## Monday, July 26, 2010

### Last Post on Blogger

So, please, visit and enjoy. All the posts from this site have been moved to the new site, which is fully searchable.

http://PerezHortinelaFamily.us

Enjoy,

Pablo

## Tuesday, July 13, 2010

### Moving the Blog

## Monday, July 12, 2010

### Exercising the Young Mind

- the determinants of dimension
- how measurements change when enlarging along one, two, or three dimensions
- the relationship between volume and area

## Sunday, July 11, 2010

### Starting a Problem Set and Resource Repository

- Problems sets I write for my daughter
- Useful math and science websites. These are resources that offer courses, tools, and other useful stuff
- Other educational resources

## Tuesday, June 29, 2010

### Difficulty vs. Acceleration in Gifted Mathematics Education

My idea of proper mathematical education may be different from that of most other people, or I may be biased because of my experience getting a Ph.D. in mathematics. Whatever the issue may be, I have an uneasy feeling that most gifted math program focus too much on acceleration and too little on difficulty and exploration. I am not against acceleration. In fact, educating a mathematically talented daughter who hates repetition and "boring" stuff, I am acutely aware of the importance of cruising through the basics as fast as possible to get to the "fun" and "interesting." Unfortunately, I have yet to find an elementary school program that uses difficulty and exploration systematically. I find this disturbing because it may be impossible to get truly good at math without solving increasingly hard problems and learning to explore by asking what-if questions. The key question here is what parents can do to foster the problem solving talent of their children? I don't have a wholly satisfactory answer. All I can do is describe the problem and offer suggestions. I am not an education expert, but I am smart and educated enough to realize that conventional methods don't work. This blog post is as much about my observations and ideas as it is about asking for suggestions from those who have gone down this path before me. I am passionate about and always happy to discuss this subject. Hit me by email at Pablo_A_Perez_Fernandez@yahoo.com if you have an answer to my question.

What do I mean by difficulty vs. acceleration? The easiest way to describe it is by example:

**The Mirman School for the Gifted –**When first looking for the right school for our daughter, my family went through the application process at the Mirman School for the Gifted. It looked like the ideal place for Paulina until we got to the last step. While she was escorted to a placement test, I waited in the library along with a group of other eager and anxious parents. Mirman's head of school took the opportunity to spend an hour or so answering questions from the group. Fifteen minutes into the Q&A, I asked how Mirman taught math to kids. I was told that kids work at their own pace. If they are ahead of their peers, they could move to more advanced "rooms." I remarked that this was great and then asked what else was done besides letting kids progress faster than their peers. I got a blank stare. I asked if the school used custom curriculum or specialized books. I was told that they used standard mathematics books but allowed kids to work faster than normal and are encouraged to participate in competitions. I did not ask any more questions.**EPGY and CTY's Distance Education Courses –**Both programs use Stanford's adaptive software for K through pre-algebra. My daughter started in K and now is in the middle of 6^{th}grade. I find EPGY to be carefully thought out, rigorous, and complete relative to the California's DEO standards. EPGY does a good job teaching concepts like variables, equations, and the representation of English statements as mathematical equations. However, I would not characterize EPGY as challenging. Paulina so far has cruised through the program, and I know that she is working below her problem solving potential. I am disturbed by this.I investigated this issue a few months ago. I discovered that EPGY provides software for non-gifted school programs, and it is identical to EPGY's gifted track. The only difference between the gifted and non-gifted tracks is a teacher-controlled flag that triggers acceleration. There is no switch for difficulty or depth.

**My Yale University Experience –**Let's fast-forward to my freshman year at Yale University. There were three tracks for freshman physics and math. Each subject offered easy, traditional, and advanced paths. The easy classes were also known as physics and math for poets. They introduced the basic ideas without the "torture" of really difficult problems. The traditional tracks resembled a traditional university course. The advanced courses were much more difficult. They covered much more material and from a far more theoretical perspective. They also required killer problem sets that few students could finish on their own and on time. Bear in mind that the students taking these advanced classes were some of the best in the world. A number of my classmates participated and won honors at the International mathematical Olympiad and other top-level competitions. The biggest difference between the advanced courses and the others was the combination of acceleration and problem solving difficulty.

I hope it is clear by now what I mean by the difference between difficulty and acceleration. Gifted kids need both. It is not okay to facilitate one but not the other. So, what can we do to help? This question is difficult to answer. Here are some suggestions:

**Use Problems to Teach Material –**There is no sense in explaining things through endless lectures. This has the potential to bore the most enthusiastic students. One learns math best by doing math. Some of the best math classes I ever took asked me to discover math my solving problems.

**Challenge the Student –**Arithmetic drills are rarely challenging for gifted students. This suggests that increasingly difficult problems are necessary to push these kids toward their potential.

**Ask Open-Ended or Broad Questions –**This is the only way to truly challenge a smart kid. Give them an open problem. See how far their minds can go. See what questions they come up along the way.

I hope this post gives you a few ideas and helps you ask the right questions when thinking about your kid's education.

## Monday, June 7, 2010

### The Outcome of Radical Acceleration: The Stanley Kids from the 1980s

## Sunday, June 6, 2010

### We Saw the Light When Our Daughter Looked into a Black Hole!

**June 2009**

- Graduates from K at a regular school
- 37% done with 2rd grade English language arts
- 74% done with 3th grade math
- Lots of children's television and little reading
- Lots of "I am bored"
- Expresses lots of interest in black holes and other space phenomena
- Math is boring

**May 2010**

- Graduates from 2nd grade from a top 5% school in California (SAS, distinguished school, API scores above 900) after skipping first grade
- 65% done with 5th grade English language arts
- Starts 6th grade math now to follow with pre-algebra in September.
- Lots of reading and virtually no television other than Friday movies and science and history programs
- Lots of arts and science experiments
- Asks an astronomer how it is possible for black holes to eject matter through its poles when nothing can escape the event horizon
- Lots of "I like interesting math and competitions like the Math Kangaroo"

## Friday, May 21, 2010

### Teaching Social Ethics

Pablo

Paulina

I don’t know. Why do you ask me so many questions?

Pablo

Well, I really care about what you think. I like talking to you, and you were glued to the computer watching the movie. Seems to me that you liked it.

Paulina

It made me mad. I don’t like the way people treated her. It wasn't fair.

Pablo

What wasn’t fair? Why wasn’t it fair? What made you mad?

Paulina

It is unfair that some people could not get a good education because it is very hard to get ahead without a good education.

Paulina left the room and came back ten minutes later. Our ensuing exchange is below. Bear in mind while reading that Paulina is an interracial child. She is 50% Spanish and 50% Filipino, with 50% of her Spanish background from Puertorican descendants of Spanish settlers. So, she is fairly light skinned but darker than Anglo-Saxon kids.

Paulina

Papa, is my skin dark or light?

Pablo

Your skin is perfect. You look like papa and mama. Your skin is beautiful. What are you worried about?

Paulina

Well, would I have had problems in Ruby’s time?

Pablo

I don’t think you would have had problems because of the color of your skin. My dad looked a lot like I do, and he went to college in Ole Miss in the 1950s. He was never asked to ride in the colored section. However, the fact that you mom and I come from different racial groups may have cause some problems. Back then, it was not common to see mixed couples, but we don’t have to worry about it because things are different today. Society still discriminates, but we are all a lot more tolerant today than we were 50 years ago.

This series of conversations took me by surprise. Race has never been an issue in my household. Traci and I are a mixed couple, and we have lived since we met in big, multi-cultural cities like Los Angeles, New York, and London. We have many friends from all over the world with whom we interact on a regular basis. Paulina knows other interracial kids. So, I was a bit surprised by her preoccupation.

I think I handled the situation well. First, I did not appear flustered or worried. Second, I brought up something people could have used to discriminate against her. Third, we talked about how discrimination is used as a tool to gain power. I explained that some people will use anything you can imagine to single out a group of people and be unfair to them. Finally, I explained to her why it is important to speak out against discrimination.

I am certain that I will have more conversations like this one soon. I am glad this one went well, and I hope I handle the next one similarly.

## Sunday, May 16, 2010

### Emotions and Teaching: Mastery Goals vs. Performance Goals-Based Teaching

What is the best way to teach highly gifted children? Some people advocate home schooling. Others posit that regular schools combined with grade and/or subject-specific acceleration is sufficient. However, I firmly believe that arguing for one approach over any other clearly misses what is almost certainly the single biggest determinant of long-term success: emotions. It is wrong in my opinion to argue that a one-size-fits-all approach is the best for gifted children. Yes, I believe that my daughter will flourish at home. However, what works for my child may not for others regardless of intellectual capacity. I just finished reading Science Education for Gifted Learners. The chapter titled The Emotional Lives of Fledgling Geniuses tackles the issue of matching educational approach and emotional personality. The key thesis is that the choice of educational approach should be dictated largely by the emotional characteristics of the student. My wife and I have chosen to home school Pauline next year. After objectively reading The Emotional Lives of Fledgling Geniuses, I feel comfortable with our decision to home school because it best matches our daughter’s personality, emotions, and approach to learning.

There are many ways to categorize teaching styles. The Emotional Lives of Fledgling Geniuses argues that one may view teaching as split into two camps:

- performance goals-based
- mastery goals-based

Let’s use my daughter as an example. Paulina does well in exams. I did too when I was a kid, but she is one of those people who seems to do well in tests without even trying. She is extremely competitive, and she has started attending contests. For instance, she participated in the Math Kangaroo this year and came out very excited, asking to do it again next year. She always wants to get the highest score in every test she takes and practices incessantly whenever she has a performance. On the other hand, she already has long-term goals. A good example is her passion for black holes. I don’t remember how this started, but she became fascinated with black holes when she was five. She would ask me to read her everything we could find on black holes. She now reads by herself everything she can find on the subject, and she has been speeding through the math curriculum as fast as possible since I explained that it is a key to understanding black holes. In the process, she has discovered probability, graph theory, and other subjects that interest her, but her goal remains to learn math because it will allow her to understand black holes and other astronomical phenomena.

## Saturday, May 1, 2010

### On the Importance of Making Math Fun

- Graphs (not to be confused with the X-and-Y variety)
- Trees
- Degree of a vertex
- Isomorphic graphs
- Circuits in graphs
- Euler circuits
- Planar graphs
- Restating problems involving maps using graphs
- The Four Color Problem

## Saturday, April 10, 2010

### A 21 Setup to Teach Probabilistic Decision Making

**Rationale for My Version of the Game of 21**

- Computing probabilities on a discrete space.
- Computing conditional probabilities when there the probability space changes.
- Determining if a bet is unfavorable, roughly fair, or favorable.
- Making decisions based on probabilities.

**Step 1**

Get a big piece of construction paper -- the kind used for school projects and presentations. Arrange a deck of cards along columns on the construction. The leftmost column should have the twos. The next column over should hold the threes and so on. Instead of using a deck of cards, you could simply draw the cards on the construction paper. You should have 13 columns total. I will call this the "board" throughout the rest of this article.

**Step 2**

- At the beginning of a hand, each player is given two cards.
- Each player gets only one card at a time after the initial two. Additional cards are only handed out when it is the player's turn and the player asks to be hit.
- The game continues until there are no cards left in the deck or there are not enough cards to start a new hand.
- There is a common pile of chocolate chips for betting. All players draw chips from the same pile.
- The winner of the game is the one who ends with the biggest pile of chocolate chips.
- A player should be dealt as many cards as desired until either he or she goes bust or decides to "stay."

**Step 3**

- Start by getting two cards.
- Continue saying "hit me" (i.e. asking for another card) until you think you will go bust.
- Say "stay" when you are done with your hand.
- Choose to hit or stay based on the probability of going bust by picking one more card. This step is crucial. One of the key reasons for the this game is to teach how to compute conditional probabilities (i.e. probability under variable change).
- If your total exceeds 21, you are out of this hand.

**Step 4**

- Points are earned or lost in each hand.
- A bet is from 0 to 4 chips.
- The person who comes closest but not over 21 wins the hand and takes all the chips bet in the round.
- If two or more players tie, they split the pile. If the pile does not split evenly, remove the smallest number of chips so it splits evenly. Add the removed chips to the next hand's pile.

**Step 5**

- Bet 0 chips if there is a high probability of losing the hand.
- Bet 1 or 2 chips if there is low probability of winning the hand.
- Bet 2 chips if the probability of winning is high.
- Bet 4 chips if you are very sure to win.

**A Few Additional Details**

The game could be played with different degrees of sophistication. For instance, students could be asked to base the size of their bets on the probability of winning the hand. However, this requires computational skills beyond the skills of most seven year olds. A better way to place bets is based on intuition.

Pablo

## Wednesday, April 7, 2010

### Vegas at Home

- Before dealing any cards, what is the chance of dealing a heart?
- Assuming that a heart is the only card that has been dealt, what is the probability of dealing another heart?
- Assuming that a heart is the only card that has been dealt, what is the probability of dealing a spade?
- Assuming that a heart has been dealt, what is a better bet for the next card? A heart or a diamond?

- a 10 of hearts and a 5 of clubs for your child
- a 2 of diamonds and 5 of diamonds for you
- a 10 of hearts and a card facing down

- To bust, you must deal a card with a value of 7 or higher.
- Given what has already being dealt, the following cards will get you busted: 4 sevens, 4 eights, 4 nines, 4 tens, 4 jacks, 4 queens, and 4 kings

## Thursday, March 25, 2010

### Teaching the Concept of Conditional Probability

Prerequisites:

- understand fractions
- understand the concept of outcomes - The possibilities for the given problem. An outcome is something that may happen. It does not necessarily mean that it has happened.
- understand the concept of sets and events - Once needs to understand what sets are to tackle events. An event is just a set of outcomes. An event is used to narrow down the set of possible outcomes in a given probabilistic universe.
- understand how to compute probabilities in finite spaces - Know how to compute probabilities by counting. We only need the very simplest computational skills at this point.

Teach your child how to draw spaces of probability outcomes using trees. I have found that they are simpler for young children to understand than simply listing out all possible outcomes.

Example 1: Assume that a family has two children. What are all the possible outcomes of pairs of children?

Moving from the top to the bottom, we can now read out the four outcomes: {BB, BG, GB, GG}.

Example 2: Assume that a family has three children. What are all the possible sets of children?

Once again, by reading from the top to the bottom of the tree, we can list all the outcomes {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}.

It is very important at this point to make sure that your child understands how to compute the number of outcomes without actually listing them out.

Example 3: Introduce an example of choosing randomly without replacement. This means that once an object is picked, it cannot be picked again. I would suggest using colored M&Ms here. Use two green and two red M&Ms. Explain that you want to build an outcome tree as follows. Pick one M&M at random. Leave it out of the box. Pick another M&M at random. Now, draw the tree of outcomes.

The key point when drawing this tree is to realize that once an M&M has been picked, it remains out of the set. Hence, there are only three choices when picking the second M&M. This means that the full set of outcomes is {GG, GR, GR, GG, GR, GR, RG, RG, RR, RG, RG, RR}.

Step 2 - Teaching the Difference Between Normal and Conditional Probability

Most people never really learn the concept of conditional probability because their teaches did not understand it either. Students are usually taught the formula P(A|B) = P(A and B) / P(B). The issue is that this makes little sense to most people. This is sad because math is about concepts, not symbolic manipulation. The only difference between regular probability and conditional probability is the set over which one computes.

Compute the probability of getting a green followed by a red in the last example. Looking at the tree corresponding to the last example, this event is equal to {RGG, RGG, RGG, RGG}. This is the case because there are four paths in the tree with RGG outcomes. Hence, the probability of one green followed by a red is 4/12 or 1/3.

Compute the probability of getting a green on the second pick given that the first pick was a red. This conditional probability is computed by counting outcomes in the sub-tree with Rs in the first pick. This is because we are told to assume that the first pick was an R.

This can be computed by looking at the correct portion of the tree, which is circled in red below.

The possible second picks when the first one is an R are {G, G, R, G, G, R}. Hence, the conditional probability is 4/6 or 2/3.

Step 3 - Repeat Many Times

The trick to teach conditional probability to a kid is to give lots of concrete examples.

I hope this post helps you in some way.

## Friday, March 12, 2010

### When Homeschooling is the ONLY Option

## Thursday, March 11, 2010

### Teaching Multiplication of Fractions Visually

While fractions are easy for most kids, multiplication of fractions can sometimes be a little tricky. What I mean is that kids learn how to multiply fractions easily without really learning why it works. I wanted to make sure my daughter understood the concept, so I resorted to a geometrical interpretation.

- Understand what a fraction is
- Understand how to compute areas. Basically, that A = L x W

Here is another example. Multiply 2/3 and 3/5.

The length is 2/3. The width is 3/5. The product is 6/15. You kid should figure this out by counting.

Finally, let me show how to illustrate multiplication of fractions involving improper fractions.

Hopefully, it is clear by now that this geometrical interpretation works equally well for heterogeneous fractions.

The point of the geometrical interpretation is that it can be used to teach multiplication right after understanding the concept of a fraction. As always, the key to learning things well is to use lots of examples and to spend enough time thinking about the concepts.

## Saturday, February 27, 2010

### Teaching Fractional Arithmetic Visually

Step 1:

Explain that fractions are the part of a whole that you are talking about. Give a ton of examples. This is a critical step before moving on. Kids must understand what a fraction is and how to read and write them using the standard notation.

Step 2: Review how to represent fractions as pictures and vice versa. For example,

Use a few more examples. Teach the kid how to draw squares divided into thirds, fifths, tenths, and a few other common denominators.

Step 3: Explain what it means to add fractions. Say that it means adding parts of a whole. Explain that if we add parts of a whole, it is easier to add pieces of the same size. Give the example of adding 1/2 and 1/4. 1/2 and 1/4 have different sizes. So, how do you tell how much of the whole you have when you put together 1/2 and 1/4? The best thing to do here is to use four blocks of the same size. Manipulate two of the four blocks as 1/2 of a whole. Give the kid enough time to realize that 1/4 is 1/2 of 1/2.

Step 4: Once the child understands that cutting a square into pieces of the same size is the key to adding fractions (i.e. making the fractions homogeneous), proceed with a few exercises such as the following. Draw the two fractions we used before.

Ask your child to figure out how to divide both fractions into pieces of the same size. Tell him that he is not allowed to erase lines already drawn. Tell him that he is allowed to draw new lines, but that the goal is both fractions to divided into pieces of the same size. Clearly, the answer here is to subdivide the 1/2 horizontally as follows:

Now (this is critical), make sure the child understands that how much is shaded in the square on the left does not change simply because we drew another line. It should be clear by now that 1/2 and 2/4 are the same fraction. Make sure this is clearly understood before proceeding.

Step 5: It is now time to learn how to convert heterogeneous to homogeneous fractions. I would suggest easy cases first, followed by slightly more complicated ones. Let's start with the following two fractions. Always draw one fraction using vertical lines and the other using horizontal lines.

The answer should be

By drawing vertical lines in one fraction and horizontal in the other, it becomes clear how to draw new lines to divide both pictures into pieces of the same size.

Let's try one more example.

The answer now is

Make sure your child understands that 3/5 is equal to 6/10. Likewise, make sure it is understood that 1/2 = 5/10.

Step 6: It is now time to introduce the pictorial representation of improper fractions. Ask your kid to draw the following fractions: 3/2, 5/4, 4/2, 5/2, etc. The answers follow:

Do as many examples as necessary until the child is proficient at drawing improper fractions.

Step 7: It is now time to bring it all together to add and subtract fractions visually. Tell the child to draw each fraction in a problem. Ask the child to complete the problem visually. Finally, ask the child to convert the drawing representing the answer to a written fraction.

As you probably realize, you can teach reduction to lowest terms visually as well.

I hope this blog entry helps you introduce fractions faster and earlier than is typically done in schools. It took only 40 minutes to teach a group of second graders in my daughter's school how to add and subtract homogeneous and heterogeneous fractions. All they knew before I taught them was what a fraction is and how to write them down in standard notation.

## Monday, February 1, 2010

### A Sample Problem Set for Second Grade

I wrote the problem set attached to the bottom of this post for my daughter's second grade class. I help the teacher once a week by splitting the classroom into three groups. I write problems sets to challenge the "gifted" group. I work with the "average" kids to make sure they are proficient on the topics mandated by the State of California -- essentially preparing them for the CST exams by reinforcing what is taught in class. Finally, I tutor the bottom third of the class to help it understand the basic concepts of arithmetic. There are clear cognitive differences between the three groups, and this makes my job quite difficult -- what works for one third of the class does not work for the other two.

The point of the including the problem set below is to give an example of how to teach the basic ideas of proof construction to kids in early elementary school. Notice the structure of the problem set. I try to emulate the way college math books are structured:

- Definitions
- Example and computational exercises
- Proof construction as a vehicle to learning math and deepen understanding

Sample Problem Set for Second Grade Statistics and Set Theory

Prerequisites: Addition, subtraction, an informal understanding of the concept of a set, and Venn diagrams. We do not assume any knowledge of multiplication, division, or fractions. We assume students do not know anything about negative numbers.

**Definition: **The **range** of a set of numbers is the distance between the biggest and the smallest.

**Exercise 1: **Find the range of the set of even numbers between 0 and 100. Assume that 0 and 100 are in the set.

**Exercise 2: **Define a set of numbers using the following four properties:

- Every number in the set is bigger than or equal to 30 and smaller than or equal to 60.
- You can get every number in this set by counting by tens.
- You can get every number in this set by counting by fives.
- You can get every number in this set by counting by twenties.

Find the set of numbers defined by the above properties. Compute the range of the set.

**Exercise 3: **Which one of the following two sets has the bigger range?

**Set 1:**The set of even numbers between 10 and 20, including 10 and 20.**Set 2:**The set of odd numbers between 10 and 22.

**Exercise 4:** What is the range of the set of numbers equal to their doubles?

**Exercise 5: **What is the set of numbers not equal to themselves:? What is the range of this set?

**Definition: **The **mode** of a set is defined as the element that appears most often. A set may have no mode, one mode, or more than one mode. We say that a set is **bi-modal** if it has exactly two modes.

**Exercise 6:** Find the mode of {100, 99, 50, 3, 2, 1, 60 ,1 ,85}

**Exercise 7: **Find the mode of {Pablo, Paulina, Alex, Kolane, Pablo, Paulina, 2, 3, 10, 1}

**Exercise 8: **Find the mode of {2, 3, 4, 5, 6, 7, 8}

**Definition: **The **median** of a set is the number for which half the elements in the set are smaller and half bigger than the mode. Sometimes, the mode is part of the set. Sometimes, the mode is not.

**Exercise 9: **Find the median of {1,2,3}.

**Example 1:** The median of the set in the exercise 9 was a member of the set. However, the median is often not a member of the set. For example, 3 is median of the {2,4}. There are only two numbers in the set {2,4}. We pick the number right in between these two. That number is 3. Half the elements in {2,4} are smaller than 3 and half bigger than 3.

**Exercise 10:** Find the median of {100, 1, 80, 52, 48, 10, 12, 15}

**Exercise 11: **If you throw away the smallest and the largest numbers in a set, does the median change?

**Exercise 12: **Assume the following things:

- There are twenty students in the classroom.
- Every student likes either apples, pears, or both.
- 15 students like apples.
- 10 students like pears.

How many students like both apples and pears? Prove your answer using a Venn diagram.

**Exercise 13: **Assume the following three things:

- There are twenty students in the classroom.
- A student likes only one type of fruit.
- 15 students like apples.
- 10 students like pears.

How many students like neither apples nor pears? Prove your answer using a Venn diagram.

**Definition: **The **mean **of a set of numbers is one that repeated as many times as there are elements in the set gives you the same number as adding all the numbers in the set.

** **

**Example 2: **Given the set {2,3,4}, computer the mean.

** **

**Solutions: **First, we add all the numbers in the set. 2+3+4 = 9. We have to find a number that added to itself three times equals 9. That number is 3, because 3+3+3 = 9. Hence, 3 is the mean of {2,3,4}.

** **

**Example 3: **Compute the mean of {1,2,3,4,5,6,7}.

** **

**Solution: **1+2+3+4+5+6+7=28. What number can I add to itself 7 times to get 28? You can find out by trial and error, but the answer is 4 because 4+4+4+4+4+4+4=28. Hence, the mean of the set is 4.

** **

**Example 4:** Show that the mean of {2,3,4,5,6,7,8}=35 is bigger than or equal to 2 without computing the sum of the numbers?

** **

**Solutions: **If the mean were smaller than 2, the first choice would be 1. This would mean that adding 1 to itself seven times would equal the sum of the elements in the set {2,3,4,5,6,7,8}. However, 1 is smaller than every element in the set. Hence, adding seven ones cannot equal the sum of the elements in the set. This implies that the mean must be larger than or equal to 2, the smallest element in the set.

** **

**Example 5:** Show that the mean of {2,3,4,5,6,7,8}=35 is smaller than or equal to 8 without computing the sum of the numbers?

** **

**Solutions: **If the mean were bigger than 8, the first choice would be 9. This would mean that adding 9 to itself seven times would equal the sum of the elements in the set {2,3,4,5,6,7,8}. However, 9 is larger than every element in the set. Hence, adding seven 9s must be larger than then sum of the elements in the set. This means that 9 must be smaller than or equal to 8, the largest element in the set.

** **

**Exercise 14:** Can the mean of a set be smaller than the smallest number in the set?

** **

**Hint: **Look at the prior two examples. This problem is solved the same way. Assume that the mean is smaller than the smallest number in the set. Compare it to every number in the set. What do you see?

**Exercise 15: **Can the mean of a set be bigger than the largest number in the set?

** **

**Hint: **Assume that the mean is bigger than the biggest number in the set. Compare it to every number in the set. What do you see?

**Exercise 16: **Show that the mean of any three numbers is both

- smaller than or equal to the largest element of the set

**Hint: **Use what you did in exercises 14 and 15.

** **

**Exercise 17:** Give an example of a set for which the mean and the median are NOT equal.