1.  Turn in homework for credit  Attendance5 min  Week 4 homework review15 min  Solving a Simpler Related Problem20 min  In-Class Practice10 min

2. DDrawing a picture or diagram MMaking an organized list MMaking a table SSolving a simpler related problem FFinding a pattern GGuessing and checking EExperimenting AActing out the problem WWorking backwards WWriting an equation Today’s Topics Tree Diagram Organized list Make a table Solving simpler…

3. Solving a Simpler Related Problem  When a problem is difficult or complicated, solve one or more simpler problems that have simpler conditions.  Solving a series of simpler problem may lead you to a pattern that provides a basis for solving the original problem

4. Luck Sevens Problem: The houses on Main Street are numbered consecutively from 1 to 150. How many house numbers contain at least one digit 7? Separate this problem into two simpler problems: 1. How many house numbers contain the digit 7 in the ones place? From 1 – 10 consecutively, there is one 7 From 1-150, there are 15 sets of 10 consecutive numbers. So, there are fifteen house numbers contain digit 7 in the ones place 2. How many house numbers contain the digit 7 in the tens place? There are ten, from 70 to 79. However, 77 was already accounted for the ones place. So there are nine. Answer: 15+9=24

5. Operation Diamond Problem: Britney’s calculator has a special ◊ key that obeys two rules: Rule 1: If the display shows a one-digit number, pressing the ◊ key replaces the displace with twice its values. Rule 2: If the display shows a two-digit number, pressing the ◊ key replaces the display with the sum of the two digits. Suppose Britney enters the value 1 on the calculator and then presses the ◊ key repeatedly. What does the display show after she presses the ◊ key 50 times.

6. Operation Diamond Rule 1: If the display shows a one-digit number, pressing the ◊ key replaces the displace with twice its values. Rule 2: If the display shows a two-digit number, pressing the ◊ key replaces the display with the sum of the two digits. Strategy: To solve this sort of problem you should generate enough numbers to see if a pattern exists. Solution: The numbers appear in pattern (sequence) are : 12 4 8 16 7 14 5 10 1 2 … (repeating 9 numbers) 45 th number would display 10. 51 st number (pressing 50 ◊ key s) would be the 6 th number 7 Answer: 7

7. In-Class Practice Problem: Suppose that the houses on Main Street are numbered consecutively from 1 to 150. How many house numbers contain at least one digit 9? 4? 1? Solution: (9) 24. 15 numbers have 9 in the ones place (9, 19, 29, …99, 109, …, 149) 10-1=9 other numbers have 9 in the tens place (90, 91, 92,…, 98). So, total is 15+9=24. (4) 33. 15 numbers have 4 in the ones place (4, 14, 24, 34, 44, …, 124, 134, 144). 20-2=18 other numbers have 4 in the tens place (40, 41, 42, 43, 45, 46, 47, 48, 49, 140, 141, 142, 143, 145, 146, 147, 148, 149). So, total is 15+18=33. (1) 70. 15 numbers have 1 in the ones place (1, 11, 21, 31, … 91, 101, 111, 121, 131, 141). 20-2=18 have 1 in the tens place (11 and 111 are not counted) 51-14=37 have 1 in the hundreds place (100, 102, 103, …, 150). 14 of the 51 numbers have been counted before. (101, 111, 121, 131, 141, 110, 112, 113, 114, 115, 116, 117, 118, 119). So, total is 15+18+37=70.

8. In-Class Practice Problem: Suppose Britney enters the value 3. How many times must she press the ◊ key in order to show the value 3 again? Solution: After the value 3 is entered, the display shows 6, 12, 3 in that order. The ◊ key is pressed three times.