Math

G. Polya, How to Solve It

Summary taken from G. Polya, "How to Solve It", 2nd ed., Princeton University Press, 1957, ISBN 0-691-08097-6.

1. UNDERSTANDING THE PROBLEMo First. You have to understand the problem.o What is the unknown? What are the data? What is the condition?o Is it possible to satisfy the condition? Is the condition sufficient to determine the

unknown? Or is it insufficient? Or redundant? Or contradictory?o Draw a figure. Introduce suitable notation.o Separate the various parts of the condition. Can you write them down?

2. DEVISING A PLANo Second. Find the connection between the data and the unknown. You may be

obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.

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

o Do you know a related problem? Do you know a theorem that could be useful?o Look at the unknown! And try to think of a familiar problem having the same or a

similar unknown.o 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?

o Could you restate the problem? Could you restate it still differently? Go back to definitions.

o 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?

o Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

3. CARRYING OUT THE PLANo Third. Carry out your plan.o Carrying out your plan of the solution, check each step. Can you see clearly that

the step is correct? Can you prove that it is correct?4. Looking Back

o Fourth. Examine the solution obtained.o Can you check the result? Can you check the argument?o Can you derive the solution differently? Can you see it at a glance?o Can you use the result, or the method, for some other problem?

An organization needs to define some standard of problem solving, so that leadership can effectively direct others in the research and resolution of issues.

In problem solving, there are four basic steps.

1. Define the problem

Diagnose the situation so that your focus is on the problem, not just its symptoms. Helpful techniques at this stage include using flowcharts to identify the expected steps of a process and cause-and-effect diagrams to define and analyze root causes.

The chart below identifies key steps for defining problems. These steps support the involvement of interested parties, the use of factual information, comparison of expectations to reality and a focus on root causes of a problem. What’s needed is to:

Review and document how processes currently work (who does what, with what information, using what tools, communicating with what organizations and individuals, in what time frame, using what format, etc).

Evaluate the possible impact of new tools and revised policies in the development of a model of “what should be.”

2. Generate alternative solutions

Postpone the selection of one solution until several alternatives have been proposed. Having a standard with which to compare the characteristics of the final solution is not the same as defining the desired result. A standard allows us to evaluate the different intended results offered by alternatives. When you try to build toward desired results, it’s very difficult to collect good information about the process.

Considering multiple alternatives can significantly enhance the value of your final solution. Once the team or individual has decided the “what should be” model, this target standard becomes the basis for developing a road map for investigating alternatives.Brainstorming and team problem-solving techniques are both useful tools in this stage of problem solving.

Many alternative solutions should be generated before evaluating any of them. A common mistake in problem solving is that alternatives are evaluated as they are proposed, so the first acceptable solution is chosen, even if it’s not the best fit. If we focus on trying to get the results we want, we miss the potential for learning something new that will allow for real improvement.

3. Evaluate and select an alternative

Skilled problem solvers use a series of considerations when selecting the best alternative. They consider the extent to which:

A particular alternative will solve the problem without causing other unanticipated problems.

All the individuals involved will accept the alternative. Implementation of the alternative is likely.

The alternative fits within the organizational constraints.

4. Implement and follow up on the solution

Leaders may be called upon to order the solution to be implemented by others, “sell” the solution to others or facilitate the implementation by involving the efforts of others. The most effective approach, by far, has been to involve others in the implementation as a way of minimizing resistance to subsequent changes.

Feedback channels must be built into the implementation of the solution, to produce continuous monitoring and testing of actual events against expectations. Problem solving, and the techniques used to derive elucidation, can only be effective in an organization if the solution remains in place and is updated to respond to future changes.

Excerpted from G. Dennis Beecroft, Grace L. Duffy, and John W. Moran, The Executive Guide to Improvement and Change, ASQ Quality Press, 2003, pages 17-19.

Basic Guidelines to Problem Solving and Decision Making

© Copyright Carter McNamara, MBA, PhD, Authenticity Consulting, LLC.Adapted from the Field Guide to Leadership and Supervision.

Much of what managers and supervisors do is solve problems and make decisions. New managers and supervisors, in particular, often make solve problems and decisions by reacting to them. They are "under the gun", stressed and very short for time. Consequently, when they encounter a new problem or decision they must make, they react with a decision that seemed to work before. It's easy with this approach to get stuck in a circle of solving the same problem over and over again. Therefore, as a new manager or supervisor, get used to an organized approach to problem solving and decision making. Not all problems can be solved and decisions made by the following, rather rational approach. However, the following basic guidelines will get you started. Don't be intimidated by the length of the list of guidelines. After you've practiced them a few times, they'll become second nature to you -- enough that you can deepen and enrich them to suit your own needs and nature.

(Note that it might be more your nature to view a "problem" as an "opportunity". Therefore, you might substitute "problem" for "opportunity" in the following guidelines.)

1. Define the problem

This is often where people struggle. They react to what they think the problem is. Instead, seek to understand more about why you think there's a problem.

Defining the problem: (with input from yourself and others)Ask yourself and others, the following questions: a. What can you see that causes you to think there's a problem? b. Where is it happening?c. How is it happening?d. When is it happening?e. With whom is it happening? (HINT: Don't jump to "Who is causing the problem?" When we're stressed, blaming is often one of our first reactions. To be an effective manager, you need to address issues more than people.)f. Why is it happening?g. Write down a five-sentence description of the problem in terms of "The following should be happening, but isn't ..." or "The following is happening and should be: ..." As much as possible, be specific in your description, including what is happening, where, how, with whom and why. (It may be helpful at this point to use a variety of research methods. Also see .

Defining complex problems:a. If the problem still seems overwhelming, break it down by repeating steps a-f until you have descriptions of several related problems.

http://www.authenticityconsulting.com/pubs/Mgmnt/MS_pubs.htm#_blank

http://www.authenticityconsulting.com/#_blank

Verifying your understanding of the problems:a. It helps a great deal to verify your problem analysis for conferring with a peer or someone else.

Prioritize the problems:a. If you discover that you are looking at several related problems, then prioritize which ones you should address first. b. Note the difference between "important" and "urgent" problems. Often, what we consider to be important problems to consider are really just urgent problems. Important problems deserve more attention. For example, if you're continually answering "urgent" phone calls, then you've probably got a more "important" problem and that's to design a system that screens and prioritizes your phone calls.

Understand your role in the problem:a. Your role in the problem can greatly influence how you perceive the role of others. For example, if you're very stressed out, it'll probably look like others are, too, or, you may resort too quickly to blaming and reprimanding others. Or, you are feel very guilty about your role in the problem, you may ignore the accountabilities of others.

2. Look at potential causes for the problem

a. It's amazing how much you don't know about what you don't know. Therefore, in this phase, it's critical to get input from other people who notice the problem and who are effected by it. b. It's often useful to collect input from other individuals one at a time (at least at first). Otherwise, people tend to be inhibited about offering their impressions of the real causes of problems.c. Write down what your opinions and what you've heard from others.d. Regarding what you think might be performance problems associated with an employee, it's often useful to seek advice from a peer or your supervisor in order to verify your impression of the problem.e.Write down a description of the cause of the problem and in terms of what is happening, where, when, how, with whom and why.

3. Identify alternatives for approaches to resolve the problem

a. At this point, it's useful to keep others involved (unless you're facing a personal and/or employee performance problem). Brainstorm for solutions to the problem. Very simply put, brainstorming is collecting as many ideas as possible, then screening them to find the best idea. It's critical when collecting the ideas to not pass any judgment on the ideas -- just write them down as you hear them. (A wonderful set of skills used to identify the underlying cause of issues is Systems Thinking.)

4. Select an approach to resolve the problem

http://www.managementhelp.org/systems/systems.htm

When selecting the best approach, consider:a. Which approach is the most likely to solve the problem for the long term?b. Which approach is the most realistic to accomplish for now? Do you have the resources? Are they affordable? Do you have enough time to implement the approach?c. What is the extent of risk associated with each alternative? (The nature of this step, in particular, in the problem solving process is why problem solving and decision making are highly integrated.)

5. Plan the implementation of the best alternative (this is your action plan)

a. Carefully consider "What will the situation look like when the problem is solved?"b. What steps should be taken to implement the best alternative to solving the problem? What systems or processes should be changed in your organization, for example, a new policy or procedure? Don't resort to solutions where someone is "just going to try harder". c. How will you know if the steps are being followed or not? (these are your indicators of the success of your plan)d. What resources will you need in terms of people, money and facilities? e. How much time will you need to implement the solution? Write a schedule that includes the start and stop times, and when you expect to see certain indicators of success. f. Who will primarily be responsible for ensuring implementation of the plan?g. Write down the answers to the above questions and consider this as your action plan.h. Communicate the plan to those who will involved in implementing it and, at least, to your immediate supervisor.(An important aspect of this step in the problem-solving process is continually observation and feedback.)

6. Monitor implementation of the plan

Monitor the indicators of success: a. Are you seeing what you would expect from the indicators?b. Will the plan be done according to schedule? c. If the plan is not being followed as expected, then consider: Was the plan realistic? Are there sufficient resources to accomplish the plan on schedule? Should more priority be placed on various aspects of the plan? Should the plan be changed?

7. Verify if the problem has been resolved or not

One of the best ways to verify if a problem has been solved or not is to resume normal operations in the organization. Still, you should consider:a. What changes should be made to avoid this type of problem in the future? Consider changes to policies and procedures, training, etc. b. Lastly, consider "What did you learn from this problem solving?" Consider new knowledge, understanding and/or skills.

http://www.managementhelp.org/prsn_prd/decision.htm

c. Consider writing a brief memo that highlights the success of the problem solving effort, and what you learned as a result. Share it with your supervisor, peers and subordinates.

Mathematical Problem SolvingThe T Puzzle

* Using the above pieces you must create a CAPITAL LETTER 'T' with no jagged bits,. no gaps, no overlaps, no bumpy bits.* the solution must be all one colour.* I use this on my first day with any class. It will show you those with gritty determination but most importantly point out the 'learned helpless' among the group.* get the kids to initial all their pieces so if you lose one... you know the drill.* have the students store their pieces in an envelope.* discuss and demonstrate the problem solving strategy of 'constant manipulation'* make some of the puzzles from wood to begin your problem solving collection that you will use for years to come.

Mathematical Problem Solving Knot Theory Problem

(a personal favourite)

Preparation:

- Over the course of the week before this activity have children bring in a tazo. (they know what they are).- cut out about 10 pieces of cardboard in the shape of a tazo as it is rather predictable that certain students will not bring in the required materials ;)- Make sure each child has a pair of scissors.- Have some wool on hand.

Method: - The Teacher Does These Steps

1. Punch two holes through a Tazo, a bayblade (or a small piece of cardboard) using a holepunch or an electric drill.

2. Loop c. 30cm of wool through the Tazo and tie it with a firm knot.

3. Tie the wool and tazo to the handle of a pair of safety scissors as shown here.

The Problem - Attempt to remove the string loop by: not bending the tazo, not cutting the string or undoing the knot and then put it back on. This is a logical thinking puzzle not a brute force one.

Just for Fun:

- Make it that the children are not allowed to use their scissors until they solve the puzzle :)

Remember:

- This is 'problem solving NOT problem showing'. Do not show the solution to your friends no matter how much they whine, you will take away the a-ha feeling from them.- when they do show their friends how to do it (and they will) discuss how it is funny that no-one got the solution and then all at once heaps of people did ;) - Discuss the taking away of the a-ha feeling as a whole group.

By placing a finger on the letter 'L' and tracing along the lines, how many differnet ways can you spell the word 'level'.

You can go forward and backwards. You can use letters twice but when you get to a letter you must use it, you can't skip over it.

I place these puzzles around the room and send them home with the kids so the whole family can have a go.

Number Theory

Start out simple...

1. List 10 possible combinations you could get with 4 darts. The numbers on the targets are 7-5-3-1. Try to establish some logical method of doing this.

2. Compare different pay scales. Decide if it is better to receive $300 a week or to be paid hourly at a rate of $7.50 per hour. What factors could affect your decision?

Now try to work this out...

3. Three watermelons and two cantaloupes weigh 32 pounds. Four watermelons and three cantaloupes weigh 44 pounds. All watermelons weigh the same and all cantaloupes weigh the same. What is the weight of two watermelons and one cantaloupe?

4. Sylvester measured his pulse and found that his heart beat at a rate of 80 beats a minute at rest. At this rate, how many days will it take his heart to beat 1,000,000 times? Show your work and be sure you explain each step.

This will really challenge you...

5. Conrad's Taxi Service charges $1.50 for the first mile and $.90 for each additional mile. How far could Mr. Kulp go for $20 if he gives the driver a $2 tip?

Measurement

Start out simple...

Rabbit's Run (taken from MATH FORUM)

6. Regina has received a pet rabbit from her neighbor Rodney who is about to move to an apartment that does not allow pets. Her father is going to help her build a run for the rabbit in their back yard, but he wants Regina to design it. Regina sits down to think about the possibilities. Her father says that the run must be rectangular with whole number dimensions. If they want to enclose 48 square feet, how many options do they have?


Area And Perimeter

7. If you fold a square paper vertically, the new rectangle has a perimeter of 39 inches. What is the area of the original square? What is the perimeter of the original square? What is the area of the resulting rectangle? Make a ratio of areas and perimeters. What do you notice?


Better Buy

8. Mr. and Mrs. Simpleton are shopping for carpet for their living room and dining room. Their living room is 21 feet by 15 feet and their dining room is 12 feet by 9 feet. They have looked at two different priced carpets. One for $14.95 a square yard installed and another for $19.99 a square yard installed. How much would they save by choosing the cheaper carpet? What are some other things besides money that they should consider before making their choice? After you find out how much carpet they need, figure out the savings in one calculation. Explain.

Geometry

Start out simple...

Where Am I?

9. Tom Terrific has a garden in the shape of a rectangle. He wanted to plant a tree in a specific spot. He wanted it to be in the exact center of the garden. What would be a way that he could find the center without using any measurement?


Reflections

10. Find 2 polygons other than a square and rectangle whose reflections are identical to the original. Draw them. Show how the reflection of a triangle can be the same as the original.


Symmetry

11. What is the largest three digit number that has both vertical and horizontal symmetry? Think of a three letter word with horizontal symmetry.

Patterns, Algebra, And Functions

Start out simple...

Going Camping

12. Groups of campers were going to an island. On the first day 10 went over and 2 came back. On the second day, 12 went over and 3 came back. If this pattern continues, how many would be on the island at the end of a week? How many would be left?


A Batty Diet

13. A bat ate 1050 dragon flies on four consecutive nights. Each night she ate 25 more than on the night before. How many did she eat each night? Solve this algebraically.


Windemere Castle (From The Problem Solver)

14. Evelyn is reading about Windemere Castle in Scotland. Many years ago, when prisoners were held in various cells in the dungeon area, they began to dig passages connecting each cell

to each of the other cells in the dungeon. If there were 20 cells in all, what is the fewest number of passages that had to be tunneled out over the years?

Data, Statistics, And Probability

Start out simple...

Bouncing Babies (Taken from MATH FORUM)

15. At a baby shower, we started discussing baby statistics. One of the women told us she had heard a report that for every 100 babies born, there were 6 more boys than girls. If we were to randomly pick a child from a representative group, what is the probability of picking a girl?


Trees

16. A team of scientists found that there were 4 oak trees for every 10 pine trees. How many oak trees were there if they counted 36 more pine than oak?


Cookies!

17. Four friends buy 36 cookies for $12. Each person contributes the following amount of money:

Tom--$2 Jake--$3 Ted--$4 Sam--$3

Each person gets the number of cookies proportional to the money paid. Draw a circle graph to represent the amount of cookies each got.Draw another circle graph to show how many each would have if Ted gives half of his cookies to Tom.

Grab Bag

Up You Go!

18. In the old days there were elevator operators to transport passengers. Don Downs always started his day in the basement. He went up 20 floors to take his boss some coffee. Then he went down 8 floors to take a Danish to his friend. He went up 7 floors to check things out. This was the halfway point in the building. How many floors are in this building? Draw a diagram to show how you would figure this out.

Remember: These are open-ended problems. There could be other solutions than the ones we are giving.

Number Theory

1. Look to see if there is some kind of logical order to the answer. For example, combinations starting with 7. There are many, many possible answers.

2. If there is no overtime and it is a 40 hour week, then there is no difference. If you can get overtime, then getting paid hourly is better. If you are working less than 40 hours, then the $300 is better. Check to see the reasoning.

3. Watermelons are 8 pounds. Cantaloupes are 4 pounds. 2 watermelons and 1 cantaloupe weigh 20 pounds.

4. During the 9th day. Check reasoning skills.

5. 19 miles

Measurement

6. 5 ways. (1,48), (2,24), (3,16), (4,12), (6,8)

7. Area of the original = 169 inchesPerimeter = 52 inchesNew Area = 84.5 inchesThe ratio of area to area is 1/2 and the ratio of perimeter is 3 to 4 new to original.

8. (Cheaper carpet is $702.65, other is $939.53.) The savings is $236.88. But, you need to consider how much dirt would show, footprints, warranty. Check for reasons.

Geometry

9. Use string to get the diagonals. Where they meet is the center.

10. Octagons and hexagons. Triangles should touch at one vertex.

11. 888; MOM


12. 77 on the island; 35 left

13. X + (X + 25) + ( X + 50) + (X + 75) =10504X + 150 = 10504X = 900X = 225

14. 190


15. 47 girls; 53 boys = .47 = 47%

16. 4/10 = 8/20 = 12/30 = 16/40 = 20/50 = 24/6060 - 24 = 36 so there are 24 oak trees

17. Check students' graphs

Grab Bag

18. 37 floors + basement

Number Theory

Start out simple...

What's the Deal?

1. Mortimer wants some doughnuts. He is very cheap and likes to save even the smallest amount of money. He found a coupon in the paper for Dunkin' Donuts.The coupon was for $1 off a dozen. This week they are on sale for $3.99 a dozen without the coupon and $.35 a piece if you use the coupon. What do you think Mortimer will do and why?

Spinning

2. Tara and Sara are going to play a spinner game. These are the rules:

When it is a player's turn, you spin both spinners. Add the 2 numbers that the spinner points to. If the sum is odd, Tara wins even if it is not her turn. If the sum is even, Sara wins even if it is not her turn.

Both girls think that they have a better chance of winning. Is either right? Justify your answer.


Going Shopping

3. Mabel and her mom are going shopping on Saturday. They bought at least one item from each of the 3 departments that they visited. Mabel gave the clerk $120 and she got back $11.76 change. What items did they buy? Think about how much they spent. NO TAX TODAY!

HOUSEWARESDishtowels: $11.38Curtain Rods: $12.98Bath Mats: $29.58

CLOTHINGShirt: $30.98Dress: $49.90Slacks: $39.90

TOOLSHammer: $17.90Saw: $23.90Drill: $25.78

What Time is It?

4. How many times in a 12 hour period does the sum of the digits on a digital clock equal 6? Try to think of a way to solve this without going through every single time. Describe what you did.


Boy Scout Hike

5. Points A and C on a map are 12 km apart if you follow a certain path. A troop of Boy Scouts leaves Point A at 11:00 a.m. They travel 3 km/hr because they have heavy packs until they reach Point B at 12:45. If they want to reach Point C by 2:00, how fast will they have to go?

Measurement

Start out simple...

Baseball Trivia

6. The greatest distance that a baseball has been thrown is 445 feet, 10 inches. Is this greater or less than the length of a football field from goal line to goal line? By how much? Tell how you made your decision.


Picture It

7. A picture that measures 12 cm by 18 cm is enlarged to 4 times its area. What are the new dimensions?


The Nutty Squirrel - From Math Forum

8. While going for my daily run, I passed a squirrel carrying a nut in her mouth. When she saw me, she ran towards a safe place and, coming to a stone wall, easily jumped up on it and disappeared. I began to wonder how high I could jump if I were a squirrel.If an average squirrel's back leg height from ground to hip is 3 1/2 inches, and that squirrel can jump a 2 foot high wall, what height wall could I jump if I were a giant squirrel? My leg height from ground to hip is 36 inches. Give your answer in feet and inches to the nearest inch.

Geometry

Start out simple...

9. Which of these choices contains the dimensions of a rectangle with the same perimeter as a rectangle whose dimensions are 5 m by 3 m.?

10 m by 8 m 7 m by 1 m 6 m by 4 m 8 m by 2 m

Show your work.


A Model House

10. Choose 3 geometric solids to build a model house. What solids did you choose? How many vertices are there? From an aerial view of your structure, how many vertices can you see? Draw a picture of the aerial view.


11. What do you need to know to solve this? You have kite ABCD. Angle B is at the top of the kite and measures 80 degrees. Angles A and C are on the sides and Angle D is at the bottom of the kite. What is the largest size that Angle A or C could be?


Start out simple...

Yum

12. Determine the number of pizza combinations you could get with 4 different toppings. Each pizza must have at least 2 toppings. Make a chart to display your results.


Ice Cream Cones

13. An ice cream stand has 9 different flavors. A group of children come to the stand and each buys a double scoop cone with 2 flavors. If none of the children chooses the same combination of flavors and every combination is chosen, how many children are there? Show how you got your answer.


Planet Krayon - (From The Problem Solver 6)

14. Zemo, Orb, Yuko, and Sam are friends who live on neighboring space stations of the planet Krayon. They commute to school every day by space shuttle. Orb's space station is one half as far from Krayon as Zemo's space station. Yuko travels as far as the total distance traveled by Zemo and Orb. Sam travels 3 times the distance that Zemo travels. How many space miles does each friend travel to school if the friends together travel 888 space miles?Write an algebraic equation.


Start out simple...

Coin Toss

15. List all the possible outcomes when four coins are tossed. Determine the theoretical probability of having exactly two heads and two tails.


Graph It!!!

16. Use the following information to make a line graph, bar graph, and circle graph. Describe how you did each one in paragraph form. Choose a title, labels, and scale.

Doritos - 20Popcorn - 15Candy - 12Chips - 10Pretzels - 3


How Old Am I?

17. The average age of a group of teachers and students is 20. The average age of the teachers is 35. The average age of the students is 15. What is the ratio of teachers to students? Express your answer as a fraction in simplest form.

Grab Bag

18. Dominic arrived at work and went behind the counter at the north end. As he faced out over the counter, north was to his right, south to his left. While he was standing at the north end, a customer ordered a sandwich. Dominic went through these steps:

3 feet to his left to pick up the bread

2 feet to his right to put bread on the plate 4 feet to his left to get mayo and pickles 2 feet to his left to pick up knife and spoon Returned to where the bread was on the plate Turned around and got the salami out of the refrigerator Put the sandwich together 3 feet to the left to serve the customer 4 feet to the left to get a soda Back to the customer to give him the soda To the south end of the counter to pick up the customer's money

Draw a diagram of the counter and its arrangement. Make suggestions for a more efficient arrangement.

October Solutions

Number Theory | Measurement | Geometry |Patterns, Algebra, and Functions | Data, Statistics, and Probability |

Grab Bag

Remember: These are open-ended problems. There could be other solutions than the ones we are giving.

Number Theory

1. He will use the coupon and pay $.35 each. That would be $4.20 - $1 = $3.20 instead of $3.99.

2. Sara is right because there are more possibilities of getting an even number answer.

3. They bought curtain rods, bath mat, slacks, and a drill.

4. 36 ways

5. Use the formula d = r t. The rate is 5.4 km/hr. Students should draw a diagram first to help.

Measurement

6. 145 ft, 10 in. Students need to remember to change feet to yards.

7. 24 cm by 36 cm. To get 4 times the area you need to double each dimension.

8. 20.57 feet or 20 ft 7 in.

Geometry

9. 7 m by 1 m

10. Answers will vary depending on which shapes they choose.

11. They need to remember that the sum of the measures of the angles in a triangle is 180 degrees and that the sum of the angles of a quadrilateral is 360 degrees. Angle A = 125 degrees.


12. Check chart for pattern. There are 11 possible pizzas.

13. 36 children. Use a tree diagram.

14. z + 1/2 z + ( z + 1/2 z) + 3 z = 8886 z = 888z = 148Zemo travels 148 space miles, Orb travels 72 space miles, Yuko travels 222, and Sam goes 444.


15. All heads; all tails; 1 head, 3 tails; 2 heads, 2 tails; 1 tail, 3 heads; the probability of 2 heads and 2 tails = 1/5 or 20%

16. There are 60 possibilities. Bar Graph-possible scale is to count by 2's or 5's. Same for Line Graph. Check for labels, title, and spacing. To make the Circle Graph, children need to make ratios.

Doritos = 20/60 = 1/3 Popcorn = 15/60 = 1/4 Candy = 12/60 = 1/5 Chips = 10/60 = 1/6

Pretzels = 3/60 = 1/20

When they make the circle graph, they can use central angles or they can estimate. That is up to the teacher.

17. 2

Grab Bag

18. Answers will vary. Check diagrams.

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