Tuesday, January 22, 2013

Dividing Fractions

In class we used models to develop an algorithm for dividing fractions.  Below is an example of  model we used.

Notice that our quotient is larger than either number in our division problem.  This because we are dividing a value less than one into another number.

Click on this link for a great site that helps you visualize operations with fractions. http://www.visualfractions.com/ When you are on the site click on "Divide" at the upper right.

Using the models we found that we can take our divisor and flip it to multiply to our dividend.  We you "flip" the second fraction this is called multiplying by the reciprocal.  The reciprocal is simply the fraction flipped.  Click the image that follows for an explanation of why multiplying by the reciprocal works.

Here are two Khan Academy videos to sum it all up.

For those of you who are brave, here is dividing positive and negative fractions.

Mid Term Study Guide

Mid-Term Review2012

Bits & Pieces II Pre-Test

Bits _ Pieces II Pre-Test 2012-2013
BITS _ PIECES II PRETEST_001

Model of Multiplying Fractions

I found this video to help explain why you need to convert mixed numbers into improper fractions to multiply.  Hope this helps.  Click here for the link to the video.

Multiplying Fractions

When you multiply fractions you do not need to find a common denominator because you are finding a part of a part.  The video below is similar to the brownie pan model which helps you understand the algorithm to multiply fractions.  To multiply you can simply multiply numerator to numerator and denominator to denominator.  Multiplying the denominators breaks the original fractions into more pieces.  Remember to always simplify the product.

A helpful strategy is to cross cancel.  When you cross cancel you are simplifying the fractions before you you multiply.  This makes the multiplication a bit easier and less cumbersome.  To cross cancel you look for a common factor that is share by any numerator and any denominator in the problem.  You can use any numerator and denominator because of the commutative property of multiplication.  Once you have "pulled out" or divided by as many common factors as you can, you can then multiply the numerators and the denominators that are left. If you have cross cancelled correctly, your product is usually simplified.

When you multiply mixed numbers you need to convert the mixed numbers to improper fractions first.  You need to do this because you need to account for the whole value of the set.  Remember 2 * 3 means three sets of two, so 2 1/3 * 3 1/4 means three and one-fourth sets of two and a third.  You can not multiply the whole numbers separately from the fractions.

Fractions and Word Problems

One of the more difficult aspects of this unit is deciphering the operation you need to do to solve a word problem.  We have talked about many phrases and words that are clues, but be careful with words like "of."  Though many times the word means to multiply, it often show up simply as a preposition to explain something.  Sometimes there are no common words or phrases in the problem and we just have to make sense of the situation.  One way to help you make sense of the situation is to substitute whole numbers or easy benchmark fractions like one-half into the problem.  Use the document below as practice making sense of word problems.
Bits Pieces II Mixed Word Problem Practicex

Subtraction Word Problem

The following problem causes confusion for many students who do not take the time to make sense of the problem and/or their answer.

In June your new puppy weighed 10 2/3 pounds and by July the puppy weighed 19 1/5 pounds.  How much weight did the puppy gain?

First makes sense of the problem by substituting easy whole numbers for the mixed numbers.  Let's make 10 2/3 simply 10 pounds and 19 1/5, 20 pounds.  Then you can make sense of the problem.  It only makes sense to subtract 10 from 20 (20 - 10 = n) to find the difference in pounds or the weight gained. Subtracting 20 from 10 (10 - 20 = n) will give you a negative integer which does not make  sense since the puppy did not lose weight from June to July.

Remember, just because a number comes first in a word problem does not mean that it will come first in a number sentence.  It is important that you stop and think about the situation and make sense of it before you begin your computation.

Subtracting Fractions

Since addition and subtraction are inverse operations, subtracting Fractions is similar to adding fractions in that you must have the same size pieces.  The video from www.khanacademy.org has a useful model to show you why simplifying is important.

Be careful when subtracting mixed numbers.  You may need to borrow from the whole.  If you borrow one whole you need to convert that to a fraction to add to the fraction portion of the mixed number.  You have not changed the value but have renamed the mixed number in a way that allows you to compute.  In the second example pictured below, the one whole was converted into 27/27 to because it needed to be added to 3/27.

The video below shows the algorithm step by step.

The key aspect to remember when you are adding or subtracting fractions is to understand that you need to have the same size pieces before you can do any computations.  Finding a common denominator is creating equal size pieces.  Without breaking your pieces (denominators) into equal sizes you are unable to compute.  Notice that finding a common denominator is the same as finding common multiples.  See the videos from www.khanacadeny.org

When adding mixed numbers you do not have to convert to improper fractions.  (I notice many students doing this and it will work but it complicates your problem and leaves you open to more errors.)  Remember the commutative property allows you to reorganize your addends in any order.  You will see this in the video below.  Therefore you can add your whole numbers and your fractions separately.  Keep in mind you still need to have the same size pieces.

Part of your assessment will be to determine what operation to use when solving word problems.  Remember we developed many examples of word phrases that give you clues.  But it is not wise to rely on memorizing them to solve all word problems.  Some word problems will not fall into any of the categories we discussed; therefore, you need to make sense of the problem.  One strategy is to substitute the mixed numbers with easy whole numbers.  This will help you determine whether or not your approach is making sense.

Thursday, January 17, 2013

Writing Equations Key

WRITING EQUATIONS KEY_001

Number Properties & Algebraic Expressions Key

NUMBER PROPERTIES _ ALGE_001

Number Properties & Aglebraic Expressions

Number Properties and Algebraic Expressions _Test Review Longer Version_

Wednesday, January 16, 2013

Combining Like Terms

Here are a few videos from Khan Academy. You will need to describe the number of terms, identify the coefficients, and name the like terms in a chart.  So you need to remember to add the opposite and complete any distribution BEFORE you complete the chart (see your pre-test)