  # Vocabulary: Say it with Symbols

 Concept Example Equivalent Expressions : are algebraic expressions that have the same value no matter what value(s) is substituted for the variable (s). Students have various ways to check for equivalence: using graphs and tables, or using Properties of Real numbers. See below.Note: Students often confuse “expression” and “equation.” “3x” is an expression. It can be evaluated for various values of x. However, “y = 3x” or “12 = 3x” are equations. These two equations have solutions . In the case of y = 3x there are an infinite number of solutions: (1, 3), (2, 6) etc. In the case of 12 = 3x there is only one solution, x = 4. Likewise: 2x2 + 10x is an expression which can be evaluated: while 2x2 + 10x = 12 is an equation which can be solved . 1. Write 2 expressions for the number of 1-footsquare tiles, N, needed to make a border around a square pool with sides of length s feet. The expression for N depends on how a student visualizes the area of the tiled border. A student might say that the border is made of 4 strips, each s tiles long, and 4 corners, as below. In this case the student will say that the number of tiles needed is N = 4s + 4.Or the student might see this as 4 strips, each (s + 1) tiles long, as below. In this case the student will say that N = 4(s+ 1) tiles are needed. 2. Determine that the two expressions for N in example 1 are equivalent. Students already know the meaning of “equivalent” in terms of functions . So they could compare the expressions “4s + 4” and “4(s + 1)” by making a table, for various values of s. It appears that these two expressions always have the same value, for any given value of s. Note: Students could also use a graph to make this comparison. However, tables and graphs are only snapshots of SOME of the values that an expression might take. To be sure that these expressions are equivalent for ALL values of the variable we need a better strategy. See below. Interpret an Expression: Students who develop familiarity with symbolic expressions can recognize what a particular expression might represent.Note: “3x” is a monomial because it has 1 term. “3x + 2y” is a binomial because it has 2 terms added. If an expression has 3 terms it is called a trinomial. An expression with more than 3 terms is called a polynomial . (You can use “polynomial” to refer to an expression with ANY number of terms.) 3. Sketch a pool whose area is given by 16π + 80 square feet.There are several possible answers, but students should be able to spot a symbolic expression related to a circle within the given binomial expression. The term“16π” is the area of a circle with radius 4 units. The other term, “80”, could be a rectangular area. One possible pool would be 4. If a student gives the number of tiles needed for the pool problem as N = (s + 2)2 – s2, how did the student visualize the problem? There are 2 squared expressions within this polynomial: (s + 2)2 and s2. The presence of the “square” implies a square area. “s2” is the area of a square with sides s; “(s + 2)2” is the area of a square with side (s + 2). On the sketch of the original pool the student could have visualized this as the outer square – the inner square. Properties of Real Numbers The Commutative Property: of addition states that the order of addition of real numbers does not matter. a + b = b + a for all real values of a and b. Note: Multiplication of real numbers also has this property. Subtraction and division of real numbers do not have this property. For example, 8 – 3 is not the same as 3 – 8. The Associative Property: of addition of real numbers states that when adding 3 (or more) real numbers you may group them in pairs and add, using any groupings. a + b + c = (a + b) + c = a + (b + c). Note: Multiplication of real numbers also has this property, but subtraction and division do not. For example, 12 ÷ 4 ÷ 2 = (12 ÷ 4) ÷ 2 = 3 ÷ 2 = 1.5. 12 ÷ 4 ÷ 2 ≠ 12 ÷ (4 ÷ 2) = 12 ÷ 2 = 6. There is a particular order of operations that we use when we have several operations to do to evaluate an expression. In this case we do the divisions in order from the left. As we have seen, if there are only additions of multiplications in the expression then we can change the order, or group them in any way. See Accentuate the Negative for more on Order of Operations. Distributive Property : • If an expression is written as a factor multiplied by a sum of two are more terms, the distributive property can be applied to multiply or expand the factor by each term in the sum. • If an expression is written as a sum of terms and the terms have a common factor, the distributive property can be applied to rewrite the expression as the common factor multiplied by a sum of two or more terms. This process is called factoring. Note: The distributive property was first introduced in Accentuate the Negative and extended in Frogs, Fleas, and Painted Cubes , to include two binomials, (a + b)(c + d). 5. Show that 4s + 4, 4(s + 1) and 2(s + 2) + 2s are equivalent by using properties of real numbers.4(s + 1) = 4s + 4 (using the Distributive property to multiply each of the terms of “s + 1” by 4.) 2(s + 2) + 2s = 2s + 4 + 2s (using the Distributive Property to multiply each of the terms of “s + 2” by 2) = 2s + 2s + 4 (Using the Commutative Property of addition to change the order) = (2s + 2s) + 4 (Using the Associative Property) = 4s + 4 (Using the Distributive Property) Note: 2s + 2s can be written as s(2 + 2) or 4s, using the Distributive Property to factor 2s + 2s. Textbooks often refer to terms like 2s and 2s as “like terms” because they have the same variable component. This permits factoring. For example, 3xy + 5xy = xy(3 + 5) = 8xy. In this example “3xy” and “5xy” are like terms. BUT, 3x + 5xy = x(3 + 5y) ≠ 8xy, because “3x” and “5xy” are not like terms. 6. Suppose a checking account contains \$100 at the start of the week. Three checks are written during the week, one for \$x and two for \$(x + 1). Write an expression for the balance in the account at the end of the week in two ways: The balance in dollars is 100 – x – 2(x + 1). This can be written as 100 – x – 2x – 2, or as 100 – 3x – 2 or as 98 – 3x, using properties of Real numbers to rewrite this expression in equivalent forms. Note: the last part of the original expression is “-2(x + 1).” We are multiplying by –2, hence the “–2x – 2” in the expanded form. 7. Write in factored form: 5x + 15 + 10z. Each of the terms of this trinomial have “5” as a factor. Therefore, we can use the Distributive Property to write this as 5(x + 3 + 2z). 8. Write in factored form: 5x + 10x2. Each of the terms of this binomial have “5” and “x” as factors. Therefore, using the Distributive Property, we can rewrite this as 5x(1 + 2x).
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