to least.
- With the cards 6, 0 and 1, all three used, what is the largest three-digit number? The smallest?
- Pick three of the cards 5, 1, 9, 6. What is the largest number you can make? The smallest?
- Worku has 238 points, Tuya 283 and Kawika 228. Who has the most? Who has the fewest? Write the three in a chain from least to greatest.
- Name every whole number that is greater than 396 and less than 401.
Explain
- Explain, with blocks, why every three-digit number is greater than every two-digit number.
- Is the number with the bigger ones digit always the bigger number? Give an example.
- Explain why comparing starts with the hundreds and not with the ones.
- Worku’s jump count is greater than Tuya’s, and Tuya’s is greater than Kawika’s. Without knowing the numbers, say whose count is the greatest, and explain with a number line.
Answers
- 452 > 425. The hundreds match; the tens decide, 5 tens against 2 tens.
- 618 < 681. The hundreds match; 1 ten is less than 8 tens.
- 799 < 801. The hundreds decide: 7 hundreds against 8 hundreds. The 9 tens and 9 ones of 799 are worth only 99, less than a hundred.
- 364 < 369. Hundreds and tens match, so the ones decide: 4 against 9.
- 760 > 706. The hundreds match; 6 tens is more than 0 tens.
- Worku has more: 391 > 319 and 319 < 391. The hundreds match, and 9 tens beat 1 ten.
- =. Seven hundreds, no tens and five ones is 705, and every column matches.
- The tens column decides. The hundreds match (2 and 2), and 4 tens is less than 7 tens, so 247 < 271.
- 96 < 196. Lined up from the right, 96 has no hundreds and 196 has 1.
- 405 > 54. 405 has 4 hundreds and 54 has none.
- 990 > 99. Lined up from the right, 990 has 9 hundreds and 99 has 0 hundreds. Three digits against two.
- Tuya. Trade ten rods for a flat: 4 flats, 5 rods and 2 small squares, which is 452. The hundreds match, and 5 tens beat 4 tens, so 452 > 445.
- =. Six hundreds and eleven ones is 600 + 11 = 611.
- Only 9. The hundreds match, so the tens must not be less than 8. With 8 tens the number is 583, and then the ones decide, 3 against 7, so 583 < 587. With 9 tens, 593 > 587.
- <. Two hundreds and nine tens is 290, and 2 hundreds is less than 3 hundreds, so 290 < 300.
- 157 < 175 < 517 < 571. Two numbers have 1 hundred and two have 5. Within each pair the tens decide: 5 tens against 7 tens.
- 460 > 406 > 64 > 46. The two three-digit numbers come first, and 6 tens beat 0 tens. Then the two-digit numbers: 6 tens beat 4 tens.
- Largest 610, smallest 106. The 6 goes in the hundreds for the largest. For the smallest, 0 cannot lead, so 1 goes in the hundreds and 0 in the tens.
- Largest 965, smallest 156. Use the three biggest cards, biggest first, for the largest; the three smallest, smallest first, for the smallest.
- Tuya has the most and Kawika the fewest: 228 < 238 < 283. All have 2 hundreds; the tens are 2, 3 and 8.
- 397, 398, 399 and 400. Count on from 396 and stop before 401.
- A three-digit number does not start with 0, so it has at least one flat, which is 100. A two-digit number has no flats, only rods and small squares, at most 9 rods and 9 small squares, which is 99. Even the largest two-digit number is less than one flat.
- No. In 519 and 602, the ones digits are 9 and 2, but the hundreds decide first: 519 < 602. The ones decide only when the hundreds and the tens both match.
- A hundred is worth more than all the tens and ones together can be, since those make at most 99. So a difference in the hundreds can never be undone by the tens or ones. Starting with the ones would mean looking at the column that matters least, and its answer could be overturned.
- Worku’s. On a number line a greater number sits further right. Worku’s dot is to the right of Tuya’s, and Tuya’s is to the right of Kawika’s, so Worku’s is the furthest right of the three.