Objective To introduce the concept of square roots and the use of the square-root key on a calculator. Assessment Management

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1 Unsquaring Numbers Objective To introduce the concept of square roots and the use of the square-root key on a calculator. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Use exponential notation to name square numbers, and explore the relationship between square numbers and square roots. [Number and Numeration Goal ] Key Activities Students investigate unsquaring numbers without using the square-root key on a calculator and use the square-root key to test their answers. They explore properties of square numbers and their square roots. Key Vocabulary unsquaring a number square root square-root key Materials Math Journal, p. Study Link 7 calculator overhead calculator (optional) 6 counters (optional) Playing Multiplication Top-It (Extended-Facts Version) Student Reference Book, p. Math Masters, p. 9 per partnership: each of number cards 0 (from the Everything Math Deck, if available), calculator Students use their knowledge of extended facts to form and compare numbers. Ongoing Assessment: Recognizing Student Achievement Use Math Masters, page 9. [Operations and Computation Goal ] Math Boxes 8 Math Journal, p. Students practice and maintain skills through Math Box problems. Study Link 8 Math Masters, p. Students practice and maintain skills through Study Link activities. ENRICHMENT Comparing Numbers with Their Squares Math Masters, p. calculator Students investigate the relationship between numbers and their squares. EXTRA PRACTICE -Minute Math -Minute Math, p. 08 slate or paper Students practice using the square root sign. Advance Preparation Familiarize yourself with the use of the square root key on your students calculators. Teacher s Reference Manual, Grades 6 pp. 7, 7, 79 8, 9 98, Unit Number Theory

2 Getting Started Mental Math and Reflexes Pose the following problems. Have students write an expression as you describe the calculation. Students are not expected to calculate the solution. Answers may vary. The sum of 9 and less than The quotient of divided by 6 / 6 7 less than the product of and 9 ( 9) - 7 Double 8 and then add 0 more ( 8) times the sum of 8 and 0 (8 + ) 8 less than the sum of 0 and (0 + ) - 8 more than triple + ( ) 0 less than triple 0 ( 0) - 0 Math Message Find the numbers that make these statements true. = = 8 Study Link 7 Follow-Up Have partners compare answers and resolve differences. Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION Interactive whiteboard-ready epresentations are available at to help you teach the lesson. Algebraic Thinking Ask a volunteer to read the first problem. Some number times some number equals. What numbers could the placeholders in this problem represent? The factors of ; and ; Write the first problem on the board or a transparency, replacing both placeholders with the letter n. Ask students what number the letter n (the variable) represents. Explain that the variable can only represent one number if the number sentence is true. Tell students that unknown numbers will be represented using variables as the placeholders. Rewrite the second problem using the variable m. m = 8 Ask a volunteer to read the problem. m squared equals 8; m m equals 8 What is the number m that makes this number sentence true? 9 Unsquaring Numbers WHOLE-CLASS Begin this activity by explaining that solving problems like the Math Message problems requires unsquaring a number. We needed to undo the operation that squared the number. If students square a number, they multiply it by itself to get the product. Given the product of a squared number, they have to undo the multiplication in order to identify the number that was squared. = p Square the number to find p. PROBLEM SOLVING n n = 6 Unsquare the number 6 to find n. Adjusting the Activity ELL Have students use counters to build the square array for 6. Note that when 6 is unsquared, the result is the same as the number of rows, or the number of columns, of the original square. AUDITORY KINESTHETIC TACTILE VISUAL The difference between squaring and unsquaring a number Lesson 8

3 Ask: What number, multiplied by itself, is equal to 89? Give students a few minutes to find the number. They may use their calculators if they wish. After a few minutes, survey the class for their solution strategies. Most students will have used one of the following approaches: The random method: Some students might have tried various numbers without using a system to guide their choices. The squeeze method: Some students might have tried various numbers, each time using the result to help select their next choice. To unsquare 89, you might: Try 0: 0 = 00; much less than 89 Try 0: 0 = 00; more than 89 Then try numbers between 0 and 0, probably closer to 0 than to 0. Try 8: 8 = ; still too large, but closer. Try 7: 7 = 89; the answer is 7. Suggested square numbers for students to unsquare Date LESSON 8 Time Unsquaring Numbers You know that The number 6 is called the square of 6. If you unsquare 6, the result is 6. The number 6 is called the square root of 6.. Unsquare each number. The result is its square root. Do not use the key on your calculator. Example: The square root of is. a. b. c. d. The square root of is. 79 The square root of 79 is.,600 The square root of,600 is. 6 The square root of 6 is.. Which of the following are square numbers? Circle them ,0,6 6,00,770 List all factors of each square number. Make a factor rainbow to check your work. Then fill in the missing numbers.. 9:. 6:. 8: ,000 00, , 8, ,89, : The square root of 00 is 0. 9 The square root of 9 is Math Journal, p. Student Page The square root of 6 is. 8 The square root of 8 is Endings and products: When students established an interval, such as the interval from 0 to 0, some might have reasoned that since 7 ends in 7, and 7 7 = 9; then 7 should be the next choice because 89 also ends in 9. If students mention using the square-root key on their calculator, acknowledge that this is an efficient way of unsquaring a number, but the focus on this portion of the lesson is to help them understand the process of squaring and unsquaring numbers before they use the calculator function. Give students a few more square numbers to unsquare. (See margin.) Challenge them to use as few guesses as possible. Tell students that when they unsquare a number, they have found the square root of the number. What number squared is 6? 8 So what is 6 unsquared? 8, because 8 8 = 6 What is the square root of 6? 8 Finding the Square Root of Numbers (Math Journal, p. ) Allow partners a few minutes to complete Problems and. Survey the class for suggestions for checking the answers in Problem. Most students will respond with the following possibilities: Multiply the square root of a number by itself. Use the square-root key, for example find the square root of a number. PARTNER ELL, on the calculator to To support English language learners, write the following on the board: 8 = 8 8 = 6. The square of 8 is 6. The square root of 6 is 8. Unit Number Theory

4 Explain that in the same way that the class has used a calculator to test the result of other computations, they will use a calculator to test that they have accurately found the square-root of a number. If available, use an overhead calculator to demonstrate how to use the square-root function key. Model for students how to test the answers in Problem. Emphasize the following points: If the display shows a whole number, then the original number is a square number. For example, 76 is a square number because using the square-root key displays a whole number. If the display shows a decimal, then the original number is not a square number. For example, 79 is not a square number because using the square-root key displays a decimal (rounded to 6 decimal places). Ask students to check the remaining numbers in Problem. Partners complete the remaining problems on journal page. Round Player >, <, = Player Game Master Round Player >, <, = Player Ongoing Learning & Practice EMcuGMM_U0_067-0.indd 9 Math Masters, p. 9 /0/0 :6 PM Playing Multiplication Top-It (Extended-Facts Version) (Student Reference Book, p. ; Math Masters, p. 9) PARTNER Students apply their knowledge of basic multiplication facts to extended facts by playing Multiplication Top-It (Extended-Facts Version). Students use the same rules as described on Student Reference Book, page ; however, they attach a zero to the first card drawn and multiply by the second card drawn. For example, suppose is the first number drawn; 7 is the second number drawn. The student would compute: 0 = 0; 0 7 = 0. Date Student Page Time Ongoing Assessment: Recognizing Student Achievement Math Masters Page 9 Use the (Math Masters, page 9) to assess students ability to solve and compare multiplication extended fact problems. Have the class record and compare 70 8 and 0 9 for the sample record. Partners record their first five rounds. Students are making adequate progress if they correctly solve and compare all five extended facts. Some students may be able to solve and compare problems with both factors multiplied by 0: [Operations and Computation Goal ] LESSON 8 Math Boxes. Write < or >. a..8 > 0.8 b. 0. > 0.0 c. 6. > 6.08 d. 0.0 < 0. e. 7. < List all the factors of 6.,,, 8, 6,, 6. Round each number to the nearest thousand. a. 8,69 9,000 b. 9,7 0,000 c.,60,8,60,000 00,000 d. 00,97 e. 99, ,000. In the morning, I need 0 minutes to shower and dress, minutes to eat, and another minutes to ride my bike to school. School begins at 8:0 A.M. What is the latest time I can get up and still get to school on time? 7:0 A.M Subtract. Show your work. Math Boxes 8 (Math Journal, p. ) INDEPENDENT 777,009 a. b. c. d ,8 8, 997 7,8 7 Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson -6. The skill in Problem previews Unit content. Math Journal, p. EMcuGMJ_U0_00-08.indd // :0 AM Lesson 8

5 STUDY LINK 8 Factor Rainbows, Squares, and Square Roots. List all the factors of each square number. Make a factor rainbow to check 7 your work. Then fill in the missing numbers. Reminder: In a factor rainbow, the product of each connected factor pair should be equal to the number itself. 8 6 For example, the factor rainbow for º 6 6 º 8 6 º 6 6 looks like this: Example: : The square root of is. : The square root of is. 9: 6: 9 The square root of 9 is The square root of 6 is 6.. Do all square numbers have an odd number of factors? Unsquare each number. The result is its square root. Do not use the square root key on your calculator..,,,, Study Link Master 0.,00 The square root of is. The square root of,00 is 0.,, 9 Yes 9,,,, 6, 9,, 8, 6 Writing/Reasoning Have students write a response to the following: Was Jason correct when he said that 6 is a prime number in Problem? Explain your answer. answer: Jason was not correct. The factors of 6 are,,, 8, 6,, and 6. Because it has more than two factors, it is a composite number. A prime number has only two factors. Study Link 8 (Math Masters, p. ) INDEPENDENT Home Connection Students list all the factors of the first square numbers, write numbers in exponential notation, and identify square roots. Practice., ,8,90 8 6,9, R 9. 9 Math Masters, p.,60 Differentiation Options ENRICHMENT PARTNER Comparing Numbers with Their Squares (Math Masters, p. ) 0 Min LESSON 8 Comparing Numbers with Their Squares. a. Unsquare the number. b. Unsquare the number 0.. a. Is greater than or less than? b. c. Is greater than or less than? 0. a. Is 0.0 greater than or less than? b. Use your calculator. 0.0 c. Is 0.0 greater than or less than 0.0?. a. When you square a number, is the result always greater than the number you started with? b. Can it be less? c. Can it be the same? Teaching Master 0 Yes Yes Greater than Greater than Less than 0. Less than No. Write true statements about squaring and unsquaring numbers. Answers vary. To further explore factoring numbers, have students investigate the relationship between numbers and their squares. Ask students to think about the following question as they work the problems for this activity: When you square a number, will the result be greater than, less than, or equal to the number? Guide students to recognize that squaring a number does not necessarily result in a number that is greater than the original number. For example, both 0 and are equal to their squares. (See Problem.) Ask students what they noticed about the numbers and relationships they found in Problem. The number 0.0 is a decimal; the square was smaller. Explain that the square of a number that is greater than 0, but less than, is always less than the original number. Ask volunteers to suggest other numbers between 0 and for partners to square. EXTRA PRACTICE SMALL-GROUP -Minute Math Min To offer students more experience with using the square root sign, see -Minute Math, page 08. Math Masters, p. 6 Unit Number Theory

6 Round Player >, <, = Player Round Player >, <, = Player Copyright Wright Group/McGraw-Hill 9

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