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Line 140: − + − + − + − + − + − + − + − + − + − + − + − This addition should be explained as. When I add three the quantity increases. When I add -3 the quantity decreases. So + 3 and -3 are the same in magnitude but do opposite things. + − For every positive integer, there is a negative integer. − Discuss examples of borrowing from the bank; someone giving a loan. − 6. Now what happens when we multiply negative numbers?+ − + − + + − + − + − - 3 x 3+ − -9 − -3 x 2 − -6 − -3 x 1 − -3 − -3 x 0 − 0 − -3 x -1 − − -3 x -2 − − -3 x -3 − − − Extend the table above. By simple pattern evaluation we see it is 3, 6 and 9. We have shown the number line above. It makes sense logically that the next number is 3 and that it becomes positive. − The best way to extend this to division is to treat this as multiplication by fraction and extend these rules. − − − Evaluation activities − Start doing this activity with objects, then numbers depending upon the level of the student. − 6 - 1 − Result − 6 - 2...continue − Result − 6 - 7 − Can I take away the objects? I start looking at these as a special kind of number and get the number line to move to -1. Extend the number line construction. What does this – 7 represent (they should say it means that when I add this it is reducing the quantity) + + + + + + + + + − + + + + + + + − +
Series of Activities on Number Systems (view source)
Revision as of 16:33, 28 January 2013
, 16:33, 28 January 2013→Lesson 1 : The idea of negative numbers and operations on negative numbers
*Small cards for writing down the numbers as well as for writing the operations<br>
*Small cards for writing down the numbers as well as for writing the operations<br>
*Pencils, etc<br>
*Pencils, etc<br>
*Two boxes – one of negative numbers and the other of positive numbers<br><br>
*Two boxes – one of negative numbers and the other of positive numbers<br>
'''How to do the activity'''
'''How to do the activity'''<br>
'''''Part 1'''''
'''''Part 1'''''
'''Part 2'''
'''Part 2'''
1. We have seen what negative numbers are. We will see how to work with them.
1. We have seen what negative numbers are. We will see how to work with them. <br>
2. We have seen that negative numbers are such that when we add them the quantity decreases. What happens when we subtract them?in
2. We have seen that negative numbers are such that when we add them the quantity decreases. What happens when we subtract them?<br>
3. Extend the activity of what should I add to 20 to make it 10, 11, 12 and so on? Extend it all the way to 30. Let the students pull the numbers out and place them along the wall/ stock on the wall etc. You will see the number line.
3. Extend the activity of what should I add to 20 to make it 10, 11, 12 and so on? Extend it all the way to 30. Let the students pull the numbers out and place them along the wall/ stock on the wall etc. You will see the number line.<br>
4. Let us pull out sets of numbers the same number but from the positive box and negative box.
4. Let us pull out sets of numbers the same number but from the positive box and negative box.<br>
3, -3 <br>
3, -3
Add them <br>
Add them
3 + (-3) = 0<br>
3 + (-3) = 0
4, -4<br>
4, -4
Add them<br>
Add them
4 + (-4) = 0<br>
4 + (-4) = 0
This addition could be explained like this. When I add three the quantity increases. When I add -3 the quantity decreases. '''''So + 3 and -3 are the same in magnitude but do opposite things. For every positive integer, there is a negative integer.''''' Discuss examples of borrowing from the bank; someone giving a loan.<br><br>
5. Now I have 25 (from the number box). I am going to subtract (-5). What will happen? When I add (-5), it becomes 20. Since negative numbers behave in this opposite way, subtracting (-5) should become 30? 25 - (-5) = 30...this is equivalent to adding 5 to 25. Hence we say (-)*(-) = +
5. Now I have 25 (from the number box). I am going to subtract (-5). What will happen? When I add (-5), it becomes 20. Since negative numbers behave in this opposite way, subtracting (-5) should become 30? 25 - (-5) = 30...this is equivalent to adding 5 to 25. Hence we say (-)*(-) = +
-3 x 3
'''Now what happens when we multiply negative numbers?'''
From the process of multiplication is repeated addition we can explain as take -3 once, take it the second time and third time. We have -3, -3, -3. We have -9.
*Let us take -3 x 3
From the process of multiplication is repeated addition we can explain as take -3 once, take it the second time and third time. We have -3, -3, -3. We have -9.<br>
3 x -3
3 x -3
Again multiplication is the process of repeated addition. Except I am multiplying it by -3. Then I have to look at the operation as opposite. I am giving away -3 once, twice and third time. We have -9
*Again multiplication is the process of repeated addition. Except I am multiplying it by -3. Then I have to look at the operation as opposite. I am giving away 3 once, twice and third time. We have -9<br>
-3 x -3. How do we do this?
-3 x -3. How do we do this? <br>
Let us look at the table below.<br>
- 3 x 3 = -9<br>
- 3 x 2 = -6<br>
- 3 x 1 = -3<br>
- 3 x 0 = 0<br>
- 3 x-1 = -3<br>
- 3 x-2 = -6<br>
- 3 x-3 = -9<br>
Extend the pattern above. By simple pattern evaluation we see it is 3, 6 and 9. We have shown the number line above. It makes sense logically that the next number is 3 and that it becomes positive. The best way to extend this to division is to treat this as multiplication by fraction and extend these rules.<br>
Yet another way of explaining could be like this. When we add -3, -3 times we are actually operating with two opposites. The (-3) times signifies the opposite of the repeated addition, think of it as repeated subtraction. I am subtracting -3 once (in effect, adding 3); -(-3) second time (adding another 3) and -(-3) for the third time -(-3) (adding one more 3). We get 9. Hence -(-) is positive.<br>
Arrange a set of randomly chosen positive and negative numbers (integers) along the number line in increasing order.
'''Evaluation activities'''
*Start doing this activity with objects, then numbers depending upon the level of the student.
*6 - 1? - what is the answer?<br>
*6 - 2? - what is the answer?<br>
*Continue this exercise until we get to<br>
6 - 7? Can we take away the objects? We start looking at these as a special kind of number and get the number line to move to -1. Extend the number line construction. What does this – 7 represent (they should say it means that when I add this it is reducing the quantity)<br>
*Arrange a set of randomly chosen positive and negative numbers (integers) along the number line in increasing order.<br>
The number line in mathematics
==The number line in mathematics==
The number line is not just a school object. It is as much a mathematical idea as functions.
The number line is not just a school object. It is as much a mathematical idea as functions.
The number line is a geometric “model” of all numbers -- including 0 1, 2, 25, 374 trillion, and -5, Unlike counters, which model only counters, the number line models measurement, which is why it must start with zero. (When we count, the first object we touch is called "one." When we measur using a ruler, we line one end of the object we’re measuring against the zero mark on the ruler.
The number line is a geometric “model” of all numbers -- including 0 1, 2, 25, 374 trillion, and -5, Unlike counters, which model only counters, the number line models measurement, which is why it must start with zero. (When we count, the first object we touch is called "one." When we measur using a ruler, we line one end of the object we’re measuring against the zero mark on the ruler.