IndexPlanningAnalysis for the student AAAAnalysis for the student BReflectionIn the first introduction, I will introduce my pre-algebra students. There are 53 students in total, divided into three classes. The students are aged 13 to 15 and are all in eighth grade. The population is made up of 33 males and 20 females. Twenty-five students are from a minority background, twenty-six students receive free or reduced-pay lunch, and seventeen students participate in the magnet program. Academically, my students are considered at or below standards for annual grade level progress. Only 18 students met the standards on our state test, the rest were below. Eighteen of the fifty-three students attend our special education program and twelve of the eighteen receive math-specific special services. In addition to their performance on standardized tests, students in this group tend to be afraid of math and have not been successful in previous years. They come to me with the understanding that they are terrible at math and will not improve. Because of this belief, they are not very motivated to get the job done. Say no to plagiarism. Get a tailor-made essay on "Why Violent Video Games Shouldn't Be Banned"? Get an Original Essay From a learning perspective, students prefer to do activities where they can work together and use manipulatives or technology to complete a task. Many are lower-level readers, so they prefer to learn by experimenting rather than taking notes or reading new vocabulary. When we have notes, students would rather simply learn about the process rather than why that process works. They struggle with any type of problem that requires application or thinking beyond mechanical processes. The topic I'm focusing on is operations with real numbers, especially integers. Students have previously learned the four basic operations with integers, but do not have a thorough understanding of them. They know how to perform tasks by following the rules, but they don't know why those rules exist, or how to explain why the answer is the one that goes beyond the list of rules. They also confuse the rules when they switch from addition and subtraction to multiplication and division. Planning This unit focuses on integers in the topic of real numbers. The first lasting understanding students should have at the end of the unit is that various models, such as drawings or manipulatives, can be used to explain operations with integers. The second enduring understanding is that correct solutions require accurate calculations. Students should be able to analyze any problem and explain why it is correct or incorrect. The final and lasting understanding of the unit is that when solving problems, students should be able to select, apply, and explain the method for calculating with integers. These objectives support students in their understanding of whole numbers. These objectives do not focus exclusively on the correct procedures for calculating integers. Instead, they focus on understanding whole numbers. They encourage students to use number sense to find an answer, not just to follow a rule. They also allow students the opportunity to think conceptually. Through problem solving, students use real-world situations not only to calculate, but to interpret whole numbers. Furthermore, objectives also develop students' critical thinking by giving them the opportunity to justify the accuracy of an answer. For this unit, here we areengaged in several activities, all focused on using manipulation as a tool to understand the calculation of integers. On the first day we started with the absolute value of a number and used number lines to see that the absolute value is about a distance, not a direction. We used real-life examples such as gaining and losing yards in a football game to connect the idea of ​​absolute value to signed numbers. One idea related to absolute value that we talked about is the size of a number. For example, using counters, we can see that there are four red counters and are more than two yellow counters, but we know that positive two is greater than negative four. On the second day we moved on to adding whole numbers. For this lesson I activated students' prior knowledge about the number line by using an image of an elevator, a vertical number line, to visually see what happens when we add positive and negative numbers. This activity is the one I'm featuring in this entry. Instead of looking at both number signs and developing "rules" for each situation, we looked at where we started on the vertical number line and which direction we moved based on the second number. We developed the models and then wrote the rule based on the models. On the third day I continued the idea of ​​addition by introducing colored counters as a way to combine whole numbers. With the colored counters, we focused on the groups of positive and negative numbers and talked about zero pairs, or opposites. I used the student groups' prior knowledge on day four when we discussed multiplying whole numbers. In previous years students have used colored counters to learn the idea of ​​multiplication as repeated addition. In this activity, Students used colored counters to form groups of positive and negative numbers. By looking at the numbers and their signs, we developed groups of red or yellow to determine a product. This activity is also described in the entry. For my described activities, I am focusing on the first lasting understanding: various models, such as drawings or manipulatives, can be used to explain operations with integers. I chose to present these two activities for two reasons. First, they are both visual models that allow students to see and manipulate numbers. With the number line they can understand what happens when they subtract a number larger than the first. I am also able to see how the size of each number affects the outcome. This activity gave them a way to understand signed numbers as a concept, not just a set of rules to follow. The colored counters also give students the opportunity to understand why the two rules of whole number multiplication exist. By creating groups and observing patterns, they can understand why two negative numbers form a positive product. Most students can follow a list of procedures over and over again. What they don't go away with, however, is the reasoning behind the rules. I want my students to understand why, so that they are able to apply what they have learned outside of my classroom. Part of this understanding comes from observing patterns, which they don't have the opportunity to do if I present them with four rules to follow and when to follow them. By giving students the opportunity to manipulate numbers on their own and see the pattern, they are developing mental habits that will always stay with them. Both of these activities are visual models that students can use to explain operations with integers. Aunderlying understanding that comes from this objective is the development of models. Both activities help students develop models for understanding the general rule. Students were able to see that mathematical rules and procedures are not just “magic,” but the result of observing patterns and creating that rule or procedure. As I worked on the first activity, I really had to push the students to notice what was developing. When they figured out the pattern, it was easy for them to tell me what the rule was and why. Putting them in that mental habit prepared them better for the second activity. From the first example, they tried to see the pattern, to make the connection between the rules they had learned previously and what was actually happening on their paper. For both activities, we worked together on several examples, each a different case relating to the rules of addition and multiplication. During the activity I asked the students to describe what was happening. I also encouraged them to compare the different examples and explain why changing the number changed the result. During this time I was able to see if the students understood the purpose of the activity. As we progressed, I was also able to clarify common misconceptions. If students had difficulty seeing what was happening in a certain case, I was able to adapt the instructions and provide them with additional examples of that case. Independently, I assigned students some reflection questions to complete. These questions required students to think about the models we had developed and to justify whether these models were always true. By reading their responses, I was able to see if my teaching had been successful in allowing my students to achieve goals, or if I needed to adjust it by re-teaching or correcting a misunderstanding. In teaching this mathematical idea, a major challenge to What concerns me is that many students prefer to apply a rule to solve a problem without understanding why. When I ask students how they got the answer, they repeat the rule they wrote in their notes and on the paper. Students can typically explain accurately if I put a problem into a real-life context (ex: if you go to the store with $10 and your bill is $12, what happens?). But if I ask them why 10 subtracted 12 is negative 2, they fail to make the connection, instead stating the rule. One challenge presented in this class is that while some students knew the rules, others did not know them at all, or confused them and used the wrong rule (e.g. three negatives plus three negatives equals six positives, because two negatives make a positive). . In the past I have taught the rules and then attempted to make connections and go deeper. Because of these challenges, this year I designed my instruction exclusively around visual activities. We didn't write the rules and then some practical examples. Instead, we used images to develop models and create overall statements. During the activities, when students tried to explain their answer using rules from previous mathematical experiences, I challenged them to go back to the pictures and use the pictures to explain why the sum or product is what it is. Analytics for Students AI chose the student Because of the challenge it gave me this year. This young man is eager to learn and works very hard. He's the kind of outside student every teacher wants. Unfortunately, no matter how hard he works, mathematics still remains a big challenge for him. He is one of our special education students, qualified inmathematics and behavioral disorders. Not afraid to ask questions or receive feedback. He will do things again and again if asked and will not give up if encouraged. However, every time I have repeated a concept or taught it in a different way, I don't always walk away with a strong understanding. He may understand it well in class, but he doesn't retain the information for homework. His goal is to always win: the only time I saw him frustrated was when he didn't win a game we were playing in class. In math class he is happy if he gets the correct answer, even if he doesn't really understand why. He is able to apply an algorithm when given a problem, but he is not always able to analyze a problem to know which algorithm to apply. When I wrote this unit plan, I knew that the students had previously been taught the rules of whole numbers. One of my goals was for them to understand why the rules work, so they don't get confused and can recreate them if they forget. Looking at Student A's work, I see that I have helped him make some improvement in his understanding, but not as much as I would have liked. Analyzing his response to the first teaching activity, I can see an improvement. When given a pretest, Student A simply added up the numbers and made them all negative (ex: -2 + 3 = -5). Using the elevator, Student A was able to visualize what was happening, and in his written explanation he showed that he understood that addition means positive movement on the number line. He also understood that the addition would always work. Unfortunately, the elevator wasn't as helpful with subtraction. Although Student A recognized that subtracting means a negative movement on the number line, in the third example he did not number his line correctly, thus getting an incorrect answer. This tells me he understands the idea of ​​motion, but is still struggling with number sense. What he should have realized is that subtracting from a negative number results in a smaller number, when in reality his result was larger. This gave me the feedback that I needed to spend more time with Student A and possibly other students to check the reasonableness of the answers and how the magnitude of the numbers affects the outcome. After analyzing his entire response, my final assessment of Task 1 for Student A is that he still has only a basic understanding of what happens when he performs addition or subtraction on signed numbers. However, he now has a tool that both he and I can use as an entry point into deeper conversations about the magnitude of quantities and direction. My goal was to move students away from rule dependency, and I feel I made progress toward this goal with Student A. By analyzing Student A's response to the second learning activity, I felt he was more successful in achieving the learning objectives. During our modeling instruction time (see Notes: Integer Multiplication) he successfully modeled products when given the case of one positive and one negative. He was also able to create his own examples and find the product. A typical misconception when multiplying two negative numbers is that the product is negative, like when adding two negative numbers. When we worked on this part of the instruction, Student A originally had a negative product. I noticed this when students were working independently and I was able to sit with him and have him think critically about his response. I asked him to set up his counter problem again and we talked about how it would be the opposite of 5groups of -3. After the feedback I gave him he understood why the answer had to be positive. He told me that it was easier for him to think of the first number simply as the “number of groups” and the second number as the “color” (red or yellow). He said that once he set up the model, he would go back and look at the first number to see if it was positive or negative. If he tested negative, then he knew he needed to take the opposite product to the product he currently had. I asked him why he liked this way of thinking and he said because “it means I only have to think one way. I don't have to know a lot of rules. When I look at his homework (see Multiplying Whole Numbers), I see that he understands the concept. However, his verbal explanation doesn't really demonstrate his level of understanding. When I asked him what he meant by "no because you have to x" in question nine, he told me that since he knew how to multiply, he could just do it instead of groups. I asked him how he would determine the mark of the product at that time, and he said he simply looked to see if there were one or two negatives. Comparing both responses to the learning activities, I have the feeling that Student A has gained in his conceptual understanding. Using manipulatives he had the opportunity to think about what was really happening when adding and multiplying whole numbers. He no longer has to rely on memorizing a set of rules and hopefully not confusing them. The analysis for student BI chose student B because, like student A, she has also been a challenge for me this year. Student B is also a hardworking student, but unlike student A he has a much higher level of understanding. He is among the 34% of students in his class who passed our state assessment and maintained an “A” for the entire year. She is a hardworking student, who completes all homework without complaint and is tenacious in finding the correct answer. Teachers also like Student B, so she often participates in class discussions. Once again, Student B from the outside is the student everyone would like to have a classroom full of. Academically, he is a star. However, she is adamant that she is not good at math. In our district we have differentiated our classes based on ability levels and Student B is at the lowest level of math we offer in eighth grade. What she struggles to understand is that even though she is in what she calls the “dumb math” class, she is in pre-algebra, which was considered the elementary level course for eighth graders until the recent reform. She only sees that there are two math lessons above her and none below. To further undermine her math self-confidence, she takes an advanced language class and an advanced science class, which makes her feel that math is her worst subject. In working with her this year, I have made many efforts to encourage more positive mathematical self-esteem. Student B's work for the first learning activity tells me that I have achieved my learning objectives. Student B was able to successfully model the addition of integers. He understood by looking at the manipulator that adding a positive number will always result in a forward or larger sum, and subtracting a positive number will result in a behind or smaller sum. He was able to do it without relying on a rule, but because he understood the direction and movement. An interesting thing in his work is that he also addressed the situation of “adding a negative”. His problem statement of "starting 2 stories below ground and going down three stories" was -2 + -3 instead of -2 - 3. Students typically.