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Why Johnny Can’t Do Mathematics DON CZARCINSKI, PH. D. LOURDES UNIVERSITY “Why Johnny Can't Read” by Rudolf Flesch Reading is Fundamental to Mathematics also. APPLICATIONS OF LINEAR EQUATIONS Word Problems I want you to estimate the answer to this problem; you do NOT have to find the exact answer. If you do, however, it will be worth an extra 2 points. Joe could paint a garage in 8 hours. Bill could paint the same garage in 6 hours. If they work together on that garage, how long would it take the two of them to paint it? 72 students None added the two rates 33 made reasonable guesses 28 averaged the two rates even though I told them they did not have to compute an answer. When a number is decreased by 12 the result is 15. Find the number. The unknown quantity is ordinarily denoted by a letter, x. “A number is decreased by 12” implies the operation of subtraction. The phrase “the result is” gives rise to an equal sign. So the problem can be translated into the equation: x – 12 = 15. John has a total of $1.60 in nickels and dimes. If he has twice as many nickels as dimes, how many of each does he have? Here the word “total” gives rise to an equation How does one arrive at this total? Number of nickels + the number of dimes There is also a relation between the number of nickels and the number of dimes. Do nothing until you finish reading the instructions completely Clap your hands Stand up If you are the first one here, announce that you are the king of following instructions. Do nothing until you finish reading the instructions completely READ REACT RESPOND Find the Greatest Common Factor of Two Numbers. Find the Least Common Multiple of Two Numbers. The Greatest Common Factor of two numbers is a divisor of both numbers. That means the GCF of two numbers must be SMALLER THAN BOTH The Least Common Multiple of two numbers is a multiple of both numbers. That means the LCM of two numbers must be LARGER THAN BOTH I teach two methods for finding the Greatest Common Factor The Prime Factors Method The Euclidean Algorithm To find the Greatest Common Factor for 42 and 70 by using the Prime Factors Method Get Prime Factorization of the two numbers 42 = 2 · 3 · 7 70 = 2 · 5 · 7 Match (and multiply) the prime factors and GCF(42, 70) = 14 To find the Greatest Common Factor for 42 and 70 by using the Euclidean Algorithm 70 ÷ 42 = 1 with remainder 28 42 ÷ 28 = 1 with remainder 14 28 ÷ 14 = 2 with remainder 0 So GCF(42, 70) is the last divisor, 14 1) Use the prime factors method to find the Greatest Common Factor of the pair of numbers 48 and 90 2) Use the Euclidean Algorithm to find the Greatest Common Factor of the pair of numbers, 35 and 42. 3) Find the Least Common Multiple of the pair of numbers 18 and 48 What is the noun in the phrase “Least Common Multiple?” Multiple “Common” and “Least” are modifying the noun 9) Find a factor common to 6 and 8. Then I ask: 10) Find the Greatest Common Factor of the following pair of numbers: 30 and 42 11) Find a common multiple of 6 and 8. Then I ask: 12) Find the Least Common Multiple of the following pair of numbers 30 and 42 Find the following: (2 points) 24 and 18 * Don’t ask, because I won’t tell. Figure it out on your own. 16 found the Greatest Common Factor 7 found the Least Common Multiple 11 found both 5 found neither 2 realized that the only mathematical interpretation of the word “and” is addition, so 24 and 18 equals 42 What are the multiples of 6? Several answer: 1, 2, 3, 6 I now ask Find five multiples of 4. Which is a better assessment of a student’s knowledge of finding Least Common Multiple? 1) Find the Least Common Multiple of 4 and 6. Or 2) One waitress has every fourth day off; her friend has every sixth day off. They are both off today. How long will it be before they have the same day off? Why do you want students to know how to find the Least Common Multiple? You leave your campsite and travel 3 miles south. You see a bear in the distance in front of you so you travel 1 mile east You then travel 3 miles north and arrive at your campsite. What color was the bear? “Mathematical Ideas” 12th Edition Miller, Hereen and Hornsby Addison Wesley, 2011 Chapter 1 The Art of Problem Solving The Art of Reading Mathematics Problems Today is your first day driving a city bus. When you leave downtown you have twenty-three passengers. At the first stop, three people exit and five get on the bus. At the second stop, eleven people exit and eight people get on the bus. At the third stop, five people exit and three get on the bus. 23 – 3 + 5 = 25 25 – 11 + 8 = 22 22 – 5 + 3 = 20 How old is the bus driver? READ REACT RESPOND What is the Least Common Multiple of 4 and 6? READ What is the noun in the phrase Least Common Multiple? Answer: Multiple REACT What are the multiples of 4 and of 6? Multiples of 4 are: 4, 8, 12, 16, 20, 24 Multiples of 6 are: 6, 12, 18, 24, 30, 36 What are the common multiples of 4 and 6? 12, 24, 36, . . . RESPOND What is the smallest of these Common Multiples of 4 and 6? 12 “Zero product property of real numbers,” That is if the product of two real numbers was 0, then one or both of them had to be 0. Solve the equation (x + 2)(x – 3) = 0 Some students a) performed the implied multiplication b) factored the resulting quadratic (often differently than was given) c) set the two linear polynomials equal to 0 and solved those equations to obtain the solutions of the given quadratic equation. See . . . Do See Think Do See × . . . Do × x2 – x – 6 = 0 x(x – 1) = 6 x = 6, 7 Six-product Property Solve the equation x2 = x Quadratic equations have two solutions. They may be different They may be the same They may not be real numbers But there are TWO. Here, 0 and 1 PEMDAS Perform the indicated operations: 5 – 3[4 – (2 – 9)] See subtraction Do subtraction 2[4 – (2 – 9)] 2[4 – (-7)] 2[11] = 22 Scorpio’s head is given by the equation 9x2 + 4y2 =36 Dirty Harry fires his .44 Magnum along the line y=½x+3 Does Dirty Harry blow Scorpio’s head clean off? Harry’s shot miss him completely, that is, the line does not intersect the ellipse Harry’s shot grazes him, that is, the line is tangent to the ellipse Or Harry blows Scorpio’s head clean off the line intersected the ellipse .44 The Toledo Blade Wednesday June 13, 2012 Services will be held at the funeral home on Saturday, June 23rd. Guess when yours truly went to the funeral home. Saturday, June 16th.