Creating Expressions & Equations from Real-Life Situations







INTRODUCTION

Firstly,
What are expressions? What are equations?
An expression is a finite combination of mathematical symbols and numbers. Symbols can designate values (constants), variables, operations, relations.
An equation (derived from the word "equal") is a mathematical statement (mathematical symbols and numbers) that asserts the equality of two expressions. Equations consist of the expressions that have to be equal on opposite sides of an equal sign, as in

Equation_1.jpg
A variable is a symbol (alphabet like "x" and "y") that represents a unknown number or quantity in an algebraic expression.
The most common variable is x.
Expressions don't have an equal sign, whereas equations do.

Translating Word Problems into Math

You can use: Problem-Solving Word Problems Using Equations
One way to problem solve a word problem-
  • It is easy to have a cross-out list, to see whether any operation goes with the word problem. If one doesn’t go with the problem, cross that out and then do it for the rest. Finally we would be left with one or two. If you remain with one operation, you would check if that goes and if it does, it is going to be easy to find the solution.
  • Check using another method

__Word problems for expressions and equations:__
Sam is 5 years older than Mary. Mary is 7 years old. How old is Sam?
Let x= Sam's Age
Mary's age: 7

5+7=x
5+7=12

Check:
12-5=7

Answer: Sam is 12 years old.

Now doing it in a different way:
(After 2 years....)
Sam is 5 years older than Mary is. The sum of their ages is 23. How old are they?
Let x= Mary's Age
5+x= Sam's Age

x+5+x=23
2x+5=23
(2x-5)+5=23-5
2x = 18
x= 9
5+x= 14

Answer: Mary is 9 years old and Sam is 14 years old.

When to use different strategies: (proportions, graphs, equations, etc)

__Proportions:__ The relationship of two variables whose ratio is constant/Two equivalent ratios.
David got 5 colored markers that cost $3. How many colored markers cost $6?
(Set up as a proportion)
Proportion1.jpg
Proportions are typically used when you want to solve for an unknown, like the above. When you get two ratios and have to equal them.
You can solve percent problems and basic rate problems by proportions. Like also in a word problem when it prompts to find when both values are equal.

Graphs: A graph is an abstract representation of a set of objects connected by links, like charts, etc. Different kinds of graphs like bar graphs, line graphs, pie charts, etc. But for equations/finding the slope we use "x-y plots." (for a cool activity go onto this site: http://www.webmath.com/plot.html)
X-Y Plot Graphs can be used for:
  • Identifying relationships between large data sets
  • Identifying trends in large data sets
Graphs are used when you want to see the frequency of values, identify relations between many groups (small or large) sets of data and seeing patterns or trends in the data. They also help when you want to check whether there is an error or different points plotted, etc.
How to create a graph on Excel?
(http://office.microsoft.com/en-us/excel-help/creating-xy-scatter-and-line-charts-HA001054840.aspx)

Graphing Equation/Slope:
You can use mx+b when you want to find the slope of a line in a graph (x-y plot).
m is the growth factor/slope while b is the y-intercept.
y= how far up
x= how far along
m= slope or gradient (how steep the line is)
b= y-intercept (where the line crosses the y-axis)
mx+b.jpg
To find the x intercept:
Example Expression: 2x+1
(Suppose you found the y-intercept, but you want to find the x-intercept)
0= 2x+1
Then you solve it and get the result which is the x-intercept.

Using complete graphs/slope:
  • Straight or curved (x is linear while X2 is parabola)
  • Y-intercept
  • Increasing or decreasing


__Fractions__
A number that represents a part of a whole.
They are related to equations because there are equivalent fractions.
Ex:
Fraction.jpeg
Fraction.jpeg
equivalent_fractions.jpg
equivalent_fractions.jpg

__Multiplying and Dividing Fractions:__
Multiplication involves cross-multiplication, while dividing involves reciprocating.
Cross multiplication-given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable.
Reciprocal-A mathematical inverse or a multiplicative inverse between two ratios.
cross-multiply.gif fra_ex14.gif

__Measuring Angles__
As a matter of common practice and convenience, it is useful to measure angles in degrees, which are defined by partitioning one whole revolution into 360 equal parts, each of which is then called one degree. In this way, one whole revolution around the unit circle measures
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radians and also 360 degrees (or
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), that is:
Trigno.jpg



__Pythagorean Theorem:__
To solve problems using the pythagorean equation-
Pythagorean_Equation_1.jpg
And use the pythagorean triangle,
Pythagorean_Equation_5.jpg

Area and Circumference of A Circle:
circle-circumference.png
Circumference of a Circle
Circumference_of_Circle.jpg
Area of a Circle
AreaofCircle.jpg


Distributive Property:
To separate or break an expression or equation into parts.
Using equations:
Example:
DistributiveProperty.jpg





Additional Sources

Translating Word Problems into Math (Numeric form, etc)
Pythagorean Theorem
Problem Solving Strategies


Translating Word Problems into Equations
Fractions
Area and Circumference of a Circle

Proportions
Probablility
Distributive Property

Multiplying and dividing fractions and mixed numbers
Rounding and estimation
Measuring Angles

Citations

http://en.wikipedia.org/wiki/Expression_%28mathematics%29
http://en.wikipedia.org/wiki/Equation
http://en.wikipedia.org/wiki/Variable_%28mathematics%29
http://en.wikipedia.org/wiki/Proportionality
http://en.wikipedia.org/wiki/Graph_%28mathematics%29
Cool Activity- http://www.webmath.com/plot.html
http://www.corda.com/docsource/doc7/Manuals/graph_ref/x-y_graphs.htm
http://www.mathsisfun.com/equation_of_line.html
http://en.wikipedia.org/wiki/Cross-multiplication
http://en.wikipedia.org/wiki/Multiplicative_inverse
http://www.sosmath.com/trig/Trig1/trig1/trig1.html
Pythagorean Triangle (Equation)
Equivalent Fraction
Circle Circumference
Cross Multiplication
Reciprocal