Curriculum Topic Study: Rational Numbers
Section I. Identify Adult Content Knowledge
"Rational Numbers" is a topic in which students will learn what a rational number is, how to express rational numbers, the difference between rational and irrational numbers, how to simplify irrational expressions, and how to graph irrational expressions. Rational numbers are used on a daily basis by students and adults alike because any whole number is a rational number since it can be written in the form x/1. In addition, many adults use rational numbers to make comparisons between quantities. Whether adults use rational numbers through ratios or even while they participate in activities like baking which has ingredient measurements in rational number form, it is a topic that occurs so often, that adults should be able to define rational and irrational numbers.
It is evident that the aim is to develop student learning in this topic through connections from previous knowledge. It is important that students build on their knowledge in this topic, progressing from learning general fractions in elementary school to fraction operation in middle school and lastly, learning about rational and irrational number, which connects to the fractions they will have previously learned, in high school. Teachers and adults should know how to define a rational and irrational number, and should recognize their common use of rational numbers in their daily life. Through this progression of knowledge, adults should be able to reflect back on fractions, fraction operation, rational numbers, and finally simplifying rational expressions as they relate to their day to day lives.
Section II. Consider Instructional Implications
As aforementioned, it is encouraged that students learn about rational numbers through a logical progression of information, relating back to previously learned topics to build new knowledge. At the earlier level, students learn about fractions in a very broad sense. They learn to relate fractions to a commonly used fraction, such as 1/2, so that if they are adding two fractions together, they can logically determine whether the sum would be greater than or less than 1. In these earlier stages of fraction instruction, students will not be instructed with fraction operations. In fact, students are not likely to learn operations with fractions until middle school. Then, once they learn about fraction operations, they progress into learning about rational numbers. This is when students can draw a connection to their previous knowledge of fractions, realizing that a rational number can be written as a fraction with both the numerator and denominator as whole numbers.
An implication of the readings is for the instructor to really remain aware of the information students need to be learning and at what stage in their process they are in. Instructors likely know all of the information the students will learn by the end of this topic, however, they need to deliver the information properly, only teaching the student what they need to learn at a given moment in time. Sometimes, teachers may get carried away with the information they know, and start diving into information their students do not actually need to know. Thus, the readings discuss the importance of instructing students by helping them make the connections to their previous knowledge in order to build their knowledge of rational numbers.
Section III. Identify Concepts and Specific Ideas
By the 9-12 grade level, students working with the topic, "Rational Numbers", should become more confident in their ability to make computations between different number systems and use arguments outside of basic arithmetic as justification to their claims. At this point, they should have worked past a main focus of being able to convert between decimals and fractions, and compute fraction and decimal operations. In addition, they should be comfortable with the inverse relationships between addition and subtraction, and multiplication and division, and they should have developed strategies to estimate result of rational number computations. It is important that they have reached the learning objectives from their previous math classes because a key concept of Rational Numbers in high school is working comfortably with arithmetic operations of rational numbers.
Language and vocabulary choice is key in this topic. As we will see with the research in the next section, it is easy for students to have misconceptions between the ideas of rational numbers and fractions. Setting expectations of the students to use the key terms within the unit, such as rational, irrational, polynomials, significant figures, and so on, can set the tone for the remainder of the lesson. In order for them to make the appropriate connections to past, present, and future knowledge, the students will need to use the appropriate mathematics terminology within this unit. The words used in our concept map represent the concepts and specific ideas mentioned in the Benchmarks and will help reach our learning goals. Some of them will prove to be more important than others within the topic, but each of them will be applicable to the topic "Rational Numbers".
"Rational Numbers" is a topic in which students will learn what a rational number is, how to express rational numbers, the difference between rational and irrational numbers, how to simplify irrational expressions, and how to graph irrational expressions. Rational numbers are used on a daily basis by students and adults alike because any whole number is a rational number since it can be written in the form x/1. In addition, many adults use rational numbers to make comparisons between quantities. Whether adults use rational numbers through ratios or even while they participate in activities like baking which has ingredient measurements in rational number form, it is a topic that occurs so often, that adults should be able to define rational and irrational numbers.
It is evident that the aim is to develop student learning in this topic through connections from previous knowledge. It is important that students build on their knowledge in this topic, progressing from learning general fractions in elementary school to fraction operation in middle school and lastly, learning about rational and irrational number, which connects to the fractions they will have previously learned, in high school. Teachers and adults should know how to define a rational and irrational number, and should recognize their common use of rational numbers in their daily life. Through this progression of knowledge, adults should be able to reflect back on fractions, fraction operation, rational numbers, and finally simplifying rational expressions as they relate to their day to day lives.
Section II. Consider Instructional Implications
As aforementioned, it is encouraged that students learn about rational numbers through a logical progression of information, relating back to previously learned topics to build new knowledge. At the earlier level, students learn about fractions in a very broad sense. They learn to relate fractions to a commonly used fraction, such as 1/2, so that if they are adding two fractions together, they can logically determine whether the sum would be greater than or less than 1. In these earlier stages of fraction instruction, students will not be instructed with fraction operations. In fact, students are not likely to learn operations with fractions until middle school. Then, once they learn about fraction operations, they progress into learning about rational numbers. This is when students can draw a connection to their previous knowledge of fractions, realizing that a rational number can be written as a fraction with both the numerator and denominator as whole numbers.
An implication of the readings is for the instructor to really remain aware of the information students need to be learning and at what stage in their process they are in. Instructors likely know all of the information the students will learn by the end of this topic, however, they need to deliver the information properly, only teaching the student what they need to learn at a given moment in time. Sometimes, teachers may get carried away with the information they know, and start diving into information their students do not actually need to know. Thus, the readings discuss the importance of instructing students by helping them make the connections to their previous knowledge in order to build their knowledge of rational numbers.
Section III. Identify Concepts and Specific Ideas
By the 9-12 grade level, students working with the topic, "Rational Numbers", should become more confident in their ability to make computations between different number systems and use arguments outside of basic arithmetic as justification to their claims. At this point, they should have worked past a main focus of being able to convert between decimals and fractions, and compute fraction and decimal operations. In addition, they should be comfortable with the inverse relationships between addition and subtraction, and multiplication and division, and they should have developed strategies to estimate result of rational number computations. It is important that they have reached the learning objectives from their previous math classes because a key concept of Rational Numbers in high school is working comfortably with arithmetic operations of rational numbers.
Language and vocabulary choice is key in this topic. As we will see with the research in the next section, it is easy for students to have misconceptions between the ideas of rational numbers and fractions. Setting expectations of the students to use the key terms within the unit, such as rational, irrational, polynomials, significant figures, and so on, can set the tone for the remainder of the lesson. In order for them to make the appropriate connections to past, present, and future knowledge, the students will need to use the appropriate mathematics terminology within this unit. The words used in our concept map represent the concepts and specific ideas mentioned in the Benchmarks and will help reach our learning goals. Some of them will prove to be more important than others within the topic, but each of them will be applicable to the topic "Rational Numbers".
Section IV. Examine Research and Student Learning
The research on the topic, "Rational Numbers", points out misconceptions which will help shape key details within the unit plan. In this topic, students tend to have difficulty with concepts about decimals and fractions. To be specific, students may have misconceptions about decimals and how they aid in representing the measurements of concrete objects. Along with the topic of decimals, research shows that students have trouble making a model of the decimal's value. In an example using the value .96, students would struggle to show, using rectangles, that 96 out of 100 rectangles would have to be shaded, or equivalently, 24 out of 25 rectangles would be shaded. A key in understanding these concepts is recognizing decimals with their fraction forms. If a student can make the conversion between a decimal to a fraction, they will be able to represent the value visually with more ease. To proceed further within decimal misconceptions, research supports the claim that students find it hard to distinguish the largest and/or smallest value in a set of decimals with varying amounts of digits to the right of the decimal point. So, given a set {.10, .101, .1001}, students may not understand which value is the largest.
Next, I got a deeper look into the misconception that students face confusion when it comes to recognizing the difference between rational numbers and fractions. While fractions are ratios of two numbers, rational numbers are to be considered as one number. This can be confusing since rational numbers are defined as the quotient of two integers, p and q, where the divisor cannot equal zero. Overall, the concept of rational and irrational numbers can even be misleading for the teachers creating lessons, and it is important that the instructors for the lesson design it with caution and the benchmarks and learning objectives in mind to avoid misconceptions and confusion.
Section V. Examine Coherency and Articulation
The readings in this section reveal the logical connections between material learned earlier in education all the way through to the 9-12 grade level. The concept map shows the early developmental blocks of learning in the topic and link it in as many ways that make sense to other aspects of the topic which come later. This can really help the teacher visualize a large amount of the interrelating topics with the specific topic they are covering and learn more about the progression of knowledge from Kindergarten to grade 12. For example, when students reach the idea that "a/b" can mean three different things (a divided by b, a as a ratio of b, or a parts of size of 1/b each), it comes from the concept learned in 3-5 that states, “fractions are numbers we use to stand for a part of something". This links to the concept from K-2, which says “sometimes in sharing or measuring there is a need to use numbers between whole numbers” which implies using fractions.
This type of development of the topic which can also be helpful to teachers because it can remind teachers the ideas and knowledge their students are coming into this topic with and help point the lesson in the right direction. It is important a teacher enters the lesson aware and prepared for the misconceptions their students may have on the topic. Through the use of the concept map and its summary, I was able to learn more about the grade range the students develop certain misunderstanding through the topic and realize that they may still be carrying some of these misconceptions. Outside of the misconceptions it points out, the concept map really gave me a sense of where my students would be conceptually in this topic when I go in to teach the lesson. For example, I know that they should understand the different meanings of "a/b" before they enter the classroom for any 9-12 instruction.
Section VI. State Standards and District Curriculum
With the help of CPALMS, we narrowed our of benchmarks for our topic, "Rational Numbers", down to four. With these four benchmarks, we used a worksheet, attached below, to unpack the benchmarks and decipher what aspects of the benchmarks were more pertinent to our own PBL. The benchmarks which most aligned with the topic we would be teaching and planning a unit for allowed us to create learning objectives for the lesson. Learning objectives and goals are integral to any lesson planning because they provide guidance to the us, as the lesson creators, on what the students should be able do by the end of the lesson (or unit). In addition, it will help us aim to develop questions and activities which cater to our learning objectives.
The learning goals created through unpacking the benchmarks include the following: Students will be able to understand and perform arithmetic operations on polynomials, identify the zeros of polynomials, factor the polynomials, construct a “rough” graph of the function, rewrite a rational expression by performing long division, simplify rational expressions, and perform arithmetic operations on rational expressions. Using the research from the CTS, we are able to support the benchmarks provided by the State Standards and the ideas from the District Curriculum by creating these learning goals which align with each of those.
The research on the topic, "Rational Numbers", points out misconceptions which will help shape key details within the unit plan. In this topic, students tend to have difficulty with concepts about decimals and fractions. To be specific, students may have misconceptions about decimals and how they aid in representing the measurements of concrete objects. Along with the topic of decimals, research shows that students have trouble making a model of the decimal's value. In an example using the value .96, students would struggle to show, using rectangles, that 96 out of 100 rectangles would have to be shaded, or equivalently, 24 out of 25 rectangles would be shaded. A key in understanding these concepts is recognizing decimals with their fraction forms. If a student can make the conversion between a decimal to a fraction, they will be able to represent the value visually with more ease. To proceed further within decimal misconceptions, research supports the claim that students find it hard to distinguish the largest and/or smallest value in a set of decimals with varying amounts of digits to the right of the decimal point. So, given a set {.10, .101, .1001}, students may not understand which value is the largest.
Next, I got a deeper look into the misconception that students face confusion when it comes to recognizing the difference between rational numbers and fractions. While fractions are ratios of two numbers, rational numbers are to be considered as one number. This can be confusing since rational numbers are defined as the quotient of two integers, p and q, where the divisor cannot equal zero. Overall, the concept of rational and irrational numbers can even be misleading for the teachers creating lessons, and it is important that the instructors for the lesson design it with caution and the benchmarks and learning objectives in mind to avoid misconceptions and confusion.
Section V. Examine Coherency and Articulation
The readings in this section reveal the logical connections between material learned earlier in education all the way through to the 9-12 grade level. The concept map shows the early developmental blocks of learning in the topic and link it in as many ways that make sense to other aspects of the topic which come later. This can really help the teacher visualize a large amount of the interrelating topics with the specific topic they are covering and learn more about the progression of knowledge from Kindergarten to grade 12. For example, when students reach the idea that "a/b" can mean three different things (a divided by b, a as a ratio of b, or a parts of size of 1/b each), it comes from the concept learned in 3-5 that states, “fractions are numbers we use to stand for a part of something". This links to the concept from K-2, which says “sometimes in sharing or measuring there is a need to use numbers between whole numbers” which implies using fractions.
This type of development of the topic which can also be helpful to teachers because it can remind teachers the ideas and knowledge their students are coming into this topic with and help point the lesson in the right direction. It is important a teacher enters the lesson aware and prepared for the misconceptions their students may have on the topic. Through the use of the concept map and its summary, I was able to learn more about the grade range the students develop certain misunderstanding through the topic and realize that they may still be carrying some of these misconceptions. Outside of the misconceptions it points out, the concept map really gave me a sense of where my students would be conceptually in this topic when I go in to teach the lesson. For example, I know that they should understand the different meanings of "a/b" before they enter the classroom for any 9-12 instruction.
Section VI. State Standards and District Curriculum
With the help of CPALMS, we narrowed our of benchmarks for our topic, "Rational Numbers", down to four. With these four benchmarks, we used a worksheet, attached below, to unpack the benchmarks and decipher what aspects of the benchmarks were more pertinent to our own PBL. The benchmarks which most aligned with the topic we would be teaching and planning a unit for allowed us to create learning objectives for the lesson. Learning objectives and goals are integral to any lesson planning because they provide guidance to the us, as the lesson creators, on what the students should be able do by the end of the lesson (or unit). In addition, it will help us aim to develop questions and activities which cater to our learning objectives.
The learning goals created through unpacking the benchmarks include the following: Students will be able to understand and perform arithmetic operations on polynomials, identify the zeros of polynomials, factor the polynomials, construct a “rough” graph of the function, rewrite a rational expression by performing long division, simplify rational expressions, and perform arithmetic operations on rational expressions. Using the research from the CTS, we are able to support the benchmarks provided by the State Standards and the ideas from the District Curriculum by creating these learning goals which align with each of those.
tucker_adamson_benchmark.docx | |
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Summary
The Curriculum Topic Study is a process that provided research, ideas, support, and information to me about the topic, Rational Numbers. Being the second CTS I have now completed, I have really grown accustomed to and appreciative of the different perspectives on student learning and development. This is especially helpful when my most recent exposure to most of the topics I will be teaching will have been when I learned it in the classroom myself. As a student, I was never able to step back and understand the learning processes taking place; I was only ever aware of the work I was completing and the information I knew before and after a lesson (not the careful placement of activities, collaborative learning strategies, and probing questions to engage students and promote higher level learning). These are the types of benefits I have gained in completing a Curriculum Topic Study for myself now that I am able to step back and view student learning as an educator.
This CTS specifically taught me a lot about misconceptions students may have. I think one of the biggest mentioned misunderstandings students have in this topic is the fact that "a/b" can have three different meanings. As mentioned above, "a/b" can mean "a divided by b", "a as a ratio of b", or "a parts of size of 1/b each". I think a great activity to include during the unit to clarify this would be to provide multiple examples of each of these meanings and have the students figure out which meaning the example follows. This would hopefully help us discover which of the meanings seems to confuse the students the most, and work to clear their confusion.
In addition, the CTS provided meaningful information about the types of connections students should be making from their previous knowledge to the new topics being learned (and to different subjects altogether that this topic can relate to). As we know, one of the most important aspects of learning relies on the students to connect different ideas together. We can use the information acquired through the CTS to create an engagement activity (or a follow-up activity that allow the students to brainstorm and discuss in groups the different applications of rational numbers in the real-world or their everyday lives. This makes the students think about the different links rational numbers have in their lives while fostering communication of different perspectives and ideas within the groups.
Lastly, the CTS provides a solid foundation for us to refer to for what the students should know about rational numbers and expressions at the point they enter the classroom for our unit. Overall, in building this unit plan, Ashleigh and I will rely on the information we assume the students to know based off of our research on this topic. We can also use this knowledge to refer to the most previous thing the students should have learned about the topic in order to refresh their memories.
Using the benchmarks and district curriculum, in addition to the information we have collected from all of the research and readings on this topic, we are able to link ideas between misconceptions, learning objectives, and previous knowledge to create a lesson and unit which will successfully teach students about Rational Numbers.
This CTS specifically taught me a lot about misconceptions students may have. I think one of the biggest mentioned misunderstandings students have in this topic is the fact that "a/b" can have three different meanings. As mentioned above, "a/b" can mean "a divided by b", "a as a ratio of b", or "a parts of size of 1/b each". I think a great activity to include during the unit to clarify this would be to provide multiple examples of each of these meanings and have the students figure out which meaning the example follows. This would hopefully help us discover which of the meanings seems to confuse the students the most, and work to clear their confusion.
In addition, the CTS provided meaningful information about the types of connections students should be making from their previous knowledge to the new topics being learned (and to different subjects altogether that this topic can relate to). As we know, one of the most important aspects of learning relies on the students to connect different ideas together. We can use the information acquired through the CTS to create an engagement activity (or a follow-up activity that allow the students to brainstorm and discuss in groups the different applications of rational numbers in the real-world or their everyday lives. This makes the students think about the different links rational numbers have in their lives while fostering communication of different perspectives and ideas within the groups.
Lastly, the CTS provides a solid foundation for us to refer to for what the students should know about rational numbers and expressions at the point they enter the classroom for our unit. Overall, in building this unit plan, Ashleigh and I will rely on the information we assume the students to know based off of our research on this topic. We can also use this knowledge to refer to the most previous thing the students should have learned about the topic in order to refresh their memories.
Using the benchmarks and district curriculum, in addition to the information we have collected from all of the research and readings on this topic, we are able to link ideas between misconceptions, learning objectives, and previous knowledge to create a lesson and unit which will successfully teach students about Rational Numbers.
Reflection
I think one of the most helpful parts of completing the CTS for this unit was that it gave us direction when we did not necessarily have any direction myself. Looking back on the time spent completing the CTS, we had only the slightest clue exactly what we were going to pick to do as the project within this unit and we had not decided yet which aspect of the topic "Rational Numbers" we were going to base the unit off of. The CTS was helpful at this point in the planning process of the unit because it left me with an infinite number of opportunities for my unit once we decided what specific concepts we would be teaching.
The most helpful aspects of completing the CTS is the perspectives on common student misconceptions because it requires me to think about the topic in a way that I may not have considered it before. It also just makes me, as an educator, more aware of the mindset my students may being coming into the classroom with because of their previous experiences with rational numbers.
CTS also does a really great job of discussing the connections between past knowledge and experiences into the newer material, and from inter-related subjects and materials. We structured a great deal of our lessons on establishing relationships to help the students learn the information. For example, we made the connection between rational numbers and rational expressions for almost every benchmark and investigative lesson we created because we really wanted the students to consider what they already knew about rational numbers and see if they could then apply that information to the rational expressions. This is the first topic I have ever taught a lesson on that flowed so well from one concept to the next so that I always had a connection to refer back to with the students, and it was extremely helpful in the planning of our unit. In addition, the CTS helps relate the topic to other subjects, which is how we discovered the physics application from rational expressions to roller coasters. At the beginning of our unit planning, we were really struggling to find a project we wanted to use that we thought we could make work well with rational expressions. Then, when we considered the physics applications with roller coasters, different aspects of the project began to piece together and we were finally able to start planning out the specific details of our project.
The most helpful aspects of completing the CTS is the perspectives on common student misconceptions because it requires me to think about the topic in a way that I may not have considered it before. It also just makes me, as an educator, more aware of the mindset my students may being coming into the classroom with because of their previous experiences with rational numbers.
CTS also does a really great job of discussing the connections between past knowledge and experiences into the newer material, and from inter-related subjects and materials. We structured a great deal of our lessons on establishing relationships to help the students learn the information. For example, we made the connection between rational numbers and rational expressions for almost every benchmark and investigative lesson we created because we really wanted the students to consider what they already knew about rational numbers and see if they could then apply that information to the rational expressions. This is the first topic I have ever taught a lesson on that flowed so well from one concept to the next so that I always had a connection to refer back to with the students, and it was extremely helpful in the planning of our unit. In addition, the CTS helps relate the topic to other subjects, which is how we discovered the physics application from rational expressions to roller coasters. At the beginning of our unit planning, we were really struggling to find a project we wanted to use that we thought we could make work well with rational expressions. Then, when we considered the physics applications with roller coasters, different aspects of the project began to piece together and we were finally able to start planning out the specific details of our project.