Developing the number-line concept while teaching addition and subtraction This Self-Sustained Educational Module developed by Dr. Victoria Kofman Fluency with number properties is one of the main prerequisites for the successful learning in Algebra class. For those who come to the Algebra class without deep understanding of the number properties in early grades, it occurs to be an unattainable task to grasp all these properties in their abstract form. To prepare our students for the challenges of Algebra, we teach number properties, concept of variable, and other algebraically relevant approaches in a course of Arithmetic where students utilize the variety of concrete approaches and semiabstract models. Here we present the games and exercises that target the development of the concept of number-line in primary grades up to the level of fluency in its application. In addition, within the presented here self-sustained educational module, students learn to add and subtract by parts, start developing the concept of addition as a reciprocal operation to subtraction, start developing the idea of substitution, and start recognizing property of identity in terms of addition and subtraction (a + = a; a =a). Playing Numberville games and performing semiabstract assignments students move from concrete to abstract internalizing the concept of number line during this process. In addition, students learn to add and subtract small quantities using that concept. (Preschool, Kindergarten, and st grade)
Numberville. Age group: Pre-school, Kindergarten, and st grade. Objectives: Students internalize the concept of number line and apply this concept to add and subtracts small quantities (±, ±). In addition students learn that subtracting a number from itself gives zero. Students create their own word problems. As an application of number-line concept, students learn to add and subtract and. Prerequisites: Students have to recognize all the numbers from to, count them forward and backwards as well as be able to tell what number goes after the given number or before the given number. In Numberville, all the houses stand in a row. Like this. Overview The inhabitants of Numberville live by the next rules. If someone has coins, he lives in a house number. If someone has coins, he lives in a house number. Illustrations Anna had coins and lived in a house number. If she will lose a coin, she will have to move to a house number. = Anna had coins and lived in a house number. If she will find a coin, she will have to move to a house number. + = Suinim has coins and live in a house number. He went for a walk. If he not lose or find any coins, he will stay at the same house. = ; + = An eraser lived in a house number. He went for a walk and lost his coins. Now he lives in a house number. =
Concrete Numberville. Game Materials: Eight houses are drawn on the board in order from to with the numbers on them; bright and big coins without numbers on them; a toy that carries coins. I. A teacher shows a toy with a given number of coins and places it into its house. II. The teacher demonstrates how the toy picks (looses) one coin. Students are trying to answer the question: Where it will live now? III. Teacher places the toy into his new house Concrete Hotel. Game Materials: Sets of number cards (red face up, green face down; each card has the same number on both sides) lay on a table representing hotel; coins; a small toy, a bell; boxes for coins. I. A client (student A) with, let say, coins comes to the hotel. II. A concierge (student B) takes coins, counts them and opens a room by flipping card number. Then he puts coins into a bank (a box) III. The client puts his toy on a green card and pretends that the toy is asleep. IV. When a teacher rings the bell, the client takes his toy, and the concierge flips the card back. Now all the cards have red side up, and the game starts all over again.
Semiabstract Number town. Game Materials: On a board, the houses with the numbers are drawn in order from to I. A teacher tells that someone has coins. Where does he live? II. The teacher tells that creature had found one more coin. Where it goes to live? III. The teacher tells that the creature had lost one coin. Where it goes to live now? If necessary, teacher adds features from the previous stage. Semiabstract Hotel. Game Materials: The set of big number cards (one face is red, and another is green. Each card has the same number on both faces. The cards are pinned to the board red face up.); flat light toy that can be pinned or hanged on the top of a card; small bell I. A teacher tell that his toy has, let say, coins and ask students to open (flip the card) a room for him. II. A teacher asks one student go to the board to flip a card and place a toy onto his room. III. After the student returns to his seat, a teacher rings the bell, and the toy awakes and goes for a walk. IV. The teacher tells that the toy found (lost) one coin, and students have to decide where to put the toy now.
Adding and subtracting or using a number line Semiabstract Abstract ) + = ) = ) = ) + = ) + = ) = ) + = ) = ) + = ) = On this stage, students have a number-line before them. Instead of manipulatives, they use their own story problems. Student s story. The eraser lived in a house number, and then the eraser found one more coin. Now it lives in a house number. Student s story. A girl lived in a house number, and then she found (lost) nothing. She stays in a in a house number.. Adding and subtracting or when a number line is not available Subtracting a number from itself No number line to be seen! ) + = ) = ) = ) + = ) = ) = ) + = ) = ) + = ) = Semiabstract Abstract On this stage, students have no number-line before them. The list of answers on this stage is a non-ordered list of numbers. Students still use the story problems that they create. Problems like = are added to the list. Student s story. Peter lived in a house number, and then lost his coins. Now it lives in a house number. Deviations Some students add and subtract by memorization and have no concept of number line formed even in rd grade. For these students the problems should include numbers up to, with addition and subtraction of. These students are older and have to write down their answers. Post-results: Abstract After a student had formed the concept of number line, he stops using a story problems since has no need in it. At that stage the student is ready to apply the learned concept. A student does not start applications until he is perfect with addition and subtraction of without using a story problem.
Applying the concept of number line Learning addition and subtraction of # Semiabstract Abstract ) + + = ) = ) = ) + = ) = ) = ) + + = ) = ) + + = Students perform two subsequent subtractions or additions of. Being challenged, some students will start using their story problems again. Students do not start next level before solving these problems without the story problems. Student s story. Kevin lived in a house number, and then he lost coin. Now he has coins, and he again lost a coin. He has coins left. # Semiabstract Abstract ) + = Being challenged, some students will start using their story problems again. See Student s story. ) = ) = ) + = ) = ) = ) + = ) = ) + = Learning addition and subtraction of Others might use fingers or two taping of fingers to help themselves with calculations. Students do not start other levels until they can add or subtract not using story problems or fingers in at least % of problems # Semiabstract Abstract ) + = Student s story. Six and two is. Eight and one is. ) + = ) = If a student is getting lost, he can add (subtract) by using ) + = ) fingers. + = ) = First, the student has to straighten fingers. After adding ) + = he bends fingers, and after adding one, he bends his ) = last straighten finger. ) + =
Learning addition and subtraction of # Semiabstract Abstract ) + = Student s story. Six and two is. Eight and two is. ) + = ) = If a student is getting lost, he can add (subtract) by using ) + = ) fingers. + = ) = First, the student has to straighten fingers. After adding ) + = he bends fingers, and after adding another, he bends ) = another fingers. ) + = Learning to solve equations by trial. The concept of substitution # Abstract ) + = ) = ) = ) + = ) = ) = ) + = ) + = ) + = In these equations the second number is missing and only three roots are possible:,, and. Student s story. Six and is. Does not work. Six and is. Does not work. Six and is. It works. If students have difficulties, tell them to try only numbers,, or. # Abstract ) + = ) = ) = ) + = ) = ) = ) + = ) + = ) = In these equations the second number is missing and only roots are possible:,,.,, and the number that we have to subtract from itself to get. Hint: If someone had coins and then nothing, how many coins he had lost? Hint: You tried,, and. Have you tried or zero? Does it work? Further practice with addition and subtraction of the numbers up to using the concept of number line as well as solving simple equations provides students with a solid foundation for mental addition and subtraction.
Taking and putting back. Age group: Kindergarten, st, and nd grades. Objectives: Students learn to find a difference between close numbers by using addition as opposite to subtraction. Student learn algebraic concept of reciprocal operations. Prerequisites: Students fluently add and subtract,, and from numbers up to. The students who can deal with the numbers up to, would have even more fun. Students can solve simple equations on addition and subtraction where the second term is missed and the roots can be,,,, and. Overview When a big number is taken from another number, it looks like that There were coins on a plate (I). Seven coins were removed (II). How many coins are left on the plate, i.e., what is the result of? +? =? By putting coins back on the plate (III), we help students understand how instead of complex subtraction problem solve a simple addition problem, + =. Illustration I have erasers. I put of them in my hat and the rest keep in my hand. How many erasers do I keep in my hand? Hint: Look, when I put the rest of erasers in the hat, how many erasers are there? Yes, But there were only erasers in a hat, so how many erasers did I put in? =? => + =
Taking and putting back Materials: coins, plastic/paper plates, pieces of paper Concrete I. A teacher writes on a board a problem, for example, =? II. Teacher asks a student to put coins on a plate. A student does it. III. Teacher takes coins from a plate covering the rest of coins with a piece of paper. IV. Teacher puts coins back on a plate taking away the piece of paper. Teacher with students count coins and see that result is. V. Teacher again takes coins from a plate and covers the rest of coins. VI. Teacher puts coins back on a plate and asks how many coins are under the paper. Semiabstract Taking and putting back Materials: coins, plastic/paper plates, pieces of paper I. A teacher writes on a board a problem, for example, =? (or =?) II. Teacher puts a pile of coins on a plate. He does not count them. He just tells that there are coins in a pile. III. Teacher moves all but coin from the plate telling that he removed from the plate coins. The last coin he keeps in his hand. IV. Teacher asks students to figure out how many coins are in his hand. V. If necessary, teacher provides hint: He returns the pile of coins ( coins) on the plate and tells: If I will add the coins that I have in my hand, we will have again our coins. How many coins do I have in my hand?
Adding and subtracting or using a number line Abstract # Semiabstract-Abstract ) = ) = ) = ) = ) = ) = ) = ) = ) = Hint: You have to take from. Let say you took it and something left. When you put back and that something you will get again. What is your number? Deviations Some students add and subtract by memorization and have no concept of number line formed even in rd grade. For these students the problems should include numbers up to, with addition and subtraction of. These students are older and have to write down their answers. Post-results: Abstract After a student had formed the concept of number line, he stops using a story problems since has no need in it. At that stage the student is ready to apply the learned concept. A student does not start applications until he is perfect with addition and subtraction of without using a story problem. Conclusion: To summarize, we analyzed here several games and exercises that help students to develop the concept of number line and apply it in solving arithmetic problems. Author believes that the proposed approach prepares students to the further challenges of mathematics by targeting students conceptual development instead of requiring them to memorize math facts. From our experience follows that the students who taught using presented here approach, became much more enthusiastic and successful in mathematics and significantly increased their problems solving skills and attention span.