* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 1.3.1 notes
Survey
Document related concepts
Large numbers wikipedia , lookup
History of mathematical notation wikipedia , lookup
Laws of Form wikipedia , lookup
System of polynomial equations wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Location arithmetic wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Transcript
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: 6(n + 5) - 9 becomes 2n +1 3 Part C: To recover the number, subtract 1 from the ending number, then divide by two Written algebraically (x is the ending #): n = x -1 2 Part D: if the third step is “divide by 2” the expression would be 6(n + 5) 2 - 9 which becomes 3n + 6 (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 5(2n + 3)- 7 which becomes 10n +8 • 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 10n +8 for “n” • Written algebraically where x is the ending number: x -8 n= 10 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 n … 5 6 7 8 … n n+1 Sn … 25 36 49 64 … n2 (n+1)2 Part B: the differences between successive square integers form the sequence of positive odd numbers 2 2 S S = (n +1) n Part C: n+1 n = (n 2 + 2n +1) - n 2 = 2n +1 #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 • 2 2 (n +1) n If you calculate for any whole number n, then the result will always be the odd number 2n +1