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Title: Estimation and Number Sense using Java applet
Submitted by: Justin Geoffrey Fong
Short description: This lesson pushes on students number sense of addition, multiplication and percentage computations. After giving students estimation strategies, they played a java-based game using applets on computers. Duration of lesson(s): 90 minutes Grade level(s) and/or target group(s): Seventh (appropriate for fifth grade and up) Subject(s): Technologies used: Java applets, iBooks
Objectives: a) Students will be able to approximate the sum of two 3-digit numbers within 10 seconds. b) Students will be able to estimate the product of two 2-digit numbers within 10 seconds. c) Students will be able to approximate the percent of a 4-digit number within a margin of 10% within 10 seconds.
Key Questions/Driving Questions: 1) What strategies can we use to find good estimates of addition, multiplication and percentage problems? 2) How do you find the percent of a number? 3) How can you find answers fast, without using paper and pencil?
Prerequisites and Sequence of Lessons in Unit: Students must have a rough understanding of percents. This lesson does not assume a great deal of knowledge about percents. Students should be comfortable performing 3-digit addition problems and 2-digit multiplication problems with the use of an algorithm.
Lesson Introduction: Students practice strategies for estimation Introduce the concept of estimation. When is it useful to practice estimation? Explain that when we are just looking for ballpark answers, it is not completely necessary to find the exact answer. Sometimes, a good estimation will get us there. Give examples of when estimation is more relevant than precision.
Offer strategies for each of the three operations they will see: 1) addition with 3-digit numbers, 2) multiplication with 2-digit numbers, 3) finding the percent of a 4-digit number, where percents are typically multiples of 5%, never breaching 100%.
For example: 1) Ignore the ones column, add the hundreds together, then add the tens. 2) Round to the nearest ten, then multiply the two digits in the tens place and add two zeroes (incorporate two factors of 10). 3) Round the percent to the nearest 10%, then find 10% of the number and multiply that by the appropriate factor.
Show students a few examples and model for them, by thinking aloud, how to estimate the answer of each one. Reveal the "precise" answer for students and show how the estimation is in the ballpark range of the precise answer. Remind them of the context for estimation, to emphasize its importance.
Distribute a practice sheet for students that contains about 10 problems they will see on the Estimator Four game (see below). Give students only 2-3 minutes to estimate each of the 10 problems. Then review answers for them, showing them a range of each problem. Allow time for discussion of methods.
Lesson Core: Students Practice estimation with Java applet, "Estimator Four" Arrange students in teams of two. (This lesson was performed with 6 laptops with a class of 24 students, meaning 4 students per laptop. Each laptop should be pre-loaded with the Java applet "Estimator Four" available from www.shodor.org.)
Have two teams match up against one another with one computer for the group of four. Begin with "easy" difficulty, "20 seconds" and all three types of problems (addition, multiplication, percent).
On an overhead LCD (if possible) demonstrate to students how the game works. (Each team gets a turn to find a reasonable estimate of one of the three types of problems, randomly selected. If a team wins, it earns a colored token (there are two teams on the applet: red and blue) to place down a column, much like the popular board game Connect Four. The first team to get four in a row wins. Teams alternate turns. Students will require technical instruction on which buttons to press, which buttons not to press, and what kind of functionality they can expect from the applet.
Allow students to play Estimator Four. Do not allow calculators, but do allow students to allow paper and pencil if it is a must. Gradually increase difficulty or reduce time for students who are advancing quickly. Increase time limit for students in need of more assistance.
Circulate to monitor student progress and diagnose any computer issues.
Lesson Closure: Students submit evidence of understanding As a closing activity, give students a timed quiz of another 10 Estimator Four-like problems. Give them only 2-3 minutes to finish. Time permitting; hold a conversation about why we estimate. When is it appropriate? When is precision important? When will they use estimation?
Evaluation/Assessments: Closing activity can serve as an informal daily assessment. Teacher should grade and analyze to see what trends are occurring among students. Some may be relying too heavily on precise operationsbasically trying to do operations faster, without the use of number sense or approximation.
Narrative of Lesson Plan Teacher: Justin Fong The Students I teach at IS 286, a middle school in Harlem with chronically low test scores. The students in my seventh math class, however, have mostly "passing" standardized test scores from the sixth grade. This led me to assume a great deal about them when I began with them this year. What I found was that they lacked many of the basic skills that I assumed "passing" students would have mastered. More importantly, their number sense was very poor. On quizzes, students commonly multiplied a number incorrectly, off by a factor of 10 or 100 perhaps, but did not catch the error. Having a familiarity with numbers is prime in mathematical success. Lesson Implementation I felt that this lesson on estimation was very important to my students. I almost wish they played Estimator Four every day for ten minutes. Their number sense would be far better for it. Introducing a game to the class is always exciting. I had a feeling they would be very excited to play, and very loud and competitive when they were in the middle of it. This was indeed the case. Most students got the point of estimation. They came up with ways to essentially round numbers, and perform mental math. (Most kids got stuck with percents, but I blame myself since we did not yet have enough exposure to percents.) Their answers were making sense, making math sense. This is an important skill that I want them to develop, and I saw it being developed as they played the game. Other students, however, did not get the point. This was evident both in their in-class worksheets and as they played the game. Some students were still going after the precise answer, seeming to be unable to "figure out" how to estimate. These students had to be pushed to veer away from their familiar operations and towards a bigger picture.
We also had a great deal of technology-related problems. The Java applets were not all running correctly. In the end, one of our main concerns was that the applet itself seemed to be flawed. Sometimes the same team would go twice, and other times, the token would fall in the incorrect column. Knowing this, it would be nice to talk to the class beforehand about the imperfections of the game (and it is a shame that there are imperfections at all, but that is life!). We could talk about what we would do if the red team ended up getting two turns in a row. (The red team could forfeit the second turn once they realized they unfairly received a second consecutive question, by deliberately entering a wrong answer, for example.) Other than that, once students get to the applets, they can fend for themselves. Because of the time-frenzy of the game, students' eyes are typically glued to the screens, even if it's just to see if their opponent will run out of time. Student Learning I feel that my students learned some quick and dirty strategies on estimation with these three types of problems (addition, multiplication and percents). Seeing a game that only asks for estimation also legitimizes my arguments for why estimation is important. Many students are resistant to the idea of imprecise answers; they need to be shown the importance of number sense. I think their number sense can begin to improve after a lesson such as this one. Number sense is not something gained overnight, however. It is a general familiarity with numbers, and it only comes from practice. This lesson focuses on that kind of practice. My learning Although most kids were very much engaged, there were certainly some groups that were dealing with high levels of frustration. We noticed that these groups were either experiencing technical difficulties, or did not have the math skills to do well enough. One group, for example, asked me if they could omit multiplication and percent problems. There are two lessons to be learned here: 1) Technical issues should be ironed out way beforehand. We had not experimented with the applets enough to know what kinds of trouble we would run into. Had we done this, we could have at least forewarned students and discussed fair troubleshooting methods with them. 2) Students need scaffolding. Students are most interested when something is at the right level. Something too hard becomes frustrating. Something too easy is boring. Something right at their level is engaging. It is often hard to judge what level might turn out to be the right one. I think I know my kids, but they can surprise me every day. (I needed to tune down the difficulty a little bit by cutting percents or by increasing time limits. This would have led to greater student learning because frustration would have been averted.) What I would do differently Next time I do this lesson, I am going to make it a lot simpler for my students. I will perhaps start slow with just addition estimation. Then we will play with the applet, and then reconvene to talk about multiplication. More applet practice, reconvene, and then work with percents. This can be spread over another period as well. Teaching percents better before this lesson would have helped also. Role of GK12 fellow My GK12 fellow, Logan Brant, was a trusty facilitator in this lesson. I had been working on percents and we had both talked about incorporating more interactive activities into the classroom. Logan suggested these applets as a way to accomplish that. Together we researched the various math-related applets available to us and settled on this one because it had good math and a fun game format. Logan helped to set up the entire computer operation. While I worked on the mini-lesson and came up with a practice worksheet, Logan's help came in handy during game time. He had the opportunity to circulate and walk students through estimation strategies, if he wasn't already tied up with a technical issue. More importantly, Logan is responsible for opening my eyes to applets and interactive games that my kids can get use. |
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