Day 7 – math 3 September 15/16, 2016

Lesson 3 – algebraic reasoning and proof (1.3.1) Lesson objectives: Use algebraic notation – letters, expressions, equations, and inequalities – to represent general patterns and relationships among variables Use algebraic transformations of expressions, equations, and inequalities to establish general propositions about quantitative relationships Look at the the yellow box on page 52 Pick an integer between 0 and 20 Add 5 to your number Multiply the result by 6 Divide that result by 3 Subtract 9 from that result

Investigation 1 Algebra and Number Magic – Problem #1 Come up with an algebraic expression Review order of operations A. write the algebraic expressions for each B. write a final, condensed expression, then simplify C. how did I find out your starting number? D. how would that change affect how I found out your starting number?

Summarizing #1 Part A and B: Part C: To recover the number, subtract 1 from the ending number, then divide by two Written algebraically (x is the ending #): Part D: if the third step is “divide by 2” the expression would be (not as easy to do in one’s head)

Investigation 1 Problem #2 – new number trick A. find the algebraic expression, and how to “decode”, or find the starting number B. why does “decoding” work?

Summarizing #2 Part A: This procedure can be represented by The decoding rule will be to subtract 8 from the end number and divide that intermediate result by 10 Part B: To find the start number, solve for “n” Written algebraically where x is the ending number:

Investigation 1 Problem #3 – your own number trick Problem #4 – Explaining number patterns A. complete the table B. calculate ALL the differences, then find a pattern C. see the algebraic pattern, then try to simplify it D. come up with an if/then statement….

Summarizing #4 Part A: fill in the table Part B: the differences between successive square integers form the sequence of positive odd numbers Part C: n … 5 6 7 8 n + 1 Sn 25 36 49 64 n2 (n+1)2

#4 continued Part D: Some examples… If you calculate differences between successive terms in the sequence of square numbers, then the result will be the sequence of odd numbers If one square number is subtracted from the next, then the result will be an odd number If you calculate for any whole number n, then the result will always be the odd number