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Manitoba Education and Training Cataloguing in Publication DataMental math : grade 12 essential mathematicsISBN: 978-0-7711-8029-3 (print)ISBN: 978-0-7711-8030-9 (pdf)1. Mathematics—Study and teaching (Secondary)—Manitoba.2. Mental arithmetic—Study and teaching (Secondary)—Manitoba.I. Manitoba. Manitoba Education and Training.510.712Copyright 2017, the Government of Manitoba, represented by the Ministerof Education and Training.Manitoba Education and TrainingWinnipeg, Manitoba, CanadaEvery effort has been made to acknowledge original sources and to complywith copyright law. If cases are identified where this has not been done, pleasenotify Manitoba Education and Training. Errors or omissions will be correctedin a future edition. Sincere thanks to the authors, artists, and publishers whoallowed their original material to be used.All images found in this resource are copyright protected and should notbe extracted, accessed, or reproduced for any purpose other than for theirintended educational use in this resource.Schools are encouraged to share this resource with parents, guardians, andcommunities, as appropriate.Any websites referenced in this resource are subject to change without notice.Educators are advised to preview and evaluate websites and online resourcesbefore recommending them for student use.Print copies of this resource (stock number 80738) can be purchasedfrom the Manitoba Learning Resource Centre. Order online atwww.manitobalrc.ca.This resource is also available on the Manitoba Education and Trainingwebsite at nible en français.While the department is committed to making its publications as accessible aspossible, some parts of this document are not fully accessible at this time.Available in alternate formats upon request.
ContentsAcknowledgementsvIntroduction1Why Mental Mathematics3Strategies4Document Features4Methodology6Assessment7Reproducible Sheets1Examples of Strategies1Mental Math Questions by Unit1Unit A: Home FinanceUnit B: Geometry and TrigonometryUnit C: Business FinanceUnit D: ProbabilityUnit E: Vehicle FinanceUnit F: StatisticsUnit G: Precision MeasurementUnit H: Review (All Units)Contentsiii
AcknowledgementsManitoba Education and Training gratefully acknowledges the contributions of the followingindividuals in the development of Mental Math: Grade 12 Essential Mathematics.WriterManitoba Educationand TrainingMichelle LevesqueIndependent ConsultantWinnipegLouise BoissonneaultCoordinatorDocument Production Services UnitEducational Resources BranchWenda DickensCoordinatorDevelopment UnitInstruction, Curriculum and Assessment BranchIan DonnellyProject LeaderDevelopment UnitInstruction, Curriculum and Assessment BranchLynn HarrisonDesktop PublisherDocument Production Services UnitEducational Resources BranchGrant MoorePublications EditorDocument Production Services UnitEducational Resources BranchAcknowledgementsv
Introduction
IntroductionMental Math: Grade 12 Essential Mathematics is a complement to the Grade 12Essential Mathematics curriculum and is intended to help students developstrategies that allow them to perform mental calculations.Why Mental Mathematics?Mental mathematics and estimation is one of the seven processes of themathematics curriculum.Mental mathematics and estimation is a combination of cognitive strategiesthat enhance flexible thinking and number sense. It involves using strategiesto perform mental calculations.Mental mathematics enables students to determine answers without paperand pencil. It improves computational fluency by developing efficiency,accuracy, and flexibility in reasoning and calculating.*It is used by individuals in their daily lives at home and at work. Mentalcalculation requires students to call on their knowledge of numbers andmathematical operations. It not only calls on memory but helps improve it aswell.Mental calculation is at the root of the estimation process. It allows us todetermine whether results obtained with a calculator are reasonable.Estimation is used for determining approximate values or quantities,usually by referring to benchmarks or referents, or for determining thereasonableness of calculated values. Estimation is also used to makemathematical judgments and to develop useful, efficient strategies for dealingwith situations in daily life.*Mental calculation is an important way of developing number sense andacquiring a better understanding of place value and mathematical operations.Students who have experience with mental math develop the ability to workwith numbers. Mental calculation can be used to prepare for written workby providing an approximate answer to a problem. Using certain mentalcalculation strategies can eliminate some steps in written calculations andhelp simplify the process. In short, mental calculation skills are at the heart ofnumeracy.* Manitoba Education and Advanced Learning. Grades 9 to 12 Mathematics: Manitoba Curriculum Framework ofOutcomes, 2014 (Winnipeg, MB: Manitoba Education and Advanced Learning, 2014) 9.Introduction3
Mental calculation is used almost daily in life. We often have to do quickmental calculations at times when we do not have paper, a pencil, or acalculator handy. Mental calculation is therefore a very practical skill.Teachers should provide opportunities for their students to use mental mathand estimation on a daily basis. They should encourage their students to findexamples of the usefulness of mental calculation in their lives, such as whenshopping, doing home renovations, estimating mileage, or working at theirjobs.StrategiesTeachers should promote a variety of mental mathematics strategies. Theyare encouraged to make students aware of the strategies described in theStrategies section of this document. The strategies that are most effectivefor mental calculation are often not the same strategies that are mosteffective for written calculation. Most students are able to develop strongmental calculation techniques, but often need help in doing so. Studentsmay discover and use some mental calculation techniques by themselvesbut need to be taught other techniques to increase their mental calculationeffectiveness. Regular mental calculation activities should be included in allmathematics curricula at all grade levels.Document FeaturesThe document includes three main sections: this introduction, a sectiondescribing strategies, and a series of mental mathematics questions organizedby units.The teacher will find mental mathematics questions relating to a specificsubstrand of Grade 12 Essential Mathematics as written in Grades 9 to 12Mathematics: Manitoba Curriculum Framework of Outcomes. The Mental Math:Grade 12 Essential Mathematics document consists of seven (7) units related tospecific substrands and one unit with a review of all substrands. Note thatthere are no mental math questions developed for the Career Life substrand.Unit4Number of PagesA: Home Finance9B: Geometry and Trigonometry6C: Business Finance3D: Probability4E: Vehicle Finance6F: Statistics5G: Precision Measurement5H: Review (All Units)14Mental Math: Grade 12 Essential Mathematics
The units may be taught in a different order as determined by the teacher.Every unit consists of several pages of mental math questions.The unit of study is identified on each page. The questions on each page aredivided into three different categories: five (5) general or review questions;three (3) questions related to the unit of study; and two (2) blanks for teachersto insert their own questions.The answers to the questions are provided in the column on the extremeright-hand side of each page. Sometimes students are asked to provide theone right answer and at other times they are to provide an estimate where arange of values would be correct.Teachers may want to prepare additional questions to better meet the needs oftheir students. A section at the bottom of each page entitled Other Questionshas been set aside for this purpose. A blank template is also provided in asection titled Reproducible Sheets. Teachers may use it to prepare additionalquestion sheets.A file in Word format is available in the Mathematics Group on theMaple (Manitoba Professional Learning Environment) site atwww.mapleforem.ca. The Word file does not contain the Strategies section ofthis document but it does contain the section with the Mental Math Questionsby Unit. It is provided to enable teachers to add or modify questions to suitthe needs of their students.Immediately following this introduction is a section describing mentalcalculation strategies along with examples. Some students may already havean inventory of strategies that they can apply; others may not. Teachers canuse the strategies information given in this document to help students expandtheir strategy knowledge.Introduction5
MethodologyGiven their usefulness, mental calculation exercises should be short and donefrequently.They should be short because they require sustained concentration. Forexample, the first five minutes or so of math class could be devoted to mentalcalculation exercises. This practice would also serve as a warm-up to theday’s lesson. Alternatively, the mental math practice could be done at theend of a class as a wrap-up to the day’s lesson. In addition, although mentalcalculation should be done within a certain period of time, it is preferablenot to emphasize speed. Although speed is a factor, it is not a primary goal.It is obviously important to ensure that time spent on mental mathematicsactivities does not infringe on the time needed for instruction and otherlearning activities.Establishing routines in the classroom also encourages students to get to workquickly at the beginning of each class or to continue to work through to theend of each class. Teachers could establish a process such as the following:QQQQQQQQQQAs soon as students arrive at the beginning of the class (or when instructednear the end of the class), they can take out a sheet of paper and write downthe numbers 1 to 10 if there are 10 questions.The teacher can project a mental math page, present questions orally, ordistribute a page with written questions.The students are given time to answer the questions.If students are unable to find an answer to a specific question, they couldleave an empty space on that question and go on to the next question. Thegoal is accuracy and the development of a bank of effective strategies.The teacher should, on occasion, spend time discussing various strategiesused by the students for one or more of the questions.To ensure students gain confidence with a new strategy, it is important thatthey are given adequate opportunity to practise it. It is up to the teacher toprovide an adequate number of exercises or problems to ensure that studentsare able to use the new strategies.6Mental Math: Grade 12 Essential Mathematics
AssessmentPrimarily, mental calculation exercises are used as assessment for learning.Mental calculation exercises should be done in a classroom environment inwhich students feel comfortable taking risks without being penalized whenthey make mistakes. However, teachers should ask students to do a selfevaluation by identifying the questions they had the most difficulty withor those they did not answer correctly. Periodically, teachers may choose touse the mental mathematics questions as assessment of learning by askingstudents to explain the strategy they used for a specific question or questions.Mental calculation can allow students an opportunity to develop a betterunderstanding of some mathematical concepts. Consequently, mentalcalculation activities should include periods for thought and discussion.During these periods, the teacher should encourage students toQQQQQQsuggest a variety of possible solutions to the same problemexplain the different methods used to come to the correct answer and theireffectivenessexplain the thought process that led to an incorrect answerThis type of discussion is very important in learning mental calculationstrategies, because it is an effective way for students to present their thinking.Questioning, reflecting, and discussing, which are integral to the activitiesof mental calculation, are excellent ways of communicating mathematicalideas. This communication requires that students be clear and concise whenexplaining their thinking to others. It is often when students describe thestrategy they used to solve questions that other students discover a newtechnique. These exchanges about the strategies as well as the results willallow the teacher to identify the difficulties encountered by some students.Subsequently, the teacher can help students discover new, relevant, useful,and important strategies.Enjoy the mental mathematics experience!Introduction7
Notes8Mental Math: Grade 12 Essential Mathematics
ReproducibleSheets
Mental MathGrade 12 Essential Mathematics (40S)Unit:General Questions1.2.3.4.5.Unit Questions6.7.8.Other Questions9.10.Answers
Mental MathGrade 12 Essential Mathematics (40S)Unit:General or Unit Questions1.2.3.4.5.6.7.8.9.10.Other Questions11.12.Answers
Examples ofStrategies
Mental MathGrade 12 Essential Mathematics (40S)Sample StrategiesS–1Begin adding from the leftWhen you do additions using paperand pencil, you usually start from theright and work toward the left.To do additions in your head, start from the left.AA1246 3840 30 706 8 1470 14 8420 10 3025.6 13.75 3 8330 8 110 39.36 7310 10 1 and 10
Mental MathGrade 12 Essential Mathematics (40S)Sample StrategiesBreak down numbers and add their partsHere’s another way of doingadditions in your head.63 28Break down the numbers, then add their parts.AA163 2863 20 883 891315 200 70 62315 276515 70 6585 6591S–2
Mental MathGrade 12 Essential Mathematics (40S)S–3Sample StrategiesFinding compatible numbersCompatible numbers are pairs of numbers that are easy to add inyour head.The following are examples of compatible numbers:8614220The sum equals 100.AA12380The sum equals 600.Find the pairs of compatible numbers that addup to 300.140851601182177321518283140 160118 182215 85217 83Find the pairs of compatible numbers that addup to 800.250175567333440467625550360250 550333 467625 175440 360
Mental MathGrade 12 Essential Mathematics (40S)Sample StrategiesS–4Create your own compatible numbersSometimes it is easier to do additions in your head by creatingyour own compatible numbers and adjusting the total.A1650 375650 350 251000 251025A1250 750 321250 7532000 32003
Mental MathGrade 12 Essential Mathematics (40S)Sample StrategiesS–5Subtract starting from the leftHere’s a technique that works wellwhen doing subtractions that do notrequire grouping.To do subtractions in your head, start from the left andthink of your answer one part at a time.A1400 300 100468– 32360 20 408 3 5100 40 5 145A29000 6000 3000500 200 3009514– 620314 3 113000 300 11 3311
Mental MathGrade 12 Essential Mathematics (40S)Sample StrategiesS–6Subtract one part at a timeWhen you do a subtraction that requires a grouping,subtract one part at a time.A1132– 59132 – 50 8282 – 9 73Check your answer by adding the following in your head:73 59 120 12 132A6.25 – 3 3.2526.25– 3.153.25 – 0.15 3.10Don’t forget to check your answer doing amental addition.
Mental MathGrade 12 Essential Mathematics (40S)Sample StrategiesS–7Balance a subtraction with whole numbersWhen you add the same number to the two elements of asubtraction, the difference between the two does not change.By adding to both elements, you balance the subtraction.That makes it easier to find the answer in your head.AA1276– 28660– 18576 2 7828 2 3078 – 30 48660 15 675185 15 200675 – 200 475
Mental MathGrade 12 Essential Mathematics (40S)Sample StrategiesS–8Balance a subtraction with decimal numbersWhen you add the same number tothe two elements of a subtraction,the difference between the twodoes not change.Adding to both elements balances the subtraction.That makes it easier to find the answer in your head.AA124.32– 1.9523.62– 15.894.32 0.05 4.371.95 0.05 24.37 – 2 2.3723.62 0.11 23.7315.89 0.11 1623.73 – 16 7.73Remember that you have to make sure the second element (notthe first) becomes a number that is easy to subtract.
Mental MathGrade 12 Essential Mathematics (40S)S–9Sample StrategiesMultiply starting from the leftIt is easier to multiply in your head when you breakdown a number and multiply starting from the left.Add in your head as you multiply each part.A1600 4 2400635 430 4 1205 4 202400 120 20 2540A500 3 15002528 320 3 608 3 241500 60 24 1584
Mental MathGrade 12 Essential Mathematics (40S)S–10Sample StrategiesCut and paste the zerosIn multiplication, whenone factor is multiplied by10, the result is alsomultiplied by 10.6 42460 424010Knowing this concept, you can easily multiplyby 10 in your head by following these steps:1. Cut all the zeros at the end.2. Multiply the remaining numbers.3. Paste all the zeros back.AA1213 706000 120013 7 919106 12 727,200,000
Mental MathGrade 12 Essential Mathematics (40S)Sample StrategiesCut and paste the zerosTo mentally divide numbers that endin zero, follow these steps:1. Cut all the zeros at the end.2. Do the division.3. Paste the zeros back.A12400 624 6 4400Check the answer by multiplying: 6 400 2400A245,000 15Check: 15 3000 45,00045 15 33000S–11
Mental MathGrade 12 Essential Mathematics (40S)Sample StrategiesS–12Cut the zeros in dividend and divisorWhen dividing the dividend and divisorin a division by the same amount, thequotient does not change.800 2080 240Knowing this concept will help you do the divisionin your head more easily when the dividend andthe divisor both end in zero.All you have to do is divide both the dividend anddivisor by the same value, 10.AA126300 904,500,000 500630 97045,000 59000
Mental MathGrade 12 Essential Mathematics (40S)Sample StrategiesS–13Work with pricesThe sale price of items is often alittle less than an even number ofdollars.To work with prices in your head, round off to thenearest dollar. Then, do the calculation required bythe problem and adjust your answer.A1 16.65 2.99 16.65 3 19.65 19.65 – 1 19.64A6 20 1202 19.98 66 2 12 120 – 12 119.88
Mental MathGrade 12 Essential Mathematics (40S)S–14Sample StrategiesCheck your changeWhen you buy something, it isimportant to check that the amount ofchange returned to you is correct.There is an easier way than subtracting in your head:add to the purchase price.A1You buy a CD for 14.35 with a 20 bill. How muchchange should you get back?Add starting from 14.3565 14.35 65 15.00A2 5 15.00 5 20.00 5 65 5.65You buy a watch for 74.15 with a 100 bill. Howmuch change should you get back?Add starting from 74.15 99.50 50 100.00 94.15 5.00 99.15 74.15 20.00 94.15 20 5 99.15 35 99.5035 50 20 5 35 50 25.85
Mental MathGrade 12 Essential Mathematics (40S)Sample StrategiesS–15Find the time differenceMental math calculation is useful tofind how much time is left beforean event.To find the difference between two given times,add by steps.A1If it is 8:27 a.m., how long do you have to wait beforelunch at noon?TO 12:00 noon3 HOURS8:27 a.m. to 8:30 a.m.3 MINUTESTO 9:00 a.m.30 MINUTESA23 HOURS33 MINUTESIf it is 9:50 a.m., how much time is there before 8:15 p.m.?TO 8:15 p.m.15 MINUTES9:50 a.m. to 10:00 a.m.10 MINUTESTO 8:00 p.m.10 HOURS10 HOURS25 MINUTES
Mental MathGrade 12 Essential Mathematics (40S)Sample StrategiesS–16Change quarter fractions to a decima
Manitoba Education and Training Cataloguing in Publication Data Mental math : grade 12 essential mathematics ISBN: 978-0-7711-8029-3 (print) ISBN: 978-0-7711-8030-9 (pdf) 1. Mathematics—Study and teaching (Secondary)—Manitoba. 2. Mental arithmetic—Study and teaching (Secondary)—Manitoba. I. Manitoba. Manitoba Education and Training. 510.712
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