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Combining Like Terms and Using Properties of Real Numbers

This lesson focuses on simplifying expressions by combining like terms and using the commutative, associative, and distributive properties of real numbers. Students will learn how to rearrange expressions and identify like terms using coefficients. The lesson also includes examples and practice problems to reinforce the concepts.

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Combining Like Terms and Using Properties of Real Numbers

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  1. Warm Up Lesson Presentation Lesson Quiz

  2. Warm Up Add. 1. 427 + 35 1.80 2. 1.06 + 0.74 462 3. 10 Multiply. 4. 25(8) 5. 1.3(22) 28.6 200 6.

  3. Sunshine State Standards MA.912.A.3.2 Identify and apply the distributive, associative, and commutative properties of real numbers….

  4. Objectives Use the Commutative, Associative, and Distributive Properties to simplify expressions. Combine like terms.

  5. Vocabulary term like terms coefficient

  6. The Commutative and Associative Properties of Addition and Multiplication allow you to rearrange an expression to simplify it.

  7. Additional Example 1A: Using the Commutative and Associative Properties Simplify. Use the Commutative Property. Use the Associative Property to make groups of compatible numbers. 11(5) 55

  8. Additional Example 1B: Using the Commutative and Associative Properties Simplify. 45 + 16 + 55 + 4 45 + 55 + 16 + 4 Use the Commutative Property. (45 + 55) + (16 + 4) Use the Associative Property to make groups of compatible numbers. (100) + (20) 120

  9. Helpful Hint Compatible numbers help you do math mentally. Try to make multiples of 5 or 10. They are simpler to use when multiplying.

  10. Check It Out! Example 1a Simplify. Use the Commutative Property. Use the Associative Property to make groups of compatible numbers. 21

  11. Check It Out! Example 1b Simplify. 410 + 58 + 90 + 2 410 + 90 + 58 + 2 Use the Commutative Property. Use the Associative Property to make groups of compatible numbers. (410 + 90) + (58 + 2) (500) + (60) 560

  12. ( ) 7 1 2 1 2 7 • 8 8•7 • • 1 2 8 • 4 7 • Check It Out! Example 1c Simplify. Use the Commutative Property. Use the Associative Property to make groups of compatible numbers. 28

  13. The Distributive Property is used with Addition to simplify expressions. The Distributive Property also works with subtraction because subtraction is the same as adding the opposite.

  14. Additional Example 2A: Using the Distributive Property with Mental Math Write the product using the Distributive Property. Then simplify. 5(59) Rewrite 59 as 50 + 9. 5(50 + 9) Use the Distributive Property. 5(50) + 5(9) 250 + 45 Multiply. 295 Add.

  15. Additional Example 2B: Using the Distributive Property with Mental Math Write the product using the Distributive Property. Then simplify. 8(33) Rewrite 33 as 30 + 3. 8(30 + 3) 8(30) + 8(3) Use the Distributive Property. 240 + 24 Multiply. 264 Add.

  16. Check It Out! Example 2a Write the product using the Distributive Property. Then simplify. 9(52) 9(50 + 2) Rewrite 52 as 50 + 2. 9(50) + 9(2) Use the Distributive Property. 450 + 18 Multiply. 468 Add.

  17. Check It Out! Example 2b Write the product using the Distributive Property. Then simplify. 12(98) 12(100 – 2) Rewrite 98 as 100 – 2. 12(100) –12(2) Use the Distributive Property. 1200 – 24 Multiply. 1176 Subtract.

  18. Check It Out! Example 2c Write the product using the Distributive Property. Then simplify. 7(34) 7(30 + 4) Rewrite 34 as 30 + 4. 7(30) + 7(4) Use the Distributive Property. 210 + 28 Multiply. 238 Add.

  19. The terms of an expression are the parts that are added together. Like terms are terms that contain the same variables raised to the same powers. Constants are also like terms. Like terms Constant 4x –3x + 2

  20. A coefficientis a number multiplied by a variable. Like terms can have different coefficients. A variable written without a coefficient has a coefficient of 1. Coefficients 1x2 + 3x

  21. Using the Distributive Property can help you combine like terms. You can factor out the common factor to simplify the expression. 7x2 + 4x2 = (7 + 4)x2 Factor out x2 from both terms. = (11)x2 Perform operations in parentheses. = 11x2 Notice that you can combine like terms by adding the coefficients and keeping the variables and exponents the same.

  22. Caution! Add only the coefficients. 6.8y2 + (-y2) ≠ 6.8

  23. Additional Example 3A: Combining LikeTerms Simplify the expression by combining like terms. 72p – 25p 72p –25p 72p and 25p are like terms. 47p Subtract the coefficients.

  24. and are like terms. Write 1 as . Additional Example 3B: Combining Like Terms Simplify the expression by combining like terms. A variable without a coefficient has a coefficient of 1. Add the coefficients.

  25. Additional Example 3C: Combining Like Terms Simplify the expression by combining like terms. 0.5m + 2.5n 0.5m + 2.5n 0.5m and 2.5n are not like terms. 0.5m + 2.5n Do not combine the terms.

  26. 3m2 + m3 Check It Out! Example 3 Simplify by combining like terms. 3a. 16p + 84p 16p + 84p 16p + 84p are like terms. 100p Add the coefficients. 3b. –20t – 8.5t –20t –8.5t 20t and 8.5t are like terms. –28.5t Add the coefficients. 3c. 3m2 + m3 3m2 + m3 3m2 and m3are not like terms. Do not combine the terms.

  27. Additional Example 4: Simplifying Algebraic Expressions Simplify 14x + 4(2 + x). Justify each step. Procedure Justification 1. 14x + 4(2 + x) 2. 14x + 4(2) + 4(x) Distributive Property 3. 14x + 8 + 4x Multiply. 4. 14x + 4x + 8 Commutative Property 5. (14x + 4x) + 8 Associative Property 6. 18x + 8 Combine like terms.

  28. Check It Out! Example 4a Simplify 6(x –4) + 9.Justify each step. Procedure Justification 1. 6(x –4) + 9 2. Distributive Property 6(x)–6(4) + 9 3. 6x – 24 + 9 Multiply. 4. 6x – 15 Combine like terms.

  29. Check It Out! Example 4b Simplify −12x – 5x + 3a + x.Justify each step. Procedure Justification 1. –12x – 5x + 3a + x 2. Commutative Property –12x – 5x + x + 3a 3. –16x + 3a Combine like terms.

  30. Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

  31. 2. Lesson Quiz: Part I Simplify each expression. 1. 165 +27 + 3 + 5 200 8 Write each product using the Distributive Property. Then simplify. 3. 5($1.99) 5($2) – 5($0.01) = $9.95 4. 6(13) 6(10) + 6(3) = 78

  32. Lesson Quiz: Part II Simplify each expression by combining like terms. Justify each step with an operation or property. 5. 14c2 – 9c 6. 14c2 – 9c 7. 301x – x 300x 8. 24a + b2 + 3a + 2b2 27a + 3b2

  33. Lesson Quiz for Student Response Systems 1. Which property states that you can add or multiply in any order? A. Associative B. Commutative C. Multiplicative D. Grouping

  34. Lesson Quiz for Student Response Systems 2. Simplify A. 5 B. 6 C. 10 D. –5

  35. Lesson Quiz for Student Response Systems 3. Which of the following are like terms? A. 3xand 2y B. 3xand 2x C. 3xand x2 D. 3xand 2x

  36. Lesson Quiz for Student Response Systems 4. Simplify by combining like terms. 2x2 + x2 A. 2x4 B. 3x4 C. 3x2 D. 4x2

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