9-30 Class notes and board shots

9-30 Class notes

Quiz on Monday, 10-3.  Some problems open notes; some problems from homework.

Homework for Monday, 10-3:  page 43; 9a, 9b and page 44; a, b, c.

Steps for graphing a one-variable inequality:

Simplify until you get the variable isolated on one side and a value isolated on the other side.
The value is the position on the number line.
Place an open dot at that position if there was no equal sign combined with the > or < symbol.
Place a closed dot if the inequality was > or < .
Shade a bar in the direction of the > (right) or < (left).

Position, type of dot, direction.

9-27 Solving multi-step equations

Answers to the homework from the weekend, page 37:

11a     -11/2
11b     10/8
11c     9/10
11d     5
12a     z = 0

In-class  problems and homework for 9-27 and 9-28:

page 37; items 12b, 12c
page 38; items 1, 2, 3, 4, 5, 6, 7, 8

9-23 Multi-step equations

There will be a homework quiz on Monday.  You may use your textbook and your notes to answer the questions.  The questions will be taken from the class notes, the homework, and the blog.

Answers for today's homework: 
10a.  x = -(11/15)
10b.  x = 8
10c.  no solution

Homework for Monday 9-26:  page 37; 11a, 11b, 11c, 11d, 12a

Hint for 12c:  Try to get a common denominator or try to clear the denominators (get rid of fractions)

After you simplify an equation, you may get to a statement that doesn't make sense.  If you have done the proper undoing, balancing, and operating equally on both sides, and you get a statement that is false,
such as 0 = 5, it means there is no solution to the equation.  There is nothing you can put in for x that will make the equation true.

If you simplify an equation and end up with a statement that is always true regardless of x, such as
5 = 5 or 2x = 2x, means the equation has infinite solutions.  No matter what you put in for x, the statement will be true.

Class notes 9-21 profit, other buttons, field trip

Homework for 9-22 is page 35, item 9, frequent visitor program



Breakeven 9-20 class notes

In the button making story, we found that when you order 200 buttons, neither company gives you a better deal.  200 buttons cost $155 whether you get them from PB or from BFY.  If you get one more button over 200 from PB, you pay 15 cents more for a total of $155.15.  If you get one more button over 200 from BFY, you pay 40 cents more for a total of $155.40.  So, for an order greater than 200 buttons, PB gives you an better deal, and that is what you recommend.

In the bellringer we solved

                                       200x = 12500 +15x

The money you get from selling x buttons at 200 cents each = the cost of buying those buttons from PB.

                              x = 68

68 is the number of buttons that makes the money coming in equal the money going out.

                             $136 = $136

68 is the breakeven number of buttons.

If you sell more than the breakeven number, you make money because revenue > cost.
For 80 buttons, 200(80) = $160 and 12500 + 15(80) = $137, so you make 160 - 137 = $23

If you sell fewer than the breakeven number, you lose money because cost > revenue.

For 40 buttons, 200(40) = $80 and 12500 + 15(40) = $131, so you make 90 - 131 = negative $41 (loss)

Profit is the money you make, calculated by subtracting cost from revenue.

Homework for 9-21 is page 34, items 6 and 7.

Homework for 9-19 page 32 items 1, 2, 3, 4, 5, 6

Homework for 9-19 page 32 items 1, 2, 3, 4, 5, 6

Use any method, show your work.  You solved some of these problems in class.

Homework and class notes

Homework for Friday 9-16:  From page 31, items b, c, d, e, f, and page 32, items a, b, c, d, e, f, choose three equations to solve.  Be prepared to show your work and explain your answer in class.

Today we worked an example of solving an equation with pictures of square and rectangle shapes ("tiles").  Pairs of positive and negative tiles were erased to simplify the picture.  In our example, one rectangle (t) equals seven squares (7).

Today students presented their small dry-erase solutions to equations.

Class notes 9-14 "UNDOing" to solve equations

Undoing an equation means using inverse operations to get back to the variable (mystery number, Who Am I?)
Look at the way an equation was built.  The last operation that built the equation is the first operation to be undone.  In most cases this means using PEMDAS in the other direction.  We use SADMRP to remember this.

Work from outer to inner, towards the grouping symbols.
First, look for Subtractions and Additions to undo. Use addition and subtraction, the inverses.
Next, look for Divisions and Multiplications to undo. Use multiplication and division, the inverses.
Next, look for Exponents to undo.  We use "R" in SADMRP because the inverse of exponentiation is Root (taking the nth root, finding the nth root).
P means it's time to work inside the Parentheses (grouping symbols).

Homework is page 29, items b, d, f.
Be prepared to talk about the steps of UNDOing, similar to the right-to-left arrows and ovals on page 29.

Homework for 9-14

Homework for 9-14:  Springboard textbook, pages 27 and 28, items 4, 5, 6, 7

Class notes 9-13

Building an equation from an unknown number ("me", "what number am I?") uses the words closest to the number to suggest the first operations and grouping.  Each word phrase adds another operation or grouping.  Finally, you translate "am I" or "I am" into an equal sign.


Solving an equation for an unknown number looks at the operation farthest away from the number so you can undo it.  Remember that when you undo something, such as when typing at a computer, you affect the last thing you did.  When you undo in math, you affect the last operation used to build the equation.


To undo a math operation, you use its inverse.  Multiplication and division are inverses. Addition and subtraction are inverses.  If the equation's last piece was adding 3 (" + 3 ") to undo it so you can make the equation simpler you should subtract 3 (" - 3") from both sides.


To evaluate an expression or to build an equation, you use the order of operations PEMDAS.


To solve an equation, you use the order of undoing operations SADMRP.
We use R instead of E because Root is the inverse of Exponent.  If raising to a power/raising to an exponent was used to build an equation, tkaing the root/finding the root is used to undo the equation

Class notes 9-08 and 9-12

Prepare for a quiz on 9-14

Wednesday's quiz will be on multiple representations of data.  We have studied tables, graphs, and expressions as ways of representing information.  You will demonstrate your ability to create different representations from an initial set of numbers, and you will be able to convert from table, to graph, to an expression and vice versa.

You will be able to see the pattern in a table of numbers.

You will be able to write an expression for a pattern of numbers.

You will be able to draw a graph from a table.

You will be able to make a table from a graph.

You will be able to write the linear equation of a straight graph.

You will be able to draw a graph from a linear equation.

You will be able to discuss differences between shapes of graphs.

After the 9-8 quiz

Given the number 23, write an equation which relates it to its reciprocal and the identity element for multiplication.

The number is 23, the reciprocal is 1/23, the identity element for multiplication is 1, because 23 * 1 = 23, the identical number.
.
23 times 1/23 is 1
23 * (1/23) = 1
1/23 is the reciprocal or multiplicative inverse of 23



Given the number 23, write an equation which relates it to its opposite and the identity element for addition.

The number is 23, the opposite is -23, the identity element for addition is 0, because 23 + 0 = 23, the identical number.

23 added to -23 is zero
23 + (-23) = 0
-23 is the opposite or additive inverse of 23

Algebra Properties 9-7-2011

Transitive - If a = b and b = c, then a = c
If two things are equal to a middle thing, they are equal to each other.
Substitute a for b into b = c.  This gives a = c.

Reflexive - A thing is equal to itself
a = a

Symmetric - You can write an equality in either order
if a = b then b = a




Identity elements -

0 is what you add to a number to get the identical number

1 is what you multiply by a number to get the identical number

To get from a number to the identity element for addition, add the opposite of the number
To get from a number to zero, add the negative of the number
To get from 7 to 0, add -7
7 + (-7) = 0

To get from a number to the identity element for multiplication, multiply by the reciprocal of the number
To get from a number to 1, multiply by the reciprocal of the number
To get from 7 to 1, multiply by 1/7
7 * (1/7) = 1

Study hints for Math 9 Quiz 9-8

Math 9 Quiz Thursday 9-8     - Study hints

Given some numbers, tell which numbers are in a category.

Which numbers are irrational?

2      pi      2.22     2/4     square root of 2    

Which numbers are rational?

2      pi      2.22     2/4     square root of 2

Which numbers are whole?

2      -2     2/4     -4/2      6/2

Be able to define real numbers with words or with a diagram

.

Be able to place numbers correctly into an N-W-I-R / Irrational and Real diagram

Be able to give an example of an algebra property (Commutative, Associative, Distributive, Reflexive, Symmetric, Transitive)

Be able to place parentheses properly into an Associative equation.

From a list of examples, be able to identify an algebra property (Commutative, Associative, Distributive, Reflexive, Symmetric, Transitive)

Explain the difference between the identity element for addition and the identity element for multiplication.

Given a number, write its opposite and its reciprocal.

Identify or give an example of the Reflexive, Symmetric, and Transitive Properties of Equality

Use Substitution to explain the Transitive Property.

Inversesand identity elements

Identity means "that which stays the same".  If you move, or grow, many things about you stay the same even though you address or your height may change.  Those things stay identical.

If you get something that's identical, you get something that's the same.  An extra picture can be identical to a first picture.

In math an identity element is a number that, when used with a stated operation, gets you the same thing.

Zero is the identity element for addition because when you add 0 to any number, you get the same thing:
97 + 0 = 97     and      2z + 0 = 2z

One is the identity element for multiplication becasue when you multiply any number by 1, you get the same thing:     86 times 1 = 86     and     7b times 1 = 7b

An additive inverse, when added to a first number, gets you 0, the identity element for addition
4 plus -4 = 0, so -4 is the additive inverse of 4

A multiplicative inverse, when multiplied by a first number, gets you 1, the identity element for multiplication
4 times  1/4 = 1, so 1/4 is the multiplicative inverse of 4

An additive inverse is a also an opposite, since you change sign to get it
4 becomes its additive inverse, or opposite, by changing sign to -4

A multiplicative inverse is also a reciprocal,since you flip a numerator and a denominator to get it
4 becomes its multiplicative inverse, or reciprocal by flipping from 4/1 to 1/4

Commutative, Associative and Distributive Properties 9-1

Commutative, Associative and Distributive Properties for addition and multiplication.
(not for subtraction and division)

a + b   =   b + a
a times b   =   b times a

When you add or multiply two things, you can do them in either order.


(a + b) + c   =   a + (b + c)
(a x b) x c   =   a x (b x c)

When you add or multiply three things, you can start with the first two or the last two.


a times ( b + c )   =   a times b   +   a times c
a x ( b + c )   =   a x b   +   a x c

When you multiply a sum, the multiplication takes place with each of the things in the sum.


Homework for 9-6:  page 20-21;  items 7, 8, 9, 10

Three inequalities form a polygonal region 9-1