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From Karnataka Open Educational Resources
Series of Activities on Number Systems (view source)
Revision as of 16:31, 27 September 2021
, 16:31, 27 September 2021→Part II – Negative Numbers
= Introduction =
= Introduction =
Mathematics expresses itself everywhere, in almost every face of life - in nature all around us, and in the technologies in our hands. Mathematics is the language of science and engineering - describing our understanding of all that we observe.Mathematics has been around since the beginnings of time and it most probably began with counting. Many, if not all puzzles and games require mathematical logic and deduction. This section uses the fun and excitement of various popular games and puzzles, and the exhilaration of solving them, to attract and engage the students to realise the mathematics in fun and games. <br><br>
Mathematics expresses itself everywhere, in almost every face of life - in nature all around us, and in the technologies in our hands. Mathematics is the language of science and engineering - describing our understanding of all that we observe.Mathematics has been around since the beginnings of time and it most probably began with counting. Many, if not all puzzles and games require mathematical logic and deduction. This section uses the fun and excitement of various popular games and puzzles, and the exhilaration of solving them, to attract and engage the students to realise the mathematics in fun and games. <br><br>
'''Descriptive Statement'''
'''Descriptive Statement'''
<br>Number sense is defined as an intuitive feel for numbers and a common sense approach to using them. It is a comfort with what numbers represent, coming from investigating their characteristics and using them in diverse situations. and how they can best be used odescribe a particular situation. Number sense is an attribute of all successful users of mathematics. Our students often do not connect what is happening in their mathematics classrooms with their daily lives. It is essential that the mathematics curriculum build on the sense of number that students bring with them to school. Problems and numbers which arise in the context of the students world are more meaningful to students than traditional textbook exercises and help them develop their sense of how numbers and operations are used. Frequent use of estimation and mental computation are also important ingredients in the development of number sense, as are regular opportunities for student communication. Discussion of their own invented strategies for problem solutions helps students strengthen their intuitive understanding of numbers and the relationships between numbers.<br><br>
<br>Number sense is defined as an intuitive feel for numbers and a common sense approach to using them. It is a comfort with what numbers represent, coming from investigating their characteristics and using them in diverse situations, and how best they can be used to describe a particular situation. Number sense is an attribute of all successful users of mathematics. Our students often do not connect what is happening in their mathematics classrooms with their daily lives. It is essential that the mathematics curriculum build on the sense of number that students bring with them to school. Problems and numbers which arise in the context of the students world are more meaningful to students than traditional textbook exercises and help them develop their sense of how numbers and operations are used. Frequent use of estimation and mental computation are also important ingredients in the development of number sense, as are regular opportunities for student communication. Discussion of their own invented strategies for problem solutions helps students strengthen their intuitive understanding of numbers and the relationships between numbers.<br><br>
In summary, the commitment to develop number sense requires a dramatic shift in the way students learn mathematics. Our students will only develop strong number sense to the extent that their teachers encourage the understanding of mathematics as opposed to the memorization of rules and mechanical application of algorithms. Every child has the capability to succeed as a user of mathematics, but the degree of success is directly related to the strength of their number sense. The way to assure that all students acquire a good sense of numbers is to have them consistently engage in activities which require them to think about numbers and number relationships and to make the connection with quantitative information encountered in their daily lives. <br><br>
In summary, the commitment to develop number sense requires a dramatic shift in the way students learn mathematics.
= Mind Map =
= Mind Map =
[[File:Number system -Resource material_html_m1be8b6d7.jpg|400px]]<br>
[[File:Number system -Resource material_html_m1be8b6d7.jpg|400px]]<br>
= Flow Chart =
= Flow Chart =
This unit on number systems is in three parts. The first part will be to develop in children a sense of numbers, the idea of quantity, counting and mathematical manipulation, collections of quantities and developing a unit of measure. The second would be to develop the ideas of the negative numbers and build an understanding based on the manipulations of numbers. The third part would be to introduce the number line as a continuum and to introduce ideas of fractions and rational numbers.While this is for classes 7and 8, we understand that students may come with very limited sense of the numbers. So the activities are designed in such a way as to help such students also.
This unit on number systems is in three parts. The first part will be to develop in children a sense of numbers, the idea of quantity, counting and mathematical manipulation, collections of quantities and developing a unit of measure. The second would be to develop the ideas of the negative numbers and build an understanding based on the manipulations of numbers. The third part would be to introduce the number line as a continuum and to introduce ideas of fractions and rational numbers.While this is for classes 7and 8, we understand that students may come with very limited sense of the numbers. So the activities are designed in such a way as to help such students also.
== Part 1: Number Sense – Counting and Operations ==
== Part I: Number Sense – Counting and Operations ==
=== Objectives ===
=== Objectives ===
1. Understand that there is an aspect of quantity that we can develop with disparate objects
1. Understand that there is an aspect of quantity that we can develop with disparate objects<br>
2. Comparison and mapping of quanties (more or less or equal)
2. Comparison and mapping of quanties (more or less or equal)<br>
3. Representation of quantity by numbers and learning the abstraction that “2 represents quantity 2 of a given thing”
3. Representation of quantity by numbers and learning the abstraction that “2 represents quantity 2 of a given thing”<br>
4. Numbers also have an ordinal value – that of ordering and that is different from the representation aspect of numbers
4. Numbers also have an ordinal value – that of ordering and that is different from the representation aspect of numbers<br>
5. Expression of quantities and manipulation of quantities (operations) symbolically
5. Expression of quantities and manipulation of quantities (operations) symbolically<br>
6. Recognizing the quantity represented by numerals and discovering how one number is related to another number
6. Recognizing the quantity represented by numerals and discovering how one number is related to another number<br>
7. This number representation is continuous.
7. This number representation is continuous.<br>
=== Lesson 1 : Quantity and Numbers ===
=== Lesson 1 : Quantity and Numbers ===
'''Teacher Note''' - this is not one period – but a lesson topic. There could be a few more lessons in this section. For example, for representing collections and making a distinction between 1 apple and a dozen apples. This idea could be explained later to develop fractions. Another activity that can also be used to talk of units of measure. Addition and subtraction have been discussed here – extend this to include multiplication and division)
'''Teacher Note''' - this is not one period – but a lesson topic. There could be a few more lessons in this section. For example, for representing collections and making a distinction between 1 apple and a dozen apples. This idea could be explained later to develop fractions. Another activity that can also be used to talk of units of measure. Addition and subtraction have been discussed here – extend this to include multiplication and division).<br><br>
'''Objectives of the activity'''
'''Objectives of the activity'''
#To develop an understanding that quantity is something we associate with objects and there are different ways of representing this and that different objects have different measures of quantity
#Represent quantity as numbers and express symbolically
#Understand the difference between the ordinal value of a number and the representation of a quantity
#Manipulate numbers, perform mathematical operations and express these operations symbolically in mathematical language.
[[File:picture1.png|400px]]<br>
<br>'''Materials needed'''<br>
<br>'''Materials needed'''<br>
*Apples in two groups (or any other fruit)<br>
*Small cards for writing down the numbers as well as for writing the operations<br>
*Pencils, etc<br>
*A collection of other objects – stones, sticks, notebooks<br><br>
'''How to do the activity'''<br>
'''How to do the activity'''<br>
#This whole activity can be done in groups with different sets of objects
#This whole activity can be done in groups with different sets of objects
#Keep two group of apples separately.
#Keep two group of apples separately.
#Keep them in physically different arrangements.
#Keep them in physically different arrangements.
#Based on visual inspection and counting, determine the number of apples and which group is larger or smaller or if both are equal.
#Based on visual inspection and counting, determine the number of apples and which group is larger or smaller or if both are equal.<br>
<br>'''Questions for discussion:'''<br>
<br>'''Questions for discussion:'''<br>
*How can you describe the two groups? [ Words – red, large, small, more, less]
*How can you describe the two groups? [ Words – red, large, small, more, less]
*Manipulating the quantities. By physically combining and representing them in numbers, introduce the idea of mathematical operations of addition and subtraction
*Manipulating the quantities. By physically combining and representing them in numbers, introduce the idea of mathematical operations of addition and subtraction
----
'''Part 1'''
'''Part 1'''
Bring the two groups of apples together. Add Group 1 to Group 2. Describe what happens [ Words – more, large, big, add]<br>
*Bring the two groups of apples together. Add Group 1 to Group 2. Describe what happens [ Words – more, large, big, add]<br>
What did this process do? I had two groups and I mixed them? What happened?<br>
*What did this process do? I had two groups and I mixed them? What happened?<br>
Can I mix them like this all the time? Why? Why not?<br>
*Can I mix them like this all the time? Why? Why not?<br>
If I have apples and notebooks can I mix them? Why? Why not?<br>
*If I have apples and notebooks can I mix them? Why? Why not?<br>
Introduce the idea of “+”. This process is called addition and is represented like this.<br>
*Introduce the idea of “+”. This process is called addition and is represented like this.<br>
Let us separate the two again. And count the number in each group. Add Group 2 to Group 1. Does it matter? Are the results the same?
*Let us separate the two again. And count the number in each group. Add Group 2 to Group 1. Does it matter? Are the results the same?
This can be written as 5 + 6 = 11<br>
This can be written as 5 + 6 = 11<br>
'''''The “+” is called a mathematical operator. The addition is called an operation. These operations have some properties and they always hold''.'''<br>
'''''The “+” is called a mathematical operator. The addition is called an operation. These operations have some properties and they always hold''.'''<br>
Separate the two groups again. Can I add only all of them together? Or can I add 1 at a time, 2 at a time, 3 at a time?
Separate the two groups again. Can I add only all of them together? Or can I add 1 at a time, 2 at a time, 3 at a time?
The ideas to be introduced would be Skip counting and quantity increase is continuous.<br>
The ideas to be introduced would be Skip counting and quantity increase is continuous.<br>
[http://karnatakaeducation.org.in/KOER/en/index.php/Addition_worksheet#worksheets addition worksheet]
'''Part 2'''
'''Part 2'''
Now from the large group take away some apples. What happens to the number? What have we done?<br>
*Now from the large group take away some apples. What happens to the number? What have we done?<br>
We have reduced the large group by 3 apples. This can be written as 11 – 3 =8.
[[File:picture2.png|400px]]<br>
When I mixed two groups what did we do? And what are we doing when we are separating the two groups? This will introduce words like increase, decrease, subtract, opposite.
*We have reduced the large group by 3 apples. This can be written as 11 – 3 =8. When I mixed two groups what did we do? And what are we doing when we are separating the two groups? This will introduce words like increase, decrease, subtract, opposite.<br>
Pose this question like this – “ I added 5 apples to six apples to make it 11 apples. What should I do to a bag of 11 apples to make it have 6 apples again”. I take away 5. This is called subtraction and is the inverse process of addition.
*Pose this question like this – “ I added 5 apples to six apples to make it 11 apples. What should I do to a bag of 11 apples to make it have 6 apples again”. I take away 5. This is called subtraction and is the inverse process of addition.<br>
'''''The “-” is called a mathematical operator. The subtraction is called an operation. These operations have some properties and they always hold''.'''
'''''The “-” is called a mathematical operator. The subtraction is called an operation. These operations have some properties and they always hold''.'''
[http://karnatakaeducation.org.in/KOER/en/index.php/Subtraction_worksheet#Subtraction subtraction worksheet]
----
'''Evaluation activities'''
'''Evaluation activities'''
*Create situations where children work in groups and compare collections of items without counting.<br>
*Create situations where children work in groups and compare collections of items without counting.
*Have groups work with 3 groups of fruits, sticks, pencils, etc. The idea that addition can be extended indefinitely can be discussed.<br>
*Have groups work with 3 groups of fruits, sticks, pencils, etc. The idea that addition can be extended indefinitely can be discussed.
*Subtraction with quantities without getting into negative numbers.<br>
*Subtraction with quantities without getting into negative numbers.
*Children must orally and in a written form represent these operations.<br>
*Children must orally and in a written form represent these operations.
*Create different sets of objects and ask the children to add [ to test for the legitimacy of addition]
*Create different sets of objects and ask the children to add [ to test for the legitimacy of addition]
*Adding to quantity in 2's, 3's - Combinations of 2 numbers that will give 10 as an answer. Can you guess the pattern for combinations of numbers that will give 20 as an answer? [ Subtraction is an inverse process of addition]
*Adding to quantity in 2's, 3's - Combinations of 2 numbers that will give 10 as an answer. Can you guess the pattern for combinations of numbers that will give 20 as an answer? [ Subtraction is an inverse process of addition]
*Worksheets of simple problem solving – with just number manipulations
*Worksheets of simple problem solving – with just number manipulations
*Worksheets of simple problem solving – with word descriptions. Children can either draw these answers or write them in statements. Evaluate the ability of the child to process written instructions.
*Worksheets of simple problem solving – with word descriptions. Children can either draw these answers or write them in statements. Evaluate the ability of the child to process written instructions.
*Worksheet: Number Stories
== Part II – Negative Numbers ==
== Part II – Negative Numbers ==
# To understand that negative numbers are numbers that are created to explain situations in such a way that mathematical operations hold<br>
# To understand that negative numbers are numbers that are created to explain situations in such a way that mathematical operations hold<br>
# Negative numbers are opposite of positive numbers; the rules of working with negative numbers are opposite to that of working with positive numbers<br>
# Negative numbers are opposite of positive numbers; the rules of working with negative numbers are opposite to that of working with positive numbers<br>
# Together, the negative numbers and positive numbers form one contiuous number line<br>
# Together, the negative numbers and positive numbers form one continuous number line<br>
# Perform manipulations with negative numbers and express symbolically situations involving negative numbers<br>
# Perform manipulations with negative numbers and express symbolically situations involving negative numbers<br>
=== Lesson 1 : The idea of negative numbers and operations on negative numbers ===<br>
=== Lesson 1 : The idea of negative numbers and operations on negative numbers ===
'''Objectives of the activity'''
'''Objectives of the activity'''
#Develop an idea that negative numbers are part of a type of numbers<br>
#Develop an idea that negative numbers are part of a type of numbers<br>
*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'''''
6. Now we transition from numbers representing some quantities to numbers being manipulated as numbers.<br>
6. Now we transition from numbers representing some quantities to numbers being manipulated as numbers.<br>
'''Part 2'''
'''Part 2'''<br>
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 -2
'''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. 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>
----
'''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>
==Part III: 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 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. <br>
[[File:Number system -Resource material_html_1de463c.jpg|400px]]<br>
-3 x -3
<br>Part of the power of addition and subtraction is that these operations work with both counting and measuring. Therefore, to understand basic operations like addition and subtraction, we need a number line model as well as counters.
=== The number line in teaching mathematics ===
One reason to use this mathematical object with students is that they need to see arithmetic in both contexts: counting and measuring. At the beginning, children may sometimes use the number line to find answers to arithmetic problems (e.g., figuring out what 3 + 10 is, before that becomes automatic to them) but that is never its purpose. We don’t rely on the number line for getting answers -- for that, we want the children to know basic facts and methods and use their heads -- but we do use the number line to understand things about the operation (addition and subtraction) and to understand what the answers mean.
For example, the answer to the subtraction problem 92 – 49 is the distance between those two numbers on the number line. That image can greatly help mental computation: 49 to 50 is one step, 50 to 90 is another forty steps, and 90 to 92 is another two steps, so altogether 43 steps. (See number line addition and subtraction below.) It also makes arithmetic with negetive numbers which has been already discussed. And the number line is essential for full understanding of fractions and decimals. In fact, a ruler (a number line!) is one of the important places students encounter and need to use fractions. <br>
'''In counting'''<br>[[File:picture3.png|400px]]<br>
Number lines are first used just to show sequence—numbers standing on line in order! At this stage, neither the straightness of the line nor the distance between numbers is mathematically important, though our images are always standard anyway. Children will look at chunks of the line, not always starting at 1, and will work forwards or backwards from some number that is placed on the line. They are learning about sequence and order, and that develops somewhat independently from counting. <br>
'''In measurement'''<br>
[[File:picture4.png|400px]]<br>
Students also learn about intervals on the number line, but just begin that process. Kids used to gain the interval idea (in a slightly different form) from their experience playing board games. They knew that when they rolled a 5, they had to count their five spaces beginning with the next space. That is, they were counting moves, not positions. Measurement depends on it. Addition and subtraction with counters does not depend on it, but those operations on the number line do depend on it. <br>
'''In addition and subtraction'''<br>
If we can add and subtract with counters, why use the number line? To connect these operations with measurement, and also because the counters no longer suffice when we get to fractions, decimals, and negative numbers. Over time, kids will connect number line images with thermometers, clocks, rulers (with fractional inches)… Coordinate graphs are based on perpendicular number lines; even bar graphs require the measurement idea more than the count idea, although they can begin with count. Addition and subtraction, or comparison, of distance is also why we use number lines. Adding distance is further developed on open number lines. Students develop many ways to subtract (in second grade, they learn to subtract 8 from anything by subtracting 10 and then compensating, and then they extend that idea to other additions and subtractions).
=== Hopping on the Number Line ===
In this lesson, students generate sums using the number line model. This model highlights the measurement aspect of addition and is a distinctly different representation of the operation from the model presented in the previous discussion. The order (commutative) property is also introduced. At the end of the lesson, students are encouraged to predict sums and to answer puzzles involving addition. <br>
'''Activity 1 - To introduce Number line'''<br>
'''Objectives'''<br>
#Represent the number line using objects.<br>
#Count each group of ones objects.<br>
#Relate the cumulative total of each group to the number on the number line. Write the cumulative total for each group.<br>
#Skip count by pointing to each group of objects.<br>
'''How to do the activity''' <br>
Encouraging students to come to the number line and skip count by ones pointing to the numbers on the number line that represent each one.<br>
Discuss a sample problem like this. Ram and his dad are driving to the city. It is 15 kms away. They have already gone 5 kms. How many more kms do they have to drive? <br>
After a minute or so, ask for a few children to share with the class.
You have to figure out how much farther they have to drive. You could keep going, like count up from 5 to 15. You could go maybe go backwards from 15 down to 5. Students will probably have a variety of ideas for solving the problem, including counting on from, or adding to 5 to reach 15, or counting backwards from 15 to find out how many kms remain. <br>
Summarize both approaches by writing the following equations below:<br>
5 + □ = 15 <br>
15 – □ = 5 <br>
'''Discussion'''<br>
We could say we just keep going from 5 up to 15, so we wrote 5 + box equals 15. What does the box mean in this equation? The students may say it means part you have to figure out. It’s where you write the answer. It’s like the problem you have to solve. 5 plus how many more to get to 15? <br>
Another interpretation is that it is like you’re finding out how far you have to go backwards to get down to 5. This is the idea that subtraction is the inverse process of addition. <br><br>
After taking students’ ideas, it is possible to introduce number lines as a new tool for solving problems like these. Number line can be drawn as a horizontal line across the board. Include an arrow on either end to show that the number line continues indefinitely in both directions. Record the smaller number (5) by marking and labeling a dot on the far left side. Then propose to move along the number line by hops greater than 1 to find the difference between 5 and 15. <br><br>
This discussion can be extended with the following question. What if Ram and his dad drive 2 more kms? How far will they be then? This can also be shown on the number line. What if they drove 3 more kms after that? How far would they be? The students should be able to say they are upto 10 miles!
They have gone 5 miles after the 5 because 2+3=5 and it is possible to know how many more miles they have to go to get to 15!
Ask students to suggest additional hops you could take along the number line to get to 15. <br><br>
You could keep going by ones, like 1 and then 2and so on more little hop up to 15. <br>
[[File:picture5.png|400px]]<br>
And this can be generalized for negative numbers also.<br>
[[File:picture7.png|400px]]<br>
'''Evaluation Questions'''
#How much farther did Ram and his dad have to drive to get to the city? <br>
#How do you know? Can you show us on the open number line? <br>
#Does this give us the answer to the problem? <br>
#Did we add or subtract to find the answer? <br><br>
'''Worksheet for hopping of numbers''' <br>
#[http://www.karnatakaeducation.org.in/KOER/en/images/f/f5/Number_line-missing_nos_-_level_1.odt click here] for level one, concept of hopping on number line.
#[http://www.karnatakaeducation.org.in/KOER/en/images/e/ee/Number_line-missing_nos_-_level_2.odt click here] for level 2 worksheet on hopping on the number line.
'''Activity2 - Sum and product of numbers'''<br>
'''Objectives'''
#Using the number line model to find sum and products <br>
#Solving and creating puzzles using the number line <br>
#Investigating the order property of addition and multiplication <br>
'''Materials'''<br>
Counters for the number line (chips, markers, etc.) <br>
'''How to do the activity(addition)'''<br>
Tell the students that they will find sums using the number line model. Then display a large number line and a 5+4 pencils , that is, a pencil with 5 spots on the left side and 4 spots on the right. Then demonstrate with a counter how a hop of 5 is taken on the number line. You may wish to encourage students to count aloud as the hop is made. Then make a hop of 4, starting at the place the counter landed. You might choose to have them record what happened using the equation notation 5 + 4 = 9, or to informally describe the moves this way: “If you take a hop of 5 spaces and then a hop of 4 spaces, you land on 9.” You may wish to highlight the fact that in this model, spaces are counted, not points on the number line.<br>
'''Questions for discussion'''
*Which number did you land on when you made a 5-hop, then a 3-hop? Could you land on the same number if you took a 3-hop first, then a 5-hop? How do you know?
The answers could be yes [ 5 + 3 = 8, and 3 + 5 = 8.]. Laws of computation can be introduced.<br>
*What sums did you model with hops? How did you record them? [Student responses will depend upon the "hops" they performed.]<br>
*Were any of the sums the same? Why? [Student responses will depend upon the "hops" they performed.]<br>
*How would you find the sum of 2 and 5? [Make a hop of 2, and then a hop of 5, to reach 7.]
*How would you tell a friend to add on the number line? <br>
*How is using a number line like measuring? How is it different?<br>
*Which students counted as they took hops and which moved directly to the number? <br>
'''Questions for teacher reflection'''
*Which students had trouble using the number line? What instructional experiences do they need next? <br>
*Did any children notice a connection with measurement? <br>
*What adjustments would we make the next time that we teach this lesson? <br><br>
'''Activity using GEOGEBRA'''
Addition using number line can be demonstrated using Geogebra.<br>
See image below<br>
Click [http://karnatakaeducation.org.in/KOER/Maths/numberlineAddition.html here] for an animation.<br>
Click [http://karnatakaeducation.org.in/KOER/Maths/numberlineAddition.ggb here] to download the Geogebra file.<br>
'''Bridge course Resource'''<br>
'''Topic:Number system'''<br>
'''Activity : Understanding numbers, properties through number line'''<br>
'''Time duration: 60 minutes'''<br>
'''Materials required: paper ,pencils/sketch pens'''<br>
'''Objectives:'''<br>
1. Understanding what is a number line<br>
2. Understanding properties(behaviour) of positive and negative numbers through number line<br>
3. Understanding number sentences /operations of numbers on the basis properties of numbers.<br>
4. Compare numbers/ordering numbers<br>
Pre-requisites:<br>
Students know about numbers,Numbers are continuous, counting, quantities for ex: difference between 2 books,2 kgs of apples or 2 litres of milk,2 mts .....<br>
'''Note for the teachers :'''<br>
Please emphasise that positive numbers increase the quantities and negative numbers decreases the quantity(amount)<br>
'''Procedure:'''<br>
[[File:Number line activity.png|200px]]<br>
'''Step 1'''<br>
Ask boys to write positive number and girls to write negative numbers on a sheet of paper. Each one should write one number. Instruct them to arrange the numbers in order on the floor.
Now ask boy and girl to come one after other to keep the number which they have written .
Suppose the first boy comes and keeps +5 then call other girl ask her to keep her number .
She can keep it anywhere she wants (imagine she has -7 on her sheet)
But the real fun starts when the third one(boy) will come(imagine he has +10 on his sheet)
You have to help him to keep the number in a proper order as he will have a positive number you should guide him where he should keep it and why?<br>
+5<br>
-7<br>
If the numbers are kept as above where +10 should be placed?why?<br>
For this we need to explain +5 has property of increasing the quantity by 5<br>
whereas -7 has a property of decreasing the quantity by -7,(+5 is more than -7) so where should be place for +10 <br>.
+10 has a nature of increasing the quantity more than what +5 does .so keep it above +5.<br>
(+10 is more than +5)<br>
+10<br>
+5<br>
-7<br>
If the next girl comes with -4 ,where she should keep it?<br>
Repeat the developmental questions like <br>
What is the behaviour of -7 and -4 <br>
(-7 decrease the quantity by 7 where as -4 decrease any quantity just by 4)<br>
Try to extract from them that -4 is more than -7.<br>
Continued pattern will be <br>
+10<br>
+5<br>
-4<br>
-7<br>
Place all the numbers what they have written on the sheet.<br>
'''Step:2'''<br>
Ask them <br>
Is there any number missed in between?<br>
If the answer is yes ask them again to write the missed numbers and repeat the activity by placing those numbers in a right place, which completes the number line. Now explain number line is a model for representing numbers .<br>
Now the number pattern will look like this after filling the gaps<br>
-9<br>
-8<br>
-7<br>
-6<br>
-5<br>
-4<br>
-3<br>
-2<br>
-1<br>
0<br>
+1<br>
+2<br>
+3<br>
+4<br>
+5<br>
+6<br>
+7<br>
+8<br>
+9 (You can also ask which is the last number?) The number arrangement need not be horizontal<br>
'''Step:3'''<br>
Now write some number sentences as below .Help them to interpret the sentences.<br>
4+(+3)=+7<br>
The number line is not just a school object. It is as much a mathematical idea as functions.
The value of 4 increased to +7 when it is added to +3 (+3 increases the value of 4 by how much?)<br>
4+(-3)=+1<br>
The value of 4 decreased to +1when it is added to -3 (-3 decreases the value of 4 by how much?)<br>
-4+(+3)=-1<br>
The value of -4 increased to +7 when it is added to +3 (+3 increases the value of 4 by how much?)<br>
-4+(-3)=-7<br>
Help them to interpret the above sentences.<br>
Give some examples to verify the resuts like<br>
-4+(-3)=-1 <br>
Do they able to check value of -4 should decrease by 3.But resultant value -1 is greater than -4<br>
So this is a wrong statement.Askthem to correct it.<br>
Prepare worksheets for practise.<br>
'''How to do the activity - Multiplication'''<br><br>
How to do the activity
On the overhead projector or chalkboard, display a large number line and demonstrate with a counter how hops of 5 can be taken on the number line. You may wish to encourage students to count aloud as the hops are made. You might choose to introduce the equation notation 4 × 5 = 20, informally reading it as "Four hops of 5, and you land on 20." After several examples with 5 as a factor, ask the students to determine what size hop to use next. Encourage the students to predict the products and to verify their predictions by moving a counter on the large numberline. You may wish to provide children with a counter and individual number lines at their desks.
On the overhead projector or chalkboard, display a large number line and demonstrate with a counter how hops of 5 can be taken on the number line. You may wish to encourage students to count aloud as the hops are made. You might choose to introduce the equation notation 4 × 5 = 20, informally reading it as "Four hops of 5, and you land on 20." After several examples with 5 as a factor, ask the students to determine what size hop to use next. Encourage the students to predict the products and to verify their predictions by moving a counter on the large numberline. You may wish to provide children with a counter and individual number lines at their desks.
parts of the curriculum for reinforcement. For example, when looking at
parts of the curriculum for reinforcement. For example, when looking at
shapes, talk about ‘half a square’ and ‘third of a circle’.
shapes, talk about ‘half a square’ and ‘third of a circle’.
A discontinuous whole is a group of items that together make up the whole. To find a fraction part of such a whole, we can divide it up into groups, each with the same number of items. We call such groups "equal-sized groups" or "groups of equal size". It is important that we always mention thathe groups are equal in size to emphasise this aspect of the fraction parts of a whole. Examples of discontinuous wholes are: 15 oranges, 6 biscuits, 27 counters, 4 new pencils, etc.
A discontinuous whole is a group of items that together make up the whole. To find a fraction part of such a whole, we can divide it up into groups, each with the same number of items. We call such groups "equal-sized groups" or "groups of equal size". It is important that we always mention thathe groups are equal in size to emphasise this aspect of the fraction parts of a whole. Examples of discontinuous wholes are: 15 oranges, 6 biscuits, 27 counters, 4 new pencils, etc.
Language patterns for a continuous whole To find 1/5 of my circular disc, I first divide the whole circular disc into 5 parts of equal size. Each part is 1/5 of the whole, and if I shade one of these parts, I have shaded of the 1/5 whole.
Language patterns for a continuous whole To find 1/5 of my circular disc, I first divide the whole circular disc into 5 parts of equal size. Each part is 1/5 of the whole, and if I shade one of these parts, I have shaded of the 1/5 whole.
find a fraction part of a unit whole, we have to cut/fold/break, etc. because the whole is a
find a fraction part of a unit whole, we have to cut/fold/break, etc. because the whole is a
single thing.
single thing.
same number as the denominator. It is thus made up of more that one item and is a
same number as the denominator. It is thus made up of more that one item and is a
discontinuous whole.
discontinuous whole.
denominator. This is also made up of more than one unit and so it is a discontinuous
denominator. This is also made up of more than one unit and so it is a discontinuous
whole.
whole.
denominator. This is also a multiple unit whole, and so it is a discontinuous whole.
denominator. This is also a multiple unit whole, and so it is a discontinuous whole.
Writing or ordering fractions in pre positioned boxes along number line. This can illustrate how a number line can be used to represent fractions of distance or length; or support the notion of a fraction being larger or smaller than another.
Writing or ordering fractions in pre positioned boxes along number line. This can illustrate how a number line can be used to represent fractions of distance or length; or support the notion of a fraction being larger or smaller than another.
Marking the fraction on an empty number line - this involves measurement and judgement of a fraction as a proportion of a length or distance.
Marking the fraction on an empty number line - this involves measurement and judgement of a fraction as a proportion of a length or distance.
the fractions with denominator equal to 5 are now displayed as shown:
the fractions with denominator equal to 5 are now displayed as shown:
To find equivalent forms of Rational Numbers
To find equivalent forms of Rational Numbers
Principal Roots and Irrational Numbers
Principal Roots and Irrational Numbers
Prerequisite Concepts: Set of rational numbers
Prerequisite Concepts: Set of rational numbers
Activity to find square root using geoboard
Activity to find square root using geoboard
Construct these triangles on a virtual geoboard. Provide students with the formula for the area of a triangle (A = ½bh) and ask them to determine the areas of the displayed triangles. [12.5 units2 and 4.5 units2.]
Construct these triangles on a virtual geoboard. Provide students with the formula for the area of a triangle (A = ½bh) and ask them to determine the areas of the displayed triangles. [12.5 units2 and 4.5 units2.]
Tell students that there are more squares that fit on a geoboard than those found during the previous lesson. Today's lesson will focus on finding some atypical squares using geoboards.
Tell students that there are more squares that fit on a geoboard than those found during the previous lesson. Today's lesson will focus on finding some atypical squares using geoboards.
7. Now, with radius AC and centre A, mark a point on the number line.
7. Now, with radius AC and centre A, mark a point on the number line.
Let the marked point is M. M represents √2 on the number line.
Let the marked point is M. M represents √2 on the number line.
[[File:Number system -Resource material_html_m489198fe.png|400px]]
TOPIC:NUMBER SYSTEM[[File:Number system -Resource material_html_m21619cb9.png|400px]][[File:Number system -Resource material_html_m6187dae0.png|400px]][[File:Number system -Resource material_html_m763a208c.png|400px]][[File:Number system -Resource material_html_m614b607b.png|400px]][[File:Number system -Resource material_html_m330e2a.png|400px]][[File:Number system -Resource material_html_m77c360a2.png|400px]][[File:Number system -Resource material_html_m7b853267.png|400px]][[File:Number system -Resource material_html_m7b853267.png|400px]][[File:Number system -Resource material_html_m6f20bce3.png|400px]][[File:Number system -Resource material_html_m3aee68a.png|400px]][[File:Number system -Resource material_html_m1f589acc.png|400px]][[File:Number system -Resource material_html_54627769.png|400px]][[File:Number system -Resource material_html_39498010.png|400px]][[File:Number system -Resource material_html_360ba54.png|400px]][[File:Number system -Resource material_html_60f99847.png|400px]][[:File:Number system -Resource material_html_58f2d11d.png]][[File:Number system -Resource material_html_33f3ff3f.png|400px]][[File:Number system -Resource material_html_4df41ad.png|400px]][[File:Number system -Resource material_html_4b50dacc.png|400px]]
== Teachers Corner ==
=== GeoGebra Contributions ===
# The GeoGebra file below demonstrates Integer Addition <br>
## Integer Addition http://karnatakaeducation.org.in/KOER/Maths/Integer_Addition.html
## Download ggb file here http://karnatakaeducation.org.in/KOER/Maths/Integer_Addition.ggb
# The GeoGebra file below demonstrates Sum of Natural Numbers
## corresponding angles http://karnatakaeducation.org.in/KOER/Maths/natsum.html
## Download ggb file here http://karnatakaeducation.org.in/KOER/Maths/natsum.ggb
## See a video for a proof by induction http://www.youtube.com/watch?v=WHVOLY9mrJ4
[[Category:Number system]]
