Operations on Decimals (A Review)

For years we have been operating on numbers that are based on powers of 10. These are called decimal numbers. For example, 231 is a decimal number because it is composed of  (2 x 100) + (3 x 10) + (1x1).

Meanwhile, there are decimal numbers that have components less than 1. To accommodate such numbers, we denote them with decimal points, and their place values are read similarly with whole number places but with "th" at the end and the decimal point is read as "and". For example, 231.045 is read as "two hundred thirty-one and forty-five thousandths".

Just as there are four basic arithmetic operations with whole numbers, there are also four basic operations with decimals. We review them here.

Addition and Subtraction
1. Align the decimal points.
2. Add normally.

Multiplication
1. Multiply normally.
2. Count the number of decimal places in both factors. Their total is the number of decimal places in the product.

Division
1. If there are decimal places in the divisor, move its decimal point to the right as many decimal places as possible to get rid of any decimal parts. For instance, if there are 3 decimal places, move the decimal point three places to the right.
2. Move the decimal point in the dividend in the same number of places as in the divisor.
3. Divide normally.


Examples.
1) 0.00056 + 0.32443 = 0.32499
2) 65.43 - 38.14 = 27.29
3) 0.3 x 0.6 = 0.18  (one decimal place in 0.3 and one in 0.6, so there are two decimal places in 0.18)
4) 24 / 0.12 = 2400 / 12 = 200 ( there are 2 decimal places in 0.12, so both dividend and divisor had their decimal points moved two decimal places)


 

On Pythagorean Triples

Recall from basic geometry that the legs and hypotenuse of a right triangle are related to each other by the Pythagorean Theorem:

a² + b² = c²

where a and b are the lengths of the legs and c is the length of the hypotenuse.

There are certain three-number sets of that satisfy the Pythagorean Theorem in such a very exact fashion that they deserve to be called Pythagorean Triples. Let us enumerate some. The examples that follow are arranged in the order "a, b, c".

3, 4, 5

5, 12, 13

7, 24, 25

9, 40, 41

11, 60, 61

13, 84, 85

15, 112, 113

17, 144, 145

19, 180, 181

 In addition to these triples, their integer multiples are also Pythagorean triples. For instance,

18, 80, 82  are integer multiples of 9, 40, 41(by a factor of 2, in this case) and so are a Pythagorean triple.

Here's a challenge to you: 

1. Aside from satisfying the Pythagorean theorem, what other properties do these numbers have?

2. Given the properties you discovered, can you name other Pythagorean triples without actually plugging them in into the Pythagorean theorem?

Let Me Go (by David Archuleta)

I remember the better days
Way before this mess we made
You were the keys to the car
Now I'm just trying to make it start
Can't you see these highs and lows
Take us down and slowly take their toll
Misguided, I don't know where we're headed
Tell me now, 'cuz

Round, round and round we go
And when it stops you say you don't know
But each time I try to stop this ride
You say it's not time
This heart, my dreams
Have been taken down too far it seems
So hold tight
Why not let me go
Why not let me go

So much I need to say
Then the truth gets in the way
You cry me another one
And watch my words just come undone
Can't you see these lows and highs
Tangled up, separating all our ties
Misguided, I can't find a way back in
So maybe it's goodbye

Round, round and round we go
And when it stops you say you don't know
But each time I try to stop this ride
You say it's not time
This heart, my dreams
Have been taken down too far it seems
So hold tight
Why not let me go
Why not let me go

When the bottom drops out
And you think you've got nowhere to go (in the cold)
But if you take a look around
You could really warm it up and you know (always told you so)

Round, round and round we go
And when it stops you say you don't know
But each time I try to stop this ride
You say it's not time
This heart, my dreams
Have been taken down too far it seems
So hold tight
Why not let me go
Why not let me go (why not let me go)

Round, round and round we go
When it stops you say you don't know
Why not let me go
Why not let me go
oh oh oh oh oh (why not let me go)

Why not let me go (yeah)
Why not let me go (yeah)
Why not let me go

Math Teaching Strategies That Add Variety and Motivation to Your Lesson Plans

Teachers, when you are considering your math teaching strategies, keep in mind that you will be able to motivate and be more successful if you can include some variety and fun into your strategies.

This applies to all subjects but especially so for math, since math can be a very abstract subject to teach and learn.
Here are several useful math teaching strategies that you should consider using.

Elementary Math Manipulatives

Math manipulatives are important and very useful educational aids that you should use as part of your math teaching strategies.
From kindergarten math manipulatives all the way to Grade 6 and even higher, math manipulatives will help you explore, discover, calculate, sort and assess. Math manipulatives can be colourful, home-made or purchased, edible, new or used.
Garage and yard sales can be a great way to find some creative manipulatives. The list of possible manipulatives is endless and so should be your creativity. Here are some of my suggestions:

Colourful

* Geometric Shapes
* Pattern Blocks
* Rainbow Dice and Dominoes
* Ten base blocks (place value)
* Mirras (used for symmetry)

Edible

* Certain Cereals (Cheerios, Fruit Loops)
* Pretzels
* Animal Crackers
* Chocolate Bars (Aero, Kit Kat, Cadbury - GREAT for FRACTIONS
Remember to check for food allergies and stay away from candy.

Other useful objects to have on hand

* Buttons
* Sea Shells
* Bingo Coins ($ Store)
* Pennies
* Dry Beans
* Playdo
* Pasta (small ones)
* Decks of Cards (add, subtract, multiply)


In the older grades, students use a compass and protractor for measurement. These can be considered math manipulatives, as well as a ruler, which is also used in the primary grades.
Even consider having your class go outside to look for and collect different math manipulatives, such as pine cones and geometric shapes.
Don't get me wrong - using the math textbook, worksheets and lecturing are all part of teaching math. But ... show your students that math can be exciting, creative and innovative.
Use multiple math teaching strategies that can reach those children who may find math boring or challenging.
And always, always, always keep in mind to set rules on how to distribute and collect these teaching aides and the proper way of using them.
Discipline and control of your class are important keys when teaching a successful, educational and fun math lesson using math manipulatives!

Singapore Math

Teaching math is all about teaching students to think logically. Singapore is and has been one of the leading countries in developing students who excel in math, so "Singapore Math" is a generic term that refers to their method of teaching math.
Singapore math teaches students to think in logical and creative ways rather than rote learning and repetitive drilling. This is why I promote these math activity books that also put some fun into learning math and will have your students asking for more.
The first is The Math Riddle Book. This book is full of math puzzles and is a great and fun way for students to build their math skills. Learning is easy, enjoyable and makes students think when they're solving puzzles!
Another source of printable math math games and fun math activities that make math exciting and easy to learn are these 4 books of Fun Printable Math Games.
Finally, The Largest Free Singapore Math Website in the World with hilarious math problems, easy-to-understand math lessons and exciting 3D clip art. Kris Murphy, the creator of this website, was educated in Singapore so he has first hand knowledge of the math curriculum there.

Make Learning Math Fun

As I said, as part of your math teaching strategies it's important to motivate your students by making learning math fun.
Here are a couple of quick game ideas that I incorporated into my math teaching strategies and have found to be quite successful. My students loved them.
1. Play "Lotto" - like Bingo but ask multiplication questions with the answers on the squares of Bingo like cards that you make and hand out.
2. Math Minute - 30 basic simple arithmetic questions to complete in one minute.
Motivate your students with these math word wall strategies.
As well as these simple math games be sure to check out my page on some of my other classroom math games .
Teach your students how to do math in their heads! Develop math confidence, overcome "math phobia". With Fun With Figures you'll show your kids exactly how to perform some amazing mental math in clear simple steps.

Math Journals

One very useful math teaching strategy for both me and my junior grade students was to have them keep a math journal. Students documented what they learned during a math lesson or wrote down the steps they took to solve a particular math problem or math exercise.
By documenting the steps the student became more aware of the process involved in solving problems.
Math journals can be of great assistance to teachers as well, by giving them a better insight into what students may or may not have learned/understood about a math lesson or math concept. The teacher can then take the appropriate action, i.e. further review, etc.

Making Math Meaningful

It can be very helpful for your students to relate math questions and problems to everyday life situations. One idea is to distribute monopoly money and use it to simulate shopping, banking and other daily life scenarios.
For example, using various food flyers, put your students in groups of 4 and give them $100 to spend (great for adding, subtracting and multiplying) OR use them to create word problems.
Discipline and the control of your class are important keys when teaching a successful, educational and fun math lesson or for that matter, any lesson. So don't miss my page on classroom management.
If you're looking for Math Tutoring and Homework Help - here you can find excellent and affordable resources for math, including a huge local tutor database, online math help, and discounts on test prep classes. Don't let your child struggle with math alone.


Source: http://www.priceless-teaching-strategies.com/math-teaching-strategies.html

On Two-Variable Linear Systems

Systems of Linear Equations in Two Variables

In an equation where there is one unknown such as x - 7 = 28, it is easy to find the single value for x. In this example, x = 35. Meanwhile, in an equation where there are two unknowns, say x - y = 1, there are infinitely many solutions. For instance, x = 9 when y = 8 is a solution. Another solution is x = 6 when x = 5. But if we have two such equations in x and y that are posed together, there will be one pair of values of x and y that will satisfy both equations. Such equations are called simultaneous because they are solved by finding those values which work for both at the same time. When these equations are plotted on a graph, the coordinates of the point where they intersect are the values of x and y that satisfy them both.

Definition: A pair of equations of the form

 a₁x + b₁y = c₁

a₂x + b₂y = c₂

is called a system of linear equations in two variables.

A solution to a system of linear equations in two variables

is an ordered pair of numbers (x, y) that satisfies all the

conditions of the system.

A linear equation ax + by = c has a straight line for its graph. Thus, for a system of two equations, two lines can be graphed on a plane. There are three possibilities for a given system of linear equations.

·         Independent and Consistent. This system has exactly one solution. When graphed, the lines intersect at one and only one point. For such system,

                              a₁/a₂ ≠ b₁/b₂ ≠ c₁/c₂

 

 

·         Dependent and Consistent. This system has infinitely many solutions. When graphed, the lines coincide. For such a system,

                  

                             a₁/a₂ = b₁/b₂ = c₁/c₂

·         Independent and Inconsistent. This system has no solution. When graphed, the lines do not intersect, and hence, are parallel. For such a system,

                             a₁/a₂ = b₁/b₂  but  b₁/b₂ ≠ c₁/c₂

 

To solve a system of equations means to determine all ordered pairs of real numbers that simultaneously satisfy all equations of the system. We can do so by using one of these methods:

·         Graphing

·         Elimination

·         Substitution

·         Cramer’s Rule (uses determinants)

Due to technical constraints, I will deal with Cramer's Rule in a separate post. 

Graphing

Recall that the graph of an equation is a drawing that represents its solution set. If the graph of an equation is a line, then every point on this line corresponds to an ordered pair that is a solution of the equation. In our first year math we have learned how to graph linear equations by:

·         Plotting points (from a table of x and y values)

·         The intercept method (finding x when y = 0, and y when x = 0)

·         The slope and y-intercept method (when we are given the slope and the y-intercept)

If we graph a system of two linear equations, the point/s at which the lines intersect will be a solution to both equations.

Steps:

1.      Graph both equations on the same coordinate plane.

2.      Find the solution to the system as follows:

a.       If the two lines intersect at one point, the solution is the ordered pair that corresponds to that point.

b.      If the two equations have the same graph, the system has infinitely many solutions.

c.       If the two lines are parallel, the system has no solution.

3.      Check the solution in both equations.

Elimination

Although graphing helps us to picture the solutions of systems of equations, it is not always fast or accurate in cases where solutions are not integers. So this time, we use algebra in finding solutions. We start with the elimination method.

As the name implies, we are going to operate on the system so as to eliminate something. In this case, we will choose to eliminate one of the two variables, which will pave the way to the solution of the other variable.

Steps:

1.      Write both equations in standard form ax + by = c.

2.      Choose and decide which variable you want to eliminate.

3.      Obtain an equivalent system where the coefficients of the variable you want to eliminate are additive inverses. If necessary, multiply one or both equations in the system by a non-zero number.

4.   Solve the equivalent system by adding or subtracting the two equations. Then solve the resulting equation for the remaining variable. Afterwards, substitute the obtained value into either of the original equations to find the value of the other variable.

5.      Check.

Substitution

In substitution, we solve one of the equations for one of the variables and plug it in the other equation to remove one of the variables. It is in fact another way of eliminating one variable first. It is important to note that substitution is convenient to use when the coefficient of the variables is 1 or -1.

Steps:

1.      Solve one equation for one of the variables.

2.      Substitute the obtained expression in the other equation and solve for the other variable.

3.      Substitute the value obtained in Step 2 and solve for the value of the other variable.

a.       If the result is a false statement, then the system has no solution.

b.      If the result is a true statement, we either have a system with infinitely many solutions or with exactly one solution.

4.      Check the obtained solutions in the original equations.