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!

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.

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!

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 pieces.....kids, 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!

Double-Trouble or Split-It!

Recently we have added a "game" to our EDC notebooks. Double-Trouble and Split-It are games that are designed to get students to strategically think about doubling and splitting numbers. These skills are crucial if a student is to build great mental number sense.

For example:

When I ask kids to split 1,700, many kids balk, because they see 17 groups of 100 as an odd number of hundreds. Most kids are not seeing this as 8 1/2 groups of 100. However, most students can readily replace 1,700 with 1,600 + 100, and they can easily split these numbers to get to 850! This still may not seem important, but since the product of 17 X 10 is so easily calculated, it seems logical that 17 X 5 (half as big as 17 X 10) should be easily found if 1,700 can be easily split.

In a similar way, being able to double numbers with ease leads to being able to double small factor pairs into larger factor pairs. If 7 X 2 = 14, then 7 X 4 is twice as big. So, 7 X 4 = 28. Then, 7 X 8 = 56, and 7 X 16 = 112...likewise, 17 X 2 = 34, 17 X 4 = 68...17 X 4 = 68, 17 X 40 = 680, and 17 X 80 = 1,360 ...all done through doubling.

When doubling a number, many students find it easiest to double the largest place value first and then work through the smaller place values.

Doubling a number like 1, 486 might look like 1,000 + 1,000 = 2,000...400 +400 = 800 ...80 +80 = 160...6 + 6 = 12...added together, the sum is 2,972.

These are VERY important concepts! Please practice. It's the kind of thing you can do in the car.

Division and Multiplication A Match Made in Heaven

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This student shows why a great understanding of basic multiplication is so vital to solving division problems. By knowing the simple factor pairs shown (15 X 1 =15 15 X 2 =30...) and knowing the relationship of these to larger landmark factor pairs (15 X 10=150 15 X 20=300), this division problem becomes an absolute snap to complete correctly. I highly advise students to write the first several multiples, and a few larger landmark multiples, of the divisor when solving almost any division problem.

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Boxes of Markers is a student sheet designed to introduce the idea of portioning items evenly as a form of division. Many students simply skip count by the number of students , in this case 23, in order to find out how many groups of 23 can be made by multiples of 70 markers. It might look like 23, 46, 69;at which point the kids realize that they have run out of markers. Most can see that each "skip count" represents a student with a marker. So, three skip counts would equal 3 markers per student.

The work above is from a student that immediately recognized this as a division problem that could be solved using easy multiples of 23 until the number of markers was exhausted. This really is the goal of this type of exercise, as it makes division and multiplication forever seen as related activities. Once the student that did this work adds sentences that explain the numerical answers, the work will be at standard to say the least!