1. BREAK THE PROBLEM DOWN INTO SUB-PARTS WE BREAK PROBLEMS DOWN INTO SUB-PARTS WHEN WE ARE GIVEN A LOT OF INFORMATION THAT NEEDS TO BE ORGANIZED AND THAT NEEDS TO BE SOLVED IN STEPS. THIS TYPE OF PROBLEM IS COMPLEX ACCORDING TO WEBB’S DEPTH OF KNOWLEDGE MODEL, SO IT HELPS US TO DEVELOP GOOD ORGANIZATIONAL AND NEATNESS SKILLS, AND IT ENCOURAGES US TO BE CAREFUL IN OUR CALCULATIONS. HERE IS AN EXAMPLE: DEREK BOUGHT 2 NOTEBOOKS FOR $1.79 EACH, 3 PENS FOR $0.89 EACH, AND A PACK OF HIGHLIGHTERS FOR $3.49. THE TAX RATE IS 8.5%. DEREK PAYS WITH A $20 BILL. THE CASHIER GIVES IN THE FOLLOWING CHANGE: A $5 BILL, 4 $1 BILLS, 2 QUARTERS, 1 DIME, 1 NICKEL, AND THREE PENNIES. IS THIS THE CORRECT AMOUNT OF CHANGE?

2. DEREK BOUGHT 2 NOTEBOOKS FOR $1.79 EACH, 3 PENS FOR $0.89 EACH, AND A PACK OF HIGHLIGHTERS FOR $3.49. THE TAX RATE IS 8.5%. DEREK PAYS WITH A $20 BILL. THE CASHIER GIVES IN THE FOLLOWING CHANGE: A $5 BILL, 4 $1 BILLS, 2 QUARTERS, 1 DIME, 1 NICKEL, AND THREE PENNIES. IS THIS THE CORRECT AMOUNT OF CHANGE? NOTE: WHEN THE PROBLEMS GET COMPLEX LIKE THIS, I WOULD RECOMMEND USING A CALCULATOR. WE DON’T WANT TO GET BOGGED DOWN IN THE COMPUTATIONS WHEN WE ARE REALLY LEARNING SOMETHING ELSE. 1)THE FIRST THING WE NEED TO FIND IS THE AMOUNT OF DEREK’S PURCHASE WITHOUT THE TAX: 2 X 1.79 + 3 X 0.89 + 3.49 = 3.58 + 2.67 + 3.49 = 9.74 2)NOW WE WILL ADD IN THE TAX. 9.74 X 0.085 = 0.8279 = 0.83 (ROUNDED) AND 9.74 + 0.83 = 10.57. 3)NEXT, WE WILL FIND OUT HOW MUCH CHANGE HE SHOULD GET BACK: 20.00 – 10.57 = 9.43. 4)NOW WE WILL CALCULATE THE AMOUNT OF CHANGE DEREK DID RECEIVE: 5 + 4 X 1 + 2 X 0.25 + 0.10 + 0.05 + 0.03 = 5 + 4 + 0.5 + 0.1 + 0.05 + 0.03 = 9.68. 5)THEN WE CAN COMPARE TO ANSWER THE QUESTION: 9.43 IS NOT EQUAL TO 9.68, SO THIS IS NOT THE CORRECT AMOUNT OF CHANGE. 6)AS AN EXTENSION ACTIVITY, YOU CAN EXPLORE WHAT WENT WRONG. AS YOU INVESTIGATE FURTHER, YOU WILL NOTICE THAT THE CASHIER GAVE HIM $0.25 TOO MUCH, SO HE GAVE DEREK AN EXTRA QUARTER.

3. NOW IT’S YOUR TURN TO SOLVE THIS TYPE OF PROBLEM. 1)FIND ALL TWO DIGIT PRIME NUMBERS WHOSE INDIVIDUAL DIGITS ARE ALSO PRIME NUMBERS. 2)HOW MANY FACTORS OF 400 ARE PERFECT SQUARES? 3)JOE HAS 20 FEET OF LUMBER. HE NEEDS TO CUT THIS INTO THREE 6 5/8 INCH PIECES. EVERY TIME HE MAKES A CUT, HE WILL LOSE AN ADDITIONAL 1/16 INCH OF WOOD. WILL HE HAVE ENOUGH WOOD? HOW MUCH WILL HE HAVE LEFT OVER OR HOW MUCH MORE DOES HE NEED? 4)FIND THE SUM OF THE WHOLE NUMBERS BETWEEN 1 AND 100 INCLUSIVE THAT ARE MULTIPLES OF 5 OR 7. 5)RENT-A-RIDE CHARGES $50 PER DAY PLUS $0.19 PER MILE TO RENT THEIR ECONOMY CAR. YOU-RENT-IT CHARGES $45 PER DAY PLUS $0.25 PER MILE FOR THE SAME CAR. IF DAWN NEED TO RENT A CAR FOR THREE DAYS AND TRAVEL 325 MILES, WHICH RENTAL COMPANY WILL COST HER THE LEAST?

4. 1) FIND ALL TWO DIGIT PRIME NUMBERS WHOSE INDIVIDUAL DIGITS ARE ALSO PRIME NUMBERS. 1)FIRST, WE WILL IDENTIFY THE ONE DIGIT PRIME NUMBERS. THEY ARE 2, 3, 5, AND 7. 2)NEXT, WE WILL LIST ALL POSSIBLE TWO DIGIT NUMBERS MADE FROM THESE DIGITS: 22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77. 3)WE CAN REMOVE ALL EVEN NUMBERS BECAUSE THEY WILL NOT BE PRIME (2 GOES INTO THEM). OUR NEW LIST IS 23, 25, 27, 33, 35, 37, 53, 55, 57, 73, 75, 77. 4)NEXT, WE CAN REMOVE ALL NUMBERS ENDING IN 5 (5 WILL GO INTO THEM) AND ALSO ALL NUMBERS WITH DOUBLE DIGITS (11 WILL GO INTO THEM). OUR LIST IS NOW: 23, 27, 37, 53, 57, 73 5)NOW WE CAN REMOVE ALL NUMBERS THAT 3 WILL GO INTO. WE CAN USE A DIVISIBILITY TEST OR JUST USE OUR CALCULATOR. OUR NEW LIST WILL BE: 23, 37, 53, AND 73. 6)WE WILL NOW CHECK TO SEE IF THESE FOUR NUMBERS ARE PRIME. WE CAN DO THAT BY CHECKING EACH NUMBER TO SEE IF 7 GOES INTO IT (WE ALREADY CHECKED FOR 2, 3, AND 5, SO 7 IS THE ONLY NUMBER LEFT. WE DO NOT NEED TO CHECK FOR 4, 6, 8, 9, OR 10 BECAUSE THE PRIME NUMBERS GO INTO THEM). 7 DOES NOT GO INTO ANY OF THESE NUMBERS, SO THEY ARE ALL PRIME. 7)OUR FINAL ANSWER IS 23, 37, 53, AND 73.

5. 2) HOW MANY FACTORS OF 400 ARE PERFECT SQUARES? 1)FIRST, WE WILL LIST ALL OF THE FACTORS OF 400. IT WILL BE EASIEST TO DO THIS IN PAIRS AND STOP WHEN WE GET TO THE MIDDLE. WE HAVE 1,400 2,200 4,100 5,80 8,50 10,40 16,25 20,20. 2)NOW WE WILL CHECK TO SEE WHICH ONES ARE PERFECT SQUARES (THE SQUARE ROOT BUTTON ON OUR CALCULATORS WILL WORK FINE HERE). THESE NUMBERS ARE 1, 400, 4, 100, 16, AND 25. (DID YOU NOTICE THAT WHEN ONE OF THE PAIR WAS A PERFECT SQUARE THEN THE OTHER ONE WAS TOO? I WONDER WHY THAT IS... ) 3)NOW WE WILL COUNT THESE NUMBERS TO ANSWER THE QUESTION. THERE ARE 6 FACTORS OF 400 THAT ARE PERFECT SQUARES.

6. 3) JOE HAS 20 FEET OF LUMBER. HE NEEDS TO CUT THIS INTO THREE 6 5/8 INCH PIECES. EVERY TIME HE MAKES A CUT, HE WILL LOSE AN ADDITIONAL 1/16 INCH OF WOOD. WILL HE HAVE ENOUGH WOOD? HOW MUCH WILL HE HAVE LEFT OVER OR HOW MUCH MORE DOES HE NEED? 1)FIRST, WE WILL CALCULATE HOW MUCH WOOD JOE NEEDS FOR THE PIECES. 3 X 6 5/8 = 19 7/8 INCHES (AGAIN, A CALCULATOR THAT DOES FRACTIONS WILL WORK WELL HERE.) 2)NEXT, WE WILL CALCULATE HOW MUCH WILL BE LOST IN THE CUTS. WE NEED TO BE CAREFUL HERE, BECAUSE ALTHOUGH THERE ARE THREE PIECES, THERE ARE ONLY TWO CUTS. 2 X 1/16 = 1/8 INCH. 3)NOW WE WILL ADD THE AMOUNTS TOGETHER TO SEE HOW MUCH TOTAL WOOD WE NEED. 19 7/8 + 1/8 = 20. 4)SINCE JOE HAS 20 FEET OF LUMBER AND HE NEEDS 2O FEET OF LUMBER, THEN HE WILL HAVE ENOUGH BUT WILL HAVE NONE LEFT OVER.

7. 4) FIND THE SUM OF THE WHOLE NUMBERS BETWEEN 1 AND 100 INCLUSIVE THAT ARE MULTIPLES OF 5 OR 7. 1)FIRST, WE WILL LIST THE MULTIPLES OF 5 THAT ARE BETWEEN 1 AND 100 INCLUSIVE. THEY ARE 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, AND 100. ALSO, THEIR SUM IS 1050. 2)NEXT, WE WILL LIST THE MULTIPLES OF 7 THAT ARE BETWEEN 1 AND 100 INCLUSIVE. THEY ARE 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, AND 98. ALSO, THEIR SUM IS 735. 3)NOW WE NEED TO THINK, BECAUSE THE NUMBERS 35 AND 70 APPEAR ON BOTH LISTS. IF WE ARE NOT CAREFUL WE WILL COUNT THESE NUMBERS TWICE. SO, WE WILL NEED TO ADD THE SUMS BUT THEN SUBTRACT 35 AND 70 TO UNDO THE DUPLICATES. 4)OUR ANSWER, THEN, IS 1050 + 735 – 35 – 70, WHICH IS 1680.

8. 5) RENT-A-RIDE CHARGES $50 PER DAY PLUS $0.19 PER MILE TO RENT THEIR ECONOMY CAR. YOU-RENT-IT CHARGES $45 PER DAY PLUS $0.25 PER MILE FOR THE SAME CAR. IF DAWN NEED TO RENT A CAR FOR THREE DAYS AND TRAVEL 325 MILES, WHICH RENTAL COMPANY WILL COST HER THE LEAST? 1)FIRST, WE WILL CALCULATE HOW MUCH DAWN WILL PAY IF SHE USES RENT-A-RIDE. 50 X 3 + 0.19 X 325 = $211.75. 2)NEXT, WE WILL CALCULATE HOW MUCH DAWN WILL PAY IF SHE USES YOU-RENT-IT. 45 X 3 + 0.25 X 325 = $216.25. 3)FINALLY, WE COMPARE THE TWO CHOICES AND CHOOSE THE ONE REQUIRING THE SMALLER AMOUNT. $211.75 IS LESS THAN $216.25, SO HER LOWEST COST OPTION IS RENT-A-RIDE.

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