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