Listen and Learn PRESENTED BY MATHEMAGICIAN Mathematics, Grade 7


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1 Number Sense and Numeration Integers Adding and Subtracting Listen and Learn PRESENTED BY MATHEMAGICIAN Mathematics, Grade 7 Introduction Welcome to today s topic Parts of Presentation, questions, Q&A Housekeeping Your questions Satisfaction meter 1
2 What you will learn At the end of this presentation, you will be able to add and subtract integers using two different models (counters and number line) Agenda Importance Definition and Terms Big Ideas Zero principle Models Addition Subtraction The Connection 2
3 Agenda Importance Definition and Terms Big Ideas Zero principle Models Addition Subtraction The Connection Importance An important concept in many areas: elevation (above & below sea level) business (income & debt) space shuttle launch acceleration & deceleration (going faster, getting slower) sports statistics hockey +/ standing, football yards gained/lost, golf above/below par 3
4 Agenda Importance Definition and Terms Big Ideas Zero principle Models Addition Subtraction The Connection Definitions and Terms Integers Integers are all positive and negative whole numbers, including 0. An example of a positive integer is (+7). An example of a negative integer is ( 5). 4
5 Definitions and Terms Addition Combining two or more numbers to get a total. Subtraction = 7 Removing numbers to get a difference. 5 2 = 3 Definitions and Terms Number Lines The whole numbers you use every day that are greater than zero are positive integers. The number line you may be familiar with begins at zero and moves to the right. 5
6 Definitions and Terms Integer Number Line The integer number line includes the negative numbers (numbers that are less than zero) Agenda Importance Definition and Terms Big Ideas Zero principle Models Addition Subtraction The Connection 6
7 The Big Ideas Every number has size and a positive or negative relationship to zero. A negative number is the opposite of a positive number of the same size. ( 5) is the opposite of (+5) Agenda Importance Definition and Terms Big Ideas Zero principle Models Addition Subtraction The Connection 7
8 Zero Principle The sum of a negative number and its opposite positive number of the same size equals zero. I owed you $2. I earned $2. After paying off my debt to you, I now have $0. Agenda Importance Definition and Terms Big Ideas Zero principle Models Addition Subtraction The Connection 8
9 Agenda Models You will see two methods for modelling integer addition and subtraction in this presentation. 1. Counters Blue represents positive. It looks like this: Red represents negative. It looks like this: Agenda Models 2. Number Lines 9
10 Agenda Importance Definition and Terms Big Ideas Zero principle Models Addition Subtraction The Connection Using the Zero Principle to Add ( 3) + (+3) =0 If red counters represent negative and blue counters represent positive, then we can model it as Recall: the zero principle states that the sum of two opposite integers ( 3) and (+3) is zero If I owe you $3 and then earn $3, this can be written mathematically as ( 3) + (+3) = $0 10
11 Using the Zero Principle to Add ( 4) + (+7) = (+7) [7 = 4 + 3] ( 4) zero (+4) + ( 4) If I owed you $4 and earned $7, I would have ( 4) + (+7) = (+3) Using a Number Line ( 4) + (+7) = (+3) Begin by indicating the start point on the number line (in this case, 4) To add a positive number you move right along the line After a movement of 7 units right, add (+7), you arrive at the sum of (+3) 11
12 Addition Model Which of the three counter models represents ( 8) + (+2)? a) b) c) How can this model be used to answer the question? Addition Model Which of the three counter models represents ( 8) + (+2)? a) b) ( 8) + (+2) = 6 c) ( 2) + (+2) can be grouped into zero How can this model be used to answer the question? 12
13 Addition Model Which of the three counter models represents ( 8) + (+2)? ( 8) + (+2) = 6 c) ( 2) + (+2) can be grouped into zero On a number line, it is modelled as Begin at ( 8) and add (+2) [ two to the right] Addition Model Which of the three models is a good representation of (+3) + ( 8)? a) b) c) 13
14 Addition Model Which of the three models is a good representation of (+3) + ( 8)? a) b) c) (+3) + ( 8) = ( 5) (+3) + ( 3) can be grouped into zero Addition Model Which of the number line models represents (+3) + ( 8)? a) b) c) 14
15 Addition Model Which of the number line models represents (+3) + ( 8)? a) b) c) Important Note Add (+ integer) Add ( integer) along line. along line (opposite of a (+ integer)) Visual Addition Summary (+) + (+) =? (+) + ( ) =? e.g. (+5) + (+3) = (+8) e.g. (+5) + ( 3) = (+2) e.g. (+3) + ( 5) = ( 2) ( ) + (+) =? ( ) + ( ) =? e.g. ( 5) + (+3) = ( 2) e.g. ( 5) + ( 3) = ( 8) e.g. ( 3) + (+5) = (+2) 15
16 Visual Addition Summary (+) + (+) =? (+) + ( ) =? e.g. (+5) + (+3) = (+8) e.g. (+5) + ( 3) = (+2) e.g. (+3) + ( 5) = ( 2) ( ) + (+) =? ( ) + ( ) =? e.g. ( 5) + (+3) = ( 2) e.g. ( 5) + ( 3) = ( 8) e.g. ( 3) + (+5) = (+2) Visual Addition Summary (+) + (+) = (+) (+) + ( ) = depends e.g. (+5) + (+3) = (+8) e.g. (+5) + ( 3) = (+2) e.g. (+3) + ( 5) = ( 2) ( ) + (+) = depends ( ) + ( ) = ( ) e.g. ( 5) + (+3) = ( 2) e.g. ( 5) + ( 3) = ( 8) e.g. ( 3) + (+5) = (+2) 16
17 Addition Rules (+) + (+) = Always positive! ( ) + ( ) = Always negative! (+) + ( ) = Depends ( ) + (+) = Depends Agenda Importance Definition and Terms Big Ideas Zero principle Addition Subtraction The Connection 17
18 Subtract Integers Subtraction is the opposite of addition Recall: Addition is combining groups of numbers (putting numbers together) Subtraction is removing groups of numbers Subtract Integers Addition: OR (+5) + (+2) Subtraction: 5 2 OR (+5) (+2) 18
19 Subtraction Model 5 2 can be written as (+5) (+2) in integer form. Model: The blue counters represent (+5). Remove two blue counters (the +2). Subtraction Model (using Number Line) (+5) (+2) = (+3) Adding a positive integer is movement right along the number line. Subtracting a positive integer is the inverse of addition. Therefore, to subtract a positive number you move left along the number line. Reminder: Add (+ integer) Subtract (+ integer) along line. along line (opposite) 19
20 Subtraction Model 8 3 can be written as (+8) (+3) in integer form. The blue counters represent (+8). Remove three blue counters (+3). Subtraction Model (+8) (+3) = (+5) Recall: To add (+ integer) move To subtract (+ integer) move along line. along line (subtraction is the opposite of addition; move in the opposite direction) 20
21 Subtraction Model ( 5) ( 2) = ( 3) Counters: The red counters represent ( 5). Subtract by removing two red counters, the ( 2). Number Line: Recall: to subtract (+ integer) move So to subtract ( integer) move opposite of (+) along line. along line, Subtraction Model ( 8) ( 3) Model: The red counters represent ( 8). Remove three red counters ( 3). 21
22 More Subtraction What about (+6) (+7)? represents (+6) There are 6 blue counters (+6). How can we remove 7 blue counters (+7)? Adding zeros still makes this (+6) still represents (+6) because zero = +1 + ( 1) (+6) + zero = (+6) Now we can remove (+7) Subtraction Model (+6) (+7) = ( 1) Number Line Notes: to add (+ integer) move to subtract (+ integer) move along line along line (subtraction is the opposite of addition; move in the opposite direction) To add ( integer) move ( ) is opposite (+) To subtract ( integer) move subtract ( ) is opposite of (+) along line; along line; 22
23 Subtraction Model Which of the three models is a good representation of (+2) (+5)? a) b) c) Subtraction Model Which of the three models is a good representation of (+2) (+5)? a) (+5) + (+2) b) (+5) + ( 2) c) This is the correct model 23
24 Subtraction Model (+2) (+5)? his represents (+2) recall: (+2) plus zeroes is still (+2) Remove (+5) from (+2) The answer is ( 3) Number Line Model: ( 3) Subtraction Model Now you try. Which of the three models is a good representation of (+3) ( 5)? a) b) c) 24
25 Subtraction Model Now you try. Which of the three models is a good representation of (+3) ( 5)? a) b) recall: start with (+3). Add 5 zeroes (+/ integers) and then subtract the ( 5) ( integers). This equals (+8). c) Subtraction Model (+3) ( 5) can also be modelled using a number line: (+8) 25
26 Visual Subtraction Summary (+) (+) =? (+) ( ) =? e.g. (+5)  (+3) = (+2) e.g. (+5) ( 3) = (+8) e.g. (+3)  (+5) = (2) e.g. (+3) ( 5) = (+8) ( ) (+) =? ( ) ( ) =? e.g. ( 5) (+3) = ( 8) e.g. ( 5) ( 3) = ( 2) e.g. ( 3) (+5) = ( 8) e.g. ( 3) ( 5) = (+2) Visual Subtraction Summary (+) (+) =? (+) ( ) =? e.g. (+5) (+3) = (+2) e.g. (+5) ( 3) = (+8) e.g. (+3) (+5) = ( 2) e.g. (+3) ( 5) = (+8) ( ) (+) =? ( ) ( ) =? e.g. ( 5) (+3) = ( 8) e.g. ( 5) ( 3) = ( 2) e.g. ( 3) (+5) = ( 8) e.g. ( 3) ( 5) = (+2) 26
27 Visual Subtraction Summary (+) (+) = depends (+) ( ) = (+) e.g. (+5) (+3) = (+2) e.g. (+5) ( 3) = (+8) e.g. (+3) (+5) = ( 2) e.g. (+3) ( 5) = (+8) ( ) (+) = ( ) ( ) ( ) = depends e.g. ( 5) (+3) = ( 8) e.g. ( 5) ( 3) = ( 2) e.g. ( 3) (+5) = ( 8) e.g. ( 3) ( 5) = (+2) Subtraction Rules (+) ( ) = Always positive! ( ) (+) = Always negative! (+) (+) = Depends ( ) ( ) = Depends 27
28 Agenda Importance Definition and Terms Big Ideas Zero principle Addition Subtraction The Connection The Connection Subtract: (+7) ( 2) =? (+7) This model represents (+7) Remove ( 2) (the two red counters) The answer is (+9) 28
29 The Connection Add: (+7) + (+2) =? (+7) (+2) The answer is (+9) Notice: Subtracting a negative = adding a positive More Connections Subtract: ( 7) (+2) =? ( 7) Remove (+2) (the two blue counters) The answer is ( 9) 29
30 More Connections Add: ( 7) + ( 2) =? ( 7) ( 2) The answer is ( 9) Notice: subtracting a positive = adding a negative To Subtract, Add the Opposite Your teacher may have said that to subtract integers you can just add the opposite. Do you see why this is the case? (+7) ( 2) = (+7) + (+2) notice instead of subtract ( 2) add the opposite (+2) ( 7) (+2) = ( 7) + ( 2) notice instead of subtract (+2) add the opposite ( 2) 30
31 Connection (Number Line) to subtract an integer, add the opposite. Try these Match up the equivalent expressions in each row ( 3) ( 9) (+3)+(+9) (+3)+( 9) ( 3)+(+9) ( 3)+( 9) (+2) (+1) (+2)+(+1) (+2)+( 1) ( 2)+(+1) ( 2)+( 1) ( 6) (+7) (+6)+(+7) (+6)+( 7) ( 6)+(+1) ( 6)+( 7) ( 2) (+2) (+2)+(+2) (+2)+( 2) (+2)+( 2) ( 2)+( 2) (+2) ( 2) (+2)+(+2) (+2)+( 2) (+2)+( 2) ( 2)+( 2) 31
32 Answers Match up the equivalent expressions in each row ( 3) ( 9) =(+6) (+3)+(+9) (+3)+( 9) ( 3)+(+9) =(+6) ( 3)+( 9) (+2) (+1) =(+1) (+2)+(+1) (+2)+( 1) =(+1) ( 2)+(+1) ( 2)+( 1) ( 6) (+7) =( 13) (+6)+(+7) (+6)+( 7) ( 6)+(+1) ( 6)+( 7) =( 13) ( 2) (+2) =( 4) (+2)+(+2) (+2)+( 2) (+2)+( 2) ( 2)+( 2) =( 4) (+2) ( 2) =(+4) (+2)+(+2) =(+4) (+2)+( 2) (+2)+( 2) ( 2)+( 2) Summary 1. The sum of two opposite integers is zero, e.g., ( 3) + (+3) = 0 2. When adding integers remember the zero principle, e.g., (+3) + ( 5) 3. To subtract integers you can add the opposite. 32
33 Resources Math League /integers.htm Squidoo HowTo Ask Dr.Math integers.html 33
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