What is the meaning of equality?

TMWYK: about solving for x


-Write a proportion
-Finding a missing # in an equation or proportion
-comp test, last problem
-You can break it down
-Do problem backwards
-Can use PEMDAS
-Guess and check
-  2 =   1
   4       2
-equal the same thing, have the same value
-2+4=3+3

Equation

R and L 64 T-shirt problem

Walkathon T-shirt problem

Given these equations, what do you think the C and n could stand for? Could you make up a problem? 

Mighty T

C= 49 + n
C=cost
n=# of students 
49 to create a design (add on)


No-Shrink T

C=4.5n
C=cost
n=number of students 
No add on, hits 0 axis


R/L 63
Here is the dilemma. I have $120. Which place should I go to buy the maximum amount of t-shirts. 

-Need two ways to justify your thinking

TMWYK: About the add on

*extra in linear problems
*not indep. or depend variable
*Found when indep (x) variable=0
*Intersecting y-axis for add-on?
*Called y-intercept because its where the line or what your graphing crosses the y-axis.
-It's the first point in the string of data

*The add on is where the intercept is

Positive and Negative Constant of Proportionality

+                          
*Add on at lowest point
*Dependent variable goes up



-
* Add on at highest point
*Dependent variable goes down

Henri and Emile

L-57 The Walking Race

Henri (younger): 1 meter per second     gets a 45 meter start   (1.5 meters slower)
Emile (older): 2.5 meters per second  (1.5 meters faster)

We want Henri to win, but not so obvious
How long/far should the race be?

*Tied at 30 seconds or 75 meters
(Race needs to be less than 75 meters)

Other Representations
-Tables
-Graphs
-Equations
-Models
-Rules

Graph
Independent Variable- Seconds
Dependent Variable- Meters

Equations
Henri    1s + 45
Emile    2.5s+0

Weeks and Money

 

Make 2 of the most relevant observations from this table. 

How is this table the same or different from the previous tables we have seen? 

-Goes down

-at 0 it is 144

L-57

144-w(12)=$

Which is the independent variable? Which is the dependent variable? 

Independent-

Dependent-

Constant of proportionality- 12

The 144 add on thing: This is the starter for the problem, It's the base. 

-Could be the head start like emile and Henri

***The add on/start is at week 0 

Where can I see the add on in this graph? 

-The highest point of the line

-The starting point

How can I find the constant of proportionality on the graph? 

-Every time is goes down by 12. 

-If you start at one point you can count the amount of spaces you go down. (The spaces between the point)

-The furthest point on the x-axis 

L-58   

What is the same and different about linear relationships that have positive and negative constant of proportionality? 

Pledge Problems

Plan A- $10 regardless of miles


Plan B- $2 per mile


Plan C-$5 plus $.50 per mile


Which plan is better? Why? Give three justifications (graph, table, equation)


Equations-

Plan A- $10      

Plan B- 2x=n   (x=miles, n=$)

Plan C- Miles x .50= $


What characteristics will be in all linear equations?
 -a constant of proportionality (multiply it by the independent variable)
-An equations that works for everything in the pattern (pos, neg, even, odd)
- Dependent variable = (indep. variable) (constant of proportionality)
-Amount $= (miles walked) x (4)
-unit rate***

Class is not entirely firm on these ideas.

Linear vs. Non-Linear and homework

R-56
Make observations on rules and graphs

Linear vs. No-linear

Linear                          Non-Linear
Straight line                 have two separate rules
Either pos or neg         negatives
up by a constant          doesn't go up by the same amount
have n                          have n















***The pattern of change is the constant of proportionality. How much something is going up by or down by. 


Homework: 7b pg 16 #3-5, 20

Graphing Patterns

Graphing Patterns
-example: x # of figures, y net value
-If it keeps going at the same rate, it will be straight.
-You need points to show where the data is
-You can use the rate to find the distance in between
-x comes first

Independent and Dependent Variables

What are the dependent and independent variables in the pattern problems?

**Independent: Doesn't change
**Dependent: depends on the independent

Independent- n or the figure #
Dependent-net value  (total neg + pos)


What kind of graphs are we going to make?

R-50 Patterns

Draw what the fourth and fifth one would look like.
1. n x 3 + 1 or n+n+n+1

**This is linear because it is going up by a constant rate of 3

2. n x 2 + -1  or n x 2-1

Dark is negative, Grey is positive

**This is linear because it is going up by a constant rate of 2

3. (n x 2) + (n x -2) +-4 or n x 0-4  or n +-n+-4

Dark is negative, grey is positive

**This is linear. It make a straight line and is going up by a constant rate of 0. The x changes, but the y remains -4.

4.even=0
odd=-1

**Not linear

5.(n x n) +1 (for negatives)
(nx-n)+-1 (for positives)

(nx2) + ((nx-n)+-1))

**Not Linear

6.Odds n+-4
Even n+4

nx2+1

**Not linear

Linear Relationships

Walking Experiment Outside-
Students were given a stop watch and meter stick and were expected to record times for walking 10 meters.

Represent your data:
Graphs
Tables
Equations
Unit rates
ratios
Proportions

**Our Best Guesses
Tell us everything you know about Linear Relationships. What generalizations can you make?

-Constant of proportionality in graph/table
-Looks like a table/graphed in a unit rate
-2 numbers in a ratio
-Table/graph should go up by a steady rate
-Graph-straight line
-Dependent and Independent variable
-Should be able to find a constant someplace

Baby and Adult Chimps

Baby Chimps
40% high fiber
60% high protein

Adult Chimps
60% high fiber
40% high protein

Tell us everything you know about this information? How can you represent it?

Tables
Graphs
Proportions
ratios


Equations: Which one works? Come up with a mathematical justification.

P=protein
F=fiber 

Baby Chimps
P/1.5=F     
F x 1.5 =P

Adult Chimps 
P x 1.5=F


L-44 New problem

The zoo keeper is making a batch of baby chimp food. He decides the food will have a total of 125 scoops. How many of those scoops will be high fiber? 
-You must use a proportion to solve the problem.

40 = x 
10    40

125   =  x  
100       40

Does this work?
40  =  50   (40 + 10=1/4 of 40)
60      75  (60 + 15=1/4 of 60)

Proportions

*Something to something else ratio
*Help you compare
*Need three pieces of info.
*Different ways to set one up
*Find a scale factor
*Use them to find a %
*Helps find mystery value

R-39 Percents

Core Math Goal: How can you use proportions to find percentages of a value when you know a certain percentage of the same value?

TMWYK: Percents


-Similar to fractions
-Show a fraction-25%=1/4
-multiply/divide turn % into decimal part of a #
-Out of 100 26/100=26%

Constant of Proportionality
-The other number in a unit rate that is not the 1
-The amount the thing goes up or down by each time
43:1

R-40
Sweater cost $36
7% sales tax

How much will the sweater cost?

Digital Camera
$249.99
4% sales tax

How much does the sweater cost?

R-34

Core Math Goal:
How can you find a unit rate/constant of proportionality in a description, an equation, a table, or graph?

Tell me what you know:
About "reading" equations

Unit rate

3/26/15 Core math goal: how can we find unit rate/ constant of proportionality in a description equation, table, graph? 

TMWYK:

Class ideas 

-at the store we see unit rate

-amount of stuff ($)/ per one 

-example $25/1hr 

-rate if single unit out of multiple units 

-ratio w/ 1 in it 

-comparing something to a whole 

There is a sale!  10 oranges for $2. Tell us everything you know about this.

3/27/15
Core math goal:

TMWYK:

*unit rate (ratio out of 1)
*chart
*graph
*equation
*ratio
*fraction
*%
*table
*decimal
*description

Mx5=O  (M=money, O=orange)
*Noticed, for every 1 dollar there were 5 oranges.

Ratios

Scale factor

What do you think scale factor means and how do we use it in math?

Corresponding Sides and Corresponding Angles

 

Corresponding Sides:

Corresponding Sides: 

BA and ED

BC and EF

AC and DF

Corresponding Angles: 

B and E

A and D

C and F

 

Corresponding Sides:

HG and LK

GI and KJ

HI and LJ

Corresponding Angles:

H and L

G and K

I and J 

Homework: L15 pg 20 # 18 and 19

Multiplication Observations and Ideas

R-11 Multiplication Generalizations 


edge x edge= array

Red (-) x Red (-)= Black (+)  Double flip

Black (+) x Black (+)= Black (+)  No Flips at all

Black (+) x Red (-) = (-) You only flip it once to red or negative

Red (-) x (+) = Red (-)You only flip once to red or negative

Multiplication arrays

The net value is 12.

The net value is 6. The red column and the black next to it zeroate.

 

The net value is -4. The bottom black row and red row next to is zeroates. 

** Students were given a chart of multiplication arrays and built them. 

* Black edge: row or column not flipped

*Red edge: row or column flipped

*2 reds: need to flip twice=black

R-7 Every Subtraction problem can be rewritten as an addition problem

R-7 Every Subtraction problem can be rewritten as an addition problem

Is this true or false?

 Can you give examples?

1st idea
7-3=4
2+2=4

2nd idea
3-8=-5
3+8=11

3rd idea
3+4=7 Fact families
7-4=3

4th idea
7+(-3)=4    Using same numbers , but changing the sign of the second one.

How can you rewrite -4-(-4) as an addition problem?

Homework: Write 4 problems that prove each subtraction rule.
Comp. Q corrections due Wed 1/14

R-6 Our final Rules for negatives

(+)-(+) and (-)-(-)

Find the absolute value of the two numbers. If the first number is larger, the answer is positive. If the 1st number is smaller, answer is negative.

(+)- (-) and (-) - (+)

Add the absolute values, if the first number is negative the answer is negative. If the first number is positive the answer is positive.

Write 4 problems that prove each subtraction rule.

Posters- 2 Subtraction rules