COMPARISON OF K- MATH IN THE U.S. & OTHER REGIONS …Chris L. Yuen, Ed.D., EOC Associate Professor of Mathematics, SUNY University at Buffalo; [email protected] ... (Module 1 or 2)

COMPARISON OF K-12 MATH IN THE U.S. & OTHER REGIONS WITH POTENTIAL IMPACT

Presenters

Chris L. Yuen, Ed.D., EOC Associate Professor of Mathematics, SUNY University at Buffalo; [email protected] Mytra Groeneveld, Professor of Mathematics & Coordinator of

Developmental Mathematics, Manchester Community College; [email protected]

Oiyin Pauline Chow, M.S., Senior Professor of Mathematics (retired),

Central Pennsylvania’s Community College, HACC; [email protected] Shane Tang, M.S., Associate Professor of Mathematics, Salt Lake Community College; [email protected]

2

Goals for our presentation

Identify the uniqueness of math education systems, assessment, and culture in various locations

Recognize how local cultures, beliefs and perceptions of parents and students can affect individual mindsets on learning mathematics.

Develop overarching themes based on the comparative findings, and discuss potential impacts for mathematics teaching and learning in the first two years of college.

Lead a discussion about mathematics teacher preparation for high school and college teachings.

3

Mathematics content and cognitive domains in Grade 8 based on TIMSS

4

Participating countries in TIMSS

5

PISA Results in 2015 – Mathematics [NEW 10/10 CY]

6

International Comparison Among 15-Year-Olds

Snapshot of Performance in Mathematics Mean Score in

PISA 2015 Rank Score

Difference OECD

average 493 –1

Singapore 564 1 1

Hong Kong 548 2 1

7

International Comparison Among 15-Year-Olds (Cont.)

Snapshot of Performance in Mathematics Mean Score in

PISA 2015 Rank Score

Difference Finland 511 13 –10

Australia 494 25 –8

United States 470 41 –2

Puerto Rico 378 - -

8

Background of our Study

Past research has shown that high failure rates exist, from 35% to over 50%, in entry-level U.S. college mathematics courses (Attewell et al., 2006; Stevenson & Zweier, 2011; Shakerdge, 2016).

To address this issue, Nagle of Penn State Behrend currently conducts and leads an NSF project, involving high school teachers and college professors. These participants are engaging in a longitudinal dialog about how to best transition high school students to college setting.

Also, based on the PISA study (2012 and 2015), Hong Kong was ranked above average (3rd in 2012 and 2nd in 2015), followed by Canada (13th in 2012 and 10th in 2015), and below average for the U.S. (36th in 2012 and 27th in 2015) in mathematics achievement among high school students.

9

Education System

Hong Kong Singapore

- Secondary School (F.1-F.6): Six years of compulsory math - Extended math classes (Module 1 or 2) for STEM students in addition to compulsory math (F.4-F.6)

- Primary Math Grades 1-6 - Lower Secondary Math

Grades 7-8 - Secondary Math Grades 1-4

10

Education System British Columbia, Canada U.S.

- Elementary School (7 years) - Secondary School (6 years) - Students must earn the

following to graduate: a Math 10 (4 credits) a Math 11 OR 12 (4

credits)

- High School (8th to 12th grade): Min. two years of math - Options for advanced / honors classes for all students - Students are assigned to the math classes based on their ability

11

Education System

Australia Finland

- Primary School – 7 or 8 years

- Secondary School – 3 or 4 years

- Senior Secondary School – 2 years

- 10 years of compulsory school starting at the age of seven

- 3 years of upper secondary school preparing for the Matriculation Test, or

- 3 years of vocational education, or

- Enter the workforce 12

Education System

Puerto Rico

- Based on American Model with Spanish as teaching language at public schools - Primary School – 1 to 6 grades - Secondary School – 7 years - Vocational Education or Tertiary Education

13

Curriculum

Hong Kong

U.S.

Grades 7 -12 Compulsory Math with three strands: Number and Algebra

Measures, Shapes, and Space

Data Handling

Grades 8 - 12: Algebra I, Algebra II,

PreCalculus, Trigonometry, Calculus I,

Statistics

Grades 10 – 12 Extended Math (by choice):

Module 1 (Calculus and Statistics) – for disciplines or

careers requiring mathematical application

Module 2 (Algebra and Calculus) – for math-related fields

and careers 14

Curriculum

Hong Kong

Grades 7 - 9 Junior Secondary Compulsory Math

5 class periods per week 40 minutes per period

480 periods available for the three years 320 hours of lesson time

15

Curriculum

Hong Kong

Grades 10 – 12

Compulsory Math with the same three strands

250 to 313 hours of lesson time

Extended Math (by choice)

Module 1 (Calculus and Statistics) or

Module 2 (Algebra and Calculus)

125 hours of lesson time 16

Curriculum - British Columbia, Canada

17

Curriculum - Australia

First 10 years, including Foundation Year 1) Number and Algebra 2) Measurement and Geometry 3) Statistics and Probability At each year level, the achievement standards focus on Understanding, Fluency, Problem-Solving, and Reasoning

18

Curriculum – Australia (Cont.)

Four senior secondary subjects for Mathematics: 1) Essential Mathematics – focus on use 2) General Mathematics – focus on solving problems 3) Mathematical Methods – focus on development in Calculus and Statistical Analysis 4) Specialist Mathematics – focus on mathematical proofs

19

Curriculum - Singapore

Three Content Strands in Primary and Lower Secondary Math Grades 1-8: -Numbers and Algebra -Geometry and Measurement -Statistics and Probability

20

Curriculum – Singapore (Cont.)

Three Content Strands and one Process Strand in Secondary Math Grades 1-4: -Numbers and Algebra -Geometry and Measurement -Statistics and Probability -Mathematical Processes

21

Singapore Education

22

• 6th graders take Primary School Leaving Exam

• assigned to a secondary school based on merit, and

then their choice.

• Gifted Education Programme

• "Express", "Normal (Academic)", or "Normal

(Technical)"

Curriculum - Finland

-Numbers and Calculations -Algebra -Geometry -Measurement -Data Processing and Statistics

23

Curriculum – Finland (Cont.)

Grades 1-2: focus on basic mathematical concepts and structures Grades 3-5: focus on developing mathematical thinking Grades 6-9: focus on depending understanding of math concepts and modeling skills

24

Curriculum – Puerto Rico

• Kindergarten – counting, basic number sense

• Grades 1 to 6 – Math progressing at each level

• Grades 7 to 9 – Intermediate School - PreAlgebra

• Grades 10 to 12 – High School – Algebra I and II, Geometry, Trigonometry, Pre-Calculus

25

Methods and Over-Arching Themes

Methods: Each presenter reviewed existing literature for each of the seven locations to find uniqueness from each. We compared and contrasted the findings, and cast them into several over-arching themes as follows:

An existing space for mathematical inquiry in a creative manner

Attitude toward Education

Connectedness of mathematical ideas in curriculum

Assessment and its access to the Cognitive Domain

26

Theme 1: An existing space for mathematical inquiry in a creative manner

27

AMPLE SPACE SOME SPACE LITTLE TO NO SPACE

Finland Hong Kong United States

Canada Australia Puerto Rico

Singapore

Theme 1: Creative Space (Ample Space) Finland

28

2002 Basic Math Exam

11. For which values of 𝑞 is the polynomial function 𝑓(𝑥) = 𝑥3 + 𝑥2 +𝑞𝑥 + 1 decreasing on some interval? Find this interval.

2000 Basic Math Exam

11. Prove that the polynomial 𝑓(𝑥) = 𝑥3 − 4𝑥 − 2 has a zero in the interval [2,3]. Find this zero using the bisection method to two correct decimals.

Theme 1: Creative Space Canada

29

A Sample Question from Canada

30

A Sample Question from Canada

31

Sample Questions from New York, United States

32

Theme 2: Attitude toward Education

33

GENERALLY POSITIVE GENERALLY NEUTRAL GENERALLY NEGATIVE

Finland Canada United States

Hong Kong Puerto Rico

Australia

Singapore

Cultural Beliefs About One’s Abilities to Learn Mathematics

Schoenfeld (1989) and Tang (2007) both characterized how learners perceive math abilities: Nature versus Nurture

Nature: the belief of the abilities in learning math comes from natural endowment and born talent.

Nurture: the belief that sufficient facilitation and the learner’s own effort can overcome difficulties in learning mathematics.

34

Manifestation of the Cultural Beliefs (U.S.)

An examination of the attitude toward homework at the high school level: Method: Focus group discussion with six PA HS math teachers in September 2016. Findings: • Homework given to students is largely procedural, similar to Schoenfeld’s (1989) description. • Minimal opportunities for learners to experience disequilibrium when doing homework.

35

Manifestation of the Cultural Beliefs (U.S.) - Continued

Findings: (Continued)

• Learners tend to resist HW when confronted with disequilibrium. • Parents also tend to resist when they observe their children “stuck” on homework problems. • School administrators do not generally support the math teachers when learners and parents are pushing back.

Interpretation:

The findings are consistent with the belief of natural endowment of math abilities for many U.S. students.

36

Manifestation of the Cultural Beliefs (Hong Kong)

There exists evidence that many learners believe in nurturing as a means to learning. Stay tuned in the later portion of the presentation about the tutoring practice in Hong Kong.

Manifestation of the Cultural Beliefs (B.C., Canada)

We will argue that while many Canadian learners believe in the nature’s disposition for math learning, there exists evidence at the policy level that B.C. system attempts to change this belief.

37

Honoring the ways of knowing from First Peoples (B.C., Canada)

Collaborate with local First Peoples Elders and knowledge keepers (B.C. Math 12) Lesson Episode from Gear (2012) Haida creation story The Raven and the First People How many children can fit in a clam shell Parental orientation

38

Finnish Attitudes toward Education

Finland felt insecured toward the end of WWII with being sandwiched between two giants, USSR and Germany. This was a motivation for them to overhaul their education system.

Contrary to the U.S. notion of “No Child Left Behind,” Finland embraces diversity that different individuals have unique talents, and each must be nurtured in a personalized way.

Play, rest, and stress management are part of the Finnish schooling components. Students enjoy nap time, ample amount of recess in school, and healthy socialization, both in primary and secondary grades.

39

Finnish Attitudes toward Education

To be a math teacher in Finland, one must possess at least a master’s degree in mathematics as well as a degree in education. Apprentice teaching lasts approximately two years. Teaching load is generally less than those of the U.S., and prep time is generally more than those of the U.S.

It is a common practice that a teacher “follows” his/her students for three to four years. The teacher is responsible for devising individualized curriculum for his/her students, as opposed to a state prescribing a curriculum for all students in each specific grade.

Those who desire to attend universities receive a more academic oriented curriculum where a large portion up to Calculus is housed. Those who prefer vocational training learn applied math that is relevant to their trade skills.

Finnish matriculation exam is a university entrance exam, and those who want to attend universities elect to take the exam. It is not a requirement to graduate from secondary schools.

40 Source: Pasi Sahlberg (2014) Finnish Lessons 2.0: What Can the World Learn from Educational Change in Finland?

Australian Attitudes toward the Math Subject

Wang and Wu (2010) found that Australian students tend to enjoy learning statistics more so than the Chinese counterparts.

Norton and Rennie (1998) found that male students tend to have a slightly more positive attitude toward the subject that female students. Overall in their findings, positivity declines in both genders as students moving up from grades 8 to 12.

De Lourdes Mata, Monteiro and Peixoto (2012) also found the attitude declination, particularly more drastic for female students. They linked attitude and achievement with a positive correlation. Interestingly, they found a strong relationship showing the importance of Australian teachers generally do have positive attitudes towards the math subject.

Pritchard (2004) found that New Zealand parents are generally positive about helping their children in mathematics.

41

Differences Hong Kong U.S.

Political uncertainty Believe that (STEM)

education is the key to success.

Believe that learning math trains the brain.

Parents generally support school teaching.

Invest additional resources and time in tutoring.

Politically stable Believe that education is one of

the ways to success. Believe that learning math is to

fulfill the requirement. Parents generally involve in

school teaching. Invest additional resources and

time in education, sports and music.

42

Differences

British Columbia, Canada Politically stable Believe that college and university education is an

investment leading to one benefit of increased employability.

Many proactive parents send their children to private schools – about 12% in B.C.

43

Singapore Math

44

• Covers fewer topics in greater depth • three-step learning process: concrete, pictorial, and

abstract

• Ready for algebra and geometry in middle school

• Learn at different paces

• Develop foundation for further math learning

• No need to reteach

Singapore Math Believe

45

• U. S.: “Some are born with naturally talented/gifted in

math, while others are not.”

• Singapore: “Effort” is the key!

hands-on group activities

pictorial phase

abstract equations.

Parental Expectations – Hong Kong (USA Today)

“Throughout much of Asia, education is seen as the only path to success. Parental demands, fear of failure, competition and pride are fueling Asia’s academic ascension. Simply put, children in Asia study with a purpose.…typical Asian Student: committed, diligent, competitive, passionate, focused and ambitious.”

46

Attitudes toward Education: Hong Kong

1,016 students from 14 primary and 27 secondary schools were polled in 2015. 67.6% of 4th grade and 5th grade students took

tutorial classes after school 40.8% of 10th grade or 11th grade students took

tutorial classes after school 47


Private tutoring in 2013: 85% of 11th grade students have private tutoring US $ 272 million private tutoring industry or

cram school Top cram schools can have 10,000 students per

month Most popular tutors can earn at least US $ 410,000 a year

48


US $ 54 per hour 10 years ago Math/Science tutors may charge

between US $108 per hour 10 years ago

49


1/3 secondary school students spent about US $ 2.58 million per month on private tutoring in 2004-2005

Private Tutoring Industry is worth at least US $ 55 million

Other reports claimed that the industry generated more than US $ 494 million

50

“In Hong Kong, the Tutor as Celebrity”

August 18, 2013

51

August 18, 2013

“Advertisements for star tutors in Hong Kong can be seen all over here: on billboards that loom over highways and on the exteriors of shopping malls. Invariably, the local teaching celebrities are young, attractive and dressed in designer outfits befitting pop stars. But beyond the polished shine, the advertisements also claim that their celebrity tutors can help students ace Hong Kong’s university entrance exam.”

54

Cultural Differences: Tutoring (United States)

“The tutoring market is fragmented. Some online tutoring marketplaces aggregate a large number of private tutors. One site has over 34,000 registered tutors in California. The hourly rate is, in average US $ 48.’’ Private tutoring is not as common in the U.S.

55

Cultural Differences: Tutoring (B.C., Canada)

In Toronto, 60% of more than 70,000 high school students have private tutors.

US$ 27-35 per hour for University student tutors US$ 39-47 per hour for certified teacher tutors US$ 217 for 2 ½ hour sessions a week + US$ 116

registration fee for children ages 3 to 6 at Tutoring Centers

At least one tutoring business per 10 blocks or 10 minutes walk in Vancouver

56

Cultural Differences: Tutoring (B.C., Canada)

Kumon has 330 centers across Canada enrolling students as young as preschool through high school in math and reading with 13% in preschool and 67% in grades 1-5.

A 2007 study by the Canadian Council on Learning stated that 1/3 of Canadian parents would hire a private tutor or a tutoring company for their children aged 5 to 24.

The global tutoring market may have surpassed US$ 107 billion this year.

57

Theme 3: Connectedness of Mathematical Ideas in Curriculum

58

Well-Designed and

Tightly Connected

Somewhat Well-Designed

Loosely Connected Generally Fragmented



Singapore

Characteristics of B. C. and Quebec Math Curricula, with comparisons to those of U. S.

American children consistently rank below most other industrialized nations on international mathematics assessments.

American students, with the spiral curriculum was merely a

fragmentation of computationally-oriented content; hence, it lacks focus – highly repetitive – and does not provide … a rigorous math and science education by international standards.

59

Some Evidence of Spiraling in B. C.’s Curriculum

In B.C., by grade eleven, 60% of instructional time was earmarked for Algebra, with 33% of objectives repeated from prior years. In contrast, under Quebec’s Data Analysis, students solved problems using correlation and probabilities. Side note: Unlike British Columbia’s mathematics curricula, Quebec's also demonstrated substantial dedication to mental calculations.

60

Inter-connectedness among knowledge in multiple subjects

A grade eleven objective, in Quebec, required that students summon all their knowledge (algebra, geometry, statistics and the sciences) and all the means at their disposal (computers, calculators, instructional materials) to solve problems. This interconnectedness was not evident in B. C.'s curriculum.

61

Influence of different educational philosophies

Quebec’s curricular documents wove together activities fostering conceptual understanding, calculations, operational applications and problem-solving. Quebec’s objectives directed teachers to assign open-ended problems thereby indicating a cognitivist learning orientation. By contrast, British Columbia’s curriculum reflected a more behaviorist view of learning. Its mathematics curriculum dealt with problem-solving as a separate topic, unlike Quebec’s where problem-solving was integrated throughout all learning objectives.

62

Australian Year 8 students

• In 2015, Australian students were outperformed by 12 other countries

• There was a dip in score in 2007 and a recovery in 2011.

• Australian students scored about the same in 1995 and in 2015.

• 64% met the Intermediate international benchmark – the proficient standard for Australia.

63

Other Observations about the Grade 8 Australian Students

• Male students performed slightly better. • Those who had more books at home perform

better. • Non-Indigenous students performed better. • Other factors: geographic location and

language spoken at home

64

Theme 4: Assessment and its access to the Cognitive Domain

65

Assessing

Deep Learning

Assessing

Surface learning



Singapore

Assessment

Hong Kong U.S.

Hong Kong Diploma of Secondary Education (HKDSE) Examination at12th Grade (exam results are generally used for local colleges/universities admission after 12th Grade)

Graduation Exam as required by each state SAT, ACT, AP etc (not necessarily compulsory for students)

Canada

Provincial Exams 12th Grade Research Projects

66

Assessment (Cont.)

Singapore - PSLE at the end of the final year of primary school, i.e. P.6 • Mathematics Paper 1 (50 minutes) • Mathematics Paper 2 (100 minutes) • Foundation Mathematics Paper 1 (60 minutes) • Foundation Mathematics Paper 2 (75 minutes) List of approved science calculator is allowed in paper 2 exams - GCE N(T), N(A), O, and A level exams.

67

Assessment (Cont.)

Australia Puerto Rico

The numeracy components

comprise 45 multiple‐choice questions.

Fourth- and eighth-grade students participated in NAEP mathematics and META assessments

68

Cultural Differences: Calculator Use (Hong Kong)

Elementary students learn how to perform basic arithmetic operations without using a calculator

When students take the Hong Kong Diploma of Secondary Education Examination (HKDSE), they can only bring a calculator from a list of permitted scientific calculators

69

Cultural Differences: Calculator Use (Canada)

Hand-held devices for mathematical computations. Students in grades 8-10 may use teacher approved scientific calculators. Students in grades 11-12 and the IB Diploma program may use graphing calculators: TI-83/84 or Nspire (non CAS). During provincial exams, a calculator inspection will be conducted.

70

Cultural Differences: Calculator Use (H.K. vs U.S.)

In 2007, 52% of 4th grade teachers in Hong Kong did not permit calculators and 31% in U.S. In 2014, use of graphing calculator is 12% in Hong Kong and 77% in U.S. In 2014, use of scientific calculator is 88% in Hong Kong and 23% in U.S. Calculators are not allowed in tests for nine- and

11-year-olds in Hong Kong 71

QUESTIONS?

72

On Test Days

1

What would a country do to help its students succeed?

Scholastic Aptitude Test

3

The South Korean government administers the College Scholastic Aptitude Test (Suneung in Korean) for high school seniors once a year. The Suneung score will determine which college they will attend, a result that some students and families believe is the primary determinant for the students' future. Since this test is administered only once a year in November, students and their families devote all the resources they can to prepare for this fateful day.


4

Since this test is administered only once a year in November, students and their families devote all the resources they can to prepare for this fateful day. Students go on special health diets, take medicinal boosters, say extra prayers, and spend more than twelve hours a day, every day for years, studying for this test.


5

In the months leading up to the test date, Buddhist temples and Christian churches are filled with parents -- usually mothers -- with a photo of their children in their school uniform, giving special prayers for their children to perform well on the test date for admission to one of the top three universities in South Korea: Seoul National University, Korea University, or Yonsei University.


9

People refer to these three top schools as the SKY universities for short. “Reach for the SKY!'' children will hear growing up. South Korea's National Youth Policy Institute and other centers that survey students found that South Korean high school students sleep, on average, just 5.5 hours every night, because they spend the remaining hours studying.


10

On exam day, the entire nation contributes to positive test-taking conditions for the 600,000 eighteen-year-olds by giving all other students the day off so that the schools are quiet. The stock market, public offices, and banks all open one hour later to keep the streets free of unnecessary traffic for students to get to their testing centers on time.


12

All planes are rerouted or grounded during the afternoon's English-language listening test so students can focus on listening.

Regular Days

13

SAT Exam Day

14


15

The Korea Electric Power Corporation, the largest electric utility corporation, commissions four thousand electricians to be on standby in case any testing center's lights go off. Police officers take emergency calls in the morning to whisk late students to their testing locations on their police motorbikes.

Comparison of K-12 Math in U. S. and Other Regions withPotential Impact

by Chris L. Yuen, Oiyin Pauline Chow, Myrta Groeneveld, and Shane Tang

1) The exponential function h(x) = Aeγx can be used to[estimate] mid-level height of the Eiffel Tower in Paris,France, where h(x) is the height of the tower at x meters fromthe left pillar, A = 15 and γ is a constant.

a) Suppose that the first level of the tower is at height of 57m and at a horizontal distance of 22 m from the left pillar.Determine γ.

b) Determine the height of the second level if it is located at ahorizontal distance of 35 m from the left pillar.

2) The point P(x, y) is equidistant from the lines y = 3 and3x +4y −18 = 0, and lies in the shaded region of the diagram.Find the equation of the locus of P.

x

y

5

5

3x+ 4y − 18 = 0

y = 3

3) A clothing company created a diagram for a vest. Toshow the other side of the vest, the company will reflect thedrawing across the y-axis. What will be the coordinates of Cafter the reflection?

x

y

−5 5

5

A

B

C

D

E

F

4) Consider a circle with center O and radius 5 cm. The lengthof the arc PQ is 9 cm. Lines drawn perpendicular to OP andOQ at P and Q respectively meets at T. Find the shaded area.

O

P

5 cm

9 cm

Q

T

CMC3 Page 1 March 3, 2018

5) What is the area, in square units, of trapezoid QRST shown?

A = 12

h(b1 +b2)Q T

SR

6

8

20

6) ABCDE is a regular pentagon.What is the measure of ∠EDB?

A B

C

D

E

7) ABCDE is a regular pentagon and ABGF is a square. EFproduced meets DC at P.Find ∠EAF, ∠DEP, and ∠DPE.

A B

C

D

E F G

P

8) Give the triangle on the right and the values: sin30◦ = 0.5and cos30◦ ∼ 0.866. Approximate the length of a.a) 30 cm

b) 17 cm

c) 13 cm

d) 7.5 cm

a

15 cm

30◦

9a) Show that the circle curve (x −10)2 + (y −12)2 = 100 touches the y-axis.b) Given the line 4y +3x = 28 is tangent to the circle with centre (10, 12). Find the equationof the circle.

10) Find an equation of a circle with (−6, 2) and (4, 4) as endpoints of the diameter.




11) C is the centre of the circle PQS. OR and OP are tangentto the circle at S and P respectively. OCQ is a straight line and∠QOP = 30◦.

a) Show that ∠PQO = 30◦.

b) A rectangular coordinate system is introduced so that thecoordinates of O and C are (0, 0) and (6, 8) respectively. Findthe equation of QR.

C

O

Q

P

R

S

30◦

12) ABCDEF is a regular hexagon.

Prove: 4AED ∼=4DBA

A B

C

DE

F P

13) 4ABC and 4DEF are drawn with Â = D̂, B̂ = Ê, and Ĉ = F̂.

Prove the theorem which states that if two triangles 4ABC

and 4DEF, are equiangular, then DEAB

= DFAC

.

A

CB

D

E F

14) The club members hiked 3 kilometers north and 4kilometers east, but then went directly home as shown by thedotted line. How far did they travel to get home?

Home

3 km

4 km

N

15) Find the range of values of p for which the equation 2px2 + (p −4)x = (2−p)x2 +2 hasno real roots.


16) Consider the two functions f (x) = x +2 and g (x) = 7−5x.Which is the composite function

(f ◦ g ) (x)?

a)(

f ◦ g ) (x) =−5x +9b)

(f ◦ g ) (x) =−5x +3 c)

(f ◦ g ) (x) =−4x +0

d)(

f ◦ g ) (x) =−5x2 −3x +14

17) ABCD is a rectangle. Find CF.

a) (a +b)sinθ cm

b) (a +b)cosθ cm

c) (a sinθ+b cosθ) cm

d) (a cosθ+b sinθ) cm

e)p

a2 +b2 sin2θ cm θA

D

B

C

F

a cm

b cm

18) What is the inverse function of f (x) = x +1?

a) f −1(x) =−x +1

b) f −1(x) = x −1

c) f −1(x) = 1x +1

d) f −1(x) = 1x −1

19) Describe and compare the key feature of the graphs of the functions

a) f (x) = x, b) f (x) = x2, c) f (x) = x3, d) f (x) = x3 +x2, e) f (x) = x3 +x.Investigate numerically, graphically, and algebraically, with and without technology, theconditions under which an even function has an even number of x-intercepts.

20) Let f (x) = log3 (1+ex) for all x. Show that f (x) has an inverse.

21) Given log9 a = log12 b = log16(a +b). Finda

b.

22a) Show that for k > 0, 1(k +1)2 −

1

k+ 1

k +1 < 0.

b) Use mathematical induction to prove that for all integers n ≥ 2,1

12+ 1

22+ 1

32+·· ·+ 1

n2< 2− 1

n




23) Find the range of values of k such that the line y = x cuts the curve y = x2+kx +1 at twopoints.

24) In the figure, the curve y = x2+bx+c meets the y-axis at C(0,6)and the x-axis at A(α,0) and B(β,0), where α> β.a) Find c and hence find the value of αβ.

b) Express α+β in terms of b.c) Using the results in (a) and (b), express (α−β)2 in terms of b.

Hence find the area of ABC in terms of b.

25) Which equation BEST represents the part of the graph shownon the right?

a) y = 1.75x b) y = 1.75x2 c) y =−1.75x d) y =−1.75x2

26) In the figure, the line y = mx +k cuts the curve y = x2 +bx + cat x = α and x = β. Find the value of αβ.

a) −b b) c c) m −b d) k − c e) c −k


27) Which of the following is the graph of y = x2 +2x −8?

28) Based on the graph on the right, which expression is a possiblefactorization of p(x)?

a) (x +3)(x −2)(x −4)b) (x −3)(x +2)(x +4)

c) (x +3)(x −5)(x −2)(x −4)d) (x −3)(x +5)(x +2)(x +4)

29) When the expression x2 −3x −18 is factored completely, which is one of its factors?

a) (x −2)b) (x −3)

c) (x −6)d) (x −9)

30) The graph of f (x) = x2 is shown on the grid. Which statementabout the relationship between the graph of f and the graph ofg (x) = 7x2 is true?a) The graph of g is narrower than the graph of f .

b) The graph of g is wider than the graph of f .

c) The graph of g is 7 units below the graph of f .

d) The graph of g is 7 units above the graph of f .


CMC3Presentation03-03-2018South KoreaInternational Math Handout

Documents

COMPARISON OF K- MATH IN THE U.S. & OTHER REGIONS …Chris L. Yuen, Ed.D., EOC Associate Professor of Mathematics, SUNY University at Buffalo; [email protected] ... (Module 1 or 2)