Describe Polya - Ppismp Sem2

10 Describe Polya’s Problem Solving Model

Meaning of Problem and problem-Solvings.

Solving mathematics problems are activities involving problems in the form of mathematics language, including mechanical problems, puzzles, quiz and the use of mathematics skills in actual situations.

According to Lester, a prominent mathematician in the 1970’s, defined problem-solving as : “ Problem involving a situation whereby an individual or a group is required to carry out the working solution. In doing so, they have to determine the strategy and method of problem-solving first, before implementing the working solution. The strategy of problem –solving needs a set of activities which will lead to the problem-solving process.”

Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing

these

situations

mathematically,

formulating

appropriate

mathematical questions, and using a variety of strategies to find solutions. By developing their problem-solving skills, students will come to realize the potential usefulness of mathematics in their lives.

Problem solving is a term that often means different things to different people. Sometimes it even means different things at different times for the same people! It may mean solving simple word problems that appear in standard textbooks, applying mathematics to real-world situations, solving non routine problems or puzzles, or creating and testing mathematical conjectures that may lead to the study of new concepts. In every case, however, problem solving involves an individual confronting a situation which she has no guaranteed way to resolve. Some tasks are problems for everyone (like finding the volume of a puddle), some are problems for virtually no one (like counting how many eggs are in a dozen), and some are problems for some people but 1

not for others (like finding out how many balloons 4 children have if each has 3 balloons, or finding the area of a circle).

Problem solving involves far more than solving the word problems included in the students' textbooks; it is an approach to learning and doing mathematics that emphasizes questioning and figuring things out. The Curriculum and Evaluation Standards of the National Council of Teachers of Mathematics considers problem solving as the central focus of the mathematics curriculum.

"As such, it is a primary goal of all mathematics instruction and an integral part of all mathematics activity. Problem solving is not a distinct topic but a process that should permeate the entire program and provide the context in which concepts and skills can be learned." (p. 23)

Thus, problem solving involves all students a large part of the time; it is not an incidental topic stuck on at the end of the lesson or chapter, nor is it just for those who are interested in or have already mastered the day's lesson. Students should have opportunities to pose as well as to solve problems; not all problems considered should be taken from the text or created by the teacher. However,

the

situations

explored

must

be

interesting,engaging,

and

intellectually stimulating. Worthwhile mathematical tasks are not only interesting to the students, they also develop the students' mathematical understandings and skills, stimulate them to make connections and develop a coherent framework for mathematical ideas, promote communication about mathematics, represent mathematics as an ongoing human activity, draw on their diverse background experiences and inclinations, and promote the development of all students' dispositions to do mathematics (Professional Standards of the National Council of Teachers of Mathematics). As a result of such activities, students come to understand mathematics and use it effectively in a variety of situations. 2

Characteristics of mathematics problem Contains elements which can be found in the environment Its solution needs proper strategy in planning, including selection of suitable methods for problem-solving. Proper strategy in planning and selection of suitable method depend on the pupils’ acquired knowledge and experience as well as understanding of the relevant problem. The ability of problem-solving is closely related to the pupils’ level of cognitive development, at least at its application level. The way used for problem-solving cannot be memorized as In the case of reciting mathematics formula or solving mechanical question by means of memorization. Every mathematics problem ought to have its own specific solution. The method of problem-solving may consist of more than one approach. The process of problem-solving needs to implement by means of a set of systematic activities. The process of problem-solving needs to apply mathematics skills, concepts or principles which have been learned and mastered.

George Polya- 1887 – 1985

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George Polya was a Hungarian who immigrated to the United States in 1940. His major contribution is for his work in problem solving. Growing up he was very frustrated with the practice of having to regularly memorize information. He was an excellent problem solver. Early on his uncle tried to convince him to go into the mathematics field but he wanted to study law like his late father had. After a time at law school he became bored with all the legal technicalities he had to memorize. He tired of that and switched to Biology and the again switched to Latin and Literature, finally graduating with a degree. Yet, he tired of that quickly and went back to school and took math and physics. He found he loved math. His first job was to tutor Gregor the young son of a baron. Gregor struggled due to his lack of problem solving skills. Polya (Reimer, 1995) spent hours and developed a method of problem solving that would work for Gregor as well as others in the same situation. Polya (Long, 1996) maintained that the skill of problem was not an inborn quality but, something that could be taught. He was invited to teach in Zurich, Switzerland. There he worked with a Dr. Weber. One day he met the doctor?s daughter Stella he began to court her and eventually married her. They spent 67 years together. While in Switzerland he loved to take afternoon walks in the local garden. One day he met a young couple also walking and chose another path. He continued to do this yet he met the same couple six more times as he strolled in the garden. He mentioned to his wife ?how could it be possible to meet them so many times when he randomly chose different paths through the garden? He later did experiments that he called the random walk problem. Several years later he published a paper proving that if the walk continued long enough that one was sure to return to the starting point. In 1940 he and his wife moved to the United States because of their concern for Nazism in Germany (Long, 1996). He taught briefly at Brown University and then, for the remainder of his life, at Stanford University. He quickly became well known for his research and teachings on problem solving. He taught many classes to elementary and secondary 4

classroom teachers on how to motivate and teach skills to their students in the area of problem solving. In 1945 he published the book How to Solve It which quickly became his most prized publication. It sold over one million copies and has been translated into 17 languages. In this text he identifies four basic principles . In How To Solve It, G. Polya describes four steps for solving problems and outlines them at the very beginning of the book for easy reference. The steps outline a series of general questions that the problem solving student can use to successfully write resolutions. Without the questions, common sense goes through the same process; the questions simply allow students to see the process on paper. Polya designed the questions to be general enough that students could apply them to almost any problem.

The four steps are: •

understanding the problem,

•

devising a plan,

•

carrying out the plan, and

•

looking back. This method is very similar to the method in Thinking Mathematically by

John Mason, except Polya separates devising a plan, and carrying out the plan. This may seem silly at first, but Polya argues that it does make a difference. By first devising a plan, students can eliminate mistakes they might make by rushing into the actual execution of the plan. When they plan it out first and then do the math, it is possible to check their work as they go along. Polya?s First Principle: Understand the Problem In the first principle, pupils would be guided to understand: (a)Variables involved in the problem; (b)Relationship between the variables which have been ascertained; and (c)Variable which needs to be thoroughly searched or answered. 5

In theother words, Understanding the Problem can be explain by this question : 1.

Can you state the problem in your own words?

2.

What are you trying to find or do?

3.

What are the unknowns?

4.

What information do you obtain from the problem?

5.

What information, if any, is missing or not needed? For the students, they should be able to state the unknown, or the thing

they want to find to answer the question, the data the question gives them to work with, and the condition, or limiting circumstances they must work around. If they can identify all of these, and explain the question to other people, then they have a good understanding of what the problem is asking. Polya suggests that students draw a picture if possible, or introduce some kind of notation to visualize the question. This seems so obvious that it is often not even mentioned, yet students are often stymied in their efforts to solve problems simply because they don?t understand it fully, or even in part. Polya taught teachers to ask students questions such as: •

Do you understand all the words used in stating the problem?

•

What are you asked to find or show?

•

Can you restate the problem in your own words?

•

Can you think of a picture or a diagram that might help you understand the problem?

•

Is there enough information to enable you to find a solution? Some techniques that may help students with this important aspect of problem solving - understanding the problem - include restating the problem in their own words, drawing a picture, or acting out the problem situation. Some teachers have students work in pairs on problems, with one student reading the problem and then, without referring to the written text, explaining what the problem is about to their partner.

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Polya?s Second Principle: Devise a plan Polya mentions (1957) that it are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. We will find choosing a strategy increasingly easy. To devise a plan, students can start by trying to think of a related problem we have solved before to help them. If the student can think of a problem they have solved before that had a similar unknown, it could also be helpful. Students can also try to restate the problem in an easier or different way, and try to solve that. By looking at these related problems, students may be able to use the same method, or other part of the plan used. After students have decided which calculations, computations, or constructions that they need, and have made sure that all data and conditions were used, they can try out their plan. A partial list of strategies is included: Guess and check We also can use these instructions to

Make an orderly list

understand more details.

Eliminate possibilities Use symmetry



unknown. You may be obliged to consider

Consider special cases

auxiliary problems if an immediate connection

Use direct reasoning

cannot be found. You should obtain eventually a

Solve an equation

plan of the solution.

Also suggested: Look for a pattern Draw a picture Solve a simpler problem Use a model Work backward Use a formula Be creative Use your head/noggin

Find the connection between the data and the

 Have you seen it before? Or have you seen the same problem in a slightly different form? 

Do you know a related problem? Do you know a theorem that could be useful?



Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.



Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible? 7

 Could you restate the problem? Could you restate it still differently? Go back to definitions.  If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?  Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

Polya?s third Principle: Carry out the plan Carrying out the plan is sometimes the easiest part of solving a problem. However, many students jump to this step too soon. Others carry out inappropriate plans, or give up too soon and stop halfway through solving the problem. To reinforce the process of making a plan and carrying it out, teachers might use the following technique: Divide a sheet of notebook paper into two columns. On the left side of the page, the student solves the problem. On the right side of the page, the student writes about what is going on in his/her mind concerning the problem. Is the problem hard? How can you get started? What strategy might work? How did you feel about the problem? This step is usually easier than devising the plan. In general (1957), all you need is care and patience, given that you have the necessary skills. Persistent with the plan that you have chosen. If it continues not to work discard it and choose another. Don’t be misled, this is how mathematics is done, even by professionals. 8

Here are some tips to use: ○ Implement the strategy in Step 2 and perform any necessary actions or computations. ○

Check each step of the plan as you proceed. This may be intuitive checking or a formal proof of each step..

○

Keep an accurate record of your work. Can you see clearly that the step is correct? Can you prove that it is correct?

Polya’s Fourth Principle: Look back This is the part of problem-solving that most people tend to ignore. One way for us to improve is to review past experiences and understand why we succeed or fail. So it is important to monitor our own performance review the whole exercise in order that we can do even better in the future. Polya mentions (1957) that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didnt. Doing this will enable you to predict what strategy to use to solve future problems. George Polya went on to publish a two-volume set, Mathematics and Plausible Reasoning (1954) and Mathematical Discovery (1962). These texts form the basis for the current thinking in mathematics education and are as timely and important today as when they were written. Polya has become known as the father of problem solving. When students look back on the problem and the plan they carried out, they can increase their understanding of the solution. It is always good to recheck the result and argument used, and to make sure that it is possible to check them. Then students should ask, "Can I get the result in a different way?"and "Can I use this for another problem?" The last chapter of the book is a very helpful encyclopedia of the terms used in the explanation of the first chapter. A partial list of strategies is included: 1. Check the results in the original problem. In some cases, this will require a proof. 9

2. Interpret the solution in terms of the original problem. Does your answer make sense? Is it reasonable? 3. Determine whether there is another method of finding the solution. 4. If possible, determine other related or more general problems for which the techniques will work. While it might seem most logical to begin problem solving with Polya's first activity and proceed through each activity until the end, not all successful problem solvers do so. Many successful problem solvers begin by understanding the problem and making a plan. But then as they start carrying out their plan, they may find that they have not completely understood the problem, in which case they go back to step one. Or they may find that their original plan is extremely difficult to pursue, so they go back to step two and select another approach. By using these four activities as a general guide, however, students can become more adept at monitoring their own thinking. This "thinking about their thinking" can help them to improve their problem solving skills.

Students move through a continuum of stages in their development as problem solvers (Kantowski, 1980). Initially, they have little or no understanding of what problem solving is, of what a strategy is, or of the mathematical structure of a problem. Such students usually do not know where to begin to solve a problem; the teacher must model the problem solving process for these students. At the second level, students are able to follow someone else's solution and may suggest strategies for similar problems. They may participate actively in group problem solving situations but feel insecure about independent activities, requiring the teacher's continued support. At the third level, students begin to be comfortable with solving problems, suggesting strategies different from those they have seen used before. They understand and appreciate that problems may have multiple solutions or perhaps even no solution at all. Finally, at the last level, students are not only adept at solving problems, they are also interested in finding elegant and efficient solutions and in exploring alternate solutions to the same problem. In teaching problem 10

solving, it is important to address the needs of students at each of these levels within the classroom. In summary, the real test of whether a student knows mathematics is whether she can use it in a problem situation. Students should experience problems as introductions to learning about new topics, as applications of content already studied, as puzzles or non-routine problems that have many solutions, and as situations that have no one best answer. They should not only solve problems but also pose them. They should focus on understanding a problem, making a plan for solving it, carrying out their plan, and then looking back at what they have done

.

20 Explain routine and Non routine Problems

Futurists continue to stress that our future is going to undergo change at a rate even greater than present generations have experienced. This implies that today’s and future problems will have a dynamic component. Such problems change or evolve as they are being studied. It is evident then that a fundamental skill for dealing with the future is active problem solving, i.e., the ability to solve problems which are undergoing change during the process of resolution. 11

Problem solving can be divided into two categories, routine and non routine which is Routine problem solving and Non routine problem solving. ROUTINE PROBLEM Routine problem solving is stresses the use of sets of known or prescribed procedures (algorithms) to solve problems. In a routine problem, the problem solver knows a solution method and only needs to carry it out. The strength of this approach is that it is easily accessed by paper-pencil tests. Since today’s computers and calculators can quickly and accurately perform the most complex arrangements of algorithms for multi-step routine problems, the typical workplace does not require a high level of proficiency in routine problem solving. However, today’s workplace does require many employees to be proficient in Non routine problem solving. Routine problems are sometimes called exercises, and technically do not fit the definition of problem stated above. When the goal of an educational activity is to promote all the aspects of problem solving (including devising a solution plan), then non routine problems (or exercises) are appropriate.

Routine problem is actually is a type of mechanical mathematic problem. It aimed at training the students for able to master basic skills, especially the arithmetic skills which involving the four operations, addition, subtraction, multiplication and division (+, -, ×, ÷), or directs applications of using mathematics formulae, laws, theorems or equations. Generally speaking, routine problems are the most basic simple type of problem-solving in mathematics, as its goal expression can be achieved by means of certain algorithm.

12

NON ROUTINE PROBLEM Non routine problem solving is stresses the use of heuristics and often requires little to no use of algorithms. The problem solver does not initially know a method for solving the problem. Unlike algorithms, heuristics are procedures or strategies that do not guarantee a solution to a problem but provide a more highly probable method for discovering the solution. Building a model and drawing a picture of a problem are two basic problem-solving heuristics. Other heuristics include describing the problem situation, making the problem simpler, finding irrelevant information, working backwards, and classifying information.

Actually, Non-routine problem is a unique problem-solving which requires the application of skills, concepts or principles which have been learned and mastered. Method for solving non-routine problem in mathematics is different from answering mechanical question. It needs systematic activities with logical planning, including proper strategy and selection of suitable method for implementation. Most of the non-routine problems required a heuristic approach such as the application of experiences and practical effort, or planned strategy, to attain its goal expression..

There are two types of non routine problem solving situations which is static and Active. Static non routine problems have a fixed known goal and fixed, known elements that are used to resolve the problem. Solving a jigsaw puzzle is an example of a static non routine problem. Given all pieces to a puzzle and a picture of the goal, learners are challenged to arrange the pieces to complete the picture. Various heuristics such as classifying the pieces by color, connecting the pieces which form the border, or connecting the pieces which form a salient feature to the puzzle, such as a flag pole, are typical ways in which people attempt to resolve such problems.

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Active non routine problem solving may have a fixed goal with changing elements, a changing goal or alternative goals with fixed elements, or changing or alternative goals with changing elements. The heuristics used in this form of problem solving are known as strategies. People who study such problems must learn to change or adapt their strategies as the problem unfolds.

According to John Dewey, learning through problem-solving includes five stages. They are briefly shown in the following figure.

PROCESS OF PROBLEM-SOLVING The process of identifying the problem involves activities to understand and ascertain important aspects contained in the problem. The stage of looking for information involves activities to collect materials or facts related to the problem. The stage that follows is the setting up of a hypothesis to suggest strategies and methods to solve the problem identified. The next stage is testing the hypothesis whereby the suggested strategies and methods are implemented in the process. Finally an evaluation is made on the techniques used during the process of solving the problem with decision on final conclusion and records.

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30 Gather information and select three non routine problems and solve each of

these problems using two or more types of problem solving strategies. Elaborate on the different strategies. Select one strategy that is deemed to be most efficient and justify selection.

Question: Amy and Judy sold 12 show tickets altogether. Amy sold 2 more tickets than Judy. How many tickets did each girl sell? Strategy: 1) UNDERSTAND: What do you need to find? You need to know that 12 tickets were sold in all. You also need to know that Amy sold 2 more tickets than Judy. 2) PLAN: How can you solve the problem? 15

a) Guess and Check You can guess and check to find two numbers with a sum of 12 and a difference of 2. If your first guess does not work, try two different numbers. 3) SOLVE: First Guess: Amy = 8 tickets Judy = 4 tickets Check 8 + 4 = 12 8 - 4 = 4 ( Amy sold 4 more tickets) These numbers do not work! Second Guess: Amy = 7 tickets Judy = 5 tickets Check 7 + 5 = 12 7- 5 = 2 ( Amy sold 2 more tickets) These numbers do work! Amy sold 7 tickets and Judy sold 5 tickets. b) Draw a Picture

12 tickets were sold Lets we divide half Each others Amy(6)

Judy(6)

16

* Amy sold 2 more tickets than Judy. Decrease 1 tickets from Judy

Now, we got the answer. Amy sold 7 tickets and Judy(5)

Judy sold 5 tickets.

Amy(7)

Justify Selection: In this question, I think that draw a picture strategy is more suitable. Students can directly see that there are how to arrange the strategy. Then, they will see how to find the answer. On the sport they can get the true answer, which is 7 show tickets is sold by Amy and only 5 show tickets is sold by Judy. Besides, this strategy is easy to carry out. It only needs to draw it on a paper. Sometimes, we are lack of ideas how can the question act. Even though we are able to get the idea, we still need to aware on the relationship on the diagram. So, I prefer the strategy of draw a picture more because it is save and easier to be carry out.

17

Question: Laura has 3 green chips, 4 blue chips and 1 red chip in her bag. What fractional part of the bag of chips is green? Strategy: 1) UNDERSTAND: What do you need to find? You need to find how many chips are in all. Then you need to find how many of the chips are green. 2) PLAN: How can you solve the problem? a) Draw a Picture You can draw a picture to show the information. Then you can use the picture to find the answer. 3) SOLVE: Draw 8 chips.

3/8 of the chips are green. 18

b) Make a diagram

3/8 of the chips are green.

Justify Selection: In this question, I think strategy draw a diagram is more suitable. This is because it is easier to understand and carry out. Besides, students can go through to the answer. . Therefore, in my opinion strategy 2 is more suitable for me because it is easier to understand and carry out.

Question: Judy is taking pictures of Jim, Karen and Mike. She asks them, " How many different ways could you three children stand in a line?" Strategy: 1) UNDERSTAND: What do you need to know? 19

You need to know that any of the students can be first, second or third. 2) PLAN: How can you solve the problem?

a) Make a List You can make a list to help you find all the different ways. Choose one student to be first, and another to be second. The last one will be third. 3) SOLVE: When you make your list, you will notice that there are 2 ways for Jim to be first, 2 ways for Karen to be first and 2 ways for Mike to be first. First

Second

Third

Jim

Karen

Mike

Jim

Mike

Karen

Karen

Jim

Mike

Karen

Mike

Jim

Mike

Karen

Jim

Mike

Jim

Karen

So, there are 6 ways that the children could stand in line.

b) Guess and Check Firstly, let say start with: Jim, follow by Karen and lastly is Mike Karen, follow by Jim and lastly is Mike Mike, follow by Karen and lastly is Jim But, there have another ways that the children could stand in line. Jim, follow by Mike and lastly is Karen Karen, follow by Mike and lastly is Jim Mike, follow by Jim and lastly is Karen. * That’s mean there are 6 ways that the children could stand in line. 20

Justify Selection:

In this question, I think strategy 1 – make a list is more suitable. This is because it is easier to understand and carry out. Sometimes, we might face on problems when use guess and check, like unable to determine the answer accurately. Then, students will waste their time to guess until get the right answer Therefore, in my opinion strategy 1 which is make a list is more suitable for me because it is easier to understand , save time and easier to carry out.

40 Create three new but similar problems and solve it using the suggested strategies.

a) Make a list 1. Doug has 2 pairs of pants: a black pair and a green pair. He has 4 shirts: a

white shirt, a red shirt, a grey shirt, and a striped shirt. How many different outfits can he put together? (Hint: Cornplete the organised list.) Make a list Pants

Shirts 21

Black—White Black—Red Black— Black— Pants

Shirts

Green— Green— Green— Green— Understanding the Problem · How many pairs of pants does Doug have? (2) · How many shirts does Doug have? (4)

Planning a Solution · Suppose Doug wears his black pants. What color shirt can he wear? (white, red, grey, or striped) · If Doug wears his striped shirt, how many different outfits can he wear? (2: striped—black and striped—green) · If Doug wears the green pants, can he wear all 4 shirts? (yes) Finding the Answer Make an Organized List Black—White Black—Red Black—Gray Black—Striped Green—White Green—Red Green—Gray Green—Striped Doug can make 8 different outfits.

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2. There’s only one bicycle at the IPGM Campus Tuanku Bainun that Sara, Tirah

and Ezzaty can borrow. How many times could each of them borrow the bicycle? Strategy: 1) UNDERSTAND: What do you need to know? You need to know that any of the students can be first, second or third. 2) PLAN: How can you solve the problem?

You can make a list to help you find all the times they can borrow the bicycle. Choose one student to be first, and another to be second. The last one will be third. 3) SOLVE: When you make your list, you will notice that there are 2 times for Sara to be first, 2 times for Tirah to be first and 2 times for Ezzaty to be first. First Second Third Sara

Tirah

Ezzaty

Sara

Ezzaty

Tirah

Tirah

Sara

Ezzaty

Tirah

Ezzaty

Sara

Ezzaty

Tirah

Sara

Ezzaty

Sara

Tirah

So, there are 6 times that the students could borrow the bicycle.

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3.The letters ABCD, can be put into a different order: DCBA or BADC. How many different combinations of the letters ABCD can you make? i. Strategy:

a) UNDERSTAND: What do you need to know? You need to know that any of the combination letters ABCD can can be first, second or third. b) PLAN: How can you solve the problem? You can make a list to help you find all the different combination letters ABCD. Choose one combination letterst to be first, and another to be second. The last one will be third. c) SOLVE: To answer this question, obviously, you have to make a list. Teach your students to make a SYSTEMATIC list. For example:

ABCD

BACD

CABD

DABC

ABDC

BADC

CADB

DACB

ACBD

BCAD

CBAD

DBAC

ACDB

BCDA

CBDA

DBCA

ADBC

BDAC

CDAB

DCAB

ADCB

BDCA

CDBA

DCBA

By making a SYSTEMATIC list, students will see every possible combination. 24

(Later, perhaps, they will learn that the number of permutations of size 4 taken from a set of 4 can be represented by the formula 4 * 3 * 2 * 1 = 24).

b)Draw a picture 1. In a sunny day, Johnny walks into the school hall, he saw 4 girls and 3 boys standing in front the gate. Each girl carrying 3 school bags and each boy carrying 2. How many shopping bags did Johnny saw?

First, draw a picture so that we can count.

Count the school bags. The four girls have 12 bags. The two boys have 4 bags. 12 + 4 = 16 Johnny saw 16 school bags in front the gate.

2. Jenny saw 2 big houses and 3 small houses are build besides the lake. Each of the big houses had 4 windows and each small houses had 2 windows. How many windows did Jenny saw?

25

First, draw a picture so that we can count.

Count the windows. Big houses have eight windows. Small houses have six windows. 8 + 6 = 14 Jenny saw 14 windows on the houses by the lake.

3.

In a restaurant, Kelly saw 6 rectangle tables and 3 triangle tables. Each of

the rectangle tables had 4 legs and the triangle tables had 3 legs each. How many table legs did Kelly see in the restaurant?

First, draw a picture so that we can count.

Count the legs of the tables. Rectangle tables have 24 legs. Triangle tables have 9 legs. 24 + 9 = 33 Kelly saw 33 table legs in the restaurant.

c) Make a Diagram 26

A coin has two face, head and tail. If you toss 3 coins together, determine that how many different combinations of heads and tails you would get.

•

There are 6 combinations of heads and tails.

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