Free Pre-Algebra Lesson 8 page 1 Lesson 8 Factor Pairs Measuring more accurately requires breaking our inches into fractions of an inch, little parts smaller than a whole inch. You can think ahead and see that once we ve measured using fractions, we ll need to be able to add and multiply the fractions in order to find area and perimeter. Fractions are useful numbers in lots of situations. Why is this Funny? Many, many people find working with fractions difficult, but bit by bit we ll overcome the reasons for this as we learn. The first reason is that using fractions requires a thinking of numbers multiplicatively, as products of their factors. Thinking multiplicatively will be a great help as we learn fraction skills. Remember that a product is the answer to a multiplication problem? The numbers that are multiplied are called factors. In the equation 2 3 = 6, 2 and 3 are the factors, and 6 is the product. Some people remember the vocabulary with the phrase the factory makes the product. After we use the terminology a bit in context it should feel more familiar. A Multiplication Table Memorization has gone out of fashion in schools, and many students are pretty rocky on their times tables. Some students have learning disabilities that make memorization difficult, but if at all possible, it s a good investment of time to refresh your multiplication memories. When you have to stop and use your calculator or think out simple multiplication and division problems in the middle of long, multi-step math problems, it s much easier to lose your place and get confused. Your brain is overloaded, because there is too much to figure out and pay attention to. Memorizing and allowing the answers to become automatic frees up your brain for the more complicated parts of the problem. Try making a table like this yourself in a spreadsheet program. You can make blank copies to fill in for practice, and you can use one while you re working on this homework. Practice NOW! Cover up the filled out table above and see how many you can fill in yourself from memory.
Free Pre-Algebra Lesson 8 page 2 Idle Observations As you re filling in your multiplication table, you might notice yourself writing certain numbers several times. Here we see all the 24s highlighted: We can write the multiplication problems that produced 24 on the table: 3 8 = 24 and also 4 6 = 24 8 3 = 24 6 4 = 24 We say, 3 and 8 are factors of 24. 24 is the product of 3 and 8. 4 and 6 are factors of 24. 24 is the product of 4 and 6. The Area is the Product, the Sides are Factors A concrete way to think about factors and products is with rectangles. A Question Are these all the number pairs that multiply to give 24 for the answer? If we were to extend the table, we d find another: If you had 24 tiles to play with, could you find other rectangles with area 24 square units? 2 12 = 24 and 12 2 = 24 If we extended it even further, we d find yet another: This is because 1 24 = 24 also. Our goal in this lesson will be to list all the pairs of numbers that multiply to make some given number. Remember that multiplication and division are related operations? We ll use that relationship to find factor pairs.
Free Pre-Algebra Lesson 8 page 3 Divisibility We say a number is divisible by another number if it divides evenly, that is, the answer is another whole number. For example, 20 is divisible by 5, but 20 is not divisible by 7. Since 20 is divisible by 5, 5 is a factor of 20, and since 20 is not divisible by 7, 7 is not a factor of 20. Divisibility Pictures Beyond the Multiplication Table Refering to a large multiplication table is an inefficient way to find factor pairs. A better way is to use the relationship between multiplication and division. If 8 3 = 24, then 24 3 = 8. If we re looking for factor pairs that multiply to make 24, we should divide 24 by every possible factor. The spreadsheet on the left divides 24 by each of the numbers 1 through 24. The highlighted rows are whole number factors. The highlighted factor pairs are: 1 24 2 12 3 8 4 6 24 1 12 2 8 3 6 4 Notice that all of the pairs are represented in the first four entries. After that we get no new pairs, only the same ones again, reversed. To be really efficient, it would be great to know when we can stop trying to find new pairs, while still not missing any.
Free Pre-Algebra Lesson 8 page 4 Checking the Fewest Possible Factors To the right is a copy of the same spreadsheet from the previous page. Look at the relationship between the left-hand numbers, in the N column, and the right-hand numbers, in the 24 N column. Notice that for the numbers highlighted in pink, the left-hand number is smaller than the right-hand number, but that beginning at 5, the left-hand number is larger than the right-hand number. Once the number on the right is smaller than the number on the left, there can be no new pairs. If the number on the left divides evenly, the other factor is smaller, so it s one we ve already seen. With this observation we find that we have to check many fewer possible factors. Steps for Finding All the Factor Pairs of a Given Number If you re using a calculator or a spreadsheet, you can list numbers until you see that the numbers on the right are larger than those on the left, then stop your columns there. Your table or spreadsheet would look like this: and you know that that s enough to give you all the pairs, because 5 is multiplied by a number smaller than 5 to make 24. But there is a way to know that you need only check 1 through 4 before you even start. 5 2 is 25, which is more than 24. Since 5 times itself is greater than 24, 5 has to be multiplied by a number less than itself to equal 24. If the square of a number is greater than the number we are factoring, that number does not need to be checked as a factor. The N column consists only of numbers with squares less than the given number. Example: Find all the factor pairs that have 15 as a product. Step 1: Figure out how many numbers you need to check by finding the first square greater than 15. Since 4 2 is 16, and 16 is greater than 15, you need only check the numbers less than 4, that is, 1 through 3. Step 2: Make a table, with the numbers 1 3 on the left. You can write the column heads as was shown in the spreadsheet, or you can just write 15 on top. Step 3: Divide 15 by each of the numbers on the left. (15 1, 15 2, etc.) If you re dividing with a calculator and the quotient has a decimal part, the number is not a factor. If you re dividing by hand (need a refresher?) and there is a remainder, the number is not a factor. Either way, just cross out the number, since it is not a factor. If the division results in a whole number answer, write that in the right-hand column. You ve found a factor pair. Step 4: Read the factor pairs from the table. The table shows: The factor pairs that have a product of 15 are: 1 and 15, 3 and 5.
Free Pre-Algebra Lesson 8 page 5 Example: Find all the factor pairs that have 32 as a product. Step 1: Since 6 2 is 36, and 36 is greater than 32, we need only check the numbers less than 6. Steps 2 and 3: See the table to the right. Step 4: The factor pairs that have a product of 32 are: 1 and 32, 2 and 16, 4 and 8. Example: Find all the factor pairs that have 79 as a product. Step 1: Since 9 2 is 81, and 81 is greater than 79, we need only check the numbers less than 9. Steps 2 and 3: See the table to the right. Step 4: The factor pairs that have a product of 79 are: 1 and 79. (Aren t you glad you didn t need to check all the way up to 79?) Divisibility Shortcuts Since you re doing a lot of dividing to find the factor pairs, here are some shortcuts to save you time. Look at the Last Digit If the last digit is even (0, 2, 4, 6, 8), the number is divisible by 2. (2 is a factor.) If the last digit is 5 or 0, the number is divisible by 5. (5 is a factor.) 378 is divisible by 2 370 is divisible by 5 Add the Digits If the sum of the digits is divisible by 3, the number is divisible by 3. (3 is a factor.) If the sum of the digits is divisible by 9, the number is divisible by 9. (9 is a factor.) Quick Eliminations If the number is not divisible by 2, it s not divisible by ANY even number. Same for 3. If it s not divisible by 3, it s not divisible by any multiple of 3. * And so on. If it s not divisible by N, it s not divisibly by any multiple of N. 42 is divisible by 3 (4+2=6) 72 is divisible by 9 (7+2=9) 37 is not divisible by 2, so I don t need to check 4, 6, 8, etc. *What s a Multiple of 3? The multiples of a number are found in the row for that number on the multiplication table. They go on to infinity. Multiples of 3:
Free Pre-Algebra Lesson 8 page 6 Example: Find all the factor pairs whose product is 103. These frames show the process step by step using the divisiblity shortcuts. Find all the factor pairs whose product is 103. Step 1: How far? 103 is greater than 10 2 = 100, and less than 11 2 = 121, so we check up to 10. Step 2: Make the table. Step 3: Check the factors. 1 is easy it always works, and the other factor is the number itself. Step 3 continued: 103 is odd the last digit is 3 so it is not divisible by 2 or any multiple of 2. Cross out 2, 4, 6, 8, and 10. Step 3 continued: 103 does not have 5 or 0 as the last digit, so cross out 5. Step 3 continued: The sum of the digits of 103 is 1 + 0 + 3 = 4, which is not divisible by 3 or 9. Cross out 3 and 9. Step 3 continued: The last number to check is 7. 103 7 = 14.714285 (or 14 remainder 5), so 7 is not a factor. Cross out 7. Step 4: Read the factor pairs from the table: The only factor pair is 1 103. Notice that we haven t actually divided anything yet. We ve just used divisiblilty shortcuts.
Free Pre-Algebra Lesson 8 page 7 Lesson 8: Factor Pairs Worksheet Name Find all the factor pairs of the given numbers. 1. These numbers are greater than 5 2 =25, but less than 6 2 =36, so we need only check the numbers 1 through 5. 2. These numbers are greater than (or equal to) 6 2 =36, but less than 7 2 = 49, so we need only check the numbers 1 through 6. 3. Find all the factor pairs that have the given number as product a. 48 b. 55 c. 70 d. 81 e. 105
Free Pre-Algebra Lesson 8 page 8 Lesson 8: Factor Pairs Homework 8A Name 1. Evaluate. a. 7 2 3 2 b. (7 3) 2 2. Find the volume of each box. a. The length is 30 inches, the width is 5 less than the length, and the height is twice the length. L = 30, W =, H = V = c. 5 7 3 d. 5(7 3) e. 52 3(14 8) b. The length is three times the width, the width is 20 cm, and the height is 2 more than the width. L =, W = 20, H = V = c. The length is 3 more than the height, the width is 3 less than the height, and the height is 13 feet. L =, W =, H =13 V = 3. Measure the lines to the nearest inch and to the nearest sixteenth of an inch. 4. Write and use a formula. a. The sea turtle swam 16 hours at 42 miles per hour. How far did it travel? a. nearest inch nearest sixteenth b. A gamer performed 305 actions per minute for 214 minutes. How many actions were performed? b. nearest inch nearest sixteenth c. The average daily temperature at the location of the Mars Rover is 35ºC. Find the temperature in degrees Fahrenheit. The formula is F = 9C/5 + 32.
Free Pre-Algebra Lesson 8 page 9 5. Write the related addition/subtraction equations with 15, 8 and 23. 6. a. Find the sum of 21 and 7. b. Find the product of 21 and 7. Write the related multiplication/division equations with 15, 8 and 120. c. Find the quotient of 21 and 7. d. Find the difference of 21 and 7. 7. Find all the factor pairs that have the given numbers as product. a. 51 b. 52 c. 53 d. 54 e. 55 f. 56 g. 57 h. 58
Free Pre-Algebra Lesson 8 page 10 Lesson 8: Factor Pairs Homework 8A Answers 1. Evaluate. a. 7 2 3 2 49 9 = 40 b. (7 3) 2 4 2 = 16 c. 5 7 3 35 3 = 32 d. 5(7 3) 5(4) = 20 e. 52 3(14 8) 52 3(6) = 52 18 = 34 2. Find the volume of each box. a. The length is 30 inches, the width is 5 less than the length, and the height is twice the length. L = 30, W = 25, H = 60 V = (30)(25)(60) = 45000 45,000 in 3 b. The length is three times the width, the width is 20 cm, and the height is 2 more than the width. L = 60, W = 20, H = 22 V = (60)(20)(22) = 26400 26,400 cm 3 c. The length is 3 more than the height, the width is 3 less than the height, and the height is 13 feet. L =16, W = 10, H =13 V = (16)(10)(13) = 2080 2,080 ft 3 3. Measure the lines to the nearest inch and to the nearest sixteenth of an inch. a. nearest inch 2 in. nearest sixteenth 1 and 11/16 in. b. nearest inch 3 in. nearest sixteenth 2 and 5/8 in. 4. Write and use a formula. a. The sea turtle swam 16 hours at 42 miles per hour. How far did it travel? d = rt = (42)(16) = 672 672 miles b. A gamer performed 305 actions per minute for 214 minutes. How many actions were performed? a = rt = (305)(214) = 65270 65,270 actions c. The average daily temperature at the location of the Mars Rover is 35ºC. Find the temperature in degrees Fahrenheit. The formula is F = 9C/5 + 32. F = 9(35)/5 + 32 = 95 95ºF
Free Pre-Algebra Lesson 8 page 11 5. Write the related addition/subtraction equations with 15, 8 and 23. 15 + 8 = 23 23 8 = 15 8 + 15 = 23 23 15 = 8 Write the related multiplication/division equations with 15, 8 and 120. 15 8 = 120 120 8 = 15 8 15 = 120 120 15 = 8 6. a. Find the sum of 21 and 7. 21 + 7 = 28 b. Find the product of 21 and 7. 21 7 = 147 c. Find the quotient of 21 and 7. 21 7 = 3 d. Find the difference of 21 and 7. 21 7 = 14 7. Find all the factor pairs that have the given numbers as product. a. 51 b. 52 c. 53 d. 54 e. 55 f. 56 g. 57 h. 58
Free Pre-Algebra Lesson 8 page 12 Lesson 8: Factor Pairs Homework 8B Name 1. Evaluate. a. 5 2 + 7 2 b. (5 + 7) 2 2. Find the volume of each box. a. The length is 12 inches, the width is 4 less than the length, and the height is 3 more than the length. L = 12, W =, H = V = c. 3 5 + 7 d. 3(5 + 7) e. 17 + 3(5 + 7) b. The length is four times the width, the width is 2 cm, and the height is 8 more than the width. L =, W = 2, H = V = c. The length is 10 more than the height, the width is 10 less than the height, and the height is 13 feet. L =, W =, H =13 V = 3. Measure the lines to the nearest inch and to the nearest sixteenth of an inch. 4. Write and use a formula to answer the questions. a. The endurance rider rode for 12 hours at an average speed of 8 miles per hour. How far did she travel? a. nearest inch nearest sixteenth b. A computer performed operations at the rate of 102 billion operations per second for 30 seconds. How many operations were performed? b. nearest inch nearest sixteenth c. Cast iron melts at approximately 1,370 C. Find the temperature in degrees Fahrenheit. The formula is F = 9C/5 + 32.
Free Pre-Algebra Lesson 8 page 13 5. Write the related addition/subtraction equations with 12, 10 and 22. 6. a. Find the difference of 12 and 3. b. Find the product of 12 and 3. Write the related multiplication/division equations with 12, 10 and 120. c. Find the sum of 12 and 3. d. Find the quotient of 12 and 3. 7. Find all the factor pairs that have the given numbers as product. a. 59 b. 60 c. 61 d. 62 e. 63 f. 64 g. 65 h. 66