NIX THETRICKS

Second Edition

byTina Cardone and the MTBoS

Updated: January 14, 2015

This work is licensed under the Creative CommonsAttribution-NonCommercial-ShareAlike 3.0 UnportedLicense. To view a copy of this license, visit Cre-ativeCommons.org

All student work was collected by classroom teach-ers during the course of regular lessons, then submit-ted to MathMistakes.org. To protect the privacy ofstudents (and in many cases to improve legibility),each example was rewritten in the authors handwrit-ing.

I would say, then, that it is not reason-able to even mention this technique. If it isso limited in its usefulness, why grant it theprivilege of a name and some memory space?Cluttering heads with specialized techniquesthat mask the important general principle athand does the students no good, in fact it mayharm them. Remember the Hippocratic oath First, do no harm.

Jim Doherty

Contents

Preface x

1 Introduction 1

2 Operations 7

2.1 Nix: Total Means Add . . . . . 7

2.2 Nix: Bigger Bottom, Better Bor-row . . . . . . . . . . . . . . . . 9

2.3 Nix: Add a Zero (Multiplyingby 10) . . . . . . . . . . . . . . 11

iv

2.4 Nix: Five or More Go up a Floor,aka 0-4 Hit the Floor, 5-9 Makethe Climb . . . . . . . . . . . . 13

2.5 Nix: Turtle Multiplication . . . 142.6 Nix: Does McDonalds Sell Cheese-

burgers, aka Dad, Mom, Sister,Brother . . . . . . . . . . . . . 16

2.7 Nix: Ball to the Wall . . . . . . 172.8 Nix: PEMDAS, BIDMAS . . . 19

3 Proportional Reasoning 223.1 Nix: Butterfly Method, Jesus

Fish . . . . . . . . . . . . . . . 243.2 Nix: The Man on the Horse . . 263.3 Nix: Make Mixed Numbers MAD 273.4 Nix: Backflip and Cartwheel . . 303.5 Nix: Cross Multiply (Fraction

Division) . . . . . . . . . . . . . 353.6 Nix: Cross Multiply (Solving

Proportions) . . . . . . . . . . . 403.7 Nix: Flip and Multiply, Same-

Change-Flip . . . . . . . . . . . 443.8 Nix: Dr. Pepper . . . . . . . . . 45

3.9 Nix: New Formulas for EachConversion . . . . . . . . . . . . 47

3.10 Nix: Formula Triangle . . . . . 513.11 Nix: Outers over Inners . . . . 52

4 Geometry and Measurement 564.1 Nix: Perimeter is the Outside . 564.2 Nix: Rectangles Have Two Long

Sides and Two Short Sides . . . 584.3 Nix: Squares Have Four

Equal Sides . . . . . . . . . . . 594.4 Nix: Obtuse Angles are Big . . 604.5 Nix: a2 + b2 = c2 . . . . . . . . 614.6 Nix: The Angle of Inclination

Is the Same as the Angle of De-pression . . . . . . . . . . . . . 63

4.7 Formula: Areas of Quadrilat-erals, Triangles . . . . . . . . . 66

4.8 Formula: Surface Area . . . . . 694.9 Formula: Volume . . . . . . . . 704.10 Formula: Distance Formula . . 72

5 Number Systems 73

5.1 Nix: Absolute Value Makes aNumber Positive . . . . . . . . 73

5.2 Nix: Same-Change-Change, Keep-Change-Change (Integer Addi-tion) . . . . . . . . . . . . . . . 75

5.3 Nix: Two Negatives Make a Pos-itive (Integer Subtraction) . . . 77

5.4 Nix: Two Negatives Make a Pos-itive (Integer Multiplication) . . 80

5.5 Nix: Move the Decimal(Scientific Notation) . . . . . . 83

5.6 Nix: Jailbreak Radicals, aka YouNeed a Partner to Go to theParty . . . . . . . . . . . . . . . 85

5.7 Nix: Exponent Over Radical . . 88

6 Equations and Inequalities 906.1 Nix: Hungry Inequality

Symbols . . . . . . . . . . . . . 906.2 Nix: Take or Move to the Other

Side . . . . . . . . . . . . . . . 926.3 Nix: Switch the Side and Switch

the Sign . . . . . . . . . . . . . 94

6.4 Nix: Cancel . . . . . . . . . . . 956.5 Nix: Follow the Arrow

(Graphing Inequalities) . . . . . 986.6 Nix: The Square Root and the

Square Cancel . . . . . . . . . . 996.7 Nix: Land of Gor . . . . . . . . 1016.8 Nix: Log Circle . . . . . . . . . 1046.9 Nix: The Log and the

Exponent Cancel . . . . . . . . 105

7 Functions 1087.1 Nix: Rise over Run as the Def-

inition of Slope . . . . . . . . . 1087.2 Nix: OK vs. NO Slope . . . . . 1107.3 Nix: What is b? . . . . . . . . . 1127.4 Nix: The Inside Does the Op-

posite . . . . . . . . . . . . . . 1137.5 Nix: All Students Take

Calculus . . . . . . . . . . . . . 1167.6 Nix: FOIL . . . . . . . . . . . . 1177.7 Nix: Slide and Divide, aka Throw

the Football . . . . . . . . . . . 1207.8 Nix: Synthetic Division . . . . . 125

8 Conclusion 127

A Index of Tricks by Standards 135

B Types of Tricks 143B.1 Imprecise Language . . . . . . . 143B.2 Methods Eliminating Options . 144B.3 Tricks Students Misinterpret . . 145B.4 Math as Magic, Not Logic . . . 145

Preface

In the beginning, there was a calculus teachercomplaining about students lack of a defi-nition of slope. Then there was a conversa-tion among my department members on trickswe hate seeing kids show up to our classeswith. I expanded the conversation to mem-bers of my online math community. We brain-stormed and debated what constituted a trickand which were the worst offenders. I or-ganized. More people joined in on the con-versation and shared better methods to em-phasize understanding over memorization. Iorganized some more. Contributions started

x

to slow down. The end result was 17 pagesin a Google Doc. I had grand dreams of abeautifully formatted resource that we couldshare with teachers everywhere. A few peo-ple shared my dream. We discussed format-ting and organization and themes. Finally,inspired by NaNoWriMo, I opened up LaTeXand started typesetting. I got some help. Thisdocument was born. I hope you enjoy it allthe more knowing how many peoples ideasare encapsulated here.

2nd Edition: This project began in Jan-uary of 2013. It is now November of 2014 andI cannot begin to describe the experience thusfar. While my primary focus is still teaching,(full time at a high need public school in Mas-sachusetts) this side project has allowed me tomake connections with educators all over theworld and pushed me to think deeply aboutmathematics at all levels. The Google Doc ofnew tricks for review has again grown to 16pages. Not to mention the additional pagesfor vocabulary and notation. Once again in-

spired by NaNoWriMo I am taking the monthto organize, edit, format and publish the sec-ond edition. In the meantime, more peoplethan I could hope to thank have contributedto Nix the Tricks in countless ways. Here arejust a few of the many people to whom I owethe existence of this text.

Thanks to all contributors:

Chuck Baker High School Math Teacher@chuckcbaker, mrcbaker.blogger.com

Ashli Black NBCT, Mathematics Consultant@mythagon, mythagon.wordpress.com

Tim Cieplowski @timteachesmath

Jim Doherty High School Math,Department Chair @mrdardy, mrdardy.wordpress.com

Mary Dooms Middle School Math@mary dooms, teacherleaders.wordpress.com

Nik Doran

Bridget Dunbar Math Instructional Resource Teacher - Middle Schoolelsdunbar.wordpress.com

Michael Fenton Nix the Tricks Editor in Chief, Mathematics Teacher,Consultant @mjfenton, reasonandwonder.com

Paul Flaherty High School Math Teacher

Peggy Frisbie High School Math Teacher

Emmanuel Garcia

Marc Garneau K-12 Numeracy Helping Teacher@314Piman, diaryofapiman.wordpress.com

John Golden Math Ed Professor@mathhombre, mathhombre.blogspot.com

Megan High School Math/Science TeacherHayes-Golding

Chris Hill Physics Teacher in Milwaukee@hillby258, hillby.wordpress.com

Scott Hills

Chris Hlas

Chris Hunter K-12 Numeracy Helping Teacher@ChrisHunter36, reflectionsinthewhy.wordpress.com

Bowen Kerins Senior Curriculum Writercmeproject.edc.org

Rachel Kernodle High School Math Teacher

Yvonne Lai

Bob Lochel High School Math@bobloch, mathcoachblog.com

TR Milne High School Math

Jonathan Newman High School Math/Sciencehilbertshotel.wordpress.com

Kate Nowak MTBoS High Priestessfunction-of-time.blogspot.com

Jami Packer High School Special Education@jamidanielle, jamidanielle.blogspot.com

Jim Pai High School Math@PaiMath, intersectpai.blogspot.ca

Michael Pershan Elementary/High School Math Teacher@mpershan, rationalexpressions.blogspot.com

Avery Pickford Middle School Math@woutgeo, withoutgeometry.com

Julie Reulbach Middle/High School Math Teacher

Gabe Rosenberg High School/College Math Teacher

Mark Sanford High School Math Teacher

Sam Shah High School Math Teachersamjshah.com

Jack Siderer

Gregory Taylor High School Math Teacher@mathtans

Sue VanHattum Math Professor, Editor, Playing with Mathmathmamawrites.blogspot.com

Lane Walker NBCT, High School Math@LaneWalker2

Lim Wei Quan

Julie Wright Middle School Math Teacher@msjwright2, sadarmadillo.blogspot.com

Mimi Yang Middle/High School Math Teacher@untilnextstop

Amy Zimmer

and many more who chose to remain anony-mous.

Each chapter follows a concept thread. Forexample, you might see how content knowl-edge should build from drawing representa-tions of fractions to solving proportions. Feelfree to read this as a book, from front to back,or jump directly to those sections that applyto your grade level of interest (see Appendix Ato find the tricks organized by Common CoreStandard).

Chapter 1

Introduction

This text is inspired by committed teacherswho want to take the magic out of mathemat-ics and focus on the beauty of sense-making.It is written for reflective teachers who em-brace the Common Core Standards for Math-ematical Practice. The contributors are peo-ple who wish for teachers everywhere to seekcoherence and connection rather than offerstudents memorized procedures and short-cuttingtricks. Students are capable of developing rich

1

conceptual understanding; do not rob them ofthe opportunity to experience the discovery ofnew concepts.

This is a hard step to take; students willhave to think and they will say they do notwant to. Students (and parents and tutors)will need to readjust their expectations, how-ever, it is in the best interest of students ev-erywhere to make the focus of mathematicscritical thinking. Will you help math reclaimits position as a creative and thought-provokingsubject?

But its just to help them remember -I taught them the concept!

SOH CAH TOA is a mnemonic device. Thereis no underlying reason why sine is the ratioof opposite to hypotenuse; it is a definition.Kids can use this abbreviation without losingany understanding.

FOIL is a trick. There is a good reasonwhy we multiply binomials in a certain way,and this acronym circumvents student under-

standing of the power and flexibility of thedistributive property. If you teach the dis-tributive property, ask students to developtheir own shortcut and then give it a name;that is awesome. However, the phrase eachby each is more powerful than FOIL sinceit does not imply that a certain order is nec-essary (my honors precalculus students wereshocked to hear OLIF would work just as wellas FOIL) and reminds students of what theyare doing. Many students will wait for theshortcut and promptly forget the reasoningbehind it if the trick comes too soon or froma place beyond their current understanding.

My students cant understand the higherlevel math, but they do great with thetrick.

If students do not understand, they are notdoing math. Do not push students too far,too fast (adolescent brains need time to de-velop before they can truly comprehend ab-straction), but do not sell your students short

either. The world does not need more robots;asking children to mindlessly follow an algo-rithm is not teaching them anything more thanhow to follow instructions. There are a mil-lion ways to teach reading and following di-rections; do not reduce mathematics to thatsingle skill. Allow students to experience andplay and notice and wonder. They will sur-prise you! Being a mathematician is not lim-ited to rote memorization (though learningthe perfect squares by heart will certainly helpone to recognize that structure). Being a math-ematician is about critical thinking, justifica-tion and using tools of past experiences tosolve new problems. Being a successful adultinvolves pattern finding, questioning othersand perseverance. Focus on these skills andallow students to grow into young adults whocan think everything else will come in time.

This is all well and good, but we donthave enough TIME!

Yes, this approach takes an initial invest-

ment of time. I would argue that we shouldpush back against the testing pressure ratherthan push back against concept development,but that is a whole other book. First, if weteach concepts rather than tricks, studentswill retain more. Each year will start with lessconfusion of keep-change-change or keep-keep-change? and any associated misconceptions,which means each year will have more time fornew content. Second, teachers have to maketough choices all the time, and choosing be-tween depth and breadth is a major choice.Every year my department gets together andtries to guess which topics in Geometry areleast likely to be on the state test. We leavethose topics for after the exam and do ourvery best to teach the other topics well. Ifstudents go into the test with solid reasoningskills they will at the very least be able to de-termine which choices are reasonable, if notreason all the way through an inscribed angleproblem (circles usually lose our lottery).

Read through these pages with an open

mind. Consider how you can empower stu-dents to discover a concept and find their ownshortcuts (complete with explanations!). I donot ask you to blindly accept these pages asthe final authority any more than I wouldask students to blindly trust teachers. En-gage with the content and discover the bestteaching approaches for your situation. Shareand discuss with your colleagues. Ask ques-tions, join debates and make suggestions atNixTheTricks.com.

Chapter 2

Operations

2.1 Nix: Total Means Add

Because:

Context matters. The phrase in all meansdifferent things in Tina has 3 cookies, Chrishas 5, how many cookies are there in all?and Cookies come in bags of 15. If you buy3 bags of cookies, how many cookies do youhave in all? Students will lose their sense

7

making skills, and their belief that math makessense, if the focus is on circling and crossingout words.

Fix:

Instead of teaching a few key words, ask stu-dents to think about all the words and whatis happening. Have students draw a modelof the situation rather than jumping directlyto the computation. Kids are able to reasonthrough problems, especially in story prob-lems where students have experiences they canapply to the context. Encourage students touse their sense-making skills (which they ar-rive to school with) both in figuring out theproblem, and in checking the reasonablenessof their answer. As a bonus, students willsimultaneously develop their reading compre-hension skills. Start with the mantra mathmakes sense and never let kids believe any-thing else.

2.2 Nix: Bigger Bottom, BetterBorrow

Because:

This trick has students sidestep thinking. Theydo not look at the numbers as a whole or rea-son about what subtraction means.

Fix:

Students must understand subtraction as aconcept and as an operation before the rulesof an algorithm can make sense to them. Ifwe try to push the algorithm before that un-derstanding is solid, then we force studentsto go against their intuition (for instance, theintuition that says you always subtract thesmaller number). When we ask children tofollow a procedure that holds no meaning forthem, they will conclude that math does notmake sense. Go back to the mantra mathmakes sense and make sure all students be-

lieve it when they learn the subtraction algo-rithm.

To achieve this: Start withadding/subtracting within 10,then within 20, then multiples of10. Combining these skills al-lows students to understand sub-traction with regrouping. Whenadding and subtracting any pair

of expressions (algebraic or otherwise), wemust add and subtract like terms. For termsto be alike, they must have the same placevalue. When subtracting 9 from 23, there areonly three ones, and so we regroup into 13ones and a ten. Before students can make thisregrouping mentally they need to experienceit with manipulatives such as ten framesand unit cubes. With enough practice, stu-dents will begin to do the computation with-out needing to count out each piece. At thispoint they will be ready for the abstraction ofan algorithm.

2.3 Nix: Add a Zero (Multiply-ing by 10)

Because:

When students study integers, multiplying by10 means to add a zero, but once they headinto the realm of real numbers the phrase changesto move the decimal point. Neither phraseconveys any meaning about multiplication orplace value. Add a zero should mean addthe additive identity which does not changethe value at all!

Fix:

Have students multiply by ten just like theymultiply by any other number. When a stu-dent discovers a pattern: first congratulatethem for noticing and sharing, then ask themwhether they think their pattern will alwayswork. If they believe it does generalize, askthem why their rule works. Finally, if the stu-

dent can justify their pattern for a particularsituation, remind them to be careful aboutmindlessly applying it in novel situations (askthem about 10 0 for example). One way tohelp students see why their pattern works isto use a place value chart to show what ishappening: each of the ones becomes a ten(1 10 = 10), each of the tens becomes ahundred (10 10 = 100), etc. The patternof shifting place value works all up and downthe decimal scale. You could name the pat-tern Student As rule for multiplying by 10if you want to be able to refer back to it.

2.4 Nix: Five or More Go up aFloor, aka 0-4 Hit the Floor,5-9 Make the Climb

Because:

When using this trick students do not real-ize that rounding means choosing the clos-est number at the next place value. Thismeans that they do not understand what todo, for example, with the ones place if theyare rounding to the hundreds place. Studentsare able to tell what is closer without a rhymeand should know the reason why we choose toround up with five.

Fix:

Begin the study of rounding using numberlines so students can see which number is closer.The problem is that five is not closer to eitherside; it is exactly in the middle (as long as itis a five with nothing but zeros after it). We

have to make a rule for which way we are go-ing to round five, because it could go eitherway. The convention in math class is to roundup, although in science they solve this prob-lem by rounding down for even digits beforethe five and up for odd digits before the 5 sothat each occurs half the time.

2.5 Nix: Turtle Multiplication

Because:

The cute story only serves tooverload the students minds and distractsthem from the task at hand multiplication.

Fix:

Just as with any other algorithm, studentsneed to build up their understanding so theprocedure is not something to be memorized,but rather a quick way to record a series ofsteps that make sense. Students should build

their understanding using area models, fol-lowed by partial products. When studentscan compute partial products fluently theyare ready for the abbreviated version (the stan-dard algorithm).

23 5 =(20 + 3) 5=20 5 + 3 5=100 + 15

=115

2.6 Nix: Does McDonalds SellCheeseburgers, aka Dad, Mom,Sister, Brother

Because:

Students should understand the process notmemorize a procedure. Not to mention thefact that it is just as hard to remember theorder of the family members in this arbitrarymnemonic as it would be to remember arbi-trary operations. Describing the process with-out understanding leads to confusion; whatmathematical operation is equivalent to bringdown the 2?

Fix:

Students cannot see place valuewhen using the standard algo-rithm for dividing. Adding somecolor and writing out the entire

number helps make the process more trans-parent. So does allowing students to takeaway less than the maximum amount eachtime - students will learn that the process goesfaster if they maximize at each step, but thereis no harm in taking two steps to do somethingif it helps students feel more confident.

2.7 Nix: Ball to the Wall

Because:

Students have no idea why theyare moving the decimal point, which meansthey are likely to misinterpret this rule andthink that 5.5 2.5 = 5.5 25 rather thanthe correct statement 5.5 2.5 = 55 25.

Fix:

If a number has digits after the decimal point,this means there is only part of a whole. Sinceit is easier to work with whole numbers when

dividing, we can write a different questionthat would have the same solution. Consider10 5 and 100 50. These have the samesolution because they are proportional:

10

5=

10

5 1

=10

5 10

10

=10 105 10

=100

50

10.3

5.2=

10.3

5.2 1

=10.3

5.2 10

10

=10.3 105.2 10

=103

52

More generally, xy

= 10x10y

, so xy = (10x)(10y). It is interesting to note that multiply-ing by 10 is not the only option, any equiv-alent ratio with a whole number divisor willwork for long division.

2.8 Nix: PEMDAS, BIDMAS

Because:

Students interpret the acronym in the orderthe letters are presented, leading them to mul-tiply before dividing and add before subtract-ing.

For example, students often incorrectly eval-uate 6 2 5 as follows:

Incorrect: 6 2 5 = 6 10 = 0.6

Correct: 6 2 5 = 62 5 = 3 5 = 15

Fix:

Students should know that mathematiciansneeded a standard order of operations for con-sistency. The most powerful operations shouldbe completed first exponentiation increasesor decreases at a greater rate than multiply-ing, which increases or decreases at a greater

rate than addition. Sometimes we want to usea different order, so we use grouping symbolsto signify do this first when it is not themost powerful operation. If students are stilllooking for a way to remember the order, re-place the confusing acronym PEMDAS withthe clearer GEMA.

G is for grouping, which is better than paren-theses because it includes all types of group-ings such as brackets, absolute value, expres-sions under radicals (ex: square roots), thenumerator of a fraction, etc. Grouping alsoimplies more than one item, which eliminatesthe confusion students experience when theytry to Do the parentheses first. in 4(3).

E is for exponents. This includes radicals asthey can be rewritten as exponents.

M is for multiplication. Division is implied.Since only one letter appears for both opera-tions, it is essential to emphasize the impor-

tant inverse relationship between multiplica-tion and division. For example, discuss theequivalence of dividing by a fraction and mul-tiplying by its reciprocal.

A is for addition. Subtraction is implied. Again,since only one letter appears for both oper-ations, it is essential to emphasize the im-portant inverse relationship between additionand subtraction. A useful definition for sub-tract is add the opposite.

Chapter 3

ProportionalReasoning

Ratios and proportions are a new way of think-ing for elementary students. Teachers oftenbemoan the difficulty that kids have with frac-tions, but students struggle because we robthem of the opportunity to develop any in-tuition with proportional reasoning. The firstexperience most people have with math is count-ing, then adding, along with additive patterns.

22

Even when students start multiplying, it tendsto be defined as repeated addition. Fractionsare the first time when adding will not work,and it messes students up. Skip the shortcutsand let your kids see that fractions, ratios andproportions are multiplicative - a whole newway to interpret the world!

3.1 Nix: Butterfly Method, Je-sus Fish

Because:

Students have no idea why itworks and there is no mathematical reasoningbehind the butterfly, no matter how pretty itis.

Fix:

If students start with visualssuch as fraction strips theywill discover the need to havelike terms before they canadd. Say a student wants toadd 1

2+ 1

4. They may start

with a representation of eachfraction, then add the frac-tions by placing them end to end. The rep-resentation is valid, but there is no way towrite this new diagram as a single fraction.

To do so, students need to cut the wholeinto equal parts. After some experience, stu-dents will realize they need common denom-inators to add. After still more experienceadding fractions with common denominators,students will realize they can simply add thenumerators (which is equivalent to countingthe number of shaded pieces) while keepingthe denominator the same (as the size of thepieces does not change).

Fractions can be compared and added orsubtracted with any common denominator;there is no mathematical reason to limit stu-dents to the least common denominator. Manyvisual/manipulative methods will not give leastcommon denominators (instead using the prod-uct of the denominators) and that is just fine!Accept any denominator that is computation-ally accurate. Students may eventually grav-itate towards the least common denominatoras they look for the easiest numbers to workwith. In the meantime, encourage studentsto compare different methods do different

common denominators give different answers?Are they really different or might they be equiv-alent? How did that happen? Fractions caneven be compared with common numerators a fascinating discussion to have with studentsof any age!

3.2 Nix: The Man on the Horse

Trick Explanation:The fraction is a man on ahorse. The man on the horse goes inside thehouse, but the horse must remain outside thehouse.

Because:

The fact that the dividend is the numeratorwhen they set up the problem to divide shouldnot be something students need help remem-bering. Plus, students are thrown off whenthere is a case where the man (numerator) is

larger than the horse (denominator) becausethat makes even less sense!

Fix:

The notation used for fractions is not entirelyforeign if students have been using either or / for division. The fraction line is just likeeach of these, except now we read top to bot-tom rather than left to right. Instead of in-troducing fraction division as a new concept,share that the fraction line is merely anotherway to write divided by.

3.3 Nix: Make Mixed NumbersMAD

Because:

This shortcut allows students to convert a mixednumber to an improper fraction without un-derstanding how wholes relate to fractions.

This process skips the important conceptualsteps that are happening behind the scenes.

Fix:

To show that 315

= 165

, students should know

that each whole is equal to 55. This process

should begin with manipulatives and diagramsso that students see that a whole is a fractionwhere the numerator is equal to the denomi-nator. Students also need to know that com-mon denominators are necessary to add frac-tions. Once both of those concepts have beendeveloped, combine the two ideas to convertthe whole numbers into fractions with com-

mon denominators:

31

5= 1 + 1 + 1 +

1

5

=5

5+

5

5+

5

5+

1

5

=16

5

After students have repeated this process manytimes they will be able to go from 31

5to 15

5+ 1

5.

Eventually they may be able to complete theentire computation mentally, but it is essen-tial that students have the understanding ofhow to write one whole as a fraction to fallback on. Repeated practice converting thewhole numbers to fractions with like denomi-nators and only then adding the fractions willsecure this concept in students memories.

3.4 Nix: Backflip and Cartwheel

Because:

There is no reasoning involvedand it is very easy to mix up.Cartwheels and backflips both involve flippingupside down in the middle; if one fraction getsflipped around, why not the other?

Fix:

See Section 3.3 for converting mixed numbersto improper fractions.

To convert improper fractions to mixed num-bers we should follow the same process in re-verse. The goal is to remove as many wholes

from the fraction as possible.

16

5=

5

5+

11

5

=5

5+

5

5+

6

5

=5

5+

5

5+

5

5+

1

5

= 1 + 1 + 1 +1

5

= 3 +1

5

After students have repeated this process afew times they will begin to take out morethan one whole at a time, going from 16

5to

155

+ 15

in one step.Representing this process as division is also

a valid approach. It will certainly help intransitioning to converting fractions to dec-imals. However, students should not need atrick to know that the denominator is the di-

visor. The denominator of a fractions tellshow many parts to divide the whole into.

Kids want to use the phrase Cross Multi-ply for everything: How do we multiply frac-tions? Cross Multiply! How do we dividefractions? Cross Multiply! How do we solveproportions? Cross Multiply! Those arethree entirely different processes; they needdifferent names. For multiplication of frac-tions, Cross Multiply means multiply across(horizontally), and there usually is not a trickassociated with this operation. By high schoolmost students have no difficulty with this op-eration it matches their intuition. In fact,they like this method so much they want toextend it to other operations (non-example:to add fractions, add the numerators and addthe denominators) a case where students in-tuition fails them. Instead of saying crossmultiply, use the precise (though admittedlycumbersome) phrase multiply numerator bynumerator and denominator by denominatorwhen students need a reminder of how to mul-tiply fractions.

To build upon and justify student intu-ition, direct students to an area model to de-termine the product.

To multiply 23 45

find the area shaded byboth that is two-thirds of four-fifths. Thefifths are each divided into thirds and two ofthe three are shaded. The resulting productis 8

15.

3.5 Nix: Cross Multiply (Frac-tion Division)

Because:

Division and multiplication are different (al-beit related) operations; one cannot magicallyswitch the operation in an expression. Plus,students confuse cross (diagonal) with across(horizontal). Not to mention, where does theanswer go? Why does one product end up inthe numerator and the other in the denomi-nator?

Please, never tell students the phrase, Oursis not to reason why; just invert and mul-tiply. A students job in math class is toreason, and a teachers job is to convince thestudents that math makes sense.

Fix:

Use the phrase multiply by the reciprocal,but only after students understand where this

algorithm comes from. The reciprocal is aprecise term that reminds students why weare switching the operation.

2

3 2

3= 1 easy!

2

3 1

3= 2 makes sense

4

5 3

5=

4

3not as obvious, but still

dividing the numerators

4

5 1

2= ? no idea!

If the last problem looked like the previousexamples, it would be easier. So lets rewritewith common denominators:

8

10 5

10=

8

5makes sense

If students are asked to solve enough problemsin this manner, they will want to find a short-cut and may recognize the pattern. Showthem (or ask them to prove!) why multiplyingby the reciprocal works:

4

5 1

2=

4 25 2

1 52 5

=4 21 5

=4 25 1

=4

5 2

1

In this case students discover that multiply-ing by the reciprocal is the equivalent of get-ting the common denominator and dividingthe numerators. This is not an obvious fact.Students will only reach this realization withrepeated practice, but practice getting com-mon denominators is a great thing for them

to be doing! More importantly, the studentwho forgets this generalization can fall backon an understanding of common denomina-tors, while the student who learned a rule af-ter completing this exercise once (or not atall!) will guess at the rule rather than at-tempt to reason through the problem.

A second approach uses compound frac-tions. Depending on what experience stu-dents have with reciprocals, this might be amore friendly option. It has the added bonusof using a generalizable concept of multiplyingby a convenient form of one which appliesto many topics, including the application ofunit conversions (Section 3.9).

To begin, the division of two fractions canbe written as one giant (complex or compound)fraction.

4

5

1

2

=

4

5

1

2

1

=

4

5

1

2

2

1

2

1

=

4

5 2

1

1

=4

5 2

1

3.6 Nix: Cross Multiply (Solv-ing Proportions)

Because:

Students confuse cross (diagonal) with across(horizontal) multiplication, and/or believe itcan be used everywhere (such as in multipli-cation of fractions).

Correct multiplication of fractions:

1

2 3

4=

1 32 4 =

3

8

Common error:1

2 3

4=

1 42 3 =

4

6

More importantly, you are not magicallyallowed to multiply two sides of an equationby different values to get an equivalent equa-tion. This process does not make sense to stu-dents. They are memorizing a procedure, notunderstanding a method; which means thatwhen they forget a step, they guess.

This student tries tomultiply fractions

using crossmultiplication:

http://mathmistakes.org/?p=476

This student usescross addition

instead ofmultiplication:

http://mathmistakes.org/?p=1320

Fix:

Instruct solving all equations (including pro-portions, they arent special!) by inverse op-erations.

3

5=

x

10

10 35

= 10 x10

10 35

= x

6 = x

Encourage students to look for shortcutssuch as common denominators, common nu-merators or scale factors. The example abovetakes fewer steps than cross multiplying, butit can be completed even faster if a studentrecognizes that ten is five doubled. Once stu-dents know when and why a shortcut works,skipping a few steps is okay, but students must

know why their shortcut is legal algebra andhave a universal method to fall back on.

Shortcuts:

3

5=x

5 x = 3 no work required,

meaning of equal

3

5=

x

10 3 2

5 2 =x

10multiply by 1 to get an

x = 6 equivalent fraction4

8=

x

10 x = 5 students recognize 1

2(any equivalent ratio works)

5

3=

10

x 3

5=

x

10take reciprocal of bothsides of the equation

3.7 Nix: Flip and Multiply, Same-Change-Flip

Because:

Division and multiplication are different (al-beit related) operations, one cannot magicallyswitch the operation in an expression. Plus,students get confused as to what to flip.

Fix:

Use the same methods as described in the fixof Section 3.5.

3.8 Nix: Dr. Pepper

Because:

Students do not need to know what a per-cent represents to use this method. Instead ofmagically moving the decimal point, use thedefinition of percent to interpret the number.

Fix:

Percent means per 100. Students who knowthat percents are out of 100 can then writetheir percents as fractions. From there theycan use knowledge of converting between frac-tions and decimals. No need to learn a newprocess (or magically move decimal points!).Percents are an excellent opportunity to workwith students on understanding the connec-tion between fractions and decimals. We readthe number 0.54 as 54 hundredths, which means54100

, which is the definition of 54%. This doesbecome slightly more complex if students have

not seen decimals in fractions. In the case of43.2% we know:

43.2% =43.2

100

=43.2

100 1

=43.2

100 10

10

=432

1000

= 0.432

Students should understand that 100% cor-responds to one whole, so that percentages0 - 100% correspond to decimal numbers be-tween zero and one. There are also percentsgreather than 100, just as there are fractionsgreater than one.

Another approach is to multiply (or divide)by 100%. This is multiplying (or dividing) by

1, which does not change the value. Bewareof students who will literally type 0.54100%into the calculator (there is a percent buttonon many scientific calculators) as this buttonis an operation not a label.

3.9 Nix: New Formulas for EachConversion

Because:

All conversions are equivalent, regardless ofunits; there is no need to give them separateformulas to make students think they shouldbe compartmentalized. Scale factor, percent,arc length and unit conversions all use thesame strategies. The more connections stu-dents can make, the more solid their under-standing will be.

Fix:

Talk about units. All the time. If studentsrecognize how to use units then each of thesesituations will be a new unit to learn, but nota new procedure to learn. When teaching cir-cles in precalculus, I assigned some problemson linear and angular speed. Most students inthe class had taken chemistry and masteredunit conversions in that course, but it did notoccur to them to apply that skill to this newcontext. We need to work with our sciencecolleagues and use units regularly.

When students convert something, showthem what they did. If there are 5 dozeneggs, students think they need to multiply5 12. If we include the units the processbecomes clearer: 5 dozen 12 eggs

1 dozen. Teach stu-

dents that they are multiplying by one theyhave not changed the value (there are stillthe same number of eggs) but instead havechanged the units (eggs vs. dozens). This willalso help students avoid the problems that

can be caused by key words (Section 2.1).If students stop thinking about what makessense, they may randomly combine numbersthat they are given. District math special-ist, Robert Kaplinsky, asked a class of eighthgraders, There are 125 sheep and 5 dogs ina flock. How old is the shepherd? and 75%of students gave a numerical answer. (Readmore about his informal research: http://robertkaplinsky.com/how-old-is-the-shepherd/.) When students do notsee what is happening in early unit conver-sion situations, they stop paying attention tothese important cues and focus solely on num-bers. Units should not magically change math makes sense.

Once they understand how to use units,students can learn to write equations in theform y = kx where k is the constant of pro-portionality. Then students can identify con-versions as a form of direct variation. To de-termine the constant of proportionality, stu-dents must recognize that a unit (of measure)is equivalent to a certain number of another

unit: 1 inch = 2.54 cm. Then, conversiontakes place by multiplying by 1, where the 1is formed from the equivalency, such as 1 in

2.54 cm.

To help students get started, ask them whatthey know about the unit. For example, withradians, ask students what angle they know inboth degrees and radians. They may tell you2pi radians = 360 degrees or pi radians = 180degrees. Then they have enough informationto set up an equation.

Proportions are the same as direct varia-tion; y = kx is equivalent to y

x= k, so equa-

tions where k is a rational number are pro-portions. Direct variation equations place theemphasis on converting from input to output,while proportions show the equivalence of twocomparisons more readily. The other benefitof direct variation is it is easy to expand. Toconvert several units (seconds to minutes tohours), extend the equation to y = k1 k2 x.

3.10 Nix: Formula Triangle

Trick Explanation:To find distance, cover D in thetriangle, rt remains. To findrate, cover r in the triangle and you see D

t.

To find time, cover t in the triangle, leavingDr.

Because:

Covering up part of a triangle is not a validmethod for solving an equation. Direct varia-tion problems are no different than any otherequation; use the same strategies.

Fix:

Students can use unit analysis (Section 3.9)to solve rate problems. It is perfectly reason-able to expect students to learn one equation,say d = rt, which is no harder to rememberthan the triangle. In fact, it is easier to write

an equation since students can check that theunits work if they forget; no such luck withthe triangle. Once they know one equationthey can use opposite operations to manipu-late it. This process involves much more sensemaking than setting up a new expression foreach variable without any understanding ofhow the parts of the triangle relate.

To find out how a middle school math teacher,David Cox, engaged his students in a discus-sion when they brought up this trick, see theconclusion, chapter 8.

3.11 Nix: Outers over Inners

Because:

Students have no idea what theyare doing, so chances are not goodthat they will remember the difference be-tween inners over outers and outers over in-ners. It seems arbitrary and has no connec-tion to the mathematical operation of divi-

sion. Remember, math makes sense.

Fix:

Complex fractions are, well, complex. Stu-dents who know how to both divide fractionsand simplify expressions get lost in the cogni-tive overload of variables combined with frac-tions! These problems do not have to be a stu-dents worst nightmare, though. While thesefractions look more complicated, they workjust like any other fraction. One way to di-vide fractions is to multiply by the reciprocal(Section 3.5)

That works well when both the numera-tor and denominator are single fractions. An-other approach, that is more versatile, is tomultiply by the common denominator, simi-lar to the first proof of (Section 3.5)

2 + x3

4 + 6x9

=2 + x

3

4 + 6x9

99

=18 + 3x

36 + 6x

=3(6 + x)

6(6 + x)

=3

6=

1

2

2 + 32x

4 53x2

=2 + 3

2x

4 53x2

6x2

6x2

=12x2 + 9x

24x2 10

=3x(4x+ 3)

2(12x2 5)

Students would do well to refer back to howthey would solve a simpler problem. Oncestudents have found a common denominator,simplifying the numerator and the denomi-nator separately can help students avoid thetemptation to cancel (Section 6.4) terms be-fore factoring.

Chapter 4

Geometry andMeasurement

4.1 Nix: Perimeter is the Out-side

Because:

Students interpret this tomean all the space outside a shape rather thanthe edge. The description should tell students

56

what the units are (length vs. area)

Fix:

The definition of perimeter is The distancearound a shape. It is important to empha-size that it is a distance, so students knowthey are looking for a one dimensional an-swer. This helps students who cannot remem-ber the difference between 2pir and pir2. Thefirst is one dimensional (r) and measures dis-tance perimeter/circumference. The otheris two dimensional (r2) and measures area.Students do not need to memorize formulasfor the perimeter of polygons. They can beencouraged to find shortcuts for computationalease (notice that all the sides of the squareare the same length so they could multiplyby four) but if a student forgets how to findperimeter they should fall back on the defini-tion rather than looking up a formula.

4.2 Nix: Rectangles Have TwoLong Sides and Two ShortSides

Because:

A rectangle is not a stretched out square.Rather, a square is a special rectangle. Thismeans that the set of rectangles needs to bea broader category with squares as a subset.Describing a rectangle as a shape with twolong sides and two short sides excludes squaresand includes parallelograms.

Fix:

The definition of a rectangle is A quadrilat-eral with four right angles. That is it! Theredoes not need to be any mention of sides,just angles. In the proper hierarchy: rect-angles are equiangular quadrilaterals, rhombiare equilateral quadrilaterals and squares areboth/regular.

4.3 Nix: Squares Have FourEqual Sides

Because:

This is the definition of a rhombus. This defi-nition does not adequately describe a square.

Fix:

Use the definition: A quadrilateral with fourequal sides AND four equal angles. Alterna-tively, A rectangle with four equal sides isan excellent description of a square as it em-phasizes the square as a subset of rectangles.

4.4 Nix: Obtuse Angles are Big

Because:

When students hear the words bigor large they are thinking about somethingtaking up a large amount of space. So stu-dents see small and large as describing thelength of the ray rather than the degree mea-sure.

Fix:

The definition of obtuse is An angle withmeasure between 90 and 180 degrees. Whenexplaining this definition or describing anglesuse wide angle not large angle. Wide vs.narrow fits the hinge analogy of an angle andfocuses student attention on the space betweenthe rays the angle rather than the raysthemselves. For students who have troubleunderstanding what an angle is, shading inthe region between two rays is a helpful way

to make an angle visible. Plus, this providesthe opportunity to discuss that shading in asmall corner vs. a wide swath does not changethe size of the original angle.

4.5 Nix: a2 + b2 = c2

Because:

The Pythagorean Theorem is a conditionalstatement, not an equation. If a2 + b2 = c2

is the only way we reference this importanttheorem teachers ask, What is c? ratherthan, What is the hypotenuse? and stu-dents yell out the formula without any con-sideration for the type of triangle needed. Re-cently my precalculus students were provingsin2(x) + cos2(x) = 1 and I had a right trian-gle with sides labeled opposite, adjacent andhypotenuse on the board. They could not fig-ure out how to write an equation from thatinformation without the a, b and c.

Fix:

The Pythagorean Theorem states If a tri-angle is right, then the sum of the squaresof the legs is equal to the square of the hy-potenuse. Students need to consider the con-ditions and use precise vocabulary. The fulltheorem is certainly a mouthful which is whya2 + b2 = c2 is what sticks in students mem-ories, but leg2 + leg2 = hypotenuse2 is rea-sonable to remember. Using the words legand hypotenuse also clues students in that ifthe triangle does not have a hypotenuse thenthe Pythagorean Theorem does not apply. Toreinforce the idea that this is a conditionalstatement, have students consider the contra-positive: if leg2+leg2 6= hypotenuse2, then thetriangle is not right. If students try this rela-tionship for both right and non-right trianglesthey will see why the condition is necessary.

4.6 Nix: The Angle of Inclina-tion Is the Same as the An-gle of Depression

Trick Explanation:Just re-draw the picture so theangle is in the bottom right corner and itworks every time!

Because:

Students should not get in the habit of chang-ing the diagram whenever they feel like it. Ifwe want students to correctly interpret situ-ations then they need to do so reliably. Bydrawing a new picture students also lose theunderstanding of what they are solving for.If the question asks for something in context,the student should give their answer in theoriginal context. Depth and height are re-lated but not the same. Not to mention thatnot all situations with angles of inclination or

depression can be solved using right triangletrigonometry, so the angle will not always fallin the bottom right corner.

Fix:

These angles must be measured from some-thing so have students start by drawing thehorizontal. Then they can draw the angle ofinclination or depression from the horizontal(going in the direction that corresponds to theword). Finally, drop a perpendicular some-where useful based on the sketch and giveninformation (assuming this is the right trian-gle trig unit, otherwise this trick did not applyin the first place!).

For students who strugglewith identifying which is the op-posite side and which is the ad-jacent side to the given angle, I

have them first label the hypotenuse (the sideopposite the right angle - which they do okaywith) then cover the angle they are workingwith. The leg that they are not touching is theopposite side. The leg that they are touchingis the adjacent side. No need to reorient thetriangle!

Frequently in geometry the lack of conceptdevelopment comes from a different angle formulas without background. These are nottricks, but if they are taught without contextthey become as magical as any other trick inthis book. Formulas can be discovered, ex-plored, derived or understood depending onthe course, but if they are merely memorizedthey are no better than a meaningless short-cut. Giving students a formula without let-ting them experience the process takes awaythe thinking involved.

4.7 Formula: Areas of Quadri-laterals, Triangles

Understanding:

Start with A = bh for rectangles.Students have seen arrays and mayimmediately make the connection to multipli-cation; for the other students, have them be-

gin with rectangles having integer side lengthsand count boxes. They will want a short-cut and can figure out that multiplying theside lengths will give the area. While lengthand width are perfectly acceptable names forthe sides, base and height are more consistentwith other shapes.

Next, explore how a parallelogram has thesame area as a rectangle with equal base andheight. Have students try some examples,then use decomposition to show that a righttriangle can be cut from one end of the par-allelogram and fit neatly onto the other endcreating a rectangle. Thus, A = bh for thisshape as well!

Now is the time to look at triangles. I amalways tempted (as are most students) to re-late the area of a triangle to the area of arectangle, but that connection only appliesto right triangles. Allow students to see thatany triangle is half of a parallelogram, thus:

A =1

2bh.

And once they know the formula for a tri-angle, students can find the area of any two-dimensional figure by decomposition. Depend-ing on the level, students could generalize theirdecompositions to find a simplified formulafor familiar shapes such as trapezoids, butmemorization of such formulas is unnecessary.

4.8 Formula: Surface Area

Understanding:

Many students do not realize that surface areais the sum of the areas of the faces of a shape.They may have been given a formula sheetwith a box labeled Lateral Surface Area andanother labeled Total Surface Area and playeda matching game:

1. Find the key words (Section 2.1) in thequestion.

2. Match those to the words in the formulasheet.

3. Substitute numbers until the answermatches one of the choices provided.

Instead of playing the game, define surfacearea as the area of the surfaces. If studentscan identify the parts of the object (and havemastered 2-D area), then they can find thesurface area. As a challenge task, ask stu-dents to draw 2-D nets that will fold into the

object they are presented with. This is a prac-tice in visualizing complex objects and under-standing their connections to simpler objects.This sounds a lot like Look for and make useof structure a.k.a. Standard for Mathemat-ical Practice 7 from the Common Core StateStandards. The task of breaking down a coneor cylinder into its parts is one that is par-ticularly difficult for students to do. Givingstudents paper nets to alternately build andflatten can help build intuition.

Depending on the level, students could gen-eralize their process to find a simplified for-mula for familiar shapes such as rectangularprisms, but memorization of such formulas isunnecessary.

4.9 Formula: Volume

Understanding:

Just as surface area is the ap-plication of familiar area formulas to a new

shape, volume formulas are not completelydifferent for every shape, in fact they con-tinue to apply familiar area formulas. Thereare two basic volume formulas: the prism andthe pyramid. The prism formula is ratherintuitive if students imagine stacking layers(or physically stack objects such as patternblocks) they will realize that volume of aprism is the area of the base multiplied bythe height. Once again, finding the value inquestion becomes a practice in visualizing What is the base? How do I find its area?What is the height?

The formula for a pyramid is not as intu-itive, but students can recognize that the areaof a pyramid is less than the area of a prismwith the same base and height. Some exper-imentation will show that the correct coeffi-cient is 1

3. This covers all shapes traditionally

taught through high school (including conesas circular pyramids).

4.10 Formula: Distance Formula

Understanding:

Please do not teach the distance formula be-fore the Pythagorean Theorem! It is muchharder for students to remember and does notprovide any additional meaning. In my every-day life I only use the Pythagorean Theorem,nothing more. And as a math teacher, find-ing the distance between points is somethingI do somewhat regularly. Manipulating thePythagorean Theorem to solve for differentvariables gives students practice solving equa-tions. That said, it does not hurt students tosee the distance formula somewhere in theirmathematical career, so long as they see howit comes directly from the Pythagorean The-orem. The Pythagorean Theorem also givesus the equation for a circle, that one smallformula turns out to be very powerful.

Chapter 5

Number Systems

5.1 Nix: Absolute Value Makesa Number Positive

Because:

This is not a definition; this is an observationthat follows from a definition. Additionally,this observation only applies to real numbers.Students are stuck when they need to solveequations using absolute value or when find-

73

ing the absolute value of a complex number:|4 i| 6= 4 + i.

Fix:

Geometrically, absolute value is the distancefrom zero. This definition is good for num-bers and still holds for equations. When wemove into two dimensions the slight tweak todistance from the origin is sufficient. Rein-force the fact that distance is positive throughconversation (going five feet to the left is thesame distance as going five feet to the right)or through the more formal algebraic defini-tion of absolute value:

|x| ={ x for x 0x for x 0

For x it is preferable to say the oppositeof x rather than negative x.

5.2 Nix: Same-Change-Change,Keep-Change-Change (Inte-ger Addition)

Because:

It has no meaning and there is no need forstudents to memorize a rule here. They areable to reason about adding integers (and ex-trapolate to the reals). A trick will circum-vent thinking, while a tool like the numberline will be a useful strategy throughout theirstudies.

http://mathmistakes.org/?p=328

This student takes Same-Change-Changeand expands it to multiplication and division,presumably thinking If you can magicallyswitch between addition and subtraction, whynot switch multiplication and division?

Fix:

Once students are comfortable adding and sub-tracting whole numbers on the number line,all they need to add to their previous under-standing is that a negative number is the op-posite of a positive number. There are manyways to talk about positive and negative num-bers, but shifting on a number line is a goodrepresentation for students to be familiar withas this language will reappear during transfor-mations of functions.

2 + 5 Start at 2, move 5 units to the right

2 + (5) Start at 2, move 5 units to the left(opposite of right)

2 5 Start at 2, move 5 units to the left2 (5) Start at 2, move 5 units to the right

(opposite of left)

Note: Some students may find a vertical num-ber line (or even a representation of a ther-mometer) a more friendly tool to start with asup/down can be more intuitive than left/right.

5.3 Nix: Two Negatives Makea Positive (Integer Subtrac-tion)

Because:

Again, it has no meaning and there is no needfor students to memorize a rule here. Crossingout and changing signs makes students thinkthat they can arbitrarily change the problem!

Fix:

The simplest approach here is to define sub-traction as adding the opposite. An addedbonus of changing from subtraction to addi-tion is that addition is commutative!

As was the case in Section 5.2, students canreason through this on the number line. Oncestudents recognize that addition and subtrac-tion are opposites, and that positive and neg-ative numbers are opposites, they will see thattwo opposites gets you back to the start. Justas turning around twice returns you to facingforward.

2 (5) Start at 2, move (5) spaces down Start at 2, move 5 spaces up

(opposite of down)

2 + 5

Therefore, 2 (5) = 2 + 5. Make surethat when you are showing this to students(and when students do their work) that theyshow each of these steps separately. Other-wise, crossing out or changing symbols seemsarbitrary when students refer to their notes.

Finally, students can see sub-traction of integers as distance, justlike they saw subtraction of wholenumbers. Using a number line,start on the second number andmove toward the first number. Thedirection tells you the sign andthe number of spaces tells you thevalue.

5.4 Nix: Two Negatives Makea Positive (Integer Multipli-cation)

Because:

Many students will overgeneralize and mis-takenly apply this rule to addition as well asmultiplication and division.

http://mathmistakes.org/?p=328

For example, this studentseems to be thinking: Nega-tive and negative = positive,right?

Fix:

Students are able to reason about multiply-ing integers (and extrapolate to the reals) in-dependently. They can look at patterns andgeneralize from a few approaches.

One option is to use opposites:

We know that (3)(2) = 6.

So what should (3)(2) equal?The opposite of 6, of course!Therefore, (3)(2) = 6.So, what should (3)(2) equal?The opposite of 6, of course!Therefore, (3)(2) = 6.

Another option is to use patterning. Sincestudents are already familiar with the num-ber line extending in both directions, they cancontinue to skip count their way right pastzero into the negative integers.

Students can use the following pattern todetermine that (3)(2) = 6:

Product Result(3)(2) 6(3)(1) 3(3)(0) 0(3)(1) 3(3)(2) 6

With (3)(2) = 6 in hand, students canuse the following pattern to determine that(3)(2) = 6:

Product Result(3)(2) 6(2)(2) 4(1)(2) 2(0)(2) 0(1)(2) 2(2)(2) 4(3)(2) 6

5.5 Nix: Move the Decimal(Scientific Notation)

Because:

Students get the answer right half the timeand do not understand what went wrong theother half of the time. Students are thinkingabout moving the decimal place, not aboutplace value or multiplying by 10.

Fix:

Students need to ask themselves, Is this a bignumber or a small number? Scientific nota-tion gives us a compact way of writing longnumbers, so is this long because it is largeor long because it is small? An importantidea of scientific notation is that you are notchanging the value of the quantity, only itsappearance. Writing a number in scientificnotation is similar to factoring, but in thiscase we are only interested in factors of ten.

Consider 6.25 102:102 is the reciprocal of 102 so,

6.25 102 = 6.25 1102

=6.25

100= .0625

To write 0.0289 in scientific notation, weneed to have the first non-zero digit in theones place, so we need 2.89 something.Compare 2.89 with 0.0289:

2.89 10x = 0.0289= 10x = 0.0289

2.89=

1

100= x = 2.

5.6 Nix: Jailbreak Radicals, akaYou Need a Partner to Goto the Party

Trick Explanation:The radical sign is a prison.The threes pair up and try to break out. Sadly,only one of them survives the escape.

Because:

It is one thing to tell a story; it is anotherfor the story to have a mystery step whereone number escapes and another disappears!This should not be a mystery. The secondthree did not disappear. When the squareroot operation was completed, 9 became 3,because 3 is the root of 9. Things happen fora reason, math makes sense.

Fix:

Show students how to identify perfect squaresthat are factors. We want them to rewrite theproblem in a way that shows the math behindthe magic:

18 =

9 2 =

9

2 = 3

2

However, many students do not see perfectsquares quickly or easily when factoring. Thereare two options here: one arithmetic and theother geometric.

Arithmetically, prime factorization helps stu-dents to find perfect squares, but they need toknow what they are finding.

18 =

2 3 3 =

2 32 =

2

32 = 3

2

This approach reinforces that square and squareroot are inverse operations. Students knowthat

32 = 3 by the defintion of square root.

Writing a prime factorization using exponents,then evaluating the square root on any perfectsquares elucidates the process.

Geometrically, thesquare root functionis asking for the sidelength of a square withgiven area. For

18 we

are looking for the side length of a square witharea 18 square units. One way to write thatside length is

18 but there might be a sim-

pler way to write that. What if we dividedthe square into nine smaller squares? Thenthe side length of each small square would be

2 and it takes three small sides to span onefull side, thus

18 = 3

2. When students en-

gage in this process, they see the squares thatare so important to finding a square root.

5.7 Nix: Exponent Over Radi-cal

Because:

The entire fraction is an exponent, not justthe numerator.

Fix:

Describe exponents as indicating the numberof factors of the base. So 23 is three factorsof two. This more clearly describes 2 2 2than the more typical Two multiplied by it-self three times (that sounds like ((22)2)2).The bonus is that rational exponents fit thisdescription.813 means One third of a factor

of eight as opposed to Eightmultiplied by itself one thirdtimes. At first glance bothphrases may seem equally nonsensical, but thefirst holds. To find a third of something you

must break your whole into three equal parts,8 = 2 2 2, then you take one of them. Sim-ilarly, 16

34 is 3

4of a factor of 16. Break six-

teen into four equal parts and take three ofthem.

Students should still learn the connectionbetween rational exponents and radicals butthis language allows the transition betweenwhole number and fraction exponents to be asmooth one.

Chapter 6

Equations andInequalities

6.1 Nix: Hungry InequalitySymbols

Because:

Students get confused with the alligator/pacmananalogy. Is the bigger value eating the smallerone? Is it the value it already ate or the one

90

it is about to eat?

Fix:

Ideally students have enough exposure to thesesymbols that they memorize the meaning. Justas they see and write = when they hear orsay equal, students should see and write 2 and2 < x are equivalent andequally valid, yet they point in opposite di-rections. Plus, not all inequalities will begraphed on a horizontal number line!

Fix:

Students need to understand what the inequal-ity symbol means. Ask students, Are the so-lutions greater or lesser than the endpoint?This is a great time to introduce test points have students plot the endpoint, test a pointin the inequality and then shade in the appro-priate direction. While this seems like morework than knowing which direction to shade,it is a skill that applies throughout mathe-matics (including calculus!).

Since the symbolic representation of an in-equality is more abstract than a number line,having students practice going from contextsor visual representations to symbolic ones willsupport student understanding of the sym-bols. While it is true that x > 2 is the morenatural way to represent the sentence the so-lutions are greater than two, students needthe versatility of reading inequalities in bothdirections for compound inequalities. For ex-ample, 0 < x < 2 requires students to con-sider both 0 < x and x < 2.

6.6 Nix: The Square Root andthe Square Cancel

Because:

Cancel is a vague term, it invokes the im-age of terms/values/variables magically dis-appearing. The goal is to make mathematicsless about magic and more about reasoning.

Something is happening here, let students seewhat is happening. Plus, the square root isonly a function when you restrict the domain.

Fix:

Insist that students show each step instead ofcanceling operations.

(5)2 6= 5 because

(5)2 =

25 = 5

x2 = 25 ; x =

25

If you cancel the square with a square rootyou miss a solution.

Instead:x2 =

25 |x| =

25 x =

25

Another approach is to have students ploty =

x2 and y = (

x)2 on their graphing

utilities and compare results.

y =x2 y = (

x)2

6.7 Nix: Land of Gor

Because:

Besides the fact that this trick does not workwhen the variable is on the right side of theinequality, the trick is unnecessary. This non-sensical phrase skips all the reasoning and un-derstanding about absolute value inequalities.Students are capable of using the definiton ofabsolute value to figure out what type of in-equality to write.

Fix:

Go back to the definition of absolute value(Section 5.1): |x + 2| > d means x + 2 isat a distance greater than d units from zero.This distance can be both to the right of zeroand to the left of zero, so students should rec-ognize that an absolute value equation or in-equality gives two options, one positive andone negative. The students should write outeach of the inequalities separately, then use agraph or critical thinking to determine if thetwo parts can reasonably be merged into onecompound inequality.

Placing the boxedexpression above thenumber line remindsstudents that those arethe endpoints for theexpression, they stillneed to solve for thevariable.

Then listen to the language: if the inequal-ity says |x + 2| > d then we say the dis-tance is greater so we shade the area fartherfrom zero. In contrast, if the inequality says|x+ 2| < d then we say the distance is lessso we shade the area closer to zero.

Since the number line is for x+ 2, we sub-tract 2 to get the graph of x instead. This re-inforces the idea that adding/sutracting shiftsa graph. (Its also a nice place to show howdividing by a negative number flips the num-ber line, which is why we have to change from> to < or vice versa.)

6.8 Nix: Log Circle

Because:

Students taught to go around the circle do notunderstand what a log is nor what it does.

Fix:

Go back to the definition of a log. Every timestudents see a log they should remember thatloga(b) asks the question: What exponent isrequired to go from a base of a in order toreach a value of b? If students continuallyrefer back to that question they will be ablerewrite a log in exponential form, but also toknow what the function means. The log hastwo inputs the value and the base andgives an output of the exponent, or, in thelanguage of Section 5.7, the log outputs thenumber of factors.

6.9 Nix: The Log and theExponent Cancel

Because:

Cancel is a vague term, it invokes the im-age of terms/values/variables magically dis-appearing. The goal is to make mathematicsless about magic and more about reasoning.Something is happening here, let students seewhat is happening!

log2 x = 4

2log2 x = 24

x = 24

If you know the definition of a log the sec-ond line is unnecessary. There is no reason tobe cancelling anything, the simplest approachis to rewrite the equation using knowledge oflogs.

If students are not thinking about the def-inition of a log, they will try to solve the newfunction in the ways they are familiar with.

Student divides bythe function:

http://mathmistakes.org/?p=95

This student dividesby the base:

http://mathmistakes.org/?p=67

Fix:

Rewrite the log in exponent form, then solveusing familiar methods. This is another rea-son to teach the definition of logarithms; thenthere is no confusion about the relationshipbetween logs and exponents.

The log asks the question base to whatpower equals value?

logbase value = power

basepower = value

log2 8 = y

2y = 8

y = 3

log2 x = 4

24 = x

16 = x

Chapter 7

Functions

7.1 Nix: Rise over Run as theDefinition of Slope

Because:

Slope is a very important con-cept and it needs a robust definition. Riseover run can describe how to find slope in agraph, but slope has a deeper meaning thancounting boxes. Additionally, some students

108

think rise means you are always going up,which causes problems.

Fix:

Slope is the rate of change. Students can findthat rate by comparing the change in y (or y-distance) to the change in x (or x-distance).When students are computing slope they arefinding the rate of something per somethingelse. Starting with a context (ex: rate of stu-dents per hour) is helpful to provide a foun-dation for understanding, but this phrasingholds in all cases, even if it means saying rateof y per each x.

Begin by analyzing slope as how much ychanges each time that x changes by one, butdo not leave that as the only option. Theslope in y = mx + b tells students for anychange in x, y changes by m times as much.For example, If I change x by one, y changesby m. is true, but the statement If I changex by 7

3, y changes by m 7

3 is equally true.

The benefit of this definition is that stu-dents can recognize slope from a table (evenif the x-difference is not one), a word problemand a graph.

7.2 Nix: OK vs. NO Slope

Because:

What is the difference between noslope and zero slope? Students frequentlyconfuse no solution with the solution is zeroand this trick does not help students to tellthe difference.

Fix:

Slope can be negative, positive, zero or unde-fined. If students struggle to remember thatdividing by zero is undefined, have them goback to multiplication and fact families. Weknow 10

2= 5 because 10 = 5 2. Now try that

with zero in the denominator. 100

= a means

10 = a 0. There are no values of a to makethat true. On the other hand, 0

10= b means

0 = b 10. In this case there is a solution,b = 0. Once students have seen why dividingby zero is undefined, do a google image searchfor divide by zero and enjoy a laugh (youmight want to turn on safe search, there aresome with inappropriate language). If you orthe students are interested in further explo-ration, try some values getting close to zero.

10

1= 10

10

.1= 100

10

.01= 1000

Imagine what would happen as the denomi-nator gets closer and closer to zero. What isthe limit?

7.3 Nix: What is b?

Because:

The answer to this question is, b is a letter.It does not have any inherent meaning; in-stead ask students exactly what you are look-ing for, which is probably either the y-interceptor a point to use when graphing the line.

Fix:

There are many equations of a line, y = mx+bis just one of them. In fact, it is not neces-sarily written that way depending on whatcountry or system you are in. Given the goalof graphing, feel free to ask what the inter-cept of the line is, but it is much easier to askfor any point. Have students pick a value forx (eventually they will gravitate toward zeroanyway since it is easy to multiply by!) andthen solve for y. This method works for anyequation and it shows why we put the value of

b on the y-axis. Remind students that, whileb is a number, we are concerned with (0, b)which is a point.

7.4 Nix: The Inside Does theOpposite

Because:

You are telling studentsthat the truth is the opposite of their intu-ition. This assumes you know their intuition,and it certainly does not help with sense mak-ing to say this makes the opposite of sense!Remember, math makes sense.

Fix:

Show students what transformations are hap-pening by showing algebraically as well asgraphically how to transform the parent func-tion into the equation. Any time we get touse change of variable you know students are

practicing seeing structure in expressions (animportant Common Core Strand).

y = (x 5)2 7y + 7 = (x 5)2

O = M2

Now students can take the values of the par-ent function and transform them accordinglyto graph the original equation.

M O-2 4-1 10 01 12 4

x M O y3 -2 4 -34 -1 1 -65 0 0 -76 1 1 -67 2 4 -3

M = x 5x = M + 5 Shift right 5.

O = y + 7

y = O 7 Shift down 7.

This is a convenient way to sketch a graphsince if students know the important pointsin a parent function (vertex, maximum, min-imum), they will get those same importantpoints on their graph! It works for shifts andstretches as well as any type of functions (in-cluding trig, which is awesome since studentsstruggle to find x values that they even knowthe sine of when making a table).

When I first taught Algebra 1, I had stu-dents graph quadratics by making a table andused the integers between 5 and 5 for x ev-ery time. This worked fine for the functionsI gave them, but when they got to Algebra2 their teacher reported back to me that notall graphs had a vertex with 5 x 5and my former students were stuck on whatto do when their graph did not look like theparabola they expected to see. Sorry first yearstudents, if only I had that PD day with theEDC sooner!

7.5 Nix: All Students TakeCalculus

Because:

Students do not need a rule to figure out whichtrig functions are positive in each quadrant.Why add an extra layer of memorization? Es-pecially when it is not even true that all stu-dents take calculus!

Fix:

Think. If a student is studying trigonometrythey know that right is positive and left is neg-ative on the x-axis. Similarly, they know thatup is positive and down is negative on the y-axis. Finally, students know how to divide in-tegers (or they will once you share Section 5.4with them). These three simple facts are allit takes to identify if a function is positive ornegative in that quadrant, and most studentsfind all three of those automatic by this point

in their mathematical careers. But, more im-portantly, we should not be askng students toonly find the sign of a trig function in a quad-rant. Instead ask them to find the value of afunction. Using the coordinates from the unitcircle for the given angle eliminates the needfor any memorization.

7.6 Nix: FOIL

Because:

It implies an order a few of my honors pre-calculus students were shocked to learn thatOLIF works just as well as FOIL does. It isalso a one trick pony. There are other waysto multiply binomials that are transferrableto later work such as multiplying larger poly-nomials and factoring by grouping.

When students memorize a rule withoutunderstanding they misapply it.

http://mathmistakes.org/?p=1180

http://mathmistakes.org/?p=1100

Fix:

Replace FOIL with the distributive property.It can be taught as soon as distribution is in-troduced. Students can start by distributingone binomial to each part of the other bino-mial. Then distribution is repeated on eachmonomial being multiplied by a binomial. Asstudents repeat the procedure they will real-ize that each term in the first polynomial mustbe multiplied by each term in the second poly-nomial. This pattern, which you might termeach by each carries through the more ad-vanced versions of this exercise.

In elementary school students learn an ar-ray model for multiplying numbers. The box

method builds on this knowledge of partialproducts.

23 45 =(20 + 3)(40 + 5)=20(40 + 5) + 3(40 + 5)

=20 40 + 20 5 + 3 40 + 3 5=800 + 100 + 120 + 15

40 520 800 1003 120 15

(2x+ 3)(x 4)=(2x+ 3)(x) + (2x+ 3)(4)=2x2 + 3x 8x 12=2x2 5x 12

2x 3x 2x2 3x4 8x 12

7.7 Nix: Slide and Divide, akaThrow the Football

Because:

Students cannot see thebig picture when they fol-low this process. Themiddle steps are notequivalent to the original expression so stu-dents cannot switch to an alternate methodpartway through. Either students are led tobelieve that 2x2 5x 12 = x2 5x 24(which is false) or they know it is not equaland they are forced to blindly trust that thefalse equivalence will somehow result in a cor-rect answer. Do not ask for students faith,show them that math makes sense.

Fix:

This trick is not so far off from a great way tofactor using chunking. Consider the trinomial

4x2 10x 24. The first term is a perfectsquare, so I could use some substitution wherea = 2x and get 4x210x24 = a25a24.This is a monic expression so it is easier tofactor. Then I can substitute a = 2x back into get the solution to my original expression.

4x2 10x 24 = a2 5a 24= (a 8)(a+ 3)= (2x 8)(2x+ 3)

You may be thinking, this is all well and good,but what do I do when the first term is nota perfect square? Thats what this trick isdoing without showing you the structure it is making the first term a perfect square foryou. If we want to factor 2x2 5x 12, thefirst term needs another factor of two. Now,this will change the value of the expression,but it is a temporary change and we are goingto make it obvious that we have changed it,

no blind trust:

2(2x2 5x 12) = 4x2 10x 24= a2 5a 24, where a = 2x= (a 8)(a+ 3)= (2x 8)(2x+ 3)= 2(x 4)(2x+ 3)

So we know that2(2x2 5x 12) = 2(x 4)(2x+ 3),thus 2x2 5x 12 = (x 4)(2x + 3). Wefactored that trinomial! (Remember - thismethod works because the problem was riggedto be factorable. If the original trinomial isnot factorable things will not fall out so nicely.)

So, now that we know why this trick works,is it worth using? I think this method is a niceway to have students practice seeing structurein expressions (an important Common CoreStrand), if they start with monics, move toquadratics with perfect squares and finally seethis process. Many teachers were raised onguess and check factoring, and were pretty

good at it but many students struggle withthat method. The box method (equivalentto factoring by parts) can help students toorganize their information.

1 xx 2x2 x

x 12

2 2x1x 2x2 x

x 121 Fill in the box just like for multiplying

polynomials Section 7.6

2 The only factors of 2 are 2 and 1, so fillin those boxes.

3 2x 21x 2x2 x6 x 12

4 2x 21x 2x2 2x6 12x 12

3 Try 2 and 6 as the factors of 12.4 The coefficient of x was supposed to be

5 but 2x 12x = 10x

5 2x 31x 2x2 x4 x 12

6 2x 31x 2x2 3x4 8x 12

5 Try again, this time using 3 and 4 asthe factors of 12.

6 Success! 3x 8x = 5x The factoriza-tion is (2x+ 3)(1x 4).

There is something to be said for practicingfactoring integers but it will depend on thestudent and the goals. Completing the squareand the quadratic formula are lovely methodsthat work for all quadratics. The methods asudent learns will depend on their level. Itis better to have a variety of options whenteaching, but that does not mean that everystudent needs to learn every option.

7.8 Nix: Synthetic Division

Because:

Students do not knowwhen synthetic division works or why itworks. The goal is to divide quickly, but usinga graphing utility or computer algebra systemis even faster! It is more important for stu-dents to understand the processs and how tointerpret an answer. Once students achievethis, then the algorithm can be carried out bya computer. Besides, the way synthetic divi-sion is usually taught only works for the lim-itied case of a polynomial divided by (x a)(it is possible to modify the process for anypolynomial but its rare to see that approachin school).

Fix:

Use polynomial long division exclusively forshowing students how to solve by hand. Stu-

dents can make the connection between poly-nomials and real numbers to see that the pro-cess is equivalent. Once students understandthe process and why it works, use technologyfor everything else. Cool fact the long divi-sion process highlighted in Section 2.6 worksjust as well for polynomials as it does for num-bers!

Chapter 8

Conclusion

Okay, Ive read the book and I want toNix the Tricks. Now what?

Start slowly. High school teacher JonathanClaydon, who blogs at InfiniteSums.com, rec-ommends picking just two tricks this year. Goback through the book, decide which trickscome up the most often in your class or whichconcepts will provide the best foundation foryour students as they continue learning math-

127

ematics. Then make a promise to try yourvery best to never use those phrases. Onetrick Im still working on nixing is Cancel (Sec-tion 6.4). We solve equations using oppo-site operations all the time in my precalculusclass and both my language and my studentslanguage defaults to cancel. Sometimes Icatch myself before the word comes out of mymouth and I congratulate myself, then say theprecise operation. Other times I say cancel,silently curse myself, then silently forgive my-self and finally follow the imprecise languagewith a precise statment. Most of the time stu-dents tell me to cancel something and I askthem how or narrate my thought process as Irecord the opposite operations. It takes manyrepetitions to undo a habit that has been in-grained in students for a long time, and in mefor at least twice as long!

What about the kids who already knowa trick?

Step one, hand a copy of this book to the

teacher, tutor or parent who taught it to them.(I am only sort of kidding.) Step two, engagethe students in some sense-making. Can theyexplain the math behind the trick? If so, itreally only counts as a shortcut, not a trick,since they understand the math. Even so, itmay trip them up later, so ask about somesituations where it applies and does not ap-ply. Push students to see if it is the most ef-ficient method in a variety of situations. Anyquestion you can ask to get students thinkingrather than blindly obeying a rule is a greatidea.

Middle school math teacherDavid Cox, who blogs at Cox-Math.blogspot.com had a bril-liant response when his studentstold him an assignment was easysince they just use the DRT tri-angle (Section 3.10). First, heasked them what it was and howit worked. Then, he told themthat he had learned the TRD

triangle. The students responded with con-flicting statements.

S1: No, that wont work. Thats not whathe told us.

S2 : He said it didnt matter how we wroteit.

Mr. Cox: So which is it; does one workor are they the same? Make your case and beready to defend it.

To decide if the new triangle worked (andthe old one for that matter!) the studentstried some examples that they understood with-out needing the triangle. They generated theirown problems, drew conclusions and formedan argument. Any day where students usesomething they understand to reason aboutsomething new is a great day for mathemat-ics. Find the complete story on Davids blog:http://coxmath.blogspot.com/2014/04/dirty-triangles.html

Will my middle schoolers/high school-ers/college students really buy into thisafter learning math with tricks for solong?

Yes! Some of my high school students arestill curious enough to ask me why a methodthat they were told to take on faith works.Other students exclaim, Ooohh! Thats whywe do that. when I show them something ina new way. I truly believe that every stu-dent would rather understand than memorizea list of steps, but some students are resis-tant because they fear that they are not smartenough to do math.

Dont take my word for it though, otherpeople are seeing success using Nix the Tricks:

Justin AionMiddle School Teacherhttp://relearningtoteach.blogspot.com/2014/04/day-143-pointed-questions-and-jelly.html

Day 120:I think I have successfully cured mypre-algebra students of using cross-multiply. When we started propor-tions, I heard it with every otherproblem. Now, I dont think Iveheard it in over a week. What Imhearing instead is dont we multi-ply 15 to both sides?

Day 143: After we completed the8 word problems, I asked a simplequestion:Me: What did I not see in any ofthese problems? What tactic wasnot used?

S: Cross-multiplication.Me: Did anyone even think to useit?S: **silence**

YES!!! One trick nixed!

Jonathan ClaydonHigh School Teacherhttp://infinitesums.com/commentary/2014/3/23/nixing-tricks

I set two goals: never say canceland never say cross multiply. Tryas hard as possible to prevent thestudents from saying those words ei-ther. If I force them to be properabout it early the growing pains willbe worth it.

Every time we needed to clear a de-nominator I showed it as an opera-tion of multiplying both sides by therelevant terms. Every time we tooka square root I mentioned the pres-

ence of two solutions. Forcing my-self to rely on the mathematicallyvalid reasoning slowly influenced thestudents. They raised the level atwhich they explained concepts to eachother. I didnt hear anyone lookingfor an easy way out. They wantedto do it my way.

This process became cyclical. As Isaw my students rise to the chal-lenge, I felt an obligation to themand the professional community tocontinue to focus on valid, universalmathematical explanations for things.

Appendix A

Index of Tricks byStandards

Sorted by the Common Core State Standards.

135

1.OA.A.2 Total Means AddSection 2.1

2.NBT.B.9 Bigger Bottom, Better BorrowSection 2.2

3.MD.D.8 Perimeter is the OutsideSection 4.1

3.G.A.1 Rectangles Have Two LongSides and Two Short SidesSection 4.2

3.G.A.1 Squares Have Four Equal SidesSection 4.3

3.OA.B.5 Add a Zero (Multiplying by 10)Section 2.3

3.NBT.A.1 Five or More Go up a Floor,aka 0-4 Hit the Floor, 5-9 Make the ClimbSection 2.4

4.G.A.1 Obtuse Angles are BigSection 4.4

4.MD.A.3 Area of Quadrilaterals, TrianglesSection 4.7

4.NBT.A.2 Hungry Inequality SymbolsSection 6.1

4.NBT.B.5 Turtle MultiplicationSection 2.5

4.NBT.B.6 Does McDonalds Sell Cheeseburgers,aka Dad, Mom, Sister, BrotherSection 2.6

5.OA.A.1 PEMDAS, BIDMASSection 2.8

5.NBT.B.7 Ball to the WallSection 2.7

5.NF.A.1 Butterfly Method, Jesus FishSection 3.1

5.NF.B.3 The Man on the HorseSection 3.2

5.NF.B.3 Make Mixed Numbers MADSection 3.3

5.NF.B.3 Backflip and CartwheelSection 3.4

6.EE.B.7 Take/Move to the Other SideSection 6.2

6.EE.B.7 Switch the Side,Switch the SignSection 6.3

6.NS.A.1 Cross Multiply(Fraction Division)Section 3.5

6.NS.A.1