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Dear Parents and Caregivers, Thank you for supporting your child to achieve success in school. We value your input and active participation in your child’s education. These letters are designed to help you understand the work your child brings home and the academic expectations of Arizona’s College and Career Ready Standards. Your child is developing the necessary skills and knowledge to help them compute, think, and reason mathematically. This letter is about expressions, equations, and inequalities in sixth grade. End-­‐of-­‐year goals In earlier grades, students have worked with numerical expressions and have been using letters to represent unknowns in problem solving situations. In sixth grade, students will work with algebraic expressions, solve one-­‐variable equations and inequalities, and determine whether a variable represents a specific number or a set of numbers. This work is a foundation for more formal work in writing and solving equations in later grades. Vocabulary •
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Variable: a quantity that can have different values; a symbol that can stand for the variable. In 5n the variable is n. Coefficient: a number used to multiply a variable, e.g., 4x; 4 is the coefficient and x is the variable Expression: a variable or combination of variables, numbers, and symbols that represents a mathematical relationship, e.g. 4r2, x + 4, 15 – m (Expressions do not have an equal sign.) Equation: a statement that two mathematical expressions are equal, e.g., n + 50 = 75 means that n + 50 must have the same value as 75 (All equations have the = sign.) Inequality: a mathematical sentence that compares two unequal expressions using the symbols • < (less than) • > (greater than) • ≤ (less than or equal to) • ≥ (greater than or equal to) • ≠ (not equal to) Exponent: the number that tells how many equal factors there are, e.g., 10 • 10 • 10 • 10 = 104; the exponent is 4; sometimes referred to as “to the power of” as in “ten to the power of 4 or ten to the fourth power” (Note that • means multiply.) Order of Operations: rules describing the sequence to use in evaluating expressions 1. Evaluate within grouping symbols. [ ] , ( ) 2. Do exponents, e.g., 62 3. Multiply or divide left to right. 4. Add or subtract left to right. Properties of Operations: Distributive property: When one of the factors of a product is written as a sum, multiplying each addend does not change the product, e.g., 3(4 + 5) = (3 x 4) + (3 x 5) Associative property of addition: The sum stays the same when the grouping of addends is changed. e.g., (6 + 4) + 2 = 12 and 6 + (4 + 2) = 12 Associative property of multiplication: The product stays the same when the grouping of factors is changed. e.g., (6 x 4)2 = 48 and 6(4 x 2) = 48 Commutative property of addition: The sum stays the same when the order of the addends is changed. 6 + 4 = 4 + 6 Commutative property of multiplication: The product stays the same when the order of the factors is changed. 6 x 4 = 4 x 6 Mesa Pubic Schools/Expressions/Equations/Inequalities/2013 Authorization to reprint or disseminate must be granted by Mesa Public Schools (February-­‐2014). Expressions Students will write and evaluate numerical expressions involving whole-­‐number exponents. For example, when given the expression 43 they can rewrite it as 4 • 4 • 4 or 16 • 4 or 64. They also will write expressions for phrases that contain a variable. For example, “Add 5 to y” can be written as y + 5; “three times p” can be written as 3p, and 20 divided by d can be written as . Students will write equivalent expressions, by applying the properties of operations. For example, to write an equivalent expression for 3(2 + x), the student could apply the distributive property that would multiply the 3 by each term inside the parentheses to show 3(2 + x) = 6 + 3x. They also learn that two expressions are equivalent if each names the same number regardless of which value is substituted. For example, s + s + s + s = 4s, when s represents any number. (5 + 5 + 5 + 5 = 4 • 5 = 20) To evaluate an expression using the order of operations, a student will follow the steps in order. (See the vocabulary section above.) For example, to evaluate 7 • 32 + (2 • 4) Evaluate within grouping symbols. 7 • 32 + 8 Find the value of exponents. 7 • 9 + 8 Multiply or divide, left to right. 63 + 8 Add or subtract, left to right 71 Equations and Inequalities To solve a one-­‐variable equation or an inequality, sixth grade students need to answer the question, “Which values from a specified set, if any, make the equation or inequality true?” They will use substitution to determine which values could work. •
Nina sold some newspapers on Saturday and 12 newspapers on Sunday. She sold 31 newspapers over these two days. This equation shows how many newspapers she sold altogether: s + 12 = 31. Which, if any of the values in this set could be the number of newspapers she sold on Saturday. (12, 19, 43) Explain how you know. “I can substitute each value in the set for s and check to see if the equation is true. 19 + 12 = 31, 12 + 12 = 24 and 43 + 12 = 55, so 19 is the only value from the set that makes the equation true.” •
Which value from the set could be the price of one baseball if Bill can buy a package of 6 for under $7.26? ($1.00, $1.21, $1.50) Graph your answer on a number line. “I can write the inequality 6x < $7.26 to show that 6 baseballs multiplied by the cost of one baseball will be less than $7.26. I can substitute each value in the set for x, multiply it by 6 and check to see if the product is less than $7.26. 6 • $1.00 is = $6.00, 6 • $1.21 = $7.26 and 6 • $1.50 = $9.00, so x < $1.21.” 0 .25 .50 .75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 How to help at home •
Encourage your child to read word problems carefully and ask them to explain the problem using his or her own words. •
Watch these videos about expressions from Learn Zillion. http://learnzillion.com/lessonsets/198-­‐write-­‐read-­‐and-­‐evaluate-­‐expressions-­‐in-­‐which-­‐letters-­‐stand-­‐for-­‐numbers •
Encourage your child to persevere, even if the problem seems difficult. Ask them to think of a strategy they already know to help solve the problem. •
Remember, making mistakes is a part of learning. Mesa Pubic Schools/Expressions/Equations/Inequalities/2013 Authorization to reprint or disseminate must be granted by Mesa Public Schools (February-­‐2014).