Combating Poor Math Scores

Combating Poor Math Scores
Studies have found that students with involved parents, no matter what their income or background, are more likely to—
· Earn high grades and test scores, and enroll in higher-level programs;
· Pass their classes, earn credits, and be promoted;
· Attend school regularly; and
· Graduate and go on to postsecondary education.
· Out perform other students
· Have fewer behavioral problems.
· Pass their classes, be promoted, and attend school regularly (Ho Sui-Chu & Williams, 1996).

While working with 6th and 7th graders at Boston Public, I realized that some of the students were lacking a lot in math. I quickly realized that they had a negative attitude toward mathematics. I quickly started trying new strategies and activities. I was blown away by how interested they were when we did fun math activities. I also learned through them that their parents don’t help them with their homework. I decided to create a website with fun math activities that both parents and kids will enjoy.

The followings are some fun math websites

www.coolmath4kids.com/
www.funbrain.com/math/
www.mathsisfun.com/
www.math.hmc.edu/funfacts/
www.coolmath.com/
www.resources.kaboose.com/games/
www.primarygames.com/math.htm
www.multiplication.com/interactive_games.htm
www.gamequarium.com/math.htm
www.math.com/parents/articles/funmath.html
www.syvum.com/math/arithmetic/level2.html
www.mathfun.com/

Armstrong Numbers
An Armstrong number is an n-digit number that is equal to the sum of the nth powers of its digits. In this lesson, students will explore Armstrong numbers, identify all Armstrong numbers less than 1000, and investigate a recursive sequence that uses a similar process. Throughout the lesson, students will use spreadsheets or other technology.

Learning Objectives



Students will:
Formulate a definition of Armstrong numbers based on several examples.
Identify all Armstrong numbers less than 1000 using appropriate technology.
Reasonably explain why the sequence of Armstrong numbers is finite.

Materials


Computer and Internet connectionArmstrong Numbers Spreadsheet (for students), and Teacher VersionArmstrong Iteration SpreadsheetStrong Arm Iteration Activity Sheet

Instructional Plan
Explain to students that they will be implementing the following process for several numbers:
Raise each digit to a power equal to the number of digits in the number. For instance, each digit of a four‑digit number would be raised to the fourth power; each digit of a five‑digit number would be raised to the fifth power; and so on.
Add the results.
Demonstrate this process for 123—because 123 is a three-digit number, raise each digit to the third power, and add: 13 + 23 + 33 = 36. If necessary, show other examples. Then, have students implement the process for the following numbers:
6
23
99
231
1634
11,111
Students are likely to notice that two numbers from this list—6 and 1634—return their original value when this process is applied; that is, 61 = 6, and 14 + 64 + 34 + 44 = 1634.
Inform students that numbers which return their original value are known as Armstrong numbers. Once you are certain that all students understand this idea, allow them to work in pairs to create a succinct definition of Armstrong numbers. After a few minutes, have each group share their definition with another pair, and then allow the class to vote on which definition is most lucid and precise. For comparison, share the following formal definition:
Armstrong number: An n-digit number equal to the sum of the nth powers of its digits.

Finding Armstrong Numbers
In examining the numbers above, students found that 6 and 1634 were Armstrong numbers. Now ask them to determine all Armstrong numbers less than 1000.
There are several strategies that students could employ to solve this problem. Allow students some time to come up with a strategy on their own. If students are stuck, suggest one of the following:
Using a list of numbers 1‑999, students can eliminate any non‑Armstrong numbers. (Many numbers can be eliminated from this list by inspection. For instance, any number ending in two 0’s, such as 100, 200, 300, …, can be eliminated, because the cube of the hundreds digit will not end in two 0’s.) While this method can produce valid results, it is susceptible to human error, and it may take a long time.
Using a programmable calculator or a computer, write a program to generate and test all numbers 1‑999. (Such a program could generate the numbers by repeating a loop of the form "For n = 0 to 9" three times.)
Using a spreadsheet, parse each number 1‑999 into its three digits, and then calculate the sum of the powers. The spreadsheet Armstrong-Students.xls can be given to kids as a starting point for their investigation; students will need to enter formulas to find the units digit in column D and the sum of the cubes in column E. Alternatively, you may choose to have students begin with a blank spreadsheet and discern a formula for parsing the hundreds and tens digits, too. The file Armstrong-Teacher.xls shows one possibility for creating formulas and testing numbers.
As it turns out, there are nine single-digit Armstrong numbers, namely 1‑9; there are no two-digit Armstrong numbers; and there are four three-digit Armstrong numbers: 153, 370, 371, and 407.

Strong Arm Iteration
The investigation of three-digit Armstrong numbers is conducted by adding the cubes of the digits. This process of cubing the digits can be continually repeated on the result to reveal some interesting patterns.
The process of Strong Arm Iteration works as follows:
Begin with an n-digit number.
Raise each digit to the nth power, and compute the sum.
Raise each digit of the result to the nth power, and compute the sum.
Repeat Step 3 until a pattern emerges.
For instance, take the number 123. Cube its digits and add them: 13 + 23 + 33 = 36. Now, take the digits of the result, cube them, and add: 33 + 63 = 243. Continue by cubing the digits and adding for each result:
243: 23 + 43 + 33 = 99 99: 93 + 93 = 1458 1458: 13 + 43 + 53 + 83 = 702 702: 73 + 03 + 23 = 351 351: 33 + 53 + 13 = 153 153: 13 + 53 + 33 = 153
Notice that the process eventually leads to 153, which gives itself when the process is continued.
Allow students to explore other three-digit numbers. As before, you may ask them to construct a computer program or spreadsheet on their own to investigate this problem; or, you might supply them with the file Armstrong-Iteration.xls to use for investigation.
To guide this investigation, you may wish to distribute the Strong Arm activity sheet to your students.

Strong Arm Activity Sheet
In particular, ask them to consider the following questions:
If you begin with 123, the sequence reaches 153, and then it begins to repeat. That is, 13 + 53 + 33 = 153. What other three‑digit numbers will eventually reach 153 and begin to repeat? Is there a pattern to the numbers that reach 153?
[There are many numbers that reach 153 and then repeat; some of the numbers are 135, 213, 369, 423, 546, 678, 775, 819, and 972. In fact, all multiples of 3 eventually reach 153.]
Other than 153, what other numbers are reached when this process is applied?
[The other three‑digit Armstong numbers are occasionally reached. For instance, 124 eventually leads to 370; 551 leads to 371; and 740 leads to 407.]
What other interesting things did you notice during this investigation? If possible, explain why these interesting things happened.
[Several numbers lead to a cycle of repetition rather than to an Armstrong number. For instance, 136 leads to 244 which leads back to 136, and this two‑number cycle repeats. Cycles of three numbers also occur.]


A Penny Saved is a Penny at 4.7% Earned
Key Economic Concepts:
Banking
Budget
Choice
Decision making
Income
Interest rate
Opportunity cost
Savings
Trade-offs
Description:
There are lots of ways to receive income, and lots of ways to spend it. In this EconomicsMinute you will develop two budgets, or plans, to help you decide how to allocate your income. Assuming you do not love making dollar bill rings.
Key Economic Concepts
Interest
Opportunity cost
Budget planning
Introduction:
Income is what gets you stuff, but how do you get income? Do you earn an allowance? Do you baby-sit? Do you mow lawns? Do you receive income as a gift? And once you have the income, what do you do with it? Do you run out and get rid of it all immediately? Do you spend some now and some later? Or do you save it all?
There are lots of ways to receive income, and lots of ways to spend it. In this EconomicsMinute you will develop two budgets, or plans, to help you decide how to allocate your income. Assuming you do not love making dollar bill rings, there are really only two things that someone like you can do with your income: spend it or save it. Spending occurs when you use your income to purchase good and services.
When you do not spend your income, you save it. You have several options when saving income. One option is to simply put it aside at home and not spend it. Another option is to deposit the money in a savings account at the bank. Why would you want to put your money in the bank? What does the bank have that your piggy bank does not? When you place money in the bank you earn interest on it. Interest is what you receive for allowing the bank to use your money. The bank pays interest because it wants to encourage people to put money in its accounts. If the bank did not pay interest to depositors, people would just keep their money in their piggy banks. Because interest is expressed as a rate, how much interest you get depends on how much money you put in the bank. The more income you save, the more interest you can expect to receive.
Below, you will find two simple budgets. The first budget assumes that you do not have a savings account. By contrast, the second budget presumes that you have opened a savings account. A budget is a plan to help you figure out what you should do with your income. What part of your income should you spend and what part should you save? What should you spend your income on?
In economics the cost of something is the value the next-best forgone alternative. This cost is referred to as opportunity cost. The highest valued alternative as a result of making a choice is its opportunity cost.
Budget #1 (no savings account)
Let's say you have five dollars. What would you like to spend it on? There are a million things you would love to use your five bucks for, but let's say there are only four things out there that you really want to buy: arcade games, gum, soda, and movie tickets. Look at the price chart in the worksheet and answer the questions that follow.
1. In terms of gum, what is the opportunity cost of two sodas, assuming that the gum is your next-best alternative? [In order to get two sodas, you give up four pieces of gum.]
2. In terms of soda, what is the opportunity cost of a movie ticket, assuming that purchasing soda is your next-best alternative? [In order to get one movie ticket, you give up five sodas.]
Budget #2 (with a savings account)
Of course you do not have to spend all of your money. You can save some of your money in your bank account. Let's say you put half of your money, $2.50, into the bank. However, you still would like to buy the same four goods at the same four prices. Fill out the following worksheet
1. Assuming soda and gum are your next best alternatives, what is the opportunity cost of putting $2.50 in the bank? [Answers will vary. Any combination of gum and soda that costs $2.50 works.]
2. If the bank pays 10 percent interest, how much interest will you earn on your $2.50 deposit? [$0.25= 2.50 x 0.10interest earned = balance in account x interest rate.]
3. With interest your $2.50 is now worth $2.75. In terms of soda, gum, and arcade games what can you buy for $2.75? [Answers will vary. The answers will be similar to those given for Question 1, except now the students will be able to purchase one additional arcade game.]
4. So, if you spend the $2.50 instead of putting it into the interest-bearing account what do you have to give up? What's the new opportunity cost of the $2.50? [By spending the money instead of putting it in the bank, you give up not only the $2.50 but the $2.50 plus any interest you will earn on that money. In terms of the example, the new opportunity cost of the $2.50 includes an additional arcade game.]
6. Compare Budget #1 and Budget #2. How have the budgets changed? Why did you change your mind? [Answers will vary. Because in Budget #2 students earn interest on their money, they probably put more in savings.]


Airport Numbers
In the following lesson, students participate in activities in which they focus on the role of numbers and language in real-world situations. Students are asked to discuss, describe, read, and write about numbers they find in familiar real-world situations. The emphasis on using components of language helps students build a broader vocabulary of numbers than the traditional symbolic representation of numbers. The activities also help develop good number sense.

Learning Objectives

Students will be able to:
§ discuss, describe, read, and write about whole numbers or decimal and common fractions

Materials

§ One reproducible "Airport Numbers" activity sheet for each student.

Instructional Plan
Background Information
The activities at this level use an airport theme to investigate numbers. Students are encouraged to relate the numbers to familiar situations, for example, to use the dimensions of the classroom to describe an airplane.
Preparing the Investigation
Reproduce a copy of the activity sheet, "Airport Numbers," for each student.
Structuring the Investigation
1. Generally discuss the airport activity sheet with the students.
Ask them to tell what they see in each of the pictures and what they think each picture is about.
Tell the students to look at the pictures and find examples of the use of numbers.
Spend enough time talking about the pictures and students' own knowledge of airports so that students describe as many different ways as possible of using numbers.
For each suggestion ask the students to describe how the number is being used.
The following observations should be made for each picture:
At the check-in counter
§ number of bags
§ weight of each bag
§ dimensions of a bag
§ cost of tickets
At the departure gate
§ time (discuss the 24-hour clock used internationally)
§ departure times of other flights
§ dates (discuss the different conventions--month/day/year and day/month/year)
§ flight numbers
§ seat-assignment numbers
Runway
§ identification of the airplane and runway
Airplane statistics
§ measurement of various attributes (length, width, height, capacity)
2. Ask the students to work in groups to write descriptions of as wide a variety of uses of numbers as possible in other areas of the airport.
They should include numbers in the following forms:
§ Whole numbers
§ Decimal fractions
§ Common fractions
§ Percents
3. The students can work independently or in small groups to complete the activities given in the directions at the bottom of the Activity Sheet.

Balancing Shapes
Students will balance shapes on the pan balance applet to study equality, essential to understanding algebra. Equivalent relationships will be recognized when the pans balance, demonstrating the properties of equality.

Learning Objectives

Students will:
§ develop an understanding of equality using shapes and a pan balance;
§ apply the properties of equality to show that the balance is maintained;
§ informally explore the Reflexive, Symmetric, Multiplication/Division, Transitive, and Addition/Subtraction Properties of Equality;
§ recognize the relationship between a pan balance and an equation;
§ draw conclusions based upon patterns in a table

Materials


Shape Pan Balance Recording SheetPan Balance - Shapes Tool Optional: Pan Balance (from science department or homemade version)Optional: Square tiles or cubes to place on balance

Instructional Plan
This lesson will enhance algebraic understanding through an informal study of equality. A two arm balance pan, shown in the classroom (borrowed from the science department, or built with a meter stick balancing on a pencil) will help students see when the pans are balanced, the left side equals the right side. This important concept in algebra will be reinforced as students manipulate shapes in the pan balance with this applet. The Properties of Equality will be identified later in the lesson.
To demonstrate how this applet works, project the Pan Balance-Shapes Tool. Place a shape on the left side of the balance; the balance tilts to the left. This is unbalanced (or an inequality). Place the same item on the right side to demonstrate equality. Next place a red square on the left, and a blue circle on the right. Show students how to place shapes in either pan until they balance, adding squares and circles, red on left, blue on right, in a varied order, until equality is reached. When they balance, show students how equivalent relationships are recorded in the table on the screen, and will remain there until you click "New Problem", which creates new weights for each of the shapes. Encourage students to use Reset Balance to keep the same weights, and keeps the table, but clears the pans. Show Array counts the contents of the pans, leading to a transition to variables, such as, 9 diamonds = 9d.

Pan Balance - Shapes
Next, provide time for students to explore this applet in pairs or groups of four as you circulate and observe the student work, asking students to explain their findings. Look for examples of the Properties of Equality (explained below) to project later for the class.
After exploration time, facilitate a discussion with the class of the discoveries (25 minutes). Students may project their discoveries by bringing their lap tops to the projector. With each one, name the property, and have students record one example of each of the properties on the Pan Balance-Shapes Recording Sheet. The teacher may provide examples of any properties students did not discover.
Examples of discoveries may include:
§ one red square = one red square, or 1s = 1s, using variables. This demonstrates the Reflexive Property of Equality, a = a.
Remind students to "Reset" the balance to show 2 pink triangles = 2 pink triangles, or one yellow diamond on each side.
This property may also be demonstrated with with a pan balance in the classroom, placing 3 blue blocks in the left pan, and 3 blue blocks in the right pan. To develop kinesthetic understanding of the Reflexive Property of Equality, have students hold 3 cubes in their left hand, and 3 cubes in their right hand.
When demonstrating to the class, you may use the following link to keep a constant set of relationships, which may be helpful when leading the discussion.
§ To demonstrate the Symmetric Property of Equality, if a = b, then b = a, place 1 red square in the left pan and 2 blue circles in the right pan. Ask students, "What if I put 2 blue circles in the left pan? What must I put in the right pan?"
[1 red square].
Reset, and demonstrate 1 pink triangle = 3 yellow diamonds. Using the Symmetric Property of Equality, ask students, "What will balance 3 yellow diamonds placed in the left pan?
[1 pink triangle in the right pan].
§ To demonstrate the Multiplication Property of Equality, if a = b, then ca = cb, place 6 blue circles in the left pan with 4 pink triangles in the right pan. Ask, "If I remove 3 blue circles in the left pan, how many pink triangles will balance in the right pan?"
[2, dividing both sides of the equality in half]. (Note: Multiplying by a fraction, one half, is equivalent to dividing by a whole number, 2. This may eliminate the need for the Division Property of Equality).
Alternatively, you may place 1 pink triangle in one pan, and 3 yellow diamonds in the other. Ask, "If I place 2 pink triangles in one pan, how many yellow diamonds will balance?"
[6, doubling each side, with the Multiplication Property of Equality].
§ To demonstrate the Addition Property of Equality, if a = b, then a+c = b+c, place combinations of colors in the left and right pans, such as 2 blue = 1 red. Add a yellow to both sides. The pans remain balanced. (Note: Removing a tile may be considered adding a negative. This may eliminate the need for the Subtraction Property of Equality).
§ To demonstrate the Transitive Property of Equality, if a = b and b = c, then a = c, use the pan balance to show if 1 red = 2 blue, and 3 blue = 2 pink, then 3 red = 4 pink.
You may discuss substitution at this time, and begin to write equations for the relationships. To do so, transition to "Count Items." For example, count it shows 4 × red squares = 3 × blue circles. Write 4r = 3b. Have students practice writing these equivalent arrays, as they are displayed on the computer.
Below is an example of one of the new problems that can result, with the equation shown using the "Count Items" feature.
Students should complete the Shape Pan Balance Recording Sheet after completing this lesson.

Shape Pan Balance Recording Sheet

Adding Fractions
There are 3 Simple Steps to add fractions:
Step 1: Make sure the bottom numbers (the denominators) are the same
Step 2: Add the top numbers (the numerators). Put the answer over the the same denominator as in step 1
Step 3: Simplify the fraction.

Squares and Square Roots
To understand square roots, first you must understand squares ...
How to Square A Number
To square a number, just multiply it by itself ...
Example: What is 3 squared?

3 Squared=3 × 3=9
Note: we write down "3 Squared" as 32, (the "2" means the number appears twice in multiplying)
Some More Squares
4 Squared=4 × 4=16
5 Squared=5 × 5=25
6 Squared=6 × 6=36
Square Root
A square root goes the other direction:
3 squared is 9, so the square root of 9 is 3
3
9
So. the square root of a number is that special value that, when multiplied by itself, gives the number.
Note: It is called a "root" because it is like the root of a tree. You can see the tree, but what is the root that produced it?
Here are some more:
4 16
5 25
6 36
Example: What is the square root of 25?
Well, we just happen to know that 25 = 5 × 5, so, if you multiply 5 by itself (5 × 5) you get 25. So the answer is 5
(Note: there is another solution! -5 × -5 = 25, so the square root of 25 is also -5. Test this idea yourself on another square root.)
The Square Root Symbol
This is the special symbol that means "square root", it is sort of like a tick, and actually started hundreds of years ago as a dot with a flick upwards.It is called the radical, and always makes math look important!
You can use it like this: (you would say "the square root of 9 equals 3")
Perfect Squares
The perfect squares are the squares of the whole numbers:

1
2
3
4
5
6
7
8
9
10
etc
Perfect Squares:
1
4
9
16
25
36
49
64
81
100
...
It is easy to work out the square root of a perfect square, but it is really hard to work out other square roots.
Example: what is the square root of 10?
Well, 3 × 3 = 9 and 4 × 4 = 16, so we could guess that the answer is between 3 and 4.
Let's try 3.5: 3.5 × 3.5 = 12.25Let's try 3.2: 3.2 × 3.2 = 10.24Let's try 3.1: 3.1 × 3.1 = 9.61
At this point, I get out my calculator and it says:
3.1622776601683793319988935444327
... and the digits just go on and on, without any pattern. So even the calculator's answer is only an approximation ! (Further reading: these kind of numbers are called surds which are a special type of irrational number)
A Special Method for Calculating a Square Root
There are many ways to calculate a square root, but my favorite method is an easy one which gets more and more accurate depending on how many times you use it:
a) start with a guess (let's guess 4 is the square root of 10)b) divide by the guess (10/4 = 2.5)c) add that to the guess (2.5+4=6.5)d) then divide that result by 2, in other words halve it. (6.5/2 = 3.25)e) now, set that as the new guess, and start at b) again
... so, our first attempt got us from 4 to 3.25Going again (b to e) gets us: 3.163Going again (b to e) gets us: 3.1623 And so, after 3 times around the answer is 3.1623, which is pretty good, because:
3.1623 x 3.1623 = 10.00014
This is fun to try - why not use it to try calculating the square root of 2?



Find at which step of the following proof, it becomes false. And give the reason.
x = y
x * x = x * y
x * x – y * y = x * y – y * y
( x + y ) ( x – y ) = y * ( x – y )
x + y = y
2 y = y
2 = 1



Find at which step of the following proof, it becomes false. And give the reason.
x = y
x + x = x + y
2 x = x + y
2 x – 2 y = x + y – 2 y
2 x – 2 y = x – y
2 ( x – y ) = x – y
2 = 1

Hope you enjoy some of these problems and remember to make math fun.