Thursday, October 29, 2009

Fraction Action in Fifth Grade

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This story about fractional numbers is told from the bottom picture up.
In our EDC notebooks, we have been tracking equivalent fractions. We started on day 1 with the fraction 1/16. All students were able to understand that there were 16 pieces in the top row and if we were to shade one of those boxes it would equal 1/16 of the whole. On day 2, most students immediately realized that when we added another 16th, we created an area equal in size to the the 8ths row immediately below. When we got to 4/16, also 2/8, students saw that this was equal to 1/4 of the whole. It was at that point that students started to fill in percentages for the fractions that we had created. Most students knew that 1/4 = 25%, and they concluded that 1/8 was half as large so it must = 12.5%. A few students then realized that 1/16 must equal half of 12.5% or 6.25%. This knowledge of fractions and their equivalent percents will be a key component in adding, subtracting and comparing fractional pieces. Most students know several landmark percents like 25%, 50%, and 75%, and they can easily shade an object like a circle graph correctly with those amounts. Soon, students learn to represent other amounts like 30% (3/10) with accuracy based on their knowledge of the landmarks mentioned. Putting fractional amounts into percentages just makes life easier.
The third picture up comes from our current investigation, and it shows how students come to see "out of" statements as fractions. One out of three is the fraction 1/3, which seems easy enough but is VERY important conceptually.
The next two pictures are the absolute most important concepts that we will cover in this unit. Students shade in portions of a 10 X 10 grid and report the percent (parts out of 100) that it took to cover the portion. One picture shows one fourth of a grid shaded (25 blocks and 25%). the shading is a bit "artistic" , and it probably led to the misconception about shading in 1/8 of the grid, which should be 12 1/2 blocks or 12.5% . The percent was listed correctly, but the grid does not have 12 1/2 blocks shaded. Still, I can tell that the idea of 1/8 being half of 1/4 is understood.
The top picture is the Holy Grail of understanding fractions. All of the fractions listed are considered landmark (IMPORTANT) fractions in our number system. Understanding the percentages that go with each of these is a very critical conceptual step.
So, know the fraction. Be The fraction!
Know the percent. Be the percent!

Wednesday, October 14, 2009

Revisions: Another Great Example!

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Doing revisions correctly takes a bunch more work than just stating a new answer. Only by clearly stating why a prior answer was wrong and providing new evidence (NEW WORK), can an instructor be sure that the concept involved has really been mastered.
In this example, every distractor (answer choice) is examined. Clear reasons are stated as to why there is only one correct answer. This is enough to convince the person in charge (ME) that the concept of what a square number really is is now understood.

Thursday, October 8, 2009

PTB Prime-Time-Battle

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Prime-Time-Battle is a really fun EDC game that we play daily. This games gets students to try and find the prime factorization of numbers, as all numbers can be expressed as the product of prime numbers (OK, not 1 or 2). Besides building automaticity with multiplication facts, PTB prepares kids for much of the learning that they will see in many advanced courses. In advanced math courses, many times LCM and GCF are found through comparing prime factors. It also seems to just be a bunch of fun, as many kids see it as a a chance to compete in friendly math endeavor!

Friday, October 2, 2009

Have I Got A Story For You!

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To leave a comment, you have to click on the word "comment" in the comment counter at the bottom of the blog :-} I smell candy!
All of these samples of student work show how creative and efficient students can be when asked to solve basic multiplication and division problems. All of these students used very efficient strategies to carry out work that is usually a very scripted set of steps (Traditional algorithms allow for very limited flexibility in mathematical thinking).
Equally important, the students were able to create their own word problems that could be answered by solving the numerical problems. This is always a very tricky situation for students of this age, (in truth, it is for students of any age) as students are so used to solving numerical problems that are written for them. Many start by imitating problems that they have read in math books. However, when students start to create problems that are really relevant to their lives, they gain a very deep insight into what multiplication and division really are.
These are works in progress, and there are some very simple errors, especially in the division problems (usually because of faulty subtraction), but the kids are showing a pretty deep understanding of what the answers really mean. In most math books, the remainders of division problems are just written as a fraction or a decimal amount, but in the real world, remainders mean something. Remainders can be valuable or inconsequential, and contrary to popular opinion, not all remainders can be split into equal, puppies, and crunchy potato chips included!
In photo number 1, there is a subtraction error. 835-690 = ?
In photo number 2, one factor pair, 26 X 20, was left out.
In photo number 3, the remaining $17 could, and probably should, become 50 cents extra per person, although giving the money to charity is admirable :-}
In photo number 4, 3135-2300 = ?
As you can see, there is a ton of input to manage, but the concepts are coming together!