Unit 2: What some are problem solving strategies?

From KNILT

Learning Objectives

·Recognize and describe different problem solving methods and understand how they can be applied to solve real-life problems.

·Apply different problem solving strategies to meet the needs of diverse learners (Howard Gardner’s Theory of Multiple Intelligences).\

Mini-lecture

For this unit, we will use the sixth grade mathematics CCSS to explore how we can use various problem solving strategies to solve problems. By exposing students to different problem solving strategies, students will become more engaged participants. Students have different preferences for the way or ways that they learn best. Since students possess different learning styles, modeling only one way to solve a problem may cause the students to lose interest. By encouraging the use of different strategies, students will be able to attempt problems in the way that most interests them. For example, visual learners would benefit from using pictures or diagrams to solve problems since this method is the way they learn best.

The graphic show below is a poster that I have hanging in my classroom which lists various problem solving strategies. Before implementing any of the strategies, the students know to first carefully read the problem and underline all of the important information. This poster serves as a reference for students to try different approaches when solving real-life mathematical problems.

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Strategy: Draw a Diagram/Make a Model

Students can solve problems by drawing a diagram to visually identify the relationships between the data that is given.

Problem: Denver, Colorado is called the “Mile High City” because its elevation is 5,280 feet above sea level. The elevation of Death Valley, California is 282 feet below sea level. How many feet higher is Denver than Death Valley?

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This graph presents a simple visual comparison of two separate elevations which gives the student enough information to correctly set up the equation to solve.

Strategy: Draw a Picture

Students can solve problems by drawing a picture to visually identify the relationships between the data that is given.

Problem: Laura has 3 green candies, 4 blue candies, 5 orange candies, and 1 red candy in her bag. What is the ratio of green candies to the total amount of candies?

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This representation of each colored candy allows for a clear, simple view of the total group.

Strategy: Make a table

Problem: Mark was mixing blue paint and yellow paint in the ratio 2:3 to make green paint. How many liters of blue and yellow paint would he need to make 45 liters of green paint?

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By setting up this table with the ratio in its simplest form first and increasing the amounts I the exact proportions to the given ratio, an answer will eventually present itself.

Strategy: Make an organized list/chart

Problem: Judy is taking pictures of Jim, Karen and Mike. She asks them, "How many different ways could you three children stand in a line?"

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This chart gives the student an explicit list of all possible solutions to the question asked.


Strategy: Write a number sentence

Problem: Kelly distributed 84 pencils into three equal groups.

We know that Kelly has a total of 84 pencils and that she would like to evenly split up or divide her pencils into 3 groups. The number sentence would be: 84 ÷ 3 = ___. The next step would be to perform the multi-digit division. The completed number sentence would be 84 ÷ 3 = 28.

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Strategy: Try simpler numbers

From my experience, I have noticed that students experience a great deal of difficulty when they are given word problems involving fractions. When the students see the fractions, they become confused and often do not understand what the question is asking or what operation is required. When working on such problems, I encourage my students to use simpler numbers. What I mean by this is that students will replace fractions with whole numbers and develop a strategy based on whole numbers in the place of the fractions. The strategy that they devise or the operation that they find appropriate can then be carried out with the fractions from the original problem.

Problem: The cattle at the Washington Farm are fed 4/5 of a bale of hay each day. The horses are fed 2/3 as much hay as the cattle. How many bales of hay are the horses fed each day?

· Replace the fractions with whole numbers - The cattle at the Washington Farm are fed 5 bales of hay each day. The horses are fed 3 times as much hay as the cattle. How many bales of hay are the horses fed each day?

·By replacing the fractions with whole numbers, it is now clearer that multiplication is needed to solve this problem. 5 x 3 = 15 bales

·Multiply the fractions that were given in the original problem.

Strategy: Guess and Check

Problem: At a library book sale, Maria used an equal number of quarters and nickels to buy a $1.20 book. How many of each coin did she use?

Guess 1: 3 quarters and 3 nickels

(3 x $0.25) + (3 x $0.05) = $0.90 -> check: the total is too low

Guess 2: 5 quarters and 5 nickels

We know that the number of quarters and nickels has to be greater based on the results of the previous guess.

(5 x $0.25) + (5 x $0.05) = $1.50 -> check: the total is too high

Guess 3: 4 quarters and 4 nickels

We know that the total from guess 2 was too high and the total from guess 1 was too low so we know that the total number of quarters and nickels is between 3 and 5.

(4 x $0.25) + (4 x $0.05) = $1.20 -> check: it works

Maria used 4 quarters and 4 nickels.

To use the guess and check strategy, it is important for students to first make an initial educated guess and then check their work. Next, the students must analyze their answer and determine how their answer will logically effect their next guess. The guess and check strategy is not always the best strategy because students who do not understand the relationship between their guess and results will spend a great deal of time trying to guess. It is beneficial for students using this strategy to have good mathematical reasoning skills so that they are able to make logical guesses based off of their previous attempts.


Howard Gardner’s Theory of Multiple Intelligences

  • Visual-Spatial - think in terms of physical space, as do architects and sailors. Very aware of their environments. They like to draw, do jigsaw puzzles, read maps, and daydream. They can be taught through drawings, verbal and physical imagery. Tools include models, graphics, charts, photographs, drawings, 3-D modeling, video, videoconferencing, television, multimedia, texts with pictures/charts/graphs.
  • Bodily-kinesthetic - use the body effectively, like a dancer or a surgeon. Keen sense of body awareness. They like movement, making things, touching. They communicate well through body language and be taught through physical activity, hands-on learning, acting out, role playing. Tools include equipment and real objects.
  • Musical - show sensitivity to rhythm and sound. They love music, but they are also sensitive to sounds in their environments. They may study better with music in the background. They can be taught by turning lessons into lyrics, speaking rhythmically, tapping out time. Tools include musical instruments, music, radio, stereo, CD-ROM, multimedia.
  • Interpersonal - understanding, interacting with others. These students learn through interaction. They have many friends, empathy for others, street smarts. They can be taught through group activities, seminars, dialogues. Tools include the telephone, audio conferencing, time and attention from the instructor, video conferencing, writing, computer conferencing, E-mail.
  • Intrapersonal - understanding one's own interests, goals. These learners tend to shy away from others. They're in tune with their inner feelings; they have wisdom, intuition and motivation, as well as a strong will, confidence and opinions. They can be taught through independent study and introspection. Tools include books, creative materials, diaries, privacy and time. They are the most independent of the learners.
  • Linguistic - using words effectively. These learners have highly developed auditory skills and often think in words. They like reading, playing word games, making up poetry or stories. They can be taught by encouraging them to say and see words, read books together. Tools include computers, games, multimedia, books, tape recorders, and lecture.
  • Logical -Mathematical - reasoning, calculating. Think conceptually, abstractly and are able to see and explore patterns and relationships. They like to experiment, solve puzzles, and ask cosmic questions. They can be taught through logic games, investigations, and mysteries. They need to learn and form concepts before they can deal with details.

Learning Activities

1. Which strategies would appeal to most students? Why?

2. Which strategy is the most difficult to demonstrate?

3. Which theory of multiple intelligences can most of these strategies be applied? Justify your answer?

Conclusion

Many of the problem solving activities that were presented appeal to visual learners. Depending upon the given problem, the complexity of some strategies would increase and students might try to seek a different, and more concrete approach. The strategies that can be used effectively depend on the student and the problem. Normal 0 7.8 磅 0 2 false false false MicrosoftInternetExplorer4