PROBLEMS, PROBLEMS:
SOLUTIONS, SOLUTIONS!
Grades 5-8

This lesson has been designed to provide students with an understanding of strategies used for effectively solving problems in various areas. Video, hands-on activities and interaction among students show the need for forming an hypothesis and to recognize correlation between an educated guess and a solution. Video and simulated situations provide opportunities for students to internalize recognition of the real problem and how to incorporate problem solving strategies to solve it successfully.

MATH WORKS
Problem Solving: Identifying the Problem
#102
Problem Solving: Simplifying the Problem
#107

GEOGRAPHY SKILLS
Problem Solving #112

Students will be able to:

• identify concrete and abstract patterns;
• discuss correlation between possible and actual outcomes;
• demonstrate logical reasoning in solving problems;
• incorporate probability reasoning and predictions to determine intended outcomes;
• list ways and methods to solve problems;
• explain importance of recognizing the real problem;
• separate unnecessary parts of a problem from those parts needed to reach a solution;
• demonstrate critical problem solving strategies.

(per class)

• 1 chalkboard, chalk, eraser
• 1 chart tablet
• 1 black marking pen
• 1 overhead projector

(per group of 4)

• 1 sheet notebook paper
• 1 pencil
• 2 grocery items or their containers/wrappers
• 1 calculator
• 1 Activity Sheet
• 1 sealable plastic bag containing 15 Red beans, 15 Lima beans and 15 Peas
• 1 paper plate

(per student)

• 2 sheets notebook paper
• 1 pencil

Say, "Today we are going to focus on how to solve a problem." Engage students in discussion pointing out: often a problem can be solved in more than one way; and the method of solution may not be obvious or readily identifiable when someone says they don't get it or they don't understand.

Divide the class into groups of four students each. Distribute a sheet of notebook paper and a pencil to each group; instruct them to select a recorder from within the group. Say, "Brainstorm within your group and decide on strategies that can be employed to figure out a problem you don't understand." Emphasize there are several things a good mathematician can do to solve a difficult problem. Say, "As your group agrees on a strategy that can be used to solve a problem, it should be written down by the recorder." Instruct students to begin the task; allow a reasonable period of time for the task to be accomplished.

Allow groups to interact as they share strategies decided upon. List strategies on chalkboard as they are suggested by individual groups, then direct discussion toward combining strategies which are similar but stated differently. Guide discussion to a logical sequencing of problem solving strategies as you ultimately conclude and record the sequenced strategies on a chart tablet as: Understand the problem. (What is being asked? What are the facts? Which infor-mation is important to solving the problem? Which is not?) Plan a solution. (What options do I have for solving the problem? Which can help? (e.g.) Draw a picture; construct a model; guess and check; work backwards; make a graph or table; or use a computation skill of addition, subtraction, multiplication or division.) Solve the problem. Check your work. After completion of the strategies list, distribute a sheet of notebook paper and a pencil to each student. Say, "Use the chart tablet list and record the strategies for your own personal use and referral as you decide how to solve various problems today."

Ask, "Why do you suppose understanding the problem is sequenced first on the list?" Allow students to tell it is necessary to know what a problem is before you can solve it. Say, "You are going to see a video about two girls attempting to figure out how much money is needed to buy their grandmother a jewelry box. The problem may appear simple to you, however, to a younger person it would not be easy." To give students a specific responsibility while viewing say, "As you watch the video, refer to your list of problem solving strategies and decide if and how the first strategy is employed by the girls, then be prepared to defend what you decide."

MATH WORKS
Problem Solving: Identifying the Problem #102
BEGIN tape immediately following opening credits. PAUSE tape on visual of narrator; audio is, "...some step to take when you're sure the clues are there but you just can't find them." Allow students to tell and defend their belief about whether the girls properly implemented the first strategy. If the following wasn't included in students' rationale, ask: "What was the mistake in computation made by the girls? What gave the mistake away?" (needed to raise more money than the jewelry box cost) Emphasize the need to always use common sense when problem solving. Write the term on chalkboard as you encourage students to share their perception of common sense and as you reinforce the need to apply logic. Say, "Refer to the list of strategies and decide which ones weren't employed by the girls." Lead discussion to conclusion that the girls didn't understand the problem (strategy #1) and they didn't check their work (strategy #4). Say, "Had the girls understood the problem and had they checked their work, what should be done would have been clear to them."

Ask, "Do you think either girl recognizes their mistake?" Allow for response. To give students a specific responsibility while viewing say, "Watch the next video and be prepared to tell what information is needed to solve the problem and which information isn't needed." RESUME tape. PAUSE tape on visual of narrator and girls; audio is, "Well, it looks like Jenny and Arlene are going to be able to buy their granny that beautiful jewelry box." Allow students to tell which information they believe is needed and which they believe isn't. To give students a specific responsibility while viewing say, "Watch as I freeze frame the video on a list of things we know about the jewelry box, then decide if your belief about needed information is correct." RESUME tape. FREEZE frame on graphic list of things known about the problem. Allow students to identify correct information as $40.00 and $19.20 difference.

Say, "Always disregard any information that isn't needed to solve a problem so it won't interfere or be confused with information you do need." To give students a specific responsibility while viewing say, "In the next video, see if you can identify which is needed information and which is not needed as the narrator states the problem to be solved." RESUME tape. PAUSE tape on graphic list of what is known and what needs to be known about the problem; audio is, "What would you do with this information?" Allow students to internalize separating needed from un-needed information as you ask, "If a farm is 120 km from where they started and the occupants of an automobile have already traveled 70 km, how many more km must they travel to reach the farm?" Allow students to discuss and answer 50 km. Ask, "How did removing unnecessary information make the problem easier to solve?" Allow students to share opinions. To give students a specific responsibility while viewing say, "Watch the next video to test the accuracy of your opinion." RESUME tape. PAUSE tape on visual of the information blinking on the screen; audio is, "...and bingo, that solves the problem." Allow time for students to tell whether their opinions were valid.

Say, "At times, a problem can appear confusing because you aren't familiar with its setting. If so and the setting can be changed to one you're more familiar with, it should be easier for you to understand." To give students a specific responsibility while viewing say, "Watch the next video which shows a problem and be prepared to tell how substituting a familiar situation for one that was unfamiliar helped." RESUME tape; PAUSE tape with audio, "...that the situation you create is parallel with the problem." Allow students to describe how bicycles were substituted for chickens, thus, eliminating an unfamiliar farm setting for city children and providing a setting they could relate to with the bicycles.

Identify a city very different from the area or city your students have grown up in and are familiar with. (e.g.) New York vs Nashville. Have students contrast how the two would differ in: recreational opportunities, housing, means of trans-portation, etc. Encourage them to consider why a youngster from Nashville might have difficulty relating to a problem about subway schedules but not the youngster from NYC; and why the youngster from NYC might have trouble relating to rabbits or squirrels playing on your lawn but not someone from Nashville. Use the discussion to reinforce the desirability of substitute situations to help solve problems.

Say, "We have discussed two important elements to help you understand problems, what are they?" List on chalkboard as students tell: eliminate unnecessary information and change an unfamiliar setting to a familiar one. Say, "A third key element to help you understand a problem is to look for a relationship between two things." To give students a specific responsibility while viewing say, "As you watch the next video, see if you can figure out which is the better buy as two children look for a contrast in price between different packages of hot dogs." RESUME tape; PAUSE tape with audio of girl, "...so two of the six packs of hot dogs would be a better buy than the package of ten." Allow for student response. Ask, "Who solved the problem but in a different way?" Discuss various methods that might have been used for solving the problem. Add number three to the list of elements to help you understand (solve) a problem on the chalkboard: find the relationship between two things.

Ask, "In what ways do many supermarkets try to help with problem solving for shoppers?" (calculators on carts to help you find better buys; items are marked with price and unit price; store brand products priced lower than a similar product with a national brand name.) Ask, "How many keys to help with better understanding a problem do you remember?" Allow students to respond. To give students a specific responsibility while viewing say, "As you watch the next video, check to see if you remembered all the keys to understanding a problem." RESUME tape. STOP tape after narrator has restated all methods. Engage students in discussion as they tell: 1. What do I know? 2. What do I need to know? 3. Change the setting. 4. Find the re-lationship by restating the problem in your own words.

Distribute to each group of four students a calculator and a set of two competing grocery items which are priced differently and are of different sizes. Suggested items are:

• 2 unlike multi-packs of bath soap
• 2 unlike sized boxes of cereal
• 6 cans of soft drink and 1 two liter bottle
• 2 unlike dish washing detergents
• 1 bottle and 1 box laundry detergent
• 1 half gallon and 1 gallon milk
• 2 varieties of cookies
• 2 rolls unlike paper towels
• 2 unlike toothpastes, etc.

Explain the items are to be used for solving a problem. (Distribute additional notebook paper as needed) Say, "Use the items distributed to your group and determine which is the better buy." Review the problem solving strategies previously presented, then instruct students to begin the task. Upon completion, allow each group to present and defend its findings as other groups use calculators to check the presenting group for accuracy.

Say, "Simplifying a problem will at times allow you to understand it more easily. In the next video two students use this strategy to determine how much candy they will have to sell in order to purchase a trampoline." To give students a specific responsibility while viewing say, "Watch and be prepared to tell how they simplified their problem."

MATH WORKS
Problem Solving: Simplifying the problem #107
BEGIN tape with visual of scene inside a gym; audio is, "Very nice." PAUSE tape with visual of scene inside an office; audio is, "... and that's how it's done." Allow students to tell how the video problem was simplified. Ask, "Did you find it easier to decide what to do with smaller numbers? How was it made easier? (easier to see relationships between smaller numbers) What are examples of situations when this might be a desirable strategy to employ?" Reinforce that an understanding of any problem can very well be the most critical of all steps in problem solving.

Say, "Just as mathematicians, scientists must also determine the problem before they can begin to solve it. Breaking a problem down to make it more specific will often create a clearer picture of exactly what the problem is." Explain; in the next video there is a very frustrated student attempting to prepare a science report. To give students a specific responsibility while viewing say, "Watch to learn why he is having difficulty developing the report and decide what he could do to make the task more manageable before the teacher shows him." RESUME tape on visual of the science topic poster; audio is, "Any problem is easier to solve if you think about a way to simplify it." STOP tape on visual of narrator; audio is, "It's a perfect way to simplify any problem." Ask, "Why was the boy having problems developing his report?" (His topic was too broad.) Ask, "When accepting a topic for research and development, why is it wise to be very specific when choosing your topic?" Accept responses. Emphasize the need to select a specific part of a topic as it permits a more in depth research strategy which will always result in a more comprehensive and thorough report. Encourage students to interact as they identify specific topics that could be selected as opposed to the following, (e.g.): Salt Water Animals; Western Ski Resorts; Balancing the National Budget; Kings and Queens of England; and The Life and Times of Galileo.

Reinforce a need to be specific as you use Galileo's theory of gravity as a specific topic that might be developed, segueing into the definition and a discussion on theory. Encourage students to share their knowledge of theory, leading to conclusion that a theory is a "best guess" based on a study of all available information about something or how something occurred. Allow students to have fun as they respond to "What is your theory about which came first, the chicken or the egg?" After students have had a moment of being frivolous, bring the responses to closure. Write the name Alfred Wagner on the chalkboard. Ask, "What famous theory is Alfred Wagner credited with developing?" Accept all responses. To give students a specific responsibility while viewing say, "Watch the next video and find out what Wagner's theory was?"

GEOGRAPHY SKILLS
Problem Solving #112
BEGIN tape with visual of the world map; audio is, "Let's compare shapes. Back in 1913..." STOP tape at end of video before closing credits. Ask, "What was Alfred Wagner's theory?" (Continental Drift) Ask, "Would you have been able to solve the problem Wagner theorized?" Allow students to respond. Ask, "What in-formation and clues were available for Wagner to study and base his hypothesis and theory on?" (Land formations on different continents that matched at the shorelines.) "Do patterns sometimes provide clues in other areas of problem solving?" Encourage students to interact and share their ideas. "What did Wagner name the super-continent?" (Pangea) Ask, "In what ways did patterning help Wagner solve the problem and develop his theory?" Guide discussion to realization that understanding the problem and his forming an hypothesis about known patterns between the continents were critically important to the development of his theory.

Distribute to each group of four students: 1 copy of the Activity Sheet; 1 sealable plastic bag containing 15 red beans, 15 lima beans and 15 peas; and 1 paper plate. Give all students in each group a piece of notebook paper and a pencil. Use an overhead projector to show the beans and a clean sheet on the chart tablet to record results and strategies which are used by students to solve the problems.

Say, "You are now going to solve problems in a hands-on process using the materials that were distributed to your group. Remember to follow the sequence of strategies discussed today in solving the problems." Explain a person familiar to the group has written a Bean Cook Book, however, the eight recipes on the Activity Sheet were returned by the publisher because some of the ingredients and amounts were not included. Say, "It is now your responsibility to solve the problems so the cookbook can be published. Each recipe must contain all three types of beans."

Instruct students to begin with problem number one and use their bag of beans to solve the problem in a hands-on manner. Say, "Then, show your calculations on notebook paper." Allow time for groups to complete the task. Select a volunteer to explain how her/his group solved the problem, then have them use the overhead projector to show all beans that were used. Next, select another volunteer from the group to demonstrate their cal-culations on the chalkboard. If other groups incorporated strategies that differed from the presenting group, have them share the strategy they used. (Ans: 2 Lima beans; 4 Red beans; 4 Peas) Record the answer on the chart tablet.

Say, "Some problems can have more than a single solution; others will have just one." Instruct students to continue and solve problem number two. Repeat the previous process as another group is selected to present. (Ans: 4 Red beans; 2 Peas; 4 Lima beans) Record results on the chart tablet.

Instruct groups to proceed with problem three. Use the established pattern for student presentation as other groups validate correctness of the presenting group's work. (Ans: 2 Red beans; 4 Lima beans; 2 Peas)

Direct groups to continue with problem number four. Again, use the previous process of presenting as another group explains and visually demonstrates their work using the overhead projector. (Ans: 5 Red beans; 5 Lima beans; 8 Peas) After presentation, validation and up-dating of the chart are completed for this problem, engage students in discussion related to examples of when concrete methods and abstract methods of problem solving have thus far been used.

Instruct students to continue with problem number five. Follow the procedure as a different group is appointed presenter. (Ans: 6 Red beans; 3 Lima beans; 3 Peas)

Have students continue with problem number six. After groups have completed the task, point out that possible solutions can be endless as the recipe stated "at least 12 beans." Any answer is correct that contains no less than a total of 12 beans and follows the other two criteria.

Direct students to compute problem number seven. (Ans: 1 Pea; 3 Red beans; 4 Lima beans)

Have students solve problem number eight. Upon completion, inform students there are several solutions to this problem. Each that does not exceed a total of 20 beans is correct.

Conduct a brief review of strategies to be used in problem solving. Have students consider the desirability for creating a mental picture to be used in visualizing important information and determining possible strategies they might use to solve a problem.

Plan a field trip and visit the Meteorology Department of a local television station. Prepare students to look for methods of problem solving used to forecast weather. As a forecast is prepared, have students look for activities which reveal problem solving procedures. They should be able to identify and describe how each is used. Invite a law enforcement officer to visit the classroom and discuss how problem solving is an on-going responsibility in her/his profession. Ask them to include how knowledge of patterns is an integral part of identifying suspects. Use the information to create a list of situations in which students solve "cases" where patterns link a suspect(s) to a crime or to one another.

Creative Writing
Have an interested group of students write a short drama about preventing poachers from destroying animals in a wildlife preserve in Kenya.

Performing Arts
Use the creative writing drama and plan a performance. Appoint students with special interests or skills to compose or select appropriate music, design a set and costumes, serve as prop and stage hands and serve as performers. Encourage them to utilize problem solving strategies from inception and planning throughout the actual time of performance.

Science/Research
Select an inventor, a doctor who discovered a cure for a disease, a great artist or performer, etc., and research their life to learn how they became successful. Apply how they succeeded to strategies for solving problems, then share findings by preparing a report to be presented to the class.

Master Teachers: James Parsons and Sharon Braden