By completing this lesson, students will be able to use a coordinate system of ordered pairs of numbers, recognize the importance of the sequence of the numbers in an ordered pair, and develop strategic thinking skills. Thus, this lesson is excellent for a class that is either starting to learn about graphing ordered pairs or for a class that is already in the process of learning about how to graph ordered pairs.
The most important thing about this lesson is that it is fun. The students will feel as if they are playing the game called Battleship. Thus, a teacher could use this game to get students to learn about plotting ordered pairs on a coordinate plane without the students even realizing that they are learning. Thus, the chances of getting students engaged in the learning experience are very high if this lesson is the one being implemented.
Another aspect of this lesson that I particularly like is the fact that students will need to use different strategies in order to "win". This is why I included questions that probe the students to think about and discuss their strategies.
Also, I practiced playing this game with my girlfriend. The game went smoothly, and all indications were, that this lesson would be an effective lesson to use with students.
This program is a lot of fun. It reminds me of the Towers of Hanoi Puzzle and the Peg Puzzle because it requires students to use their problem solving solving skills. In addition, it requires students to use their measurement and arithmetic skills.
A teacher could easily spark interest in this program by showing students a clip from a Die Hard movie where the police officers had to solve a riddle that involved getting a certain amount of water in two jugs. However, the program is interesting even without this hook.
While some of the virtual manipulatives are not such a "big deal" because one might be better off using real manipulatives, this program allows students to do problems that could be very messy if they were solved in "real life". In fact, most math classrooms do not have sinks, so to do these problems in "real life" in a math classroom would be very challenging. One would have to go to a science classroom, an art classroom, or a home economics classroom..
This program can be found under the measurement tab for grades six through eight. However, this could certainly be a fun program that a math teacher could give to lower level high school math classes.
As teachers, it is important to select activities that will get the students to be engaged. Since students are stimulated by "exciting" video games, computer programs, electronic equipment, and television shows, it is important to make class activities and homework assignments as much fun as possible. This program does just this by creating interesting problem-solving scenarios.
After completing this lesson, students will be able to determine the mean, median, and mode for a data. They will also be able to draw conclusions about a data set and recognize patterns in data set. In addition, students will have to make and test hypotheses in this activity, which reinforces what they learn in eighth grade science. As a result, this lesson is interdisciplinary in nature. If other teachers decide to use this lesson then they should consider reaching out to the eighth grade science teachers in their respective schools. It is possible that an interdisciplinary assignment could be developed as an extension to this activity.
The best part about this activity is that it is fun. Students tend to enjoy making predictions and testing them to see if they were right, and this is at the heart of this activity. Thus, when implementing this lesson teachers should find it relatively easy to get students engaged in the learning experience.
Also, this lesson is designed for a sixty-eight minute period. However, it could easily be broken up into two shorter lessons. One lesson could focus on the Grab Bag Activity and the other lesson could focus on the Grab Bag Extension Activity.
A few of the applications of this program could be used in middle school; however, this program is best suited for high school level mathematics. It can be used to help students learn how to solve for a missing angle or a missing side in a right triangle. It also can be used to reinforce the students' understanding of sine, cosine, and tangent. Thus, this program would be great for independent practice in the classroom, and it could also be used as a review activity.
This program is great because it allows students to solve for multiple missing measurements for a right triangle. In step one, students must pick the variable that they would like to determine. In step two, students must determine the method that they need to use in order to solve for the variable that they selected. In step three, students must actually solve for the variable that they selected using the method that they chose in step two.
I particularly like that students must decide what method to use to solve for a particular variable. This prevents students from simply "plugging and chugging". In other words, it prevents students from simply using a given formula and plugging in numbers without thinking deeply about what anything in the problem means.
Another aspect of this program that I like is the fact that if you select an incorrect method to use, the program gives the user some information that is meant to help him or her understand why his or her chosen method would not work. Also, if step three is not done correctly then the program gives the user a helpful hint. Of course, the program also informs students about correct answers. Thus, students can get instant feedback from this program. If this type of assignment were provided to students in a paper and pencil format then it would be much harder to give students immediate feedback. Remember, there is usually only one or two teachers per classroom.
This lesson plan requires the use of Geoboards. After completing this lesson, students will be able to find the fractional parts of a whole, represent fractions spatially, add fractions, and find the areas of irregular shapes. There is no doubt that some groups will make this lesson harder than others. When I had my girlfriend complete the lesson she broke the "peanut brittle" into some very complicated shapes. As a result, it was very challenging for her to complete the lesson. However, after an extensive amount of time, she was able to do it. Thus, my advice for those of you who want to use this lesson in the classroom is to allow for some groups to take longer than others.
"Box plots and histograms are both used to summarize data graphically.
A box plot shows the minimum data value, the lower quartile, the median, the upper quartile, and the maximum data value on a number line. A box is drawn from the lower quartile to the upper quartile. The median is marked inside the box.
A histogram divides the range of values in a data set into intervals. Over each interval is placed a block or rectangle whose area represents the percentage of data values in the interval."
This program can be used to introduce box plots. Students can analyze the data that is provided when the program opens. They can also make changes to some of the data or delete all the data and enter their own. The program then graphs then automatically creates a box plot with the data.
This program can also be used to check one's work. Students can create a box plot by hand and then use this program to see if their work is correct.
This lesson plan is appropriate for eighth grade. Here is the New Jersey Core Curriculum Content Standard that this lesson plan meets:
4.2.8 E. Measuring Geometric Objects 1. Develop and apply strategies for finding perimeter and area.
By completing this lesson, students will be able to devise methods for finding areas, formulate and test generalizations, learn about and apply the Pythagorean Theorem, and use mathematical reasoning to solve a real-world problem.
Lastly, this lesson plan helps make the Pythagorean Theorem more "real" for the students. By completing this lesson, students will see that a squared, b squared, and c squared actually represent the areas of squares. Thus, after completing this lesson, students will be able to picture in their minds what the numbers in the Pythagorean Theorem actually mean.
This is a very simple virtual manipulative. However, it is very effective for giving students an opportunity to practice depicting the union and intersection of sets on venn diagrams. If a teacher were to give students several problems of this nature, it would be challenging for that teacher to give each student individual feedback. However, this program makes this possible. Students can click on the "check" button to see if their answer is correct. They can also click on the "show solution" button to get the answer to a problem that they cannot solve. These two options make it possible for students to get immediate feedback, which is something that a teacher likely cannot provide without the help of technology.
Another nice feature of this program is that the problems it creates get progressively more difficult. Thus, more advanced students can work quickly and get to the harder questions, and students who are struggling with either the concept of venn diagrams or unions/intersections of sets can take their time and work on the easier problems.
However, there is one thing that could improve this program. This would be the addition of a "hint" button. While I like that students can check their answers and also get solutions to the problems that they cannot solve, I would like there to be an option that helps students without giving them the answer. I suppose this option could be provided by the teacher; however, I think students might jump to click on the "show solution" button too quickly. In essence, the desire for immediate feedback could get in the way of true learning by struggling to solve difficult problems.
The formula A(P,T) = 0.5P + 1T - 1, where A stands for area, P stands for the number of posts, and T stands for the number of trees, still works for all of these examples.
This lesson is great if you are trying to get students to develop a deeper understanding of the properties of right triangles or trying to get students to use more than one method for finding the area of right triangles. It is also very good for getting students to think deeply about proving that they have found all of the solutions. It is not an easy task to develop and present a convincing argument that one has found all possible solutions to a particular task. This lesson requires that students do this.
Once again, I tested this lesson out on my girlfriend. It worked very well. However, in my opinion, her argument that she had found all possible solutions was weak. Thus, I think the discussion on proving that one has found all possible solutions could take a considerable amount of class time. Also, if students have not been trained on how to develop such an argument throughout the course of a school year then this might be challenging. Thus, in the future, I will try to train my students on how to prove that they have all possible solutions starting in September. Lastly, I feel that requiring students to prove that they have found all possible solutions is very important because it forces students to think like mathematicians.
This is a great virtual manipulative to use if you are going to teach students to graph on a coordinate plane. Even though this virtual manipulative is very simple, it is an excellent learning aid because it gives students immediate feedback as to whether they are correct or not. In order to get this feedback, all students have to do is plot the points and click on the check answer button. The program will tell a student whether he or she got all the points correct, some of them correct, or none of them correct. It will even indicate which specific points are correct and which ones are not. The ones that are not correct will show up red, and the ones that are correct will show up green.
Another nice thing about this program is that it makes it easy to provide differentiated instruction. For the students who are good at graphing, they can do the program without the grid values. However, the teacher can set the program to show grid values for the students who are not as strong at graphing. Also, the program contains many different problems, so students who are good at graphing can do additional programs while the students who are not as good at graphing can complete fewer problems.
This lesson plan requires the use of Geoboards. I tried out this lesson with a family member, and it certainly is an effective lesson. The reason that I feel that it is effective is because it requires students to play a game in order to meet the objectives of the lesson. In other words, the lesson is fun, and when students do something that they consider to be fun they are more likely to be engaged in the learning experience.
This virtual manipulative is simple, but it has excellent applications. The number of spins can be changed, and the spinner can be altered in a number of ways. The number of colors, the colors themselves, and the size of the colored regions can all be changed. The particularly nice thing is that you can record the results of the spins. These results are presented in a graphical form, which students can copy and transfer onto a sheet of graph paper. Thus, this virtual manipulative can be used to teach graphing skills. In addition, the virtual spinner has applications with regard to probability and percentage calculation. Students can calculate the theoretical probability and then compare it to the experimental probability. In addition, the bar graph results can be converted to percentages.
Obviously, "real-life" spinners could be used in a classroom. However, it is not likely that the color choices and the size of the colored areas on the spinner could be altered. In addition, it would take a long time to spin a spinner hundreds of times and record the results. Thus, for many reasons this virtual manipulative is better than a "real-life" manipulative.
CLICK HERE for the lesson plan and the supporting documents.
The objectives of this lesson are to identify and extend patterns, explore the sequence of perfect squares, make predictions based on patterns, and express relationships algebraically.
I had four eighth grade students do the activities in this lesson at lunch on Thursday. They enjoyed the lesson, but they found the part where they had to come up with algebraic expressions to be very challenging. This is likely because the students that volunteered to do the activities are in the lowest eighth grade math level. If they were in algebra then these activities would have been a lot easier for them because they would have been used to expressing problems algebraically. However, with a lot of assistance, the students were able to come up with algebraic expressions to represent the patterns they saw.
On another note, when the students were working on the activities from this lesson I could see that they were very good at identifying patterns and relationships. Thus, they found all aspects other than the part that specifically relates to algebra to be relatively easy.
Also, after the students completed the activities, I felt a sense of accomplishment because I was able to reinforce concepts related to patterns and relationships and help students with the difficult task of learning algebra. In essence, I was able to help students discover math patterns and express them algebraically.
Speaking Standard 3Speaking Standard 3The idea of using real-life problems in mathematics is particularly important.Students often cannot relate to discussions involving mathematics.However, by using real-life situations and problems, students can see real-life reasons why they should learn mathematics.
Speaking Standard 3It is not uncommon for mathematics students who are not planning on majoring in mathematics or business to ask why they need to learn high-level mathematics.However, by using a real-life example like calculating the speed of a cat, students can “relate” to the problem at hand.In addition, the speed of the cat problem is a very good introduction to calculus.It provides students with the opportunity to analyze changes in speed.
Speaking Standard 3I really liked how the students were asked to model the actions of a cat.By having the students actually move along a line in the same way that the cat moved, students were able to develop a deeper understanding of what the dots on the graphs that they created actually meant.Honestly, I feel that this deep understanding of what a graph actually means is often lacking for many mathematics students.It is evident in the fact that there is math phobia, and it is even more evident in the fact that many students have trouble interpreting graphs and charts.Basically, what this video shows is that the solution to graphical and tabular interpretation problems that students have is to have them model what is going on in the graphs and tables.In essence, we need to let students bring graphs and tables “to life”.This will allow students to make sense of mathematical information.
Speaking Standard 3In my opinion, the “Rutgers Way of Teaching” is certainly an excellent teaching strategy.Basically, it allows for teachers to be facilitators of learning because it is an inquiry-based approach.By using this way of teaching, students discover new concepts, and they become the teachers for other students.As many people know, if one can teach something that means that they truly understand it.The “Rutgers Way of Teaching” also allows students to develop “ownership” of a problem.It allows students to develop a sense of confidence that they are able to solve a problem on their own.In addition, it allows for students to explain to their peers how to solve a problem.This is particularly important because sometimes the way students go about teaching their peers is more effective then if a teacher shows a student how to do something.Lastly, this strategy of teaching allows for students to see that there are multiple ways to solve a problem.Thus, I feel that the “Rutgers Way of Teaching” is much more effective then a teacher simply telling students information.By telling students information, the chances of them retaining it are not nearly as high.In addition, simply lecturing to students can be boring for everyone involved.
Speaking Standard 3I also agree with the narrator that having students justify their answers is very important.If students can justify their answers then they really understand a topic.This is a great test for educators to use when determining if their students have developed a deep understanding of a particular topic.
Speaking Standard 3The Private Universe Project video series makes it clear that students learn best by working with manipulatives, being challenged, being asked to justify their answers, being required to find their own way of answering a question, modeling mathematical information, and working with real-life problems.If teachers take the lessons learned from this video series and apply them to their own classrooms then student interest levels in mathematics should rise.Ultimately, student test scores should follow suit.
Virtual Manipulative - Savings Calculator Grade Level: 6 - 8 Category: Data Analysis & Probability Link: Savings Calculator
When I was searching for a virtual manipulative I wanted to select something that could help students in the "real world". Obviously, when I came across this savings calculator I knew that students would learn something that they would need to know for banking purposes. Thus, this virtual manipulative has practical applications, and it would likely spark interest in many students. As a result, this is a "must use" application when the topic of compound interest is discussed.
Developing an understanding of compound interest is important. However, since it is a challenging topic, it is important to clearly demonstrate the results of compound interest. Without a clear demonstration of the results of a compound interest account, students may develop the misconception that whether or not interest is compounded is insignificant.
This application is perfect for demonstrating the significance of compound interest. Not only can students see the result of investing money in an account with compounding interest, but they also can see the difference between interest that is compounded daily, weekly, monthly, quarterly, and annually. In fact, I know many adults who do not understand the different types of compound interest, and this program demonstrates to students the difference in terms of the outcome.
My main recommendation to teachers that they enter large numbers for the deposits. This will lead to a more dramatic difference in interest earned for the different types of compound interest. I also suggest that teachers utilize this application in an introductory lesson as park of a "hook" to grab students' attention. In addition, this application can be used later on in a unit on compound interest to demonstrate the difference between different types of compound interest.
By using an application like this one, it is possible to spark student interest in mathematics. Today's students really need to know that what they are learning could impact their lives. This application can be used to prove to students that what is being taught in math class can affect their lives.
I was particularly impressed by Melissa Sharp, a first year second grade teacher in the Englewood Public School District.She was able to get second grade students to think at a higher level.In the video, she stated that she does not accept students simply giving an answer.She wants them to be able to convince her that they have come up with the correct solution or all of the solutions.In essence, she wants every student to develop a deep understanding of the problems they are working on so that they can build on what they learn in future grades.If more teachers develop this philosophy of teaching then I suspect that the retention of ideas learned in a particular grade will increase.
After listening to some of Arthur Powell’s ideas, I can see the importance of getting students to “own a problem”.Getting students to think deeply about a problem is so important, and this can be done by getting them to be engaged in working with a problem.In addition, it is important to have students share ideas.Obviously, this requires students to get used to the idea of working together and discussing problems, but doing this process has value in and of itself.The reason that I feel so strongly about this is because so many of my students are able to solve a problem, but they do not completely understand what the answer means.Sometimes they come up with answers that do not make any sense, and they do not realize why their answers are not realistic.For example, they may come up with answers regarding time and speed that are negative.How is this possible?Thus, it is important to get young students to work with problems that are hands-on so that they develop a deep understanding of what numbers actually mean before moving on to more advanced abstract problems in later grades.For the same reason, it is also important to press students to provide a convincing argument for why their answers are correct.In this way, students are required to think like mathematicians, and if they are able to prove that their solution or solutions are correct then they have obviously developed “ownership” of the problem at hand.
I think Gina Kiczek, a teacher in the Jersey City Public School District, made an important realization, which is that students often do not see things the same way that the teacher does.Often there is a “disconnect” between students and their teacher because the students do not think the same way that the teacher does about a problem.This is why it is very important to give students the opportunity to share their ideas about problems.In the type of learning environment where students are given the opportunity to work together and discuss problems, they can not only develop their own line of thinking, but they can also learn from their peers.This is much more valuable then the outdated teaching model where a teacher is a “sage on a stage” and students simply absorb knowledge from him or her.By giving students the opportunity to work together and discuss what they are thinking, there is a much greater chance that students will develop a deep understanding of a problem and what the answer to a problem means.In essence, this style of teaching allows for more students to develop “ownership” of a problem.
I was particularly impressed by the ice cream lesson.This lesson is a testament to the value of a lesson that allows students to work together and solve problems.The reason that I say this is because the lesson was planned for an eighty minute block period.Based on the video, it appeared that students were engaged in the problems provided to them for the full eighty minute block.This is quite an achievement!
I thought the World Series problem was a good problem to use with older students.It is important to choose topics that fit students’ interests.Many students are obviously interested in the World Series, and thus, when one gives students this type of problem they are more likely to become engaged in what they are assigned to do.
Name of Virtual Manipulative: How High? Category: Geometry Grade Level: 6 - 8 (probably best for grade 8) Link: How High? Virtual Manipulative
This virtual manipulative is great because it forces students to try to understand spatial relationships. The truly analytical student will figure out very quickly that it is necessary to calculate the volumes of both figures and then compare them. By no means is this an easy activity. The beauty of this program is that the learning curve for an eighth grader would be very steep. At first, they will likely start out making wild guesses. Then, they will make more educated guesses. Finally, they should come to the realization that if they calculate the volume of the liquid and compare it to the volume of the container on the right they should be able to accurately predict how high the liquid will go when poured into the container on the right. Thus, this activity forces students to also study the volume of figures.
The really nice thing about this virtual manipulative is that it is applicable to the real world. It is very common for people to transfer a liquid from one container to another in order to save space in a refrigerator or cupboard. Thus, students will likely see "the point" of doing this activity.
I had my girlfriend do this virtual manipulative. She started out making wild guesses. Then, she graduated to educated guesses, and finally, she began calculating volumes and comparing them. Thus, it is no surprise that, at first, she thought the activity was very difficult. However, by the end, she had changed her opinion because she had developed a mathematical strategy for determining the answers.
I tried this lesson with four of my eighth grade students at lunch. It is very interesting to note that their predicted probabilities did not match up with either their experimental probabilities or the theoretical probabilities. Based on the discussion that the students and I had, it became apparent that they had trouble understanding the concept of not replacing a Color Tile. In essence, if there are six tiles for the first draw then there are only five tiles for the second draw. Obviously, this caused many of the students to come up with predicted probabilities that were not close to what one would expect them to be.
On the other hand, students developed a deep understanding of the idea that as the number of trials increases the experimental probability becomes closer to the theoretical probability. This is a very important concept for students to learn because it is something that can be applied to the real world quite often. In addition, students must use this concept quite often in the scientific laboratory setting.
One very important suggestion that was made in the video with regard to the Tower of Hanoi Problem was to try a simpler problem.This is a very important strategy to teach students because often problems in mathematics look very difficult.Trying to solve what appears to be a very difficult problem can quickly become frustrating.However, if students can simplify a problem, learn something new from the simpler problem, and then apply what they learned to the original more difficult problem then the problem becomes easier to solve.The nice thing about having students use a strategy like this is that it forces them to look for patterns in simplified problems that can be applied to more challenging problems.Thus, this strategy encourages students to think creatively because it requires students to create new simplified problems that they can solve to determine if there is a pattern that can be applied to a more difficult problem.It also encourages students to think critically because students need to determine what the pattern is.
The narrator of the video points out that the thought process that the students were using is much like the thought process of a mathematician.This is because mathematicians typically analyze simpler problems in order to develop a pattern that can be applied to more complex problems.Since this is what “real world” mathematicians do, I feel that this is the type of mathematical thinking that students really need to learn in school.Thus, it is important that videos like this one exist to show prospective teachers how they should teach mathematics to meet the needs of twenty-first century learners.
I particularly liked the teaching strategies that Janet Walter employs.This is because I believe that it is important to encourage students to become active learners.When students become active learners they rarely become distracted.Also, by becoming a part of the learning experience, students generally develop a deeper understanding of the material.Unfortunately, I have experienced many mathematics courses that are not engaging.In the type of mathematics class that I am describing, the teacher typically acts like a “sage on a stage”.He or she gives students the information necessary to solve problems, and then the students are asked to take notes and solve problems using those notes.In this type of mathematics courses, classroom discussions about mathematics questions and problems are a rarity.In essence, students are not asked to think deeply about problems in this type of class.They are simply asked to do a process that was clearly mapped out by the teacher.Since this style of teaching simplifies the job of a student, it is not common to hear of a student who complains about this method of instruction.However, this style of teaching is not the most effective way to get youngsters to learn mathematics.Thus, I am glad to see that this video encourages new teachers to break this mold and get students to think about mathematical problems.
Lastly, I think the most important thing that this video demonstrates is that if teachers give students the opportunity to become active learners then there is a good chance that the students will have fun.I have heard many students tell me that math is boring, math is hard, and math is inapplicable to their daily lives.This is likely because students are often not asked to participate in mathematics lessons in ways that encourage them to think deeply about a problem.I think the strategies presented in this video have the potential to change this type of reaction to mathematics.By employing the strategies demonstrated in the video, teachers can get students to think like mathematicians, develop a deep understanding of the material, and enjoy what they are learning.
I started teaching in 2006, and I knew education was the right profession for me right away.
I love the outdoors. I spend much of my free time camping and hiking.
