# Difference between revisions of "What are negative numbers"

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− | =Activity - | + | =Activity - The idea of negative numbers and operations on negative numbers = |

+ | ==Objectives of the activity== | ||

+ | #Develop an idea that negative numbers are part of a type of numbers<br> | ||

+ | #That negative numbers are continuous<br> | ||

+ | #Repesent the number line with zero as a place holder<br> | ||

+ | |||

==Estimated Time== | ==Estimated Time== | ||

+ | This is an activity for a lesson to introduce negative numbers; can be more 2-3 period | ||

==Materials/ Resources needed== | ==Materials/ Resources needed== | ||

+ | *More apples (or any other fruit)<br> | ||

+ | *Small cards for writing down the numbers as well as for writing the operations<br> | ||

+ | *Pencils, etc<br> | ||

+ | *Two boxes – one of negative numbers and the other of positive numbers<br> | ||

+ | |||

==Prerequisites/Instructions, if any== | ==Prerequisites/Instructions, if any== | ||

==Multimedia resources== | ==Multimedia resources== | ||

==Website interactives/ links/ simulations/ Geogebra Applets== | ==Website interactives/ links/ simulations/ Geogebra Applets== | ||

==Process (How to do the activity)== | ==Process (How to do the activity)== | ||

+ | Use the same example as before ([[Quantity_and_Numbers|apples and counting]]) to discuss add, take away etc. This activity is in multiple part. | ||

==Developmental Questions (What discussion questions)== | ==Developmental Questions (What discussion questions)== | ||

− | == | + | ===Part 1=== |

− | + | #By the previous activity we have 11 apples. Now we add a few more. Let us say we add 4 more. We have 15 apples. | |

− | + | <b>11 + 2 = 13 ( + 2 signifies increase in apples) | |

− | + | <br>13 + 2 = 15 Write the expression. Again + signifies increase | |

− | + | #2. Let us say I ask the question – what should I add to 15 to make the number of apples 10? They will say ”take away”. Let us say we cannot use the word “take away”. <br> | |

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− | 11 + 2 = 13 | ||

− | 13 + 2 = 15 Write the expression. Again + signifies increase | ||

− | |||

− | 2. Let us say I ask the question – what should I add to 15 to make the number of apples 10? | ||

− | They will say ”take away”. Let us say we cannot use the word “take away”. <br> | ||

Write the expression like this 15 + (-5) = 10. | Write the expression like this 15 + (-5) = 10. | ||

− | + | #The numbers that when added to a number increase the original quantity are called positive numbers. The numbers that when added to a number decrease the original quantity are called negative numbers. The negative number is thus an opposite of the positive number.<br> | |

− | + | #Now frame the question as follows | |

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*What do I have to add to 15 to make it 7? The answer is (-8). | *What do I have to add to 15 to make it 7? The answer is (-8). | ||

*What do I add to 15 to get 8? (-7) | *What do I add to 15 to get 8? (-7) | ||

*What do I add to 15 to get 9? (-6) | *What do I add to 15 to get 9? (-6) | ||

*What do I add to 15 to get 10? (-5) | *What do I add to 15 to get 10? (-5) | ||

− | + | *When I add (-7). I get 8. For me to get 9, I have to add a number greater than (-7) and I have added (-6). Similarly (-5) is greater than (-6). So the larger negative number is actually smaller''.'''<br> | |

− | + | # Now we transition from numbers representing some quantities to numbers being manipulated as numbers.<br> | |

− | + | ===Part 2=== | |

− | + | #We have seen what negative numbers are. We will see how to work with them. <br> | |

− | + | #We have seen that negative numbers are such that when we add them the quantity decreases. What happens when we subtract them?<br> | |

− | + | #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> | |

− | + | #Let us pull out sets of numbers the same number but from the positive box and negative box.<br> | |

− | + | 3, -3 <br> | |

− | + | Add them <br> | |

− | + | 3 + (-3) = 0<br> | |

− | + | 4, -4<br> | |

− | + | Add them<br> | |

− | + | 4 + (-4) = 0<br> | |

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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> | 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 (-)*(-) = + | ||

− | + | ===Part 3=== | |

'''Now what happens when we multiply negative numbers?''' | '''Now what happens when we multiply negative numbers?''' | ||

− | + | #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 . 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 . 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? <br> | + | *-3 x -3. How do we do this? |

− | Let us look at the table below.<br> | + | <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> | 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> | 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> | ||

− | |||

− | + | ==Evaluation (Questions for assessment of the child)== | |

− | + | #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 6 - 7? | |

− | 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> | + | #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> | |

+ | #See attached workheets | ||

+ | |||

+ | ==Question Corner== | ||

+ | ==Activity Keywords== | ||

+ | Negative numbers | ||

+ | '''To link back to the concept page''' | ||

+ | [[Numbers]] |

## Revision as of 18:19, 7 July 2016

# Activity - The idea of negative numbers and operations on negative numbers

## Objectives of the activity

- Develop an idea that negative numbers are part of a type of numbers
- That negative numbers are continuous
- Repesent the number line with zero as a place holder

## Estimated Time

This is an activity for a lesson to introduce negative numbers; can be more 2-3 period

## Materials/ Resources needed

- More apples (or any other fruit)
- Small cards for writing down the numbers as well as for writing the operations
- Pencils, etc
- Two boxes – one of negative numbers and the other of positive numbers

## Prerequisites/Instructions, if any

## Multimedia resources

## Website interactives/ links/ simulations/ Geogebra Applets

## Process (How to do the activity)

Use the same example as before (apples and counting) to discuss add, take away etc. This activity is in multiple part.

## Developmental Questions (What discussion questions)

### Part 1

- By the previous activity we have 11 apples. Now we add a few more. Let us say we add 4 more. We have 15 apples.

**11 + 2 = 13 ( + 2 signifies increase in apples)
13 + 2 = 15 Write the expression. Again + signifies increase
**

- 2. Let us say I ask the question – what should I add to 15 to make the number of apples 10? They will say ”take away”. Let us say we cannot use the word “take away”.

Write the expression like this 15 + (-5) = 10.

- The numbers that when added to a number increase the original quantity are called positive numbers. The numbers that when added to a number decrease the original quantity are called negative numbers. The negative number is thus an opposite of the positive number.
- Now frame the question as follows

- What do I have to add to 15 to make it 7? The answer is (-8).
- What do I add to 15 to get 8? (-7)
- What do I add to 15 to get 9? (-6)
- What do I add to 15 to get 10? (-5)
- When I add (-7). I get 8. For me to get 9, I have to add a number greater than (-7) and I have added (-6). Similarly (-5) is greater than (-6). So the larger negative number is actually smaller
*.'*

- Now we transition from numbers representing some quantities to numbers being manipulated as numbers.

### Part 2

- We have seen what negative numbers are. We will see how to work with them.
- We have seen that negative numbers are such that when we add them the quantity decreases. What happens when we subtract them?
- 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.
- Let us pull out sets of numbers the same number but from the positive box and negative box.

3, -3

Add them

3 + (-3) = 0

4, -4

Add them

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.

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 (-)*(-) = +

### Part 3

**Now what happens when we multiply negative numbers?**

- 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.

- 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.
- -3 x -3. How do we do this?

Let us look at the table below.

- 3 x 3 = -9

- 3 x 2 = -6

- 3 x 1 = -3

- 3 x 0 = 0

- 3 x-1 = -3

- 3 x-2 = -6

- 3 x-3 = -9

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.

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.

## Evaluation (Questions for assessment of the child)

- Start doing this activity with objects, then numbers depending upon the level of the student.
- 6 - 1? - what is the answer?
- 6 - 2? - what is the answer?
- Continue this exercise until we get to 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)
- Arrange a set of randomly chosen positive and negative numbers (integers) along the number line in increasing order.
- See attached workheets

## Question Corner

## Activity Keywords

Negative numbers
**To link back to the concept page**
Numbers