Enriching Math Using Chess

Vector �

The Official Journal of the British Columbia

Association of Mathematics Teachers

Vector

Summer 2006�

Vector is published by the BC Association of Mathematics Teachers.

Articles and Letters to the Editors should be sent to:

David Tambellini, Vector Editor John Kamimura, Vector EditorBox 445 Kwantlen Park Secondary SchoolChristina Lake, BC 10441 132nd StreetV0H 1E0 Surrey, BC V3T [email protected] [email protected]

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Summer �006 • Volume 47 • Number �

Vector �

Inside this issue…

Vector

Official Journal of the BC Association of Mathematics Teachers

4 BCAMTExecutivefor2006-2007

6 Summer2006Puzzles

7 LetterfromthePresidentoftheBCAMT9 LettertotheEditors

11 MultipleIntelligencesintheClassroom:ReachingAllLearners

15 GradeK/1–NumberConcepts

25 Let’sLearnTogether

31 ReducingFractionsanditsApplicationstoRationalExpressions

36 ACoupleofLittleSongstoShare

40 Summer2006WebSites

43 EnrichingMathUsingChess

59 Calculus12:TheUltimatePre-calculuscourse

74 SolutionstotheSpring2006Puzzles

RobertSidley

DarlenePolishak

CandyCheuk

AndreaEaster

DuncanE.McDougall

FrankHo

Summer 2006 Volume 47 Issue Number 2

JarrettSpannier

VeselinJungicKarlKraemer

Summer 20064

Your 2006 - 2007Elementary School Representatives

Janice NovakowskiSessional Instructor (UBC - Curriculum Studies)School: (604) 822-5056Fax: (604) 822-4714E-mail: [email protected]

Selina Millar(see above)

BCAMT President & Newsletter Editor

Marc GarneauMathematics Helping Teacher (Surrey)School: (604) 592-4220Fax: (604) 590-2588 E-mail: [email protected]

Past President

Selina MillarScience Helping Teacher (Surrey)School: (604) 592-4322Fax: (604) 590-2588Email: [email protected]

Vice-President

Robert SidleyH. J. Cambie Secondary School (Richmond)School: (604) 668-6430Fax: (604) 668-6132BCAMT Hotline: (877) 888-MATHE-mail: [email protected]

Secretary and TreasurerCarole SaundryDistrict Curriculum CoordinatorSchool District 38 (Richmond)School: (604) 668-6079Fax: (604) 668-6191E-mail: [email protected]

Membership

Dave EllisEric Hamber Secondary School (Vancouver)School: (604) 713-8927Fax: (604) 713-8926E-mail: [email protected]

Middle School Representative

Andrea HelmerNumeracy and Gifted Support Teacher(S.D. #33)School: (604) 824-0702Fax: (604) 824-0711E-mail: [email protected]

Secondary School Representatives

Robin DeleurmeFraser Lake Elementary-Secondary SchoolSchool: (250) 699-6233Fax: (250) 699-7753E-mail: [email protected]

Regional Representatives

Chris BeckerPrincess Margaret Secondary School, PentictonSchool: (250) 770-7620Fax: (250) 492-7649E-mail: [email protected]

David Sufrin (Vancouver Island)Ballenas Secondary School (Parksville)Tel: (250) 248-5721Fax: (250) 954-1531E-mail: [email protected]

Vector �

BCAMT ExecutiveMinistry of Education RepresentativeRichard DeMerchantMinistry of Education Office: (250) 387-4416Fax: (250) 356-2316E-mail: [email protected]

Irene PercivalSimon Fraser UniversityE-mail: [email protected]

Justin GrayMathematics DepartmentSimon Fraser UniversityE-mail: [email protected]

Post-Secondary Representatives

Marc Garneau(see above)

NCTM Representative

Webmaster

John PusicLangley Instructional Services (SD#35)Tel: (604) 530-2711Fax: (604) 530-2906E-mail: [email protected]

Listserv

Brad EppNorKam Secondary School (Kamloops)School: (250) 376-1272Fax: (250) 376-3142E-mail: [email protected]

Canadian NCTM Regional Representative (Zone 2)

Carol Matsumoto

Email: [email protected]

Private Schools Representative

Position is presently vacant.

Summer 20066

Summer 2006 Puzzles

Please mail or e-mail your completed answer(s) to:

David Tambellini, Vector Editor Box 445Christina Lake, BC V0H [email protected]

Puzzle 1

Puzzle 2

For each correct solution to these problems that you send in, your name will be entered in a draw to win a BCAMT designer T-shirt.

Find all integer solutions of

y x2 3 432= −

If porkchop

c=

and c is greater than 2, what different numbers do pork and chop represent? None of he letters stands for zero.

Vector 7

Letter from the President of the BCAMT

Vector Feedback

December 16, 2006.

In our endeavors to improve the service to our members, the BCAMT is piloting a change in the format of its award-winning journal Vector. All three editions for 2006 will be published in electronic form and your feedback will be invited.

The rationale for this change is twofold. First, the cost of publishing an electronic version is significantly less than the cost of publishing a traditional, paper version. This cost savings will improve the BCAMT’s ability to provide support to its members in a variety of ways. Second, the electronic format has a great deal of potential for improving Vector. In electronic form, we can embed hyperlinks, applets, and video clips into the articles.

As a member of the BCAMT, your feedback on this change is very important to us. If you like the change, wish to suggest ways we can improve Vector, or would prefer us to return to the paper format, please let us know. You can email your comments to us at:

[email protected]

Thank you,

Robert Sidley

President,BC Associaton of Mathematics Teachers

Summer 2006�

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Vector �

Letters to the Editors

Please mail or e-mail your letters to:

David Tambellini, Vector Editor Box 445Christina Lake, BC V0H 1E0

[email protected]

Dear Editors

The letter to the editors by Marcus Barnes includes the point that “we must try our best to write what we mean.” If we do not write (and say) what is correct, “can we blame our students for getting confused?”

I agree with Barnes and I believe our attempts to avoid confusion require correctness and specificity. [See “Watch Your (Our) Language,” Vector, summer 2002.] A few examples as well as corresponding questions and comments based on the content from the spring 2006 edition of Vector can be used to illustrate some of my concerns.

To avoid confusion and incorrect learning, we need to distinguish between figure and shape. Students encounter 2-dimensional figures and 3-dimensional figures or objects. Shape is a characteristic of a figure or an object. I believe it is helpful for students if we talk about a triangular shape, rather than a triangle. Students need to know that it is not possible to “pick up a triangle (circle, square, etc.).” Without that distinction, how will students ever get to know what a triangle is?

I think many of the assessment suggestions that are included in the issue lack the specificity that is required. Statements of a general nature require examples before any possible agreement of the intent is reached and before many of the suggested strategies can be employed. Consider the following examples (phrases that are part of suggested assessment strategies) and questions.

- most basic requirements (How does this differ from a basic requirement? What is considered to be “basic”?)- minor errors (What are the differences between errors, minor errors and major errors?)- simple extension (What is an example of an extension? What are the different types of extensions? What is a

simple extension? Simple according to whom?)- efficient work (How is it possible to assess efficiency? What are some examples of efficiency?)- sound relevant assignments (How are sound and relevant defined? How do they differ?)- some illogical and or irrelevant mathematical ideas (How is illogical defined? Illogical according to whom? What is an example of irrelevancy?)- complex and connected mathematical ideas (How are complex and connected defined?)- clarity and logic (What is meant or what did the authors have in mind? Are both terms really needed?)

Summer 2006�0

The list of examples could go on. Some of the statements sound impressive, to say the least, but my question is, “How much agreement of interpretation and implementation would there exist between different teachers and teachers with various mathematics education backgrounds? I am very curious indeed, because I know the lack of agreement that exists when teachers-to-be attempt to give meaning to statements of this type.

For the sake of our students, I hope that the language we use when we implement the WNCP mathematics framework will be not only specific, but also correct.

Werner Liedtke

Professor Emeritus University of Victoria

Vector ��

Multiple Intelligences in the Classroom: Reaching All Learners

Darlene Polishak is a pre-service teacher in the two-year elementary program at the University of BC.

Darlene Polishak

During my short practicum, I taught a unit on transformational geometry to a split Grade 5/6 class. Although I eventually was able to teach the majority of the students the key concepts of the unit, I had trouble reaching one boy in particular. This troubled me because on a previous visit, he had been identified to me as an exceptionally agile student in mathematics. The class Special Ed Assistant shed some light on the subject when she told me that the boy was bright but was the closest to an absolutely non-visual learner she had ever seen. It made me wonder how I could best teach students who do not easily access mathematics according to traditional methods.

As a result of my research, I believe that the pedagogy with the greatest chance for success in teaching all learners is that of Howard Gardner’s multiple intelligences theory (MI). This essay will examine the case that mathematics is becoming a more visual medium, the theory behind MI, some problems with the application of MI and give examples of MI applications to the subject of mathematics.

To assess the need to incorporate multiple intelligences methodology into the current curriculum, I will first examine the increasing importance of visual media in mathematics teaching. Life in our everyday world is putting increasing importance on the ability to process visual information swiftly and accurately (Arcavi, 2003). Also, the school curriculum is being revised to reflect that change with greater textbook emphasis on visual methods of instruction. Whereas visualization was once frowned upon as a means of reasoning, today it is becoming more accepted as a means of problem solving and even proving problems (Arcavi, 2003).

Visualization can perform three main functions for mathematicians:

As (a) support and illustration of essentially symbolic results, (and possibly providing a proof in its own right)…(b) [as] a possible way of resolving conflict between (correct) symbolic solutions and (incorrect) intuitions…and (c) as a way to help us re-engage with and recover conceptual underpinnings which may be easily bypassed by formal solutions. (Arcavi, 2003, p.223)

With visualization serving these functions, mathematics has become more accessible than ever before to visual learners, the traditional mode of education. One key problem with the current trend of reliance on visualization, however, is the fact that the ability to visualize varies widely among the population (Arcavi, 2003). For students in visually rich cultures, it is a boon, allowing them to rely on their visual intelligences as a means of countering any deficits in their learning of concepts (Arcavi, 2003).

Since not all students come from visually-rich backgrounds, however, it is necessary to find additional means of educating all those who cannot, for whatever reason access this means of teaching and explaining mathematics. This is where Howard Gardener’s theory of multiple intelligences comes in.

Summer 2006��

In MI theory, there are eight recognized intelligences, or biological and psychological potentials/capacities (Gardner, 1995). They are called logical-mathematical, naturalistic, bodily-kinaesthetic, linguistic, spatial, interpersonal, intrapersonal and musical (Hill, 2004). These human intelligences consist of three components:

[a] set of skills that enables an individual to resolve problems; the ability to create an effective product or offer a service that is of value; and the potential for finding or creating problems, thereby laying the groundwork for the gaining of new knowledge. (Hill, 2004,132)

The main switch between an MI approach and a traditional school system approach to education involves a broadening of the “acceptable” methods of learning and instruction. Traditional schools focus on only two intelligences: logical-mathematic and verbal-linguistic (Stone, 2004). While this method works quite successfully for some students, it fails to recognize the abilities of many and can consequently lead to poor results in the classroom. By broadening the possible methods of instruction, teachers can also curtail a student’s frustration with being forced into operating in an intelligence that may be the weakest of his/her eight.

In one study on the use of MI methods in the math and science classroom, educators adopted a MI methodology in order to promote learning (Hill, 2004). By co-opting each student’s natural inclinations towards certain intelligences into the educational program instead of fighting them as a traditional school would have done, the teachers saw rapid growth in student comprehension and achievement levels (Hill, 2004).

While there is no set multiple intelligences pedagogy as such, teachers’ general application of the theory has been to adapt or create a curriculum that students can master by means of one of the eight intelligences, often using several different intelligences in teaching the same particular unit or lesson (Gardner, 1995). For instance, individuals could study multiplication by using the bodily-kinaesthetic, verbal-linguistic and spatial intelligences in addition to the logical-mathematical intelligence (Willis & Johnson, 2001). This could entail having the students practise the times-table by doing a grouping exercise during gym class, writing a poem to help them learn math facts, having them create models that represent the times-table, or using problems or logic puzzles to help them master their new knowledge (Willis & Johnson, 2001).

In addition to broadening the points of entry into a topic for the students thereby allowing for a greater number of students to be reached (Gardner, 1995), the use of multiple intelligences over the course of a unit gives students greater repetition in a topic, increasing their mastery of the subject (Willis & Johnson, 2001). By allowing the students to “play to their strengths,” MI decreases students’ frustration and likely increases students’ self-esteem as they have more opportunities for success and creative experimentation with mathematical concepts (Willis & Johnson, 2001).

Despite the many bonuses that multiple intelligences bring to the classroom, there are some caveats that must be observed if MI is to work successfully. Superficial usage of an intelligence must be avoided, as it offers no opportunity for mastery. For instance, having the students do jumping jacks while you recite the times-table does not offer any bodily-kinaesthetic intelligence building to take place (Gardner, 1995). Instead, one needs to utilize the intelligence as the means of understanding the concepts (Gardner, 1995).

Similarly, intelligences are not to be used as mere mnemonic devices or background noises as may be the case with some applications of musical intelligence (Gardner, 1995). Further damaging is the pigeonholing of students as having a certain intelligence to the exclusion of all others (Hatch, 1997). In doing so, one assumes that the child’s interests and abilities will not change throughout his/her life. It denies the fact that “young children often display their strengths in specific activities

Vector ��

or roles, rather than in all activities related to a particular intelligence”(Hatch, 1997). Instead, Hatch advocates that children learn using the intelligences they are adept in as a means of broadening their range of abilities across all the skill areas rather than narrowly focusing in on the ones they are declared best at.

MI is a valuable concept in providing teachers with a means to reach more students that the traditional math and linguistics focuses allow. By giving students greater opportunities and directions from which to approach subject matter, multiple intelligences affirm the importance of all types of learning for all types of learners. While there are some potential hazards to be avoided in the application of the theory to teaching practice, there is much more to be gained in the practice of the theory than from adherence to strictly traditional methods.

Summer 2006�4

Please share your exPertise with Vector’s readers.

Getting published in Vector is a relatively easy procedure: simply describe a lesson or an idea that has worked well in your class. Maybe you have developed an open-ended problem-solving session, or perhaps you have re-organized a unit from the regular curriculum that your students have enjoyed.

Your article need not be as formal as the lesson plans that you wrote during your pro-fessional development at university. All we need is an outline of the steps you took with your students, and your impressions of the results. The editors will then “massage” your contribution into an article format. In addition, you might consider sending us samples of your students’ work to illustrate the lesson.

How could it be easier to add “published author” to your curriculum vitae? And as a bonus, you will receive a free one-year membership in the BCAMT.

Share and

Learn

Vector ��

Grade K/1 - Number Concepts

Candy Cheuk is a pre-service teacher in the two-year elementary program at the University of BC.

Candy Cheuk

Rationale:The class will be studying the theme “My Body,” which provides rich material for learning math concepts such as recognizing parts to a whole (head, arms, legs make up parts of the body); counting (e.g. 1 nose, 2 eyes); similarities and differences in body sizes, shapes, and colours; and measurement (e.g. weight, height, length). For this unit we will focus on counting, sorting, comparing, and estimating. The Body Book, by Shelley Rotner and Stephen Calcagnino will be used as an introduction to the theme. There are many wonderful books that connect number to reality and according to Van de Walle (2005), involving children in books in different ways “makes [math] a personal experience, and provides ample opportunities for problem solving” (p. 110). Using students provides opportunities for children to get to know one another, recognize similarities and differences in one another, and learn how to work cooperatively in groups. Moreover, using children themselves as material for counting, sorting, and comparing allows for opportunities to move around in the classroom and to increase physical activity. It also provides concrete links to the real world for what they are learning.

The activities teach daily classroom routines and familiarize students to those routines and the classroom itself through daily use. Using concrete manipulatives such as buttons, marbles, and counters for counting, making sets, and sorting engages students in hands-on sensory and tactile learning activities, and helps them learn responsibility for classroom materials. Allowing time for free play with materials helps children explore and become familiar with the objects they will be using for the year. They also get to practise the routines for getting materials and putting them away. Providing opportunities to work in large and small group settings helps children learn to work cooperatively. The interaction is necessary to engage students in meaningful dialogue, and to encourage peers to scaffold one another in their learning. As a conclusion to the unit, we will use the information obtained to complete a class book about the students.

Prescribed Learning Outcome: It is expected that students will recognize, describe, and use numbers from 0 to 100 in a variety of familiar settings.

General Statement: Students will explore and become familiar with numbers through hands-on activities that engage students in learning about themselves. Using students as “material” for counting, sorting, and comparing will provide them with relevant and concrete connections as they continue to make sense of the world around them. At the end, we may use the information gathered to create a “Book About Us” as a class.

Summer 2006�6

Journal writing is an important tool to gain insight into the different ways students solve problems. It also teaches that math is a different language that should be conveyed and communicated to others. In writing, students are able to reflect and have time to organize their thoughts before they express their thoughts, unlike in oral communication. It also gives students privacy and frees up any feelings of inadequacies they may have when speaking in front of a class. Students may journal their thoughts and responses with only the teacher having access to them. Moreover, Stiles (1992) points out that journals “offer students an opportunity to respond, practise, record, gain awareness of themselves as writers, make observations and predictions, collect information, and share with others” (Van de Walle & Folk, 2005, p. 78). By journaling, they learn to think critically about problem solving, as well as about connecting math to other subject areas such as language arts.

The assessments used in the unit are mainly performance tasks to find out what concepts students have acquired, aside from procedural knowledge. Other assessments include frequent informal observations that evaluate students’ progress according to simple rubrics.

Pre-Assessment Activities

Need to explore students’ number readiness…• asking how high they can count• asking what is the biggest number they know• asking students to show or point to numbers they know on a number line• asking whether they can show me what the number represents: if they can count out 5 objects

or maybe show 1 finger, 3 fingers, 5 fingers or draw 2 objects to demonstrate they understand the concept and quantity

• asking whether they are able to draw a numeral

Vector �7

LESSON 1

Prescribed Learning Outcome: It is expected that students will recognize, describe, and use numbers from 0 – 100 in a variety of familiar settings.

Objectives: • Kindergarten students will be able to count orally by 1s up to 20.• Grade 1 students will be able to count orally by 1s up to 20.

Procedure: Introduction The Body Book by Shelly Rotner and Stephen Calcagnino.

• Have the kids gather in a circle. Read the book about our bodies. Facilitate a discussion about the different parts of a body that make it a whole, and about differences in sizes, shapes, and colours. We will carry this theme on into later units.

• Ask students to count from 1 up to 20. • Ask if students can count how many “bodies” are in the classroom.• Together, count the number of children in the class. As you count, ask a child to come up

to the front until you have finished counting. This allows for a visual representation of understanding the 1 to 1 correspondence between students and the list of counting words they are saying aloud.

• Explain the “Friends Three Game”. Show how students can group themselves by number. (e.g. If I ask Alana, Roland, and Mark to come up, how many children are there in this group?)

Lesson Focus Children must count how many “bodies” are in their group and arrange themselves into groups

of different sizes so as to recognize that the numeral corresponds to the quantity.• Call out “friends three” and students should place themselves in groups of that size. • Have students count their group to check. • Count how many groups there are.• Record this on the board. • Repeat with different numbers.

Closure • As a class, discuss how children were able to know they had enough people in their group.

How did they count? Explain the recordings on the board.• Journals: Ask students to draw the different groups they came up with from the activity,

referring to what was recorded on the board. (E.g., if we had 3 in a group, how many groups in the class were there?)

Summer 2006��

Things to Consider:a. Adaptations

• Students may complete worksheets that ask them to count by 1s. For more advanced students, you may ask them to count by 2s.

1 + 1 + 1 + 1 + 1 + 1

1 2 3 4 __ __ __ __ __ 2 4 6 8 __ __ __ __ __

b. Assessment • Observe students counting behaviours as they are counting out the children. Use a full-

class observation checklist (Van de Walle, p.78) with a 3-point general rubric that shows “not yet on target,” “on target,” or “Wow!” and that allows for comments.

Topic: Counting

Not YetNo 1-1

correspondence concept

On TargetAble to count in

sequence to 20 by 1s

Wow!Can count beyond

rote and create different groups with

multiple numbers

Comments

Names:

• Ask children to count by rote starting at 1. Can they start at 5 and count? • Provide a worksheet and have children colour ‘x’ number of objects: e.g. 3 balloons, 2

suns, 5 stars.• Matching a sequence of objects. Set out a pile of 8 or 9 blocks and ask students to count

the blocks. Alternatively, have them match your grouping of blocks using their own set. Observe whether they line up the blocks in one-to-one correspondence. Also, do they count one set and count out a matching set, or use guesses/estimates?

c. Student work examples • Collect their journals and worksheets.d. Cross Curricula Connections • Physical activity and language arts.e. Classroom Set-Up • Students will be working as a class, in groups, and then individually.f. Materials/Resources • The Body Book, worksheets, counters or blocks, observation sheets

Vector ��

LESSON 2

Prescribed Learning Outcome: It is expected that students will recognize, describe, and use numbers from 0 to 100 in a variety of familiar settings.Objectives:

• It is expected that students will be able to use a calculator or computer to explore and represent numbers up to 100.

Procedure:Introduction Free-Play• Allow students to play with the standard 8-space calculator for 10 minutes. • Ask them to observe what happens when they press certain buttons. How do they turn it on?

What do the buttons feel or look like? How are they placed on the calculator? Do they know what the symbols on the buttons stand for? Can they guess what a calculator is used for?

• The teacher should then demonstrate how to use the calculator either on an overhead projector calculator, poster, or by drawing it on the board.

Lesson Focus Students will explore patterns on the calculator and make predictions about the digits that

appear on the calculator. Also, they will practice their printing of numerals.

Calculator Activity – How many digits? 1) Have students enter C then press 1 key twelve times on a standard 8-space calculator. Ask

if they notice anything special or unusual.2) Have students count how many digits show on the screen and copy them. 3) Now individually or in pairs, ask students to predict what they may see when they use the

other digits 2 through 9.4) Use the 0 key last, as the pattern is broken by 0, which only appears once on the display.

Closure • Ask students what observations they have made. Did they notice any patterns? Ask why they

think 0 only appears once.• Journals: Draw something they learned about calculators. Grade 1s may write a brief sentence

about something they observed.

Things to Consider:a. Adaptations

• Working in pairs may help those who need extra assistance or who are very unfamiliar with using technology. An added challenge for gifted students may be to ask them to find different ways in representing double-digit numbers.

b. Assessment • Observe whether students are able to follow directions and locate the appropriate buttons

on the calculator. Observe their counting behaviour as they count the digits shown on the calculator.

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c. Student work examples • samples of their printing of the digits and journalsd. Cross Curricula Connections • Technology and language arts.e. Classroom Set-Up • Students will work individually or in pairs if they choose.f. Materials/Resources

• Class set of calculators, overhead projector calculator, paper and pencils, large number line or provide individual number lines for students to use.

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LESSON 3

Prescribed Learning Outcomes: It is expected that students will recognize, describe, and use numbers from 0 to100 in a variety of familiar settings.

Objective: It is expected that students will be able to recognize and compare sets of objects (0 to 50) using both comparative and numerical terms.

Procedure:Introduction Likeness and Differences:• Discuss how we are similar in some ways and different in others ways. • Explain how they will make a chart to show the number of boys and girls in the class and

what the chart represents.

Lesson Focus Children will build and compare sets of boys and girls, and recognize that boys and girls make

up a set of children.Boys-Girls Picture Chart: • Have the children divide into a boy group and a girl group.• Ask the children to compare which section they think has more or less children. • Have students draw portraits of themselves and attach them to the correct sections on a chart

(boy versus girl). • Once the pictures are all up, have the students count the pictures in each section and the total

number in all.

Closure • Discuss their answers. Were their predictions about which section has more or less children

correct? How did they know? Did they count? Guess? • Ask children to record this information into their math journals and perhaps draw something

that makes them unique from their classmates.• Grade 1: Write a sentence about their similarities or differences.

Things to Consider:a. Adaptations – An added challenge may be to now think of other ways they may divide the

class, whether by hair or eye colour, and to compare those sets.b. Assessment

• Observe how students match counters to numerals on their mats and their ability to assign a numeral to a set of objects.

• Sorting – Provide a piece of paper separated into 4 quadrants and provide counting objects. Give students simple directions. Ask them to sort in different quadrants 3 bread tags, 4 milk caps, 10 pebbles, etc. After, check counting by asking questions like, “How many milk caps do you have?”

c. Student work examples • Collect math journals.

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d. Cross Curricula Connections • language arts and art

c. Classroom Set-Up • Students work as a class and then individually.

d. Materials/Resources • Chart paper, tape, art supplies, drawing paper, counting objects

Vector ��

LESSON 4

Prescribed Learning Outcome:It is expected that students will recognize, describe, and use numbers from 0 to 100 in a variety of familiar settings.Objectives: • Kindergarten students will be able to read number words up to 10. • Grade 1 students will be able to read and order number words up to 10. Procedures:

Introduction – acting out counting rhymes. • Have the children crouch down as small as they can for the first verse, and become taller and

taller for each successive verse.

The End

When I was one, I had just begun. When I was two, I was nearly new. When I was three, I was hardly me. When I was four, I was not much more. When I was five, I was just alive. But now I’m six, I’m as clever as clever. So I think I’ll be six now for ever and ever.

A. A. Milne

• Follow-up activity: Ask students to find examples of each number around the classroom: e.g., 2 students, 3 blocks, 6 desks.

Lesson Focus – Number words:1) You may use the numbers rhyme from the Introduction to teach children to read number

words. 2) Write the rhyme on chart paper, or use a pocket chart.3) Have the number words written on separate cards that may be removed or attached.4) Practise the rhyme with children and as they begin to recognize the words, you may ask

them to help you fill in the blanks and put the words in the correct places.

Closure • Ask children to draw in their journals what they think they look liked or did from ages one

up to six. • Grade 1: Ask them to write a short sentence to describe what they have drawn or ask them to

create their own rhyme using the number words.

Summer 2006�4

Things to Consider:a. Adaptations For those who have trouble reading, you may want to write the numeral digit on the back

of the number word to facilitate recognizing the word. For the more advanced, you may choose to add words higher than ten or have them order the number words from greatest to least and vice versa.

b. Assessment Have a paper made up to assess students’ ability to read word numbers up to 10 and to

make matching sets. Give each student small objects such as beans, beads, or buttons and have them build a set to match the number. They may also draw to match.

Three Ten

Five Eight

Four One

Two Seven

Six Nine

c. Student work examples • Have students choose their favourite numbers. Use materials like small pebbles, macaroni,

or rice so they can count while they make the numbers.d. Cross Curricula Connections • Drama and arts.e. Classroom Set-Up • Work together in a large group and individually.f. Materials/Resources • Large chart paper or pocket chart to write the rhyme, paper clips or magnets or Velcro

to attach/detach the word number cards to the chart, assessment worksheets, small counting objects.

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LESSON 5

Prescribed Learning Outcome:It is expected that students will recognize, describe, and use numbers from 0 to 100 in a variety of familiar settings.

Objective: • It is expected that students will be able to estimate and count objects in a set (0 to 50) and

compare estimates to the actual number.Procedure:

Introduction Hand and Feet• Ask students, “How many hands do you think there are in the classroom?”• Discuss and explain different ways they can figure out the number of objects.• Explain the importance of estimating.• Have children make guesses and record these on the board. • Ask them to count to check their estimates.

Lesson Focus Fingers and Toes1) In pairs, have children estimate (guess) how many fingers there are in the room.2) Record their answers on hand cutouts. 3) Gather and count together and compare their estimates to the actual count.4) Repeat with how many toes there are in the classroom.5) Ask how many fingers and toes there are in total.

Closure – • Ask whether knowing the answer for the number of hands helped in estimating the number

of feet. Why or why not?• Did knowing the number of fingers help in estimating the number of toes? • Write brief sentences about the number of hands and feet in the class.

** Ask students to think about all the information they have learned about themselves and brainstorm as a class what they would like to include in the class book. Have them think about what samples of work they would like to volunteer to put into the book. **

Things to Consider:a. Adaptations • Include a mystery jar of marbles or jellybeans for advanced children to estimate. At the

end of the day, you may open the jar and have the children sort the jellybeans by colour and count in varying ways.

b. Assessment • Have children explain their estimates. Ask, “Why did you pick that number? How do you

know it will be more or less than the last one?”c. Student work examples • Collect their recordings of their estimates compared to the actual number. Take a photo of

the chart to put into their class book.

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d. Cross Curricula Connections • Language arts

e. Classroom Set-Up • Students will work in large groups and in pairs.

f. Materials/Resources • Chart paper, art supplies, glue or tape, prepared cutouts of hands and feet

References

British Columbia Ministry of Education. (1996). Integrated Resource Package: Mathematics. Victoria, BC.

Kelleher, H. J. (1992). Mathworks, Book A. Toronto, ON: Houghton Mifflin Canada.

Moomaw, S., & Hieronymus, B. (1995). More than counting. St. Paul, Minnesota: Redleaf Press.

Rotner, S., & Calcagnino, S. (2000). The body book. New York, NY: Orchard Books.

Van de Walle, J. A., & Folk, S. (2005). Elementary and middle school mathematics: Teaching developmentally. (Canadian ed.). Toronto, ON: Pearson Education Canada.

Woolfolk, A. E., Winne, P. H., & Perry, N. E. (2005). Educational psychology. (3rd Canadian ed.). Toronto, ON: Pearson Education Canada.

Wortzman, B. K. R., Cornwall, J., Kennedy, N., Maher, A., & Nimigon, B. (1996). Mathquest One: Teacher’s edition. Toronto, ON: Addison-Wesley Publishers.

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Let’s Learn Together

Andrea Easter recently completed the 12 month elementary program at the University of BC. She is currently working as a teacher-on-call for the Vancouver School District.

Andrea Easter

Until recently, “sharing” answers or working as a group on a math problem bordered on the line of cheating. Remember the kids who “worked away at problems behind a cupped hand, like used car salesmen trying to massage the numbers before quoting a price” (Artzt & Newman, 1990, p. 29)? Those days have come to an end, as an era of cooperative learning has emerged in the elementary classroom.

“Cooperative learning” is a generic term used to describe an instructional arrangement for teaching academic and collaborative skills to small, heterogeneous groups of students (Sharan, 1980). Cooperative learning is a successful teaching strategy in which small teams, each with students of different ability levels, use a variety of learning activities to improve their understanding of a subject. Each member of the team is responsible not only for learning what is taught, but also for helping their teammates learn, thus creating an atmosphere of achievement (OERI, 1992). Cooperative learning is deemed highly desirable because of its tendency to reduce peer competition and isolation, as well as its ability to promote academic achievement and positive interrelationships among children of all ages. Research supports cooperative learning as an effective approach for including all students in classroom group work and promoting peer acceptance (OERI, 1992).

It should be noted that some researchers caution teachers to integrate both direct instruction and cooperative learning in their classrooms. However, for the purpose of this paper I will discuss the elements of cooperative learning, how teachers effectively implement this learning into the elementary classroom, and the positive effects such learning has on motivation, achievement and social relationships.

Elements of Cooperative Learning

There are several elements and approaches to the cooperative learning strategy. Cooperative-learning groups are relatively small and as heterogeneous as circumstances allow. This small size promotes individual accountability for all the group members. This feature stipulates that each member of a group has to make a significant contribution to achieving the group’s goal. Since cooperative groups are heterogeneous with respect to ability, and because their success depends on positive interdependence, promotive interaction, and individual accountability, it is important to ensure that all students have an opportunity to contribute to their team.

Cooperative learning also promotes interaction among students. This element is made necessary by the existence of positive interdependence. Positive interdependence refers to students seeing the importance of working as a team and realizing that they are responsible for contributing to the group’s effort. Students are shown how to help each other overcome problems and complete assigned tasks. This may involve episodes of peer tutoring, temporary assistance, exchanges of information and material, challenging of each other’s reasoning, feedback, and encouragement to keep one another highly motivated (Biehler & Snowman, 1997).This interaction among the students leads to the development of stronger interpersonal skills. Positive interdependence and interaction are not likely

Summer 2006��

to occur if students do not know how to make the most of their face-to-face interactions. As a result of working together in groups, students are taught such basic skills as leadership, decision making, trust building, clear communication, and conflict management.

A final element of cooperative learning is that it promotes team competition. This may sound contradictory, but the main problem with competition among students is that it is rarely used appropriately. When competition occurs between well-matched competitors, is done in the absence of a norm-referenced grading system, and is not used too frequently, it can be an effective way to motivate students to cooperate with each other (Biehler and Snowman, 1997).

How Cooperative Learning Works

Cooperative learning is part of classroom management (Glosser, 2005). To use it effectively, educators must train their students to work effectively in groups (Glosser, 2005). Group categorization is crucial to the outcome of cooperative-learning activities. Groups, according to research, should have approximately four or five students and should be made up of differing abilities, as well as personalities (Rivera, 1996). For example, one group may have one top level, two middle level, and one struggling student (Rivera, 1996). Designing math activities for cooperative learning groups requires consideration of both the instructional objectives and the purposes for having children work in a cooperative instructional arrangement (Rivera, 1996). For cooperative learning to work effectively within an elementary classroom, educators should design activities that promote math understanding by having students practise, experiment, manipulate, reason, and problem solve, both individually and within a group.

During cooperative-learning, teachers should circulate among groups monitoring the students’ ability to complete the assigned mathematics activity and their ability to demonstrate the intended collaborative skills. The teacher should also facilitate group work by asking questions to help students redirect their work, by providing additional instruction to some students who may be struggling with the task, and by reinforcing students’ efforts for working collaboratively and seeking solutions to problems (Rivera, 1996). When the cooperative learning activity is finished, teachers may want to administer an individual posttest to determine how well each student has mastered the mathematics content (Rivera, 1996). With careful planning, implementation, and evaluation, most students can succeed in cooperative learning.

Positive Effects of Cooperative Learning

Various research has shown that the different forms of cooperative learning have been more effective than non-cooperative structures in raising the levels of variables that contribute to motivation, raising achievement, and in producing positive social outcomes (Biehler and Snowman, 1997). These are only a few of the positive effects that cooperative learning has on student success.

A strong indicator of motivation is the actual amount of time students spend working on a certain task. Most studies have found that cooperative learning students spend significantly more time on-task than do non-cooperative learning students (Slavin, 1995). In addition, cooperative learning contributes to high levels of motivation in students because of the pro-academic attitudes that it fosters among group members. Slavin (1995) cites several studies in which students in cooperative learning groups felt more strongly than did other students that their group members wanted them to come to school every day and work hard in class. Furthermore, these students were more likely to attribute success to hard work and ability than to luck (Slavin, 1995).

Cooperative learning also has positive effects on student achievement. Slavin (1995) examined several dozen studies that lasted four or more weeks and that used a variety of cooperative learning methods.

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Overall, students in cooperative learning groups scored about one-fourth of a standard deviation higher on achievement tests than did students taught traditionally. Other studies have yielded similar results.

The Office of Educational Research and Improvement (OERI) found that cooperative learning also leads to improved academic achievement: “when two necessary elements-group goals and individual accountability are used together, the effects on achievement are consistently positive” (OERI, 1992). Furthermore, OERI argues that cooperative learning also increases the achievement of students with learning disabilities. This is because students with learning disabilities are “mainstreamed,” leading to significant improvements in relationships between these students and other children (OERI, 1992). Furthermore, cooperative learning provides students with learning disabilities, math disabilities, and social interaction difficulties, an instructional strategy that fosters the application and practice of mathematics and collaborative skills within a natural setting: i.e., group activity. Thus, cooperative learning has been used extensively to promote mathematics achievement of students both with and without a learning disability (Rivera, 1996).

Cooperative learning also has a positive effect on social relationships amongst children. Documented results show that cooperative learning leads to an “increased liking of school and classmates” (OERI, 1992). Furthermore, most studies illustrate that students exposed to cooperative learning were more likely than students who learned under competitive or individualistic conditions to name a classmate from a different race, ethnic group, or social class as a friend or to label such individuals as “nice” or “smart” (Biehler and Snowman, 1997).

As a result of cooperative learning, students tend to be more highly motivated to learn because of increased self-esteem, the pro-academic attitudes of group members, appropriate attributions for success and failure, and greater on-task behavior. They also score higher on tests of achievement and problem solving and tend to get along better with classmates of different racial, ethnic, and social class backgrounds. This last outcome is particularly important in today’s society where schools are marked by cultural diversity.

Conclusion

Cooperative learning is an effective teaching strategy for all subjects; however, it is particularly effective in teaching elementary mathematics. Small groups offer a “social support mechanism,” thus allowing students to feel more comfortable asking their peers questions that they may not have asked their teacher (Rogers, et al., 2001). Furthermore, in a group, students may be able to solve problems that are more complex and thought provoking than the problems they can solve individually (Rogers, et al., 2001).

According to the National Council of Teachers of Mathematics (NCTM, 1991), learning environments should be created that promote active learning and teaching, classroom discourse, and individual, small-group, and whole-group learning. Cooperative learning is one example of an instructional arrangement that can be used to foster active student learning, which is a crucial element of mathematics learning. Furthermore, cooperative learning activities can be used to supplement textbook instruction by providing students with opportunities to practise newly introduced skills and concepts. Teachers can use cooperative learning to help students make connections between the concrete and abstract level of instruction through peer interactions and carefully designed activities.

Cooperative learning offers many educational benefits for both students and teachers. But most of all, it engages students and offers them a fun approach to math… and what could possibly be wrong about that?

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Bibliography

Artzt, A.F. & Newman, C.M. (1990). Cooperative learning in math class raises ability levels, enhances critical thinking. Curriculum Review, 30 (4), p. 29.

Biehler, R.F. & Snowman, J. (1997). Psychology applied to teaching, 8th edition. Boston, MA: Houghton Mifflin.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Retrieved November 1, 2005 from http://www.nctm.org

Office of Educational Research and Improvement (OERI). (1992). Education consumer guide: Cooperative learning. Retrieved October 30, 2005 from http://www.ed.gov/pubs/OR/ConsumerGuides/cooplear.html

Pedrotty Rivera, D. (1996). Using cooperative learning to teach mathematics to students with learning disabilities. LD Forum: Council for Learning Disabilities.

Rogers, E., Reynold, B., Davidson, N., and Thomas, A. (2001). MAA Notes, Vol. 55. Mathematical Association of America.

Slavin, R. (1995). In cooperative learning: Theory, research, and practice, 2nd edition. Boston MA: Allyn and Bacon.

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Reducing Fractions and its Applications to Rational Expressions

Duncan McDougall teaches at TutorFind Learning Centre in Victoria.

Duncan E. McDougall

The following method is designed for alternative thinkers, or for the type of student who may not follow or use conventional methods or algorithms. Perhaps a student won’t use a prescribed algorithm demonstrated in a textbook because he or she doesn’t like it or understand it. The author may have a particular style that the student simply doesn’t relate to and this may also be true of his or her classroom teacher. Regardless of the dominating hindrance, the student needs a method that he or she can understand.

Accordingly, if a different method isn’t in the textbook, then it’s up to the teacher to find one. Of course, interaction with the student is vital as it often leads to fruitful insights as to how that particular person approaches a problem. Alternative thinking, or “thinking outside of the box,” has distinct advantages. For example, if the idea or concept on the part of the learner is deemed “legal,” “useful,” or even “practical,” he or she becomes proud of the contribution made. In turn, this process adds greatly to a student’s self-confidence. In fact, this newly discovered method might appeal to others in the classroom as well as the teacher. Indeed, it may even be better than the prescribed method in the textbook, and therefore worth publishing.

Imagine the plight of the Math 10 or Math 11 student whose factoring skills are less than adequate. That person may find factoring a chore because of a lack of success in the past. To increase his or her comprehension, a student needs a method to make the task easier. For example, in the fraction 39/65, if we knew that 13 was a factor of both 39 and 65, then we could write as

39

65=

13× 3

13× 5=

3

5.

The educator knows that 13 is the greatest common factor. But how would the g.c.f. be apparent for

a student in something like x xx x

2

2

2 3

7 12

− −− +

?

For reference and clarity I would now like to refer to my publication in the article “Reducing Fractions,” published in the Scientific Journal of Junior Math and Science, London, England 1990. It demonstrates the premise that the only possible factors available to reduce a fraction to lowest terms, come from the difference between the numerator and the denominator.

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Numerically, it looks like this.

Reduce

39

65

Steps: (1)65 – 39 = 26 the difference between the numerator and the denominator

(2)Factors of 26 1, 2, 13, 26

(3) Disregard 1, 2, and 26 because they are all even

(4) Try 13. If 13 doesn’t work, then nothing else will.

(5)

39

65÷

13

13=

3

5

To transfer this concept to rational expressions, examine the following two examples.

Example:

Reduce x x

x x

2

2

2 3

7 12

− −− +

Steps:(1) ( ) ( )x x x x2 22 3 7 12− − − − +

(2) Disregard 5, consider (x – 3) .

(3) x x x x2 2 3 3 1− − = − +( )( ) and x x x x2 7 12 3 4− + = − −( )( )

(4) x x

x x

x

x

2

2

2 3

7 12

1

4

− −− +

=+−

= − − − + −

− −

= −

x x x x

x

x

2 22 3 7 12

5 15

5 3( )

At this stage, we can factor more easily because we know that (x – 3) is one of the desired factors; failing that, use long division.

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Example:

Reduce x

x

3 1

1

−−

Steps:(1) (x3 − 1) − (x − 1)

(2) Disregard x and x + 1. Consider x – 1.

(3) x x x x3 21 1 1− = − + +( )( )

(4) x

x

x x x

xx x

3 221

1

1 1

11

−−

=− + +

−= + +

( )( )

What I also like about this method is that we can discover which factors will not work in a given situation.

Consider x x

x x

2

2

12

6

− −− −

Steps:(1) ( ) ( )x x x x2 212 6− − − − −

As we can see, there is no point in factoring and looking far a common term if indeed none exists to begin with.

Now it’s no secret that a method for finding the g.c.f. of two polynomials does exist, but it does involve long division, and therefore, it would look something like this.

Example:

Find the g.c.f. for x x2 2 3− − and x x2 7 12− + , or g.c.f. ( , )x x x x2 22 3 7 12− − − + .

Divide one into the other, and keep track of the remainder. Now divide the remainder into the previous divisor, and again, keep track of the remainder. Continue this last step until the remainder is zero. The divisor, which gives zero as a remainder, is our greatest common factor.

= − − +

= −

= −

= − +

x x

x x

x x

x x x

3

3

2

1 1

1

1 1

( )

( )( )

Assuming that the algebra is done correctly, we can see immediately that there cannot be a common factor of the form (x +a).

= − − − + +

= −

x x x x2 212 6

6

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This means we would have the following.

)x x x x

x

x x

2 22 3 7 12

5 15

1

2 2 3

− − − +

− +

− −

Take only (x – 3), as –5 is not a factor of the form (x + a)

x x x

x x

x

x

− − −

−−−

3 2 32

2 3

3

3

)

0

x + 1

Actually, finding the g.c.f. in this manner is part of the reason why the above method of subtraction works. The teacher now has more than one way of presenting this material to various types of learners, and can provide alternatives for the reluctant student.

A welcome application of this approach is the calculation of limits for the calculus student. In general, we have the following.

lim( )

x a

x x a b ab

x a→−

+ + ++

2

Instead of evaluating directly, and giving the indeterminate form

0

0, we can subtract the two

polynomials, factor this difference, and then try to reduce it to its lowest terms.

This would create the following.

( ( ) ) ( )x x a b ab x a x ax bx ab x a2 2+ + + − + = + + + − −

This reveals first that the expression can be reduced, and second that (x+a) is the common factor. A numerical example would look as follows.

limx

x x

x→−

+ ++2

2 3 2

2

( ) ( )x x x2 3 2 2+ + − +

= − −5 3( )x

= + + + − +

= + + −

x x a b x a x a

x a x b

( ) ( ) ( )

( )( )

1

1

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= + + − −

= +

= +

x x x

x x

x x

2

2

3 2 2

2

2( )

= lim( )( )

x

x x

x→−

+ ++2

1 2

2 Disregard x, consider (x + 2)

= +→−lim ( )x

x2

1

= − += −

2 1

1

In summary, some students see math as a necessary evil. However, now they can “get into it” a bit more because someone has found a method that makes sense to them.

Summer 2006�6

A Couple of Little Songs to Share

Jarrett Spannier teaches senior mathematics and physics in Revelstoke.

Jarrett Spannier

Last September when I saw my Mathematics 12 class list, I knew I was in for a great semester. The 24 young people in this class were absolutely wonderful students. They definitely did not let me down as they proved to be the hardest working, best mannered and most talented Grade 12 class that I have ever had.

So as the high school Christmas Follies concert approached, we as a class decided to show our love of mathematics somehow. The students were willing to get up in front of the entire school and sing if I wrote a song that contained the word logarithm. But before I show you the song, here are some definitions.

A Math Party: Where a group of students get together to do their math homework (I’m not sure how much homework gets done but... .)

Spanny: That’s me!

Kai: The kid in the class getting 99%.

Without further ado, in honour of 2003 Revelstoke Semester 1, here is the Math 12 class’ “Lets Sing a Jingle Bell” tune.

Math 12 is the Best! A Christmas Jingle.(performed live on 19 December 2003)

(Verse1)

Dashing off to class Our brains ready to play We can’t wait to seeWhat Spanny has for us today.

Probability Maybe a log or twoGive us a geo-sequence,Or a conic will do.

(Chorus)

Oh… Ma-ath 12 Ma-ath 12This class is the best.We solve problems that Are too hard for the rest.Oh… Ma-ath 12 Ma-ath 12Give us a problem or two.We will take them to a math party And solve them for you!

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(Verse2)

A day or two agoWe had a quizzy-pooTrigonometryThese graphs are so darn new

Sinusoidal waves With period/amplitudeGraphing is just so darn fun – Transformations are too. Oh… .

(Chorus)

(Verse3)

Bring on the statisticsWe are not afraid of you.A couple z-scoresOr a binomial too

Let me count the waysTo line you up just soOr calculate the chances You are standing next to Moe –Oh…..

(Chorus)

(Verse4)

But our days are numberedThey count only a fewThen we have that testThat can be so hard to do

Oh me Oh my Oh me Oh my Oh me oh I’ll just sighI’ll answer every thing I know And try to beat Kai. Oh….

(Chorusandend)

We think this could easily go to the Top 10.

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A second song I like to use is on the day where my Math 12s write their mock-provincial exam. They always need some kind of stress relief as I think the mock provincial stresses them out more than the actual provincial exam does. I have been using this song for four years now and they seem to like it.

THE EXAM (Sung to: The Gambler)

On a cold January morningIn a study hall bound for trouble,I met up with the SpannyWe were both too tired to sleep.

So we took turns a starin’At a couple unsolved problems‘Til frustration over took us,And he commenced to speak

He said, “Kid I made life, Out of solving number problems. Knowing when to use radians, Knowing geometric sums.”

“If you don’t mind me saying, You just added unlike bases,For a taste of your Pepsi,I’ll give you some advice!”

So I handed him my bottle,And he drank down my last swallow.Then he bummed an eraserAnd scratched out all my work.

Then the room got kinda quiet,His face lost all expression,“If your gonna pass the provincial exam, Let me give you some advice”

(Chorus)

“You gotta know trig identities Know log restrictions Know graphing conics Know the binomial pdfYou never cancel your x’sWhen they are sitting in a fractionThere will be time enough for slashingWhen the factoring is done!

‘Cuz every student knows,That the secret to survivingIs knowing when to move aheadAnd not get stuck in place.

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‘Cuz every question’s worth A bit more of your mark nowSo the best you can hope for is to Answer them or guess!”

(Verse)

And when he finished speaking,He turned back toward the doorBroke of his pencil leadAnd slurped back a cup of joe.

And somewhere in the darknessThat Spanny he went crazy.But in his final words I foundThe will I need to pass!

(Chorus)

By the time I hit that last chorus, they are usually clapping along to the song and not thinking about the exam much anymore. My feeling is that a little humour can go along way in stressful courses like Principles of Mathematics 12, Calculus 12 and Physics 12.

Perhaps next time I’ll submit the poems I have for before my Calculus 12 midterms and finals. This is goofy humour with the same purpose: more fun and less stress.

So laugh it up and sing along!

Summer 200640

Summer 2006 Web Sites

I would highly recommend the Millennium Math Project at

http://www.mmp.maths.org.uk/links/first.html

Follow this link through to NRICH: there are a myriad of activities there, including games, math problems, and articles. Also, you will find some fantastic Web sites after a simple Google search for “history of mathematics.”

This Web site provides a great “list” of mathematics literature for many different grade levels.

http://www.sci.tammuc.edu/~eyoung/literature.html

From the Freudenthal Institute in Holland, try this Web site:

http://www.fi.uu.nl/rekenweb/en

It has proven to be one of the most favourite interactive Web sites of students I have worked with,

Enjoy these great opportunities.

Submitted by PamHagen

Listen to “Pi” sung as a duet at

http://pi.ytmnd.com

Also, try ON-Math (Online Journal of School Mathematics):

http://my.nctm.org/eresources/journal_home.asp?journal_id=6

ON-Math, a peer-reviewed journal from the National Council of Teachers of Mathematics (NCTM), was launched in 2002. Until recently, it was available only to NCTM subscribers. All issues are now freely accessible, including:

• Current Issue• Fill ‘n Pour• Shear Mathematics or Euclid’s Paradoxicals Applied• What Does Teaching Look Like Around the World?• Using Technology in Your Classroom• Hinged Geometry• Back Issues• Articles by Grade

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ON-Math presents a broad range of ideas for teaching and learning mathematics at any level, from early childhood to young adult. The journal capitalizes on the unique opportunities afforded by electronic media.

About a year ago, I put some provincial exam review assignments onto the BCAMT Web site. I’ve updated them, and I have also posted solution keys for some of them. I’ll be posting solution keys for the rest as the semester progresses. If you want to download them, go to:

http://www.pittmath.com/PrMa12/UnitReviews.html

Feel free to distribute them to any of your students, or to modify them in any way you wish. Hopefully Principles of Math 12 teachers will find the reviews useful.

Those of you who are interested in the history of calculating machines might find the following site worth a look. The author has built a calculating machine out of Lego pieces.

http://acarol.woz.org

Some of your students might like this site (or, if you’ve ever played Monopoly, and wondered…)

http://www.tkcs-collins.com/truman/monopoly/monopoly.shtml

Finally, for those of you with computer projectors in your rooms, visit

http://www.fractal-recursions.com/anim.html

Submitted by KelvinDueck

Check out “Math on the Simpsons” at

http://100cia.com/noticias/index.php?subaction=showfull&id=1149928198&archive+&startfrom=&ucat=12

Submitted by JonathanKung

Here are a few sites to visit for number facts.

Seasonalfacts…BythenumbersDo you want some quick facts related to a current season or special day? Look in the “By the numbers” area, located in the right-hand corner of the teacher’s page. You’ll find back to school numbers-on enrolment, teachers, cost of school supplies, university tuition fees, etc.

http://www42.statcan.ca/smr08/smr08_060_e.htm

See also the “Spotlight on public schools” articles on declining enrolment.

http://www42.statcan.ca/smr04/2006/08/smr04_23306_04_e.htm

and on increasing expenditures per student

http://www42.statcan.ca/smr04/2006/08/smr04_23306_05_e.htm

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Watch for “Halloween... by the numbers” coming later in October at

http://www.statcan.ca/english/edu

Go to “Teachers” and click on “Pumpkin Time.”

Submitted by DenisTanguay

PrinciplesofMath12

Check out the new Web site

http://www.math12.com,

which features an online study guide for Principles of Math 12 students. All lessons and practice exams are free for students and teachers to download, print, and photocopy.

Submitted by BarryMabillard

I loved the pi duet that someone posted a while ago. This is along the same line. Click the link...

http://www.vvc.edu/ph/TonerS/mathpi.html

Submitted by JamesPengilly

This link provides a few interesting properties of some familiar real numbers.

http://www.archimedes-lab.org/numbers/Num1_69.html

Submitted byBrucePayan

MonthlyPIMSProblems

Have your students try the monthly problems from PIMS.

http://www.pims.math.ca/education/math_problems

with a mirror at

http://www.math.ubc.ca/~adler/problems/

You may want to draw the problems to the attention of students who are in Grade 10 or below (though possibly more advanced in mathematics) who are looking for somewhat of a challenge.

Submitted by AndrewAdler

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MathDataSets

In SD 67 Okanagan we have been building Data Sets (organized in IRP categories) and are trying to produce or gather as many as possible to give teachers as many tools as possible. The students enjoy using them and they get it more often than if I am teaching it.

http://www.sd67.bc.ca/instruction/browse/browser.asp

Check out the page that has Grades 6-9 so far.

Submitted by AaronGrant

Summer 200644

Enriching Math Using Chess

Frank Ho is a teacher at the Ho Math and Chess LearningTM Centre in Vancouver, BC.

Frank Ho

Background

I started to teach chess to my son when he was five years old and soon noticed that the relationship between mathematics and chess is one of those generally presumed truisms. However, I was not able to find a book of junior level math and chess hybrid problems for my son to work on. In 1995, after seriously looking into the possibility of writing one myself, I pioneered the idea of integrating chess symbols and their values into math. For the first time, elementary students were able to visualize how concepts in both math and chess are related. The idea of using chess symbols and their point values and chess moves directly in puzzles characterizes what I believe is a breakthrough that permits a true integration of math and chess, since the puzzles are no longer limited to only traditional chess puzzles.

Later, I created worksheets with a 2-column system: chess on one side and math puzzles on the other. Students are given the opportunity to see the relationship between math and chess side by side. I also used chess symbols directly in arithmetic operations. The effect is that simple, one-step questions instantly become abstract and symbolic multi-step questions that require children to analyse the problem and take necessary steps to understand the abstract concept before coming up with a solution: excellent training for critical thinking and problem solving.

I have been fortunate to have the opportunities to personally field-test these problems at my own learning centre since I have been teaching math from Kindergarten to Grade 12 over the past ten years. This unique experience has allowed me to obtain a wide spectrum of feedback from different student backgrounds. The article will attempt to show how different kinds of mathematical chess puzzles can be produced and how they can potentially benefit learning outcomes.

Chess Benefits Children

Why does chess fascinate children? Dr. Montessori observed that younger children were intensely attracted to sensory development apparatus. Chess being hands-on and multi-sensory, involves coordination between eyes, brain and hands in multi-direction, and embodies concepts that are non-linear when compared to most video and computer games. As young as five, my son was thrilled to be able to capture my chess pieces as his reward: perhaps this sensory experience was what kept him hungry for more games.

In addition to having fun, playing chess has been shown to improve cognitive and critical thinking skills, reasoning and problem-solving abilities, focusing, visualizing, analytical and planning skills. Educational research papers have backed up these conclusions. (1)

Howaremypuzzlesdifferentfromothers?

In the past, many chess puzzles have been published. As for strictly chess problems, the standard has long been set by Sam Loyd, the “Puzzle King,” composer of some of the most paradoxical,

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almost phenomenal, chess problems. The most recent mathematics and chess related book entitled Mathematics and Chess has 110 entertaining problems and solutions. (2) Almost all of these published puzzles, old and new, are related to the moves of chess pieces and the majority of them are considered too difficult for most elementary students.

A math textbook series called Challenging Mathematics (3) has chess as part of a logic section but the chess content itself is stand-alone and is not integrated with any math concepts or math problems.

I used chess symbols, chess values, chess moves, chessboard, algebraic notation, chess set-up, attacking and defending counting, order of exchanges, and chess rules to create mathematical chess puzzles. The fundamental difference between my mathematical chess puzzles and those traditionally published is that chess symbols, moves, and values are integrated into math to create problems in patterns, logic, geometry, counting paths, relations, arrangement, numeration, and even data management. In this way children from pre-Kindergarten through elementary school, while learning chess, are provided an opportunity to explore mathematical puzzles by making use of the very basics of chess knowledge.

The puzzles are designed to enhance math ability using chess as a teaching tool. It is not intended to substitute for instruction in school math but rather to serve as enrichment or supplemental material. Children learn best by playing games: Math + Chess = a fun way of learning math.

Howchessandmathareintegrated

The creation of math and chess problems requires one to have a thorough understanding of chess knowledge and of the school mathematics curriculum at each grade level. Only with these qualifications and with a creative mind, can a meaningful mathematical chess puzzle be created. Chess and math problems are integrated using the following principles.

(1) Chessboard and chess pieces (See figure 1.)

The chessboard is symmetric in main diagonals in terms of its colour. The chessboard is made of four identical small boards if it is divided by one horizontal line and one vertical line going through the centre. The set-up position of chess pieces is symmetric between Black and White. The chess pieces set-up position on either side is palindromic, excluding the King and Queen.

The ranks and files are related to coordinates. When a piece is being attacked or defended, it requires some arithmetic calculations in terms of the number of attacking and defending pieces. This is the first lesson a child would learn in counting.

(2) Chess moves

A Rook’s move is a slide motion (left/right, up/down) in geometry. The between moves of the Rook before reaching a destination are similar to the commutative concept. For example, the move from a1 to h1(7) = a1 to c1 (2) + d1 to h1 (5) = h1 to d1 (5) + c1 to a1 (2).

To figure the “best” move, one needs to find out all possible paths. A similar concept in math would be to use a factor tree to find the prime factors of a number.

How does one checkmate an opponent? If a Rook is at a1, and is free to move along file a and rank 1, what does one have to consider before moving? To see if there any opponent’s pieces intersecting with the Rook would be like finding what y would be when x = 1. The checkmate positions are actually the intersections of ranks or files, which are very similar to the concept of solving equations by the graph method.

Finding the common squares (squares both pieces could control) is similar to the idea of finding common factors of two numbers.

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8

7

6

5

4

3

2

1

a b c d e f g h

Figure 1

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One would think that chess perhaps has nothing to do with fractions since all moves are all in whole numbers. Why is the Queen is the most powerful piece and why do we usually move pieces toward the centre? They all have something to do with the ratio a/64 where a is the number of squares under control.

When chess players check possible moves, the view encompasses a circular locus through 360°. For example, when checking a Rook’s possible moves, a player scans the following angles: 0°, 90°, 180°, 270°. In other words, the Rook’s move is equivalent to rotating the Rook in four directions. The same rotation concept is true for the Bishop, Queen, and King.

(3) Chess Symbols and Chess Values

The Use of Chess Symbols as Constants

Roman letters such as x, y, and z are normally used to represent unknown numeric values. These unknown letters are also called variables and they normally do not have singly defined values. On the other hand, each chess symbol has a defined meaningful point value that is related to each piece’s strength in the chess game. The point system is the static value of a piece and generally serves as a guide to making trades with the opponent. Take a look at the following example.

Let x = 1, and y = 3. Then x + y = 1 + 3 = 4.

In the above particular example, x is 1 and y is 3. But x does not always have to be 1 nor does y always have to be 3. They are variables. If we use chess symbols in the above example, we get

p+ b = 4

The above pawn and bishop have specifically defined values 1 and 3 respectively. They will not change their values just because the problem is different: in other words, they are constants. This is intuitive for children since the value of each chess symbol is pre-defined and hence has an implicit meaning to them.

Using chess symbols teaches children how to transfer from a concrete object to an abstract concept.

For example, a concrete object (like a Bishop chess piece) could be associated with a symbol b, which in turn could be substituted with a value of 3. This process of learning and thinking is in line with the concept of Montessori teaching.

When compared to using chess symbols, using animal figures or any other symbols such as x, y and z in mathematical chess puzzles would be less meaningful to children. Children do not get confused with chess symbols, nor will they become handicapped when learning variables in secondary school just because they learned substitution at a young age. The chess symbols are used only as “pictograms” or representations.

The other reason for using chess symbols in mathematical chess puzzles is that chess symbols represent movements. Coincidently, some of the directions of movements resemble some arithmetic operators: for example a Rook can move up and down and left right and thus its trace of capable moves looks like a + sign.

Each chess symbol has a specially defined direction of movement and these directions are “embedded” within each piece. I have taken advantage of chess pieces’ moves and defined them as follows.

Addition/Subtraction = Rook (Could also be Queen or King)Multiplication = Bishop (Could be also be Queen or King)Division = King (Opposition of two Kings)

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Chess Values Used

The values of chess symbols are the same as the ones used in the Chess Teaching Manual published by the Chess Federation of Canada. (4)

The cancellation technique for counting the points of chess pieces would be very similar to the concept of the subtraction property of an equation.

Chess symbols and values are integrated in arithmetic operations to create a new type of problems. My purpose of using chess symbols is to create more interesting questions and to encourage children to think ahead and use spatial-temporal reasoning to solve problems.

Each chess piece has been assigned a certain point value, as outlined in the following table.

k (king) = 0 point h (knight) = 3 points r (rook) = 5 points

p (pawn) = 1 point b (bishop) = 3 points q (queen) = 9 points

My experience in using chess values to teach arithmetic operations has been very positive. Elementary students who have not learned variables but who have worked on my worksheets using chess symbols have absorbed the concept of algebraic variables or substitution in a natural and intuitive way. There is no need to explain the concept of variable other than to mention the values of chess pieces. Consider the following example.

r + 5 = _____

Examples

Listed below are a few examples I created to show the relations between math and chess.

Example 1. Addition and Subtraction

– r ________________ __________________

+ b q q

– r ______ ______

+ r 6

6

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Example 2. Multiplication

rX

25 5×=

r25 5×=

r □

% | | &= □ =

l | | m

r □

□

)5 25r

□

□

)5 25

| | | |

r ) 25 □

r) 25 □

I created the above problem with the view that in real life, children do not really learn addition, subtraction, multiplication, or division sequentially. So here I incorporate the idea of multiplication using different formats of computing. The purpose of this worksheet is not only to learn multiplication but also to expose children to how multiplication could be written in different ways.

x x

x

x

x x

x

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Example 3. Logic

ChessSymbol LogicTraining

New chess symbols are defined as follows. In the following equation, observe the chess symbols on the left and fill in each ○� with a number.

If + + + = 10

then ÷ + + = ○�

+ ÷ ○� ÷

Chess figurines

k

r

h

b

q

p

Chess symbols

÷ (Opposition)

The above problem, suitable for Grade 3 and above, demonstrates to me that children can be led to correctly solve problems using additional “creative” chess symbols.

Using the above Chess Symbol table, find the following pattern.

Z, ÷, O, , T, , T, , F, , _____ ,

Using the above Chess Symbol table, find the following pattern.

0, ÷, 1, , ___, , 3, , 5, , ____,

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Most students could not solve the two puzzles above. However, they appreciate the sophistication of the problems when I explained to them the logical relationships. These puzzles use a combination of chess symbols and their values.

The use of chess values is much like the use of monetary values. When seen by children, chess or money figures both represent pre-defined, meaningful values. The following is an example where the values of chess pieces could be monetary values and “Total Points” could be a sum of money.

Example 4 Table Values

Fill in the different number of chess pieces to come up with each total.

Number of p Number of h Number of r

1 1 1 93 2 0 90 3 0 9□ □ □ 10□ □ □ 10□ □ □ 11□ □ □ 12□ □ □ 13□ □ □ 14□ □ □ 15

Example 5. Equations

The following examples demonstrate how chess symbols and chess values are integrated with arithmetic operations.

q + h + x = 54

x = ____

Total points

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Example 6 Addition and Subtraction, If Then - Else

10 – k = □ 5 + k = □ +

□□ □

If 10 + r = □, then 9 + r must be □. If r+ 10 = □, then r + 9 must be □.

Example 7 Cross Multiplication

p| |

8| |

□ □

× m l l m ×

□ □

□ + □ = 6

×

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Example 8Multiplication and division

q × 2 ______________ 18 ÷ 2 = □

q × r

_______________ □ ÷ 5 = □

q × r ______________ □ ÷ 9 = □

q × q ______________

□ ÷ q = □

q × h ______________

□ ÷ h = □

q × h ______________

□ ÷ q = □

r × h ______________

□ ÷ h = □

r × 8 ______________

□ ÷ r = □

The above operations should not confuse children. The expression qxh will not make sense if it is explained literally as a Queen times a Knight. However if it is translated into numerals, children will understand that they are computing 9 x 5.

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Example 9. The following puzzle requires a knowledge of chess moves.

Filling in by a chess piece Geometric shapes

p

r

□

b

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Example 10 Use chess moves to solve the following puzzle.

At first glance, many students are not able to solve this puzzle. Why? Students are so used to do computation from left to right, but this question has to be solved in an unconventional direction.

Example 11 Use chess symbol moves to solve the following.

If 2 r3 = 5 then 2 b 3 is = ____Surprisingly, some of my students have no trouble solving the above puzzle.

Example 12

Use chess symbol moves to solve the following puzzle.

? 14

? 21

h? 28

42 ?

=

? ∆

brb

bh

rt

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Example 13

If q ÷ b = r

Then what is k ÷ r = ?The above problem cannot be solved using chess values; students tried it and knew it. So what is the trick behind the idea of this puzzle?

Example 14

Rook Path

Cross mark (X) the common square(s) that all rooks could share.

Find the common factors of the following numbers.

12, 24

13, 26

8

7 t

6

5

4

3 t

2

1

a b c d e f g h

Cross mark (X) the common square(s) that all rooks could share.

Find the common factors of the following numbers.

11, 121

3, 26

8

7

6

5

4

3

2 r r

1

a b c d e f g h

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Chess move problems can be compared to math concepts. I give the above examples to demonstrate the idea with chess problems on one side and math problems on the other.

Example 15

Find values to replace ? or fill in the empty box.

Summary

I have found the idea of using chess symbols very helpful for elementary students: they can learn to solve mathematical chess puzzles using their chess knowledge.

Many cannot play chess well when against an opponent, but feel very proud that they can solve mathematical chess puzzles. Mathematical chess puzzles provide some children additional opportunities that can challenge them. For this reason, I give prizes to chess winners and also to puzzle solvers.

The most interesting aspect of using chess symbols is that the chess symbols themselves not only possess pre-defined values but also have the implied meaning of movements. These two special characteristics allow me to create some very interesting mathematical puzzles with pizzazz.

By using chess symbols, a simple one-step arithmetic problem could become a multi-step problem. As a result, chess symbols and values offer children more opportunities to work on other types of questions that can simulate children’s minds and improve their problem-solving abilities. So the benefit of working on these types of problems is double edged: improving chess knowledge and also mathematical problem-solving ability.

My mathematical chess puzzles do not involve just mechanically substituting numbers with chess symbols. Many mathematical chess puzzles also involve patterns, sequences, geometry, set theory, and logic. In other words, the integration is very diversified and also involves multi-direction visualization.

To conclude this article, I give the following pattern-like puzzles demonstrating how chess symbols and values are presented in a multi-directional and multi-sensory approach.

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101 11

999?4

qp

h

r

k p

References

Ferguson, R. (n.d.). Chess in education research summary. Retrieved from http://www.quadcitychess.com/benefits_of _chess.html.

Petkovic, M. (1997). Mathematics and chess. Dover Publications: Mineola, N.Y.

Lyons, R. and Lyons, M. (1995). Challenging mathematics. Cheneliere/McGraw-Hill: Montreal, QC.

O’Donnell, T. (n.d.). School training manual. The Chess Federation of Canada: http://www.chess.ca.

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Calculus 12: The Ultimate Pre-calculus Course

Veselin Jungic is with the Department of Mathematics at Simon Fraser University. Karl Kraemer teaches mathematics at Burnaby North Secondary School.

Veselin Jungic and Karl Kraemer

Abstract

For many students, the first calculus course is the biggest challenge at the beginning of their post-secondary education. The paper discusses questions related to the preparation of secondary school students for this challenge. In particular, the authors are interested in the role of Calculus 12 in the light of the recent changes in the curriculum of Principles of Math 12. Analyses conducted in recent years at both Simon Fraser University and the University of British Columbia indicate that Calculus 12 is the best preparation for university level calculus courses. The paper looks into why this is so, and how awareness of this influences teaching calculus both at both the university and secondary school levels.

1. Introduction

The main question that we are interested in is the following. What is the role of Calculus 12 as a link between secondary school mathematics and the first calculus course at the post-secondary level? In an attempt to answer this question, we give the definition of Calculus 12 as the Ministry of Education gives it, together with various practices in teaching Calculus 12 and different ways that post-secondary institutions in British Columbia handle freshmen who come with a secondary school calculus course. Our objective is to give one possible view of the current situation and to offer potential answers rather than to look for definite solutions of the problem of harmonizing teaching of secondary and post-secondary mathematics.

The first year calculus course is the biggest academic challenge for many students at the beginning of their post-secondary education. The course is usually topic-packed and fast-paced, and taught in large lecture halls or in a multiple section setting. The typical course covers limits, continuity, differentiation and its application. The type of applications and the level of complexity depend on the stream in which the course is. For example, a calculus course for engineering and science students might include an epsilon–delta definition of continuity, while a course for business students might not.

Common to all calculus courses at this level is the expectation that the incoming students have a clear idea what a function is and a fair knowledge about several types of functions, their basic properties, and their graphs. This includes linear and quadratic functions, polynomials in general, rational, exponential, logarithmic, and trigonometric functions. Students’ confidence in manipulating algebraic expressions and familiarity with elementary geometry is also expected. In BC, these topics are covered in Principles of Mathematics 11 and Principles of Mathematics 12. It should be clear that the expectations are quite high and to assure that those expectations are met, post-secondary institutions require a relatively high mark in Principles of Mathematics 12. For example, Simon Fraser University requires at least a B grade in Principles of Mathematics 12 for all first year calculus courses. Many institutions offer pre-calculus courses. These courses give an overview about functions and their properties, with attention to the families of functions that are listed earlier in this paragraph.

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Therefore, for success in a first calculus course, an incoming student does not need any previous experience with calculus. In other words, a secondary school calculus course is not a necessary condition for understanding all the material taught in the university course. This said, a couple of questions arise. The first question is, “Why do secondary school students take Calculus 12 if they neither need this course for a prerequisite nor get a university credit for it?” The other question goes in the opposite direction. Faced with the reality that some incoming students have previous calculus experience, how can universities and colleges best organize the teaching of calculus courses? In this article, we offer possible answers to both questions.

The paper is organized in the following way.

In Section 2, we give the description of Calculus 12 as the Ministry of Education sets it. Also, in this section we compare the list of topics in a standard pre-calculus course with the curricula of Principles of Mathematics 11 and 12. We use segments from two recent surveys to illustrate the role of the Principles of Mathematics pathway in preparing secondary students for calculus.

In Section 3, we describe how Calculus 12 has been advertised in BC secondary schools and we list different practices in teaching Calculus 12. In this section, we also describe the Calculus Challenge Exam.

In Section 4, we give data obtained by the University of British Columbia and Simon Fraser University about students’ performance in first year calculus courses. We list current offerings of calculus courses, stressing the differences between courses for students with previous calculus experience and courses for those without it. We also include information about practices at a few BC colleges.

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2. Calculus 12 – A Member of the Pack

The structure of secondary mathematics courses in British Columbia is given by Figure 1. In the BC curriculum model, math topics are re-visited in successive grades as students learn the material in greater depth and to an enhanced extent.

In the present form, Calculus 12 has been offered in BC secondary schools since the year 2000. As it is shown in figure 1, Calculus 12 is an addition to the Principles of Mathematics pathway. It is intended for “students who have completed (or are concurrently taking) Principles of Mathematics 12 or who have completed an equivalent college preparatory course that includes algebra, geometry, and trigonometry.” (Source: http://www.bced.gov.bc.ca/irp/math1012/mapath.htm)

Figure 1

Source: http://www.bced.gov.bc.ca/irp/math1012/marat.htm

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The course has been developed under the assumption that teachers have 100 instructional hours available to them. Table 1 shows the estimated instructional time in hours by curriculum topics as suggested by the BC Ministry of Education. As a comparison, in the most right column of Table 1 we give estimated instructional times for the same or similar topics in SFU Math 151 – Differential Calculus and Math 152 – Integral Calculus (Summer Semester 2004). We note that Math 151 and Math 152 each run for thirteen weeks, with three hours of lectures per week. In the 2004 summer semester, about 80% of the instructional time in Math 151 was spent on the topics that were listed by the Ministry of Education as topics in Calculus 12. It is important to observe that the Calculus 12 “teachers may freely adjust the instructional time to meet their students’ diverse needs.” (Source: http:// www.bced.gov.bc.ca/rp/math1012/calc12est.html .)

Topic Calculus 12 SFU Math 151/152

Functions, Graphs, and Limits 10 – 15 hours 4

The Derivative (Concept and Interpretations) 10 - 15 2

The Derivative (Computing Derivatives) 15 - 20 6

Applications of Derivatives (Derivatives and the Graph of the Function)

15 - 20 8

Applications of Derivatives (Applied Problems) 20 - 25 7

Antidifferentiation (Recovering Functions from their Derivatives)

5 - 10 10

Antidifferentiation (Applications of Antidifferentiation) 10 - 15 12

Table 1

Source: http://www.bced.gov.bc.ca/irp/math1012/calc12est.htm

A comparison of the instructional time for the topics in the first 5 rows in Table 1, which are common for Calculus 12 and Math 151, suggests that students in Calculus 12 have been given more than double the instructional time than students in Math 151 to learn the same material. So, why does Calculus 12 not transfer at least as the first (differential) university calculus course?

It is important to note that the curriculum of the first university calculus course does not require any previous experience with calculus and that a fair knowledge of topics from Principles of Mathematics 11 and 12 should be enough to assure students’ success in the calculus course. We illustrate this fact by Table 2 where we compare topics from Math 100, a standard pre-calculus course offered by Simon Fraser University, and topics from Principles of Mathematics 11 and 12.

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Topic Math 100 Math 12 Math 11

Graphs, Functions and Models + + +

Polynomial and Rational Functions

+ +

Exponential and Logarithmic Functions

+ +

Trigonometric Functions + +

Analytic Geometry + + +

Table 2

(The source for topics in SFU Math 100 was the course outline posted on http://www.math.sfu.ca/courses/next/math100.shtml . The source for topics in Principles of Mathematics 11 and 12 was http://www.bced.gov.bc.ca/irp/math1012/mathtoc.htm ).

Ms. Rose Albiston, a math teacher from Terry Fox Secondary, Port Coquitlam, BC, compares the current secondary school math curriculum with the curriculum before 2000 in the following way.

… the high school program … is not as much of a ‘spiral’ in presentation as it used to be before the current program. In the old program geometry was introduced in Math 8 and revisited subsequently in 9, 10, 11 and 12. In the current program the students ‘officially’ get it (possibly) in Grade 8 (middle school in Coquitlam) and again in Grade 11 only. Similarly in the old program trigonometry was introduced in Grade 9 and developed through Math 10, 11 and 12 as opposed to the students getting trigonometry only in Math 10 and 12 in the current. In addition ‘officially’ topics previously not even part of the Grade 12 course have been relegated to Math 11 where the students are anything but ready to do justice to the topics of rational, radical and absolute value inequalities and analytic geometry (with virtually no geometry background). In addition advanced algebra, which was part of Math 12, was moved down to Math 11 to make room for the finite math topics at the Grade 12 level. In other words Math 11 and 12 combined do not adequately develop significant portions of the algebra and geometry concepts that would be very important for a student to be comfortable in a post secondary calculus course. (Not to mention that some topics have simply been left out...factoring of the sum and difference of cubes, inverse variation...)”

(Rose Albistone, Personal communication)

In “Mathematics Proficiency for Post-Secondary Mathematics/Statistics Courses,

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Project Report” by L. G. Neufeld (1999), the author gives the ranking of the order of importance of the proficiency categories for calculus. In Table 3 we give some of the results obtained by Neufeld.

Proficiency Category Description Overall Importance Rating (Out of 4)

Understand and use the Function Concept 4.00

Understand and use Polynomial Expressions 3.96

Understand and use Exponential Expressions 3.96

Understand and use Straight Line and Linear Functions 3.96

Solve Equations and Inequalities 3.92

Understand and use Circular Trigonometric Functions 3.92

Understand and use Rational Expressions 3.83

Understand and use Triangle Trigonometry 3.83

Understand and use the Quadratic Function 3.83

Understand and use the Logarithmic Function 3.79

Understand and use Radical Expressions 3.75

Understand and use the Geometry of Lines and Points 3.50

Understand and use Polynomial Functions 3.42

Understand and use Quadratic Relations 3.29

Understand and use Sequences and Series 2.75

Understand and use the Geometry of Circles 2.65

Understand and use some Concepts of the Calculus 1.06

Table 3

Source: http://members.shaw.ca/bccupms/document/mathematics%20Proficiencies%20Project.pdf (p. 17)

We note that Neufeld’s 1999 Report was one the documents that determined the curriculum changes of the secondary mathematical courses that were made in 2000. Consequently, since “one of the primary purposes of Principles of Mathematics will be to develop the formalism students will need to continue on with the study of calculus” the current lists of topics in Principles of Mathematics 11 and 12 mirror in a great extent Neufeld’s findings. (http://ww.bced.gov.bc.ca/irp/math1012/mapath.htm )

As a curiosity we note that “Understand and use some Concepts of the Calculus” has a rating 1.06 (out of 4) which brings us to the following question. If Calculus 12 does not transfer to the first year calculus course and it is not necessary for success in the university course, what motivates students to take it and, as we will see later, universities to distinguish freshmen with Calculus 12 from those without it?

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A possible answer to this question is offered by the “Western and Northern Canadian Protocol, Consultation with Post-Secondary Institutions, Business and Industry Regarding Their Requirements for High School Mathematics: Preliminary Report and Findings” prepared by System Improvement Group Alberta Education and distributed on April 30, 2005.

Pathway 1 has too many outcomes to be taught in one 5-credit course per year, within the number of hours allocated to 5-credit courses. It has 153 outcomes, as compared to the 115 outcomes currently included in Principles of Mathematics pathway, which itself is considered too packed. Other options will need to be explored for delivering these outcomes. One possibility is to have students aiming for the Science take two mathematics courses in one or more of the high school grades, and the Science major areas would ask for completion of both grade 12 courses for admission. Another possibility is o divide the outcomes into those that will be taught in high school and those that will be taught in the first year post-secondary year. (p. 48)

Pathway 1 – If an outcome was identified as “master” by at least 50% of “Science” (calculus-based) respondents and/or as “master” or “expose” by at least 70% of such respondents, it was included herein. (p.4)

In our opinion, Calculus 12 follows the spirit of the above quote. This course is especially useful for secondary students because it gives them a breathing space, an opportunity to review, or to learn, the pre-calculus material and, at the same time, links them directly with the material that they will learn in the post-secondary institution of their choice.

3. Calculus 12 – In Practice

In this section we describe how Calculus 12 has been advertised in the British Columbia secondary schools. We list different practices in teaching Calculus 12 and we offer a few reasons why students take Calculus 12. Finally, we describe the Calculus Challenge Exam.

The Ministry of Education proposes the outcome for students completing Calculus 12 in the following way.

Students taking Calculus 12 should be prepared to write the UBC - SFU - UVic - UNBC Challenge Examination if they choose to do so. ( …) Some schools may choose to develop articulation agreements with their local colleges. Students under these agreements may receive credit for first-term calculus (depending upon the particular agreement).

(Source: http://www.bced.gov.bc.ca/irp/math1012/mapath.htm )

Thus the intention of the creators of the current concept of Calculus 12 in British Columbia was to allow secondary school students to obtain a university credit for first-term calculus through the Calculus Challenge Exam or through articulation agreements.

Our research of the yearly calendars and course outlines shows that, in the BC secondary school system, Calculus 12 is advertised mostly as a transition course, a course that prepares students for the first-term post-secondary calculus course. In other words, Calculus 12 is advertised as a “pre-calculus” course. We support this claim with the following two quotes.

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Calculus [12] will introduce the student to the fundamentals of differentiation and integration along with applications. Topics include graphing, maxima and minima, related rates, area, volumes, and exponential functions. This course is a good introduction to university level calculus.

(Source: http://cariboo.sd41.bc.ca/departments/math/index.html )

This course [Calculus 12] is intended as an introduction to calculus for students who intend to take calculus at university. It is not an “advanced placement” course designed to replace first year calculus. It is rather intended to show you what calculus is, in order to ease your transition into first year calculus.”

(Source: http://magee.vsb.bc.ca/dsheldan/calculus12/pdf/calculus12outline.pdf )

In our opinion there are at least two reasons for this discrepancy between the outcome suggested by the Ministry (“… should be prepared … may receive credit …”) and the practice (“… a good introduction … to easy your transition…”) One is that some secondary schools offer so-called AP (Advanced Placement) Calculus, a course that under certain grade conditions all BC universities transfer as a first year calculus.

(Note: For more about theAdvanced Placement program see http://en.wikipedia.org/wiki/Advanced_For more about the Advanced Placement program see http://en.wikipedia.org/wiki/Advanced_Placement_Program )

Another and, for the authors of this note, more important reason is that schools recognize that an additional math course helps students to achieve the level of mathematical knowledge and maturity necessary for success in post-secondary math courses.

Currently Calculus 12 is taught as a mixture of a remedial course and a preview course for first year Calculus at University/ College. The degree of mixing these two extremes depends on the school, the instructor, and the particular group of students.

We categorize teachers of Calculus 12 into four groups when considering the content of their Calculus 12 courses. (This categorization is based on the experience of the second author as a Calculus 12 teacher and his communication with other Calculus 12 instructors.)

• Teachers who generally follow Ministry of Education guidelines At the same time they do much more filling in of gaps from the secondary school program.

• Teachers who spend little time on any calculus topics and most of the time on the pre-calculus topics such as lines, functions and graphs.

Since the pre-calculus skills learned here are asked upon throughout first year college calculus, their mastery is expected.

• Teachers who focus on all the fundamentals of calculus (differentiation and integration) and remove the aspects of algebra, such as simplifying, that increase the difficulty of the questions without adding to the understanding of calculus.

The algebra that is taught/reviewed includes the common techniques embedded in the solving of calculus, such as use of the conjugates. The algebra in simplification is not stressed as the students often do not see and understand the difference in their answers.

• Teachers who teach Calculus 12 as if it were a first year post-secondary calculus course

One of the implicit factors facing all Calculus 12 teachers is that there is more to teach than there is time allowed. A slower pace brings most of the class to a better understanding of the topics covered, but not as many topics can be covered. The alternative has most, or all of the topics covered, but numerous students not completely understanding. It should be noted that students taking Calculus 12

Vector 67

are good but usually not the elite math students as the latter are most often taking AP Calculus. Most teachers aim to prepare their students for post-secondary Calculus. What they teach, or do not teach can be justified, as there is little time and so much more that could be taught.

Calculus 12 is a unique secondary school course: it is rarely taken by students not going on to university/college, yet it gives no assistance in gaining entry into post secondary institutions in British Columbia. (We note that students from British Columbia applying at Canadian universities outside British Columbia are often asked to have Calculus 12 as one of admission requirements.)

Students who take calculus at the secondary school level are preparing themselves for first year calculus at post-secondary schools. These students are all typically good at mathematics, having obtained A’s and B’s in Principle of Mathematics 11 and 12. Some of them have already completed Principle of Mathematics 12, while others are taking Principle of Mathematics 12 concurrently with Calculus 12. It is our experience that all of the students are familiar with the fact that BC universities will not look at the course as an entrance requirement. When one considers the pressure of getting high marks on numerous Grade 12 provincial exam courses and the difficulty of Calculus 12, a student should have good reasons for taking Calculus 12. What are these reasons? We suggest a few of them.

Secondary school students have heard that the first year university calculus course is extremely difficult and unlike anything they have yet learned. Topics from Mathematics 8 to Principles of Mathematics 12 have little connection to the fundamentals of calculus. Students who go through one of the mathematics streams in secondary school have had the same topics slowly taught to them over 5 years. Mathematics 8 teaches the basic skills that will be built upon in Mathematics 9 to 12. For example, the trigonometry covered in Principle of Mathematics 12 is needed in calculus, but the basics of trigonometry are first taught in Principles of Mathematics 10. Manipulation of exponents and powers is taught in Grade 8 and expanded upon in later years to include radicals and other manipulation methods such as conjugates. Each of the mathematical skills a student gains from Mathematics 8 to 12 is taught on an overlapping gradual agenda. Calculus on the other hand is a completely new mathematical concept and students use Calculus 12 to discover some of the mystery that surrounds it.

First year university calculus students often regret not taking Calculus 12 in secondary school. (This claim is based on the numerous conversations that both authors have had with their current and former students.) Those students felt they were at a disadvantage learning calculus for the first time. Having heard these regrets, secondary school students are motivated to take Calculus 12.

Some of the best secondary students want to get as high a mark as possible in the first year of university. By taking and working hard in Calculus 12, students increase their chance of maintaining a high GPA during the first year, which enables them to receive financial rewards and bursaries.

We complete this section with some facts regarding the Calculus Challenge Examination.

(Source: http://www.math.sfu.ca/outreach/schools/challenge )

The exam is organized jointly by Simon Fraser University, the University of British Columbia, the University of Northern British Columbia, and the University of Victoria. It is based on the Calculus 12 curriculum and it could be taken by students who have studied calculus in school and have not yet started college or university. Only one attempt is permitted. Writers scoring over 50% are entitled to request credit for the first term of calculus at any of the four universities listed above. The exam score will be shown as a course grade from the university they attend. We note that the choice of claiming the credit and grade is up to the student. Students who pass have the option of ignoring their exam score and taking first-year calculus for credit instead.

The cost of the exam in 2006 is $ 88. About 200 students took the exam in 2004.

Summer 20066�

4. Calculus 12 – The Day After

In this section, we give data obtained by the University of British Columbia and Simon Fraser University about students’ performance in first year calculus courses. We list current offerings of calculus courses, stressing the differences between courses for students with previous calculus experience and those without it. We include information about practices at a few BC colleges.

In the 1993 BCAMT Fall Conference, Dr. George Bluman presented the following table. (Source: G. Phyllips and C. Koe, Making Time for Mathematics, Vector 35:2, Spring 1994, pp. 18-23.)

YearNumber enrolled in first-year

calculusatUBCFailurerate

1976 1713 20%

1977 1599 29%

1991 1701 15%

1992 1622 10%

Table 4

Bluman explained the dramatic drop in the failure rate from the mid seventies to the early nineties. “In the ‘70s, less than 10% of the (secondary) students were taking one semester of calculus, and less than 10% were taking a year-long calculus course. In the ‘90s, over 65% have at least one semester of calculus in addition to Mathematics 12.”

In September 2004 at Simon Fraser University, 946 BC secondary students took one of the first semester calculus courses. Among those students, 191 came with Calculus 12, which made up about 20% of this segment of the freshmen population. Table 5 gives the percentage breakdown by class and grade of B.C. high schools graduates that were enrolled in one of the first semester calculus courses at SFU in fall 2004. We note that Table 5 supports the claim that the majority of B.C. secondary school students who took Calculus 12 obtained an A in Math 12. (The data in Table 5 and Table 6 is provided to the authors by Dr. Malgorzata Dubiel, Simon Fraser University.)

Math 12 Calculus 12 %

A Yes 16.2

A No 44.7

B Yes 3.6

B No 32.6

Table 5

The performance of the SFU Math 151 class is summarized in the following table. (SFU Math 151 – Calculus I is a class for students with intended major in mathematics, computing science or engineering.)

Vector 6�

Math 12 Calculus 12Math 11

A B

A

A

Yes

No

3.23

2.63

2.67

1.92

B

B

Yes

No

2.22

1.98

2

1.43

Table 6

For example, under column A, the grade point average in Math 151 for students who obtained an A grade in both Principles of Mathematics 12 and Principles of Mathematics 11and who took Calculus 12 was 3.23 (out of 4). We note that, for this particular group of students, there was a significant difference in performance in Math 151 depending on two parameters.

First, we observe that students with no Calculus 12 underperformed in Math 151 comparing with students with the same grades in Math 12 and Math 11 but with Calculus 12. In Table 7 we show the relative changes in the average marks in Math 151 for students with the same grades in Math 12 and Math 11 but with and without Calculus 12.


A B

A

A

Yes

No

1

0.81

1

0.72

B

B

Yes

No

1

0.89

1

0.72

Table 7

Secondly, Table 6 suggests the importance of Math 11 in preparation for a university calculus course. In Table 8 we show the relative changes in the average marks in Math 151 for students with the same grades in Math 12 and the same Yes/No status in Calculus 12 but with different marks in Math 11.

Summer 200670


A B

A Yes 1 0.83

A No 1 0.73

B Yes 1 0.90

B No 1 0.72

Table 8

Faced with the fact that a growing number of freshmen in calculus classes have previous experience with calculus, the University of British Columbia and Simon Fraser University introduced separate calculus classes for the two groups.

In fall 2001, UBC introduced new calculus courses for students without a previous calculus course. The main change was that the newly created courses had an additional credit/weekly lecture hour. The following example, a Calendar description for UBC Math 100 and Math 180, illustrates the change.

MATH 100 (3) Differential Calculus with Applications to Physical Sciences and Engineering: Derivatives of elementary functions. Applications and modeling: graphing, optimization. [3-0-0] Prerequisite: A score of 64% or higher in Principles of Mathematics 12 and high-school calculus.

MATH 180 (4) Differential Calculus with Physical Applications: Topics as for Math 100; intended for students with no previous knowledge of Calculus. Not for credit for students with High School Calculus, AP Calculus AB, AP Calculus BC, or a passing score on the UBC-SFU-UVIC-UNBC Calculus Challenge Examination. [4-0-0]Prerequisite: A score of 64% or higher in one of Math 099, Principles of Mathematics 12.

(Source: http://students.ubc.ca/calendar/courses.cfm?code=MATH )

Both classes write the same final exam, and the rest of the courses in the calculus sequence are same for the both groups.

The following table compares results of the two classes in December 2004 for students from the graduating classes of BC secondary schools. Results for 2003 are given in brackets.

Math 100 Math 180

# of students 839 (972) 460 (423)

% with A standings 37 (27) 18 (9)

% passing 93 (87) 76 (74)

Average school mark91 (91) 87 (86)

Average UBC mark 73 (68) 62 (57)

Table 9

(Source: http://www.math.ubc.ca/Schools/FirstYearcalculus/index.shtml )

Vector 7�

We note that the results for previous years (2001 – 2003) are similar to those given in Table 9 and that, together, they confirm that generally speaking, students with a secondary school calculus course perform better in the first university course than students without a secondary school calculus course, even if the latter group gets an additional weekly hour of lectures.

In 2005, Simon Fraser University decided to go with a similar concept and to offer Math150, a 4-credit calculus course for freshmen with no previous calculus experience. In the rational for introduction of Math 150 it was said that the proposed course would be an “alternate choice for Math 151, intended for students with a somewhat weaker high school background”.

It is important to repeat that the curriculum of the first university calculus course does not require any previous experience with calculus and that a fair knowledge of topics from Math 12, and as we have seen in Table 2, from Math 11 should be enough to assure students’ success in the calculus course. In our opinion, the fact that two major BC universities opted to offer an additional weekly hour for students with Math 11 and Math 12, but with no Calculus 12, shows that there is a gap between the projected and actual outcomes in math knowledge for graduates from B.C. secondary schools.

Community and university colleges play a significant role in the post-secondary education system in British Columbia. Traditionally, the colleges offer courses that are equivalent to BC Math 11, BC Math 12, pre-calculus courses, and a full array of university calculus courses. Typically, all of those courses are taught with 4 hours of lectures per week in classes with less than 40 students.

Mr. Wesley Snider, a math instructor at Douglas College, New Westminster, B.C., describes the current state of teaching calculus classes at Douglas.

It is not feasible for us to have a different stream for students with Calc 12. I presume SFU is thinking of adding an additional hour for students without Calc 12. All of our Calculus I (for science) students have 4 hours of lecture plus another 2 hours of tutorial each week. Even at that the success rates are not very good. Therefore we cannot increase the time any longer for the weaker students nor decrease the time for the stronger students (given the success rates). The only discrimination I can see at the moment may be in terms of prerequisites for Calc I. We currently take students with a Math 12 A or B grade. It could be that in future we take a Math 12 A, or a Math 12 B plus Calc 12 B (or C). I think these discussions will be coming up shortly here at Douglas.

(Wesley Snider, personal communication)

On the other hand, Vancouver Community College (VVC) in the sequence “Upgrading Courses,” offers a couple of courses that together match Calculus 12. The objectives of these two courses are given in the following way.

Math 096 and Math 097 are introductory calculus courses designed to ease the transition from Math 12 to 1st year calculus at college/university. Students completing both Math 096 and 097 are eligible to write BC University calculus Challenge Examination. Students who pass this exam and go on to a BC University may claim credit and exemption from the first semester of university calculus.

(The course content of VVC Math 096)… covers a solid review of Math 12 topics required to succeed in 1st semester Calculus as well as the fundamentals of differential calculus: the limit concept, the concept of continuity, the derivative and rate of change, basic differentiation rules, derivatives of algebraic functions, maxima and minima, applied optimization problems and curve sketching.

Source: http://upgrading.vcc.ca/math/coursedes097.cfm

Summer 20067�

We note that VVC offers a standard pre-calculus course Math 1020. We quote the course description for Math1020 as a comparison with the course content for VCC Math 096.

Pre-calculus is intended for students planning to take calculus for science, business, commerce and social science programs. Emphasis is placed on the extensive study of polynomial, rational, logarithmic, exponential, trigonometric functions, their inverse and applications. The objective of the course is to provide a solid foundation for the development of calculus.

(Source: http://www.vcc.ca/programs/detail-course.cfm?WPGM_PROGRAM_ID-155&WC2P_COURSE_ID =2682&DIVISIONID=16 )

As a curiosity we mention that there is a college in British Columbia that lists Calculus 12 as a recommended course for a one hundred level physics course.

(Source: http://web.mala.bc.ca/hearnd/Courses/Phys121/Course%20Outline/default.htm )

We conclude this section with a quote from Dr. Lin Hammill, a math instructor from Kwantlen University College, Richmond, BC. Dr. Hammill describes her experience with students coming to Kwantlen with Calculus 12.

Now, what do I think of the high school calculus? My experience has not been very positive. I find that the quality of learning varies greatly depending, it seems, on the individual teacher. Overall, I find that the students have been taught differentiation formulae and techniques and are able to find critical points. However, they really do not understand what a derivative is, what critical points really are and how to use the information supplied to solve problems (as opposed to completing exercises). They come in over-confident and so many of them do not settle down to work right away and this can be very damaging to them. They sometimes find it difficult to catch up once they realize that they actually do not know everything already. I would prefer they have no calculus at all, but have a much better grounding in the basics: functions, graphs, solving equations (without the use of a graphing calculator) and doing multi-step problem solving. This will serve them far better than a catalogue of differentiation formulae. On the other hand, since only the more able students are likely to opt for calculus in high school, perhaps this is a way to screen for those who are more likely to do well in calculus.

(Lin Hammill, personal communication)

We note that Dr. Hammill’s comment is in the line with Neufeld’s findings and recommendations given by “Western and Northern Canadian Protocol, Consultation with Pos-Secondary Institutions, Business and Industry Regarding Their Requirements for High School Mathematics: Preliminary Report and Findings”. At the same time, the experience from the University of British Columbia and Simon Fraser University shows that the majority of students with Calculus 12 perform better in a first post-secondary than their peers without Calculus 12.

Vector 7�

5. Conclusion

In British Columbia secondary schools, Calculus 12 is advertised and taught as a course that will ease students’ transition into first year university calculus. Calculus 12 is taken mostly by students planning to continue their schooling at post-secondary institutions. The University of British Columbia and Simon Fraser University have introduced separate courses for students with and without Calculus 12.

The main benefit for students taking Calculus 12 is that an additional math course in Grade 12 helps them to reach the mathematical maturity level needed for success in a post-secondary calculus course. This is obtained by reviewing the material learned in other secondary school math classes, filling in of gaps from the secondary program, and learning the basics of calculus.

Acknowledgments: The authors thank to Ms. Rose Albiston, Dr. Malgorzata Dubiel, Dr. Lin Hammill, Mr. Ozren Jungic, and Mr. Wesley Snider for their help, contributions, comments, and suggestions.

Summer 200674

Solutions to the Spring 2006 Puzzles

(Contributed by John Assadi of North Vancouver.)Puzzle 1A farmer has a field in the form of a right isosceles triangle with legs equal to 100 m as shown in the diagram. (The area is thus ½ hectare.) He wants to divide the field into 2 parts of equal areas by installing a straight fence inside the field. What is the shortest length of such a fence, and where should it be installed?

At first glance, the problem seems to be quite simple, and one of the following 3 diagrams seems to be the answer. However, none of the diagrams below illustrates the shortest distance.

Our thanks go to Colin Sherk of Invermere, John Assadi of North Vancouver, and Wolfgang Hennig of Delta for submitting their solutions to the spring 2006 puzzles. As a result of our draw, Wolfgang will receive a BCAMT t-shirt.

Solution By the web site author

Vector 7�

Let MN be the fence of the shortest length and the area of r MNB be one half of the area of r ABC. Let BM = x and BN = y.

Area of ∆MNB x y xy= ⋅ ⋅ =12

45 24

sin

Area of ∆ABC = × =12

100 100 5000( )

24

12

5000

5000 2

xy

xy

= [ ]∴ =

Let MN = l. Cosine Law in rMNB :

l x y xy2 2 2 2 45= + − cos

= + − ×

= + −

= + −

x y

x yx y xy

2 2

2 2

2 2

2 5000 2 22

100002

( )

+ −2 10000xy add oppposite terrms[ ]

= − + −

= − + −

( )

( ) ( )

x y xy

x y

2

2

2 10000

2 5000 2 100000

10000 2 12= − + −( ) ( )x y

This expression is minimum iff x=y

lmin ( )2 10000 2 1= − → = −

lmin 100 2 1 64.4 metres≅

To determine x and y

x y

x x

⋅ =

⋅ =

5000 2

5000 2 m→ = = ≅x y 50 8 84 14 . eetres

rMNB is in fact an isosceles triangle.

Summer 200676

Solution

Given triangle ABC of side-lengths 13, 20 and 21 and point D in the interior such that the areas of triangles ADB, ADC and CDB are equal, determine the exact lengths of segments AD, BD and CD.

Contributed by Gary Tupper of TerracePuzzle 2

Contributed by Wolfgang Hennig

Vector 77

Use “Heron’s Formula” to find the area of r ABC s =+ +

=13 20 21

227

Area of ABC

∆ = − − −27 27 13 27 20 27 21( )( )( )

=

15876 =126

Therefore, each smaller triangle has an area equal to 1/3 of 126 = 42

Let such that

Let b such t

a mDF DF AC

mDG

= ⊥

=

,

, hhat

Let c such that

DG BC

mDE DE AB

⊥

= ⊥,

∆CADa

a's area 212

42= ==

=

4

202

42∆BCDb's area b

s area

=

=

4 2

132

42

.

'∆ABDc c = 84

13

Let

Let

Let

z mCD

q mCF

r mFG

=

=

=

By the Cosine Law

13 21 20 2 21 200 8

1

2 2 2= + −=

= −

( )( )coscos .

sin

CC

C 00 8 0 62. .=

Let so cos cos(180- cos

m FDG T CT C C∠ = = −

= = −180

) == −

= − − =

0 8

1 0 8 0 62

.

sin ( . ) .and T

Summer 20067�

r T

r

2 2 24 4 2 2 4 4 2

60 52

= + −

=

. ( )( . )cos

.

Again, by the Cosine Law

Let m DGF Q∠ =

By the Sine Law

sin sinTr

Q=

4

So sin . ( ).

Q =0 6 460 52

Let m CGF W Q∠ = = −90

Then sin W Q Q= − = = − sin( ) cos ( . )( ).

90 1 0 6 460 52

2

By the Sine Law

sin sinWq

Cr

=

= 0 9048248613.

q

z q

= =

∴ = +

12 3 37342 2 2

.

mCD z= =1513

3

mAD mBD= =23

205 13

697 and

Likewise, it can be shown that

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