This document defines integers and the rules for adding and subtracting them. It defines positive and negative numbers, opposites, and absolute value. It then presents two rules for adding integers: 1) if signs are the same, add the numbers and use that sign, and 2) if signs are different, subtract the smaller number from the larger and use the sign of the number with the larger absolute value. It also explains that subtracting a negative number is the same as adding a positive number. Examples are provided to illustrate the rules and a number line model is presented. The document checks understanding through practice problems.
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This will be used as part of your Personal Professional Portfolio once graded.
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Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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http://sandymillin.wordpress.com/iateflwebinar2024
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Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
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Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
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Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
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The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
2. What You Will Learn
Some definitions related to integers.
Rules for adding and subtracting
integers.
A method for proving that a rule is true.
Are you ready??
9. Negative Numbers Are Used to
Measure Under Sea Level
0
10
20
30
-10
-20
-30
-40
-50
10. Negative Numbers Are Used to
Show Debt
Let’s say your parents bought a car but
had to get a loan from the bank for $5,000.
When counting all their money they add
in -$5,000 to show they still owe the bank.
11. Hint
If you don’t see a negative
or positive sign in front of a
number it is positive.
9
+
12. Integer Addition Rules
Rule #1 – If the signs are the same,
pretend the signs aren’t there. Add
the numbers and then put the sign of
the addends in front of your answer.
9 + 5 = 14
-9 + -5 = -14
16. Integer Addition Rules
Rule #2 – If the signs are different
pretend the signs aren’t there.
Subtract the smaller from the larger
one and put the sign of the one with
the larger absolute value in front of
your answer.
-9 + +5 =
9 - 5 = 4
Larger abs. value
Answer = - 4
20. One Way to Add Integers Is
With a Number Line
0 1 2 3 4 5 6
-1
-2
-3
-4
-5
-6
When the number is positive, count
to the right.
When the number is negative, count
to the left.
+
-
21. One Way to Add Integers Is
With a Number Line
0 1 2 3 4 5 6
-1
-2
-3
-4
-5
-6
+
-
+3 + -5 = -2
22. One Way to Add Integers Is
With a Number Line
0 1 2 3 4 5 6
-1
-2
-3
-4
-5
-6
+
-
+6 + -4 = +2
23. One Way to Add Integers Is
With a Number Line
0 1 2 3 4 5 6
-1
-2
-3
-4
-5
-6
+
-
+3 + -7 = -4
24. One Way to Add Integers Is
With a Number Line
0 1 2 3 4 5 6
-1
-2
-3
-4
-5
-6
-
+
-3 + +7 = +4
25. Integer Subtraction Rule
Subtracting a negative number is the
same as adding a positive. Change
the signs and add.
2 – (-7)
is the same as
2 + (+7)
2 + 7 = 9!
26. Here are some more examples.
12 – (-8)
12 + (+8)
12 + 8 = 20
-3 – (-11)
-3 + (+11)
-3 + 11 = 8
27. Solve the problems on
Part C of your worksheet
now. Click to the next slide
when done.
29. How do we know that
“Subtracting a negative number
is the same as adding a positive”
is true?
We can use the same
method we use to check our
answers when we subtract.
30. Suppose you subtract a – b
and it equals c:
a – b = c
5 – 2 = 3
To check if your answer is
correct, add b and c:
a = b + c
5 = 2 + 3
31. Here are some examples:
a – b = c a = b + c
9 – 5 = 4 9 = 5 + 4
a – b = c a = b + c
20 – 3 = 17 20 = 3 + 17
32. If the method for checking
subtraction works, it should
also work for subtracting
negative numbers.
33. If a – b = c, and….
2 – (-5) is the same as
2 + (+5), which equals 7,
Then let’s check with the
negative numbers to see if it’s
true…
34. a – b = c a = b + c
2 – (-5) = 7 2 = -5 + 7
It works!
a – b = c a = b + c
-11 – (-3) = -8 -11 = -3 + -8
YES!
35. Solve the problems on
Part D of your worksheet
now. Click to the next slide
when done.
36. Check Your Answers
1. Solve: 3 – 10 = 7
Check: 3 = 10 + (-7)
2. Solve: 17 – ( 12) = 29
Continued on next slide
Check: 17 = -12 + 29
38. You have learned lots of things
About adding and subtracting
Integers. Let’s review!
39. Integer Addition Rules
Rule #1 – If the signs are the same,
pretend the signs aren’t there. Add
the numbers and then put the sign of
the addends in front of your answer.
9 + 5 = 14
-9 + -5 = -14
40. Integer Addition Rules
Rule #2 – If the signs are different pretend
the signs aren’t there. Subtract the smaller
from the larger one and put the sign of the
one with the larger absolute value in front
of your answer.
-9 + +5 =
9 - 5 = 4
Larger abs. value
Answer = - 4
41. One Way to Add Integers Is
With a Number Line
0 1 2 3 4 5 6
-1
-2
-3
-4
-5
-6
When the number is positive, count
to the right.
When the number is negative, count
to the left.
+
-
42. Integer Subtraction Rule
Subtracting a negative number is the
same as adding a positive. Change
the signs and add.
2 – (-7)
is the same as
2 + (+7)
2 + 7 = 9!
43. How do we know that
“Subtracting a negative number
is the same as adding a positive”
is true?
We can use the same
method we use to check our
answers when we subtract.
44. a – b = c a = b + c
2 – (-5) = 7 2 = -5 + 7
It works!
a – b = c a = b + c
-11 – (-3) = -8 -11 = -3 + -8
YES!
45. Discuss with a partner ways
that you know that that is
problem is solved correctly.
6 – (-9) = 15