Tiêu đề Word Problems Explained.pdf ✅

73 Pre-Activity Preparation Do you remember this old nursery rhyme riddle? Ask this riddle of any second grader and be sure that he or she will furiously begin to multiply seven by seven by seven... After all the multiplying and adding, the second grader is usually dismayed to find that only one was going to St Ives. Word problems, though often tricky, are not meant to be riddles. One of the goals of math class is to use the skills learned in the course to solve problems that are presented verbally rather than symbolically. This activity highlights the process for translating from English to mathematical symbols. • Translate key English words to symbolic notation • Translate a symbolic presentation into English • Use a symbolic representation to define a problem statement Interpreting Word Problems Section 1.5 New Terms to Learn See the key word chart on the following pages. Previously Used even odd operation proportion ratio simplify Learning Objectives Terminology
74 Chapter 1 — Whole Numbers Building Mathematical Language Working a word problem consists of two parts—translating English words and their underlying meaning into a mathematical expression or equation, and then simplifying the expression or solving the equation. Students are commonly taught to look for key words or phrases when working word problems. The process is to translate each key word or phrase into its corresponding math operation symbol and to work with whatever expression or equation results. The key words usually translate to operations on numbers or phrases. Unfortunately, the key word method does not always take into account the meaning or context of the problem. Scanning the problem for numbers and key words alone may miss the whole point. In the riddle, for example, the man was going to St. Ives; nowhere does it state that those whom he met were also going to St. Ives. None of the math words were relevant to the actual riddle. ** **Historical note: This is a very old riddle; notes about it can be found on the web. (Try searching by the first line of the poem.) Most accounts of the modern wording date to the 17th century, but the first account of solving the mathematics (7 × 7 × 7 × 7) dates to about 1600 BC. Key Words English Phrase Symbolic Phrase For addition altogether If there are four boys and seven girls, how many are there altogether? 4 + 7 or 7 + 4 together Joe has some dollars and Joan has $7; together, they have what amount? $x + $7 or $7 + $x in all Joe has $4 and Joan has $7, what do they have in all? $4 + $7 or $7 + $4 more than seven more than four 4 + 7 sum of the sum of a number and seven x + 7 or 7 + x total What is the total of four and some number? 4 + x or x + 4 comparatives (phrases such as taller than, bigger than, faster than, etc.) If Jan is 64 inches tall and John is 6 inches taller than Jan, how tall is he? 64 ̋ + 6 ̋ Note that it is not always immediately obvious whether a comparative phrase indicates addition or subtraction; be sure to check the context of the problem. For subtraction difference Find the difference of 4 and 7. Find the difference of 7 and 4. 4 – 7 7 – 4 fewer than four fewer than some number x – 4 how many more How many more is seven than four? 7 – 4 less than four less than some number x – 4 Notice which number comes first!
Section 1.5 — Interpreting Word Problems 75 For subtraction (continued) left/left over What is left if I take 4 from some number ? x – 4 minus seven minus four 7 – 4 remains What remains if I take some number from 7? 7 – x decreased by seven decreased by four 7 – 4 comparatives (phrases such as shorter than, smaller than, slower than, etc.) If Pete drives at 80 miles per hour and Steve drives 15 miles per hour slower than Pete, how fast does Steve drive? 80mph – 15mph Note that it is not always immediately obvious whether a comparative phrase indicates addition or subtraction; be sure to check the context of the problem. For multiplication times four times some number 4x every four in every row of seven 4 × 7 at the rate of at the rate of 4 for every number 4x each Everyone got four each. 4x product the product of seven and a number 7x twice/double twice a number; double a number 2x For division quotient Find the quotient of 7 and 4. Find the quotient of 4 and 7. 7 ÷ 4 4 ÷ 7 equal pieces a number is cut into four equal pieces x ÷ 4 split a number is split into four equal pieces x ÷ 4 average Find the average of four numbers whose sum is known. x ÷ 4 half Find one half of 18. 18 ÷ 2 shared equally 18 pieces of candy are shared equally among 3 people. 18 ÷ 3 divided equally A 12 ̋ sub sandwich is divided equally among 4 people. 12 ̋ ÷ 4 Other key words If the problem says: Represent the numbers with two consecutive numbers Find two consecutive numbers... x and x + 1 consecutive odd Find two consecutive odd numbers... x and x + 2 consecutive even Find two consecutive even numbers... x and x + 2 the sum of two numbers The sum of two numbers is 7. x and 7 – x Notice which number comes first!
76 Chapter 1 — Whole Numbers Some statements may sound very similar, but convey very different meanings. For instance: “The sum of 4 times a number and 5” The sum of 4 times a number and 5 4 • n + 5 “4 times the sum of a number and 5” 4 times the sum of a number and 5 4 • (n + 5) Sum signifies addition and times signifies multiplication. To understand what we are multiplying by 4, underline it. Parentheses are needed to show that the key word “times” refers to the sum, not just the number. Punctuation can clarify the meaning of a phrase in a world problem. “Three minus a number plus seven” can be interpreted in two different ways: (3 – x) + 7 or 3 – (x + 7) If we let x = 2, we see that these two interpretations give very different results: (3 – 2) + 7 = 8 or 3 – (2 + 7) = –6 If a comma is inserted in the phrase, the correct interpretation is clear: “Three minus a number, plus seven” can only mean (3 – x) + 7. Another Classic Riddle A farmer in California owns a beautiful pear tree. He supplies the fruit to a nearby grocery store. The store owner has called the farmer to see how much fruit is available for purchase. The farmer knows that the main trunk has 24 branches. Each branch has exactly 12 boughs and each bough has exactly 6 twigs. Since each twig bears one piece of fruit, how many apples will the farmer be able to deliver?