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Expressions and Equations The Relationships of the Operations Common Core: Engage New York 6.EE.1, 6.EE.2, 6.EE.3 and 6.EE.4 Lesson 5: Special Notations of Operations Focus Standard: 6.EE.A.2c Write, read, and evaluate expressions in which letters stand for numbers. c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas π=π 3 and π΄=6π 2 to find the volume and surface area of a cube with sides of length π =1/2. What does 6.EE.A.1 cover? Write and evaluate numeric expressions involving whole-number exponents. What does 6.EE.A.2 cover? β’ Write, read, and evaluate expressions in which letters stand for numbers a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation βSubtract π¦ from 5β as 5βπ¦. b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8+7) as a product of two factors; view (8+7) as both a single entity and a sum of two terms. c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas π=π 3 and π΄=6π 2 to find the volume and surface area of a cube with sides of length π =1/2. What does 6.EE.A.3 cover? Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2+π₯) to produce the equivalent expression 6+3π₯; apply the distributive property to the expression 24π₯+18π¦ to produce the equivalent expression 6(4π₯+3π¦); apply properties of operations to π¦+π¦+π¦ to produce the equivalent expression 3π¦. What does 6.EE.A.4 cover? Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions π¦+π¦+π¦ and 3π¦ are equivalent because they name the same number regardless of which number π¦ stands for. Table of Contents Date Title Page 2/6/14 NEW Focus 8- Algebraic Expression Scale and Chart Fresh Left 2/6/14 F8 Engage NY Lesson 1 β Add and subtract Relationship Fresh Left 2/7/14 F8 Engage NY Lesson 2 β Multiplication and Division Relationship Fresh Left 2/10/14 F8 Engage NY Lesson 3Multiplication and Addition Relationship Fresh Left 2/11/14 F8 Engage NY Lesson 4Division and Subtraction Relationship Fresh Left 2/18/2014 F8 ENY L5- Exponents Fresh Left Focus 8 Algebraic Expression Learning Goal I am able to work with numerical expressions and use letters to represent unknowns in problem solving situations, investigate and apply properties of operation in numerical contexts, such as the associative, distributive, and commutative properties. I can build on my understanding of inverse operations to solve algebraic expressions. Today, my learning target is toβ¦ β’ Discover that 3π₯=π₯+π₯+π₯ is not the same thing as π₯ 3 which is π₯·π₯·π₯. β’ Understand that a base number can be represented with a positive whole number, positive fraction, or positive decimal and that for any number π, we define ππ to be the product of π factors of π. The number π is the base and π is called the exponent or power of π. MY PROGRESS CHART Before we start the Learning Target Lesson, think about the Learning Target for todayβ¦. How much prior knowledge do you have regarding that goal? Chart your prior knowledge using your pre-target score icon. Lesson Notes & Preparation Lesson Notes β’ In 5th grade, students are introduced to exponents. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10 (5.NBT.A.2). β’ In this lesson, students will use new terminology (base, squared, and cubed), and practice moving between exponential notation, expanded notation, and standard notation. The following terms should be displayed, defined, and emphasized throughout Lesson 5: base, exponent, power, squared, and cubed. Opening Exercises (2 minutes)As you evaluate these expressions, pay attention to HOW you arrived at your answers. Socratic Discussion (15 mins) β’ How many of you solved the problems by βcounting onβ? That is, starting with 4, you counted on 4 more each time (5,6,7,π,9,10,11,ππ,13,14,15,ππ...) β’ If you did not find the answer that way, could you have done so? β’ Yes, but it is time-consuming and cumbersome. Addition is a faster way of βcounting on.β β’ How else could you find the sums using addition? β’ Count by 4, 9, or 10. β’ How else could you solve the problems? β’ Multiply 4 times 10; multiply 9 times 5; multiply 10 times 5. Multiplication is a faster way to add numbers when the addends are the same. Socratic Discussion β’ When we add five groups of 10, we use an abbreviation and a different notation, called multiplication. β’ 10+10+10+10+10=5×10. β’ If multiplication is a more efficient way to represent addition problems involving the repeated addition of the same addendβ¦ β’ Do you think there might be a more efficient way to represent the repeated multiplication of the same factor, as in 10×10×10×10×10= ? β’ Allow students to make suggestions; some will recall this from previous learning. 10×10×10×10×10=105 β’ We see that when we add 5 groups of 10, we write 5×10, but when we multiply 5 copies of 10, we write 105 . So, multiplication by 5 in the context of addition corresponds exactly to the exponent 5 in the context of multiplication. Socratic Discussion Be aware of the correspondence between addition and multiplication because what you know about repeated addition will help you learn exponents as repeated multiplication as we go forward. β’ The little 5 we write is called an exponent and is written as a superscript. The numeral 5 is written only half as tall and half as wide as the 10, and the bottom of the 5 should be halfway up the number 10. The top of the 5 can extend a little higher than the top of the zero in 10. β’ Why do you think we write exponents so carefully? β’ It reduces the chance that a reader will confuse 105 with 105 . Student Practice #1-5 (5minutes) Record your answer in your math workbook. Student PracticeRecord your answer in your math workbook. β’ Go back to Examples 1 β 4 and use a calculator to evaluate the expressions. β’ What is the difference between 3π and π3 ? β’ Take time to clarify this important distinction. β’ The base number can also be written in decimal or fraction form. Try Examples 6, 7, and 8. Use a calculator to evaluate the expressions. β’ The repeated factor is called the base and the exponent is also called the power. Say the numbers in examples 1β5 to a partner. β’ Check to make sure students read the examples correctly: β’ Five to the sixth power, two to the fourth power, eight to the third power, ten to the sixth power, and π to the third power. SP #1-5 Answers Student Practice #6-8 (4 minutes) Record your answer in your math workbook. The base number can also be a fraction. Convert the decimals to fractions in Examples 7 & 8 and evaluate. Leave your answer as a fraction. Remember how to multiply fractions! SP#6-8 ANSWERS The base number can also be a fraction. Convert the decimals to fractions in Examples 7 & 8 and evaluate. Leave your answer as a fraction. Remember how to multiply fractions! Thursday- 5th Period Start Here Student Practice #9-10 (1 minute) Record your answer in your math workbook. Student Practice #9-10 (1 minute) Record your answer in your math workbook. Squared vs. Cubed There is a special name for numbers raised to the second power. When a number is raised to the second power, it is called squared. Remember that in geometry, squares have the same two dimensions: length and width. For π>0, π2 is the area of a square with side length π. β’ What is the value of 5 squared? β’ 25 β’ What is the value of 7 squared? β’ 49 β’ What is the value of 8 squared? β’ 64 β’ What is the value of 1 squared? β’ 1 There is also a special name for numbers raised to the third power. When a number is raised to the third power, it is called cubed. Remember that in geometry, cubes have the same three dimensions: length, width, and height. For π>0, π3 is the volume of a cube with edge length π. β’ What is the value of 1 cubed? β’ 1×1×1=1 β’ What is the value of 2 cubed? β’ 2×2×2=8 β’ What is the value of 3 cubed? β’ 3×3×3=27 Thursday- 1st, 2nd Period β’ START HERE! Exercises #1-4 (8minutes) Record your answer in your math workbook. 2. Write βfive cubedβ in all three forms (exponential form, written as a series of products, standard form) 3. Write βfourteen and seven tenth squaredβ in all three forms. 4. One student thought two to the third power was equal to six. What mistake do you think they made and how would you help them fix their mistake? Exercises #1-4 ANSWERS Closing (2 min) β’ We use multiplication as a quicker way to do repeated addition if the addends are the same. We use exponents as a quicker way to multiply if the factors are the same. β’ Carefully write exponents as superscripts to avoid confusion. Student Exit Ticket for Lesson 5 Exponents Complete the Handout and Turn In for a GRADE Exit Ticket - Lesson 5 ANSWERS Today, I achieved my learning target byβ¦ β’ Discovering that 3π₯=π₯+π₯+π₯ is not the same thing as π₯ 3 which is π₯·π₯·π₯. β’ Understanding that a base number can be represented with a positive whole number, positive fraction, or positive decimal and that for any number π, we define ππ to be the product of π factors of π. The number π is the base and π is called the exponent or power of π. MY PROGRESS CHART Before we start the Learning Target Lesson, think about the Learning Target for todayβ¦. How much prior knowledge do you have regarding that goal? Chart your prior knowledge using your pre-target score icon. The End of Lesson 5 Exponents ο
