Sums of Consecutive Counting Numbers

lilijia Post in AoPS_Number_Theory
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Sums of Consecutive Counting Numbers

by Mr. John Bookston

Definition:  A sum of consecutive counting numbers can start with any counting number add at least the next bigger counting number and stop adding after any number of consecutive integers.

Examples:  3+4 = 7                   7 is a consecutive sum of counting numbers

   1+2+3+4+5 = 15    4+5+6 = 15 and 7+8 = 15     15 is a consecutive sum in 3 distinct ways.

Investigation: 

For numbers that are consecutive sums, investigate patterns to predict the number of distinct ways larger numbers can be written as consecutive sums.  

Starters:       1 and 2 are not consecutive sums

Every odd integer starting with 3 can be written as a consecutive sum of two counting numbers:

                                      (n-1)/2  +  (n+1)/2  =  n       for every odd number, n.

Make a conjecture about which numbers are expressible as “the sum of 3 consecutive counting numbers”. 

Extension: Do the same for numbers that can be expressed as sums of 4,5 and 6 consecutive counting numbers.

Extension:  Consider the prime factorization of the counting numbers that cannot be expressed as a consecutive sum.  Make a conjecture as to which counting numbers less than 1000 fall into that category.

Enjoy.             

Good luck. 

SAT Math Practice 3 Answer

lilijia Post in SAT math
Comments Off on SAT Math Practice 3 Answer

Dear all,


In the email please find the answer key to SAT Math Practice 3. Please do not hesitate to reach out if you have any questions. Have a nice weekend and see you on 3/1/2020!


Best regards,

Lijia

SAT Math Practice Test 3

lilijia Post in SAT math
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Dear all,

In the parents’ email please find SAT Math Practice Test 3 if you need some more practice during the spring break. I will email the answer key next weekend.


There is no school for the following two weeks. Please feel free to contact me if you have any questions regarding our previous exercises. I will be very glad to help.

Enjoy a safe and lovely spring break, and I look forward to meeting you on 3/1!

Best,

Lijia

Answer to the bank question

lilijia Post in AoPS_Number_Theory
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Dear all,

Enclosed please find the excel file with the answer to the bank question in our brain stretcher last week. Many thanks to Mr. Zhang (Jason’s dad) for sharing!

Q4: Bank of America has money in the forms of $1, $5, $10, $20 and $100. How many different combinations can Tony withdraw $100 dollars at the counter?

https://docs.google.com/spreadsheets/d/1_GSWdR27K8XLUscFJAIIvpCTKWQFYNtYVfrSxpwv00s/edit?usp=sharing

There is no school for the following two weeks, but please feel free to contact me if you have any questions regarding our previous chapters/exercises. I will be very glad to help.

Enjoy the spring break, and I look forward to meeting you all when we are back!

Best,

Lijia

SAT Math 2/9 Online Class

lilijia Post in SAT math
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Thank you for attending the online class and bringing the math questions to the class discussion today! I enjoyed working with you all.

Here are the questions that we went through:

Abby’s question, Practice 1, Section 3, Question 15
Danielle’s question: Sample test (20 questions), Question 9
Emma’s question: Practice 1, Section 4, Question 38
Ms. Li’s Question: can SSA work for congruent triangles? No.
Answers

Please try to finish the two practice tests that I shared last week and this week. Look forward to seeing you all soon.

Take care,

Lijia

AoPS_Number_Theory 2: Primes and Composites

lilijia Post in AoPS_Number_Theory
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AoPS_Number_Theory 2020 Spring Week 2:  Primes and Composites

Welcome new friend Quinn! It’s great fun to try our first online class today. The kids did a great job participating in the class activities!

Lesson Review:

  • Prime number & composite number
  • 0,1 (special numbers)
  • Methods to identify prime numbers:
    • Sieve of Eratosthenes
    • Test prime factors from 2 to root square of q
  • Perfect number (a positive integer that is equal to the sum of its proper divisors): 6, 28, 496, 8128…

Homework:

  • Page 31: Ex 2.3.2
  • Page 35: Ex 2.4.1, 2.4.2
  • Page 37: Ex 2.7, 2.9 (a) (c) (e), 2.10. 2.11
  • Page 38: Ex 2.14, 2.15

Other Resources:

https://www.mathsisfun.com/numbers/prime-number-activity.html

https://www.britannica.com/science/perfect-number

No school for the following 2 weeks. No school for the following 2 weeks. Hope the Coronavirus gets controlled soon. Best wishes to our motherland China!

AoPS_Number Theory Online Class 2/9

lilijia Post in AoPS_Number_Theory
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Lijia Li is inviting you to a scheduled Zoom meeting.

Topic: AoPS: Number Theory
Time: Feb 9, 2020 12:30 PM Eastern Time (US and Canada)

Join Zoom Meeting
https://us04web.zoom.us/j/4977920611

Meeting ID: 497 792 0611

One tap mobile
+16465588656,,4977920611# US (New York)
+14086380968,,4977920611# US (San Jose)

Dial by your location
+1 646 558 8656 US (New York)
+1 408 638 0968 US (San Jose)
Meeting ID: 497 792 0611
Find your local number: https://us04web.zoom.us/u/fbEYdCPF9e

SAT Math Practice & Online Class 2/9

lilijia Post in SAT math
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Dear parents,


In your email please find the practice test for your kids this month. Our next on-site class will be on 3/1/2020. 
Again, welcome to the online class this Sunday afternoon. Please let me know if your kids will attend. 

Topic: SAT Math
Time: Feb 9, 2020 03:30 PM Eastern Time (US and Canada)

Join Zoom Meeting
https://us04web.zoom.us/j/4977920611

Meeting ID: 497 792 0611

One tap mobile
+16465588656,,4977920611# US (New York)
+14086380968,,4977920611# US (San Jose)

Dial by your location
        +1 646 558 8656 US (New York)
        +1 408 638 0968 US (San Jose)
Meeting ID: 497 792 0611
Find your local number: https://us04web.zoom.us/u/fbEYdCPF9e

Best regards,

Lijia

No School 2/9

lilijia Post in AoPS_Number_Theory, SAT math
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接学校通知,2月9日继续停课,改为4月19日补课。欢迎各位家长去学校网站subscribe The NCLS Newsletter,以方便在第一时间收到学校通知。

非常时期,Take care!

https://mailchi.mp/e569cf6da705/ncls-newsletter-recruitment-notice-4745105?e=81d19efa09

2/2 AoPS_Number Theory Brain Stretcher

lilijia Post in AoPS_Number_Theory
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Dear students and parents,

Here is the Brain Stretcher worksheet for you to work with this weekend. Some may be new to you, some functions as a review. Please feel free to reach out if you have any questions.

Take care in this special season!


Best,
Lijia