Integers

Table of Contents

Integers

One of the aims of math in school is for students to understand math concepts and their interrelations and understand the skills in a flexible, efficient, and practical way to solve daily problems, employing both previously taught and upcoming concepts. Students will use concept understanding to plan for the next course of learning and solve problems in their daily lives. learning integers will also serve as a basis for higher abilities such as the ability to apply mathematical concepts, use mathematical reasoning, analyze critically, and solve problems. Consequently, one of the mathematical ability requirements that must be satisfied is the student’s understanding of mathematical ideas.

Integers are made up of positive, negative, and zero numbers. Integers are an important topic to teach since they are one of the most beneficial tools for students. By learning integers, students can solve problems and use their understanding of numbers in real-life situations (Shlapentokh, 2011). Furthermore, when advanced pupils need to control their understanding of numbers to comprehend algebra, this is compatible with the +remark that integer comprehension must be acquired first to comprehend the subject algebra.

In mathematics, an integer is a collection of whole numbers and negative numbers which does not include fractions. As a result, integers may be defined as a collection of positive, negative, and zero values denoted by the letter “Z” on a number line (Fuadiah & Suryadi, 2019). Zero is a number without a sign and no negative or positive value. Whole numbers are represented on the number line by the value “Z+” on the right side of the zero, such as Z+ 1, 2, 3, 4, 5, while negative numbers are symbolized by the value “Z-” on the left side, such as Z- -1, -2, -3, -4, -5. The operations of integers encompass the integer procedures of addition, subtraction, multiplication, and division. Chamberlin & Powers (2010) noted that introducing the notion of signed numbers to a child’s mathematics education is critical. Integers, like fractions, expand the range of numbers in children’s mathematical worlds, resulting in a host of challenges. Some of the issues children experience in understanding negative numbers are comparable to those that mathematicians have had in the past. Most children’s experiences learning about numbers today, on the other hand, are very different from those of ancient mathematicians (Chamberlin & Powers, 2010). Diophantus, after all, had no instructor or textbook to contradict him when he rejected the possibility of harmful solutions to linear equations. This position was very conceivable given his number of conceptions. The history of integers inspired our review of current textbook strategies for integer teaching. The employment of conventional textbook methodologies is called into question by juxtaposing these two groups of data. Simultaneously, we have seen children effectively engage with positive and negative numbers in interviews before formal instruction. These findings suggest some potentially fruitful new directions for integer textbook instruction.

The discrepancy between magnitude and order that develops with negatives was a source of consternation for ancient mathematicians: -7 is more significant than 3. Yet, it appears before 3 on the number line, so we use the word “less than” to refer to this connection. Children exposed to negative numbers have the same problems (Chamberlin & Powers, 2010). However, our society has progressed to the point that today’s mathematically knowledgeable folks may accept negative numbers as a given. Thus, instructors may consider the introduction of negatives as a matter of fact when it is a mental revolution for youngsters.

It took a tremendous intellectual battle for great mathematicians like Descartes and Newton to accept negatives as integers. Therefore, it is no surprise that kids struggle to understand negative numbers and procedures utilizing them. At the same time, experts have shown that kids, especially in the lower elementary grades, can think about numbers in quite complex ways. Bishop et al. discovered, for example, that first graders who had never heard of negative numbers learned to construct and reason effectively about them while playing a number line game and completing open number sentences.

We utilize integers for a variety of purposes, including counting. The basic idea is that if three items are counted, the cardinal value obtained will be precisely equivalent to that obtained by counting them twice. Two parameters of the same ongoing physical quantity, on the other hand, will only produce the same answer twice due to error (for example, rounding error) because determining the value of an ongoing (that is, real-valued) quantity like length or timeframe with great precision is unthinkable in principle (zero residual uncertainty). Counting the representatives of a set, on the other hand, necessitates the use of integers, which means that recurring counts should yield accurate equality unless there is a counting mistake, such as when an object is jumped or double recorded (Chamberlin & Powers, 2010). It has been suggested that children understand a bidirectional mapping between preverbal magnitudes generated by the accumulator and the number of words. They claim that children understand the formal resemblance between the non-verbal and verbal counting processes presented. Both provide a one-to-one connection between a stably ordered set of symbols in one instance, consecutive magnitudes in the other, and the objects to-be-counted in the set. They both utilize the end symbol to denote the set’s cardinality. Categorical values are susceptible to mathematical rules, as Gelman has often stated. The interpretation and application of symbols are susceptible to mathematical computation in human and animal contexts. When counting is integrated into activities that require arithmetic processing, children who are still studying to count and do perform better and show more indication of grasping the numerical signifiers of the count words.

Bryant et al. (2020) suggested that those young toddlers still taught to count may reason about discontinuous numbers in a substitutability-aware way. They seem to understand the concept of additive inverses. When “added” to a set whose numerosity has been changed by “addition,” the additive inverse of a value recovers the set’s numeric values. We use the terms ‘addition’ and ‘added’ in scare quotes because we are using them in the technical meaning that includes subtraction. Since +3 and –3 are additive inverses, adding one negates the impact of adding another one; when a set is decreased by adding –3 to it (that is, removing 3), its numerical value may be restored by adding 3 to it.

Children’s conduct in the magic paradigm was initially revealed when they were faced with a dish from which one mouse had been secretly added or withdrawn, making it no longer a “winning” plate. The number shift caught the children’s attention, and they considered it crucial to determining whether the plate was a winner or not (unlike, for example, changes in item identities, which were not seen as critical). The most significant outcome is that even the youngest children thought that by adding one thing to the plate, it might be returned to winning status (not some items, but one item). This implies that they believed that adding one might cancel out the impact of deleting one and vice versa. It’s unclear why they believe this if their logic is entirely based on operations with distracting magnitudes. Adding a magnitude from a spread centred on one does not always reverse the impact of deleting a magnitude from within the identical distributions (Leslie, Gallistel & Gelman, 2007). Indeed, as previously stated, it is unclear how one could even certify a return to the status quo ante in a system that solely processed loud magnitudes.

“Numerical value” or “magnitude” are other terms for “absolute value.” Basing on a number line, the absolute value of a number is the distance from zero. With the values between them, the modulus symbol, ‘| |’, represents the absolute value. For instance, the absolute value of 7 is written as |7|. The absolute value is the distance between a number and the start on the number line; as it shows the polarity of the number, whether positive or negative. Because it indicates distance, and distance can never be negative, it can never be negative. As a consequence, it’s always a positive situation. Because the absolute value is a distance measurement, it can never be negative. A sign is occasionally applied to a numeric number in addition to the value to identify the direction. To describe a rise or decline in quantity, values surrounding the mean value, profit, or loss in a transaction, a positive or negative value is sometimes attributed to a numeric number (Hertel & Wessman-Enzinger, 2017). In absolute value, the sign of the numeric value is ignored, and just the numeric value is considered. The absolute value just represents the numeric value and does not include the sign of the numeric value. There is no absolute value for zero since the absolute value turns the sign of the integers into positive, while zero has no sign. If the number is positive, it will only be returned as a positive number. If a number is negative, the modulus of that number will also be positive. The modulus of any vector quantity is always positive and equal to the absolute value. Absolute values also represent quantities like distance, cost, volume, and time. An example is the absolute value of |+4| = |-4| = 4. There is no symbol associated with absolute value.

Lesson 3:  Adding Integers

The technique of getting the sum of two or more numbers is known as adding integers. The value may change depending on whether the integers are positive, negative, or a mixture. The sum of integers of the same or different signs is found using the addition of integers arithmetic procedure (Stephan & Akyuz, 2012). A number of rules to follow when adding two or more integers. Integers are whole numbers that aren’t split into fractions. Positive integers, zero, and negative are all included.

Things become a little complicated when we speak about negative numbers. A “-” symbol appears in front of a negative number. -4, for example, denotes a negative number. Let’s look at a scenario involving your bank account. If you have $10 in the bank and the bank unintentionally takes $5 out, your balance will be $10 – $5 = $5. You’d have $5 in your bank account. Because the bank made a mistake and wants to deduct a mistake (the -$5), the equation will be $5 – (-$5) = $5 + $5 = $10. They will delete the negative (in this example, the error), which will result in a positive result, and you will have $10 in your bank account once again. When we talk, we may see an example of two negatives in real life by using double negatives. We would tell you not to cross the road if we said so. However, if we instructed you not to cross the road, that implies you should. You may use this to assist you in similarly adding negative numbers. When you see two similar signs, add the numbers together (two “-” signs, or two “+” signs, for example). If you observe two opposite signs (for example, one “-” and one “+”), it becomes a negative, and you must subtract it.

The purpose of subtracting integers is to find the difference between two numbers. Whether the integers are positive, negative, or a combination may culminate in a value gain or loss. The difference between two numbers of similar or different signs is calculated using the arithmetic process of integer subtraction. Integer subtraction has a set of criteria that must be followed to obtain the desired results, such as the fact that subtracting 0 from any number yields the integer itself. We may derive the additive inverse or counterpart of any number by subtracting it from zero. By altering the sign of the subtrahend, you may subtract integers. If both numbers have the same sign, we sum the absolute values and impose the common sign after this step. If the signs of the two numbers vary, we compute the difference and use the more significant number’s sign to substitute the smaller number’s sign. By studying the subtraction of the positive integer and the flags that we perceive subtraction with, we can explain the basic rules by using patterns to support the expected standards (Fuadiah & Suryadi, 2019). For example, in (-4) (2), you start with four negative counters and delete two. Just two negative counters are remaining.

Multiplication is a fundamental mathematical operation. Integers are a broad category of numbers containing positive and negative numbers. The process of repeatedly adding integers while conforming to a set of rules and qualities is known as integer multiplication. The recommendations will help you deal with a variety of mathematical problems efficiently. Multiplication’s commutative, associative, distributive, identity, and zero properties help solve complex mathematical problems. Please continue reading to discover more about Integer Multiplication and how to solve problems with it.

The first rule of dividing two integers, both positive or negative, is that the division of their fundamental values is a positive integer equal to the quotient of their fundamental values. Consequently, when dividing two integers with similar signs, we divide the numbers without respect to sign and add a plus sign to the quotient.

The other condition of a positive and a negative integer quotient is a negative integer. The absolute value is equal to the quotient of the corresponding absolute values of the integers. When dividing integers with opposite signs, we divide their values regardless of their sign and assign a negative sign to the quotient (Fuadiah & Suryadi, 2019). As with multiplication, divide the integers without the sign and give the symbol as per the table’s rule. Divide two integers with like signs to get a positive quotient, and divide two integers with unlike signs to get a negative quotient.

Mixed operations combine whole numbers and negative numbers, while integers are the set of whole numbers and negative numbers. The characters (I) or (Z = left – 3, 2, – 1,0,1,2,3… right) signify the set of numbers. All of the basic mathematical operations on whole numbers or natural numbers may be carried on integers. In this lesson, we will look at how to add and subtract integers using a number line. We will also look at mixed-integer operations and the rules for executing them. The PEMDAS or BODMAS rule is utilized (Hill & Charalambous, 2012). Certain basic concepts must be followed in a precise sequence when doing many operations, such as addition, subtraction, multiplication, and division. First, solve the numbers enclosed inside the parenthesis or bracket to use the expression. From the inside out, we cluster the operations and solve the parenthesis. We should also take notice of the phrase’s bracket pattern. The round bracket (()), followed by the curly bracket ({}), and lastly, the box (square) bracket is all solved ([]). The order of operations should adhere to the brackets. After working on parentheses, we look for exponents. Resolve any issues that may have arisen. We’ve narrowed our attention to the four core businesses. This step will look for numbers that can be multiplied and divided. If they are inexistent, we solve them from left to right. The last operations to finish are addition and subtraction, which must be solved from left to right.

Integers are similar to whole numbers in that they include negative values but do not include fractions. Integers are employed in various applications, particularly in the business world, where profits are represented by positive numbers and losses by negative integers. In addition, if positive numbers indicate the height above sea level, negative integers represent the depth below sea level.

Both instructors and students have difficulties when it comes to the expansion of natural numbers. On the path to real numbers, children are encouraged to broaden their mathematical horizons to append negative integers and (non-negative) rational numbers. These extensions test students’ preconceptions, which often contain overgeneralizations of their natural-number experiences. In the rational number domain, there is a lot of study on teaching and learning. However, few papers address the difficulties that pupils have while learning numbers. Students struggle with integers and integer operations on a conceptual level. In view of the world of mathematics, when mathematicians battled with paradoxical conceptions related to negative numbers, these issues are understandable.

 

 

References

Bryant, D. P., Bryant, B. R., Dougherty, B., Roberts, G., Pfannenstiel, K. H., & Lee, J. (2020). Mathematics performance on integers of students with mathematics difficulties. The Journal of Mathematical Behavior, 58, 100776.

Chamberlin, M., & Powers, R. (2010). The promise of differentiated instruction for enhancing the mathematical understandings of college students. Teaching Mathematics and Its Applications: An International Journal of the IMA, 29(3), 113-139.

Fuadiah, N. F., & Suryadi, D. (2019). Teaching and Learning Activities in Classroom and Their Impact on Student Misunderstanding: A Case Study on Negative Integers. International Journal of Instruction12(1), 407-424.

Hertel, J. T., & Wessman-Enzinger, N. M. (2017). Examining Pinterest as a curriculum resource for negative integers: An initial investigation. Education Sciences, 7(2), 45.

Hill, H. C., & Charalambous, C. Y. (2012). Teacher knowledge, curriculum materials, and quality of instruction: Lessons learned and open issues. Journal of Curriculum Studies44(4), 559-576.

Leslie, A. M., Gallistel, C. R., & Gelman, R. (2007). Where integers come from. The innate mind: Foundations and the future3, 109-149.

Shlapentokh, A. (2011). Defining integers. Bulletin of Symbolic Logic, 17(2), 230-251.

Stephan, M., & Akyuz, D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education, 43(4), 428-464.

 

UNIT PLAN

Unit Title or Key Question

 

 

Lesson Number Lesson Title/ Intention/ Enquiry Question Learning Objectives/ Success Criteria

 

Activities Resources Assessment/ Checking Learning
1ST lesson  

An introduction to integers

 

·         To be able to understand the meaning of integers.

·          To be able to identify positive and negative integers on a number line

·         To be able to identify real-world examples of integers

·         order integers from least to greatest and most significant to least

  • Create an entry in their math journals titled ‘Ordering Integers.’
  • students attempt to put the following numbers in order
  • engage students in a discussion
  • Share and discuss answers to exercise
  • standard deck(s) of cards
  • chalkboard/whiteboard
  • marker pen
  • number line

 

  • Ask students what they generally know about integers.

 

  • Students working in a group to try order integers on a number line

 

  • Giving students feedback about their work.

 

 

2nd lesson Absolute value ·         To be able to determine the absolute.

·         To be able to apply absolute values in real-world contexts.

·         compare numbers using absolute value

·         to be able to apply absolute values in real-world contexts

·         use a number line to represent values on a number line

·         response to questions concerning absolute values

·         compare numbers using absolute value

 

·         Blank index cards

·         Index cards with absolute values written on them in sets

·         Counters for the number line

·         compare numbers using absolute value

 

·               Write a number line on the board with a few missing numbers and ask the student to complete them

·         Ask student

compare numbers using absolute value

·         Discuss responses to the question: “What is the difference between taking the opposite of an integer and finding its absolute value?”

3rd lesson Adding integers
  • To be able to understand the basic definitions of addition
  • To be able to estimate the sum of numbers
  • To be able to add integers with like and unlike signs
  • Add positive and negative integers together by using a visual aid.

 

  • Use two different types of cereal to practise adding positive and negative integers
  • Answer some practice questions on the addition of integers
  • white butcher paper
  • A worksheet was created using the questions from the associated lesson quiz.
  • Box of rice Chex cereal
  • Students will be given an exercise on addition and marked
  • Feedback will be given back to students on their score
  • Students will be asked to brainstorm what they learn on the addition of integers
4th lesson Lesson 4: Subtracting Integers
  • To demonstrate their understanding of integers by extending their knowledge of adding integers to subtracting integers
  • To understand why a negative number subtracted from a negative number changes the operation to addition
  • To be able to outline the necessary steps for subtracting integers
  • Learning steps to subtract integers
  • Learn real-life examples of subtraction
  • A worksheet with 15 integer subtraction problems
  • Number line
  • marker pen
  • whiteboard

 

  • Asking the students to solve two subtracting problems and share the answer with the class
  • Giving out homework assignments to gauge if the students  have grasped the concept of subtracting integers
5th  Lesson Division of Integer
  • Able to divide positive and negative integers,
  • Able to solve real-world problems that involve dividing integers.
  • The student will be able to restate that the quotient of two integers with unlike signs is a negative integer
Learn how to divide integers

Learn basic operations in divisions

Learn real-life examples of division

  • whiteboard
  • use a worksheet
  • marker pen
  • Integer division activity sheet
  • Provide students with multiplication/division problems
  • Ask students to explain their answers by talking through the analogy.
6thLeson Multiplying integers
  • The student will be able to restate that the product of two integers with unlike signs is a negative integer.
  • Understand the rules necessary to solve problems with integers using multiplication
  • Being able to use multiplication in a real-life situation
  • Find the products of integers
  • Learn real-life examples of multiplication
  • Math notebook/journal
  • Class set of number lines (laminated)
  • Counter chips
  • Integer multiplication activity sheet
•           Asking students to solve two multiplication questions and share the answer with the class
7th lesson Mixed operations
  • appreciate the order of operations, including brackets addition, subtraction, multiplication, and division
  • Evaluate expressions and solve problems using integer operations
  • Learn BODMAS rule
  • solve problems with integers using addition, subtraction, multiplication, and division
  • Evaluate expressions and solve problems using integer operations
  • Learn real-life examples where mixed operations are used
  • standard deck(s) of cards
  • chalkboard/whiteboard
  • Marker pen
  • Sets of playing cards
  • BODMAS chart
  • Evaluate expressions and solve problems using integer operations by asking questions
  • Giving students feedback about their work

 

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