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Math Lesson: Real World Problems / Grade Level: 6
Lesson Summary: Students practice identifying extra and missing information in word problems, planning and carrying out multistage problems, and “sanity checks”. Advanced learners use dimensional analysis, and struggling learners work on simple problems.
Lesson Objectives:
The students will know…
· that real-world problems may not be presented as neatly as in a math text.
· that it may not be immediately clear what to do when solving these problems.
The students will be able to…
· analyze a problem to determine the appropriate approach.
· make and carry out a plan with several steps.
· evaluate the solution for reasonableness (“sanity check”).
Learning Styles Targeted:
Visual / Auditory / Kinesthetic/Tactile
Pre-Assessment: Use this activity to introduce real world problems and the importance of understanding what you are trying to find out and what information you need before calculating.
1) Read a problem out loud: “Martha bakes cookies for a soccer team fundraiser. She needs to bring at least 80 cookies to the fundraiser. She uses 4 cookie sheets that are each 18” x 24”. She puts 24 cookies on each sheet. She plans to pack the cookies into boxes that are 6 ½” wide, 13” long, and 4” high. The cookies are 3” in diameter and each weigh about 1 ½ oz. How should she pack the boxes so that there are the same number of cookies in each?”
2) Ask students what they notice about the problem. (It contains a lot of extra information.) Ask the students what information they are missing to be able to complete the problem. (How many boxes does she want to use? If there are 6 boxes, 16 cookies can go in each box, 2 layers of 8. If there are 12 boxes, there will be only one layer of 8, etc.)
3) Point out that when making calculations for real life situations, you often need to sort through what you know to identify and seek out extra or missing information.
Whole-Class Instruction
Materials Needed: PowerPoint Presentation*, several sets of multistage problem index cards
Procedure:
Presentation
Discuss each of the following problem-solving strategies.
1) Making a plan: Don’t just start in crunching numbers. Know where you are going with the operations. Divide the problem into steps. Example:
“Sophia wants to have a lemonade stand. She plans to sell the lemonade for 25 cents for an 8-ounce cup. She has enough lemonade mix to make 5 gallons of lemonade. If she makes and sells all the lemonade, how much money will she make?”
Before you start in calculating, make a plan. (Multiply to find how many ounces in 5 gallons. Divide that to find the number of cups needed. Multiply to find the money earned. Notice that the information was not given in the order it was used.)
2) Carry out the plan: Keep track of where you are in the problem. Don’t write down an intermediate answer as the final one. Don’t lose track of what operation you are doing.
128 ounces per gallon x 5 gallons = 640 ounces
640 ounces / 8 ounces per cup = 80 cups
80 cups x $0.25 per cup = $20.00
3) Evaluate the solution for reasonableness. Suppose in the lemonade problem you multiplied by 25 instead of 0.25, resulting in an answer of $2,000.00. Does it make sense that someone could earn $2,000 by selling 5 gallons of lemonade? (no) This is an example of an order of magnitude error, being off by a factor of 10, 100, 1,000, etc.
4) Again in the lemonade problem, suppose you got called away from your homework partway through and when you got back, you thought you were done and took the 80 cups as your final answer. When you check by reading the problem again, you should notice that the problem asks for money units, while you have found units of cups. This is an example of a units error—reporting an answer using incorrect units. To fix it, you would NOT just change the units. For example, changing the lemonade problem answer to $80 instead of 80 cups would be incorrect. Use a discrepancy with the units to alert you to a potential problem.
Guided Practice
5) Divide the class into groups. Give each group a set of index cards showing the multistage problem below. The group arranges the cards in order and figures out the problem.
Write each of these bullets on one card: (Answers are: 300 working hours, 90 hours remaining, 10 working days left, ending on Tuesday the 31st.)
· Tony wants to buy an ATV, which costs $4,500. He makes $15 an hour, and has worked 210 hours so far. He can work 9 hours a day. Today is Tuesday the 17th. Tony does not work Saturdays or Sundays. If he continues working tomorrow, on what date will he finish earning the money for the ATV?
· How many working hours are needed to pay for the ATV?
· How many working hours does Tony still need to do?
· How many days will it take Tony to work the remaining hours?
· On what date does the last of the working days fall?
Independent Practice
6) Do the first four slides of the PowerPoint presentation (after the title slide), identifying missing and extra information. Answers:
Slide 2: Extra is the cost of gas and the number of days. Answer: 2 tanks of gas.
Slide 3: Extra are the number of children and hours worked after school. Answer: 32 more hours
Slide 4: Missing is the number of tests left in the school year. If there is only 1, a 90% average is not possible.
Slide 5: Extra is the job for 2 years and Troy’s 3 friends. Missing is Troy’s (or Jasmine’s) age. Example: If Troy is 2, then Jasmine is 6 and Janet is 18.
Closing Activity
7) Do the last six slides of the PowerPoint presentation, sanity checks. Answers:
Slide 6: 307 is not a possible bowling score (max = 300). The addition is incorrect, carrying a 2 instead of a 1. The correct answer is 207.
Slide 7: $425 is too low for a three-year-old car. Even if students do not recognize that, it is also too low to be half of nearly $18,000. The division problem started at the wrong place value, as if the 17 were not there. The correct answer is $8,925.
Slide 8: $50 million is far more than the original prize fund. Also 50 people getting $50 million each would be a lot more than what the lottery can reasonably give out. The operation used is incorrect—division should have been used instead of multiplication. Correct answer: $2,000.
Slide 9: $740 for gas is excessive. The problem is in the placement of the decimal point in the multiplication. The correct answer is $74.
Slide 10: $2.44 is too low for the tax on such a large item. The problem is the common error of using both the decimal equivalent for percent and dividing by 100. The correct answer is $243.75.
Slide 11: No one can run a mile in 48 seconds. This is another decimal-point-placement problem, in the division. The correct answer is 8 minutes, or 480 seconds, per mile.
Advanced Learner
Materials Needed: paper and pencils
Procedure:
1) Introduce the topic of dimensional analysis, which is an extremely useful “sanity check” used in science for making sure extended calculations result in the expected units.
Example: If a car gets 20 miles per gallon of gas, and is going 60 miles per hour, how many gallons per minute are being used?

Check that units of hours and units of miles cancel, leaving only gallons in the numerator and minutes in the denominator. Then perform the calculations with the numbers.
Struggling Learner
Materials Needed: sample problems, paper, and pencil
Procedure:
1) Build up from simple examples: provide students with problems where only one piece of information is extra, one necessary piece is missing, etc. Order stages of problems of only two or three steps.

*see supplemental resources