Lesson 1.1 Reteach Divide Multi-Digit Numbers When you divide multi-digit whole numbers, you can estimate to check if the quotient is reasonable. Divide 399 4 42. Step 1 Estimate, using compatible numbers. 400 and 40 are compatible numbers because 400 4 40 5 10 400 can be shared among 40 groups evenly. Step 2 Divide the original numbers. 9 r21 42 q w 399 2378 21 Step 3 You can write the remainder as a fraction. Use the remainder for the numerator, and the divisor for the denominator. Simplify if possible. Step 4 Compare the quotient with your estimate. 21 4 21 42 4 21 5 1_ 2 399 4 42 5 9 1_ 2 Since 9 1 _ is close to 10, the 2 quotient is reasonable. Estimate. Then find the quotient. Write the remainder, if any, with an r. 1. 17 q w 965 2. 29 q w 4,380 3. 62 q w 1,178 Estimate. Then find the quotient. Write the remainder, if any, as a fraction. 4. 836 4 32 5. 1,392 4 18 6. 2,518 4 48 Houghton Mifflin Harcourt ublishing Company 1-21 Reteach
Lesson 1.1 Enrich Division Bugs Scientists have discovered a new insect! Use the features of the new insect that are listed below to solve the problem. Write a division number sentence for each. The insect has: 6 wings 15 legs 32 eyes 116 scales 213 hairs 1. A scientist found a colony of the new insects. If she counted 165 legs, how many insects are in the colony? 2. Dr. Wilburn is studying a group of insects. There are 2,784 scales in the group. How many insects are in Dr. Wilburn s group? 3. In a group of insects, there are 264 wings. How many insects are there? 4. Name the new insect and give it another feature (such as number of antennae). Write a division sentence using the insect s new feature. 5. Stretch Your Thinking Carla sees a group of insects. She counts 190 eyes. Is this possible? Explain. 6. Explain how knowing simple multiplication facts can help you with division with greater numbers. Houghton Mifflin Harcourt ublishing Company 1-22 Enrich
Lesson 1.2 Reteach rime Factorization A number written as the product of prime numbers is called the prime factorization of that number. To break a number down into its prime factors, divide it by prime numbers. The first eight prime numbers are listed below. 2, 3, 5, 7, 11, 13, 17, 19 You can use a factor tree to find the prime factorization of a number. You can use a ladder diagram to find the prime factorization of a number. 165 3 55 5 11 Divide the number by the least prime factor possible. Try 2, 3, 5, and so on. Break 55 down because it is not a prime number. The numbers at the bottom of the branches are all prime. 5 165 3 33 11 11 1 165 ends in 5, so it is divisible by 5. Divide 165 by 5. Write the quotient below 165. The sum of the digits in 33 is divisible by 3, so divide 33 by 3. 11 is prime. Divide 11 by itself. The bottom number is 1 and all the numbers to the left are prime. Write the number as a product of prime factors. The factors should be in order from least to greatest. So, the prime factorization of 165 is 3 3 5 3 11. Find the prime factorization of the number. 1. 21 2. 130 3. 84 Houghton Mifflin Harcourt ublishing Company 1-23 Reteach
Lesson 1.2 Enrich Symmetric Factor Trees Both factor trees below show the prime factorization of 40. The factor tree on the right is symmetric because both branches have the same shape. Not Symmetric 40 Symmetric 40 8 5 4 10 4 2 2 2 2 5 2 2 Create a symmetric factor tree for each number. 1. 90 2. 16 3. Stretch Your Thinking Create a symmetric factor tree for a number that has more than two prime factors and is greater than 100. 4. Can you make a symmetric factor tree for every composite number? Explain. Houghton Mifflin Harcourt ublishing Company 1-24 Enrich
Lesson 1.3 Reteach Least Common Multiple The least common multiple, or LCM, is the least number that two or more numbers have in common in their list of nonzero multiples. Find the LCM of 3 and 9. List the first ten nonzero multiples of each number: Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 The first three nonzero multiples that 3 and 9 have in common are 9, 18, and 27. So, the LCM of 3 and 9 is 9. Find the LCM. 1. 4, 10 List the first ten multiples for each number. Multiples of 4: 4, 8,, 16,, 24, 2. 6, 8 List the first ten multiples for each number. Multiples of 6:,, 36, Multiples of 10: 10,, 30,, 50,, 70,,, 100 List the numbers that appear in both lists. Multiples of 8: List the numbers that appear in both lists. Common multiples: and Common multiples: The LCM of 4 and 10 is. The LCM of 6 and 8 is. 3. 5, 20 4. 6, 15 5. 12, 30 6. 7, 14 7. 10, 15 8. 6, 18 Houghton Mifflin Harcourt ublishing Company 1-25 Reteach
Lesson 1.3 Enrich Mix and Match In the table below, draw lines from each LCM in the first column to two numbers in the second column that share the LCM. nce a number in the right column is used, it cannot be used again. Example: LCM: 12 Numbers: 3 and 4 1. 2. 3. 4. 5. 6. 7. LCM 12 18 42 60 72 160 176 Numbers 3 4 6 9 14 15 16 18 20 21 22 24 32 40 8. Stretch Your Thinking ut together a new pair of numbers from the table, whose LCM is not in the table. Write the numbers and the LCM. Houghton Mifflin Harcourt ublishing Company 1-26 Enrich
Lesson 1.4 Reteach Greatest Common Factor A common factor is a number that is a factor of two or more numbers. The greatest common factor, or GCF, is the greatest factor that two or more numbers have in common. Find the common factors of 9 and 27. Then find the GCF. Step 1 Step 2 List the factors of each number. Identify the common factors. Factors of 9: 1, 3, 9 Common factors of 9 and 27: Factors of 27: 1, 3, 9, 27 1, 3, 9 The greatest of the common factors is 9. So, the GCF of 9 and 27 is 9. You can use the GCF and the Distributive roperty to express the sum of two numbers as a product. Write 9 1 27 as a product. Step 1 Step 2 Write each number as the product of Write an expression multiplying the GCF the GCF and another factor. and the sum of the two factors from Step 1. 9 5 9 3 1 27 5 9 3 3 9 3 (1 1 3) The product 9 3 (1 1 3) has the same value as 9 1 27. So, 9 1 27 5 9 3 (1 1 3). Find the GCF. 1. 18, 45 2. 33, 66 3. 72, 96 4. 50, 80 Use the GCF and the Distributive roperty to express the sum as a product. 5. 18 1 24 6. 15 1 75 7. 36 1 54 8. 16 1 20 Houghton Mifflin Harcourt ublishing Company 1-27 Reteach
Lesson 1.4 Enrich Greatest Common Factor Match Match the set of numbers inside each large triangle with its greatest common factor inside a small triangle. 1. 98 76 64 66 2. 84 108 60 2 3. 330 462 66 30 4. 570 390 1,380 12 5. Explain how you found the greatest common factor for three numbers. Houghton Mifflin Harcourt ublishing Company 1-28 Enrich
Lesson 1.5 Reteach roblem Solving Apply the Greatest Common Factor Use the Distributive roperty and a diagram to solve. Bethany is packing cookies for her drama club s bake sale. She has 28 oatmeal cookies and 36 peanut butter cookies to pack. Each bag will contain only one kind of cookie, and every bag will have the same number of cookies. What is the greatest number of cookies she can pack in each bag? How many bags of each kind will there be? Read the roblem What do I need to find? I need to find the cookies for each greatest bag number of and the number of each kind of cookie bags for. What information do I need to use? I need to use the number of cookies butter cookies and the number of How will I use the information? oatmeal peanut First, I can find the GCF of 28 and 36. Then I can draw a diagram showing bags of cookies the.. Step 1 Find the GCF of 28 and 36. Use prime factorization. 28 5 2 3 2 3 7 36 5 2 3 2 3 3 3 3 Multiply common prime factors: 2 3 2 5 GCF: 4 Solve the roblem Step 2 Write 28 as a product 28 5 4 3 of the GCF and another factor. Write 36 as a product of the 36 5 4 3 GCF and another factor. Step 3 Use the Distributive 28 1 36 5 roperty to write 28 1 36 4 ( 7 1 9 ) as a product. Step 4 Use the product to draw a diagram of the bags of cookies. Write for each oatmeal cookie and for each peanut butter cookie. 4 7 9 So, each bag will have cookies. There will be bags of oatmeal cookies and 4 9 7 bags of peanut butter cookies. 1. Jacob is putting 18 nonfiction and 30 fiction books on bookshelves. Each shelf will have only fiction or only nonfiction, and every shelf will have the same number of books. What is the greatest number of books for each shelf, and how many shelves will there be for each type of book? Houghton Mifflin Harcourt ublishing Company 1-29 Reteach
Lesson 1.5 Enrich Shelving Fun The Mega-Super-Mart gets shipments of all kinds of products every day. Unpack the shipping boxes and put the products in the bins. Make sure to only put one kind of product in each bin and put the same number of items in each bin. The bins should also hold the greatest number of items possible. 1. CNTENTS 9 baby dolls (B) CNTENTS 12 dress-up dolls (D) 2. CNTENTS 18 juice glasses (J) CNTENTS 24 water glasses (W) 3. CNTENTS 15 miniature toy cars (C) CNTENTS 20 miniature toy trucks (T) 4. CNTENTS 24 storybooks (S) CNTENTS 40 coloring books (C) Houghton Mifflin Harcourt ublishing Company 1-30 Enrich
Lesson 1.6 Reteach Add and Subtract Decimals Estimate 84.9 1 0.463. Then find the sum. 84.9 1 0.463 85 1 0 _ 85 Round each number to the nearest whole number. So, a good estimate is 85. Now line up the decimal points. Then add. 84.9 1 0.463 Write zeros as placeholders. 84.900 1 0.463 Regroup. 1 84.900 1 0.463 85.363 The answer is close to the estimate. So, the answer is reasonable. Evaluate 45.2 2 (27.93 2 10.84) using the order of operations. erform the operations in parentheses. 27.93 2 10.84 17.09 Subtract. So, 45.2 2 (27.93 2 10.84) is 28.11. 45.20 2 17.09 28.11 Estimate. Then find the sum or difference. 1. 62.38 + 26.92 2. 48.28 2 9.41 3. 81.04 52 + 16.44 4. 27.29 2 19.39 5. 743.5 2 462.87 6. 98.01 1 52.003 7. 74.9 2 16.227 Evaluate using the order of operations. 8. (235.152 1 77.12) 2 46.326 9. 11.024 2 (1.518 1 1.7) Houghton Mifflin Harcourt ublishing Company 1-31 Reteach
Lesson 1.6 Enrich Digit Logic Each symbol stands for a different digit from 0 9. The dark dots represent decimal points. Use the addition and subtraction problems to discover which digit is represented by each symbol. 5 5 3 5 5 5 5 0 5 4 5 5 5 5 1 2 1. Stretch Your Thinking Create your own subtraction problem with decimals using the symbols. Your problem should include regrouping. Houghton Mifflin Harcourt ublishing Company 1-32 Enrich
Lesson 1.7 Reteach Multiply Decimals When multiplying decimals, you can estimate to help you place the decimal point in the product. Estimate $32.05 3 7.4. Then find the product. Step 1 Estimate. Round each factor to the nearest ten or the nearest whole number. $32.05 is about $30, and 7.4 is close to 7. So, the product should be close to $210. 32.05 3 7.4 _ 30 3 7 _ 210 Step 2 Multiply. 32.05 3 7.4 12820 224350 237170 Step 3 lace the decimal point. Remember, the product is estimated to be 210. lace the decimal point so that the product is close in value to 210. So, the product is $237.17. 237.17 Estimate. Then find the product. 1. 8.6 3 4.1 _ 2. 12.8 3 2.21 3. $8.65 3 9.2 Evaluate using the order of operations. 4. 19.5 3 (21.04 2 18.7) 5. 11.7 1 (7.92 3 8.5) Houghton Mifflin Harcourt ublishing Company 1-33 Reteach
Lesson 1.7 Enrich lace the Decimal oint Each answer shows the decimal point in the wrong place. Estimate each product. Then make sure the decimal point is put in the correct place. 1. 1.6 3 0.9 14.4 Estimate: Correct answer: 3. 2.7 3 18.1 4.887 Estimate: Correct answer: 5. 0.9 3 57.9 5.211 Estimate: Correct answer: 7. 34.5 3 12.2 42.09 Estimate: Correct answer: 2. 4.6 3 3.2 147.2 Estimate: Correct answer: 4. 9.6 314.7 1,411.2 Estimate: Correct answer: 6. 0.75 3 4.22 316.5 Estimate: Correct answer: 8. 28.8 3 15.5 4.464 Estimate: Correct answer: How can estimating the product help you place the decimal point? Houghton Mifflin Harcourt ublishing Company 1-34 Enrich
Lesson 1.8 Reteach Divide Decimals by Whole Numbers When you divide a decimal by a whole number, place the decimal point in the quotient directly above the decimal point in the dividend. Estimate 12 q w 60.84. Then find the quotient. Step 1 Estimate the quotient, using compatible numbers. 60 and 12 are compatible numbers because 12 divides evenly into 60. 60 4 12 = 5 Step 2 Use long division to divide. 5.07 12 q w 60.84 260 8 20 84 284 0 lace the decimal point in the quotient directly above the decimal point in the dividend. Since 8 tenths cannot be shared among 12 groups, write 0 as a placeholder in the tenths place. So, 60.84 412 5 5.07. Estimate. Then find the quotient. 1. 16.48 4 8 2. 191.7 4 9 3. 4 q w 21.64 4. 14 q w 41.44 5. 21 q w 49.14 6. 6 q w 3.78 7. 92.8 4 16 8. 5 q w 1.725 9. 11 q w 135.3 10. 9 q w 7.29 Houghton Mifflin Harcourt ublishing Company 1-35 Reteach
Lesson 1.8 Enrich Decimal Match-Up Draw lines to match each expression in the left column to a value in the right column. 1. 169.12 4 14 39.7 2. 186.24 4 12 16.08 3. 158.8 4 4 147.156 4. 122.23 4 17 12.08 5. 385.92 4 24 24.03 6. (300 2 100.9) 4 10 3.94 7. (189.6 1 74.73) 4 11 15.52 8. 102.756 1 (710.4 4 16) 8.88 9. 122.14 4 31 7.19 10. 781.44 4 88 19.91 Houghton Mifflin Harcourt ublishing Company 1-36 Enrich
Lesson 1.9 Reteach Divide with Decimals When dividing a decimal by a decimal, rewrite the divisor as a whole number. To keep the problem equivalent, move the decimal point in the dividend the same direction and the same number of places. Rewrite the problem so that the divisor is a whole number. 300.7 4 1.24 300.7 is the dividend and 1.24 is the divisor. Change the Divisor 1.24 3 100 5 124 Change the Dividend 300.7 3 100 5 30,070 So, 300.7 4 1.24 is equivalent to 30,070 4 124. Find the quotient. 0.55 q w 24.2 Step 1 Step 2 Rewrite the problem so that the divisor is a whole number. Divisor Dividend 0.55 3 100 5 55 24.2 3 100 5 2,420 So, 24.2 4 0.55 5 44. Multiply 1.24 by 100 because 1.24 has two decimal places. To keep the problem equivalent, multiply the dividend by the same number, 100. Divide. 44 55 q w 2420 2220 220 2220 0 Rewrite the problem so that the divisor is a whole number. 1. 8.9 4 0.62 2. 21.05 4 0.2 3. 512.3 4 2.71 4. 18.62 4 0.02 Find the quotient. 5. 8.75 4 0.7 6. 72.24 4 5.6 7. 0.21 q w 1.3545 8. 2.17 q w 18.228 Houghton Mifflin Harcourt ublishing Company 1-37 Reteach
Lesson 1.9 Enrich Find the Decimal oint Circle the correct answer without actually dividing. 6.9 1. 3.45 4 0.5 5 0.69 0.069 19 2. 1.52 4 0.08 5 1.9 0.19 6.0 3. 1.08 4 1.8 5 0.6 0.06 9 5. 0.009 4 0.1 5 0.9 0.09 51,000 7. 30.6 4 0.006 5 5,100 510 251.3 9. 50.26 4 2 5 25.13 2.513 16,007 11. 64.028 4 0.4 5 1,600.7 160.07 100 4. 2.3 4 0.23 5 10 0.1 100.2 6. 2.004 4 0.02 5 10.2 1.2 4,008 8. 2.004 4 0.0005 5 408 48 2 10. 0.00006 4 0.3 5 0.002 0.0002 20 12. 4.1 4 2.05 5 2 0.2 161,000 13. 480.3 4 0.03 5 160,100 16,010 15. How many places should you move the decimal point in the dividend if the divisor is 0.000000009? 500,000 14. 5 4 0.0001 5 50,000 5,000 16. Stretch Your Thinking If the dividend is a whole number and the divisor is 3.68, how do you move the decimal point for the dividend? Houghton Mifflin Harcourt ublishing Company 1-38 Enrich