Factoring Trinomials Using the Box Method

Factoring Trinomials Using the Box Method

A fairly new method, or algorithm, called the box method is being used to multiply two binomials together. When a trinomial of the form ax² + bx + c can be factored into the product of two binomials, the format of the factorization is (dx + e)(fx + g) where d x f = a and e x g = c.

The box method enables you to fill in a two-by-two square to create the desired factorization. You still need to know the factors of a and c, but the box method gives you a more systematic process for determining which factors and terms to choose.

A common method for multiplying the two binomials together is called FOIL, and the factoring of the resulting trinomial is often referred to as unFOIL. In traditional algebra classes the unFOIL method was taught, having students find the factors of the coefficient, a, and the factors of the constant, c, and then filling in parentheses to get the desired product.


The following shows the factorization of 3x² + 10x – 8 using the box method.

1. Draw a two-by-two square.
   
2. Put the first term of the trinomial in the upper-left corner and the last term in the lower-right corner.
   
3. Multiply the first and last terms: 3x²(–8) = –24x².
4. Find two factors of the resulting product whose sum is the middle term, 10x.
   Possible factors of 24x² (these are without the negative sign) are x(24x), 2x(12x), 3x(8x), and 4x(6x).
   Using 2x(12x) and making the 2x negative, you have –2x(12x) = –24x². The sum of the two factors is –2x + 12x = 10x, the middle term of the trinomial.
5. Place the factors in the two remaining squares.
   
6. Find the greatest common factor (GCF) of each row and each column. Write them to the side and below.
   
7. The two entries along the right side are the terms needed in one of the binomials, and the terms along the bottom are in the other factorization. So, 3x² + 10x – 8 = (3x – 2)(x + 4).
   It doesn’t matter where you place those two factors in Step 5. If their positions had been reversed, the square would look like this:
   
   
   And the GCFs:
   
   
   You get the same entries for the binomials.

Algebra Tiles "Hands-On Algebra"

Algebra tiles are mathematical manipulatives that allow students to better understand ways of algebraic thinking and the concepts of algebra.  Algebra tiles are good manipulatives to give students a hands-on approach to be used in addition to the normal way it's taught in the classroom. Using Algebra Tiles You Can “Feel” the Mathematics.

• 
• Pictures of Algebra Tiles

The algebra tiles are made up of small squares, large squares, and rectangles. The number one is represented by the small square, which is also known as the unit tile. The rectangle represents the variable x and the large square represents x². The length of the side of the large square is equal to the length of the rectangle, also known as the x tile. When visualizing these tiles it is important to remember that the area of a square is s², which is the length of the sides squared. So if the length of the sides of the large square is x then it is understandable that the large square represents x². The width of the x tile is the same length as the side length of the unit tile. The reason that the algebra tiles are made this way will become clear through understanding their use in factoring and multiplying polynomials.

 These tiles have proven to provide concrete models for elementary school, middle school, high school, and college-level introductory algebra students. T Algebra tiles allow both an algebraic and geometric approach to algebraic concepts. They give students another way to solve algebraic problems other than just abstract manipulation.^[2] The National Council of Teachers of Mathematics (NCTM) recommends a decreased emphasis on the memorization of the rules of algebra and the symbol manipulation of algebra in their Curriculum and Evaluation Standards for Mathematics. According to the NCTM  standards relating models to one another builds a better understanding of


                                    

Algebra Tiles

Tiles 02

Tiles

   

Measuring Metrically with Maggie

An Introduction to Metric Units

Wow, I just flew in from planet Micron.  It was a long flight, but well worth it to get to spend time with you!
My name is Maggie in your language (but you couldn't pronounce my real name!)
When I first arrived I couldn't understand how you measure things, but my friend Tom taught me all about measurement, and I am going to share with you everything he taught me.
The first thing Tom told me was that you can measure things using two different systems: Metric and US Standard.
Today is my day to learn Metric !
Tom says that if I understand 10, 100, and 1000 then I will have a very easy time learning the metric system. I wish I had ten fingers!

Liquids

Since it was such a long flight, the first thing I could use is something cold to drink.  
But I want to know how much to ask for! So I can get a drink that is not too big or too small.
Tom says I only need to know about:

• Milliliters
• Liters

A milliliter (that is "milli" and "liter" put together) is a very small amount of liquid.
Here is a milliliter of milk in a teaspoon.
It doesn't even fill the teaspoon!
Tom says if you collect about 20 drops of water, you will have 1 milliliter:

20 drops of water

make about 1 milliliter

And that a teaspoon can hold about five milliliters:

1 full teaspoon of liquid

is about 5 milliliters

Milliliters are often written as ml (for short), so "100 ml" means "100 milliliters".
But a milliliter is definitely not enough for someone who is thirsty! So Tom told me about liters.

A liter is just a bunch of milliliters put all together. In fact, 1000 milliliters makes up 1 liter.

1 liter = 1,000 milliliters

This jug has exactly 1 liter of water in it.
Liters are often written as L (for short), so "3 L" means "3 Liters".
Milk, soda and other drinks are often sold in liters.
Tom says to look on the labels, so the next time you are at the store take a minute and check out how many liters (or milliliters) are in each container!
Now I know that a milliliter is very small, and a liter is like a jug in size, I think I will ask for half a liter of juice!
So this is all you need to know:

1 Liter = 1,000 Milliliters

Mass (Weight)

Next I wanted to eat some chocolate ... so I should learn about mass. You often call it "weight", but it is only because of the gravity on your planet that items have weight! 
Tom tells me that to understand mass, I should know these three terms:

• Grams
• Kilograms
• Tonnes

Grams are the smallest, Tonnes are the biggest.
Let’s take a few minutes and explore how heavy each of these are.

Grams

A paperclip weighs about 1 gram.
Hold one small paperclip in your hand. Does that weigh a lot? No! A gram is very light. That is why you often see things measured in hundreds of grams.
Grams are often written as g (for short), so "300 g" means "300 grams".

Tom tells me a loaf of bread weighs about 700 g

Kilograms

Once you have 1,000 grams, you have 1 kilogram.

1 kilogram = 1,000 grams A dictionary has a mass of about one kilogram.

Kilograms are great for measuring things that can be lifted by people (sometimes very strong people are needed of course!).

Kilograms are often written as kg (that is a "k" for "kilo" and a "g" for "gram"), so "10 kg" means "10 kilograms".
Scales measure our weight using kilograms. Tom weighs about 40 kg. How much do you weigh?
But when it comes to things that are very heavy, we need to use the tonne.

Tonne

Once you have 1000 kilograms, you will have 1 tonne.

1 tonne = 1,000 kilograms

Tonnes (also called Metric Tons) are used to measure things that are very heavy.
Things like cars, trucks and large cargo boxes are weighed using the tonne.
This car has a mass of about 2 tonnes.
Tonnes are often written as t (for short), so "5 t" means "5 tonnes".
Final thoughts about mass:

1 kilogram = 1,000 grams

1 tonne = 1,000 kilograms

Length

Measuring how long things are, how tall they are, or how far apart they might be are all examples of length measurements.
Tom says I should know about:

• Millimeters
• Centimeters
• Meters
• Kilometers

Small units of length are called millimeters.
A millimeter is about the thickness of a plastic id card (or credit card).
Or about the thickness of 10 sheets of paper on top of each other.
This is a very small measurement!

Centimeters

When you have something that is 10 millimeters, it can be said that it is 1 centimeter.

1 centimeter = 10 millimeters

A fingernail is about one centimeter wide.
We can use millimeters or centimeters to measure how tall we are, or how wide a table is, but to measure the length of football field it is better to use meters.

Meters

A meter is equal to 100 centimeters.

1 meter = 100 centimeters

The length of this guitar is about 1 meter
Meters might be used to measure the length of a house, or the size of a playground.

Kilometers

When you need to get from one place to another, you will need to measure that distance using kilometers. A kilometer is equal to 1,000 meters.
The distance from one city to another or how far a plane travels are measured using kilometers.

Final thoughts about measuring length:

1 centimeter = 10 millimeters

1 meter = 100 centimeters

1 kilometer = 1000 meters

Temperature

I was feeling a bit hot, so I asked Tom how to measure temperature.
So he showed me a thermometer. But I saw 2 sets of numbers!
Tom explained that a thermometer measures in degrees (°) of either Celsius or Fahrenheit.
"Why two scales?", I asked.
Tom said that some people like one scale and some like the other, and that I should learn both!
He then gave me an example: when water freezes the thermometer shows:

• 0 degrees Celsius on the left side,
• but on the right side it shows 32 degrees Fahrenheit.

So there can be two numbers for the same thing!
He gave me more examples.

• A hot sunny day might have a temperature of 30 degrees Celsius, which is 86 degrees in Fahrenheit.
• Water boils at 100 degrees Celsius or 212 degrees Fahrenheit.
• And you can bake cookies in your oven at a temperature of 180 degrees Celsius, which is 356 degrees Fahrenheit.

I decided to get my own thermometer, so I would learn about all this.

I hope you enjoyed learning all about metric measurement. Now I must return home. Keep measuring until I see you again!!!!!!!!!

An Introduction to US Standard Units

Also known as "English Units" or "US Customary Units"

Wow, I just flew in from planet Micron.  It was a long flight, but well worth it to get to spend time with you!
My name is Maggie in your language (but you couldn't pronounce my real name!)
When I first arrived I couldn't understand how you measure things, but my friend Tom taught me all about measurement, and I am going to share with you everything he taught me.
The first thing Tom told me was that you can measure things using two different systems: US Standard Units and Metric.
Today is my day to learn US Standard Units !

Liquids

Since it was such a long flight, the first thing I could use is something cold to drink.   But I want to be sure how much to ask for! So I can get a drink that is not too big or too small.

The first thing I need to know when asking for a drink is the types of units used to measure liquids, which are:

• Fluid Ounces
• Cups
• Pints
• Quarts
• Gallons

Fluid Ounces (oz) are small. About how much fits into a small medicine cup ... but that isn't enough for someone who is thirsty!

Then Tom showed me a small carton, and told me it held 8 fluid ounces, which is also called 1 cup. But I wanted more!

So Tom showed me a pint, which is equal to 2 cups. That seemed about right for someone who was very thirsty!

(Tom also told me that I can measure things using measuring cups)

To measure many cups of liquid all put together you can use quarts. A quart (qt) is the same thing as 4 cups or 2 pints.

If you still need more liquid you may want to switch to using gallons. A gallon (gal) is the same as 16 cups or 8 pints or 4 quarts.  It is the largest liquid measurement. (Oh wow! A quart is a quarter of a gallon!)

So now I know that 1 ounce is too small for me, but 1 gallon is too much. I think I will ask for a pint of juice!
Final thoughts about measuring liquids: 

1 Gallon = 4 quarts = 8 pints = 16 cups= 128 fluid ounces

Mass (Weight)

Next I wanted to eat some chocolate ... so I should learn about mass. You often call it "weight", but it is only because of the gravity on your planet that items have weight! 
Tom says I need to know:

• Ounces
• Pounds
• Tons

Ounces Again! One thing that really confused me is that when I asked for a drink I could use ounces, but ounces are also used for mass ... ... the same word can be used in two different measuring systems!  How amazing is that? But they are really different.

Tom says:

If you mean an ounce of fluid say "fluid ounce" ("fl oz")
Otherwise ounce usually means mass.

So we are not talking about fluids! I already had a drink. I need to know about mass.

The smallest unit of mass is ounces (oz).  A slice of bread is about one ounce.  It is very light.

But when you add up the ounces you get a new label for mass:

If you have 16 ounces, it can also be called a pound (lb).  Typically, this is the unit that you use to measure your own weight.  1 pound = 16 ounces Pounds are used to measure lots of things from people to food to animals.  Tom says he weighs 90 pounds.

But if something weighs many pounds we use yet another label

2,000 pounds is also one ton.  1 ton = 2,000 pounds That is really heavy!  Trucks, ships and heavy equipment are measured using tons instead of pounds. An elephant has a mass of about 8 tons!

So now I know that 1 ounce of chocolate is too small for me, but 1 ton is way too much. I think I will ask for a pound of chocolate!
Final thoughts about mass:

1 pound = 16 ounces
1 Ton = 2,000 pounds = 32,000 ounces

Length

The last kind of measurement we will explore is length.  This is important for lots of different reasons.  Measurement of length helps you to know how far you have traveled, how far you have left to travel, how tall you are and many other things.

I need to know about:

• Inches
• Feet
• Yards
• Miles

Small units of length are called inches.  The last joint of your finger or thumb is about 1 inch (depending on how big your fingers are!).

Lots of things are measured in inches from rainfall to paper length.
Measuring in inches gives us a way for everyone to understand the size of something.

When we have 12 inches together, it is known as a foot. 

1 foot = 12 inches

A long time ago, people used their feet to measure things. But everyone has different sized feet so it did not work very well.
Using 12 inches put together to make one foot lets everyone have an accurate picture of what exactly a "foot" of length is.

When 3 feet are together, this is called a yard.  (This isn't the same thing as a lawn, though they are both referred to as a "yard"!)

1 yard = 3 feet

The length of this guitar is about 1 yard.

When you put together 1,760 yards, you have a mile.  1 mile = 1,760 yards = 5,280 feet Miles are long distances and are mostly used to measure the distance between places which are far away from each other.  Most people refer to miles when they are driving, biking or jogging.

Final thoughts about measuring length:

1 foot = 12 inches

1 yard = 3 feet = 36 inches

1 mile = 1,760 yards = 5,280 feet = 63,360 inches

Temperature

I was feeling a bit hot, so I asked Tom how to measure temperature.
So he showed me a thermometer. But I saw 2 sets of numbers!
Tom explained that a thermometer measures in degrees (°) of either Celsius or Fahrenheit.
"Why two scales?", I asked.
Tom said that some people like one scale and some like the other, and that I should learn both!
He then gave me an example: when water freezes the thermometer shows:

• 0 degrees Celsius on the left side,
• but on the right side it shows 32 degrees Fahrenheit.

Two numbers for the same thing!
He gave me more examples.

• A hot sunny day with a temperature of 30 degrees Celsius is 86 degrees in Fahrenheit.
• Water boils at 100 degrees Celsius or 212 degrees Fahrenheit.
• And you can bake cookies in your oven at a temperature of 180 degrees Celsius, which is 356 degrees Fahrenheit.

I decided to get my own thermometer, so I can learn all about this.

Bye for Now!

I hope you enjoyed learning all about measurement.  Now I must return home.  Keep measuring until I see you again!!!!!!!!!