# 1.1.3: Comparing Whole Numbers - Mathematics

Learning Objectives

• Use > or < to compare whole numbers.

There will be times when it’s helpful to compare two numbers and determine which number is greater, and which one is less. This is a useful way to compare quantities such as travel time, income, or expenses. The symbols < and > are used to indicate which number is greater, and which is less than the other.

When comparing the values of two numbers, you can use a number line to determine which number is greater. The number on the right is always greater than the number on the left. In the example below, you can tell that 14 is greater than 8 because 14 is to the right of 8 on the number line.

Example

Which number is greater, 8 or 14?

Solution

14 is to the right of 8, so 14 is greater than 8.

14 is greater than 8.

In the example below, you can determine which number is greater by comparing the digits in the ones place value.

Example

Which number is greater, 33 or 38?

Solution

In both 33 and 38, the digit in the tens place is 3.

Because they have the same number in the tens place, you can determine which one is greater by comparing the digits in the ones place.

In the number 38, the digit in the ones place is 8.

In the number 33, the digit in the ones place is 3.

Because 8 is greater than 3,38 is greater than 33.

38 is greater than 33. This answer was obtained from comparing their digits in the ones place value, which are 8 and 3, respectively.

Exercise

Which number is greater, 17 or 11?

1. 17
2. 11
1. 17

Correct. The number 17 is 6 units to the right of 11 on the number line.

2. 11

Incorrect. The number 11 is to the left of 17 on the number line, so 17 is greater. The correct answer is 17.

If one number is significantly greater than another number, it may be difficult to compare the numbers effectively with a number line. In general, whole numbers with more digits are greater than whole numbers with fewer digits. For example, 542 is greater than 84 because 542 has the digit 5 in the hundreds place. There are no hundreds in 84.

Exercise

Which number is greater, 71 or 710?

1. 71
2. 710
1. 71

Incorrect. You can see that there is no digit in the hundreds place, which means that 71 is less than 710. The correct answer is 710.

2. 710

Correct. The number 710 has 7 hundreds, but 71 has no hundreds.

An inequality is a mathematical sentence that compares two numbers that aren’t equal. Instead of an equal sign (=), inequalities use greater than (>) or less than (<) symbols. The important thing to remember about these symbols is that the small end points towards the lesser number, and the larger (open) end is always on the side of the greater number.

There are other ways to remember this. For example, the wider part of the symbol represents the jaws of an alligator, which “gobbles up” the greater number. So "35 is greater than 28" can be written as 35 > 28, and "52 is less than 109" can be written as 52 < 109.

Example

Replace the question mark with < or > to make a true sentence: 180 ? 220.

Solution

180 is to the left of 220, so 180 < 220. The symbol points at 180, which is the lesser number.

180 < 220

Exercise

Which expression correctly compares the numbers 85 and 19?

1. 85 < 19
2. 19 = 85
3. 85 > 19
4. 19 > 85
1. 85 < 19

Incorrect. The symbol should point at the lesser number, 19. On a number line, 85 is to the right of 19, so 85 is greater than 19. The correct answer is 85 > 19

2. 19 = 85

Incorrect. This symbol says that 85 is equal to 19, which is false. On a number line, 85 is to the right of 19, so 85 is greater than 19. The correct answer is 85 > 19.

3. 85 > 19

Correct. The open part of the symbol faces the larger number, 85, and the symbol points at the smaller number, 19.

4. 19 > 85

Incorrect. The symbol should point at the smaller number, 19. On a number line, 85 is to the right of 19, so 85 is greater than 19. The correct answer is 85 > 19.

Many times, an answer needs to be a range of values rather than just a single value. For example, you want to make more than $22 an hour. This can be expressed as all numbers greater than 22. See the example below. Example ? > 22. What whole number(s) will make this statement true? Solution The symbol points at 22, so the numbers you want to replace the question mark with are greater than 22. There are many numbers that work. 23, 24, 25, 26 and any additional whole numbers that are greater than 26 make this statement true. Exercise A farmer has produced 230 pumpkins for the autumn harvest. Last year, he produced 198. Write an expression that compares these two numbers. 1. 230 > 198 2. 230 < 198 3. 198 = 230 4. 198 > 230 Answer 1. 230 > 198 Correct. 230 is greater than 198, and this is reflected in the symbol because the open part of the symbol faces 230. 2. 230 < 198 Incorrect. 230 is greater than 198, and the symbol is pointing in the wrong direction, with the open part facing the lesser number. The correct answer is 230 > 198. 3. 198 = 230 Incorrect. This statement says that 198 is equal to 230, which is incorrect. The correct answer is 230 > 198. 4. 198 > 230 Incorrect. 230 is greater than 198, and the symbol is pointing in the wrong direction, with the open part facing the lesser number. The correct answer is 230 > 198. To compare two values that are not the same, you can write an inequality. You can use a number line or place value to determine which number is greater than another number. Inequalities can be expressed using greater than (>) or less than (<) symbols. Do not use a calculator at all in section 1.1. ### Build numbers up and break numbers down Write each of the following sums as a single number: The word sum is used to indicate two or more numbers that have to be added. ( 800 000 + 70 000 + 3 000 + 900 + 2) The answer obtained when the numbers are added, is also called the sum. We say: 20 is the sum of 15 and 5. What is the sum of (8 000) and (24)? Write each of the numbers below as a sum of units, tens, hundreds, thousands, ten thousands and hundred thousands, like the numbers were given in question 1(e) and (f). When a number is written as a sum of units, tens, hundreds, thousands etc., it is called the expanded notation. Arrange the numbers in question 3 from smallest to biggest. Write the numbers in expanded notation (for example, (791 = 700 + 90 + 1)). Arrange the numbers in question 5 from biggest to smallest. Write each sum as a single number. (600 000 + 40 000 + 27 000 + 100 + 20 + 34) (320 000 + 40 000 + 8 000 + 670 + 10 + 5 ) ( 500 000 + 280 000 + 7 000 + 300 + 170 + 38) Write each sum as a single number. (300 000 + 70 000 + 6 000 + 400 + 80 + 6) (400 000 + 20 000 + 2 000 + 500 + 10 + 3 ) ( 500 000 + 40 000 + 7 000 + 300 + 60 + 6) (800 000 + 90 000 + 7 000 + 800 + 90 + 8) (300 000 + 110 000 + 12 000 + 400 + 110 + 3) In each case, add the two numbers. Write the answer in expanded form and also as a single number. (a) The number in 8(a) and the number in 8(b) The number in 8(c) and the number in 8(b) The number in 8(c) and the number in 8(a) The number in 8(d) and the number in 8(a) Subtract the number in 8(b) from the number in 8(d). Are the numbers in 8(b) and 8(e) the same? Subtract the number in 8(a) from the number in 8(b). Write each of the following products as a single number: The word product is used to indicate two or more numbers that have to be multiplied. (2 imes 3 imes 5 imes 7) (2 imes 3 imes 5 imes 7 imes 2) The answer obtained when numbers are multiplied, is also called the product. We say: 20 is the product of 2 and 10. (2 imes 3 imes 5 imes 7 imes 2 imes 2) What is the product of 20 and 500? Write 1 000 as a product of 5 and another number. Write 1 000 as a product of 50 and another number. Write 1 000 as a product of 25 and another number. What is the product of 2 500 and 4? What is the product of 250 and 40? In the table on the right, the number in each yellow cell is formed by adding the number in the red row above it to the number in the blue column to its left. Write the correct numbers in all the empty yellow cells. The table below is formed in the same way as the table on the right. Fill in all the cells for which you know the answers immediately. Leave the other cells open for now. ### Multiples In the arrangement below, the blue dots are in groups like this: The red dots are in groups like this: How would you go about finding the number of blue dots below, if you do not want to count them one by one? Implement your plan, to find out how many blue dots there are. Suppose you want to know how many black dots there are in the arrangement on page 6. One way is to count in groups of three. When you do this, you may have to point with your finger or pencil to keep track. The counting will go like this: three, six, nine, twelve, fifteen, eighteen . . . Another way to find out how many black dots there are is to analyse the arrangement and do some calculations. In the arrangement, there are ten rows of threes from the top to the bottom, and three columns of threes from left to right, just as in the table alongside. One way to calculate the total number of black dots is to do (3 imes 10 = 30) for the dots in each column, and then (30 + 30 + 30 = 90). Another way is to add up in each row ((3 + 3 + 3 = 9)) and then multiply by 10: (10 imes 9 = 90). A third way is to notice that there are (3 imes 10 = 30) groups of three, so the total is (3 imes 30 = 90). When you determined the number of blue dots in question 1(b), did you count in fives, or did you analyse and calculate, or did you use some other method? Now use a different method to determine the number of blue dots and check whether you get the same answer as before. Describe the method that you now use. The numbers that you get when you count in fives are called multiples of five. Draw circles around all the multiples of 5 in the table below. How many red dots are there in the arrangement on page 6? Describe the method that you use to find this out. Underline all the multiples of 7 in the table in question 3. A number that is a multiple of 5, and also a multiple of 7, is called a common multiple of 5 and 7. Which multiples of 5 in the table are also multiples of 7? How many yellow dots are there in the arrangement on page 6? Describe the method that you use to find this out. Which numbers in the table in question 3 are common multiples of 7 and 9? Fill in all the cells for which you know the answers immediately. Leave the other cells open for now. Write down the first thirteen multiples of each of the numbers in the column on the left. The multiples of 4 are already written in, as an example. Complete this table. For some cells, you may find your table of multiples above helpful. Go back to the table in question 8(b). If you can easily fill in the numbers in some of the open spaces now, do it. Suppose there are 10 small black spots on each of the yellow dots in the arrangement on page 6. How many small black spots would there be on all the yellow dots together, in the arrangement on page 6? ### Multiples of and How many spotted yellow dots are there on page 11? Explain what you did to find out. How many learners are there in your class? Suppose each learner in the class has a book like this. How many spotted yellow dots are there on the same page (that is, on page 11) of all these books together? Each yellow dot has 10 small black spots, as you can see on this enlarged picture. How many small black spots are there on page 11? How many small black spots are there on page 11 in all the books in your class? Here is a very big enlargement of one of the black spots on the yellow dots. There are 10 very small white spots on each small black spot. How many very small white spots are there on all the black spots on page 11 together? How many very small white spots are there on 10 pages like page 11? How many very small white spots are there on 100 pages like page 11? 10 tens are a hundred: (10 imes 10 = 100) 10 hundreds are a thousand:( 10 imes 100 = 1 000) 10 thousands are a ten thousand: (10 imes 1 000 = 10 000) 10 ten thousands are a hundred thousand: (10 imes 10 000 = 100 000) 10 hundred thousands are a million: (10 imes 100 000 = 1 000 000) Write (7 000 + 600 + 80 + 4) as a single number. Write 10 times the number in (a) in expanded notation and as a single number. Write 100 times the number in (a) in expanded notation and as a single number. Write each of the following numbers in expanded notation: Write 10 000 as a product of 10 and one other number. Write 10 000 as a product of 100 and one other number. Write 100 000 as a product of 10 and one other number. Write 100 000 as a product of 1 000 and one other number. Write 1 000 000 as a product of 1 000 and one other number. In the table below, fill in all the cells for which you know the answers immediately. Leave the other cells open for now. Fill in all the cells in the table for which you know the answers immediately. Leave the other cells open for now. How many multiples of 10 are smaller than 250? You may make an estimate, and then write the multiples down to check. In each case first estimate, then check by writing all the multiples down and counting them. How many multiples of 100 are smaller than 2 500? How many multiples of 250 are smaller than 2 500? How many numbers smaller than 2 500 are multiples of both 100 and 250? How many numbers smaller than 2 500 are multiples of both 250 and 400? In each of the tins below, there are three R10 notes, three R20 notes, three R100 notes and three R200 notes. Zain wants to know what the total value of all the R10 notes in all the tins is. He decides to find this out by counting in 30s, so he says: thirty, sixty, ninety . . . and so on while he points at one tin after another. Complete what Zain started to do. Count in 300s to find out what the total value of all the R100 notes in all the tins is. How much money is there in total in the eight yellow tins in question 13? Join with two classmates and tell them how you worked to find the total amount of money. 1. Investigate what is easiest for you, to count in twenties or in thirties or in fifties, up to 500. 2. Many people find it easier to count in fifties than in thirties. Why do you think this is so? What do you expect to be the most difficult, to count in forties or in seventies or in nineties? Investigate this and write a short report. Here is some advice that can make it easier to count in certain counting units, for example in seventies. It feels easier to count in fifties than in seventies because you get to multiples of 100 at every second step: fifty, hundred, one hundred and fifty, two hundred, two hundred and fifty, 300, 350, 400, 450, 500 . and so on. When you count in seventies, this does not happen: seventy, one hundred and forty, two hundred and ten, two hundred and eighty . It may help you to cross over the multiples of 100 in two steps each time, like this: In this way, you make the multiples of 100 act as "stepping stones" for your counting. Count in forties up to 1 000. Try to use multiples of 100 as stepping stones. You can write the numbers below while you count. Write down the first twenty multiples of 80. Write down the first twenty multiples of 90. Write down the first ten multiples of 700. ### Doubling and halving Write the next eight numbers in each pattern: Which pattern or patterns in question 1 are not formed by repeated doubling? The pattern 3, 6, 12, 24, 48 . may be called the repeated doubling pattern that starts with 3. Write the first nine terms of the repeated doubling patterns that start with the numbers in the left column of the table. The pattern for 13 has been completed as an example. Doubling can be used to do multiplication. For example, (29 imes 8) can be calculated as follows: 8 doubled is 16, so (16 = 2 imes 8) (step 1) 16 doubled is 32, so (32 = 4 imes 8) (step 2) 32 doubled is 64, so (64 = 8 imes 8) (step 3) 64 doubled is 128, so (128 = 16 imes 8) (step 4). Doubling again will go past (29 imes 8). (16 imes 8 + 8 imes 8 + 4 imes 8 = (16 + 8 + 4) imes 8 = 28 imes 8). So (28 imes 8 = 128 + 64 + 32) which is 224. So (29 imes 8 = 224 + 8 = 232). Work as in the above example to calculate each of the following. Write only what you need to write. Continue each repeated halving pattern as far as you can: 64 000, 32 000, 16 000, 8 000 Halving can also be used to do multiplication. For example, ( 37 imes 28) can be calculated as follows: (100 imes 28 = 2 800). Half of that is (50 imes 28) which is half of 2 800, that is 1 400. Half of (50 imes 28) is half of 1 400, so (25 imes 28) is 700. (10 imes 28 = 280), so (25 imes 28 + 10 imes 28 = 980), so (35 imes 28 = 980). (2 imes 28 = 2 imes 25 + 2 imes 3 = 56), so (37 imes 28) is (980 + 56 = 1 036). (80 imes 78 = 6 240). Use this information to work out each of the following: If chickens cost R27 each, how many chickens can you buy with R2 400? A way to use halving to work this out is shown on the next page. 100 chickens cost (100 ( imes) 27 = R2 700. That is more than R2 400. 50 chickens cost half as much, that is R1 350. So I can buy 50 chickens and even more. Half of 50 is 25 and half of R1 350 is R675. So 75 chickens cost R1 350 + R675, which is R2 025. So there is R375 left. 10 chickens cost R270, so 85 chickens cost R2 025 + R270 = R2 295. There is R105 left. 3 chickens cost 3 ( imes) R25 + 3 ( imes) R2 = R81. I can buy 88 chickens and that will cost R2 376. Use halving as in the above example to work out how many books at R67 each a school can buy with R5 000. ### Using multiplication to do division R7 500 must be shared between 27 netball players. The money is in R10 notes, and no small change is available. How much money will be used to give each player R100? Do you think there is enough money to give each player R200? Do you think there is enough money to give each player R300? How much of the R7 500 will be left over, if each player is given R200? Is there enough money left to give each player R50 more, in other words a total of R250 each? What is the highest amount that can be given to each player, so that less than R270 is left over? Remember that you cannot split up the R10 notes. Work like you did in question 1 to solve this problem: There is 4 580 m of string on a big roll. How many pieces of 17 m each can be cut from this roll? Hint: You may start by asking yourself how much string will be used if you cut off 100 pieces of 17 m each. Work like you did in questions 1 and 2 to solve this problem: A shop owner has R1 800 available with which he can buy chickens from a farmer. The farmer wants R26 for each chicken. How many chickens can the shop owner buy? What you actually did in questions 1, 2 and 3 was to calculate (7 500 div 27), (4 580 div 17) and (1 800 div 26). You solved division problems. Yet most of the work was to do multiplication, and a little bit of subtraction. When you had to calculate (1 800 div 26) in question 3, you may have asked yourself: With what must I multiply 26, to get as close to 1 800 as possible? Division is called the inverse of multiplication. Multiplication is called the inverse of division. Multiplication and division are inverse operations. ## Direct Instruction I use this site to teach the skill of comparing and ordering numbers: This site allows for interaction during the lesson. Students enjoy when they have instructional video lessons. What I like about this site is that it gives the students a chance to practice the skill as it teaches the lesson. In this particular lesson, the students must figure out if Mia is improving on her speed in the track meets. Mia has the following times for her track meets: 25 seconds, 27 seconds, 22 seconds, 23 seconds, and 24 seconds. This video points out to the students that all of the numbers have a 2 in the tens place. Therefore, we must use the ones place. It goes on to explain to the students that we must use the ones place to compare the numbers. I stop the video and ask the students to tell me which numbers are in the ones place. The students call out 5, 7, 2, 3, and 4. I ask, "Which number is the largest?" The students tell me 7. I continue on with the video so that the students can see how the numbers are put in order from greatest to least, with 27 being the greatest and 22 being the least. (I let the students know that the numbers can also be put in order from the smallest to the largest.) The students can see from ordering the numbers that Mia's time has improved since the first track meet. However, her best time was at track meet 3. Mia will have to continue improving to beat her best time of 22 seconds. Upon completion of the video lesson, the students understand that numbers can be put in order from least to greatest or greatest to least. The students should also know that you compare numbers depending on their place value (4.NBT.A2). ## 1.1.3: Comparing Whole Numbers - Mathematics Comparing two Whole Numbers In this page, comparison of two whole numbers is explained in the following two forms. Whole numbers are used to represent count or measure of quantities. Comparing the number of fish and cats in the figure, it is found that, number of fish is more than number of cats. That is 10 10 is larger than 5 5 . . number of cars is equal to the number of cats. number of cats is lesser than number of fish. That is 5 5 is less than 10 10 Whole numbers are used to measure quantities. An example is height of a tree. The pine tree, shown in picture, is taller than the palm tree shown in the picture. Such measurement of quantities or numbers can also be compared for smaller and equal. One of the following is true while comparing two quantities. • A quantity is lesser than another. • A quantity equals another. • A quantity is more than another. For example, • 4 4 is lesser than 5 , which is given as 4 < 5 • 5 equals 5 , which is given as 5 = 5 • 5 is more than 4 , which is given as 5 > 4 When compared to a number, another number can only be one of the three, (1) smaller, (2) equal, or (3) larger The symbol < is used to denote the number on the left side is less than the number on the right. In this case, 3 is less than 7. The symbol > is used to denote the number on the left side is greater than the number on the right. In this case, 7 is greater than 3. The symbol = is used to denote the number on the left side equals the number on the right. In this case, 3 equals 3. Comparison in First Principles Considering numbers 7 and 8 . There are 7 burgers and 8 dogs in the figure. Consider comparing 8 and 7 . The comparison of numbers as quantities is illustrated in the figure. In first principles, two quantities are matched one-to-one. The quantity that has excess is greater. The other quantity is smaller. If the two quantities match to the last count, then they are equal. Number of dogs have one excess, so number of dogs is greater than the number of burgers. Consider comparing 8 and 7 . The comparison of numbers as quantities is illustrated in the figure. The quantity that has excess is greater. And The the other quantity is lesser. Comparison of two numbers 19 and 17 is illustrated in the figure. Which one is smaller? On comparing the quantities, 17 is found to be smaller. Comparing 19 and 17 . By first principles: The quantities are matched. The tens is made of 10 units. So first the tens, given in purple bar, are matched one-to-one. Then the units, given in blue cubes, are matched and found that 19 has excess. So 17 is found to be smaller. Compare numbers 47 and 53 . Which one is larger? The answer is " 53 ". Simplified Procedure : Comparison by place-value Comparing the numbers 47 and 53 . By first principles : The quantities are matched. A ten is made of 10 units. The tens, given in purple bar, are matched one-to-one. From this, 53 is found to be larger. As a simplified procedure : Comparison by place-value, the numbers in tens place value can be compared. 4 in 47 and 5 in 53 . On comparing, 53 is found to be larger. Note that once it is found that the tens place are not equal, the larger number is decided. The units place does not need to be compared. Compare numbers 214 and 132 . Which one is larger? The answer is " 214 " Comparing numbers 214 and 132 . By First principles, the two quantities are matched. The hundreds are the largest among the three and made of 10 tens. So the hundreds are matched. From this, 214 is found to be larger. As a simplified procedure: Comparison by place-value, the numbers in hundreds place are compared first. 2 from 214 is compared with 1 from 132 . Since 2 is greater than 1 , it is concluded that the 214 is larger. Comparing numbers 214 and 132 . The numbers are given in the place-value form. Comparing the hundreds place, 2 is greater than 1 , it is concluded that the 214 is larger. Compare numbers 156 and 88 . Which one is smaller? The answer is "Comparing 1 of 156 and 0 of 088 , 88 is smaller" Comparing 156 and 88 . It is noted that the hundreds place is 0 for 88 . By the simplified procedure, the digits in the hundreds place are compared and 0 is smaller than 1 . It is concluded that 88 is smaller. As a Simplified Procedure : Comparison by place-value , the numbers in tens place value can be compared. 1 in 19 and 1 in 17 . Both digits in tens place are equal. Then the numbers in the units position are compared. 9 in the 19 and 7 in 17 . On comparing, 17 is found to be smaller. Which of the following is larger than the other? 23 or 61 . The answer is " 61 " or 61 > 23 Which of the following is smaller than the other? 2183 or 2661 . The answer is " 2183 " or 2183 < 2661 . Which of the following is greater than the other? 341 or 341 . The answer is "the numbers are equal". 341 = 341 Comparison of whole numbers : Two numbers can be compared to find one of them as When compared to a number, another number can only be one of the three, Comparison by First Principle: To find if one number is larger or smaller than another number, the quantities represented by them are compared. Example: Comparing the numbers 47 and 53 . The quantities represented by them is compared in the figure. It is found that 47 is smaller than 53 . Comparison by Place-value -- Simplified Procedure to Compare Large Numbers: To find if one number is larger or smaller than another number, the digits at the highest place value are compared and if they are equal, then the digits at next lower place value are compared. For example: Let us take the numbers 2471 and 2437 . Largest place value is 1000 and the digits are 2 and 2 , which are equal. Next smaller place value is 100 and the digits are 4 and 4 , which are equal. Next smaller place value is 10 and the digits are 7 and 3 , in which, 7 is greater than 3 . So 2471 is greater than 2437 . Which of the following step help to find the larger between 23 and 156 ? • compare first digits 2 and 1 OR • compare the 100 s place 0 and 1 The answer is "compare the 100 s place 0 and 1 ". The numbers are equivalently given as 023 and 156 . Comparison : Two numbers can be compared to find one of them as Comparison by First Principle: Two quantities are matched one-to-one and compared in the count or magnitude of the quantities. As a result one of them is smaller or equal to or larger to the other. Example: Comparing the numbers 7 and 8 . The quantities represented by them is compared in the figure. It is found that 7 is smaller than 8 . Simplified Procedure -- Comparison by ordered-sequence: To find if one numbers is larger or smaller than another number, the numbers are compared using the order 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , ⋯ . Example: Comparing the numbers 4 and 7 . 4 is on the left-side to 7 in the order 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 and so 4 is smaller than 7 . Comparison by Place-value -- Simplified Procedure to Compare Large Numbers: To find if one number is larger or smaller than another number, the digits at the highest place value are compared and if they are equal, then the digits at next lower place value are compared. For example: Let us consider the numbers 2471 and 2437 . Largest place value is 1000 and the digits are 2 and 2 , which are equal. Next smaller place value is 100 and the digits are 4 and 4 , which are equal. Next smaller place value is 10 and the digits are 7 and 3 , in which, 7 is greater than 3 . So 2471 is greater than 2437 . A set of symbols < = > are introduced to represent the comparison of two numbers. Consider 2 and 3 ## Weightlifting: Comparing Multi Digit Whole Numbers Activity Teams of students will work to hang weight plates and write multi digit numbers. 1. Student teams cut out (and color) the weight plates. They should have 10 of each size. See pages 5-10 on the worksheet. 2. Place tape or a thin strip of paper across the desk and have a cardboard divider or book to prevent teams from seeing the other end. 3. Teams read the category, discuss what the category description means, and hang their weights so their number fits the category description. 4. Students need to have the weights and record the number in standard form. 5. When both teams are ready, remove the divider and compare the weights. 6. First compare the visual weights and see which is heavier then compare the standard form to determine which is heavier. 7. Record BOTH numbers on the worksheet. 8. Circle the place value that determined which side was heavier. The team that has the heavier number earns 1 point. 9. If the numbers are EQUAL, then both teams earn 10 points. 10. Continue play, making sure to follow the category name, record numbers and compare. ## Comparing Bits and Pieces Problem 1.3 (equivalent fractions): On the Line 1. Use strips of paper 8 1 /2 inchs long. Each strip represents 1 whole. Fold the strips to show halves, thirds, fourths, fifths, sixths, eighths, ninths, tenths, and twelfths. Mark the folds so you can see them easily, as shown below. 2. What strategies did you use to fold your strips? 1. How can you use the halves strip to fold eighths? 2. The picture below shows a student's halves, fourths, and eighths strips. How does the size of one part of a halves strip compare to the size of one part of an eighths strip? 3. What fraction strips can you make if you start with a thirds strip? 4. Which of the fraction strips you folded have at least one mark that lines up with a mark on a twelfths strip? What equilvalent fractions do the matching marks on the strips suggest? ## B) 710 Correct. The number 710 has 7 hundreds, but 71 has no hundreds. An inequality is a mathematical sentence that compares two numbers that aren’t equal. Instead of an equal sign (=), inequalities use greater than (>) or less than (<) symbols. The important thing to remember about these symbols is that the small end points towards the lesser number, and the larger (open) end is always on the side of the greater number. There are other ways to remember this. For example, the wider part of the symbol represents the jaws of an alligator, which “gobbles up” the greater number. So “35 is greater than 28” can be written as 35 > 28, and “52 is less than 109” can be written as 52 < 109. Replace ? with < or > to make a true sentence: 180 is to the left of 220, so 180 < 220. The symbol points at 180, which is the lesser number. Which expression correctly compares the numbers 85 and 19? Incorrect. The symbol should point at the lesser number, 19. On a number line, 85 is to the right of 19, so 85 is greater than 19. The correct answer is 85 > 19. Incorrect. This symbol says that 85 is equal to 19, which is false. On a number line, 85 is to the right of 19, so 85 is greater than 19. The correct answer is 85 > 19. Correct. The open part of the symbol faces the larger number, 85, and the symbol points at the smaller number, 19. Incorrect. The symbol should point at the smaller number, 19. On a number line, 85 is to the right of 19, so 85 is greater than 19. The correct answer is 85 > 19. Many times an answer needs to be a range of values rather than just a single value. For example, you want to make more than$22 an hour. This can be expressed as all numbers greater than 22. See the example below.

? > 22. What whole number(s) will make this statement true?

The symbol points at 22, so the numbers you want to put in the brackets are greater than 22. There are many numbers that work.

23, 24, 25, 26, and any additional whole numbers that are greater than 26 make this statement true.

A farmer has produced 230 pumpkins for the autumn harvest. Last year, he produced 198. Write an expression that compares these two numbers.

Correct. 230 is greater than 198, and this is reflected in the symbol because the open part of the symbol faces 230.

Incorrect. 230 is greater than 198, and the symbol is pointing in the wrong direction, with the open part facing the lesser number. The correct answer is 230 > 198.

Incorrect. This statement says that 198 is equal to 230, which is incorrect. The correct answer is 230 > 198.

Incorrect. 230 is greater than 198, and the symbol is pointing in the wrong direction, with the open part facing the lesser number. The correct answer is 230 > 198.

To compare two values that are not the same, you can write an inequality. You can use a number line or place value to determine which number is greater than another number. Inequalities can be expressed using greater than (>) or less than (<) symbols.

## Lesson Worksheet: Comparing and Ordering Numbers up to 1�� Mathematics

In this worksheet, we will practice comparing and ordering numbers up to one million by comparing place values or using number lines.

• A 574,200 , 574,700
• B 574,400 , 574,800
• C 573,300 , 573,600
• D 573,400 , 573,700
• E 573,300 , 573,700

Jackson has 159,357 stamps, while Matthew has 199,357 stamps. Which of them has a smaller number of stamps?

Charlotte ordered the following numbers from least to greatest, as shown.

Did she order them correctly?

The area of Nigeria is nearly 910,770 km 2 . The area of Egypt is nearly 995,450 km 2 . Which country has a greater area?

Arrange the numbers 222,202 ,

, 936,479 , and 814,606 from greatest to least.

A student makes a survey that orders countries by forest area. He finds that there are 376,309 km 2 of forest area in Zambia, 591,040 km 2 in Angola, 184,180 km 2 in Spain, and 675,420 km 2 in Sudan. Order these countries from least to greatest according to the size of their forest area.

## 1.1.3: Comparing Whole Numbers - Mathematics

### Counting and Comparing Difficulties

Subitizing is the ability to recognize a number of briefly presented items without actually counting.

A common response to students who are having counting problems is to simply have them do daily counting practice however, students with counting and comparing difficulties also benefit from practice that utilizes patterns and relationships. These strategies improve their ability to conceptualize and compare numbers without counting. Data in a study of dyslexic students who had difficulty with basic arithmetic skills (Fischer B., Kongeter A., Hartnegg K., 2008) showed that dyslexic children could also improve subitizing and visual counting through daily practice. It is important to distinguish the whole-to part process involved with this training. Not all daily counting practice is created equal. These dyslexic students did not achieve their gains in arithmetic merely through the process of counting. They were taught counting strategies for many years to add and subtract numbers with little benefit to their overall concept of number. Students made their gains because they were supplied with a whole- or gestalt then they combined subordinate parts to reconstruct the image. Over time they improved their ability to match quantity with successively larger patterns.

(See the example strategy below.)

### Example Strategy: Using Icons of Quantity To Teach Whole-to-Part Relationships

Woodin: The ability to identify a subordinate quantity in relation to a whole enables these quantities to be seen in a relational context that fosters comparison without employing the inefficient and often inaccurate process of counting. The following exercise explains a whole-to-part procedure that is driven by a concrete visual model. Using a whole-to-part model, place five pieces of cereal on a table in a canonical “: • :” pattern. Model a subtraction event by removing pieces. Label the process by making a subtraction sentence, then let the student replace the pieces, and label this action with a related addition fact:

Teacher: “How many are you starting with?”

-The teacher removes the center piece of cereal to leave a square arrangement.

(: • : – • = : :).

Teacher: “Tell me what happened.”

Student: “We had five, you took away one and four are left.”

Teacher: “Say the same thing with a number sentence.”

-The teacher hands the piece of cereal to the student.

Teacher: “Put this back and make a number sentence.” (: : + • = : • :).

### Example Strategy: Using Patterns To Support Number Comparisons

Woodin has seen impressive results from instruction that incorporates visual patterning. Consistent graphic organizers that relate quantities to both five and ten provide the structure necessary to develop cardinality with numbers one through ten, and then extend this knowledge to the base-ten system. Consider the following patterns that relate to the gestalts of five and ten.

Each of these icons of quantity is subordinate to the gestalt of the Ten Icon (Woodin, 1995). Using the Ten Icon as a reference, the other icons represent a quantity of items that are visible, as well as an identifiable void that is the missing addend to make 10. Additional shading allows comparison to 5. For example, the Six Icon is made with a blue component of 5 and one red block: (6 = 5 + 1). In reference to the Ten Icon, the 6 is missing a square arrangement of 4 red blocks: (10 = 6 + 4). All of the missing addends to 10 may be driven by flashing an Icon card, asking the name of the card and the missing shape or number needed to make ten. See an example of this technique by viewing the video below.

### Strategy Demonstration: Prompt the Missing Addends to 10

Quantities that are presented in concrete form, or represented by a diagram, are not subject to reversals. With the circles/dots at the left, there’s no change of misreading the quantity they represent. Consider an Arabic “4,” which can be incorrectly written in its mirrored form, versus a square pattern with an identical mirror image. These patterns can be used to diagram addition or subtraction problems. An “X” is used as an efficient way of producing a group of five in the following to make icons of 6 and 7. When addition problems are diagramed with Icon patterns, the sum emerges from the two addends. The following video clip illustrates the Icon addition process:

### Patterning Should Be Extended to the Multiplication Process

Generate the 2x facts by stamping finger patterns on a template like the one pictured above. The teacher should write the number of fingers to be stamped in the top circle. The student then holds out that number of fingers, dips them in paint, and stamps this quantity onto the template “two times.” The student should first stamp his fingers at the top of the template, then duplicate the same pattern again below it. From this student-centered location the student produces a concrete diagram of multiplication from his primary frame of reference. The process of stamping the quantity “two times” provides the student with a concrete definition of the “times: X” multiplication operation and sign.

### Strategy I: Stamp 3 two times

Use the student’s right hand to produce patterns of two times: one, two, three, or four fingers. Have the student dip his right-hand fingers in red paint, then “stamp” the pattern twice on the red portion of the template. The example shown above depicts stamping 2 x 3 = 6. Dip three fingers on the right hand in red paint and stamp them twice.

The six red dots can be smeared into an Arabic “6” when done.

To stamp numbers 5 and larger “two times,” the left-hand fingers will be needed. Dip the left-hand fingers in blue paint, then extend additional right (red) fingers necessary to match the number. The stamped quantities of fingerprints display a chunked depiction of the product that aligns with base ten place value. A blue pattern of ten will be produced on the left in the ten’s place. Additional red prints will be produced on the right.

### Strategy II: Stamp 8 Two Times

Extend all left-hand fingers and dip them in blue finger paint. Extend three right-hand fingers and dip them in red paint. Stamp both hands (two times) on the paper. By stamping all eight fingers two times, the student will create the product: 1 group of ten blue fingerprints on the left–in the ten’s place, and six red prints on the right–in the ones place: “2 x 8 = 16.” The following movie demonstrates the process of creating the 2x products with the finger-stamping technique.

### Using a Clock Face To Chunk and Organize Information

Chris Woodin has graciously allowed us to offer PDFs of some of the graphic organizers he uses in his classroom. To see more organizers and information, click here.

The 12 clock positions of the analog clock may be used to learn and compare the 5x facts. This is accomplished through a series of gross motor kinesthetic activities that initially place the student at the center of a large clock dial, facing the 12 position. From this student-centered location the student internalizes the relative number locations from his primary reference frame. Minute values are then associated with these positions. Clock positions are merged with minute values to create 5x multiplication facts in a relational context. As the student points to the 3 position of the clock, both the number 3 and minute value of 15 may be simultaneously held in memory. The internalized structure of the clock also provides a student with the ability to simultaneously access several facts so they may be compared. This ability is particularly useful when dividing. For instance, if a student were able to visualize 37 between the benchmark minute values of 35 and 40, he would be able to determine that 37 cents would be made with 7, not 8, nickels.

In the following video, students will use the familiar structure of the clock to learn about the 5x fact family within a relational context. Students learn to interact with the analog clock dial from within their primary reference frame, and then externalize this structure to a paper-and-pencil task. Ultimately, they are empowered to create and compare 5x facts.

One educator in Connecticut suggested this for learning the 9 facts: When multiplying the numbers 2 – 9 by 9, the two digits in the resulting answer will add up to nine. So, for instance, 5 x 9 = 45 (4 plus 5 adds up to 9). It’s a good way to check your answers.

Christopher Woodin is a specialist in the field of mathematics and learning disabilities. A graduate of Middlebury College and Harvard Graduate School of Education, he has taught extensively at Landmark School in Massachusetts. At Landmark School’s Elementary/Middle School Campus, he holds the Ammerman Chair of Mathematics. Christopher served on the Massachusetts Department of Education’s Mathematics 2011 Curriculum Framework Panel, and teaches graduate-level professional development courses during the summer through Landmark’s Outreach Program. Chris was the 1997 Massachusetts Learning Disabilities Association (LDA) Samuel Kirk Educator of the Year. He has presented at numerous international LDA and International Dyslexia Association (IDA) conferences, and has led math workshops to audiences across the country.

Christopher has published The Landmark Method of Teaching Arithmetic ©1995 and several journal articles. His latest project, Multiplication and Division Facts for the Whole-to-Part, Visual Learner: An Activity-Based Guide to Developing Fluency with Math Facts, is currently in press and due to be released in 2012. This comprehensive text features the methodologies and many of the activities that are described on The Yale Center for Dyslexia & Creativity’s website. To learn more about Mr. Woodin and his work, please visit his page on the Landmark School website and his own website.

In 2010, Sacramento, California, received 23 inches in annual precipitation. In 2011, the city received 17 inches in annual precipitation. In which year was there more precipitation ?

Locate the two whole numbers 23 and 17 on a number line and mark them.

23 is to the right of 17 on the number line.

This means that 23 is greater than 17.

We can write the above situation in terms of inequality as 23 > 17.

17 is to the left of 23 on the number line.

This means that 17 is less than 23.

We can write the above situation in terms of inequality as 17 < 23.

There was more precipitation in 2010.

John recorded the following golf scores during his first week  at a golf academy. In golf, a lower score beats a higher score.

Graph John’s scores on the number line, and then list the numbers in order from least to greatest.

Graph the scores on the number line.

Read from left to right to list the scores in order from least to greatest.

The scores listed from least to greatest are

After having gone through the stuff given above, we hope that the students would have understood "Compare whole numbers".