Upload File

mathematical thinking 2 · 2018. 8. 28. · specializing and generalizing •the table shows the...

  • Home
  • Documents
  • Mathematical Thinking 2 · 2018. 8. 28. · Specializing and Generalizing •The table shows the last digits in the powers of 1, 2, 3, …9. •The last digits repeat in cycles. •The
40
Mathematical Thinking 3 SEAMEO RECSAM 19 April 2017

Upload: others

Post on 27-Jan-2021

1 views

Category:

Documents


0 download

Report

TRANSCRIPT

  • Mathematical Thinking 3

    SEAMEO RECSAM

    19 April 2017

  • 18-Apr-17 Assoc. Prof. Dr Kor Liew Kee 2

  • What is Mathematical Thinking?

    • http://study.com/academy/lesson/critical-thinking-math-problems-examples-and-activities.html

    http://study.com/academy/lesson/critical-thinking-math-problems-examples-and-activities.htmlhttp://study.com/academy/lesson/critical-thinking-math-problems-examples-and-activities.htmlhttp://study.com/academy/lesson/critical-thinking-math-problems-examples-and-activities.htmlhttp://study.com/academy/lesson/critical-thinking-math-problems-examples-and-activities.htmlhttp://study.com/academy/lesson/critical-thinking-math-problems-examples-and-activities.htmlhttp://study.com/academy/lesson/critical-thinking-math-problems-examples-and-activities.htmlhttp://study.com/academy/lesson/critical-thinking-math-problems-examples-and-activities.htmlhttp://study.com/academy/lesson/critical-thinking-math-problems-examples-and-activities.htmlhttp://study.com/academy/lesson/critical-thinking-math-problems-examples-and-activities.htmlhttp://study.com/academy/lesson/critical-thinking-math-problems-examples-and-activities.htmlhttp://study.com/academy/lesson/critical-thinking-math-problems-examples-and-activities.htmlhttp://study.com/academy/lesson/critical-thinking-math-problems-examples-and-activities.htmlhttp://study.com/academy/lesson/critical-thinking-math-problems-examples-and-activities.html

  • 4 fundamental processes on how to think mathematically

    • Specializing – trying special cases, looking at examples

    • Generalizing - looking for patterns and relationships

    • Conjecturing – predicting relationships and results

    • Convincing – finding and communicating reasons why something is true.

  • Task 1

  • Exponents

  • Exponents

    https://aiminghigh.aimssec.ac.za/grades-8-10-powerful-thinking-2/

  • Questions

    • What pattern can you see in the last digits of the powers of 2?

    • What about the last digits of the powers of 3?

    • What patterns can you see in the last digits for the numbers in the picture?

    • Can you explain why all these numbers are multiples of 5?

  • Further questions

    • What about the 99th powers 299+399?

    • What can you find out about 199+299+399+499?

    • What other patterns with powers can you find that have special properties?

  • Possible solutions

  • Explanation

    • The table shows the last digits in the powers of 1, 2, 3, …9.

    • The last digits repeat in cycles. • For example the last digits in the powers of 2 are

    2,4,8,6,2,4,8,6,2… and • the last digits in the powers of 3 are 3,9,7,1,3,9,7,1,3…

    both cycles of length 4. • Adding the sums of the last digits of the numbers with

    odd powers: 21 + 31, 23 + 33, … are always 5 for all odd powers.

    • So these numbers are divisible by 5.

  • Exploring 199+299+399+499

    • The last digits in the powers of 4 repeat in a 2-cycle 4, 6, 4, 6,… with the last digit for odd powers always equal to 4.

    • Also 1 raised to any power equals 1 so the last digit of 199 + 299 + 399 + 499 ends in 0.

    • Hence, this number is divisible by 10 (also by 5).

  • What about the 99th powers 299+399?

    • As 99 is an odd number we know that 299 + 399 is divisible by 5 (is a multiple of 5) because it has 5 as the last digit.

  • Key Questions

    • What do you notice?

    • Can you describe a pattern that you see?

    • What is special about the powers 1, 3, 5, 7 and 9 in this question?

    • What do you notice about the last digits?

    • Can you explain how you know that?

    • How do you know when a number is divisible by 5?

    • How do you know when a number is a multiple of 5?

    • Can make up your own pattern of numbers involving sums of powers that share the same property?

  • Possible support

  • Task 2

  • Exponents with nth power

    https://aiminghigh.aimssec.ac.za/grades-8-10-powerful-thinking-2/

  • Questions

    • Will the first number 2n+3n be a power of 10 for any value of n?

    • What can you say about the values of n that make the second number 3n+7n a power of 10?

    • Are there any other pairs of integers between 1 and 10 which have similar properties?

  • Specializing

  • Specializing and Generalizing

    • The table shows the last digits in the powers of 1, 2, 3, …9. • The last digits repeat in cycles. • The last digits of 2n are 2, 4, 8, 6, 2, 4, 8, 6, … repeating the

    cycle 2, 4, 8, 6 over and over again • The last digits of 3n are 3, 9, 7, 1, 3, 9, 7, 1 … repeating the

    cycle 3, 9, 7, 1 over and over again … • So adding 2n + 3n gives the last digit 5, 3, 5, 7, 5, 3, 5, 7 …

    repeating the cycle 5, 3, 5, 7 over and over again. • The numbers 2n + 3n are divisible by 5 for all odd values of

    n. • The last digit is never zero so 2n + 3n is never a power of 10

    for any value of n.

  • Conjecturing and Convincing

    • The last digits in 31 + 71 , 33 + 73, 35 + 7 5, 37 + 77, … and so on for all odd powers all end in 0.

    • So these numbers 3n + 7n are divisible by 10 (multiples of 10) for all odd values of n.

    • Similarly the last digits in 13 + 93, 15 + 95, 17 + 97, … and so on for all odd powers all end in 0

    • So these numbers 1n + 9n are divisible by 10 (multiples of 10) for all odd values of n.

    • Similarly the last digits in 23 + 83, 25 + 85, 27 + 87, … and so on for all odd powers all end in 0.

    • So these numbers 2n + 8n are divisible by 10 (multiples of 10) for all odd values of n.

    • Similarly the last digits in 43 + 63, 45 + 65, 47 + 67, … and so on for all odd powers all end in 0.

    • So these numbers 4n + 6n are divisible by 10 (multiples of 10) for all odd values of n.

  • Key Questions

    • What do you notice?

    • What do you notice about the last digits?

    • Can you explain how you know that?

    • Can you describe a pattern that you see?

    • Can you use algebra to make what you are saying into a general statement?

  • Possible support

  • Task 3

  • https://aiminghigh.aimssec.ac.za/wp-content/uploads/2016/09/Teacher-Notes-POWERFUL-THINKING3.pdf

  • Questions

    • Imagine the first number 34 × 45 × 56 is written out in full. How many zeros would there be at the end?

    • Why?

    • Can you find the answer without doing the whole calculation?

  • Possible solution

    • 34 × 45 × 56

    • = 34 x 210 x 56

    • = (3 x 2)4 x 26 x 56

    • = (3 x 2)4 x 106

    • So there will be 6 zeros at the end.

  • Questions

    • Work out a few values of the number 1n + 2n + 3n .

    • What can you say about the sum for all values of n?

    • For what values of n will 1n +2n +3n + 4n be even?

  • Possible solution

    • 1 + 2 + 3 =6, • 12 + 22 + 32 = 14, • 13 + 23 + 33 = 36, • 14 + 24 + 34 = 98 … All these are even. • For all values of n, 1n = 1 is odd, 2n is even and 3n

    is always odd. • The sum of two odd numbers is always even and

    the sum of two odds and an even number is even.

    • So 1n + 2n + 3n is always even for all values of n.

  • Questions

    • What about 1n +2n +3n +4n +5n ?

    • What can you say about the sums of powers of the counting numbers 1, 2, 3, 4, 5, 6, 7, …p?

  • Possible solution

    • As 4n is even for all values of n it follows that 1n +2n +3n +4n is even for all values of n.

    • As 5n is odd for all values of n it follows that 1n +2n +3n +4n +5n is odd for all values of n.

    • All nth powers of odd numbers are odd and all nth powers of even numbers are even.

    • So 1n + 2n + 3n + … + pn is even if p is even and odd if p is odd.

  • Key Questions

    • Is 4 a prime number? How could you write it as a power of a prime number?

    • What do you notice? • What do you notice about the last digits? • Can you explain how you know that? • Can you describe a pattern that you see? • Is the sum of 2 odd numbers odd or even? How do you

    know? • Is the sum of 2 even numbers odd or even? How do you

    know? • Is the sum of an odd number and an even number odd or

    even? How do you know?

  • Possible extension

  • Further Exploration

    • Is this number 3444+4333 divisible by 5?

    • Investigate other big powers.

    • Make up some similar numerical expressions involving powers that have interesting properties.

    • What about 2666 + 6222 ,7888 + 8777?

  • Key Questions

    • What do you notice? • What do you notice about the last digits? • Can you explain how you know that? • Can you describe a pattern that you see? • Can you see a repeating pattern? How many

    digits are repeated? • What is the length of the cycle of days of the

    week? (or months of the year?) • How many cycles of length 4 would you complete

    before you get to the 666th power (or the 444th)?

  • Solution

    • https://aiminghigh.aimssec.ac.za/wp-content/uploads/2016/09/Teacher-Notes-POWERFUL-THINKING4.pdf

    https://aiminghigh.aimssec.ac.za/wp-content/uploads/2016/09/Teacher-Notes-POWERFUL-THINKING4.pdfhttps://aiminghigh.aimssec.ac.za/wp-content/uploads/2016/09/Teacher-Notes-POWERFUL-THINKING4.pdfhttps://aiminghigh.aimssec.ac.za/wp-content/uploads/2016/09/Teacher-Notes-POWERFUL-THINKING4.pdfhttps://aiminghigh.aimssec.ac.za/wp-content/uploads/2016/09/Teacher-Notes-POWERFUL-THINKING4.pdfhttps://aiminghigh.aimssec.ac.za/wp-content/uploads/2016/09/Teacher-Notes-POWERFUL-THINKING4.pdfhttps://aiminghigh.aimssec.ac.za/wp-content/uploads/2016/09/Teacher-Notes-POWERFUL-THINKING4.pdfhttps://aiminghigh.aimssec.ac.za/wp-content/uploads/2016/09/Teacher-Notes-POWERFUL-THINKING4.pdfhttps://aiminghigh.aimssec.ac.za/wp-content/uploads/2016/09/Teacher-Notes-POWERFUL-THINKING4.pdfhttps://aiminghigh.aimssec.ac.za/wp-content/uploads/2016/09/Teacher-Notes-POWERFUL-THINKING4.pdf

  • Resource Book


Air Cover 2005 · The last three digits represent the last digit of the year the boat was built and the final two digits of the boat model year. A boat built in July 2004 is actually

Employee Site Web Address: User ID: First initial of first name and complete last name with last four digits of SSN (JDoe-Smith5626)

Name Period Primes Number Theory - Edl - if the number formed by the last 3 digits is divisible by 8. (ask me why this works) 9 - if the sum of the digits is divisible by 9. 10 - if

doc.westmedica.comdoc.westmedica.com/52RU/сертификаты/ce_77e.pdf · 1116 Buda, Last two digits of the year in which the CE marking was affixed: 02 Budapest, 2003-06-19

MATH FORMULAS - MyNotesAdda · Last 2 digits Last two digits of a1^b will be [last digit of a*b]1 Last two digits of 24^Odd will be 24 and 24^Even will be 76. We can use this to find

EMPLOYEE INSTRUCTIONS - Quality Investigations, … INSTRUCTIONS QUALITY ... A second window will open for the Installer software. Select Run ... the last four digits of your Social

2020 SPRING TURKEY - gfp.sd.govgfp.sd.gov/userdocs/docs/2020SpringTurkey-app.pdfSocial Security Number The last four digits of your Social Security Number are required. The Social

Multi-Subject CST Mathematics Preparation › mypersonalinfo › ... · 1. Take the digits that repeat and make those digits the numerator 2. The number of digits that repeat denote

FUSE LINKS - WESCO · 2019-11-22 · For 26”: *Solid head K or T links, change the last two digits from 23 to 26. †Removable head K or T links, drop the last two digits. Application

Monday – Multiply 4-digits by 2-digits

Rules of DIVISIBILITY - Metcalfe County Schools rules have to work. the last three digits in the number are divisible by 8. 1. Add all the digits. remaining numbers. 2. If the sum

Name Period Primes Number Theory - AGMath.comagmath.com/media/DIR_57727/NumberTheory.pdf3 - if the sum of its digits is divisible by 3. 4 - if the number formed by the last 2 digits

Significant digits Objectives: State the purpose of significant digits State and apply the rules for counting and doing calculations with significant digits

To Cross-Validate or Not? · 2016-09-13 · Company Department Account Product Line Intercompany 3 digits Numeric 4 digits Numeric 5 digits Numeric 3 digits Alphanumeric 3 digits

BE SERIESThe last 2 digits of the room # MUST start at "01". NO room # can end with "00" The last 2 digits of the room # MUST start at "01". NO room # can end with "00" 5225 Sheridan

Scantron Directions Write your last name, first name, middle initial, company and platoon AND name of the class Fill in the last nine digits of your EDIPI

  · Web viewDivisor Divisibility Test 2 The number must be even (end in a 0, 2, 4, 6, or 8) 3 The sum of the digits must be divisible by 3 4 The number formed by the last two digits

Guida Rapida Resist SA TAG 10 SA - FAAC · 0000 The first two digits are the enabling of the outputs for reading on A, whereas the last two digits are the enabling of the outputs

Index of contents - I.E.S. LLANES. Sevilla The last digit is even (0,2,4,6,8) 3 The sum of the digits is divisible by 3 4 The last 2 digits are divisible by 4 5 The last digit is 0

Normality and the Digits of - Computational Research Divisioncrd-legacy.lbl.gov/~dhbailey/dhbpapers/normality-digits-pi.pdfNormality and the Digits of ... The rst computation on ENIAC

SECTION 1 - catalogimages.wiley.com · ing at the last two digits of the number. If the number formed by the last two digits can be divided by 4, the number is divisible by 4. Which

Name:&& &&&&&&Last&4&Digits&of&Student&I.D.#:& & CBE140

CHAPTER 6 – Expenditure Accounts · Chapter 6 – Expenditure Accounts 6-2 Effective Date: 9-1-18 The program codes are the last two digits of the revenue codes. The first two digits

The Muscular System Presenter: Chung-Bang Weng Last 4 digits of SSN: 9142 Presentation date: September 5 2007

mysticva.entest.org Grade 3 MATH... · last digit is 5. The number 148,567,896,094 has twelve digits, and the last digit is 4. Example 3 The number 186,000 has how many digits? Solution

MANUFACTURING Catalog FOR INDUSTRIAL AUTOMATIONFEMA · Catalog 2015-2016 09 3 Digits 3 1/2 Digits 4 Digits 4 Digits 5 Digits 6 Digits 14 mm (digit height) OEM 72 x 36 mm (front size)

One Touch Y580 Quick Start - Blink · SSID：BLINK-POCKET-WIFI_XXXX, XXXX is the last four digits of MAC-Address Key: Last eight digits of IMEI Step 1: Use the Wi-Fi manager on the

Standards Matrix › Standards › media › pdf › ... · The first four digits identify the work group that maintains the standard or initiated the ballot and the last four digits

BBRRDD@@ffffiiccee - brdoffice.ro · Foreign exchange ... foreign exchanges and deposits cancelation or the last 6 digits of the

STUDENT’S WORKBOOK - XTEC · Student’s workbook Page 4 3.2 Boolean logic. Logic gates. In the last lesson you used BINARY DIGITS to represent NUMERIC VALUES. BINARY DIGITS can

Table of Contents - Virginia Dept of Accounts · Cardinal removes the first 2 digits and sends the last 8 digits of the Cardinal ID (e.g., 00000001) *Note: Cardinal allows an interfacing

Take a seat and sign in to your chrome book with the following formula: first letter of your first name, entire last name, last 4 digits of your student

User Guide · Web viewClick the Last 4 SSN field and enter the last 4 digits of the SSN. Click Medical Record Number and enter number. Click the Medicaid ID field and enter number

Maryknoll Email for Students. Step 1: YOUR INFORMATION! Your User Name: Last name + 1 st name initial + year graduating (last 2 digits) Example: marsb

3. 4. Log x Log y MATHEMATICS Log z then xyz = The last digit in 7300 is How many numbers of 6 digits can be formed from the digits of the number 112233 ? 1) 30 3) 90 2) 60 4) 120