Positive-negative

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

You must build before you draw!!!

What is the rule for a positive - a negative? What is the justification?  

Class Ideas: 
* Add the two absolute values
*Why: Because you have to add zeros in order to take away negative
*Rewrite the problem into an addition problem.

R-10 All Subtraction Rules

(+) - (+)
Find the difference between the 2 numbers and  if the number you are subtracting from is larger and answer is (+).

(-) -(+)
Add the absolute values of 2 #'s make answer negative.
Justification: Because you add the same # of zero's as the second # because there are no (+) to take away. After you take (+) away, you total (-) and increase by the amount of the second #. 

(+)-(-)

* Add the two absolute values
*Why: Because you have to add zeros in order to take away negative
*Rewrite the problem into an addition problem. 

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

How are the rules similar? Can you generalize?  

Rule for Negative - Positive

 How can we figure out -768+324= without using the chips. What rule can we make every time we have a negative and subtract a positive.

Class Rule: Add the absolute value of 2 #'s and make answer negative. 

Rule for Subtracting 2 positives

Positive - Positive

Find the difference between the two numbers and if the number you are subtracting is larger the answer is -, if the number you are subtracting from is larger answer is +.

negative-positive

What do you predict will happen if you have a negative- positive problem?
-closer to zero
-Answer will be negative 
-sometimes +/-
-Depends on absolute value
-Can be anything?

Must build and draw
-4-5

 

-7-3
-9-2
-8-9
-1-5
-7-3

***Cancellation isn't taking away. We need to take away. Not cancel and make sure that we don not change the net value. 


What did we notice? 

* keep spending and your debt gets bigger. 


HW: 
 Must DRAW a model and solve each!!! No model=no credit
L3 pg. 44 
#1-13, #17-19, 27, 31, 35, 36, #48 a, b, c, e, g

Due on Monday 12/15!!!

How are cancellation and subtraction different? (Examples of subtraction problems with chips)

L1 Zeroation/Cancel-Best Definition 
-When a +/- are put into a collection and it's net value doesn't change. The problem's net value doesn't change.


How is this helpful to us? 
-Clear zeros to simplify problem 



When you have zeroation, what does it do to the netvalue?

-It doesn't do anything. It does nothing to the value, it is just zero.


2-7=

Way 1

 

          Step 1                                      Step 2                                     Step 3

Way 2

            Step 1                                                                              Step 2



147-132=

How would I figure out how many zeros I need to add to the pot?

367-147=220
I need 220 zeros

Subtracting Integers

R-65 Subtraction

What are all the types of subtraction problems that we will need to test? ex. for addition its +/+, -/-, +/- 


Must Build and draw first. 

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

What is the difference between subtraction and cancelling out (zeroation) 

subtraction- taking away
Cancelling-there needs to be a match between a positive and a negative 

Why can I cancel out a positive and negative? But not a positive and positive or negative and negative? 

-They are opposites. They are the same distance away from zero with the same absolute value. They have the same absolute value of zero. 

Adding Integers

What do we think the word zeroation means?
Cancelling out, putting something together to make zeros
examples:
-7+7=0
-8+8=0

Can also be 5+-3

L-62
5+-2=3
5-2=3

How are these the same and different? 
-they both got the same answer.
-for the negative 2, we needed to think of a problem in the real world where you would start with a negative
- -2 will always represent in debt
-when we subtract 2 they don't

example: Noah has 10 candy bars and I take 3 away from him, Mack as 4 candy bars and I give him 3.
-they both have 7.
-Noah feels bad because you took them away
-Is what happened with Zach and Noah different? yes. Did they get the same number?

It matters in real life whether we are adding or subtracting.

Make a prediction...

What happens when you add:

+/+

-/-

+/-



These problems matter in real life and it does make a difference.


2+3=
-3+2=
5+-2=
-4+-5=
-4+9=
-10+3=
2+8=
-3+-7=

Class Public Record:
+/+: Get a bigger positive 

-/-: Get a smaller # and it will always be negative

+/-: 
a. If neg #'s absolute value is bigger answer is negative, if positive #'s absolute value is bigger answer is positive.

b. Answer is The difference between the 2 absolute values

**Need to do both parts a and b

Black and red chips

Observations: 

-black and red equal or more black than red 

-net value is the sum (reds+black=net value) 

-net value=blacks-reds 

-blacks =red ---net value =0

-odd total #of tiles net value not equal to 0. 

-if one or more red than black NV is -1 

-if one more black than Ed is 1

What is addition and what is subtraction? 

Class definitions 

Addition: putting stuff together

Subtraction: difference between 2 things-- take away 

put in other picture

We had a confusion: we had a total number of 7 tiles. 2 red tiles. 5 black tiles. a net value of +3. 

The class had 2 different equations:

Advik: 5-2=3 Took away (Subtraction)

Corinna: -2+5=3 Added Negatives (addition) 

What is the same? The answer. 

Does it matter which math sentence? 

yes. Why? 

Two different things are happening. In one place you are gathering something and in the other you are taking them away. 

L63 Homework: 

Write a good word problem to go with each of those equations.

5+-2=3

5-2=3

Make a prediction: 

what happens if I add 2 positive numbers? 

what happens if I add 2 negative numbers? 

what happens if I add 1 positive and 1 negative number?

Homework: 

L62 pg. 22 #23-35 

Thermometer and Number Line 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

What is a rational number?

Class definition:
-Integer divided by integer(cannot be 0) equals a rational number

-Integer shown in 2 different ways (Not equal to 0)


What is an example of something that is a rational number?  isn't a rational number?
Rational- 7/8, 1/2, 9/11, .799, 799/1000, 10/8, 3
Not rational or irrational: π, √2 

(A number that doesn't terminate and it doesn't repeat)

Integer Models

R-56 Private Think Time: When you think about positive and negative numbers what models come to mind?

-Number lines
-Coordinates graph (positive and negative numbers)
-absolute value. Distance from zero. [-5] 5 units from zero
-regular graph

Where in the real world do you see positive and negative numbers?
-temperature: thermometer
-Bank account (owe 21, give 20, -1)
-receipts: extreme coouponers
-Test scores
-Fantasy sports (points, touchdown, error, tunr over)




Similarities and differences?

-Both have degrees (number lines can), both measure in something
-thermometer-to tell temperature
-number lines-all numbers\
-thermometer is a number line flipped (negative on bottom, positives on top)






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.

R-54 Transversals

R-54 Transversals





















Things we notice:

-appear the same degree
-could make angles-which are sturdy.
-maybe ,'s a + b arent equal to b + c
-a + b+ c+ d= 360
-looks like a previous strategy when we broke off the corners and made a circle of 360

Supplementary<'s=180
1. a and b
2. c and d
3. a and d
4. c and b

Are a and c equal? Are d and b equal?

-Some think a and c equal each other, but they are not supplemental.

Can you prove a=c and b=d?

Math sentence
a+b=180
c+d=180
a+d=180
c+b=180

How can you use these mathematical sentences to prove that a=c and d=b? 

 Idea:
a+b=c+d

 Idea:
c+d=180
a+d=180
so 
c+d=a+d
c=a 
How does this idea make sense? 

Big Idea: 
These are your supplementary angles
a=c
b=d
we can prove it! 

 R-55

What do you think a,b, and c is? How can you justify it?
180-150=30 degrees
a and c are both equal to 30.
4 angles equals 360 degrees. If you add them all together and subtract them it equals 180
150+30+30=210+150=360

The shape in the middle is a parallelogram.

How is that helpful to us?
I know its a quadrilateral, 2 sets of parallel lines, 2 angles the same and opposite of each other.

60/2=30+30 (there were 2 equal angles left)

How do I now know that m is 30?
l=150
n=150
210+150=360


L-55
What observations can you make about the relationships between angles with a transversal?

-With a transversal through parallel lines, the opposite angles and are always going to be equal.
- All angles in a circle=360
-adjacent (<'s next to each other) <'s=180)
-Exact same number of 150's as there are 30's
-You can make the exterior angles in there to represent 180.

Projects due tomorrow. Justifications are due on Friday.

Quadrilaterals

R-53 What do we know about triangles?

-Exterior + interior
-Angle sum =180 degrees
*at least 3 pieces of info needed for duplicating a triangle
-They can tile and tessellate 
*2 small sides greater>biggest side

Quadrilaterals -How many pieces of information do we think we would need to give? 
-4 pieces of information needed (3 angles, 1 side or 4 sides)
-2 for some where2 sides are the same (if equilateral)
-3 (side, side, angle)
*Class still is not sure how many pieces of info. would be needed

Side Lengths            Did we create a unique quad?                    Sketch
6,10,15,15                yes-unique-only one can be made
3,5,10,20                  no-not unique-multiple quads can be made
8,8,10,10                  Impossible-can't make a quad
12,20,6,9

Make up your own

L-53 Make 5 observations about your chart
-No unique quads
-were repeated #'s in side lengths
-may need < to make unique(conjecture)
-both angles and side lengths are relevant for unique shapes
-these smallest +2 >largest to make quadrilateral (because of extra bend in the shape)
-All could shift around to make lots of quads

Students were given a challenge. They needed to make a quadrilateral. Then, needed to make a structure so that it would be stable and not move because there are no unique triangles.

Chris discovered that if you put a poly strip in the middle it creates two triangles. Triangles are unique.

Creating triangles creates stability.

Many builders create triangles. There are many structures that have triangles throughout.

L-53 2 Big Ideas:
 1st Big Ideas
You can't make a unique quadrilateral because it is flexible.
To make a quad stable, create 2 triangles with a diagonal


2nd Big Idea:
Their side lengths have to be bigger than the biggest one.
Their smallest +2 >largest to make quad

Debate! 3 sides vs. 2 sides

The class had a debate!

They needed to prove whether or not 3 sides works or 2 sides works. 

3 side idea
-If you just had 2 angles, there are so many different ways to make it. 
-multiple angles from 4 cm and 74 degrees

2 sides

-Angle BCA=60 degrees
Side CA =3.6 cm
Side CB=4cm

Draw angle first

L-52 Notes of text Presentations

L-52 Notes of text Presentations

What do you think is the least amount of information you could use in a text?

Class came up with different ideas:

3 pieces of info- side, side, side (Noah) or side, side, angle (Bella) or angle, angle, side (Gabby)
2 pieces of info-side, angle (Mac)

Text triangle

What information would you text to someone in 7A so that they could build this triangle? 

1. angle A is 74 degrees, angle B is 60 degrees, angle C is 46 degrees.

-The length is between A and B is 3 cm and B and C is 4 cm and C and A is 3.6 cm.  Draw this triangle.

2. Angle ACB is 46 degrees, angle CBA is 60 degrees, Angle BAC is 74 degrees and side A-C is 3.6 cm. Side C-B is 4 cm and side B-A is 3 cm. This is a triangle. Draw it.

3. Angle A is on top. B is to the left and lower. C is to the right and lower

4. AB is 3 cm.

5. It's a scalene triangle.

Lars thinks there is a more efficient way to write this.

R-52 Is there a shorter way to write this? Is it possible for me to give a little bit less information and get the exact triangle? What is the shortest possible text I could send?

1. Abby2

AC=3.6 cm

BC=4cm

BA=3cm

2. Abby 1

Angle ACB= 46 degrees

Angle CBA=60 degrees

Angle BAC=74 degrees

2. Abby1/Bella

Top C=74 degrees

Bottom Left=60 degrees

Bottom Right= 46 degrees

3. Ursula and Lauren's' Text

Angle ACB=46 degrees

CBA= 60 degrees

BAC=74 degrees

AC=3.6cm

BC=4 cm

BA=3cm

4. Advik

Angles= 46, 60, 74 degrees

Sides= 3.6cm, 4 cm, 3cm

5. Lars

Side AB=3cm

Side BC=4cm

Angle BCA=46 degrees 

6. Noah

Angle ABC=60 degrees

Angle BCA=46 degrees

BC=4cm 

Make a triangle

Make 5 triangles.

Class Observations:

Yes

*equilateral triangles

*#'s close together

*Larger

No

*Smaller

*#'s farther apart

Aiden's idea: *two smaller #'s if they don't add up to the biggest # or higher they don't make a triangle.

(ex. 4,6, 3   4+3)


There was some disequilibrium in the room.

How do we know for sure its a no? How do we know for sure its a yes?

Word Problems from Class: Reflection and Edit

L-47


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?








L-46


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 the problem so that you can make generalizations and look for patterns and regularity.
**This is a finished project. Would be published in a math journal.



Due Thursday 10/23

The Classroom: What does it look like and sound like?

-Direct instruction
      -Quiet
      -Eyes on instructor, board, Elmo
      -follow instructions
      -Giving information like an answer
      -Copy/write notes
      -Check work
      -Listen to understand
      -Ask questions

-private think/ write time

       -giving ppl space, quiet, focused on the problem, write alone and not with partner, develop individual ideas. If I have a question I can write it down (instead of raise my hand)

-partner work

-plenary-whole class discussion 

        -listen to your peers, share your ideas, don't interrupt, ask questions. Raise hands: so we take turns and are repsectful. 

        -should be quiet, except for one voice. 

        -we need to stay relevant to the math 

        -collaborating 

        -teacher: selects speakers to record ideas. Connect speakers to connect ideas. 

        -students ideas only 

        -teachers push on your thinking 

-homework
      -Complete: Best of your ability (Ask family for help, use your notebook, online notebook, phone a friend, ask Mrs. O'Toole or in class, take a break and come back later)
      -If none of the above work: Show your evidence of an attempt or ask questions in notebook
      -Finish your HW on time
      -Quiet work space
      -FADAF (Frustration and difficulty as feedback) Write about the feedback in your notebook.
      -Answer all the questions and explain
      -Email Mrs. O'Toole
      -Lots of Room

 -Small group work

         -not commenting about partners

         -equity among partners (equal or fair amount of time to share/speak) 

         -need to make decisions together, equal share work
         -Volume should be in the middle: people can hear you, but you aren't yelling 
         -If no one is in your group, you can do pieces of the group work
         -Include everyone by asking for their opinion and making the work equitable.

-Debate


        -Listen to each other
        -Prove/explain your idea and how it is correct
        -Share with equity--choose from multiple ideas
        -Write questions down
        -Debating ideas, NOT people
        -Prove why other idea doesn't work--explain with steps


       
   

Exterior and Interior Angles

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


























R-49 Draw 4 different polygons. Measure the exterior angle and compute the sum.

Tiling

Tiling

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

Think about what you know for tessellations:

I know a shape tessellates if it's angles are factors of 360.

 

Observations

*When I look at their vertices (where the shapes come together) it equals 360. 

*Angle measure is always 360 (with no overlaps or gaps)

*Can draw triangles in hexagon. 60+60=120

*I can tell they tile if the angle measures added together equal 360. 

*The shapes make the angle for the other shape.

Tessellations

After watching the bee video, the hexagons in the honeycomb tessellate. What does that mean?

Which shapes tessellate?

What do we notice about the shapes that tessellate?

Observations:

*All the angles are the same

*Single angle measures are factors of 360

(90 degrees goes into 360 4x)

(60 degrees goes into 360 6x)

(120 degrees goes into 360 3x)

Class rule: I will know a shape tessellates if...

I know a shape tessellates if it's angles are factors of 360.

Making a Rule for Irregular Polygons

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. 

Are regular polygons the same or different from irregular polygons? 




Casey:


Any polygon will have the circle in the middle subtract out the value of the circle which is 360.


The reason for the # of triangles is to count the # of sides.


The 360 is already include in the whole shape and we don't need it so we subtract.


Rule for regular polygons works for irregular polygons.


Look back to R-43.

What is a rule for finding the sum all angles? What about a single angle measure?

Homework: 
Comp Corrections Due Wed

Due Tuesday 
L-40 pg. 53 #3-10
pg. 58 #23 



R-43 Write a Rule: If I know the sum of all the angles how can I figure out the single angle measure?

Sum of all angles Divided by # of sides = single angle measure



1. Devon's work: I created some irregular polygons and ripped off the sides. Is there a relationship between the # of sides and the angle of degree?

2. Trevor's work: I took some polygons, some are regular and some are irregular. Cut them in thirds, halves, fourths. Trying to make other shapes or triangles. (Angle measure for a triangle is always 180)

3. Casey's work: I made each side the base of a triangle. I also made the same number of triangles to match sides.

R-44 Write as many observations as you can. Can you find the relationship between the # of sides and degree angle?

Polygon chart

This week we were given 6 polygons and were asked to count the sides, find the angle measure, and the sum of all the angles. We also had a discussion about precise measuring and how measuring is not perfectly accurate. 

We also looked for patterns on this chart and used what we knew about the patterns to determine the degree of a Decagon (10 sides) and a nonagon (9 sides). 

R-43 write a rule that you could use to find the sum of all the angles for any number of sides polygon. 

For example: 112 sided polygon 

What is a Polygon?

What Observations can you make? 


Based on the information above, what do you think is a Polygon?