Lesson plan - Study Island
|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: |
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|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. |
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|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: |
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|Visual |
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|Auditory |
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|Kinesthetic/Tactile |
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|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. |
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|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?” |
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|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.) |
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|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. |
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|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: |
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|“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?” |
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|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.) |
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|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. |
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|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 |
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|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. |
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|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. |
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|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. |
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|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.) |
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|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? |
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|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. |
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|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. |
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|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? |
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|[pic] |
|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
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