Chapter 13 - Building Strategies for Fraction Computation (Addition &…
Chapter 13 - Building Strategies for Fraction Computation
Understanding Fraction Operations
A problem-based, number sense approach
1 - Use tasks with meaningful contexts (have students create own problems).
2 - Explore each operation incorporating a variety of models (have students defend their solutions using models).
3 - Include estimation and invented methods as the foundation for strategy development (keep focus on numbers and meaning of operations.)
4 - Address common misconceptions (discuss openly why some approaches lead to right answers and why other approaches don't).
Grade 3 - add equal sized units.
Grade 4 - students add and subtract fractions with like denominators and multiply fractions with whole numbers. Grade 5 - Students add and subtract fractions fluently with like & unlike denominators, multiple and divide fractions. Grade 6 - Students divide fractions by fractions.
Addition & Subtraction
Contextual examples & models
area models: Circldes are an effective model for adding and subtracting fractions because circles allow student to develop mental images of the sizes of different pieces.
length models: Cuisenaire rod, rulers, and number lines are all linear or length models. One advantage of number lines are they can be easily connnected to the ruler, which is one of the most common real contexts for adding or subtracting fractions.
Most students first see examples of addition and subtraction in real life by measurement. (how tall they are, time, picture frame, how much they have to run, etc.) Making recipes is a common intro to these fractions
Estimation & Invented Strategies: Estimation is a “Thinking tool” that students should use to build the meaning of addition and subtraction with fractions.
Developing the Algorithms: The “Common Core State Standards” suggest that 4th graders should add and subtract fractions with like denominators. If students have a good foundation with fraction concepts, they should be able to add or subtract like fraction immediately.
Fractions Greater Than One: Include mixed numbers in all activities with add and subtract and let the students solve these that make sense to them. Students will tend to add and subtract whole numbers as they are more femilar with them.
Misconceptions: Explicitly discuss misconceptions with your students as fraction operations make students incorrectly generalize rules from whole-number operations.
fraction by whole number: Notice the skip counting, called iterating, is the meaning behind a whole number times a fraction
whole number by fraction: Students’ next experinces with fraction multiplication should involve finding fractions of whole numbers. Although multiplication is commutative, the thinning involved in this type of multiplication involves partitioning (not iterating). Counters (a set model) is an effective tool for finding parts of a whole.
fractions of fractions: Once students have had experiences with fractions of a whole or whole times fractions, a next step is to introduce finding a fraction of a fraction, but to carefully pick tasks in which no additional partitioning is required.
area model: Area model for modeling fraction multiplication has several advantages. 1. Works for problems in which partitioning alength can be challenging. 2. Provides a powerful visual to show that a result can be quite a bit smaller tha either of the fractions used or that if the fractions are both close to 1. 3. It is a good model for connecting to the standard algorithm for multiplying fractions.
Estimating & Invented Strategies: In the real world, there are many instances when whole number and fractions must be multiplied and mental estimate are quite useful. You can mentally calculate products of fractions and whole numbers by thinking of the meanings of the numerator and denominator.
Developing the Algorithms: With enough experiences (several weeks) using multiple representations, students will begin to notice a pattern. Then, the standard multiplication of fractions algorithm will logically develop from those ideas.
Misconceptions: 1. Treating the denominator the same as in Addison/subtraction problems. 2. Inablitiy to estimate approximate size of the answer. 3. Matching multiplication situations with multiplication (and not division). 4. Using the deadly key word strategy.
factors greater than one: Once students have explored products with both factors less than one, include tasks in which one factor is a mixed number. The more integrate these problems, students will have opportunities to think about the impact of multiplying a number less than one by a number greater than one.
Contextual Examples & Models
Whole number divided by whole number: A partition or sharing contact is helpful in interpreting division of a whole number by a whole number.
fraction divided by whole number: These problem types are introduced in the CCSS-M in grade 5 unit fractions and in 6th grade for no unit fractions.
whole numbers divided by fractions: This problem type lends to a measurement interpretation (also called repeated sub ration or equal groups).
fractions divided by fractions: Over time, using various contacts and numbers that vary in difficulty, students will be able to take on problems that are more complex both in the context and in the numbers involved.
Estimating & Invented Strategies: Use estimation to suppor understanding of division of fracitions. This can help them think about the meaning of division and then develop a reasonable estimate.
Developing the Algorithms
Common-denominator algorithm: Relies on the measurement or repeated subtraction concept of division. Also it links to what students have already learned in adding and subtracting fractions with common denominators and is aligned with whole-number division
Invert-and-multiply algorithm: To invert the divisor and multiply may be one of the most commonly taught, but poorly understood, mathematical procedures in the elementary curriculum
Thinking the answer should be smaller because they are 'higher numbers'
Students confuse the representation with the answer. when counting parts they say 6 but don't know how to explain as a fraction
Units must be emphasized otherwise they don't understand what they're fraction is a part of
Knowing what the unit is, is critical for interpreting the remainder. Students may also use the original unit instead of the denominator of the divisor