Perfect Squares, Square Roots, and Equations of the form x² = p

Transcription

1 Perfect Squares, Square Roots, and Equations of the form x² = p LAUNCH (8 MIN) Before Why might someone think that these expressions are equivalent? During Can these two expressions ever be equal? Explain. After If the variable x were a length in inches, what units would you most likely use for each expression? KEY CONCEPT (4 MIN) Remind students that the square of a negative number is positive, such as ( 2) 4. Note that the exponent here applies to the entire number 2, which is not equal to 2 4. PART 1 (7 MIN) How many solutions will each equation have? What number times itself is 4? How can you check your answer? PART 2 (7 MIN) How is this equation different from the one in Part 1? Andie Says (Screen 1) Use the Andie Says button to help students realize that both the numerator and denominator are perfect squares. Can a proper fraction be a perfect square? What is the first step in solving this equation? How do you find the square root of a fraction? PART 3 (6 MIN) About how big are most stamps? Why does this stamp seem larger? Andie Says (Screen 1) Use the Andie Says button to remind students that this equation has two solutions, but a negative side length does not make sense. While solving part (b) Is 1,200 a perfect square? How does that change how you solve the problem? All positive numbers have square roots. Do you think most square roots are integers? CLOSE AND CHECK (8 MIN) How can you compare the solutions of two equations of the form x p? For x p, p cannot be negative because a nonzero number squared is always positive. Is the same true for x p?

2 Perfect Squares, Square Roots, and Equations of the form x² = p LESSON OBJECTIVES 1. Use square root and cube root symbols to represent solutions to equations of the form x p and x p, where p is a positive rational number. Know that 2 is irrational. 2. Evaluate square roots of small perfect squares. FOCUS QUESTION How can you apply what you know about squares and square roots to write and solve equations of the form x p? How can you use equations in that form? MATH BACKGROUND In previous grades, students studied area and surface area and worked with square units. Previous topics in this course focused on exploring and approximating irrational numbers and solving linear equations in one variable. In this lesson, students combine their work with exponents and roots and solving equations. They learn about squares and square roots and apply this knowledge to solve equations of the form x p, where p is a positive rational number. Although students are familiar with squaring numbers, they will benefit from being reminded that the square of a negative number is positive, such as 2x 2 4. Therefore, every positive number has two square roots, as shown in the figure below. The radical symbol, by convention, refers to only the nonnegative (positive and zero) square root. So, the negative square root of 100 is written as 100 and the two roots of 100 can be written as 100. Students should know that the square of an integer is called a perfect square. The problems in this lesson may be easier to solve if students have memorized the perfect squares up to 144. Because all positive numbers have square roots, however, most square roots are not integers. When p is not a perfect square, students will use guess and check to estimate the square root. They will apply what they learned about approximating square roots in previous lessons to write the solutions in decimal form. When solving equations such as x 4 and x 16, students should understand that 25 taking the square root of each side of an equation uses inverse operations. Since the variable is squared, you can isolate the variable by taking the square root, and the equation is still balanced. One important difference is that there are two square roots of a number, so you have to include on one side of the equation. In future lessons, students will learn about cubes and cube roots, and solve equations of the form x p, where p is a positive rational number. They will learn the properties of exponents, including negative and zero exponents, in order to write equivalent expressions. In later topics, they will use exponents to express very large and small numbers in scientific notation. Make sure students can read the exponents correctly on the whiteboard. Students may not treat them as exponents or may ignore them because the exponents are

3 smaller than most numbers on the board. If students do have a hard time reading the exponents, you may want to explain that the word exponent is derived from Latin, meaning out of place. Since the exponent is written smaller and above its base, it looks out of place. LAUNCH (8 MIN) Objective: Apply knowledge of square numbers to compare algebraic expressions involving exponents. Students examine and compare two algebraic expressions, one linear and one exponential, that are sometimes equal but not equivalent. Instructional Design Consider recording a class vote before solving the problem. Have each student pair up with a neighbor to discuss the problem and to reduce the anxiety about giving a wrong answer. Before Why might someone think that these expressions are equivalent? [Sample answer: The expressions look similar because they both include the variable x and the number 2.] During What is different about the role the number 2 plays in each expression? [Sample answer: In the first expression it is an exponent. In the second expression, it is a coefficient.] How do you say each expression using words? [Sample answer: 2x is two x or two times x ; x is x squared or x raised to the power of 2.] Can these two expressions ever be equal? Explain. [Sample answer: Yes; for x-values of 0 and 2] After If the variable x were a length in inches, what units would you most likely use for each expression? [Sample answer: 2x would be in inches and x would be in square inches, or in. ] Is 2x the same as (2x)? Explain. [Sample answer: No; (2x) 2x 2x 4x, which is not equivalent to 2x.] Solution Notes Make sure students recognize that both expressions have the same variable x and number 2, but 2 is multiplied by the variable in 2x and 2 is the exponent in x. Some students may set the two expressions equal to solve for x as shown below. 2x = x 2 2x x = x2 x 2 = x Note that this only identifies one possible solution: x 2. When you divide each side of an equation by x, you are assuming that x is nonzero (since division by zero is undefined). However, you are also saying that x 0 satisfies the equation.

4 Connect Your Learning Move to the Connect Your Learning screen. In the Launch, students compared two algebraic expressions using substitution. Have students explain that some values of the variable make the expressions equal, but not all values. Discuss why this means the expressions are not equivalent. Use the Focus Question to get students thinking about solving equations that involve exponents and how square roots may be useful. KEY CONCEPT (4 MIN) Teaching Tips for the Key Concept Students are familiar with squaring numbers. However, they will benefit from being reminded that the square of a negative number is positive, such as ( 2) 4. Note that the exponent here applies to the entire number 2, which is not equal to 2 4. How many solutions does an equation of the form x p have? [Sample answer: two solutions:, for p 0] Does p stand for perfect square? [No; p can be any number.] Why it Works To solve an equation of the form x p, you are looking for a number x, that when squared, equals p. This is the definition of a square root, so squaring a number and taking the square root of a number are inverse operations. Therefore, you can solve for x by taking the square root of each side of the equation. PART 1 (7 MIN) Objective: Solve equations of the form x p, where p is a positive perfect square rational number. Students use square roots to solve equations of the form x p where p is a perfect square. They will write the solutions using the new notation, How many solutions will each equation have? [Sample answer: Two; every positive number has two square roots.] In what other situations have you found more than one solution? [Sample answer: solving inequalities.] What number times itself is 4? [ 2] How can you check your answer? [Sample answer: You can substitute the solution for the variable and check that the equation is true. When you have two solutions, you have to check each answer separately.] Solution Notes Many students will be able to solve these problems mentally. Encourage students to show their work and check their answer. Stress that they will not always be able to find square roots using mental math, so they will benefit from the experience.

5 Differentiated Instruction For struggling students: To help students better remember perfect square numbers, have them physically build squares using square tiles in the Square Numbers & Square Roots mode of the Area Models tool. Students should then create a list of commonly used perfect squares and memorize them. Error Prevention Students may only express the positive square root in the solution. Emphasize that students must write the symbol when they take the square root of each side. Got It Notes Unlike in the Example, students probably cannot solve this problem using mental math. Have students state how many solutions they expect this equation to have. If they agree that it has two solutions, they can eliminate choices A and B. Since they know 625 is not 5 squared, the answer must be D. Alternatively, students could use the answer choices to solve the problem. They can square each answer choice to see whether it equals 625. Some students get carried away when finding square roots. 5 and 5 are the results of taking the square root of 625 twice. PART 2 (7 MIN) Objective: Solve equations of the form x p, where p is a positive rational number but not a perfect square. Students use square roots to solve equations of the form x = p where p is a rational number but not a perfect square. Both the numerator and denominator of p are perfect squares. Instructional Design This problem includes a blank Think-Write organizer as a problem-solving tool. You can call on different students to fill in each box on the whiteboard, or you can have each student record one step of the solution. How is this equation different from the one in Part 1? [Sample answer: The number on the right side of the equation is a fraction.] Andie Says (Screen 1) Use the Andie Says button to help students realize that both the numerator and denominator are perfect squares. Can a proper fraction be a perfect square? [Sample answer: No; the square of an integer is another integer.] What is the first step in solving this equation? [Sample answer: Take the square root of each side of the equation.] How do you find the square root of a fraction? [Sample answers: You can find the square root of the numerator and denominator separately. You can look for a number that, when multiplied by itself, equals the fraction.]

6 Solution Notes If you do not show the provided solution, it is still important to show students how to check their solutions by substituting them and to review squaring fractions. Differentiated Instruction For struggling students: You may need to remind students why the square root of a fraction is equivalent to the square root of the numerator over the square root of the denominator. For advanced students: Have students explore square roots of proper fractions and compare each square root to the original number. They may predict that the square root is less than the number, but they will realize that the square root of a fraction between 0 and 1 is actually greater than the fraction. For example, Got It Notes If you show answer choices, consider the following possible student errors: Students that select A or B are finding the square root of only the numerator or denominator, respectively. Students that choose C are finding only the positive square root. PART 3 (6 MIN) Objective: Evaluate square roots of small perfect squares less than 1,000. Students use square roots to solve real-world problems using equations of the form x p, where p is not a perfect square. They estimate x to one decimal place either by using a strategy from a previous topic or using a calculator. The numbers chosen are great enough that students cannot use mental math to estimate the square roots. About how big are most stamps? Why does this stamp seem larger? [Sample answer: Most stamps fit in your hand. It seems large because the area is 1,200, but the units are in square millimeters, and millimeters are a very small unit of measure. (There are over 25 mm in an inch.)] Andie Says (Screen 1) Use the Andie Says button to remind students that this equation has two solutions, but a negative side length does not make sense. Both answers make the equation true. Why else might an answer be unreasonable? [Sample answer: You cannot have a negative length for a stamp. There is only one answer, the positive square root.] While solving part (b) Is 1,200 a perfect square? How does that change how you solve the problem? [Sample answer: No; you need to estimate the side length to the nearest tenth of a millimeter.] All positive numbers have square roots. Do you think most square roots are integers? [Sample answer: No, most square roots are not integers. There are only 11 perfect squares from 0 to 100 inclusive.]

7 Solution Notes Show students how to identify the two perfect squares that 1,200 is between. Students are likely to guess and check or use a calculator. The provided solution shows both methods for part (b). You can use the method of approximation that students learned in a previous topic and then have students compare the result to the one they get when using a calculator. Differentiated Instruction For struggling students: Some students may benefit from using a calculator to test numbers and to check their work. Make sure students understand how to square a number and find square roots using a scientific calculator, a graphing calculator, or the Calculator tool. Got It Notes You can show students an example of a square painting, such as the paintings of Piet Mondrian or Paul Gauguin s Head of a Young Peasant Boy. Because each answer choice is between a distinct pair of consecutive integers, you may want to have students solve before showing answer choices. After solving the problem, you can have students square each answer choice to make sure that their answer is the closest. CLOSE AND CHECK (8 MIN) Focus Question Sample Answer Equations in the form x p can be used to find the side length of a square when given its area. You can solve equations in this form using square roots. Focus Question Notes Student answers should mention that two solutions exist when p 0 and that you can use the notation Stress that taking the square root is the inverse operation of squaring a number. You can use the square root to undo an exponent of 2 and isolate a squared variable. Essential Question Connection Students used exponents not only to solve equations but also to approximate or rewrite the solutions. The Essential Question asks, How can you make such measurements easy to use and compare? Use the questions that follow to help students connect approximating irrational numbers and solving equations. How can you compare the solutions of two equations of the form x p? [Sample answer: The greater p is, the greater the absolute value of the solutions is. If p is not a perfect square, you do not have to approximate the square roots to compare the solutions. You can find the solution with the greatest radicand.] For x p, p cannot be negative because a nonzero number squared is always positive. Is the same true for x p? [Sample answer: No; if you cube a negative number, you get another negative number. So a negative number has a cube root even though it cannot have a square root.]