Direct Translation is the process of translating English words and phrases into numbers, mathematical symbols, expressions, and equations.

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1 Section 1 Mathematics has a language all its own. In order to be able to solve many types of word problems, we need to be able to translate the English Language into Math Language. is the process of translating English words and phrases into numbers, mathematical symbols, expressions, and equations. Translating words into math is the foundation to understanding and successfully solving word problems. In this section, you will learn your new Math Language. Learning this language will make solving word problems easier than you would have ever imagined. Basic Operation Words: Sum, Difference, Product, Quotient Basic Operation Words tell you what operation to perform. An operation refers to a mathematical calculation such as adding, multiplying, or using exponents. Basic Operation Words are always accompanied by the word and. The operation word tells you what operation to perform and which symbol to use; the word and tells you where to put the symbol. Basic Operation Words What They Mean Symbol Used In Place Of And The sum of To add + The difference of/between To subtract The product of To multiply or ( ) The quotient of To divide 1

2 Word Problem Workbook For example, in a word problem using the phrase: the sum of 7 and 4, the word sum tells you that you are using addition. The word and tells you where to put the symbol of addition, the plus sign (+). In this case, the phrase the sum of 7 and 4 in English would translate to in Math Language. TRANSLATION EXAMPLES English: The difference between x and 5 English: The quotient of 16 and r Math: x 5 Math: 16 r? Reverse Operation Words: More Than, Added To, Less Than, Subtracted From Reverse Operation Words also tell you what operation to perform and which symbol to use. But Reverse Operation Words reverse the order of the numbers, variables, or expressions. Reverse Operation Words What They Mean Symbol Used With Reversed Order Of Numbers, Variables, Or Expressions More than To add + Added to To add + Less than To subtract Subtracted from To subtract 2

3 Word Problem Workbook For example, in a word problem using the phrase: 6 less than 8, the phrase less than tells you that you are using subtraction so you will use a minus sign. But also, because less than is a Reverse Word phrase, you need to change the order of the numbers. So in this case, the English phrase 6 less than 8 would translate to 8-6 in Math Language. TRANSLATION EXAMPLES English: y added to 10 English: 6 subtracted from x Math: 10 + y Math: x 6? Miscellaneous Operation Words: Increased By, Decreased By, Twice, Squared These operation words are not accompanied by the word and. The Miscellaneous Operation Words will tell you what operation to perform, which symbol to use, and how a specific English word translates into Math Language. When using Miscellaneous Operation Words, the order of the numbers, variables, or expressions remain the same. Miscellaneous Operation Words What They Mean How To Translate -- Represents Number, Variable, Or Expression Increased by To add + Decreased by To subtract Twice Two times the number 2 Squared To use an exponent of 2 2 Cubed To use an exponent of 3 3 TRANSLATION EXAMPLES English: 14 decreased by y Math: 14 y English: Twice m Math: 2 m English: w squared Math: w 2 3

4 Word Problem Workbook Translation Words: Is, The Result Is, A Number These words are the missing links that will help you fully translate English sentences into Math Language. Translation Words do not give you a specific operation to perform, but they must be translated in order for you to change your word problem into a math equation you can solve. Miscellaneous Translation Words What They Mean How To Translate Is Is equal to = The result is Is equal to = A number An unknown (variable) x or n or any letter TRANSLATION EXAMPLES English: Four times five is twenty English: Seven decreased by a number Math: 4 5 = 20 Math: 7 n Important Multiplication Translation Rules Sometimes in a word problem you will see a phrase that tells you to multiply an amount times a number, such as three times a number. This means you would multiply three times a single item. In this case, it would be 3 times the variable x, or in Math, 3x. 4

5 Word Problem Workbook However, when you see a phrase that tells you to multiply an amount times an operation, such as three times the sum of y and 5, this means you would be multiplying three times the result of the operation. In this case, you would first have to translate the operation words into Math Language, and then put that resulting expression in parentheses to be multiplied by 3, such as 3(y + 5). This pattern would hold true no matter what type of operation it would be. The examples below illustrate this important concept. TRANSLATION EXAMPLES English: Five times the difference of a number and three English: Six times a number Math: 5(x 3) Math: 6n Putting It All Together With the information you now have, it is time to put everything together. By being able to translate English word sentences into Math Language, the dreaded word problems will become normal math equations. HELPFUL HINTS HELPFUL HINTS Do not look at the problem as one long sentence. Look for keywords and break down Do the not sentence look at into the problem separate as phrases. one long sentence. Look for keywords and break down the sentence into separate phrases that are more manageable. The first thing to look for is a word or phrase that indicates where the equal sign will The be. first thing to look for is a word or phrase that indicates where the equal sign will be, because that will break up the long sentence into two separate phrases. Whatever words are in front of the phrase that means is equal to will translate into Whatever the math words expression are in that front goes of before the phrase the equal that means sign. is equal to will translate into the math expression that goes before the equal sign. Whatever is after that phrase will go af Whatever is after that phrase will translate into the expression that goes after the equal sign. 5

6 Word Problem Workbook Word Problem Workbook Table Key Words What They Mean How To Translate The sum of To add + The difference of/between To subtract The product of To multiply or ( ) The quotient of To divide More than To add + Added to To add + Less than To subtract Subtracted from To subtract Increased by To add + Decreased by To subtract Twice Two times the number 2 Squared To use an exponent of 2 2 Is Is equal to = The result is Is equal to = A number An unknown (variable) x or n or any letter Cubed To use an exponent of 3 3 6

7 Word Problem Workbook 7

8 Word Problem Workbook EXAMPLES EXAMPLE 1: Six subtracted from a number is 5. SOLUTION English Bold Type Math Translation Six subtracted from a number is 5. Six subtracted from a number = 5 Six subtracted from a number is 5. Six subtracted from x = 5. Six subtracted from a number is 5. x 6 = 5 EXAMPLE 2 : If 7 is added to 3 times a number, the result is the difference of the number squared and 8. SOLUTION English Bold Type If 7 is added to 3 times a number, the result is the difference of the number squared and 8. If 7 is added to 3 times a number, the result is the difference of the number squared and 8. If 7 is added to 3 times a number, the result is the difference of the number squared and 8. Math Translation If 7 is added to 3 times a number, = the difference of the number squared and 8. 7 is added to 3x = the difference of x 2 and 8. 3x + 7 = x 2 8 EXAMPLE 3: Twice the sum of a number and 6 is equal to one more than 10 times the number. SOLUTION English Bold Type Twice the sum of a number and 6 is equal to one more than 10 times the number. Twice the sum of a number and 6 is equal to one more than 10 times the number. Twice the sum of a number and 6 is equal to one more than 10 times the number. Twice the sum of a number and 6 is equal to one more than 10 times the number. Math Translation Twice the sum of a number and 6 = one more than 10 times the number Twice the sum of x and 6 = one more than 10 times x 2(the sum of x and 6) = one more than 10x 2(x + 6) = 10x + 1 8

9 Word Problem Workbook 9

10 Word Problem Workbook : Exercise Set Translate the following statements: 1. The sum of 12 and a number is The difference between a number and 6 is Twice a number decreased by 2 is Three times a number increased by 7 is Six less than a number cubed is A number squared added to 8 is Five times the sum of 6 and a number is Twice the sum of a number and 10 is If 3 is subtracted from 4 times a number, the result is the sum of 5 times the number and If a number is added to 6, the result is the difference between twice the number and Four times the sum of a number and 10 is Six times the difference of 7 and a number is If a number is subtracted from 94, the result is 19 more than the product of 5 and the number. 14. If 3 is added to twice a number, the result is the difference of 4 times the number and Eight subtracted from 9 times a number is equal to the sum of 6 and the number squared. 16. The sum of 12 and a number is equal to the number cubed added to Eleven times the difference of a number and 42 is equal to the sum of the number cubed and The product of 6 and a number added to 3 is equal to the number squared increased by 8. 10

11 Word Problem Workbook 19. If 7 is subtracted from a number squared, the result is twice the sum of 5 times the number and 20. Six times the difference of twice a number and one is equal to the number cubed less than

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Solve each equation. Check your solution. 13x + 2 = 4x + 38 Page 1 6(n + 4) = 18 7 = 11 + 3(b + 5) Page 2 6(n + 4) = 18 7 = 11 + 3(b + 5) 5 + 2(n + 1) = 2n Page 3 5 + 2(n + 1) = 2n Since, this equation

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles November 4/5, 2014 Exponents Quick Warm-up Evaluate the following: 1. 4 + 4 + 4 +

( ) FACTORING. x In this polynomial the only variable in common to all is x.

FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated

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Sect 1.6 - Properties of Real Numbers and Simplifying Expressions Concept #1 Commutative Properties of Real Numbers Ex. 1a.34 + 2.5 Ex. 1b 2.5 + (.34) Ex. 1c 6.3(4.2) Ex. 1d 4.2( 6.3) a).34 + 2.5 = 6.84

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Expressions Chapter Questions 1. Explain how distribution can simplify a problem. 2. What are like terms? 3. How do you combine like terms? NJ Center for Teaching and Learning ~ 1 ~ www.njctl.org Expressions

Common Core Standards for Fantasy Sports Worksheets. Page 1

Scoring Systems Concept(s) Integers adding and subtracting integers; multiplying integers Fractions adding and subtracting fractions; multiplying fractions with whole numbers Decimals adding and subtracting