# SQUARES AND SQUARE ROOTS

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1 1. Squares and Square Roots SQUARES AND SQUARE ROOTS In this lesson, students link the geometric concepts of side length and area of a square to the algebra concepts of squares and square roots of numbers. They create a table of perfect squares. They use the table to find square roots of perfect squares, and they approximate the square root of a whole number. This lesson is the first of a lesson cluster where students develop and practice rules for exponents. In future lessons, students will use the definition of exponents and inductive reasoning to make conjectures about rules for exponents. Then the rules for exponents will be formalized, and students will simplify expressions that include exponents and roots. Math Goals (Standards for posting in bold) Understand geometrically and numerically the connection between squaring a number and finding the square root of a number. (ALG.0) Approximate a square root by locating it between two consecutive integers. (Gr7 NS.4; Gr7 MR.7) Use fractions and decimals to approximate square roots. (Gr7 NS1.; Gr7 NS1.3; Gr7 MR.1; Gr7 MR.7) Summative Assessment Future Week Week 5: Exponents and Roots (Gr7 NS1.; Gr7 NS.4) Week 1 TP11

2 1. Squares and Square Roots PLANNING INFORMATION Student Pages Estimated Time: Minutes Materials Reproducibles * SP10: Ready, Set, Go * SP11: Table of Squares * SP1-13: Estimating Square Roots SP14: More Square Root Estimates Calculators (optional) Homework Prepare Ahead Management Reminders SP14: More Square Root Estimates Assessment * SP5: Knowledge Check 1 R86: Knowledge Challenge 1 A101-10: Weekly Quiz 1 Strategies for English Learners Give a visual review of why square numbers are called square numbers. 5 units = 5 Strategies for Special Learners Refer often to a number line with numbers and their square roots so that students see why the linear interpolation method makes sense. * Recommended transparency: Blackline masters for overheads and 51 can be found in the Teacher Resource Binder. Week 1 TP1

3 1. Squares and Square Roots exponential notation square of a number perfect square square root of a number linear interpolation THE WORD BANK n The exponential notation b (read as b to the power n ) is used to express n the product of b with itself n times: b = b b... b (n times). The number b is the base, and the natural number n is the exponent. Exponential notation is extended to arbitrary integer exponents by setting b 0 = 1 and - n b = 1. Example: 3 = = 8. (The base is and the exponent is 3.) = = 1,15. (The bases are 3 and 5.) 0 = = =. The square of a number is the product of the number with itself. Example: The square of 5 is 5, since is also 5, since (-5) = (-5) (-5) = 5. b n 5 = 5 5 = 5. The square of -5 A perfect square, or square number, is a number that is a square of a natural number. Example: The area of a square with integral side-length is a perfect square. The perfect squares are 1 = 1, 4 =, 9 = 3, 16 = 4, 5 = 5,. A square root of a number n is a number whose square is equal to n, that is, a solution of the equation x = n. The positive square root of a number n, written n, is the positive number whose square is n. Example: Both 5 and -5 are square roots of 5, because 5 = 5 and (-5) = 5. The positive square root of 5 is 5. Linear interpolation refers to a method of approximating the values of a function f at points of an interval a < x < c by the values of the linear function that coincides with f at the endpoints a and c. y y = f(x) linear approximation a c x Example: We may approximate x for 5 < x < 36 by the linear 1 y = 5 + ( x 5), which has values y = 5 = 5 function ( ) 11 at x = 5 and y = 36 = 6 at x = 36. Week 1 TP13

4 1. Squares and Square Roots MATH BACKGROUND Approximating Square Roots by Linear Interpolation Linear interpolation is a method by which the values y = f(x) of a function f on an interval x 1 < x < x are estimated by the values of the linear function y = mx + b that matches the values of f at the endpoints of the interval. The parameters m and b satisfy the two equations f(x 1 ) = mx 1 + b and f(x ) = m x + b. The graph of the linear approximation y = mx + b is then the straight line segment joining the points (x 1, f(x 1 )) and (x, f(x )) on the graph of f. The linear approximation can be found directly through proportional reasoning, without writing down any equations of lines. To illustrate, we approximate values of x. To find an approximate value for 7 : First find the closest perfect square (5) that is less than 7 and the closest perfect square (36) that is greater than 7. Then 5 < 7 < 36. The points 5 and 36 will be the endpoints of the interval of interpolation. Take the square root of each number: 5 = 5, 36 = 6. We aim to approximate 7 using the y-coordinate of the straight line through (5,5) and (36,6). y Math Background 1 Teacher Mathematical Insight y = x (5, 5) (36, 6) x Now 7 is units larger than 5, and 36 is 11 units larger than 5. Thus 7 is two of the distance from 5 to 36. We approximate 7 by the number that is elevenths ( 11) two elevenths of the distance from 5 to 36, that is, two elevenths of the distance from 5 to 6 (proportional reasoning!!). This number is 5 + = Summary: On a number line, 7 is of the distance from 5 to 36. Therefore, 11 7 is approximately equal to 11 of the distance from 5 = 5 to 36 = 6, which is The approximation to 7 is By linear interpolation to the nearest thousandth: Accurate to the nearest thousandth: Week 1 TP14

5 1. Squares and Square Roots PREVIEW / WARMUP Whole Class SP10* Ready, Set, Go Introduce the goals and standards for the lesson. Discuss important vocabulary as relevant. Students draw several squares and record both their side lengths and areas. Share and discuss. Stress the relationship between side length and area, the definitions of squares and square roots (numerically and geometrically), and the notations for squares and square roots. INTRODUCE Whole Class SP11* Table of Squares SP1-13* Estimating Square Roots Calculators Students complete the table of perfect squares. Encourage students to keep this table handy (or even memorize some of the perfect squares) because it will make computations that involve square roots easier. Students put the square roots of perfect squares on a number line. Why is the distance between 5 and 36 the same as the distance between 1 and 4? The distance between 5 and 6 ( 5 and 36 ) is the same as the distance between 1 and ( 1 and 4 ). Have students estimate the location of 7 on the number line. 7 is between what two perfect squares? 5 and 36. How can we use this information to approximate 7? Since 5 = 5 and 36 = 6, 7 is between 5 and 6. Since it is somewhat closer to 5, a reasonable approximation might be 5.1, 5., or 5.3. Show students the process of linear interpolation in order to find fraction and/or decimal approximations of the square roots of non-perfect squares. On a number line, 7 represents what fraction of the distance from 5 to 36? Since the distance from 5 to 36 is 11 units, and 5 to 7 is units, the fraction is 11. Since 5 = 5, we estimate 7 by 5 + = 5. Since = the estimation above, that is approximately 5., is supported. Week 1 TP15

6 1. Squares and Square Roots EXPLORE Pairs/Individuals SP1-13* Estimating Square Roots Calculators Students continue to estimate the square roots of non-perfect squares, first by finding the whole numbers between which they lie on the number line, and then by using linear interpolation. Use calculators as a check for reasonableness. PRACTICE Individuals SP14 More Square Root Estimates Use for additional practice or homework. In order to help students practice number sense and estimation skills, stress that the calculator should be used for checking results only. SUMMARIZE Whole Class SP1-13* Estimating Square Roots SP14 More Square Root Estimates Discuss the problem below. How do you know that 0 is between 4 and 5? The two perfect squares closest to 0 below and above are 16 and 5, and we know that 16 = 4 and 5 = 5. CLOSURE Whole Class SP10* Ready, Set Review the goals and standards for the lesson. Week 1 TP16

7 1. Squares and Square Roots SP11 Table of Squares 1 = 1 6 = 36 SELECTED SOLUTIONS = 4 3 = 9 7 = 49 8 = 64 4 = 16 9 = = 11 1 = = = = = = = = 441 = = 59 4 = = 5 10 = = 5 0 = = < 7 < 36, but 7 is closer to 5. 5< 7 < 6, but 7 is closer to = Therefore, 7 is about 5 or 5. (number will vary) (calculator check: 5.) 11 SP1-13 Estimating Square Roots < 40 < 49, but 40 is closer to 36. 6< 40 < 7, but 40 is closer to = Therefore, 40 is about 6 4 or 6.3 (number will vary) (calculator check: 6.3) < 30 < 36, but 30 is closer to 5. 5 < 30 < 6, but 30 is closer to = Therefore, 30 is about 5 5 or 5.5. (calculator check: 5.5) < 77 < 81, but 77 is closer to < 77 < 9, but 77 is closer to = Therefore, 77 is about 8 13 or 8.8. (calculator check: 8.8) 17 Week 1 TP17

8 1. Squares and Square Roots Number Between square roots of perfect squares: Between two integers: About (fraction): About (decimal): Calculator check (to the nearest tenth) SP14 More Square Roots Estimates and 9 & and and 5 & & and 81 8 & and 5 4 & and & and & Estimates are generally accurate to the nearest tenth. Then method is useful if a calculator is not available. Estimation develops number sense. Week 1 TP18

9 STUDENT PAGES

10 1. Squares and Square Roots SQUARES AND SQUARE ROOTS Ready (Summary) We will find squares and square roots of numbers, and approximate square roots that are not perfect squares. Set (Goals) Understand geometrically and numerically the connection between squaring a number and finding the square root of a number. Approximate a square root by locating it between two consecutive integers. Use fractions and decimals to approximate square roots. Go (Warmup) Draw several squares of different sizes on the grid paper. Record the side length and area for each. Unit 6: Conjecture and Justification (Student Packet) Week 1 SP10

11 1. Squares and Square Roots TABLE OF SQUARES Complete the table. 1 = = 3 = 4 = 5 = 6 = 7 = 8 = 9 = 10 = 11 = 1 = 13 = 14 = 15 = 16 = 17 = 18 = 19 = 0 = 1 = = 3 = 4 = 5 = Unit 6: Conjecture and Justification (Student Packet) Week 1 SP11

12 1. Squares and Square Roots ESTIMATING SQUARE ROOTS 1. Locate the following numbers on the number line below: Use the values on your number line to estimate the location of 7. 5 < 7 < 36, but 7 is closer to. 5 < 7 <, but 7 is closer to. Estimate the decimal part of 7 as a fraction = Therefore, 7 is about 5. (calculator check: ) 3. Use the values on your number line to estimate the location of 40. < 40 <, but 40 is closer to. < 40 <, but 40 is closer to. Estimate the decimal part of 40 as a fraction. = Therefore, 40 is about. (calculator check: ) Unit 6: Conjecture and Justification (Student Packet) Week 1 SP1

13 1. Squares and Square Roots ESTIMATING SQUARE ROOTS (continued) 4. Use the values on your number line to estimate the location of 30. < 30 <, but 30 is closer to. < 30 <, but 30 is closer to. Estimate the decimal part of 30 as a fraction. = Therefore, 30 is about. (calculator check: ) 5. Use the values on your number line to estimate the location of 77. < 77 <, but 77 is closer to. < 77 <, but 77 is closer to. Estimate the decimal part of 77 as a fraction. = Therefore, 77 is about. (calculator check: ) Unit 6: Conjecture and Justification (Student Packet) Week 1 SP13

14 1. Squares and Square Roots MORE SQUARE ROOT ESTIMATES Use fractions and decimals to approximate each square root. A B C D E F Number Between square roots of perfect squares: Between consecutive integers: About (fraction): About (decimal): Calculator check (to nearest tenth) and 9 and Compare your square root estimates in columns C and D to the rounded answer in column E. What are some advantages and disadvantages of each? Unit 6: Conjecture and Justification (Student Packet) Week 1 SP14

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