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Divisibility Objectives To introduce divisibility rules for division by 2, 3, 5, 5 6, 9, and 10; and how to use a calculator to test for divisibility by a whole number. www.everydaymathonline.com ePresentations eToolkit Algorithms Practice EM Facts Workshop Game™ Teaching the Lesson Key Concepts and Skills • Use divisibility rules to solve problems. [Number and Numeration Goal 3] • Explore the relationship between the operations of multiplication and division. [Operations and Computation Goal 2] Key Activities Students use a calculator to test for divisibility by a whole number. They learn and practice divisibility rules. Key Vocabulary factor rainbow divisible by quotient divisibility rule Materials Math Journal 1, pp. 13 and 14 Study Link 1 4 calculator overhead calculator (optional) Family Letters Assessment Management Common Core State Standards Ongoing Learning & Practice 1 2 4 3 Playing Factor Captor Student Reference Book, p. 306 Math Masters, pp. 453 and 454 counters or centimeter cubes calculator Students practice finding factors of a number. Math Boxes 1 5 Math Journal 1, p. 15 Students practice and maintain skills through Math Box problems. Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 4. [Number and Numeration Goal 1] Study Link 1 5 Math Masters, p. 15 Students practice and maintain skills through Study Link activities. Curriculum Focal Points Interactive Teacher’s Lesson Guide Differentiation Options READINESS Practicing Divisibility with Counters per partnership: 66 counters, 3 dice Students use dice and counters to predict divisibility relationships between 2 numbers. EXTRA PRACTICE Practicing Multiplication Facts Math Journal 1, p. 9 Math Masters, p. 11 Students use a multiplication facts routine. ENRICHMENT Exploring a Test for Divisibility by 4 Math Masters, p. 16 Students use place-value concepts to investigate a test for divisibility by 4. ELL SUPPORT Building a Math Word Bank Differentiation Handbook, p. 142 Students add the terms divisor, dividend, quotient, and remainder to their Math Word Banks. Advance Preparation Teacher’s Reference Manual, Grades 4–6 pp. 79–83, 267–269 Lesson 1 5 037_EMCS_T_TLG1_U01_L05_576825.indd 37 37 2/1/11 8:55 AM Getting Started Mental Math and Reflexes Math Message Pose basic and extended multiplication/division facts. Have students write the answers for each set of problems. At the end of each set, ask students to describe the patterns. Suggestions: 5 ∗ 5 25 5 ∗ 50 250 5 ∗ 500 2,500 5 ∗ 5,000 25,000 6 ∗ 3 18 60 ∗ 3 180 600 ∗ 3 1,800 6,000 ∗ 3 18,000 8 ∗ 4 32 80 ∗ 40 3,200 800 ∗ 400 320,000 8,000 ∗ 4,000 32,000,000 Solve Problems 1 and 2 at the top of journal page 13. Study Link 1 4 Follow-Up Have partners compare answers. Ask the class how they know that all possible factors have been listed. Have volunteers model using a factor rainbow to pair factors for 25, 28, 42, and 100. If there is an odd number of factors, the middle factor is paired with itself. Explain that this only happens with square numbers. 1 NOTE Some students may benefit from doing the Readiness activity before you begin Part 1 of each lesson. See the Readiness activity in Part 3 for details. 2 3 4 6 8 12 16 24 48 1 Teaching the Lesson ▶ Math Message Follow-Up WHOLE-CLASS DISCUSSION (Math Journal 1, p. 13) NOTE If possible, use an overhead calculator to model the keystrokes and calculator displays for lesson examples. Interactive whiteboard-ready ePresentations are available at www.everydaymathonline.com to help you teach the lesson. Students share solution strategies. Use students’ responses to emphasize to the class that even numbers are numbers that are divisible by 2. ▶ Using a Calculator to Test INDEPENDENT ACTIVITY for Divisibility by a Whole Number (Math Journal 1, p. 13) NOTE Factor rainbows are introduced in the Study Link Follow-Up. This tool helps students identify all of the factors for a given number. The rainbow is a visual representation of the factor pairs and provides a way to check if the factor list is complete. Share the factor rainbow in the Study Link Follow-Up with the class. Factor rainbows will be used again in Lesson 1-6. Recall for students the class discussion on the review of divisibility in Lesson 1-4. Remind students that a whole number (the dividend) is divisible by a whole number (the divisor) if the remainder in the division is zero. The result or quotient, must be a whole number. If the remainder is not zero, then the number being divided is not divisible by the second number. If your students use calculators that display answers to division problems as a quotient and a whole number remainder, you might want to demonstrate the procedure. With the TI-15 calculator, this is done by pressing the Int÷ key instead of the ÷ key. With the Casio fx-55, use the key. For example, If you press 27 Int÷ 5 , or 27 5 , the display will show a quotient of 5 with a remainder of 2. 38 Unit 1 Number Theory 038-041_EMCS_T_TLG1_U01_L05_576825.indd 38 1/24/11 5:28 PM Student Page Date Adjusting the Activity Time LESSON 䉬 Math Message 1. Circle the numbers that are divisible by 2. 2. What do the numbers that you circled have in common? 28 dividend divisor quotient remainder K I N E S T H E T I C T A C T I L E 57 33 112 123,456 211 Allow 5 to 10 minutes for students to complete Problems 3–10 on the journal page 13. 705 Example 2: Is 122 divisible by 5? To find out, divide 122 by 5. 135 / 5 ⫽ 27 122 / 5 ⫽ 24.4 The answer, 27, is a whole number. So 135 is divisible by 5. The answer, 24.4, has a decimal part. So 122 is not divisible by 5. Use your calculator to help you answer these questions. 3. Is 267 divisible by 9? No 5. Is 809 divisible by 7? 7. Is 4,735 divisible by 5? No Yes 9. Is 5,268 divisible by 22? No Yes 4. Is 552 divisible by 6? 6. Is 7,002 divisible by 3? 8. Is 21,733 divisible by 4? Yes No 10. Is 2,072 divisible by 37? Yes Math Journal 1, p. 13 WHOLE-CLASS ACTIVITY PROBLEM PRO PR P RO R OB BLE BL L LE LEM EM SOLVING SO S OL O LV VING VIN IIN NG Ask: How can you know that a number is divisible by 2 without actually doing the division? Numbers that end in 0, 2, 4, 6, or 8 are divisible by 2. Can you tell whether a number is divisible by 10 without dividing? Yes; numbers that end in 0 are divisible by 10. Can you tell whether a number is divisible by 3 without dividing? Allow students to explore this question before continuing. There are rules that let us test for divisibility without dividing or using a calculator. 1. Go over the divisibility-by-3 rule on journal page 14: A number is divisible by 3 if the sum of its digits is divisible by 3. 2. Illustrate by using the rule to test several examples. ● 399 V I S U A L When testing for divisibility with a calculator that does not display remainders, the first number is not divisible by the second number if the quotient has a decimal part. Ask students to use their calculators to test whether 27 is divisible by 9. 27 is divisible by 9 because the result is 3—a whole number. Test whether 27 is divisible by 5. 27 is not divisible by 5 because the result is 5.4— not a whole number. (Math Journal 1, p. 14) 900 Suppose you divide a whole number by a second whole number. The answer may be a whole number, or it may be a number that has a decimal part. If the answer is a whole number, we say that the first number is divisible by the second number. If the answer has a decimal part, the first number is not divisible by the second number. Example 1: Is 135 divisible by 5? To find out, divide 135 by 5. ▶ Introducing Divisibility Rules 5,374 They are all even numbers. 135 ÷ 5 = 27 R0 A U D I T O R Y Divisibility 15 Write the number model from the first example on journal page 13 on the board with each number appropriately labeled, including a remainder of zero. Is 237 divisible by 3? Yes. 2 + 3 + 7 = 12, and 12 is divisible by 3. Student Page Date Time LESSON 15 䉬 Divisibility Rules For many numbers, even large ones, it is possible to test for divisibility without actually dividing. Here are the most useful divisibility rules: 䉬 All numbers are divisible by 1. 䉬 All even numbers (ending in 0, 2, 4, 6, or 8) are divisible by 2. 䉬 A number is divisible by 3 if the sum of its digits is divisible by 3. Example: 246 is divisible by 3 because 2 + 4 + 6 = 12, and 12 is divisible by 3. 䉬 A number is divisible by 6 if it is divisible by both 2 and 3. Example: 246 is divisible by 6 because it is divisible by 2 and by 3. 䉬 A number is divisible by 9 if the sum of its digits is divisible by 9. Example: 51,372 is divisible by 9 because 5 + 1 + 3 + 7 + 2 = 18, and 18 is divisible by 9. 䉬 A number is divisible by 5 if it ends in 0 or 5. 䉬 A number is divisible by 10 if it ends in 0. ● Is 415 divisible by 3? No. 4 + 1 + 5 = 10, and 10 is not divisible by 3. 1. Divisible. . . Number Students complete Problems 1–3 independently. Have them check each other’s work. by 2 ? 7,960 384 by 3 ? by 6 ? by 9 ? ✓ 75 3. Ask students to provide examples of a number that is divisible by 3 and a number that is not. Encourage them to apply the divisibility-by-3 test first. Then have them check that it works by carrying out the division on their calculators. Assign small groups to present examples for the remaining divisibility rules (5, 6, or 9). Test each number below for divisibility. Then check on your calculator. ✓ ✓ ✓ ✓ ✓ ✓ ✓ 3,725 90 36,297 ✓ ✓ ✓ 2. Find a 3-digit number that is divisible by both 3 and 5. 3. Find a 4-digit number that is divisible by both 6 and 9. by 5 ? by 10 ? ✓ ✓ ✓ ✓ ✓ ✓ Sample answers: 735; 540 Sample answers: 1,800; 5,454 Math Journal 1, p. 14 Lesson 1 5 EM3cuG5TLG1_038-041_U01L05.indd 39 39 11/5/10 7:20 PM Student Page Date Time LESSON 15 1. Circle the numbers that are divisible by 3. 221 381 474 922 2. 726 Round 3,045,832 to the nearest… a. million. b. thousand. c. ten-thousand. 3,000,000 3,046,000 3,050,000 11 3. 4 249 Complete the table. Fraction 4. Decimal Percent 0.60 0.25 60% 0.50 50% 70% 3 ᎏᎏ 5 ᎏ1ᎏ 4 ᎏ1ᎏ 2 7 ᎏᎏ 10 25% 0.70 0.85 85 ᎏᎏ 100 Write an 8-digit numeral with 5 in the hundredths place, 8 in the tens place, 3 in the ones place, 8 in the thousands place, 4 in the hundreds place, and 6 in all other places. 6 6 8, 4 8 3 . 6 5 85% Complete. a. b. c. d. e. PARTNER ACTIVITY ▶ Playing Factor Captor (Student Reference Book, p. 306; Math Masters, pp. 453–454) Students practice finding factors of a number by playing Factor Captor. Students have the option of playing any of the two Factor Captor grids. If students are using Grid 2 for the first time, suggest that they omit the last two rows of the gameboard. 4 30 31 80 90 5. 2 Ongoing Learning & Practice Math Boxes 6. 56,000 400 5,000 ⫽ 2,000,000 70 6,300 ⫽ 90 300 21,000 ⫽ 70 900 720,000 ⫽ 800 70 800 ⫽ Pencils are packed 18 to a box. How many pencils are in 9 boxes? 162 pencils (unit) INDEPENDENT ACTIVITY ▶ Math Boxes 1 5 (Math Journal 1, p. 15) 18 19 20 Math Journal 1, p. 15 NOTE As students continue to develop their strategies for Factor Captor, they will find that as more numbers are used, the scoring rules increasingly reward a player for planning ahead and anticipating an opponent’s moves. Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 1-7 and 1-9. The skills in Problems 5 and 6 preview Unit 2 content. Writing/Reasoning Have students write a response to the following: Explain how you solved Problem 6. Sample answer: Because there are 18 pencils per box and 9 boxes total, I multiplied 18 ∗ 9: 10 ∗ 9 is 90 and 8 ∗ 9 is 72; 90 + 72 = 162. There are 162 pencils in all. Ongoing Assessment: Recognizing Student Achievement Date Use Math Boxes, Problem 4 to assess students’ understanding of place value. Students are making adequate progress if they are able to correctly position and identify digits and their values in whole numbers through the hundred-thousands and decimals through the hundredths. Study Link Master Name Math Boxes Problem 4 [Number and Numeration Goal 1] Time Divisibility Rules STUDY LINK 15 䉬 䉬 All even numbers are divisible by 2. 11 䉬 A number is divisible by 3 if the sum of its digits is divisible by 3. ▶ Study Link 1 5 䉬 A number is divisible by 6 if it is divisible by both 2 and 3. 䉬 A number is divisible by 9 if the sum of its digits is divisible by 9. INDEPENDENT ACTIVITY (Math Masters, p. 15) 䉬 A number is divisible by 5 if it ends in 0 or 5. 䉬 A number is divisible by 10 if it ends in 0. 1. Use divisibility rules to test whether each number is divisible by 2, 3, 5, 6, 9, or 10. Number 夹 998,876 夹 36,540 5,890 Divisible… by 2? by 3? by 6? by 9? by 5? by 10? ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ 33,015 1,098 Home Connection Students use divisibility rules to test whether numbers are divisible by 2, 3, 5, 6, 9, or 10. They learn the divisibility rule for 4 and recheck the numbers for those that are also divisible by 4. A number is divisible by 4 if the tens and ones digits form a number that is divisible by 4. Example: 47,836 is divisible by 4 because 36 is divisible by 4. It isn’t always easy to tell whether the last two digits form a number that is divisible by 4. A quick way to check is to divide the number by 2 and then divide the result by 2. It’s the same as dividing by 4, but is easier to do mentally. Example: 5,384 is divisible by 4 because 84 / 2 ⫽ 42 and 42 / 2 ⫽ 21. 2. Place a star next to any number in the table that is divisible by 4. 3. 250 º 7 ⫽ Practice 5. 1,750 (20 ⫹ 30) º 5 ⫽ 4. 250 6. 1,931 ⫹ 4,763 ⫹ 2,059 ⫽ 78 ⫼ 6 ⫽ 8,753 13 Math Masters, p. 15 40 Unit 1 Number Theory EM3cuG5TLG1_038-041_U01L05.indd 40 11/5/10 7:20 PM Teaching Master Name 3 Differentiation Options READINESS ▶ Practicing Divisibility LESSON 15 䉬 PARTNER ACTIVITY 15–30 Min with Counters To explore the concept of divisibility using a concrete model, have students use counters to determine whether a number is divisible by the numbers 1–6. Partners take turns rolling three dice. Make a two-digit number with two of the dice, and count out that number of counters. They predict whether the number of counters is divisible by the number on the third die. Then partners check the prediction by dividing the counters into the number of groups indicated on the third die. EXTRA PRACTICE ▶ Practicing Multiplication Facts SMALL-GROUP ACTIVITY Date 2. 10 cubes 100 cubes 1,000 cubes 1. Time Divisibility by 4 What number is shown by the base-10 blocks? 1 cube 1,111 Which of the base-10 blocks could be divided evenly into 4 groups of cubes? The groups of 1,000 cubes and 100 cubes 3. Is the number shown by the base-10 blocks divisible by 4? 4. Circle the numbers that you think are divisible by 4. 324 5,821 7,430 No 35,782,916 Use a calculator to check your answers. 5. Use what you know about base-10 blocks to explain why you only need to look at the last two digits of a number to decide whether it is divisible by 4. Sample answer: Because 1,000 and 100 are divisible by 4, the numbers that the thousands place and the hundreds place represent are always divisible by 4. So you have to look at only the number formed by the tens and ones digits. Math Masters, p. 16 5–15 Min (Math Journal 1, p. 9; Math Masters, p. 11) To provide additional practice with basic multiplication facts, have students use the facts routine introduced in Lesson 1-3. See Teacher’s Lesson Guide, pages 28 and 29 to review the procedure. ENRICHMENT ▶ Exploring a Test PARTNER ACTIVITY 5–15 Min for Divisibility by 4 (Math Masters, p. 16) To further explore divisibility, have students use place-value concepts to investigate why only the last 2 digits in a number determine whether the number is divisible by 4. ELL SUPPORT ▶ Building a Math Word Bank SMALL-GROUP ACTIVITY 5–15 Min (Differentiation Handbook, p. 142) To provide language support for division, have students use the Word Bank Template found on Differentiation Handbook, page 142. Ask students to write the terms divisor, dividend, quotient, and remainder; draw a picture representing each term; and write other related words. See the Differentiation Handbook for more information. Lesson 1 5 EM3cuG5TLG1_038-041_U01L05.indd 41 41 1/3/11 2:15 PM Name LESSON 13 Date Multiplication Facts Copyright © Wright Group/McGraw-Hill A List 3 ∗ 6 = 18 6 ∗ 3 = 18 3 ∗ 7 = 21 7 ∗ 3 = 21 3 ∗ 8 = 24 8 ∗ 3 = 24 3 ∗ 9 = 27 9 ∗ 3 = 27 4 ∗ 6 = 24 6 ∗ 4 = 24 4 ∗ 7 = 28 7 ∗ 4 = 28 4 ∗ 8 = 32 8 ∗ 4 = 32 4 ∗ 9 = 36 9 ∗ 4 = 36 5 ∗ 7 = 35 7 ∗ 5 = 35 5 ∗ 9 = 45 9 ∗ 5 = 45 6 ∗ 6 = 36 6 ∗ 7 = 42 7 ∗ 6 = 42 6 ∗ 8 = 48 8 ∗ 6 = 48 6 ∗ 9 = 54 9 ∗ 6 = 54 7 ∗ 7 = 49 7 ∗ 8 = 56 8 ∗ 7 = 56 7 ∗ 9 = 63 9 ∗ 7 = 63 8 ∗ 8 = 64 8 ∗ 9 = 72 9 ∗ 8 = 72 9 ∗ 9 = 81 Time B List 3 ∗ 3= 9 3 ∗ 4 = 12 4 ∗ 3 = 12 3 ∗ 5 = 15 5 ∗ 3 = 15 4 ∗ 4 = 16 4 ∗ 5 = 20 5 ∗ 4 = 20 5 ∗ 5 = 25 5 ∗ 6 = 30 6 ∗ 5 = 30 5 ∗ 8 = 40 8 ∗ 5 = 40 6 ∗ 10 = 60 10 ∗ 6 = 60 7 ∗ 10 = 70 10 ∗ 7 = 70 8 ∗ 10 = 80 10 ∗ 8 = 80 9 ∗ 10 = 90 10 ∗ 9 = 90 10 ∗ 10 = 100 Bonus Problems 11 ∗ 11 = 121 11 ∗ 12 = 132 5 ∗ 12 = 60 12 ∗ 6 = 72 7 ∗ 12 = 84 12 ∗ 8 = 80 9 ∗ 12 = 108 10 ∗ 12 = 120 5 ∗ 13 = 65 15 ∗ 7 = 105 12 ∗ 12 = 144 6 ∗ 14 = 84 11 EM3cuG5MM_U01_002-032.indd 11 12/30/10 5:10 PM