Negative-Negative

What do you think will happen when you have a negative- negative?



You must build, draw, and then come up with an answer.

• -3 - (-7)
• -9 - (-4)
• -2 - (-5)
• -6 - (-1)
• -8 - (-7)
• -4 - (-7) 

Public Record: Change the #'s to the absolute value and find their difference. If the larger absolute value is first the answer is negative.

R-10 All the Subtraction Rules!

R-10 All the Subtraction Rules!

 (+)-(+)
Find the difference between two #s and if the # you are subtracting from is larger, answer is +.


 (+)-(-)
Add together the absolute value of the 2 numbers to get the answer.
-Justification: Why does this work?

* I you do not have negatives to take away, you have to zeroate so you do not change the net value, but you get the negatives you need.

(-)-(-)

Change the #'s to the absolute value and find their difference. If the larger absolute value is first the answer is negative. If the larger absolute value is second the answer is +. 
Justification: If you don't have enough (-) to take away add parts of zeros until you do. Then take away (-). Remaining is the answer.  

(-)-(+)
Add absolute value of #'s and convert answer to negative. 

The class made 2 Rules: 

Different Sign
1.  (-)-(+) and (+)-(-) Add the absolute value. If the first # is (-), make the answer negative.

Same Sign
2. (+)-(+) and (-)-(-) Find the difference of the absolute value if the bigger # is first, the answer is positive.
 

Positive -negative

8-7=1
7-8=-1

Difference between the numbers
1st numbers smaller=>-
1st numbers bigger =>+

10-(-3)
2-(-7)
3-(-10)
7-(-2)
9-(-4)
4-(-9)


Class Rule: 

What is the difference between subtracting and zeroation?

What is the difference between subtraction and zeroation/cancellation? 


4-7= 




















3-9=

















***The difference between each subtraction problem was how many zeros we added 

154-268=

268-154=

154+114=268
Now I have enough to take away. 

Subtracting Integers

R-67 Possible Subtraction Problems
(big -) - (small +)=
(-) - (-)=or (big -) - (big -)

Multiple numbers to subtract: 
(Small +) - (bigger +)=
(big +) - (small +)=

(+) - (-)=

positive - (opp)


L Page
(Small +) - (bigger +)=
(big +) - (small +)=

10-3=
2-7=
3-10=
7-2=
9-4=
4-9=

Adding Integers

Class definition for Zeroation:

-When +/- integers cancel each other out and doesn't affect the net value
-The chips being cancelled out have to have the same absolute value
-A way to solve a problem visually-doesn't affect answers.


The class made predictions:

What happens if you add:

2 positives
*Bigger number  +
*Greater absolute value

1 negative and 1 positive
*Maybe 0
*Depends if + or - has greater absolute value which ever is greater is the sign of the answer.
*First zeroation than leftovers is answer

2 negatives
*negative answer
*answer smaller
*absolute value bigger
(Answer further from 0)

Show with the red and black chips:
2+3=
-3+2=
5+-2=
-4+-5=
-4+9=
-10+3=
2+8=
-3+-7=

Let's make some generalizations. Could we write a rule?

1. The sign of the number that has the larger absolute value
2. Difference between those 2 positive numbers

Public Record for Adding

positive  plus positive
-Get a bigger positive

negative plus negative
-Get a bigger negative or smaller number or bigger absolute value. 

positive plus negative 
-convert both numbers to positive and find their difference. The answer ad the sign of the # with the longer absolute value. 

-4+3=>3+-4
             3-4

Commutative property-

Black and red chips

What do we think zeroation is?
-They cancel each other out

Hans's Zeroation-































The value of the bottom collection is -5. What happens after I do zeroation? There are a certain number of blacks and reds that get removed. It takes it away and has no effect on the net value.

L page

Write your own personal definition for zeroation?

Thermometer Problems

Thermometer
1. Figure out which is further from - 2 degrees F. Explain/Show how you know.
   a. 6 degrees F or -6 degrees F
   b. -7 degrees F or 3 degrees F
   c. 2 degrees F or -7 degrees F
   d. -10 degrees F or 7 degrees F

2. What temperature is halfway between the given two??
   a. 0 degrees F and 10 degrees F
   b. 5 degrees F and -15 degrees F
   c. -5 degrees F and 15 degrees F
   d. -8 degrees F and 8 degrees F

Number Line
1. Graph all the possible solutions for x using a number line.  Explain your thinking.
   a. x is positive
   b. x < -7
   c. 6 < x
   d. x is less than or equal to -5.
   e. x > or  = 5
   f. -1 < or = x

2. What are the values for x that make these sentence true??  Show using a number line.
   a. x + 5 > 0
   b. x - 1 < and = 0
   c. 3x < 9

Rational and Irrational Numbers

Based on the definitions from the book, how can we put the definition for a rational number in our own words?

What do we think it means?
-The answer to a division problem. A number that you divide by another number (can't be zero)  like 2/1 (the 1 is the divisor)
-A number that isn't 0
-maybe 0 is the mid part and rational numbers are positive numbers

Can we give some examples?
Class thinks:
1/2, 9/11, -.3, 3/4, -7/5, .799, 21, 1.3, 22

Are rational numbers positive or negative?

6/3 is rational then is 2 rational?

Maybe whole numbers are rational and decimals and fractions are irrational?

Idea: If we go by the first definition, then rational #=everything

Can we give an example of an irrational number?

We now think negatives can be rational.  -6/-3=rational -6/0=irrational

Now we know 2 is rational.

Can every single number be rational?
yes, can terminate and repeat

Can you think of an irrational number?
-π (it never ends, it never repeats) 

R-62 Rational Numbers 
Integer ÷ Integer (≠0)=Rational #

An integer is a whole number either positive or negative. 

Integer Models

R59-Private Think Time: When you think about positive and negative numbers what models come to mind?
-below 0
-temperature outside (thermometer)
-Number lines (0 is in the middle, right is positive, left is negative)
-Any negative number plus
(-5+5=0)
-  -5 and 5 is the same distance (units) from zero
-color strips (red was positive, black was negative)
-Distance from 0 can be shown as absolute value [5]
-Is a positive plus a negative a negative?
-When you add a negative is it subtracting?


Similarities and differences?



Private Think Time: It is 2 degrees outside. At sunset the weatherman predicts  a five degree drop in temperature. Create a number sentence that matches the problem.

PTT Math Jeopardy- If a team of students played this type of game and the Super Brains had a score of -300 and the Rocket Scientists had a score of 150 and the Know-It-Alls had a score of -500 make some observations.

-The rocket scientists have more points than Super brains or No It Alls. 
-Rocket Scientists are the only team  whose score could have been gotten in 1 round (1 question) 

Quiz Review

What do I expect will be on tomorrow's quiz? 

-Transversals: angle measures, what they are and why they have that relationships 
-Tiling-what shapes tile 
-Quadrilateral-what info. do I need to build one?
  * side lengths and angle measures 
  *Four sides doesn't make it unique (to reinforce it, will make it unique)
-Triangles-What info do I need to build?
  * side lengths and angle measures
 *3 pieces-3 sides, 1<2 sides or 1 side 2<'s
-Build and construct several shapes on this quiz
-Exterior and interior angles 
-Complimentary (2 angles make 90)
-Supplementary (2 angles make 180) 
-What is the difference between a regular and irregular polygon? 
  *Regular: every angle and side the same

How do you study for a math test? 
-Look back through your math notebook
-Give yourself problems, practice them (from your notebooks so you have the answers to check back) 
-If you forget your notebook, you can go on the blog. 

How long? 
- 20 to 60 min 
-If you don't have a lot of time focus on things you aren't as confident on. 
-Break up the time into sections 

Transversal 2

R-57 Transversals

Find all the angles and label them. You need to justify how you came up with every single angle.
Should have 15 math sentences to go with your idea.

One idea:
The middle shape is a parallelogram.
h=150
o=g

L-57 Transversal Definition

*A line that intersects two other lines

a,b,c
a=30
b=150
c=30

d,e,f
d=150
e=30
f=150

h,i,j
h=150
i=30
j=150

l,m,n
l=150
m=30
n=150

Justify
150+150=300
360-300=60/2=30

360 is the sum of all angles measured for a quadrilateral.

Justify
150+a=180
150+c=180
150+30+30+6=360

R-56 Transversals

R-56 Transversals

What do we notice?



















*Supplementary <'s-2 ,<'s add to 180

1. b and c make up line 1
2. a and d make up other sides of line 1
3. a and b make up line 2
4. d and c male up line 2

-all ,'s together=360 degrees
-Does A=C??
-Does B=D??
The class thinks they look the same. Someone thinks because the lines intercept
-If I moved one of the lines, they would still be the same.
-Line 1 is splitting like a circle in half, 180+180=360
-They look like interior <'s
-They make 4 different triangles or 2 different triangles. 

Class is wondering: Why does a=c and b=d? How can we prove/justify?

Line 1=180
Line 2=180

If b+c=180 and a +d=180
a+b=180 and c+d=180

a+b=180
a+d=180 so b=d

a+d=180
80+100=180

c+d=180
80+100=180

Quadrilaterals

Triangles Recap

-Sum of all angles= 180
-3 vertices
-3 sides
-regular angle =60 degees
-2 short sides have to be bigger than the longest side
-exterior sum of the angle is 360 degrees


Exploring Quadrilaterals



Side Lengths             Does it make a unique Quad.                  Sketch
1. 6, 10, 15, 15
2. 3,5,10,20
3. 8,8,10,10
4. 12,20,6,9

*Can't make a unique quad. because too flexible. 
*We decided to add a piece in the middle for stability (diagonal) 
*Creates triangles which are more stable shapes

Where do we see triangles in our everyday lives?
-Construction (saw: 2 ends are triangles, stable enough to hold pieces of wood)
-Buildings, roofs
-Tips of a saw, ridges

2 Big ideas about Quadrilaterals:

1) Can't make a unique quad. because too flexible. We decided to add a piece in the middle for stability (diagonal). It creates triangles which are more stable shapes

2)  3 smaller sides need to be greater then the biggest side

Text triangle posters

The class is determining how many pieces of information to give to the other 7th grade class. They tested the text messages by trying them out on their own.


L-55 Notes of text Presentations

Some ideas:
Amount of info given
4 pieces=3 sides, 1 angle
3 pieces=2 angles, 1 side or 3 sides or 2 sides, 1 angle
2 pieces= 1angle, 1 side

 Students are trying to prove how 2 angles and one side can work.

If the angles are different it will go up by a different amount

Math Problem Assignment

 Name of the problem (Hindu Dilemma, Minted Coins, Newspaper Ads)
1. Type your response (Shared Google Doc or word doc printed out)
2. Three parts to your response
       a. Answer the prompt or question (Draw visually and show work)
          -attach any paper w/drawings or math as needed
      b. Explain your strategy. What was your thinking? Think about your process of solving the       
      problem. (We want to know the messy stuff. Tell us everything you tried, your thought process)
         -Needs to be thorough
      c. How could you challenge yourself with this problem?
         -Is there another question I could ask? Is there another part of the problem I could solve?


Edit Problem Paper
1-Read Rubric on your Google Doc
2-Reread your problem
3-Assess your work by highlighting in yellow the correct space in the rubric. Please use yellow.
4-Edit your work.


**Ms. O'Toole is only assessing the edited work.
**Push yourself so that you can make generalizations and look for patterns and regularity.
**This is a finished project. Would be published in a math journal.

Triangles

We are going to make several polygons and record data.


In a table:


Make 5 triangles.


1. Side lengths (Between 2 and 20)
2. If a triangle could be created

3. Space to sketch the triangle

L-52

Rules for Knowing if 3 Side Lengths Will make a Triangle

**Two short sides added together must be longer than the longest side

Text Messages to send to 7A

-3 corners A, B, and C. A is at the top

-B is at the bottom left, C is at the bottom right.

-A's measure is 74 degrees

-B's angle measure is 60 degrees and C's angle measure us 46 degrees.

-Side AB is 3 cm. Side BC is 4 cm and side AC is 3.6 cm.

-Build this triangle: angle A(top)=74 degrees

-Angle B (bottom left)=60 degrees

-Angle C (Bottom right)= 64 degrees

-Line segment AB=3 cm, BC=4 cm, CA=3,6 cm

Could I minimize this data and still produce the triangle?

* Don't put anything about top, bottom, left, and right. Does it have to be in the same position as what is on the board?

Delaney and Caitlyn's Message

-AB=3 cm

-BC=4 cm

-AC=3.6 cm

and angle A=74 degrees

Gillian's Message

AB=3cm

BC=4 cm

AC=3.6 cm

A=74 degrees

B=60 degrees

C=46 degrees

        

Nick's Message

A=74 degrees

B=60 degrees

C=46 degrees

Hans's Message

AB=3cm

A=74 degrees

B=60 degrees

Noah's Message

AB= 3cm

A=74 degrees

C=46 degrees

Jack's Message

A=74 degrees

BC=4 cm


What is the least info we need to text to get the same triangle?

What is happening in the classroom?

What does it look like and sound like?



*Direct Instruction

     -Listen
     -Look at board/Elmo
     -No side conversations
     -Thinking quietly or ideas to remember
     -No doodling if distracting
     -No fidgeting
     -Jot down notes 



*Private think/write time




*Partner Work



*Plenary-whole class discussion


    -Don't blurt out
    -Write stuff down
    -Raise your hand-so you don't interrupt
    -Everyone should participate 
    -No side conversations
    -Ah..ha moments
    -Relevant MATH ideas
    -Ask genuine questions if you get stuck
    -Use specific language 




Teacher Role:
Plenary- Guiding: Information from the students, Select people to call on, record ideas, revoice ideas, keep on topic, ask questions/prompts 
Direct Instruction:Teacher is telling you stuff




*Homework
    -Use full ability
    -All done
    -Ask for help
    -Email Ms. O'Toole
    -Call a Friend
    -Hand in on time
    -Use Planner
    -Check Work
    -Read instructions carefully



*Small group work

    -Flexible- listen to others ideas
    -Only talking to people in your group
    -Be specific and complete
    -Don't interrupt
    -Talk equally (Mrs. O'Toole could assign A,B,C to rotate tall time)
    -Relatively quiet
    -Always do observations
    - The group needs you
    -Give them a job to boost confidence
    -Share equally all the work
    -explain/have conversation
    -Give examples
       -Work on individual papers
       -Ask for observations
       -Encourage them and show them steps
    
   

*Debate
     -Show/justify your thinking
     -Don't interrupt
     -Test all ideas--Keep open mind
     -Be respectful--we are just talking about ideas not people
     -Get everyone on board
     -Don't argue
     -Listen to understand what they were thinking
     -Ask for clarification
     -Stay on topic
     -Don't bring personal stuff in

The ins and outs of polygons

Interior angle: The angle inside a polygon formed by 2 adjacent sides of a polygon. (the angles inside the figure)
-Jacks ideas: (# of sides-2) x 180= sum of anterior angles 
-Delany's: (# of sides x 180) -(180x2)= sum of anterior angles 

Exterior angle: Are measured by extending a side of a convex polygon and measuring the angles that lie outside the line. 

-The sum of all exterior angle measures is 360. 

*What is the relationship between interior and exterior angles? 
-They are supplementary: If you add them together they equal 180. 


Label the exterior angles and the interior angles. 

Predict: 
What would be the sum of all the angles? 

Track problem: 
What is the sum total of those left turns? No math, make a reasoning as to what it is going to be. 

As there are more sides, the shapes become more circular.

Tiling

Tiling

You can use 2 shapes, but you can't have any overlap or gap.

Think about what you know for tessellations:

The shapes that tessellate make 360 degrees. The shapes that do not make more than 360 degrees.

Tesselate

We watched this video in class.


http://www.youtube.com/watch?v=X9lJhDLtFeM




The hexagons in a beehive all tessellate. What do we think the definition of tessellation is?

Which of these regular polygons can tessellate and which can not?

Why do certain polygons tessellate? Make 5 observations.

What do you notice this student did?

The shapes that tessellate make 360 degrees (or are factors). The shapes that do not make more than 360 degrees.

Irregular Polygon Observations and Rules

In your group, choose three strong observations. Then, make a rule that can be used for all irregular polygons to find the sum of all the angles.(Devon, Casey, Trevor ideas)

Devon

-angle measure=180

-quad measure=360

-added 180 each time

1. (# sides x 180)- 360   Guess and Check!

2. (# of sides -2) x 180   When you change some get smaller and bigger--total same. 

Trevor 

Casey

 

Finidng the Sum of all angle measures.

The Sum of all angle measures for polygons

Jack's: ( # of sides-2) x180= sum of angle measure



Delany's: (# of sides x 180) -(180 x 2) 



360/4= 90 (square) 
1440/10=144 (Decagon)

To find the single angle measure
Sum of the angle measures 

sum of the angle measures/# of sides= single angle measure 
 
1. Devon: Tore off the corners and put them together. 

2. Trevor:Tried to cut them up into equal pieces. Used the lines to create certain degrees and added them together. Maybe tried to cut the shapes into triangles to find the sum of the angles.

 3. Casey: Put a dot near the center of the shape and made them into triangles. Number of sides of the shape was the same as the #of triangles.

*Make as many observations as you can. Is there a relationship between the number of sides in the polygons and the angle measure?