algebra: linear functions, solving formulas, solving radical equations, solving rational equations...
TRANSCRIPT
Algebra:Linear Functions, Solving Formulas,
Solving Radical Equations,Solving Rational Equations
ELEMENTARY LEVEL
Session #1 Presented by: Dr. Del Ferster
Immaculata Week 2015July 27—July 31, 2015
Sometimes, we need to be inspired
How do we present the idea of function in the elementary grades?
At what point in the elementary curriculum are students introduced to the idea of a “variable” or “letter”?
How soon is too soon to introduce elementary students to “graphing technologies (apps, programs, graphing calculators)?
Some questions to get us started
We’ll be considering the idea of linear function.◦The key idea here is that each input corresponds to an accompanying output.
Will take a look at manipulating formulas.◦This is a lot like solving equations, but they have more letters and fewer numbers.
◦Some of the formulas have a nice science application, too.
What’s in store for today?
We’ll consider radical equations◦You remember square roots, don’t you?
◦Radical equations contain a square root (or maybe something even uglier.
◦We’ll also encounter the notion of EXTRANEOUS SOLUTIONS
What’s in store for today?
We’ll wrap it up by solving some rational equations.◦A rational equation is simply an equation that involves fractions.
◦Again, we must be aware of extraneous solutions, in some cases.
What’s in store for today?
Linear FunctionsAlthough our focus will stay on LINEAR functions, these ideas are easily extended to more complex functions
At first thought, you might say, Del, we don’t really deal with the idea of functions at the elementary level, but I might point out, that functions are a natural extension of one thing that I’m sure that you consider in the elementary grades—Patterns, and predictions.
Sure, you might not get too bogged down with the notion of VARIABLES, and words like DOMAIN and RANGE, but I feel that one of the most powerful things that we can accomplish in early math instruction, is the idea of pattern.
First, a word from your presenter
Conor loves building with LEGO blocks. First he made a tower 2 blocks high. Next to it he made a tower 4 blocks high. Next to those he made a tower 6 blocks high. If Conor continues this pattern, how many blocks in all will he have used after he has completed a tower 10 blocks high?
Go ahead, get an answer.
Let’s look at a problem
30
Conor loves building with LEGO blocks. First he made a tower 2 blocks high. Next to it he made a tower 4 blocks high. Next to those he made a tower 6 blocks high. If Conor continues this pattern, how many blocks in all will he have used after he has completed a tower 10 blocks high?
How might an elementary student solve this problem?
By using actual blocks?By making a chart?By creating a
rule that compares blocks used in each tower?
Suppose we let B= the number of blocks in the tower
And suppose we let n = the tower number (tower 1, tower 2, etc.)
Our function would be…
Of course, we want students to work comfortably with both letters.
How might we approach that problem using a function?
2B n
One way to introduce young students to the idea of a function is to have them “guess” the rule that “turns” one number into another number.
For instance…what rule would turn
Functions: The Guess My Rule game
INTO
3 75 110 1
50 101100 201
Double it, then add 1
This is a nice visual representation that aids students to understand that a function consists of an INPUT, some RULE, and a subsequent OUTPUT.
The Function Machine
Consider the function machine below. For the function named what happens when the numbers 0, 1, 3, 4, and 6 are input?
Examples: The Function Machine
f
x
( )f x
30add0 301 313 334 346 36
x ( )f x
Jimmy Johnson and Kyle Busch are having fun driving in a desert baja race. Jimmy has a fast car, and arguably is the better driver, so he gives Kyle a 60 mile head start. After t hours, Kyle’s distance from the starting line is given by the formula and Jimmy’s distance from the starting line is given by the formula
A. Determine each racer’s distance from the starting line at
Let’s consider this problem
5t
8 60K t
10J t
Jimmy Johnson and Kyle Busch are having fun driving in a desert baja race. Jimmy has a fast car, and arguably is the better driver, so he gives Kyle a 60 mile head start. After t hours, Kyle’s distance from the starting line is given by the formula and Jimmy’s distance from the starting line is given by the formula
B. Determine each racer’s distance from the starting line at
Let’s consider this problem
8 60K t
10J t
10t
Jimmy Johnson and Kyle Busch are having fun driving in a desert baja race. Jimmy has a fast car, and arguably is the better driver, so he gives Kyle a 60 mile head start. After t hours, Kyle’s distance from the starting line is given by the formula and Jimmy’s distance from the starting line is given by the formula
C. How far ahead of Jimmy is Kyle after hours?
Let’s consider this problem
8 60K t
10J t
t
Jimmy Johnson and Kyle Busch are having fun driving in a desert baja race. Jimmy has a fast car, and arguably is the better driver, so he gives Kyle a 60 mile head start. After t hours, Kyle’s distance from the starting line is given by the formula and Jimmy’s distance from the starting line is given by the formula
D. Does Jimmy ever catch Kyle? If so, when?
Let’s consider this problem
8 60K t
10J t
Again, it’s all about how one thing compares to another.
Don’t get too hung up on math words like domain, range, dependent variable, independent variable, etc.
Have your students practice writing “rules” for problems that you create
Experiment with Function machines
Wrapping up functions, for now
Manipulating Formulas
Like solving equations, but with more letters!
It is often necessary to rewrite a formula so that it is solved for one of the variables.
This is accomplished by isolating the designated variable on one side of the equal sign.
It’s worth noting, that this process involves the same properties that we’ve always used when solving equations.
Solving Formulas
If needed, multiply to clear fractions. Use distributive property to remove
grouping symbols. Combine like terms to simplify each side. Get all terms containing specified variable
on the same side, other terms on the opposite side.
Isolate the specified variable.
The procedure for isolating a given letter of a formula
It can get messy!!To solve a formula, you do what you've done all along to solve equations, except that, due to all the variables, you won't necessarily be able to simplify your answers as much as you're used to doing
Solving formulas (continued)
Let’s look at some examples
m m
Fa
m
F m a
Fa
m
F is given in terms
of m and a.
a is given in terms of
F and m.
Example 1: Solve for a F m a
A is given in terms
of b and h.
1
2A b h
1(2) (2)
2A b h
2A b h b b
2Ah
b
2Ah
b
h is given in terms of A and b.
Example 2: Solve for h
V is given in terms
of r and h.
21
3V r h
21(3) (3)
3V r h
23V r hr2
2
3Vh
r 2
3Vh
r
h is given in terms
of V and r.
r2
Example 3: Solve for h
T Pr
P t
T P
rPt
Example 4: Solve for rT P P r t
T P P r t
P tP t
1
TP
r t
1
TP
r t
Example 5: Solve for PP P rT t
P(1 )T r t (1 )r t (1 )r t
The area A of a triangle is given by the formula A = ½bh where b is the base and h is the height.
h
b
Use formula above to find the height of the triangle shown, which has a base of 5 cm and an area of 50 cm.
b.
Solve the formula for the height h.a.
2Ah
b
20 .h cm
5
Celsius vs. Fahrenheit Celsius: used in most countries around the world.
Fahrenheit:used in the USA only.
Celsius and Fahrenheit are different scales used to measure temperature.
You are visiting a foreign country over the weekend. A local website tells you that the weather will be 27⁰C on Saturday and Sunday. •Should you pack a Packers sweatshirt or should you pack a bathing suit for your trip? 9
325
C F
Our solution
932
5C F
5 9 32C F
5 9 288C F
5 288 9C F
5 288
9
CF
5 27 288
9F
135 288
9F
Better Bring this !!
Our solution (continued)
135 288
9F
423
9F
47oF
TOO COLD FOR
Solving Radical Equations
Time to get RADICAL !!!!
To solve Radical Equations (equations that contain one or more radical), we’ll make use of INVERSE OPERATIONS.
While we might have radical equations that contain cube roots, or fourth roots, etc., the emphasis on this topic for our purposes will be restricted to SQUARE ROOTS.
First, a word from your instructor
How would you solve the following equation?
Solve by taking the square root of both sides, why?
Square roots and Squaring are inverse operations… they “undo” each other!
Using Inverse Operations
2 9x
How would might we approach this one?
If we use the idea of inverse operations, wouldn’t we SQUARE BOTH SIDES?
Using Inverse Operations
5x
2 2
5x 25x
Inverse of Multiply is ____________ Inverse of Add is ____________ Inverse of Divide is ____________ Inverse of Subtract is ____________ Inverse of squaring is ____________ Inverse of taking the square root is ____________
A Refresher on Inverse Operations
RADICAL EQUATIONS
A radical equation is an equation with a radical in it.
204 x
Isolate the radical – get the radical on one side everything else to the other side.◦Note: when the problems get a bit more
complex, this step might have to be repeated.
Square both sides of the equation Solve for x CHECK YOUR ANSWER
◦The process of squaring both sides may introduce EXTRANEOUS SOLUTIONS—solutions that DO NOT solve the original equation.
A Procedure for Solving Radical Equations
Solve for x
Example #1
181326 x2 13 3x 2 13 9x
2 22x 11x
OUR CHECK
6 2(11) 13 18
6 22 13 18
6 9 18
18 18
So, our solution is a good one and
11x
Example #2
3 5 5 15x
5 5 5x 5 5 25x
5 20x
OUR CHECK
3 5(4) 5 15
3 20 5 15
3 25 15
15 15
So, our solution is a good one and
4x
2 5 5 5 5 5 15x x
4x
Solve for x
Example #3
2 24 16x x x 4 16 0x
4 16x 4x
OUR CHECK
24 4 4 16 4
16 16 16 4
16 4
4 4
So, our solution is doesn’t check—IT’S EXTRANEOUS
NO SOLUTION
2 4 16x x x
Solve for x
Example #4
2 4 12x x 2 4 12 0x x
6 2 0x x
4 12x x
Solve for x
6 0 or 2 0 x x
6 or 2x x
Checking our solutions
OUR CHECK
6 4 6 12
6 24 12
6 36
6 6
4 12x x
-2 is extraneous, so our solution is
6 x 2x 2 4 2 12
2 8 12
2 4
2 2
6 x
A graphical approachSolving Radical Equations
via a graphing utility
As it turns out, graphing technologies allow students to solve many “difficult” equations (including radical equations) without a lot of algebraic knowledge.
Don’t get me wrong, I think that knowing how is important, but you might benefit from this look, too.
Suppose that we want to solve this equation
Let’s take a minute and explore
2 5 8x
Let’s take a look at this one via a graphing calculator.
We’ll put 2 graphs on the screen
Solution via use of technology
2 5 8x
1
2
2 5
8
y x
y
Our solution to the equation is the x coordinate of the point of intersection
11x
Of course you can always use your trusty graphing calculator.
If you are an ANDROID user, you might like
WABBITEMU
If you want something good for your ipad, you might enjoy:QUICK GRAPHFREE GRAPHING CALCULATOR
Some good graphing utilities that you might find helpful
Solving Rational Equations
Don’t be a fraction hater!!
RATIONAL EQUATIONS are equations that contain FRACTIONS.◦Yes, it’s true, we’re going to allow fractions in our equations.
You’ve met the idea when you solved proportions.◦So, yes, the idea of “cross multiplying” works quite well if we have an equation that says one fraction is equal to another fraction.
Rational Equations (an overview)
The tricky rational equations will have variables on the denominators.◦This introduces the possibility of EXTRANEOUS SOLTUIONS, again, because we must make sure that the bottom of any fraction isn’t ZERO.
Rational Equations (an overview)
Is it a proportion?◦If so, just cross multiply and proceed to solve.
◦Some solutions will be relatively easy, while others might involve factoring.
◦In any case, be aware of EXTRANEOUS SOLUTIONS
A Procedure for Solving Rational Equations
If the equation is more “cluttered”.◦Find a LEAST COMMON DENOMINATOR for all of
the fractions. If needed, look to factor the given denominators,
first.◦Eliminate all of the fractions by multiplying both
sides of the equation by your LCD.◦Now, cool things happen…all of the fractions are
eliminated, and we’re left to solve the remaining equation. As before, this can be easy, or might be more
complicated. Also, as before, we must be on the look out for
EXTRANEOUS SOLUTIONS.
A Procedure for Solving Rational Equations
Solve for x
Example #1
7 6
2 5x x
7 5 6 2x x
7 35 6 12x x 47x
does 47 make
any of the denominators
equal to 0
NOPE !!
So, our solution is a good one and
47x
Solve for x
Example #2
2
3 1
4 4x x x
21 4 3 4x x x 2 4 3 12x x x
2 12 0x x
4 3 0x x
4 0
3 0
x
or
x
4 3x or x
Checking our solutions
OUR CHECK
4 make
any of the denominators
equal to 0?
does
YES
-4 is extraneous, so our solution is
4 x 3x
3 x
2
3 1
4 4x x x
3 make
any of the denominators
equal to 0?
!
does
NOPE
Solve for x
Example #3
3 1 12
2x x
2LCD x
3 1 122 2
2x x
x x
3 1 122 2 2
2x x xx x
6 24x
18x
18x
Checking our solution
OUR CHECK
18 make
any of the denominators
equal to 0?
!
does
NOPE
so our solution is
18 x
18x
3 1 12
2x x
Solve for x
Example #4
2
3 2 61
2 4
x
x x
FACTOR FIRST
3 2 6
12 2 2
x
x x x
2 2LCD x x
#4 Continued
3 2 6
2 2 2 2 12 2 2
xx x x x
x x x
3 2 62 2 2 2 2 2 1
2 2 2
xx x x x x x
x x x
2 3 2 6 2 2x x x x 2 23 2 6 4 6 2 2 4x x x x x x
#4 Continued
2 23 4 4 2x x x 22 4 6 0x x
2 2 3 0x x 3 1 0x x
2 23 2 6 4 6 2 2 4x x x x x x
3 or 1x x
Checking our solutions
OUR CHECK
3 make
any of the denominators
equal to 0?
does
NOPE
Both Solutions are GOOD
3 x 1x
3,1
1 make
any of the denominators
equal to 0?
!
does
NOPE
2
3 2 61
2 4
x
x x
Here’s a cool picture of Morgan at the beach!!
THAT WAS A LOT OF WORK!!!
Questions or Concerns?
Next time we’ll be looking at:
1. We’ll be working on quadratic functions
2. We’ll explore ways to solve quadratic equations that have real solutions.
Looking Ahead