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TIMSS Standard Indicators NAEP 8.7.1-8.7.12 Don’t Break the Law (of Exponents) Purpose Students will solve problems by choosing strategies, explaining their extending reasoning, making calculations, and checking results. THE ACTIVITY Materials Create a matching For the teacher: chalk, chalkboard game for groups to For each student: paper, pencil, copies of Black Line Masters (BLMs) play. On one set of Discovering Laws of Exponents and Understanding the Law, calculator cards, write exponential (optional) expressions. On the Activity second set of cards, write the simplified A. Introducing the Problem versions of those 1. Tell students that today they will be discovering patterns that can expressions. Ask groups be used to simplify expressions containing exponents. to use the laws of 2. Explain that, in math, we frequently use laws found by other exponents to match mathematicians to shorten the time it takes to solve problems. equivalent expressions. Give examples such as the Pythagorean Theorem or the Quadratic Formula. 16 × c ÷ (b ) 2 5 −2 2 3. Write the following expression on the chalkboard: cb 2 3 4. Call on volunteers to help you solve the problem at the chalkboard. Solve the problem by writing in expanded form. 16 × 16 × c × c × c × c × c × b × b × b × b [ c×c×b×b×b ] 5. Explain to students that they will discover laws of exponents that will help simplify expressions like this one. B. Solving the Problem 1. Divide the class into groups of three students. Distribute one copy of the BLM Discovering Laws of Exponents to each student. 2. Tell students to look at the first column of each row. Explain that students may use their scratch paper and/or their calculators to substitute values into the expressions for the terms a and b. 3. Ask students to look for patterns and write algorithms based on those patterns. 4. Tell students to test their algorithms for several different values. Once groups are convinced the algorithm is correct, write it in the second column on the BLM. 5. Allow ample time for groups to complete their charts. Standard 7 (continued) Standards Links 8.1.4, 8.1.5 Standard 7 / Activity 8 Indiana Mathematics Grade 8 Curriculum Framework, October 2002 page 199 Activity (continued) 6. Discuss the results with the class. Write algorithms on the chalkboard. Make sure students correct any mistakes on their chart before going on to section C. (Correct laws are found in the answer key on the back side of the BLM.) C. Understanding the Laws 1. Tell students that now they have discovered the Law of Exponents, it is time to show they understand what the laws mean. 2. Distribute one copy of the BLM Understanding the Law to each student. Allow students to remain in groups of three to complete the BLM. 3. Tell students to copy the correct algorithm for each law onto the BLM. Ask the students to then verbalize the law. Refer students to the example for Law One. 4. Allow students time to work on the BLM and then ask students to share the results with the class. Classroom Assessment Basic Concepts and Processes While groups work on solving the problem, ask students the following questions to gauge their understanding of the Standard Indicator: What patterns did you find for law [insert law number here]? How did you go about finding the algorithm for law [insert law number here]? Which values did you use to test the law [insert law number here]? Standard 7 Standard 7/ Activity 8 page 200 Indiana Mathematics Grade 8 Curriculum Framework, October 2002 Names: Discovering Laws of Exponents Work with your group to find a rule to simplify each exponential expression. Write the algorithm for each rule in the second column. bm × bn bm ÷ bn (bm)n (a × b)m (a ÷ b)m b−m a −n ( )b Standard 7 / Activity 8 Black Line Master 1 Indiana Mathematics Grade 8 Curriculum Framework, October 2002 page 201 Discovering Laws of Exponents Teacher Directions Distribute one copy of the BLM Discovering Laws of Exponents to each student. Have students work in groups of three. Ask groups to substitute values for the variables and look for patterns in the solutions. Tell students to write an algorithm for the pattern and test the algorithm using several different values. Have students write the algorithm in the second column on the BLM. Discuss the results with the class. Make sure the students have the correct algorithms, as shown in the answer key, before continuing on to the next section of the activity. Answer Key bm × bn bm+n bm ÷ bn bm-n (bm)n bmn (a × b)m am × bm (a ÷ b)m am ÷ bm 1 b−m bm a −n b n ( ) b ( ) a Black Line Master 1 Standard 7 / Activity 8 page 202 Indiana Mathematics Grade 8 Curriculum Framework, October 2002 Names: Understanding the Law Copy the rule you found for each law. Underneath each rule, write a verbal description of the law. Use Law One as an example. Law One: Product Law bm × bn = bm + n The product of two exponents of the same base is equal to the base raised to the sum of the exponents. Law Two: Quotient Law bm ÷ bn = Law Three: Power of a Power (bm)n = Law Four: Power of a Product (a × b)m = Law Five: Power of a Quotient (a ÷ b)m = Law Six: Negative Power b−m = Law Seven: Inverse Powers a −n (b) = Standard 7 / Activity 8 Black Line Master 2 Indiana Mathematics Grade 8 Curriculum Framework, October 2002 page 203 Understanding the Law Teacher Directions After groups complete the BLM Discovering the Laws of Exponents, review the laws with the class. Distribute one copy of the BLM Understanding the Law to each student. Allow them to work in their groups to copy the correct law from their earlier work and write a verbal expression for each law. Read the example given for Law One in class. Regroup and discuss the results with the class. Answer Key Law One: Product Law bm × bn = bm+n The product of two exponents of the same base is equal to the base raised to the sum of the exponents. Law Two: Quotient Law bm ÷ bn = bm−n The quotient of two exponents of the same base is equal to the base raised to the difference of the exponents. Law Three: Power of a Power (bm)n = bmn The power of a power is equal to the product of the powers. Law Four: Power of a Product (a × b)m = am × bm The power of a product is equal to the product of the powers of the individual terms. Law Five: Power of a Quotient (a ÷ b)m = am ÷ bm The power of a quotient is equal to the quotient of the powers of the individual terms. Law Six: Negative Power b−m = 1mb A base number raised to a negative power is equal to the reciprocal of the base number raised to the positive power. Law Seven: Inverse Powers −n n a b (b) = (a) The negative power of a fractional expression is equal to the reciprocal of the fraction raised to the positive power. Black Line Master 2 Standard 7 / Activity 8 page 204 Indiana Mathematics Grade 8 Curriculum Framework, October 2002