2.1 Making generalizations (continued)

• Wiskunde
• WO
• Studiejaar 4

Good morning!

Schedule: 

• Learning goals
• Upcoming assessment
• Revision 
• Generalizing specific problems
• To generalize or not to generalize 
• Homework 

Learning goals 

At the end of this lessons I can:

• Identify patterns in number problems
• Solve complicated problems by looking at a more general case
• Make generalizations from a given pattern

Upcoming assessment

Criterion A: Knowledge and understanding 

Equivalence and inequalities 

Chapters 2.1 - 2.3 in your book

November 10 (3 weeks after the break) 

Give a generalization of something you have observed today

Generalization with specific problems

Don't use a calculator!!

What is the specific problem? 

What is the general problem? 

How did solving the general problem make it easier to solve the specific problem?

To generalize or not to generalize

RSA-encryption is used to encrypt and decrypt messages. 

Using prime numbers in this system makes it secure and the safest way to encrypt messages.

 

Banking details, Chat applications, web browsers, etc. 

To generalize or not to generalize

Now consider the expression: 

n​2​​+n+41

Try a few more numbers and give a generalization...

Generalization

To generalize or not to generalize

Now consider the expression: 

n​2​​+n+41

Now calculate for n = 40

Homework

HOMEWORK:

P. 98: Practice 2

P. 100: Practice 3

Due

Level 1-2

Let a = odd integer and b = even integer

Calculate (a x b) multiple times with different numbers

Generalize and suggest a conjecture.

Level 3-4

 Without using a calculator, find the value of the following expression:

20192 − 2021 × 2017

Level 5-6

(Exploration 1, P. 99)

a. Choose a positive integer as your number. Square your number and subtract your number from the squared number. Try this multiple times. 

Generalize and suggest a conjecture.

b. Use specific generalization to prove that your conjecture holds for every positive integer. 

Hint: use a variable for your chosen number. 

Level 7-8

Let S = sum of all integers 

so S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + ..... 

You will find a exact (finite) answer for S with generalization. 

Starting from 2, add 3 consecutive numbers together which you will then again add all together: 

S = 1 + (2 + 3 + 4) + (5 + 6 + 7) + (8 + 9 + 10) + .....

Use generalization to find an exact, finite value for S

Hint: