tag:blogger.com,1999:blog-90708528720581434872020-02-27T13:09:14.798-08:00AdjectiveNounMathUnknownnoreply@blogger.comBlogger4125tag:blogger.com,1999:blog-9070852872058143487.post-3588516488544609712010-02-10T13:08:00.000-08:002010-02-10T17:08:33.850-08:00The Adjective/Noun Theme and Signed NumbersWhat follows is a joint effort by Judy and Herb.<br /><br />To compute the sum of signed numbers, for example -8 + (+5), we first use the fact that a negative number such as -8 can be expressed as the sum of two negative numbers (appealing to the adjective/noun theme) and rewrite -8 as -3 + -5. Then we can make use of the fact that the sum of a number and its opposite has been defined to equal 0 to do the following:<br /> -8 + (+5)<br />= ( -3 + -5) + (+5)<br /> = -3 + ( -5 + (+5) )<br /> = -3 + 0<br /> = -3.<br />To see this result more concretely, we can use the profit/loss model.<br />---------------------------------------------- Profit----------Loss<br />First Transaction ............................................................ 8<br />Second Transaction ...................................5<br />___________________________________________________<br />Net <br />___________________________________________________<br /> <span style="font-size:85%;">Figure 1: -8 + (+5) = ?<br /><br /><span style="font-size:100%;"><span style="font-size:130%;"></span></span></span> We know that if the situation had been a $5 loss and a $5 profit, the net would have been $0. This is simply the profit/loss model for saying that -5 + (+5) = 0. That is:<br />---------------------------------------------- Profit----------Loss<br />First Transaction ............................................................ 5<br />Second Transaction ...................................5<br />___________________________________________________<br />Net..............................................................0<br />___________________________________________________<br /> <span style="font-size:85%;">Figure 2: -5 + (+5) = 0<br /></span><br />Combining the idea of rewriting -8 as -3 + -5 with the profit/loss model, we can view the $8 loss as the sum of a $5 loss and a $3 loss. Then we can rewrite Figure 1 in the form:<br /><blockquote></blockquote>---------------------------------------------- Profit----------Loss<br />First Transaction ............................................................ 5<br />Second Transaction ...................................5<br /> ..........................................................................................3<br />___________________________________________________<br />Net ....................................................................................3<br />___________________________________________________<br /> <span style="font-size:85%;">Figure 3: -8 + (+5) = -3<br /><span style="font-size:100%;"><br /></span></span> <span style="font-size:100%;">The $5 profit and the $5 loss then "cancel" one another and we see that the net result is a $3 loss. This result is the same result that is gotten by applying the usual rule for adding two numbers that have opposite signs.<br /><br /><span style="font-weight: bold;">Rule: </span>Subtract the smaller magnitude from the greater magnitude and keep the sign of the number that has the grater magnitude.<br /><br />In terms of the profit/loss model, the smaller magnitude (5) is in the "profit" column so we subtract it from the greater magnitude (8), which is in the "loss" column. We then place the difference (3) in the "loss" column.<br />---------------------------------------------- Profit----------Loss<br />First Transaction ............................................................ 8<br />Second Transaction ...................................5<br />___________________________________________________<br />Net ....................................................................................3<br />___________________________________________________<br /><br />However, since $5 is modifying profit and $8 is modifying loss, it </span><span style="font-weight: bold;font-size:100%;" >appears</span><span style="font-size:100%;"> that we cannot apply the adjective/noun theme that says that we cannot subtract $5 from $8 and obtain $3 as the answer because $5 and $8 are modifying different nouns (</span><span style="font-size:100%;">Footnote 1). We are faced with a dilemma.</span><br /><br /><div style="text-align: center;"><span style="font-weight: bold;">Resolving The Dilemma</span><br /></div>First, let’s remove the problem from the profit/loss model but retain the visual aid. We can do this by replacing the headings “Profit” by “Positive”, “Loss” by “Negative” and “Transaction” by “Number”. The display now becomes:<br /> ------------------------------------------ Positive----------Negative<br />First Number ....................................................................8<br />Second Number ...................................5<br />___________________________________________________<br />Net ....................................................................................3<br />___________________________________________________<br /><span style="font-size:85%;"><span style="font-size:100%;"></span><br /></span> The subtle point is this: even though we <span style="font-weight: bold;">subtracted</span> 5 from 8 we <span style="font-weight: bold;">added</span> the two signed numbers. In order to <span style="font-weight: bold;">add</span> -8 and +5 we found we had to subtract one adjective (5) from another (8), but we did <span style="font-weight: bold;">not</span> subtract -8 and +5 . In other words, when we add two signed numbers that have different signs, we subtract the lesser adjective from the greater adjective and keep the noun that is modified by the greater adjective. To show that this rule is consistent with the adjective/noun theme, we can put the rule we have just discovered into the following form known as the “subtract the opposite” rule.<br /><br /><div style="text-align: center;"><span style="font-weight: bold;">Subtract the Opposite Rule: </span><span style="font-style: italic; font-weight: bold;">a + b = a – (-b)</span> <span style="font-size:78%;">Footnote 2<br /></span><div style="text-align: left;"><div style="text-align: left;">To apply the “subtract the opposite” rule to our original problem, replace <span style="font-style: italic;">a</span> by - 8 and replace <span style="font-style: italic;">b</span> by +5 . Since <span style="font-style: italic;">b</span> is equal to +5 , the opposite of <span style="font-style: italic;">b</span> (in symbols (-<span style="font-style: italic;">b</span>) ) is -5. Making those substitutions in the rule <span style="font-style: italic;">a + b = a – (-b)</span>, the expression <span style="font-style: italic;">a + b</span> becomes -8 + (+5 ) and <span style="font-style: italic;">a – (-b)</span><span style="font-style: italic; font-weight: bold;"> </span>becomes -8 – (-5). Therefore, our original problem now takes the form -8 – (-5). In this form both 8 and 5 are modifying “negative”. Since both adjectives (8 and 5) modify the same noun (negative), by the adjective/noun theme the answer is -3. Note that this is the same result we got by using the profit/loss model.<br /></div><br />We have just illustrated that the subtract the opposite rule we discovered is consistent with the rule that says to add two signed numbers, subtract the smaller adjective (in this case, 5) from the greater adjective (8) and keep the noun (sign) of the term with the greater adjective (negative). But, and this brings us to the “punch line” of this note, we have also illustrated that <span style="font-weight: bold;">t</span><span style="font-weight: bold;">he adjective/noun theme does hold for signed numbers</span>. To make use of the adjective/noun theme to add two numbers that have different signs, just do the following:<br /> Step 1:<br /> Write the sum with the number having greater magnitude placed first.<br /> In this example: -8 + (+5).<br /> Step 2:<br /><div style="text-align: left;"> Change the addition sign to a subtraction sign and replace the second number with its opposite (i.e., change its sign). In this example: -8 – (-5).<br /></div> Step 3:<br /> The two adjectives are now modifying the same noun so we can do the “usual”<br /> arithmetic. In this example:<br /> -8 – (-5) = -3.<br /><br /><span style="font-size:85%;">Footnote 1: Using the example that 3 dimes + 2 nickels = 40 cents, we motivated the rule that we add adjectives only if they modify the same noun. Similarly, the fact that 3 dimes – 2 nickels = 20 cents motivates the rule that we subtract adjectives only if they modify the same noun.</span><br /><br /><span style="font-size:85%;">Footnote 2: This rule is analogous to dividing two fractions. Namely, to divide by a fraction we multiply by the reciprocal of the fraction. Our rule says to add two signed numbers, subtract the opposite of the second number (formally known as the subtrahend). Also, when dividing by a fraction we are NOT multiplying by the original fraction. Similarly, when adding two signed numbers we are NOT subtracting the original ‘subtrahend”.</span><br /><br /><br /><br /></div></div><br /><span style="font-size:85%;"> </span>Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-9070852872058143487.post-49794737146057887512010-01-28T09:34:00.000-08:002010-01-28T09:36:23.886-08:00A message from Herb: Arithmetic as a Gateway to AlgebraIt is not uncommon for a first or second grade teacher to say such things as “Why do I have to go to a workshop on fractions? I just teach whole number arithmetic”. Hopefully our workshop on January 25th demonstrated that by using the adjective/noun theme and having a thorough understanding of whole number arithmetic, the study of fractions became much simpler.<br /><br />More generally, the more you know about what lies ahead for the students, the better you can motivate your course material in ways that are more relevant to them. With this in mind, you might like to view Lesson 1 of the algebra course that is posted on our website. I believe it will show you why understanding arithmetic is a very important prerequisite for any students who is enrolled in an algebra class.<br /><br />For example, even though one does not (consciously) teach algebra in the first grade, the concept is already present when we ask students such fill-in-the-blank questions as<br /><br />5 + ____ = 8.<br /><br /> In essence it is a much less threatening form of the equivalent problem:<br /><br />For what value of x does 5 + x = 8?<br /><br />Lesson 1 of our algebra course explores this idea in much greater detail. I know you all are very busy and do not need to have more put on your plate. However I believe that if you look at Lesson 1 you will see the very close connection between arithmetic and algebra. And if you already teach algebra, you might enjoy seeing what might be a point of view that is new to you.<br /><br />If you do find the time to look at this lesson, our Team will be pleased to read your reactions to it.Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-9070852872058143487.post-91140716468177641432010-01-27T06:17:00.000-08:002010-01-27T06:24:57.087-08:00Easton/Redding Workshop on January 25, 2010. Our team (Herb, Rick and I) and, of course, Louise (Herb's bride of 56 years) arrived at Samuel Staples Elementary School in the middle of a monsoon. Happily the Easton/Redding people are hardy souls and everyone arrived safely.<br /><br />To quote Herb:<br /><b><span style=";font-family:Arial;font-size:100%;" ><span style="font-weight: bold;font-family:Arial;font-size:12;" >"I would like the teachers to know that I (as well as Rick and Judy) was very impressed by how courteous and attentive they were throughout the entire day. There was a minimum of walking “in around and out” of the room nor did we discern any disconcerting “table talk” during the presentations. In short, the teachers as well as any other attendees were a pleasure for us to work with.</span></span></b><b><span style=";font-family:Arial;font-size:100%;" ><span style="font-weight: bold;font-family:Arial;font-size:12;" > Hopefully the positive effects will not wear off and the teachers will keep in touch with us and our website whenever they feel that our input might be helpful to them. "<br /></span></span></b><br />We hope that we hear from many of the teachers with feedback and more problems to work on together!Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-9070852872058143487.post-55231060638227816492010-01-26T06:10:00.000-08:002010-01-26T06:23:39.940-08:00Day 1 of Blogging about Math EducationToday is day 1 of the AdjectiveNounMath Blog. We hope you will post questions, comments, ideas and suggestions related to the teaching and learning of the fundamentals of mathematics. It is our passion and we are committed to offering an innovative and intuitive way to understand the basic operations of mathematics.<br /><br />Please read more about us at www.adjectivenounmath.com and we look forward to hearing from you.<br /><br /><br />Unknownnoreply@blogger.com1