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Expressions and Equations The Relationships of the Operations

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 6 : Order of Operations. Focus Standard: 6.EE.A.2c Write, read, and evaluate expressions in which letters stand for numbers.

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Expressions and Equations The Relationships of the Operations

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  1. Expressions and EquationsThe Relationships of the Operations Common Core: Engage New York 6.EE.1, 6.EE.2, 6.EE.3 and 6.EE.4

  2. Lesson 6: Order 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 𝑉=𝑠3 and 𝐴=6𝑠2 to find the volume and surface area of a cube with sides of length 𝑠=1/2.

  3. What does 6.EE.A.1 cover? Write and evaluate numeric expressions involving whole-number exponents.

  4. What does 6.EE.A.2 cover? • 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 “Subtract 𝑦 from 5” as 5−𝑦. 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 𝑉=𝑠3 and 𝐴=6𝑠2 to find the volume and surface area of a cube with sides of length 𝑠=1/2.

  5. 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+𝑥) to produce the equivalent expression 6+3𝑥; apply the distributive property to the expression 24𝑥+18𝑦 to produce the equivalent expression 6(4𝑥+3𝑦); apply properties of operations to 𝑦+𝑦+𝑦 to produce the equivalent expression 3𝑦.

  6. 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 𝑦+𝑦+𝑦 and 3𝑦 are equivalent because they name the same number regardless of which number 𝑦 stands for.

  7. Table of Contents

  8. Focus 8 Algebraic Expression Learning Goal I am able towork 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.

  9. Today, my learning target is to… • Evaluate numerical expressions • Recognize that in the absence of parentheses, exponents are evaluated first.

  10. MY PROGRESS CHART Before we start the Learning Target Lesson, think about the Learning Target for today…. How much prior knowledge do you have regarding that goal? Chart your prior knowledge using your pre-target score icon.

  11. Opening Exercises (5 minutes)- Evaluate this expression, record your answer in your math notebook. Share your answer with your partner, discuss how you evaluated the expression.

  12. Discussion (5 minutes) • How could you evaluate the expression 3+4×2 ? • 3+4 could be added first for a sum of 7; then, 7×2=14. • 4×2 could be multiplied first for a product of 8; then, 8+3=11. • Only one of these can be correct. When we evaluate expressions, we must agree to use one set of rules so that everyone arrives at the same correct answer. • During the last lesson, we said that addition was a shortcut to “counting on.” How could you think about subtraction? • Subtraction is a shortcut to “counting back.” • These were the first operations that you learned because they are the least complicated. Next, you learned about multiplication and division. • Multiplication can be thought of as repeated addition. Thinking back on Lesson 4, how could you think about division? • Division is repeated subtraction.

  13. Discussion (5 minutes)- Reason for The Rules Multiplication and division are more powerful than addition and subtraction, which led mathematicians to develop the order of operations in this way. When we evaluate expressions that have any of these four operations, we always calculate multiplication and division before doing any addition or subtraction. Since multiplication and division are equally powerful, we simply evaluate these two operations as they are written in the expression, from left to right. Addition and subtraction are equally powerful in the order of operations and are evaluated from left to right in an expression. Now that these rules of Order of Operations are clear, can you go back and evaluate the expression 3+4×2 as 11? The diagram correctly models the expression 3+4×2 With addition we are finding the sum of two addends. In this example the first addend is the number 3. The second addend Happens to be the number that is the value of the expression 4×2, so before we can add we must determine the value of the second addend.

  14. Example 1 (5 minutes): Expressions with Only Addition, Subtraction, Multiplication, and Division • What operations are evaluated first? • Multiplication and division, from left to right. • What operations are always evaluated last? • Addition and subtraction, from left to right.

  15. Exercises (5minutes)Record your answer in your math workbook.Evaluate the Expression

  16. In the last lesson, you learned about exponents, which are a way of writing repeated multiplication. So, exponents are more powerful than multiplication or division. If exponents are present in an expression, they are evaluated before any multiplication or division. • When we evaluate expressions, we must agree to use one set of rules so that everyone arrives at the same correct answer. These rules are based on doing the most powerful operations first (exponents), then the less powerful ones (multiplication and division, going from left to right), and finally the least powerful ones last (addition and subtraction, going from left to right). • Evaluate the expression 4+6×6÷8. • 4+(6×6)÷8 • 4+(36÷8) • 4+4.5 • 8.5 • Now evaluate the expression 4+÷8. • 4+36÷8 • 4+(36÷8) • 4+4.5 • 8.5 • Why was your first step to find the value of ? • Because exponents are evaluated first.

  17. Example 2 (5 minutes) Expressions with Four Operations and ExponentsRecord your answer in your math workbook.

  18. ANSWERS

  19. Exercises #4-5Record your answer in your math workbook.

  20. Example 3 (5 minutes) Expressions with ParenthesesRecord your answer in your math workbook. The last important rule in the order of operations involves grouping symbols (usually parentheses). These tell us that in certain circumstances or scenarios, we need to do things out of the usual order. Operations inside grouping are always evaluated first, before exponents. • Consider a family of 4 that goes to a soccer game. Tickets are $5.00 each. The mom also buys a soft drink for $2.00. How would you write this expression? • How much will this outing cost?

  21. Example 3 (5 minutes) Expressions with ParenthesesRecord your answer in your math workbook. • Consider a different scenario: the family goes to the game like before, but each of the family members wants a drink. How would you write this expression? • Why would you add the 5 and 2 first? • How much will this outing cost? • How many groups are there? • What is each group comprised of?

  22. Answers

  23. Exercises #6-7 Expressions with ParenthesesRecord your answer in your math workbook. The last complication that can arise is that if two or more sets of parentheses are ever needed, evaluate the innermost parentheses first, and work outward.

  24. Closing (5 min)

  25. Student Exit Ticket for Lesson 6Order of Operations Complete handout and turn in for a grade

  26. Exit Ticket - Lesson 6 ANSWERS

  27. Today, I achieved my learning target by… • Evaluating numerical expressions • Recognizing that in the absence of parentheses, exponents are evaluated first.

  28. MY PROGRESS CHART Before we start the Learning Target Lesson, think about the Learning Target for today…. How much prior knowledge do you have regarding that goal? Chart your prior knowledge using your pre-target score icon.

  29. The End of Lesson 6Order of Operations

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