Answer Key:

今天的期末考试有些难（AoPS系列课程属于相对Advanced level，加上这门课与下周即将开始的Number Theory一样，许多知识点在日常课程教学大纲之外，属于数学竞赛内容），所以孩子们的分数若不理想，不要太灰心。更重要的是做好订正（正如我要求学生日常做好每次作业的订正一样道理），查漏补缺。

学习数学最主要的是锻炼数学思维，激发钻研的兴趣，以及和诸多其他项目学习一样: 培养孩子们迎难而上的韧劲与永不言败的毅力。 一纸分数是眼前的，一颗种子是一辈子的^_^

再次感谢所有孩子与家长这一学期来的努力！愿大家收获一颗热爱探索的种子，在合适的时机与适宜的土壤，发芽、长叶、开花！

我们新学期见！

Dear all,

Please bring a calculator to school this Sunday for the final exam.

1/18 will be the first day of the Spring Semester. We have finished AoPS: Counting & Probability in fall, and will have AoPS: Number theory in the Spring.

Course Description: http://www.newtonchineseschool.org/news/AOPSNumbertheoryCourseDescription.pdf

Textbook: AOPS – Introduction to Number Theory, by Mathew Crawford
Purchase online: https://artofproblemsolving.com/store/item/intro-number-theory

Best,
Lijia

AoPS_CP Week 14: The Binomial Theorem

Lesson Review:

• The first term’s exponents start at n and go down
• The second term’s exponents start at 0 and go up
• Coefficients are from Pascal’s Triangle, or by calculation using  n!/k!(n-k)!

Homework:

• Page 214: Ex 14.4.1, 14.4.2, 14.4.4, 14.4.5, 14.4.6
• Page 216: Ex 14.8 (e), 14.9, 14.10, 14.11

Other Resources:

• https://www.mathsisfun.com/algebra/binomial-theorem.html (highly recommended to read this one!)
• https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:polynomials/x9e81a4f98389efdf:binomial/v/binomial-theorem
• https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:polynomials/x9e81a4f98389efdf:binomial/v/coefficient-in-binomial-expansion

Something Else:

Next week we are going to have our final exam. (Ch 1-6 counts 30%, Ch 7-14 counts 70%). Please be prepared.

AoPS_CP Week 12: Pascal’s Triangle

Lesson Review:

Pascal’s Triangle (also known as Yanghui Triangle in China) is a triangular array constructed by summing adjacent elements in preceding rows. It is widely used in counting and polynomial expressions.

The entries of Pascal’s Triangle are the combinations (n choose r).

The numbers of Pascal’s Triangle also count the number of paths from the top of the triangle to the point where the number appears.

The sum of each row in Pascal’s Triangle = 2^n.

Pascal’s Identity:

Homework:

• P180 Ex 12.3.1, 12.3.3
• P184 Ex 12.4.1 (a)
• P186 Ex 12.5.1
• P187 Ex 12.12
• P189 Ex 12.19, 12.20

Other Resources:

https://artofproblemsolving.com/wiki/index.php/Pascal%27s_triangle

https://www.mathsisfun.com/pascals-triangle.html

https://medium.com/i-math/top-10-secrets-of-pascals-triangle-6012ba9c5e23

https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:polynomials/x9e81a4f98389efdf:binomial/v/pascals-triangle-binomial-theorem

https://artofproblemsolving.com/wiki/index.php/Pascal%27s_Identity

Something Else:

*I will be back to China from 12/10 to 12/27. Next Sunday (12/15) we will have Mr. Xu (the teacher of AoPS: Precalculus) to teach Ch.11 Expected Value and Ch. 13 The Hockey Stick Identity. Thank you, Mr. Xu!

There is no school on 12/22 and 12/29. Merry Christmas and Happy New Year, everyone! See you on 1/5/2020 for our last class before the final exam on 1/12!

Here is what we have met in Problem 10.3 yesterday, about the “greatest integer less than or equal to x”. The mathematical name is “floor function”. On the other hand, we have a “ceiling function”, which is the “smallest integer greater than or equal to x”.

Please visit this website for detailed information: https://www.mathsisfun.com/sets/function-floor-ceiling.html

Lesson Review:

Geometric Probability can be used in situations where we don’t have a set of individual items to count; rather, we have infinitely many outcomes which can be nicely represented using some geometric object (a line segment, a region of a plane, etc). Or say, we use geometric probability when our outcomes are continuous rather than discrete.

We learned probability using lengths and areas. (In the future, you may meet some problems that require volumes.)

P (success) = Size of successful region/size of possible region

Homework:

• P156 Ex. 10.2.1, 10.2.3, 10.2.5
• P161 Ex 10.3.1-10.3.4
• Bonus: P162 Ex 10.7, P163 Ex 10.16

Other Resources:

https://www.khanacademy.org/math/ap-statistics/random-variables-ap/geometric-random-variable/e/geometric-probability

*Next Sunday we have no school. Wish everyone a great Thanksgiving holiday! See you on 12/08!

Lesson review：
Today we have finished the second half of Chapter 8: Basic Probability Techniques and Chapter 9: Think about it!

1. Addition or Multiplication
   We add probabilities of exclusive events: P(A or B) = P(A) + P(B);
   and multiply probabilities of independent events: P(A and B) = P(A) X P(B)
2. Independent VS Dependent Events
   Independent Events: the probability that one event occurs in no way affects the probability of the other event occurring.
   Dependent Events: the probability of one event occurring influences the likelihood of the other event.
3. Avoid unnecessary computation:
   1) take advantage of symmetry
   2) ignore the irrelevant information
   3) consider multiple events at once instead of dealing with them separately
   4) be flexible, think out of the box!

Homework:
Page 135: Ex 8.4.1, 8.4.4, 8.5.1, 8.5.4, 8.5.6
Page 151: Ex 9.8, 9.10, 9.12, 9.14

Additional Resource:
https://www.siyavula.com/read/maths/grade-11/probability/10-probability-02

https://www.wyzant.com/resources/lessons/math/statistics_and_probability/probability/further_concepts_in_probability

https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:prob-comb/x9e81a4f98389efdf:dependent-events-precalc/v/independent-events-1

https://www.khanacademy.org/math/ap-statistics/probability-ap/probability-multiplication-rule/v/compound-probability-of-independent-events

See you next Sunday and I look forward to exploring the geometric probability with you! Have fun!

Flexibility Assessment:

Thanks to everyone for participating in the pilot Flexibility Assessment from Harvard Graduate School of Education. Your feedback will help to revise this assessment. This study will be carried out in Finland, Sweden, Spain, America, and China.

It’s interesting to see that most of the students in this class solved the problems in a traditional way in Part I, but developed better (more innovative) methods in Part II for the same set of problems. Perhaps because of the time constraint, or being used to the more traditional ways. We learned that for some problems, there are some “innovative strategies”. Be familiar with the properties and principles in math, you can solve problems faster, and better:)

Lesson Review:

8.2 Probabilities and Addition
If events A and B are mutually exclusive, then P(A or B) = P(A) + P(B)
More generally, if A1, A2, …, An are mutually exclusive, then P(A1 or A2 or A3 … or An) = P(A1) + P(A2) + … + P(An)

If the events are not mutually exclusive, you have to correct for the overcounting:
P(A or B) = (P(A)+P(B)-P(A and B)
(think about the Venn Diagram)

8.3 Complementary Probabilities
P (A does not occur) = 1 – P(A occurs)

Sometimes it is easier to calculate the “opposite” events, so the complementary method can work better than the constructive method. (Refer to Example Problem 8.6)

Homework:
Page 128: 8.2.1-8.2.3, 8.2.6
Page 130: 8.3.1-8.3.4

Next week we are going to finish Chapters 8 and 9. Please be prepared and look forward to seeing you soon!

Lesson review:

1. Midterm Exam Paper Correction
2. Chapter 7: Introduction to probability
   1)P(A) = Number of outcomes for A / Number of possible outcomes
   or P(A) = Number of successful outcomes/Number of possible outcomes
   2). The range of P(A) is [0,1]

Homework:
Page 114: Ex 7.2.1-7.2.2
Page 117: Ex 7.3.1 -7.3.3
Page 120: Ex 7.4.1- 7.4.5
Page 121: Ex 7.12 (Classwork)

Annabelle gave a close guess to the “Shooting star” warm-up exercise: around 80%. The accurate probability is 84%. We will discuss it in class next Sunday.

See you next week!