Integers: Part Two, Multiplication & Division

Multiplication and Division of Integers, especially when mixing  positive and negative numbers can be tricky!  Luckily mathematics provides us with some rules that always prove true and can help to guide us through this, sometimes confusing process.

Just as subtraction is the opposite of addition, division is the opposite of multiplication.  When we multiply we combine groups together, when we divide we split them apart.

The first step:  Learn your Multiplication facts!  I cannot stress the importance of memorizing these enough.  You will use them from early elementary school right through advanced math in high school, and on into adulthood for practical, real-life situations.  Knowing these simple facts will save you valuable time that can be spent on the more complex aspects of solving multiple step problems.

 

It seems that students often have the most difficulty with the sevens and eights.  Rhymes or other tricks to remember these facts can be helpful ("You can’t have 7 and 8 before you have 5 and 6… 7 x 8 = 56", "I know now and you do too! 6 x 7 is 42", or "Say this when you're in a fix. 7 x 8 is 56"), it may seem silly, but it can help!

Basic Rules for Multiplying & Dividing:  

When working with positive and negative numbers and trying to determine whether the product or quotient will be negative or positive remember this; same signs result in positive, different signs result in negative answer.  The chart below illustrates this rule:

Rule for the relationship between Multiplication Division:

a / b = c
if and only if
c x b = a
(when be does not = 0) 

4 Properties that Guide Multiplication Process: 

(Click heading for Helpful Link:)

Closure Property of Mulitiplication  when dividing a and b, the product ab is a unique integer (neither a or b), unless a or b is the integer 1 or zero, see below.

Multiplicative Identity Property  any integer that is multiplied by the integer 1 does not change, it keeps it's "identity"!

Commutative Property  the order in which the multiplication is preformed does not effect the product at all, for example 3x2=2x3.

Associative Property  similarly to how you order integers in the commutative property, the associative property tells us that how you GROUP integers also has no effect on the product, for example (2x3)4=2(3x4)

More information on the rules governing muliplication and division, as well as many links to fun games and toutorials can be found on the MindMap I created on the subject, visit mindomo.com

 

***Especially for Parents***

Sometimes as our children get into the higher grades and begin exploring more advanced mathematical concepts we can feel inadequate providing effective homework help.  Multiplication and division are concepts that we, as adults are generally comfortable with, in the beginning.  Once addition of unknown variables, multiple steps and opperations, exponents, parentheses and brackets that call for knowledge of Order of Opperations , formulas and rules.  Fortunate for us the internet has supplied us with unlimited resources for learning and knowledge.  This too can be a bit intimidating so I have put together a short list of the most helpful sites I have found.

Khan Academy  over 3000 educational videos!  Concise and easy to follow.

Purple Math  from beginning to advanced algebra lessons & practice.

Math.com  homework help, calculators and tools.

IXL.com  pre-k to 8th grade, links to standards, practice, track & progress reports available.

Integers: Addition & Subtraction

Addition & Subtraction of Integers

What is a Number?

A mathematical symbol of value.  Numbers are used to measure, count, and label.

What is an Integer? 

Integers are positive whole numbers, negative numbers, and zero.

Is Zero a Number?  

It has NO VALUE... yes it does, it has a value of NONE!

MindMap

The above link will bring you to a "Mind Map" that I created on the topic of inters.  This map is full of links to websites containing videos, lesson plans, games for students, problem examples, and helpful tips on how to explain some of the concepts included in the idea of numbers.

The Addition of Integers 

Video Explanation of Integer Addition

THE RULES

• (+/-) When adding two integers with unlike signs (one positive and one negative) look at their absolute values and subtract the smaller one from the larger, the answer will have the same sign as the larger.
• (-/-) Add the absoute values and attach a negative sign (-) to the sum.
• (+/+) Just add together, the answer will be positive.
• In adding integers, it does not matter which you add first, second, third, etc. you will get the same sum in the end no matter what order you add them in.

Addition is the combining of two or more values into one.  In early mathematics students may benefit from using manipulatives such as counting blocks or other objects, dice, dominoes, or their fingers.  Color coded counters or charged-field modeling becomes more useful once students are dealing with negative numbers.  A good resource for buying these math manipulatives for use at school or home can be found at www.learningresources.com

The Subtraction of Integers

Every positive number has an opposite, or negative number, for 3 it is -3, for 297 it is -297 and for -8 it is 8.  When we subtract we essentially add the opposite: a-b=a+(-b).  Subtraction means to take a value away from an original value, to remove some.  A good video that provides a step-by-step example of the process of subtracting can be found at MathHelp.com

 

Using Number Lines to Add & Subtract

Number lines can provide a great visual representation of where a number lays in regard to value compared to other numbers.  It is helpful to be able to count forward (right) or backward (left) as needed.  Numbers increase in size, or value, as you move right on the number line.  ALL numbers have their own special, unchanging, place on the line.  In the center of this infinite line of numbers is zero.  Zero is at the center because it has a value of none and is neither positive or negative.  All numbers to the right of zero are positive, and to the left are negative.

Educational & Fun Games for Addition, Subtraction and Numberline Practice can be found at:

Fun 4 the Brain Adding 

Fun 4 the Brain Subtracting 

Line Jump Brain Fun 

 

Teaching Elementary Estimation Skills

Estimation 

"We always need more people who can think," were the closing words of an article I just read by
Robert J. Doman Jr.  Who can argue with that?!?  Estimation in math does just that, helps people think.  When you estimate you go beyond rote memorization, you are not just applying a meaningless formula.  What you are doing is asking, "what is the value of these symbols," and, "does this make sense?"  It helps you to understand the concept and it's applicable meaning.  

Two other benefits of estimation, Dorman points out, are visualization and conceptualization.  
When a student is completing a math problem often it is helpful for them to visualize or conceptualize it in some sort of real-world situation, forming a mental picture.  Associating meaning to math is very helpful to children, particularly in the primary grades.  For example, if I ask a 2th grader what 5-7=?  I may be met with a blank stare.  If I phrase the question as a familiar situation,"If you are at the toy store with your mom and you see a toy that costs $7.00, but you only have $5.00 with you, how much do you owe your mom when you get home?" The child will quickly respond, "Oh, that's simple, I had to borrow $2.00 from her," they understand better when it is a situation that they can relate to, one that they can envision.

Teaching Estimation

Estimation by rounding

When a student is asked to estimate in a simple computation problem, often what the question is asking them to do its round.  Rounding is also a handy skill when checking over problems.  If a student ends up with 97 when subtracting 406 from 603 he or she can quickly assume that is not a reasonable answer.  This quick check prompts them to take a closer look, noticing that they simply forgot to bring complete the last step and place a one in the hundreds place.  

Estimating Length or Distance

Now, here's a concept best taught hands-on.  Get out the rulers & meter sticks!  Comparisons of things of different lengths, which is larger or smaller, taller or shorter will give them a real-world idea of what these measurements actually mean.  

Maps are another helpful tool when teaching distance estimation.  Plotting multiple points on a map and asking a question such as, "The Jones family is planning a trip.  They have a budget of $290 to spend on gas for their vehicle that gets approximately 30 per gallon.  If the price of gas is $2.89 per gallon which destinations on their wish list are options for their trip? (Don't forget, they have to get back home!)"

Estimating Numerical Value

What is 76?  Well, it's more than 50, which would be half way between 0 and 100, that's a good start.  Conceptualizing where a number lies in relation to other numbers.  This concept is helpful in teaching anything from beginning addition to decimals and fractions!

Useful Links in Estimation:

http://www.math.com/

math.com offers some of the best, concise, and easy to understand explanations of mathematical concepts that I have come across so far.  They also offer practice problems. 
  

http://nacd.org/newsletter/0309_estimation.php 

Dorman's article, Estimation: How to Accelerate the Learning Process with Math and Build Visualization and Conceptual Skills Simultaneously can be found in NACD's Newsletter Archive at: 



Some estimation and rounding online practice can be found at:

http://www.aaamath.com/est.htm

Visit:
http://www.transum.org/  For a list of Maths lesson starter activities and interactive exercises for students on the topic of Estimating.  Free for students and teachers!

And, finally, the most useful article I found:
Developing Estimation Skills in the Primary Grades

Fantastic teaching strategies and activities!