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Grade 7:  The Learning Equation Math

11.09 Introduction to Integers

Number Concepts

Refresher pp 18-19

Learning Outcomes:

The student will:

tile_neg_1.gif (852 bytes) + tile_pos_1.gif (852 bytes) = 0

Sum of Opposite Integers Explanation
tile_neg_1.gif (852 bytes) tile_pos_1.gif (852 bytes) 1 + -1 = 0    The positive tile cancels the negative tile.
tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) 2 + -2 = 0    The 2 positive tiles cancel the 2 negative tiles.
tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) 3 + -3 = 0    The 3 positive tiles cancel the 3 negative tiles.
tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) 6 + -6 = 0    The 6 positive tiles cancel the 6 negative tiles.
tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)

tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)

12 + -12 = 0    The 12 positive tiles cancel the 12 negative tiles.

 

source:  Thermometer Applet Demo

The initial temperature displayed below the thermometer is 32oF.    If you follow the directions carefully you will see what you need displayed in the liquid level of the thermometer and may not have to wait for the temperature to be displayed(unfortunately this part of the applet runs slowly). 

Click on 1109_h2.jpg (815 bytes) three times and 1109_h1.jpg (793 bytes) twice to reduce the temperature by  32o.    The level of the liquid in the thermometer should bewpeB.jpg (1097 bytes) and the temperature displayed should be 0oF.

Click on wpe82.jpg (851 bytes) .  The liquid level should be:  wpe5.jpg (1199 bytes)Now click on 1109_h2.jpg (815 bytes).  The liquid level should be: 1109.h5.jpg (1097 bytes)and the temperature displayed should be 0oF.

Click on 1109_h3.jpg (797 bytes) and check the temperature displayed.  It should be zero.

Try other combinations to demonstrate the sum of opposite integers is zero.

 

Decimal/Integer Calculator (do not leave spaces):

Expresion:

Other expressions to enter:

Result: -4+4 -34+34 125+-125

 

Click Reload/Refresh if calculator is not functional.

Add the following opposite integers to see if the sum is equal to zero:

Opposite Integers Method Sum
4, -4 4 wpeC0.jpg (771 bytes) 4wpeC1.jpg (801 bytes) wpeC2.jpg (787 bytes) 0
-12, 12 12 wpeC1.jpg (801 bytes) wpeC0.jpg (771 bytes) 12 wpeC2.jpg (787 bytes) 0
55, -55 55  wpeC0.jpg (771 bytes) 55 wpeC1.jpg (801 bytes) wpeC2.jpg (787 bytes) 0
-111, 111 111 wpeC1.jpg (801 bytes) wpeC0.jpg (771 bytes) 111 wpeC2.jpg (787 bytes) 0
331, -331 331  wpeC0.jpg (771 bytes) 331 wpeC1.jpg (801 bytes) wpeC2.jpg (787 bytes) 0
-501, 501 501 wpeC1.jpg (801 bytes) wpeC0.jpg (771 bytes) 501 wpeC2.jpg (787 bytes) 0

Click on the following to see the the groups of integers:

 

Adding Integer Explanation
tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)
3
tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)
tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)      
13
tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) -7
tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)
tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)     
-13

FUNBRAIN - Number Line

Lesson on Introduction to Integers

Number Line

Integer Bars: Learning Fractions and more with On-line Manipulatives

Cartesian Coordinates

Study the following animated gif to learn the parts of the Cartesian Plane.
 
cartdef.gif (12584 bytes)
Now let us investigate ordered pairs.  An ordered pair is the location (x, y) of a point on the grid.  In this case the x- and y-axis both extend from -5 to 5.  The integers for this range are:

{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}

Move the mouse over the applet from -5 to 5 over the x-axis.  Notice the value of the integer gets larger.  The value of the integer gets smaller when you move your mouse from 5 to -5 on the x-axis.  Similarly the value of the integer gets smaller as you move the mouse for -5 to 5 on the y-axis and smaller when you move the mouse from -5 to 5. The applet is pretty sensitive so you will need to be patient.

source:  http://www.coe.tamu.edu/~strader/Mathematics/Algebra/Cartesian/cartesian.html

Lesson on Comparing and Ordering Integers

 

Rules for Ordering Integers

Directions Example(s)

3458 and 3472

If  both numbers are positive:

Line up the ones digit for the numbers.
Compare the place value for each digit starting from the left.  When you find the first difference, the number with the largest digit is the largest number.

 

3458
3472

The ones digits are lined up.  Starting from the left, the first difference is in the tens column.

Since 7>5, then 3472 > 3458

 

If one number is positive and the other is negative, then the positive number has the greatest value

34 > -3458

-3458 and -3472

If both numbers are negative:

Line up the ones digit for the numbers.
Compare the place value for each digit starting from the left.  When you find the first difference, the number with the largest digit is the smallest number.

-3458
-3472

The ones digits are lined up.  Starting from the left, the first difference is in the tens column.

Since -7<-5, then -3472 < -3458

 

Comparing Integers

Integer Inequalities

 

Algebra Tiles

Both Postitive Integers Both Negative Integers One Positive and One Negative Integer
tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)          tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)

3 < 4

tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)          tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)

-3 > -4

tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)          tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)

3 > -4

tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)           tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)
tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)           tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)      

6 > 5

tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)           tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)
tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)           tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)     

-6 > -5

tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)          tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)
tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)          tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)     

6 > -5

tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)           tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)
tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)           tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)
tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)           tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)

9 = 9

tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)           tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)
tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)           tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)
tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)           tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)

-9 = -9

tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)           tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)
tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)           tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)
tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes) tile_pos_1.gif (852 bytes)           tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes) tile_neg_1.gif (852 bytes)

9 > -9

 

Which number is bigger?

Instructions

  • Select value range and difference between values.
  • Click to get each pair of integers.
  • Click the Correct button below the largest value.
  • After 10 tests the average will be calculated and recorded in the table to the right.
  • If you want to start over before a trial is completed, click .

Value range:    to 

Difference between values:



Click Corect  below the largest integer.

Times:

Average

Trial# Average Time
1
2
3
4

source:  Reaction Timer

 

Fun with integers

Source:  http://cs.georgefox.edu/~ewheeler/310hw/1.20/

 

Review:

Lesson on Comparing and Ordering Integers

Integer Inequalities

Comparing Integers

 

Enrichment:

 

integer, negative, positive, whole number, average, infinite

Prerequisite Skills:

..., 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, ...

The counting(natural) numbers are:

positive whole numbers (1,2,3...)

An integer is:

a positive whole number (1,2,3...),
a negative whole number (-1,-2,-3...)
or zero (0).
The whole numbers are:

positive whole numbers and zero (0,1,2,3...)

Watch what happens to the liquid level in the thermometer when you increase (+) or decrease (-) the temperature.  The new temperature will be shown eventually, but unfortunately this part of the applet runs slowly.

source:  Thermometer Applet Demo

The Applet of Thermometer shows minus quantities by comparing the difference in temperature between two days.

Applet of Thermometer

 

 

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