Action Algebra[1]

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A M f M

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Royal Lyon Publications

Klamath Falls, Oregon


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Copyright 2010 by Ed Lyons

All rights reserved.No part of this publication may be reproduced, stored ina retrieval system, or transmitted in any form or by anymeans, electronic, mechanical, photocopying, recording, orotherwise, without the prior written permission of the author.

Printed in the United States of America

ISBN: 1453612122

First printing, June 2010Second printing, September 2010

Corrections and additions are posted at our website: ActionAlgebra.com

Acti A g b a A Model for Math and a Handbook for Arithmetic to Algebr


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Tab f C t t

IntroductIon Arithmetic to Algebra In Just Ten Minutes! . . 8

B asIc P rIncIPlesEquality. . . . . . . . . . . . . . . . . . . . 11

Value Change Needs Counterchange . . . . .12Priority . . . . . . . . . . . . . . . . . . . .13First Calculate the Complicated. . . . . . . .13Insight . . . . . . . . . . . . . . . . . . . .15Changing Looks does not Change Value . . .15

a ctIon c hart

Zoom l evels

a rIthmetIc : n umBers1) Benefits of a Number System Line. . . . . 202) Translating Numbers and Words . . . . . . 213) Numbers are arrows . . . . . . . . . . . .214) Comparing numbers. . . . . . . . . . . .235) Kinds of numbers . . . . . . . . . . . . . 256) Parts of compound numbers . . . . . . . .26

a rIthmetIc : c omBIne7) Adding on a Number Line . . . . . . . . .308) Subtracting on a Number Line. . . . . . . 319) Adding Stacked Numbers . . . . . . . . .32

10) Subtracting Stacked Numbers . . . . . .3311) Combining Stacked Integers . . . . . . .3412) Series of Signs . . . . . . . . . . . . . .3713) Combining Series of Signs . . . . . . . .3714) Combining Big Integers. . . . . . . . . .3815) Combining Decimals . . . . . . . . . . .3816) Combining Tags . . . . . . . . . . . . . 39

a rIthmetIc : m ultIPly17) Multiplying on a Grid. . . . . . . . . . .4218) Learning the Times Table. . . . . . . . .45

19) Learning Multiples . . . . . . . . . . . .4720) Negative Numbers on a Grid . . . . . . .4721) Multiplying Big Numbers . . . . . . . . .5122) Multiplying Bigger-Smaller . . . . . . . . 5323) Multiplying Decimals . . . . . . . . . . . 5324) Multiplying Fractions . . . . . . . . . . . 5425) Multiplying Tags by Merging . . . . . . . 5526) Finding Common Multiples. . . . . . . .56

a rIthmetIc : d IvIde27) Dividing on a Grid . . . . . . . . . . . .5828) Learning How to Shift . . . . . . . . . .5929) Dividing and Bigger-Smaller . . . . . . . 6230) Speed Division . . . . . . . . . . . . . . 6231) Long Division . . . . . . . . . . . . . . 6332) Long Division with Decimals . . . . . . .6433) Factoring . . . . . . . . . . . . . . . . . 6534) Prime Factoring. . . . . . . . . . . . . .6635) Finding Common Factors. . . . . . . . .6736) Reducing Fractions . . . . . . . . . . . .6837) Dividing Fractions Using Reciprocals . . . 7038) Making Like Fractions . . . . . . . . . . 7139) Combining Fractions . . . . . . . . . . .7240) Canceling Tags . . . . . . . . . . . . . . 73

P re -a lgeBra : e xPonents41) Basics . . . . . . . . . . . . . . . . . . 7642) Negative Exponents . . . . . . . . . . . 7843) Multiplying Bases. . . . . . . . . . . . .8044) Dividing Bases . . . . . . . . . . . . . . 8145) Zoom Levels . . . . . . . . . . . . . . . 8246) Groups with exponents . . . . . . . . . .8447) Exponents in Fractions . . . . . . . . . . 8448) Scientific Numbers . . . . . . . . . . . . 8649) Adjust Scientifics . . . . . . . . . . . . .8750) Multiply Scientifics . . . . . . . . . . . .8851) Powers of Scientifics . . . . . . . . . . .8952) Dividing Scientifics . . . . . . . . . . . . 9053) Combining Scientifics. . . . . . . . . . .90

P re -a lgeBra : m orPhs54) Fractions and Mixed Numbers . . . . . .9255) Rounding . . . . . . . . . . . . . . . . . 9356) Fractions and Decimals . . . . . . . . . .9457) Fractions and Percents . . . . . . . . . . 9558) Decimals and Percents . . . . . . . . . . 9659) Units . . . . . . . . . . . . . . . . . . .9760) Metric Units. . . . . . . . . . . . . . . 100

P re -a lgeBra : c alculate61) MUD before COLT . . . . . . . . . . . 10462) FUN before MUD. . . . . . . . . . . . 10563) IN before FUN . . . . . . . . . . . . . 10664) Order with fractions . . . . . . . . . . 107


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65) Nesting . . . . . . . . . . . . . . . . . 10866) Absolute Value . . . . . . . . . . . . . 10867) Formulas . . . . . . . . . . . . . . . . 10968) Units in Formulas. . . . . . . . . . . . 11069) 2D Shapes . . . . . . . . . . . . . . . 11270) 3D Shapes . . . . . . . . . . . . . . . 11371) Averages . . . . . . . . . . . . . . . . 114

72) Rates . . . . . . . . . . . . . . . . . . 11473) Ratios . . . . . . . . . . . . . . . . . 116

P re -a lgeBra : r oots74) What is a Root?. . . . . . . . . . . . . 11875) Reducing Roots . . . . . . . . . . . . . 11976) Combining Roots . . . . . . . . . . . . 12077) Multiplying Roots . . . . . . . . . . . . 12178) Fractional Exponents . . . . . . . . . . 12279) Rationalize Roots . . . . . . . . . . . . 12380) Roots with Same Base . . . . . . . . . 124

a lgeBra : P olynomIalsThinking in Algebra . . . . . . . . . . . . . 126

As Few Variables as Possible . . . . . . . . 12981) Distribution . . . . . . . . . . . . . . . 13082) FOIL . . . . . . . . . . . . . . . . . . 13183) Rationalize with conjugates. . . . . . . 13284) Common Factoring. . . . . . . . . . . 13385) Bifactoring . . . . . . . . . . . . . . . 13386) Bifactor when a>1 . . . . . . . . . . . 13587) Bifactor- other . . . . . . . . . . . . . 13688) Squares. . . . . . . . . . . . . . . . . 13789) Double factoring . . . . . . . . . . . . 13790) Quadratic Formula . . . . . . . . . . . 13891) Reduce fractions . . . . . . . . . . . . 14092) Multiply fractions . . . . . . . . . . . . 14193) Combine fractions . . . . . . . . . . . 14194) Long Division . . . . . . . . . . . . . 142

a lgeBra : l Inear e quatIons95) Recognize equation types. . . . . . . . 144

96) Answer!. . . . . . . . . . . . . . . . . 14697) Break variable term. . . . . . . . . . . 14798) Combine like terms. . . . . . . . . . . 14899) Decouple like terms. . . . . . . . . . . 149100) Eliminate decimals . . . . . . . . . . 150101) Eliminate fractions . . . . . . . . . . 151102) Fill Parentheses . . . . . . . . . . . . 152103) Flip complex fractions . . . . . . . . . 153

104) Figure functions . . . . . . . . . . . . 154105) Proportions . . . . . . . . . . . . . . 155

a lgeBra : q uadratIc e quatIons106) Fill, Flip, or Figure. . . . . . . . . . . 156107) Eliminate fractions or decimals . . . . 157108) Descending order = 0. . . . . . . . . 158109) Common factor . . . . . . . . . . . . 159110) Bifactor . . . . . . . . . . . . . . . . 160111) Answer formula . . . . . . . . . . . . 161

a lgeBra : o ther e quatIons112) Linear . . . . . . . . . . . . . . . . . 162113) Rational . . . . . . . . . . . . . . . . 163114) Multi-variable . . . . . . . . . . . . . 165115) Exponential . . . . . . . . . . . . . . 166116) Inequalities . . . . . . . . . . . . . . 167117) Radical . . . . . . . . . . . . . . . . 168System of Equations . . . . . . . . . . . . 169118) Systems by Substitution . . . . . . . . 169119) Systems by Elimination . . . . . . . . 172120) Systems of Three . . . . . . . . . . . 175

a ctIons e xPlaIned

r ule s heet

g oals & m ethodsEncrypted Education . . . . . . . . . . . . 194What Is Understanding? . . . . . . . . . . 195

Readiness. . . . . . . . . . . . . . . . . . 196Resources. . . . . . . . . . . . . . . . . . 197My Student Is Stuck! . . . . . . . . . . . . 199Pre-Formal Math . . . . . . . . . . . . . . 200

Activities . . . . . . . . . . . . . . . . . . 200Whiteboards and Vinyl . . . . . . . . . . . 202

g rade s heets


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Action Algebra8

InTroduCTIon

This book is written for teachers and parents who want to understand the big pictureof arithmetic to algebra so they can intelligently explain it to students in a connectedframework. 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 ina new way and you have the ability to reason. I also assume that you took standardarithmetic and algebra courses sometime in the distant, hazy past and you may or maynot 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 toadvanced 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 theirfirst presentation and finish them later after the additional principles are introduced. Forexample, in the second chapter on combining, only fractions with common denominatorsare 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 ashomeschoolers 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 thecommon thinking processes connecting the huge variety of problems in the textbooks willbe of help to any teacher in any situation. For example, many parents teaching Saxon arelost when trying to explain previous concepts more than a few lessons back.

This handbook covers arithmetic, pre-algebra, and algebra in 40 lessons each. Thefocus is on the math, but with the worksheets and videos, many word problems are alsocovered. 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 laysa solid, complete system of understanding that will fully prepare a student for all their othercourses. 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 thisbook. The only thing "lacking" is the duplication of topics that the teach-reteach textbookshave made popular by their grossly inefficient methods.

Years ago I figuratively started with grade 2 math and worked my way up to Algebra2. I kept every new topic, but ripped out the pages dealing with a repeat or slight revisionof the topic. At the end I had enough pages left to make two textbooks. That discoveryspawned the development of this curriculum.


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Introduction 9

The Action Algebra worksheets can be used as a supplement to any curriculum, butthe full power and time-savings will be realized when they are used as the curriculumitself. Because of their almost limitless variety and ability to be customized, students canstudy 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 endof 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 knowin why and what they are doing. Students are not left to randomly discover principles, butneither are they engaged in almost mindless drill. They are guided to understand conceptsand 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 Americanexcellence, 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 themnow is that it is possible to cut the usual seat time in half or in third. For a student toaccomplish this seemingly miraculous feat only means that they understand and review asthey proceed. The consequences of this is that there will be more time to apply math bothin math and science classes. It also means that schools will not only raise their graduationrates, 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 teachersI recommend reading the arithmetic and pre-algebra chapters. This will give you anunderstanding of the next level for which you are preparing your students. Just like withthem, understanding the next level seals in the current level. For you, understandingthe process of combining with negative numbers is crucial. If you feel you need moreexamples, please look at the worksheets and videos.

For middle and high school teachers, the whole book is necessary, especiallyunderstanding 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 seeminglyrandom steps. Also, the FA method of solving equations is very simple to teach as a unitoutside 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.

One last note before launching into the math. You may want to look at the last chapters


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Action Algebra10

of the book on Goals and Methods, and on Pre-Formal education. Math is the mostabstract of all classes and we must realize who the young students are that we teach asmuch as knowing how to teach a topic. Also, if you are thinking of using Action Algebra asyour 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.

A it m tic t A g b a I J t T Mi t !

Once upon a time little Einstein stuck his finger in an olive and then in another oliveand 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 fingersin them I was counting .

Then he thought again. What if I want to count more olives than I have fingers? Iguess 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 mothers fingers. The budding scientist was too smart to put them onhis 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 fathers orchard, his neighbors orchard, and his unclesorchard across town. In fact, Einstein knew the number of olives in all the orchards aroundtown. He also knew the numbers of cats, dogs, and horses. Yet little Einstein wanted tosomething 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 theorchards 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 numberstogether 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.


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Introduction 11

Little Einstein was starting to catch on to the power of numbers and so it wasnt longbefore he discovered he could multiply and divide the rows and columns of trees in theorchards to quickly find out how many were in each.

As he shared his knowledge with the townspeople, they soon began to ask him questionsand 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 histen 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 wholeolives like the others. Thats when he realized he needed a way to keep track of partialthings. Thus, fractions were born. He used a slash or a horizontal line because it remindedhim of a cut. The top number represented how many pieces he ate and the bottom numberrepresented the total number of pieces the olive had broken into. The number on top wasusually smaller than the number on the bottom because some pieces fell on the floor.

To save himself some time, little Einstein put adecimal at the end of the whole numberof olives, then started counting tenths and tenths of tenths on the right side. That way hedid not have draw a slash and put a bunch of zeroes below it. It was a special, convenientfraction that always meant tenths.

Then, because he had whole numbers and tenths in a decimal number, he put a wholenumber in front of a fraction and called it a mixed number . Then, because people useddollars so much and were always figuring prices as some part of 100 pennies, he inventedthe percent symbol to make everybodys 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 olivesthere might be in the whole world. Einstein replied that his friends question was notelementary. He would need to invent another kind of number to handle the enormoustask of multiplying all the olives on all the trees in all the orchards of all the towns of all thcountries of the world. So he made scientific numbers with a handy little device called anexponent , which compressed the multiplying of many numbers down into one.

After all their research and calculations, Einstein and Sherlock discovered that somepieces 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 Einsteins 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 itscircumference, which was always just about 3.14. Curious, eh?


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Right after eating pie and discovering pi, Einstein discovered c. This was the speedof light that he and Sherlock measured every time they took a flash picture of olives atnight. Shortly thereafter, Einstein muttered E=mc 2 as he tried to think of creative ways todestroy all the olives in a country in a very short time.

But Einsteins greatest number was still waiting to be discovered.

In the laboratory, Sherlock was deep in calculations and very frustrated when Einstein walked in. Whats 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 mysteriesthat are really just a repeat of the same kind of problem.

Well now, Einstein exclaimed, Isnt solving mysteries your cup of tea? Why shouldsome 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 dont know exactly which one. The numbers just vary from time totime. Lets call them variables and learn how to do adding, subtracting, multiplying, anddividing with variables!

Sherlock looked at thescientist with his mouthagape and jaw dropped.

After a bit, he raised hisindex finger like he waschecking the wind, and

declared, I think youvegot an idea!If you could do that,

we could make formulasand equations that holdthe spots for our numbersbefore we get them fromthe orchards. We could dosome of the calculations

only once and never haveto do them again! Instead of re-inventing the wheel and figuring out what to do with eachnumber every time, we would have a template we could use over and over again. Thats

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.


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Basic Principles 13

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 thecomplicated; and, changing looks does not change value.

eq a it 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 wealso need time with family and friends. We cant be alone and with a group at the sametime, so we have to balance our time between the two. Sometimes we might split ourtime between two different activities, like watching TV and doing homework. Yet, onestill affects the other. We cant do whatever we want whenever we want. Humans requirebalance or else we get sick or get a hangover. One way or another our lives demand, andget, balance.

Balance is necessary because of limits. Unless the parents are infinitely wealthy, if sistegets more allowance, then brother gets less. If there are more eagles above the river, thenthere are less fish in the river. If you have driven more miles down the road, then there is

less gas in the tank. These arelike the Law of Conservationof Energy: Energy can neitherbe created nor destroyed, it

just changes form. Causmatches effect. Action equalsreaction. Input equals output.In other words, the pot of soupnever grows, it just gets stirred

A math problem is thsame. The answer must equalthe problem. It is no differentin value, just in format. Forexample 5 truckloads of 10


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Action Algebra14

crates of 200 boxes of cereal is a problem, not an answer to my question. I want to knowhow 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 theproblem, 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 bea 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 doesnt upset the balance so it is fine. However, removing a sockon one side requires the same kind of sock to be removed from the other side. Balancebefore = balance after.

This principle of equality and balance seems to be telling us what we cant do andtherefore limits our options. However, it actually opens the door to two powerful Actions.

Va C a g n C t c a gBecause 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 leftby 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, sothat 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 onlyone side of an equation. For example, if I have two water balloons hanging from the leftside 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 gotbigger 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 10dimes in return. I have more things, but each item has less value.

Another example from real life is air conditioning. To cool off my house I must heat


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Basic Principles 15

up the outside. Think of this Action as the principle of the Up and Down. My house goesdown in temperature, while the outside goes up in temperature. If you dont believe this,go stick your hand over the exchanger!

P i it 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 thesand 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 gravelaround them. Then she poured the sand in and shook the jar until it all fit. She succeededby 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 orcollapse under a load. Life is filled with priorities. Starting the day with a good breakfasmakes us healthier. To eat a good breakfast we have to wake up early, which means weneed to go to bed on time. Getting our homework done on time gives us privileges likegoing outside to play and getting good grades. This gives us feelings of accomplishmentand happiness. That makes us better, kinder people. Paying our bills before blowing ourmoney on extras is another priority that wise people adhere to. Keeping one priority oftenhelps us keep other priorities straight. As we figure out our priorities and follow them, thahelps us achieve a balanced life.

Fi t Ca c at t C mp icatMath 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 donot 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 muscalculate functions, because then I will know the final number to multiply. And in themidst of all that, I must pay attention to parentheses, because they can override anythingat anytime.

A pretty good rule of thumb is to first calculate the things you learned last. In other


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Action Algebra16

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 aproblem. Next comes exponents and other functions like roots and trigonometry. Theorder of learning these things will vary a little depending on the textbook, but all of theseare on the same level of importance which is above multiplication.

A simple real-life situation will illustrate the meaning. Lets say it is your task to countthe total production of toy blocks on a certain day. There is a pile of blocks waiting to bepackaged. There are boxes of blocks stacked on pallets, and there is a machine crankingout blocks constantly.

You can count the blocks in the pile easy enough, but to count the blocks in the boxesyou 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 youcannot 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 themcompletely separate from the output of the machine while you wait. Either way you aregiving priority to the machine before calculating your total.

This simple illustration shows that functions (machines that, in professorial terms, mapa set of numbers to another set) must be considered before boxes before loose items oryou must have a way of separating them. Likewise, roots andlogs 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 calculatecorrectly. (IN FUNny MUD is aCOLT)

In, or Inbox, means I should work inside parentheses first.Parentheses ( ) and brackets[ ] and braces { } all act likemathematical boxes to group

what is inside them. We must find

a single value for the whole groupbefore we can add or times itby what else is there.

Fun is short for function.Exponents, logarithms,and trig functions are the


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Basic Principles 17

common functions to be encountered in pre-algebra and algebra. They are little mysteriesthat 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 ibasically multiplication in reverse. So almost all rules that apply to multiplication apply todivision, 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 theall-in-one method of adding and subtracting that is covered in the second chapter. It isthe one way that works for all of arithmetic and all of algebra. SSADDL is the how-toprinciple that goes with COLT, because every colt needs a saddle!

I ig t

We hope to raise our children with enough insight to know that changing costumesdoes not really change the actor. It is still the same person behind the mask. Similarly, wetry to teach them that beauty is more than skin deep and that the value of persons doesnot depend on the color of their skin. Also, an old dollar is worth as much as a new dollar.

C a gi g l k t C a g VaMost 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 getforgotten.) This group of Actions are called Insight because it takes looking at the problemand your options in a slightly different way to figure out that if you used one of these, youcould make things easier.

None of these Actions changes the value of numbers, so counterchanges are notneeded. The Show Action hides or unhides what is already there. Sort re-arranges whatis 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 the10 Actions. As we proceed through the lessons I will amplify the Actions at appropriatepoints, then use them to explain the current problem.


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Fi t ca c at t c mp icat

IN You must rst work inside the boxes( ) [ ] { } and fraction bars to gurethe answer you need to work with. Think of unwrapping a present from the inside out.

FUN Then you must feed raw numbersinto the mouth of the function(funnel, get it?!) to gure the answer you need towork with. Function processes input, you use onlythe output.

MUD Then you can MUltiply andDivide all kinds by merging tags.Figure the sign by using NOPE- Negative OddPositive Even. (MUD can get on all things, but dowe like it? NOPE!)

COLT Then Combine Only LikeThings (Terms, Tags) by usingSSADDL- Same Signs Add, Differents Destroy,Largest sign is answer sign. (SSADDL your COLT)

ACTIONS


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Va c a g m t av c t c a g

C a gi g k t c a g va

SYNCYou may do the same thingonce to each whole side of anequation. 1 effect, 2 opposite sides.

SHIFT You may change the value ofan object at any time if youcounter it with an equal, opposite change withinthat object. 2 opposite effects, 1 side.

SORT You may re-arrange theobjects in a level at any time,but never change a division part.

SHOW You may show or hideinvisible objects at any time

MORPH You may convert anobject from one formatto another at any time.

SUB You may replace object A withobject B at any time, if A=B.


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Action Algebra20

ZooM leVels

Zoom levels are not needed to teach or learn arithmetic, but these next two pagesare 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 mathproblem. 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 becomesincreasingly important to be aware on what level we are on. Any given step of any problemtakes place on only one level. The level can change from step to step, but it will neverchange within a step. For example, we dont do something on the term level, then try tobalance it on the factor level.

As you can see in the diagram, factors multiply (or divide) to make a term (calledcompound 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 groupfactors into superfactors and terms into superterms. That is when our abstractingabilities really get tested. It happens a little bit in pre-algebra, but mostly in algebra.(Arithmetic teachers, you can breathe a sigh of relief!)


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Zoom Levels 21

6^9factor factor factor factor factor

8xy -2(5+6xfactor f a c t o r

Now lets 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 ofnumbers 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. Notonly are they numbers or variables, but they are also factors because they multiply eachother. Also, as factors, they bond themselves into packages called, terms.

The example on the right is interesting because of the grouping. The whole example isone big term made of -2 ^ ( ). However, inside the ( ) factor are two subterms of 5 and6x. 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 meansthat a single number can be a single factor (times an invisible 1) making a single termmaking an expression. It all depends from which zoom level you choose to look at it.

9-7(4+9y)9-7(4+9y4 +9y

3*8-5*73*8-5*7

6x+0-24y6x+0-24y

A useful, and often challenging, exercise I do with algebra students is to give themrandom algebra expressions to be split into terms. I use a double slash or squiggly lineso that it is not confused with some other symbol. The point is that students must seealgebra.

What helps me visualize this is I think of expressions like railroad trains. The termare the cars coupled together with + and - signs. Inside the cars are boxes of stuff calledfactors. We can split trains apart at the couplings between the cars, but we never split thecars 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 awareof as we progress through math. We will work with the objects in only one level at atime. That means factors are the objects in terms. Terms are the objects in expressions.Expressions are the objects in an equation.


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Action Algebra22

ArIThMeTIC: nuMBers

This chapter covers numbers and the number system. It shows how numbers arearrows from the number line, which is an infinite arrow. We then compare numbers toeach 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 clearon a childs level.

1) B fit f a n mb s t m liOf 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 requiresthe idea of place value, numbers would just be an endless invention of names. Nottoo helpful.

If we did not organize numbers into an orderly sequence on a line, we wouldthink 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 thenumber farthest to the right is greater. If the number line is turned vertically, then thehighest number is greatest.

The worksheet about locating numbers is a good time to point out that thenumber line starts at 0 and goes endlessly in both directions. It should also be pointedout that all counting begins at 0 and we count steps. Adults sometimes make themistake of counting the starting point. Instead, we start at 0 and count the first stepto 1, the second step to 2, and so on. This is like counting on our fingers and we

already have 3 fingers up. We dont start at 3 and count 1. We start at 3, move over

0

...,-3,-2,-1,0,1,2,3,...


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Arithmetic: Numbers 23

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, dont 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 dont have our abstract understanding of them. More details willbe covered in lesson 3.)

2) T a ati g n mb a W

From pre-formal activities, the student should already be familiar with both the ideaand 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 largemay make a critical difference.

3) n mb a a w

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 havesigns. Sign + digits = number. If you do not see a sign that means there is an invisiblepositive. 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. ThinkI have $5 or I owe $5. I walk up 8 stairs or I walk down 8 stairs. The temperature is 15degrees above zero or 15 degrees below zero. Some things may not be negative, but thenumbers used to describe them can be. For example, can you eat -3 pieces of cake?! Canyou be -5 feet tall? Of course not. My height cant be negative. But I am always 5 feet tall

whether I am climbing up a cliff or hanging upside down. Numbers always have size anddirection.


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Action Algebra24

6 feet tall

6 feet tall

6 feet abovethe oor=+6

6 feet belowthe oor=-6

oor = 0

W at i a a w? 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 thearrow 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 causesus to misunderstand adding and subtracting, which then causes us more problems whenit comes time for pre-algebra. Even when we dont see a sign in front of the digits, that

just means there is an invisible + sign there. All plain numbers are positive. A number isnegative only if there is a - sign in front.

n gativ a ma

Lets look at the numbers that strike fear in the heartsof many. A negative number (or, a minus number, as somecall them) is just a regular number that goes left on thenumber line instead of right. If the number line is in the

vertical position then negatives go down. This is just likea thermometer that is minus 5 degrees below 0 when it is


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Action Algebra26

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 meanhigher 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 -8is bigger than +4 because

its arrow is longer Bigger, smaller, larger

only see the length of thearrow, not its directionGreater than and less than

include size and directionwhich gives a nal position

on the number line

When doing the worksheets it may help to put your finger over the signs when usingbigger, larger, smaller. Now it is like both numbers are positive and pointing upward. Thatis 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 itis easier to see that any positive number, no matter how small, is greater than any negativenumber, no matter how large. Also, any negative number is less than (lower) than anypositive number.

You should find students readily grasping the concepts separately, but may get mixedup when all the words are used on the all comparisons sheet. I wish I knew of an easy,

obvious memory device here, but I dont.


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5) Ki f mb

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 theylook 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, including0. Integers are a subset of decimals.

+5 -2 16 -39 +82) 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 theanswer keys and give credit for that answer as well. For further study you could look uprational and irrational numbers in Wikipedia in case you come across it on standardizedtests or textbooks, but the distinction is not central to arithmetic and algebra.

7.25 1.33... 0.09 .13) 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% 91/ 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 intohaving only integers as soon as possible.

1/2 4/ 13 -3/7123

5) Mixed number, or simply, mixed. A visible integer next a fraction with only integeris a mixed. I rarely work with mixed numbers. Instead, I turn them into fractions, solve theproblem, then re-convert back to a mixed.

5 1/2 -94/ 13 -11 3/7 6 126) Scientific. A decimal times 10 to a power is the basic form of a scientific number.


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Action Algebra28

Technically, the decimal must be between 1.0 and 9.99999... so that there is only a onesdigit followed by decimal places. However, at this point, if any decimal followed by ^10N is called scientific, that is good enough.

2.7^104 9.003^10-15 7) 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 forarithmetic 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 toadd or not.

x y apple a8) 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 constantlythe same value in every problem.

& e i

6) Pa t f c mp mb 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 childs 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 nextchapter 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 amountof 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 andstill finish algebra years early. This leaves plenty of time for side topics, applications, andother 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


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(denominator) are individual numbers, but stepping back and looking at the numbers witha 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 bya 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 musthave matching tags before we can add or subtract two numbers, and we must know whatis in the tag so we know what to multiply.

5 players

-1 xy

7 /92 /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 bottomnumber (or the right hand number if written sideways) is the tag. The fraction bar isincluded.

(Algebra teachers: A tag is all factors in a term except the coefficient and/or numerator.)C mp n mbVery 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 optionalthird part is attached to some numbers called the Unit (miles, feet, meters, pounds, etc...),but it is really part of the tag, also.


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Action Algebra30

Compound Numbe

FrontnumTagThe algebra word for compound number is T

In short, the front number is the first part of every compound number and is thequantity 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 andstrange to you because they are not standard vocabulary. However, they are labels forstandard math items that you learned when you were in school that were left unnamed inthe lower grades or not named at all.

Young children can easily identify the parts of a compound number, even if they dontyet understand everything those parts do. Rather than use a strange word like coefficientthat still makes no real sense to me (a math teacher), they can easily and visually relate tofront 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 unlabeledso we have nothing to lose there. The main point is that children learning math for the firsttime will accept whatever words you use. What might feel strange to you will be acceptedas normal by them.


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Arithmetic: Combine 33

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 biggernegative number.

So you see, it is not because numbers go up that they add, it is because they go inthe 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. Theymistakenly 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 someoneelse $2, how much do you owe altogether? Of course, you owe $6 total. In your head youknew that owing $4 was bad and so was owing $2. So putting them together meant thatthings were going to get worse. You added, not subtracted, the debts. Your answer gotfarther from, not closer to, 0.

Again, lets 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 anegative 5, because it is below 0, which is ground level.

Teach adding this way to prepare the student to understanding subtraction correctly.

8) s bt acti g a n mb li

Adding lines up arrows in the same direction, so subtraction puts them together inopposite directions. Subtraction is not always taking away, but taking away is alwayssubtraction. Taking away only deals with size, but of course, numbers have size anddirection. Therefore, subtraction must take into account size and direction, which is boththe number and its sign.

0

+6

+4

-2

-12-8

+4

Subtracting lines up arrows in opposite directions, head


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Action Algebra34

This is critical for us as adults to understand before teaching our students. We havebeen conditioned to think that subtraction is only taking away, but this leads to a mentalroad block. For example, if I have a stack of 3 books on the table, how can I take away5? If I think that subtractions is only taking away, then this problem is impossible. Whena 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 theirthinking.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 landme 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 BOTHof their signs. If the signs are different, subtract, but if they are the same, add. The answermight be positive or it might be negative, but it will always be closer to zero than thebiggest 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 peekat 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 forthe answer. When using pencil and paper with just the numbers, notice that -2 and +6have different signs. +4-12 means you right 4 and then go left 12 to end at -8 for the finalanswer. Again, notice the different signs and so the arrows go different directions.

9) A i g stack n mb

This lesson is the standard lesson which teaches students to add numbers vertically.This will be a real test of a childs abstract abilities. Some may need to wait, some maytake 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.


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+8+6

+14

1 0+37+13+50

+72+291190

+101

+858+245

1390

1000+1103

1 0+79+76

+155

To prepare the way for combining, the worksheets put the biggest number on top andall 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 somestudents. Instead of writing the carry above the column to the left, the answer is written inone 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) s bt acti g stack n mb

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 workhere. (Take away is not wrong, it is just incomplete.)

As you can see in the examples, the standard way of subtracting, with all the borrowingand slashing can be quite messy. If we as adults dont like it, we can imagine the troublethis mess causes young children. So in the beginning you might want to break down thesteps 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 placefrom the place to the left and temporarily squeeze it in. What we are really doing when wesqueeze 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 dont needto, nor to borrow more than 10. In either case we will make a problem that results in twodigits. And as Hardy use to say to Laurel, Thats another fine mess youve gotten meinto.

Lets look at the 93-27 example in the middle and sort of dialog our way through it.


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Action Algebra36

+15 -7+8

+24-19

+5

/ 1 1 8 0

+93-27+66

/ 1 +72-59+13

/ 6 1 +858

-269+589

/ 7

/ 141

I am going to work right to left, because along the way the top digit might be smallerthan the bottom digit. To solve that problem I need to have a big, rich neighbor on my leftthat 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 answerI 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 placeso 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 finalanswer of 66.

When demonstrating that to students you have two options. Slash and write the borrowreal 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 puttheir 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 usingtheir place value skill and bigger/smaller skill from the Numbers chapter. Nothing a studentlearns is extra or useless. It all leads to something in a later chapter, or even in the verynext lesson, which is about to happen!

11) C mbi i g stack I t gStarting 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 studentis able to do all the skills to that point. It will be a good review spot. However, we must firststart with combining plain integers (blank tags).


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Before beginning let me clear up some terminology. I have seen some books usecombining the same way I do to mean either adding or subtracting. I have also seen somebooks use the word adding in the same way I use combining. I am comfortable withboth usages, but in this book, I will use adding in only the way I have already describedit. 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 wilcombine (no pun intended) adding and subtracting. Mastering this method, a student isset to conquer arithmetic, word problems, and algebra.

C mbi i g c a c f iNow 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-stylesubtraction and what the sign of the answer will be. You dont need to memorize specialcases 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 +03+18

+15 -03+12

-15 +03-12

-15 -03-18

Before I explain, just study the examples to see if you can find a pattern. Did younotice that the biggest number is always on top? Did you notice that the answer alwayshas the same sign as the biggest number? In other words, the top sign is always the answersign. Did you notice that when the signs are the same, the numbers add to get a biggernumber farther from zero? Did you notice that when the signs are different, the numberssubtract 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 andnegative numbers. Merge the two processes with correct ideas of bigger/larger into the


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Action Algebra38

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 ColT Most of the time numbers are not stacked vertically and you dont want to take the time

to re-write them that way. Here are the similar steps to combine positive and negativenumbers 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. Dont 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 wordas difference. The word, difference, is used in word problems as a clue to subtract. Soyou could say Different Difference to keep things straight, but people may look at you alittle funny as if you are a verbal photocopier! (but you wont forget!)

1) Find sign of largest number 2) Copy it to answer sign3) Same Signs Add, Differents Destroy

-7+9=-7+9=+-7+9=+2

1

23

-6-4=-6-4=--6-4=-10

1

23


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12) s i f sig

This lesson really belongs in the chapter on multiplication, but I need to insert it herebecause 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. Thisis not necessary once you know combining and you will later have to unlearn the crutchof depending on an extra sign. (I have seen more students confused by this device thanhelped.)

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 matterif 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.Ill explain why this works in the next chapter.

13) C mbi i g s i f sig

Now that the student knows how to condense a series of signs into one, he will be ableto 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 totalis 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--NegativeOddPositiveEven


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Action Algebra40

If you think this looks like standard pre-algebra, you are right. If you think it is tooearly to introduce it to students, just remember that we have arrived here in a smoothprogression. If the student has successfully handled the previous lessons, there is no reasonto assume she will not handle this one successfully. Dont let your fears and biases get inthe way of the students blank slate! Also, the earlier something is learned, the more it is

reviewed to the point of becoming automatic.

14) C mbi i g Big I t g

This lesson introduces no new concepts. Rather, it consolidates previous learning andextends 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) C mbi i g d cima

Combining decimals is no different than combining integers, except that a decimalpoint is visible. Sure the decimals must be lined up, but we lined up the invisible decimal


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points in integers when we lined up the ones, tens, and so on. Why must we always linethings up this way? Because of COLT, Combine Only Like Things. We add pennies withpennies and dimes with dimes, so we also add ones with ones and tenths with tenths.There is nothing magical about the decimal point. It is the place values that must be linedup.

The icon looks like it is lined up on the right side. This is a small visual reminder to linup numbers correctly to the right side. If you fill in the invisible zeros past the decimal, thenall numbers line up to the right, but the key is lining up the decimal so that place valuesmatch above and below.

Notice that all the numbers above have blank tags. Because they are blank we donteven need to draw or label the tags. If you put them in, you wont be wrong, but combiningblank tags gives you a blank tag.

16) C mbi i g Tag

COLT says Combine Only Like Things. We know two things are alike if they havethe same tag. So you could also say Combine Only Like Tags. Also, when you combineidentical tags, you get the same tag for an answer.

Now the question arises, What do I do if I need to combine things with different tags?Dont! You cant! Just stop and do nothing. You are done! (Multiplying can work withdifferent tags, but combining cannot. Sometimes multiplying can change the tags so that

combining can work.)

-400.- 75.-475.

-28.+ 5.-23.

+5.10+ .26+5.36

-75.30- 2.04-77.34

Think about it. You have 3 apples in your left hand and 5 oranges in your right hand.How many do you have altogether? Did you say 8? You should have asked me, Howmany what? Do you have 8 apples? No. Do you have 8 oranges? No. You have 8 fruits,but was that what I was asking for? You dont know. Therefore, you cant answer.

Math is exact. The question and the numbers you have, must ALL match exactly.Otherwise, dont answer the question, because you cant. This may sound nitpicky and


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Action Algebra42

hard for students to understand, but actually it is easy. The rule is simple: tags matchexactly or do nothing. No exceptions. Not a lot of thinking. Combine ONLY Like Tags.

+3 apples+5 apples+8 apples

+3 apples+5 orangesSTOP

+3 miles+5 miles+8 miles+3 miles+5 booksSTOP

Frontnum Tag Frontnum Tag

-9.2 xy+3.7 xy -5.5 xy

Frontnum Tag

+17 a+24 a+41 a

Frontnum Tag

-8 /3+3 /3-5 /3

Frontnum Tag

+7 /4-1 /4+6 /4

Frontnum Tag

+17 &+24 &+41 &

Frontnum Tag

-9.2 %+3.7 % -5.5 %

Frontnum Tag


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Action Algebra44

ArIThMeTIC: MulTIPly

In this chapter we will learn how to multiply compound numbers. We will see that thetags do not have to be the same, and in fact, multiplying makes them different.

The icon gives hints that multiplying and dividing go together. Dividing is justmultiplication in reverse. That is why we call this Action, MuD, for MUltiply and Divide.However, we will look at dividing in its own chapter.

The icon looks like a small grid and reminds us that multiplying is based on a grid, while the stairs of the COLT icon were like a single number line. The icon also suggests what do with the + and - signs.

17) M tip i g a G i

Multiplying is putting two arrows together, tail to tail, at right angles. The answer is thearea of the rectangle that they make. Heres why.

We have already seen that combining is counting numbers one after another withoutrestarting the count. This is visually represented by arrows lined up head to tail on a single

number line. What we want to do now is repetitive counting. We want to copy a numbersome number of times and get that total.If you think about that last sentence closely you will realize that we are introducing a

new idea that we dont yet have words for. The words I am about to use are not officialas if they must be memorized. I am simply trying to describe an idea using whatever words

are available.Here is the new idea that multiplying introduces: number

role. Each of the two numbers in a multiplying problem havedifferent roles. One number is the original number and the

other number is the copy number. The original number getsduplicated according to the copy number. This is like puttingthe original number on a piece of paper on a copy machine.Then we punch in the copy number and we get that manypieces of paper on the output tray that each have the originalnumber. Then we add up all those pages and get the total.


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Arithmetic: Multiply 45

Do you see how this is different from combining where the numbers just sit there waiting for us to count them? The numbers in a combining problem have no roles. They

are inert and lifeless. In a multiplying problem, however, theyhave different roles to perform. They have different meanings.Even though we can interchange the original number and make

it the copy number (which makes the copy number into theoriginal number) and still get the same answer, whichever way we solve it, we give the numbers different roles. This subtle idealeads to new ideas.

Notice that length ^ length = area. Length ^ length doesnot equal another length. Multiplying makes a new thing, a

different thing. Now compare this to combining. Length + length = length. Combiningkeeps everything the same. Multiplying makes things different.

This is easy to forget when we work with plain numbers for so long and forget about

the real things that they count. Numbers dont exist to count themselves. They exist tocount real things. When you combine real things you get the same kind of thing as ananswer. When you multiply real things, you get a different thing as an answer.

This common sense pattern of life can be used when solving word problems. Forexample, you are told that your room is 10 feet long by 12 feet wide. Then you are askedto find the area of the room. In the information and the question you have feet, feet, andarea. You have more than one kind of thing. You can know automatically that combiningthe numbers will not give you theanswer, because feet + feet = feet.

So, you must multiply.Multiplying literally adds anotherdimension that combining does notknow exists. Two number lines puttogether on a grid lets us multiply twodifferent things at once. If we knowthat different things are involved, then

we know multiplication was used. If we use multiplication, then we know

different things will result. Apply that idea to this question.If 5 trucks each have 1000 cookies,how many total cookies are there?We have trucks and cookies andmust find cookies. What I am not sure

original c o p

i e s

total

length=9

l e n g

t h = 6

area=549*6=54

Combining works ona number line, whilemultiplying works on a gri


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length

l e n g t

h

AREA

Simply put, division is reverse multiplication where the order matters. Never changethe order of a division problem or the division part of a main problem!

18) l a i g t Tim Tab

Before getting too deep into multiplication, the students will need to know the timestable. Sure a calculator can do the raw calculation, but then the students will never gain

automaticity, which means they will mentally stumble every time they face a conceptbased on multiplication. (This applies the same to combining and dividing.) They shouldbe able to work mentally with at least single digits and small double digits in all four of thbasic operations.

Memorizing the times table need not appear as a giant task if we cut out the duplication.We can further trim it by cutting out the 2s and 9s because there are better ways of dealing with them. So what is left is a smaller, easier to manage group of numbers. (A sidebenefit is that the squares become obvious along the diagonal.)

This times table follows the good study habit of not studying the same thing twice.

Students do not need to memorize 3^4 and 4^3, they understand that is the sameproblem. By making them study the same thing twice, or learning what they already know,they begin to doubt if they know it and so start slipping backwards. Review is good, butintensive study should focus only on what is not yet understood or memorized.

So now here is why 2 and 9 are left off the table.


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Action Algebra48

You can figure 2 ^ a number by doubling the number in your head. Just add number+ number. 2^8=8+8=16 5^2=5+5=10

You can figure 9s by either of two neat little tricks. Lets solve the problem 4^9Method A) Subtract 1 from 4 to get 3. That is your 10s digit. Subtract the 3 from the

9. That is your 1s digit. Your answer is 36. Always subtract 1, then subtract from 9.

^ 3 4 5 6 7 83 9 12 15 18 21 24

416 20 24 28 32

5 25 30 35 406 36 42 487 49 568 64

Method B) Hold your hands in front of you and curl your 4th finger (count left to right).Now count the fingers to the left of your curled finger, 3. This is your 10s digit. Count thefingers to the right of your curled finger, 6. That is the 1s digit. Your answer is 36.

Try both of these tips with all the 9s to prove it to yourself.Before leaving this lesson, look at the times table above one more time. Doesnt that

look like a less daunting task than memorizing the full table? If it looks easier to us, then it will also look easier to children!


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19) l a i g M tip

Multiples are very much related to the times table and they can help a student learntheir multiplication facts. A list of multiples is generated by starting with any number then

repeatedly adding that number to itself. Sometimes this is called skip-counting.

2, 4, 6, 8, 10, 12, 14, ...5, 10, 15, 20, 25, 30, ...7, 14, 21, 28, 35, 42, ...

11, 22, 33, 44, 55, 66, ... You can also start multiplying the number by 1, then by 2, then by 3, and so forth:

4^1=4; 4^2=8; 4^3=12; 4^4=16; 4^5=20; 4^6=24; etc... This is where the wordmultiple comes from. The reason to find multiples is to find common multiples, which

will then be used by fractions and other problems later on.

20) n gativ n mb a G i

It is time to complete our understanding of negative numbers. We have a system that works for all adding and subtracting called, combining. Now we need a system that worksfor multiplying, and along with it, dividing. We also need to know how to tell the twosystems apart.


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Action Algebra50

+16

-16+16

-16

+4

+4

-4

-4

+

+

-

-

12

3 4

+4*+4=+16-4*+4=-16-4*-4=+16+4*-4=-16

Lets return to our copier illustration, but this time we will copy money instead of plain

paper.I want to solve the problem 8^10. In other words, I want to make 8 copies of a $10bill. I put the bill on the glass, punch in 8, and out comes 8 bills. I add them up and get$80. Sweeeet!!

Translating this problem to a grid I can put 10 on the horizontal axis (the X axis) and8 on the vertical axis (Y axis). Since the 10 is positive it goes to the right. Since the 8 ispositive it goes up. Therefore, the rectangle that they make is in the upper right quadrant(quadrant 1). Since the answer is positive (I have $80 in cash, not debt) we say that anyanswer in quadrant 1 is positive.

Now lets modify the problem. Instead of a $10 bill, I now have a $10 IOU. When Imake 8 copies of the IOU, I will be in debt $80. That is -80.Going to the grid, my 8 is still positive and up, but the 10 is negative, so it goes left,

not right. In which quadrant is my answer rectangle? In the upper left, quadrant 2. So anyanswers in the northwest quadrant are negative.

Notice that so far the answer sign is following the NOPE rule that was introduced in


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the Combining chapter. Negative Odd Positive Even, and the first problem had 0 negativenumbers, and 0 is even, and the answer was +. The second problem had 1 negativenumber, which is odd, and the answer was -.

Now things get interesting because we must imagine a negative copier. Lets pretendthat we have a copier destroys existing copies instead of creating new ones. When I ask

for 3 negative copies of my original, this strange machine destroys 3 originals. Negativecopies, get it?!So now I put my $10 bill on the glass and punch in -8 copies. Rather than spitting

out $80 for my spending pleasure, it reaches into my wallet and shreds 8 $10 bills intooblivion. Obviously, I am not happy, but the copier did what I told it to do!

So now I am $80 poorer. The answer to -8^10 is -80. I show this on the grid witha +10 to the right and a -8 down. Therefore, my answer rectangle is in the lower right(southeast), which is quadrant 4. Thus, we can conclude that any answers in quadrant 4are negative.

And here comes the final mindbender. I put an IOU of $10 on the copier and punchin -8 copies. The copier goes into my wallet, pulls out 8 IOUS of $10 each and destroysthem! Weird, but quite nice! Better than a bailout! I am $80 less in debt. A positive thing

just happened to me because something bad got destroyed. A negative got negated. Adouble negative is positive. We will overlook the moral implications of arguing that two

wrongs make a right and instead focus on a reversed reverse is forward. Does all thatmake sense?! If my debt (-) is destroyed (-) then that is a + result for me.

On the grid that problem looks like this: the -8 goes down and the -10 goes left.Therefore, the answer rectangle is in the third quadrant (southwest), and so quadrant 4 is

positive.In summary, multiplying is two arrows making a rectangle in one of four quadrants.Which quadrant the rectangle is in determines the sign.

T i g t iff cHow do we tell the difference between a multiplying problem with + and -, and a

combining problem with + and -? Multiplying always has * or ^ or ( ). If you see nothingthat says to multiply, then combine.

-6+(-4) becomes -6-4-6^+(-4) becomes -6^-4

8-(+5) becomes 8-5 8*-(+5) becomes 8*-5


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Action Algebra52

Sometimes a sign touches the ( ) instead of a number. In that case, only the signmultiplies the sign inside. This is part of the general math notation that says that thingsthat touch multiply each other. For example, 5x means 5 times x, and 7b means 7 times b.

A couple of exceptions to this general way of writing are digits and mixed numbers. If twodigits are next to each other, then they are spelling a number using place value. 25 means

twenty five, not 2^5. Also, mixed numbers put a whole number next to a fraction. In thatcase, it means add, not times.

-2+5 = -7 but -2*+5 = -10+1-8 = -7 but +1^-8 = -8

-3-6 = -9 but -3(-6) = -18-4+9 = +5 but (-4)(+9) = -36

12

3 4

+

+

-

-


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21) M tip i g Big n mb

Now that we have a system in place for multiplying numbers with signs, we can nowattack 2 and 3 digit numbers. This will introduce the distributive property and carrying.

(Note: This explanation of distributive property is for you as a teacher. I have not foundit necessary to add to the load of young minds with another new technical phrase. Whenthe student encounters it again in pre-algebra, then the wording will be more appropriate.Right now your goal is to teach common sense.)

You may not be familiar with the name distributive property, but you have usedit every time you multiply by more than one digit. For example, 3^15 is the same as3^(10+5) which is the same as 3^10 + 3^5. The 3 was distributed to each number in the( ). Using the distributive property I can sometimes multiply up to 2 digits ^ 2 digits in myhead.

12^34 = (10+2)(30+4)10^30 + 10^4 + 2^30 + 2^4

300 + 40 + 60 + 8 = 408

29^75 = (20+9)(70+5)20^70 + 20^5 + 9^70 + 9^5 1400 + 100 + 630 + 45 = 2175

Distribution is used a lot in algebra, so it is good to introduce the concept with plainnumbers to make it easier to grasp. The basic idea is that distribution lets you split the

main problem into smaller problems, solve them, then combine their subtotals into a finalanswer. For example, 8^12 = 8^(10+2) = 8^10 + 8^2 = 80 + 16 = 96The key point not to miss here is that distribution works only when the group of

numbers in the ( ) are being combined. If the problem looks like this: 3^(10^5) thendistribution does not apply because it is not needed.

I really dont think of the distributive property as another rule to memorize. It is really


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Action Algebra54

just a common sense interpretation of what is written. It is always wise to break down bigproblems into smaller ones and the distributive property gives us a label to help correctlydescribe what we are doing. I use the word fill when I teach equations and sometimesthe visual of a sprinkling can helps students see what is happening no matter what theycall it.

So here is the complete idea. Every number in one group must multiply every numberin the other group. Look at the examples.Of course, if one is careful, the part with the parentheses can be skipped, and the four

subproblems can be written out in any form, as long as the student remembers to combinetheir subanswers.

P w f 10When multiplying big numbers much time can be saved if the student knows how

handle powers of ten. 0s at the end of the numbers (not in the middle) can be writtendown as part of the answer right away. Then normal multiplication can be performed

on the remaining digits and their answer written on the left side of the answer 0s. Thisshortcut works because of the distributive property. For example, 5^100 is 5^1 and two0s for an answer of 500. 20^30 is 2^3 and two 0s for an answer of 600.

V tica M tip icatiWhether you have your students break up the problem as in the examples or use the

standard vertical method, the underlying principle is the distributive property--everything^ everything.

Now lets look at vertical multiplication and a couple different ways to do it. Thenormal carrying can be used, or else the full subanswers can be written out in the middle

area. I recommend the latter method because it is cleaner and easier to read. Youngchildren often get confused about where and why to put their carries and which carry isthe current one to use.

15 ^12102050

100180

2^5

2^1010^5

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Action Algebra[1]