and a Handbook for
Arithmetic to Algebra
Royal Lyon Publications
Klamath Falls, Oregon
Copyright ©2010 by Ed Lyons All rights reserved.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the author. Printed in the United States of America
ISBN: 1453612122 First printing, June 2010
Second printing, September 2010
Corrections and additions are posted at our website: ActionAlgebra.com
Action Algebra
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ntroductIonArithmetic to Algebra In Just Ten Minutes! . . 8
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asIcP
rIncIPles Equality. . . . 11Value Change Needs Counterchange . . . . . 12
Priority . . . . 13
First Calculate the Complicated. . . . 13
Insight . . . . 15
Changing Looks does not Change Value . . . 15
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umBers 1) Benefits of a Number System Line . . . . . 202) Translating Numbers and Words . . . . 21
3) Numbers are arrows . . . . 21
4) Comparing numbers . . . . 23
5) Kinds of numbers . . . . 25
6) Parts of compound numbers . . . . 26
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omBIne 7) Adding on a Number Line . . . . 308) Subtracting on a Number Line . . . . 31
9) Adding Stacked Numbers . . . . 32
10) Subtracting Stacked Numbers . . . . 33
11) Combining Stacked Integers . . . . 34
12) Series of Signs . . . . 37
13) Combining Series of Signs . . . . 37
14) Combining Big Integers. . . . 38
15) Combining Decimals . . . . 38
16) Combining Tags . . . . 39
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ultIPly 17) Multiplying on a Grid . . . . 4218) Learning the Times Table. . . . 45
19) Learning Multiples . . . . 47
20) Negative Numbers on a Grid . . . . 47
21) Multiplying Big Numbers . . . . 51
22) Multiplying Bigger-Smaller . . . . 53
23) Multiplying Decimals . . . . 53
24) Multiplying Fractions . . . . 54
25) Multiplying Tags by Merging . . . . 55
26) Finding Common Multiples. . . . 56
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IvIde 27) Dividing on a Grid . . . . 5828) Learning How to Shift . . . . 59
29) Dividing and Bigger-Smaller . . . . 62
30) Speed Division . . . . 62
31) Long Division . . . . 63
32) Long Division with Decimals . . . . 64
33) Factoring . . . . 65
34) Prime Factoring. . . . 66
35) Finding Common Factors. . . . 67
36) Reducing Fractions . . . . 68
37) Dividing Fractions Using Reciprocals . . . 70
38) Making Like Fractions . . . . 71
39) Combining Fractions . . . . 72 40) Canceling Tags . . . . 73
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xPonents 41) Basics . . . . 76 42) Negative Exponents . . . . 78 43) Multiplying Bases . . . . 80 44) Dividing Bases . . . . 81 45) Zoom Levels . . . . 8246) Groups with exponents . . . . 84
47) Exponents in Fractions . . . . 84 48) Scientific Numbers . . . . 86 49) Adjust Scientifics . . . . 87 50) Multiply Scientifics . . . . 88 51) Powers of Scientifics . . . . 89 52) Dividing Scientifics . . . . 90 53) Combining Scientifics. . . . 90
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orPhs 54) Fractions and Mixed Numbers . . . . 9255) Rounding. . . . 93
56) Fractions and Decimals . . . . 94
57) Fractions and Percents . . . . 95
58) Decimals and Percents . . . . 96
59) Units . . . . 97
60) Metric Units. . . . 100
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alculate 61) MUD before COLT . . . 10462) FUN before MUD. . . . 105
63) IN before FUN . . . 106
67) Formulas . . . 109 68) Units in Formulas. . . . 110 69) 2D Shapes . . . 112 70) 3D Shapes . . . 113 71) Averages . . . 114 72) Rates . . . 114 73) Ratios . . . 116
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oots 74) What is a Root?. . . . 118 75) Reducing Roots. . . . 119 76) Combining Roots . . . . 120 77) Multiplying Roots . . . . 121 78) Fractional Exponents . . . . 122 79) Rationalize Roots . . . . 12380) Roots with Same Base . . . 124
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olynomIals Thinking in Algebra. . . . 126As Few Variables as Possible . . . 129
81) Distribution . . . . 130
82) FOIL . . . 131
83) Rationalize with conjugates . . . . 132
84) Common Factoring . . . . 133
85) Bifactoring . . . 133
86) Bifactor when a>1 . . . 135
87) Bifactor- other . . . 136 88) Squares. . . . 137 89) Double factoring . . . 137 90) Quadratic Formula . . . 138 91) Reduce fractions . . . 140 92) Multiply fractions . . . 141 93) Combine fractions . . . 141 94) Long Division . . . 142
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quatIons 95) Recognize equation types. . . . 14496) Answer!. . . . 146
97) Break variable term. . . . 147
98) Combine like terms. . . . 148
99) Decouple like terms. . . . 149
100) Eliminate decimals . . . 150
101) Eliminate fractions . . . 151
102) Fill Parentheses . . . 152
103) Flip complex fractions . . . . 153
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quatIons 106) Fill, Flip, or Figure. . . . 156107) Eliminate fractions or decimals . . . . 157
108) Descending order = 0. . . . 158
109) Common factor . . . 159
110) Bifactor . . . 160
111) Answer formula . . . 161
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there
quatIons 112) Linear . . . . 162 113) Rational . . . . 163 114) Multi-variable . . . . 165 115) Exponential . . . 166 116) Inequalities . . . 167 117) Radical . . . 168 System of Equations . . . 169 118) Systems by Substitution . . . . 169 119) Systems by Elimination . . . 172 120) Systems of Three . . . 175a
ctIonse
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ules
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oals& m
ethods Encrypted Education . . . 194 What Is Understanding? . . . 195 Readiness. . . . 196 Resources. . . . 197 My Student Is Stuck! . . . 199 Pre-Formal Math . . . 200 Activities . . . 200Whiteboards and Vinyl . . . 202
InTroduCTIon
This book is written for teachers and parents who want to understand the big picture of arithmetic to algebra so they can intelligently explain it to students in a connected framework. At the end of the book I show that a connected framework is understanding.
The only assumptions I bring to this book is that you have a desire to see math in a new way and you have the ability to reason. I also assume that you took standard arithmetic and algebra courses sometime in the distant, hazy past and you may or may not have passed those classes. Some of you are teaching math whether you like it or not.
Action Algebra covers the foundation or core of math from beginning numbers to advanced equations. I proceed in a logical, step-by-step manner in the same order of lessons 1-120 as with the students. Therefore, I leave some topics incomplete at their first presentation and finish them later after the additional principles are introduced. For example, in the second chapter on combining, only fractions with common denominators are used. Later, in the divison chapter, fractions with different denominators are covered.
Some of you are in a position to teach your students from the very beginning, such as homeschoolers with young children. Others of you have some flexibility, but your student(s) already have years of habits (for better or for worse) ingrained in them. Still others of you are in a classroom with many students and a fixed curriculum. Understanding the common thinking processes connecting the huge variety of problems in the textbooks will be of help to any teacher in any situation. For example, many parents teaching Saxon are lost when trying to explain previous concepts more than a few lessons back.
This handbook covers arithmetic, pre-algebra, and algebra in 40 lessons each. The focus is on the math, but with the worksheets and videos, many word problems are also covered. Topics that are applications or electives of math, such as statistics, geometry, trigonometry, and science are planned to be covered in future classes. Action Algebra lays a solid, complete system of understanding that will fully prepare a student for all their other courses. If possible, have your student(s) master these lessons before any other math.
Believe it or not, every essential topic from arithmetic to algebra is covered in this book. The only thing "lacking" is the duplication of topics that the teach-reteach textbooks have made popular by their grossly inefficient methods.
Years ago I figuratively started with grade 2 math and worked my way up to Algebra 2. I kept every new topic, but ripped out the pages dealing with a repeat or slight revision of the topic. At the end I had enough pages left to make two textbooks. That discovery spawned the development of this curriculum.
The Action Algebra worksheets can be used as a supplement to any curriculum, but the full power and time-savings will be realized when they are used as the curriculum itself. Because of their almost limitless variety and ability to be customized, students can study a topic in an organized, focused context, then practice it until it becomes automatic, then they can return to it for review as needed and at scheduled review points. There is a lot of time wasted in "gear changing" and the mind loses focus. (See the section at the end of the book on how to use the grade sheets.)
So the full Action Algebra approach combines the best of both worlds. Repetitive drillwork is combined with a constructivist approach that results in students really knowing why and what they are doing. Students are not left to randomly discover principles, but neither are they engaged in almost mindless drill. They are guided to understand concepts and procedures in connection with each other.
If American high school math students are ever to regain the top spot in the world, we must combine both approaches that are fighting with each other in the education arena. American ingenuity and American hard work are compatible, resulting in American excellence, quality, and performance.
The 40 lessons (roughly one per week) in each class are not magic numbers. They could easily increase or decrease as time goes on. The point I am making with them now is that it is possible to cut the usual seat time in half or in third. For a student to accomplish this seemingly miraculous feat only means that they understand and review as they proceed. The consequences of this is that there will be more time to apply math both in math and science classes. It also means that schools will not only raise their graduation rates, but their average levels of achievement will raise much higher. Homeschoolers, of course, will cut down their time even further.
But now the present lies nearer than the glorious future. For lower grade teachers I recommend reading the arithmetic and pre-algebra chapters. This will give you an understanding of the next level for which you are preparing your students. Just like with them, understanding the next level “seals in” the current level. For you, understanding the process of combining with negative numbers is crucial. If you feel you need more examples, please look at the worksheets and videos.
For middle and high school teachers, the whole book is necessary, especially understanding how the Shift Action is involved in so many problems and steps. This is the single biggest concept that students need so they can tie together so many seemingly random steps. Also, the FA method of solving equations is very simple to teach as a unit outside of any textbook, then your curriculum can proceed with much greater ease. As with the elementary teachers, you may need more examples, so look at the worksheets (many of which have step by step solutions) and videos.
of the book on Goals and Methods, and on Pre-Formal education. Math is the most abstract of all classes and we must realize who the young students are that we teach as much as knowing how to teach a topic. Also, if you are thinking of using Action Algebra as your main curriculum, the Grade Sheet section will give you a good introduction.
Now that we have addressed some technicalities, I hope you will find many Aha! moments as you begin studying this book. To help us get started with the big picture, Einstein will semi-seriously take us from arithmetic to algebra in ten minutes.
Arithmetic to Algebra In Just Ten Minutes!
Once upon a time little Einstein stuck his finger in an olive and then in another olive and another and another until he had an olive on each finger.
“Hmmm,” he thought, “There is something similar between the olives and my fingers. I have the same (what shall I call it?) number in both groups. As I was sticking my fingers in them I was counting.”
Then he thought again. “What if I want to count more olives than I have fingers? I guess I should invent a symbol for each number and a way of re-using those symbols when I run out of fingers.”
So little Einstein invented the number system with ten digits and place value. He was pretty clever about it, because his first digit, 0, represented having no olives on his fingers. Ten, 10, represented having his fingers completely full without any extra and ready to start putting olives on his mother’s fingers. The budding scientist was too smart to put them on his toes because he knew he would get them dirty and squish them sooner or later.
Some days later, little Einstein started to get bored with his number system. He had counted all the olives in his father’s orchard, his neighbor’s orchard, and his uncle’s orchard across town. In fact, Einstein knew the number of olives in all the orchards around town. He also knew the numbers of cats, dogs, and horses. Yet little Einstein wanted to something more, something new. He sat down in the dirt road and thought and thought.
Then it came to him! What if he could find out the number of all the olives in all the orchards together! Why not do something with the numbers he had already collected so that he could figure the answer without recounting?!
Of course, that was a brilliant idea. He came up with a process of putting numbers together that he called adding. In no time flat, little Einstein knew the total of all the olives, animals, houses, and people in the town. Not long after that, he invented something called subtraction so he could accurately change his total when olives were eaten or exported. Sometimes, an animal died and he needed to take that into account as well.
Little Einstein was starting to catch on to the power of numbers and so it wasn’t long before he discovered he could multiply and divide the rows and columns of trees in the orchards to quickly find out how many were in each.
As he shared his knowledge with the townspeople, they soon began to ask him questions and wanted to learn what he was doing. This forced little Einstein to invent symbols for each of his ideas so it could be written down and made permanent. So, in addition to his ten digits, he made symbols for his four operations that he could do with those numbers.
One day at supper time he was struck with a puzzle. One of the olives he put on his finger split into pieces. He could not count them as 1, 2, 3 because they were not whole olives like the others. That’s when he realized he needed a way to keep track of partial things. Thus, fractions were born. He used a slash or a horizontal line because it reminded him of a cut. The top number represented how many pieces he ate and the bottom number represented the total number of pieces the olive had broken into. The number on top was usually smaller than the number on the bottom because some pieces fell on the floor.
To save himself some time, little Einstein put a decimal at the end of the whole number of olives, then started counting tenths and tenths of tenths on the right side. That way he did not have draw a slash and put a bunch of zeroes below it. It was a special, convenient fraction that always meant tenths.
Then, because he had whole numbers and tenths in a decimal number, he put a whole number in front of a fraction and called it a mixed number. Then, because people used dollars so much and were always figuring prices as some part of 100 pennies, he invented the percent symbol to make everybody’s life just a little bit easier.
But it was his friend, Sherlock, who prompted Einstein to make some of his bigger discoveries. One day, Sherlock asked Einstein if he had any idea how many olives there might be in the whole world. Einstein replied that his friend’s question was not elementary. He would need to invent another kind of number to handle the enormous task of multiplying all the olives on all the trees in all the orchards of all the towns of all the countries of the world. So he made scientific numbers with a handy little device called an exponent, which compressed the multiplying of many numbers down into one.
After all their research and calculations, Einstein and Sherlock discovered that some pieces of their data never changed and other data changed a lot.
Einstein called the data pieces that stayed constant--get this--constants. Sherlock thought that bit of naming was too elementary, but could not argue with the logic.
One of Einstein’s first constants was something he called “pie.” Actually, he spelled it “pi” because he did not want to offend any of his Greek neighbors. Pi was the ratio of the diameter of an olive to its circumference, which was always just about 3.14. Curious, eh?
Right after eating pie and discovering pi, Einstein discovered “c.” This was the speed of light that he and Sherlock measured every time they took a flash picture of olives at night. Shortly thereafter, Einstein muttered E=mc2 as he tried to think of creative ways to
destroy all the olives in a country in a very short time.
But Einstein’s greatest number was still waiting to be discovered.
In the laboratory, Sherlock was deep in calculations and very frustrated when Einstein walked in. “What’s the matter, old boy?” Einstein asked.
Sherlock replied, “I go through the same long process over and over again as new numbers come in from the orchards. There must be a better way of solving these mysteries that are really just a repeat of the same kind of problem.”
“Well now,” Einstein exclaimed, “Isn’t solving mysteries your cup of tea? Why should some unknown numbers stop you--.”
Just at that moment, Einstein had an incredible insight.
“Unknown numbers!” he cried.“They are not totally unknown. After all, we know they are numbers, we just don’t know exactly which one. The numbers just vary from time to time. Let’s call them variables and learn how to do adding, subtracting, multiplying, and dividing with variables!”
Sherlock looked at the scientist with his mouth agape and jaw dropped. After a bit, he raised his index finger like he was checking the wind, and declared, “I think you’ve got an idea!”
“If you could do that, we could make formulas and equations that hold the spots for our numbers before we get them from the orchards. We could do some of the calculations only once and never have
to do them again! Instead of re-inventing the wheel and figuring out what to do with each number every time, we would have a template we could use over and over again. That’s just as good as recycling all the olive boxes!”
Einstein raised his hand in the air as if he was posing on the steps of the Acropolis and pronounced, “We shall call doing math with variables, Algebra.”
BAsIC PrInCIPles
Three basic principles upon which math is founded are equality, priority, and insight. Applying them to math gives us: value change needs counterchange; first calculate the complicated; and, changing looks does not change value.
equality
Life is a constant balancing act. We have to balance work and play with rest and sleep. We need to get enough time alone to think for ourselves and do our own things, but we also need time with family and friends. We can’t be alone and with a group at the same time, so we have to balance our time between the two. Sometimes we might split our time between two different activities, like watching TV and doing homework. Yet, one still affects the other. We can’t do whatever we want whenever we want. Humans require balance or else we get sick or get a hangover. One way or another our lives demand, and get, balance.
Balance is necessary because of limits. Unless the parents are infinitely wealthy, if sister gets more allowance, then brother gets less. If there are more eagles above the river, then there are less fish in the river. If you have driven more miles down the road, then there is less gas in the tank. These are like the Law of Conservation of Energy: Energy can neither be created nor destroyed, it just changes form. Cause matches effect. Action equals reaction. Input equals output. In other words, the pot of soup never grows, it just gets stirred.
A math problem is the same. The answer must equal the problem. It is no different in value, just in format. For example 5 truckloads of 10
crates of 200 boxes of cereal is a problem, not an answer to my question. I want to know how many boxes of cereal are coming to my store. 10,000 is an acceptable answer. 5 ^ 10 ^ 200 is accurate, but not acceptably simple enough. Now in my attempts to solve the problem, I cannot arbitrarily inflate a number or remove a number. I can do many things, but one thing I can never do is create or destroy value. Before must equal after at every step from beginning to end.
Imagine a math problem taking place on a balance scale. (A very simple one can be a hanger with clothes pins holding different items in balance.) You can do whatever you want to items on one or both sides as long as your actions leave the hanger in balance. Folding a hanging sock doesn’t upset the balance so it is fine. However, removing a sock on one side requires the same kind of sock to be removed from the other side. Balance before = balance after.
This principle of equality and balance seems to be telling us what we can’t do and therefore limits our options. However, it actually opens the door to two powerful Actions.
Value Change needs Counterchange
Because I cannot create or destroy but must maintain equality, I must make a counterchange for every change of value that I introduce. (Notice that I said, “I introduce.” I am not talking about the calculations that the problem tells me I must do.) For example, if I subtract 3 from one side, then I must subtract 3 from the other side. If I multiply the left by 28, then I need to multiply the right by 28. These are examples of the Sync Action. My change tips the scale out of balance, but my counterchange brings it back into balance, so that is perfectly “legal.” In other words, it really works.
Another Action based on the principle of balance is Shift. It is used when I am dealing with only one side of an equation, or with an expression, which is a problem that is only one side of an equation. For example, if I have two water balloons hanging from the left side of my hanger and I want to take 6 oz. of water from one, then I must add 6 oz. of water to the other. In math this looks like 10 + 15 becoming 4 + 21.
If I squeeze a balloon so that the top has less water the bottom automatically has more water. (Popping balloons not allowed!) In math this happens when we reduce fractions. 6/8 becomes 3/4. There are less pieces of the pie on top, but the size of the pieces got bigger on the bottom. This may not be readily apparent to you, but we will look at this Action many times with fractions and other examples. It is used a lot!
Another example of Shift happens with units. When I change a 1 dollar bill I get 10 dimes in return. I have more things, but each item has less value.
up the outside. Think of this Action as the principle of the Up and Down. My house goes down in temperature, while the outside goes up in temperature. If you don’t believe this, go stick your hand over the exchanger!
Priority
Years ago I saw a simple demonstration that I have never forgotten. A lady had a jar with several large rocks beside it. There was also a pile of gravel and sand. She put the sand in the jar, followed by the gravel, but only one rock would fit. Then she emptied the jar and started over from scratch. This time she put the rocks in first, and poured the gravel around them. Then she poured the sand in and shook the jar until it all fit. She succeeded by starting with the biggest stuff.
Likewise, life is filled with order. You build a house from the foundation up and then from the outside in. Order matters or else the house will be ruined by the weather or collapse under a load. Life is filled with priorities. Starting the day with a good breakfast makes us healthier. To eat a good breakfast we have to wake up early, which means we need to go to bed on time. Getting our homework done on time gives us privileges like going outside to play and getting good grades. This gives us feelings of accomplishment and happiness. That makes us better, kinder people. Paying our bills before blowing our money on extras is another priority that wise people adhere to. Keeping one priority often helps us keep other priorities straight. As we figure out our priorities and follow them, that helps us achieve a balanced life.
First Calculate the Complicated
Math also has its priorities. The important things must be calculated first, and that which is most complicated is most important. Why? You cannot count that which you do not know.
Very loosely speaking, the simple goal of much of math is to count. We want a number, a value, which is a count of miles, hours, dollars, items, or other things. Before I can count, I must add, because adding is counting two or more groups of things. Before I can add, I must multiply, because multiplying is repetitive adding. Before I can multiply, I must calculate functions, because then I will know the final number to multiply. And in the midst of all that, I must pay attention to parentheses, because they can override anything at anytime.
words, solve a problem in reverse order of when you learned the parts of it. For example, everyone learns adding before multiplying, but we should multiply before adding in a problem. Next comes exponents and other functions like roots and trigonometry. The order of learning these things will vary a little depending on the textbook, but all of these are on the same level of importance which is above multiplication.
A simple real-life situation will illustrate the meaning. Let’s say it is your task to count the total production of toy blocks on a certain day. There is a pile of blocks waiting to be packaged. There are boxes of blocks stacked on pallets, and there is a machine cranking out blocks constantly.
You can count the blocks in the pile easy enough, but to count the blocks in the boxes you must first count how many are in one box, then multiply by the number of boxes, then multiply by the number of pallets. You must go inside a box and count because you cannot count that which you do not know.
Now you have a choice. You can wait for the machine to stop making blocks and let them get boxed up to do your count, or you can count what is available and keep them completely separate from the output of the machine while you wait. Either way you are giving priority to the machine before calculating your total.
This simple illustration shows that functions (machines that, in professorial terms, map a set of numbers to another set) must be considered before boxes before loose items or you must have a way of separating them. Likewise, roots and
logs come before multiplication which comes before addition, or you must have some way of completely separating them.
Four Actions- In, Fun, MuD, COLT- in that order- help us to calculate correctly. (IN FUNny MUD is a COLT)
In, or Inbox, means I should work inside parentheses first. Parentheses ( ) and brackets [ ] and braces { } all act like mathematical boxes to group what is inside them. We must find a single value for the whole group before we can add or times it by what else is there.
Fun is short for function. Exponents, logarithms, and trig functions are the
common functions to be encountered in pre-algebra and algebra. They are little mysteries that must be unraveled before we know the final value we have to work with.
MuD is short for Multiply and Divide. They are of equal importance because division is basically multiplication in reverse. So almost all rules that apply to multiplication apply to division, also. When talking about multiplication, keep division in mind. The NOPE trick of figuring negative and positive signs applies to both.
COLT is short for Combine Only Like Terms (or Tags or Things). Combining is the all-in-one method of adding and subtracting that is covered in the second chapter. It is the one way that works for all of arithmetic and all of algebra. SSADDL is the how-to principle that goes with COLT, because every colt needs a saddle!
Insight
We hope to raise our children with enough insight to know that changing costumes does not really change the actor. It is still the same person behind the mask. Similarly, we try to teach them that beauty is more than skin deep and that the value of persons does not depend on the color of their skin. Also, an old dollar is worth as much as a new dollar.
Changing looks does not Change Value
Most math steps depend on the Balance and Priority Actions, but the Insight Actions are nice helper tools. They are easy to use, but not needed as often. (So they tend to get forgotten.) This group of Actions are called Insight because it takes looking at the problem and your options in a slightly different way to figure out that if you used one of these, you could make things easier.
None of these Actions changes the value of numbers, so counterchanges are not needed. The Show Action hides or unhides what is already there. Sort re-arranges what is there. Morph changes the form of a number into another equal form. Sub trades one value with another equal value. All of these Actions change only the looks of a number, but do not change the value of the number. It is like putting a new paint job on a car without changing the car itself.
So that is a real quick introduction and overview of the 3 basic principles and the 10 Actions. As we proceed through the lessons I will amplify the Actions at appropriate points, then use them to explain the current problem.
Action chArt
First calculate the complicated
IN
You must first work inside the boxes
( ) [ ] { } and fraction bars to figure
the answer you need to work with. Think of
unwrapping a present from the inside out.
FUN
Then you must feed raw numbers
into the mouth of the function
(funnel, get it?!) to figure the answer you need to
work with. Function processes input, you use only
the output.
MUD
Then you can MUltiply and
Divide all kinds by merging tags.
Figure the sign by using NOPE- Negative Odd
Positive Even. (MUD can get on all things, but do
we like it? NOPE!)
COLT
Then Combine Only Like
Things (Terms, Tags) by using
SSADDL- Same Signs Add, Differents Destroy,
Largest sign is answer sign. (SSADDL your COLT)
Changing looks does not change values
SYNC
You may do the same thing
once to each whole side of an
equation. 1 effect, 2 opposite sides.
SHIFT
You may change the value of
an object at any time if you
counter it with an equal, opposite change within
that object. 2 opposite effects, 1 side.
SORT
You may re-arrange the
objects in a level at any time,
but never change a division part.
SHOW
You may show or hide
invisible objects at any time.
MORPH
You may convert an
object from one format
to another at any time.
SUB
You may replace object A with
object B at any time, if A=B.
ZooM leVels
Zoom levels are not needed to teach or learn arithmetic, but these next two pages are inserted as an overview for teachers and they show what I meant by “objects” on the Action pages. Even teachers of basic math will benefit from this because they can see where their topics fit into the big picture.
factor ^ factor
term + term
expression = expression
equation
term + term
factor ^ factor factor ^ factor factor ^ factor Zoom levels is a phrase I coined to describe the varying levels of focus in a math problem. Sometimes we are working on the factors within a term, while at other times we are working with the terms in an expression. As we advance through math it becomes increasingly important to be aware on what level we are on. Any given step of any problem takes place on only one level. The level can change from step to step, but it will never change within a step. For example, we don’t do something on the term level, then try to balance it on the factor level.
As you can see in the diagram, factors multiply (or divide) to make a term (called “compound number” in arithmetic). Terms add (or subtract) to make up an expression. Expressions are linked with an equal sign to make an equation. So we have four levels where Actions happen.
Groups, ( ) and [ ] and {} and fraction bars, can be used at any level. They can group factors into “superfactors” and terms into “superterms.” That is when our abstracting abilities really get tested. It happens a little bit in pre-algebra, but mostly in algebra. (Arithmetic teachers, you can breathe a sigh of relief!)
6^9
factor factor factor factor factor
8xy -2(5+6x)
factor f a c t o r Now let’s see what these things look like in real life.Any two things that multiply each other are factors. (Division is included, because division is reverse multiplication.) So all the different kinds of numbers and groups of numbers can be factors. It all depends on how they are connected.
In the above examples you can see how the numbers and letters have a dual role. Not only are they numbers or variables, but they are also factors because they multiply each other. Also, as factors, they “bond” themselves into packages called, terms.
The example on the right is interesting because of the grouping. The whole example is one big term made of -2 ^ ( ). However, inside the ( ) factor are two “subterms” of 5 and 6x. The 6x term has its own factors of 6 and x. Do you see why I call this “zoom levels?” You zoom in from the problem as a whole until you get down to the individual parts.
One more note about the above examples: Each one is an expression, because an expression can be made of 1 term, just as a term can be made of only 1 factor. This means that a single number can be a single factor (times an invisible 1) making a single term making an expression. It all depends from which zoom level you choose to look at it.
9-7(4+9y)
9¤-7(4+9y)
4 ¤ +9y
3*8-5*7
3*8¤-5*7
6x¤+0¤-24y
6x+0-24y
A useful, and often challenging, exercise I do with algebra students is to give them random algebra expressions to be split into terms. I use a double slash or squiggly line so that it is not confused with some other symbol. The point is that students must “see” algebra.
What helps me visualize this is I think of expressions like railroad trains. The terms are the cars coupled together with + and - signs. Inside the cars are boxes of stuff called factors. We can split trains apart at the couplings between the cars, but we never split the cars themselves because that would make a mess on the tracks.
So the summary of the matter is that there are four zoom levels we need to be aware of as we progress through math. We will work with the objects in only one level at a time. That means factors are the objects in terms. Terms are the objects in expressions. Expressions are the objects in an equation.
ArIThMeTIC: nuMBers
This chapter covers numbers and the number system. It shows how numbers are arrows from the number line, which is an infinite arrow. We then compare numbers to each other. Finally, we identify the types of numbers and the parts of compound numbers. All of this is approached from a concrete, rather than abstract, perspective to make it clear on a child’s level.
1) Benefits of a number system line
Of course, we just call it “number line,” but I am trying to capture the complete wisdom of the idea by saying, number system line. Without a number system, which requires the idea of place value, numbers would just be an endless invention of names. Not too helpful.
If we did not organize numbers into an orderly sequence on a line, we would think of numbers in random order and places. Like counting the pennies in a jar, it would be too hard to precisely compare piles (sets) of things. Lining things up and comparing the lengths of the lines at a glance is the easiest way.
If we are comparing two numbers and the number line is horizontal, then the number farthest to the right is greater. If the number line is turned vertically, then the highest number is greatest.
The worksheet about locating numbers is a good time to point out that the number line starts at 0 and goes endlessly in both directions. It should also be pointed out that all counting begins at 0 and we count steps. Adults sometimes make the mistake of counting the starting point. Instead, we start at 0 and count the first step to 1, the second step to 2, and so on. This is like counting on our fingers and we already have 3 fingers up. We don’t start at 3 and count 1. We start at 3, move over
0
to the fourth finger, then count 1. We are counting steps, not marks on the line.
If the number is negative, we go down or to the left. This is just like a thermometer getting colder, or going down the stairs to the basement.
(If you have anxieties about negative numbers, don’t show them. Children have not seen them before and so they have no hang-ups about them. Six year old children can easily do this even if they don’t have our abstract understanding of them. More details will be covered in lesson 3.)
2) Translating numbers and Words
From pre-formal activities, the student should already be familiar with both the idea and the wording of place value. Orally s/he should be able to count to 100, but now the transition to the written form needs to take place. This is one lesson where writing large may make a critical difference.
3) numbers are arrows
What was implied in lesson 1 is now clearly stated.
A number is an arrow. It has both size and direction. It is a piece of a number line. To exactly describe the size of an arrow we use the digits 0 through 9 to “spell” numbers. The digit part of a number tells us the size of the number, which is the length of the arrow.
Like an arrow, a number always has direction. In math, there are two basic directions: positive or negative. Arrows always have direction so numbers always have signs. Sign + digits = number. If you do not see a sign that means there is an invisible positive. You will never be wrong if you want to write it yourself. 8=+8 3=+3 +6=6
Numbers are not just bars. They are arrows. Think of their size AND direction. Think, I have $5 or I owe $5. I walk up 8 stairs or I walk down 8 stairs. The temperature is 15 degrees above zero or 15 degrees below zero. Some things may not be negative, but the numbers used to describe them can be. For example, can you eat -3 pieces of cake?! Can you be -5 feet tall? Of course not. My height can’t be negative. But I am always 5 feet tall whether I am climbing up a cliff or hanging upside down. Numbers always have size and direction.
6 feet tall
6 feet tall
6 feet above
the floor=+6
6 feet below
the floor=-6
floor = 0
What is an arrow?An arrow is a line with a pointer on one end to tell us what direction it is going. The pointer end is called the head. The end without the pointer is the tail. This is where the arrow begins. Every arrow starts at 0 length and stretches out to its end where the pointer is. This is exactly what numbers do. They start at 0 on the number line and stretch a certain distance. The digits part of the number tells us the size, and the sign part of the number tells us the direction. + is up or to the right, and - is down or to the left. Just like an arrow is a line with a pointer, a number is a digit part with a sign part. We often leave out the sign which gives a number direction. Forgetting about it causes us to misunderstand adding and subtracting, which then causes us more problems when it comes time for pre-algebra. Even when we don’t see a sign in front of the digits, that just means there is an invisible + sign there. All “plain” numbers are positive. A number is negative only if there is a - sign in front.
negatives are normal
Let’s look at the numbers that strike fear in the hearts of many. A negative number (or, a minus number, as some call them) is just a regular number that goes left on the number line instead of right. If the number line is in the vertical position then negatives go down. This is just like a thermometer that is minus 5 degrees below 0 when it is
really cold outside. It is also like being 1200 feet below sea level in Death Valley. There is nothing bad or different about negative numbers. In fact, they are really good when you are keeping score in golf!
The real source of our anxiety about negative numbers comes from trying to add and subtract them. The typical way adding and subtracting is explained breaks down when it comes time to introduce negative numbers and it is this breakdown that is the real cause of confusion. The next chapter on combining will fix this problem.
4) Comparing numbers
This lesson is not mathematically hard, but the language can be a little subtle. In normal life we use bigger, larger, and greater in similar ways to mean the same thing. However, in math we make a definite distinction between them. Bigger and larger mean the same thing: the size of the number, which is the length of the arrow. Greater, however, means position on the number line. Let me explain.
Bigger, larger, smaller only want to know the size of the number, not the sign. Bigger and larger want to know which of two arrows stretches furthest from 0. The direction in which they stretch does not matter. Smaller does not care about direction either. It just want to know which arrow stretches the least from 0.
These distinctions are useful when we subtract, because we want to make sure the bigger number is on top. We don’t care if it is positive or negative, only the size of the number determines if it is bigger, or larger.
If you are familiar with absolute value, bigger/smaller is exactly the same idea.
When we wonder if a number is greater than another, we are wondering if it is higher on the number line. As in the picture below, the greater number may actually be smaller, but because it ends at a higher spot, it is greater. So greater than and less than mean higher and lower. When you see these symbols < and > they are referring to greater/less than, not bigger/smaller.
-8
+4
+4 is greater than -8,
because it is higher, but -8
is bigger than +4 because
its arrow is longer
Bigger, smaller, larger
only see the length of the
arrow, not its direction
Greater than and less than
include size and direction
which gives a final position
on the number line
When doing the worksheets it may help to put your finger over the signs when using bigger, larger, smaller. Now it is like both numbers are positive and pointing upward. That is exactly what absolute value will do later on, so this is not wrong or a “get by” trick.
When working with greater/less than use a number line in the vertical position. Now it is easier to see that any positive number, no matter how small, is greater than any negative number, no matter how large. Also, any negative number is less than (lower) than any positive number.
You should find students readily grasping the concepts separately, but may get mixed up when all the words are used on the all comparisons sheet. I wish I knew of an easy, obvious memory device here, but I don’t.
5) Kinds of numbers
This is another memory lesson. All the students need to do is recognize the types of numbers. They do not need to do any comparison or math with them. This is like bird identification. The student does not need to know how they fly, just recognize what they look like.
There are 8 basic kinds of numbers we use in arithmetic and algebra:
1) Integer. I use the more technical word instead of “whole number.” Whole number is not used consistently. Integers are simply positive and negative whole numbers, including 0. Integers are a subset of decimals.
+5 -2 16 -39 +8
2) Decimal. Anytime a number has a visible decimal point, I consider it a decimal. Technically, a number like 3.0 could also be considered an integer, so you can override the answer keys and give credit for that answer as well. For further study you could look up rational and irrational numbers in Wikipedia in case you come across it on standardized tests or textbooks, but the distinction is not central to arithmetic and algebra.
7.25 1.33... 0.09 .1
3) Percent. Usually it is the integers and decimals that have a % tacked on them, but any number type can be made into a percent. So I consider the % symbol trumps all else.
8.9% 6% 9
1/
5
% 30.18%
4) Fraction. Any kind of number can appear in a fraction on either the top or the bottom, or before or after the slash. However, we try to convert (Morph) the fraction into having only integers as soon as possible.
1/2
4/
13
-3/7
23
1
5) Mixed number, or simply, mixed. A visible integer next a fraction with only integers is a mixed. I rarely work with mixed numbers. Instead, I turn them into fractions, solve the problem, then re-convert back to a mixed.
5 1/2 -9
4/
13
-11 3/7 6
1
2
Technically, the decimal must be between 1.0 and 9.99999... so that there is only a ones digit followed by decimal places. However, at this point, if any decimal followed by ^10N
is called scientific, that is good enough.
2.7^10
49.003^10
-157) Variable. Just an introduction is necessary here. Variables are letters that wrap mystery numbers within them. Algebra will tell us how to solve mystery numbers, but for arithmetic all we need to know is that they are shorthand numbers for things like apples, boxes, and miles. We need to know a little bit about the things so we know whether to add or not.
x y apple a
8) Constant. There are just two special decimals that we abbreviate to letters in standard elementary math. They are & and e. A calculator will give you the long decimal values if you want them, but for now all the student needs to know is that & and e are constantly the same value in every problem.
& e i
6) Parts of compound numbers
As I am sure you have already noticed, this first chapter on numbers has not been standard. While I have not relied on a young child’s inability to comprehend deep concepts, nevertheless a complete foundation has been laid for all the rest of arithmetic and algebra. There will be no need to teach, unteach, and then reteach. Starting with the very next chapter you will see the advantages of introducing all the details of numbers right up front. One continuous system and framework can be built that cuts out a tremendous amount of duplication and work arounds. Using the endless supply of Action Algebra worksheets, the student can progress at his or her own pace in a simple, straightforward fashion and still finish algebra years early. This leaves plenty of time for side topics, applications, and other investigations!
I needed to say that to prepare you for wording new to you.
Just like we have compound words, we have compound numbers. Fractions are good examples because they are numbers within numbers. The top (numerator) and the bottom
(denominator) are individual numbers, but stepping back and looking at the numbers with a bar in between we see a fraction. Thus, we have a compound number.
A compound number is made up of a simple number (integer, decimal) followed by a tag.
Tag
“Tag” is not a regular math word. It is a word I made up to help you see the parts of a number and what they do. The tag always comes after the regular number and tells us what kind of compound number we are looking at. This is important because we must have matching tags before we can add or subtract two numbers, and we must know what is in the tag so we know what to multiply.
5 players
-1
xy
7 /9
2 /3&
8
.3
Regular numbers like 2 or 7.4 have blank tags. Sometimes the tag can be a variable, like x, or it can be an item from daily life, like shoes or books.
Since fractions are compound numbers, they have tags you can see. The bottom number (or the right hand number if written sideways) is the tag. The fraction bar is included.
(Algebra teachers: A tag is all factors in a term except the coefficient and/or numerator.)
Compound number
Very simply put, a compound number, like a compound word, is made up of more than one part. Be mindful that one of the parts might be invisible. All compound numbers have at least two parts called the Front Number (frontnum, for short) and the Tag. An optional third part is attached to some numbers called the Unit (miles, feet, meters, pounds, etc...), but it is really part of the tag, also.
Compound Number
Frontnum Tag Unit
The algebra word for compound number is TERM.
In short, the front number is the first part of every compound number and is the quantity part. It tells us how many tags we have. The frontnum is always an integer, decimal, or top of a fraction. Once in a while it is invisible, but we will talk about that later.
The tag is everything after the frontnum that is attached by multiplication or division. This includes other numbers and all letters. Multiplication and division signs are included.
These labels, “compound number” “frontnum” and “tag,” should seem new and strange to you because they are not standard vocabulary. However, they are labels for standard math items that you learned when you were in school that were left unnamed in the lower grades or not named at all.
Young children can easily identify the parts of a compound number, even if they don’t yet understand everything those parts do. Rather than use a strange word like “coefficient” that still makes no real sense to me (a math teacher), they can easily and visually relate to “front number.” Term vs. compound number is a toss-up. If you want to skip the baby word and go right to “term” that would make sense to me, also. Tag labels the unlabeled so we have nothing to lose there. The main point is that children learning math for the first time will accept whatever words you use. What might feel strange to you will be accepted as normal by them.
ArIThMeTIC: CoMBIne
In this chapter we will learn how to add and subtract compound numbers. Adding and subtracting are just pieces of an overall process that is called “combining.” It is easier to learn how to combine positive and negative numbers right from the beginning.
This chapter introduces our first Action, which is called COLT- Combine Only Like Things (like things have like tags). As you learn about adding and subtracting, you will begin to see that they are just like walking up and down the stairs of the icon. Above the water line is positive and below the water line is negative.
We will start with a proper understanding of adding and subtracting, but quickly move to the all-in-one method of combining. This method will not only work for arithmetic, but it is also the far better way of adding and subtracting terms in algebra.
7) Adding on a number line
We usually think of adding this way: Put two numbers together to get a bigger number. If you use “bigger” the same way we used it in the previous chapter, then you are right. However, most people don’t use it that way or teach it that way. They fear negative numbers, so they think of only the up direction for bigger. They don’t realize that numbers also get bigger as you go farther from 0 in the down direction. But the previous chapter showed us they do!
0 +5 +6 +3 +4 -4 -4 -2 -6 -8 -8 +11 +7
Arrows lined up in the same direction, head to tail, is adding
Let’s use arrows to help us make a better definition. Adding is lining up two arrows together in the same direction (head to tail). On the number line this means that adding
is putting two numbers together so that the answer is farther away from zero. Therefore, two positive numbers add up to a bigger positive, while two negatives add up to a bigger negative number.
So you see, it is not because numbers go up that they add, it is because they go in the same direction, even if that direction is down. Once again, if you understand the difference between “bigger” and “greater than” you are farther ahead than many. They mistakenly think that every time they add the answer must be higher on the number line, but really, the answer must be farther from zero, up or down.
Think of this in practical terms it will make sense. If you owe someone $4 and someone else $2, how much do you owe altogether? Of course, you owe $6 total. In your head you knew that owing $4 was bad and so was owing $2. So putting them together meant that things were going to get worse. You added, not subtracted, the debts. Your answer got farther from, not closer to, 0.
Again, let’s say you dig a hole 3 feet deep, then you dig another 2 feet. How deep is the hole? It only makes sense that if you go down, then down some more, you end up with a deeper hole, which means you must add the 3 and the 2 to get 5. Of course, it is a negative 5, because it is below 0, which is ground level.
Teach adding this way to prepare the student to understanding subtraction correctly.
8) subtracting on a number line
Adding lines up arrows in the same direction, so subtraction puts them together in opposite directions. Subtraction is not always “taking away,” but “taking away” is always subtraction. “Taking away” only deals with size, but of course, numbers have size and direction. Therefore, subtraction must take into account size and direction, which is both the number and its sign.
0 +6 +4 -2 -12 -8 +4
This is critical for us as adults to understand before teaching our students. We have been conditioned to think that subtraction is only taking away, but this leads to a mental road block. For example, if I have a stack of 3 books on the table, how can I take away 5? If I think that subtractions is only taking away, then this problem is impossible. When a student believing this enters pre-algebra and negative numbers, all sorts of mental difficulties and confusion arise. Some never get over it. Many take months to expand their thinking.
To subtract 5 books from 3, I need to see that the original stack goes up 3 from 0, which is the tabletop. Then I need to see that I must go down 5, which of course will land me in negative territory below the tabletop at -2. I owe the table 2 books.
The only correct way to tell if you need to subtract two numbers is by looking at BOTH of their signs. If the signs are different, subtract, but if they are the same, add. The answer might be positive or it might be negative, but it will always be closer to zero than the biggest number. Therefore, the usual advice to put the bigger number on top when setting up a subtraction problem is always correct. (At this point, you might want to take a peek at lesson 11 so you can see where this is all going, which is to the all-in-one method of combining.)
Now look at the examples. -2+6 means you go left 2 then go right 6 to end at +4 for the answer. When using pencil and paper with just the numbers, notice that -2 and +6 have different signs. +4-12 means you right 4 and then go left 12 to end at -8 for the final answer. Again, notice the different signs and so the arrows go different directions.
9) Adding stacked numbers
This lesson is the standard lesson which teaches students to add numbers vertically. This will be a real test of a child’s abstract abilities. Some may need to wait, some may take many weeks to master it to the level of being automatic. Again I advise not to push. Challenge, but not push. There is a significant jump from concrete, pre-formal thinking to juggling the abstract idea of numbers in the head.
+8
+6
+14
1 0+37
+13
+50
+72
+29
11
90
+101
+858
+245
13
90
1000
+1103
1 0+79
+76
+155
To prepare the way for combining, the worksheets put the biggest number on top and all the signs are written. The process of adding and carrying the spillover is the same as what you are familiar with. However, I have seen a variation that could be helpful to some students. Instead of writing the carry above the column to the left, the answer is written in one place below. It is a little bit more writing, but the place values are made plain all the way through. Notice also that the problem can be done either right to left or left to right.
10) subtracting stacked numbers
With the exception of showing all the signs, this is a standard lesson on subtraction. The big number is on the top, so even the standard “take away” explanation will work here. (Take away is not wrong, it is just incomplete.)
As you can see in the examples, the standard way of subtracting, with all the borrowing and slashing can be quite messy. If we as adults don’t like it, we can imagine the trouble this mess causes young children. So in the beginning you might want to break down the steps for them more clearly to aid their understanding.
The cause for borrowing (as well as for carrying in addition) is place value. Because we cannot always store enough value in the top digit, we must borrow 10 times that place from the place to the left and temporarily squeeze it in. What we are really doing when we squeeze in extra value is making a new problem within the main problem.
Look at the 24-19 example. After borrowing 10 from the 20 we can look at it as two problems, 14-9 and 10-10. Each of those problems only have 1 digit for an answer, which fit fine in one place. So we want to make sure not to borrow when we don’t need to, nor to borrow more than 10. In either case we will make a problem that results in two digits. And as Hardy use to say to Laurel, “That’s another fine mess you’ve gotten me into.”
+15
-7
+8
+24
-19
+5
/
1 1 8 0+93
-27
+66
/
1+72
-59
+13
/
6 1+858
-269
+589
/
7/
14 1I am going to work right to left, because along the way the top digit might be smaller than the bottom digit. To solve that problem I need to have a big, rich neighbor on my left that has not spent all her money yet.
So the first digit I will work with is the 3 and I see it must subtract a 7. For a final answer I can go into debt, but not in the middle of a problem, so I must borrow. The 9 is the big, rich neighbor and she is happy to loan me 1. But guess what?!! The 9 is in the tens place so it is really a 90 and the 1 she will loan me is really a 10. That will make my subtraction work!
10+3 is 13, so I now have 13 squeezed into the ones place that can easily have 7 taken away from it. 13-7 is 6, so I write a 6 in the ones place of my answer.
Now I move to the second column and the 8 that remains from the 9 can subtract the 2 beneath it. 8-2 is 6, so I write a 6 in the tens place of my answer. I now have a final answer of 66.
When demonstrating that to students you have two options. Slash and write the borrow real tiny, or make separate problems. (Here is where a big whiteboard can come in handy. Next to the main problem, you can write the two (or three) smaller problems, then put their answers back in place under the main problem.)
Either way you do it, be sure to note to yourself and the students that they are using their place value skill and bigger/smaller skill from the Numbers chapter. Nothing a student learns is extra or useless. It all leads to something in a later chapter, or even in the very next lesson, which is about to happen!
11) Combining stacked Integers
Starting with this lesson and before we complete the chapter, we will roll all the previous lessons into one. Being able to combine tags at the end of the chapter means the student is able to do all the skills to that point. It will be a good review spot. However, we must first start with combining plain integers (blank tags).
Before beginning let me clear up some terminology. I have seen some books use combining the same way I do to mean either adding or subtracting. I have also seen some books use the word “adding” in the same way I use “combining.” I am comfortable with both usages, but in this book, I will use adding in only the way I have already described it. Adding is two arrows in the same direction head to tail. This translates to numbers on paper as I showed two lessons previous.
Combining I will use only to describe the process I am about to show you, which will combine (no pun intended) adding and subtracting. Mastering this method, a student is set to conquer arithmetic, word problems, and algebra.
Combining clears confusion
Now lets put adding and subtracting together into one new process called, “combining.” Combining will tell you when to do the old-style adding and when to do the old-style subtraction and what the sign of the answer will be. You don’t need to memorize special cases and what to do in case a number is negative. Everything is all wrapped up into one overall process.
1) Always write largest number on top
2) Answer sign is Largest sign (top)
3) Same Signs Add, Differents Destroy
+7
-2
+5
+7
+2
+9
-7
-2
-9
-7
+2
-5
+15
+
0
3
+18
+15
-
0
3
+12
-15
+
0
3
-12
-15
-
0
3
-18
Before I explain, just study the examples to see if you can find a pattern. Did you notice that the biggest number is always on top? Did you notice that the answer always has the same sign as the biggest number? In other words, the top sign is always the answer sign. Did you notice that when the signs are the same, the numbers add to get a bigger number farther from zero? Did you notice that when the signs are different, the numbers subtract to get a number closer to zero?
This is the process you should have been taught starting in first grade. With combining, there is no need to learn, unlearn, and then re-learn separate processes with positive and negative numbers. Merge the two processes with correct ideas of “bigger/larger” into the
one process of combining that ALWAYS works, even in algebra.
(Remind the student that bigger and larger mean the same thing, just as they learned in the Numbers chapter. I use the word Larger here because it fits into a mnemonic.)
ssAddl your ColT
Most of the time numbers are not stacked vertically and you don’t want to take the time to re-write them that way. Here are the similar steps to combine positive and negative numbers when they are in normal, sideways format.
Same Signs Add. -8-5 means combine by adding. +9+2 means combine by adding. 18+7 means combine by adding. Don’t forget the invisible + in front of 18!
Differents Destroy. This is a shorthand way of saying different signs destroy each other. This goes back to the old Pacman game. The + are like cherries and the - are like Pacmans who eat cherries. Put a - and + together and they destroy each other like matter and anti-matter. Poof! This is the same as a hole and a pile of dirt. If you fill the hole with the pile, then both the hole and the pile are gone. Poof! Positives and negatives destroy each other, when combined.
Another little tip to fix this in the memory is that “different” is basically the same word as “difference.” The word, difference, is used in word problems as a clue to subtract. So you could say Different Difference to keep things straight, but people may look at you a little funny as if you are a verbal photocopier! (but you won’t forget!)
1) Find sign of largest number
2) Copy it to answer sign
3) Same Signs Add, Differents Destroy
-7+9=
-7+9=+
-7+9=+2
1 2 3-6-4=
-6-4=--6-4=-10
1 2 312) series of signs
This lesson really belongs in the chapter on multiplication, but I need to insert it here because some students will be confused by their textbook. Because they teach adding and subtracting separately, some books insert an extra + or - sign intending to be helpful. This is not necessary once you know combining and you will later have to unlearn the crutch of depending on an extra sign. (I have seen more students confused by this device than helped.)
So here is what to do. Count all the signs that look like - that are in front of the number. If the count is odd, the number is -. If the count is even, the number is +. It does not matter if you call the - sign a negative sign, a minus sign, or subtraction. Count all the - signs.
This is the NOPE rule that goes with the MuD Action. I’ll explain why this works in the next chapter.
13) Combining series of signs
Now that the student knows how to condense a series of signs into one, he will be able to combine any number of numbers with any number of signs. (No need to go crazy here. Every problem can be broken down into combining two numbers at a time until the total is reached.)
4--7 = +4+7
-8+-4 = -8-4
--++-2-+-6=-2+6
9-(+7)=+9-7
-3+(-5)=-3-5
11+(+2)=+11+2
-=---=+
---=---=+
---=---=+
---=-
NOPE--Negative
Odd
Positive
Even
If you think this looks like standard pre-algebra, you are right. If you think it is too early to introduce it to students, just remember that we have arrived here in a smooth progression. If the student has successfully handled the previous lessons, there is no reason to assume she will not handle this one successfully. Don’t let your fears and biases get in the way of the student’s blank slate! Also, the earlier something is learned, the more it is reviewed to the point of becoming automatic.
14) Combining Big Integers
This lesson introduces no new concepts. Rather, it consolidates previous learning and extends it to large numbers written sideways. It is up to the student to find the largest/ biggest number, put it on top, then add or subtract according to the signs.
4--7 = +4+7 = +11
-8+-4 = -8-4 = -12
--++-2-+-6 = -2+6 = +4
9-(+7) = +9-7 = +2
-3+(-5) = -3-5 = -8
11+(+2) = +11+2 = +13
15) Combining decimals
Combining decimals is no different than combining integers, except that a decimal point is visible. Sure the decimals must be lined up, but we lined up the invisible decimal