Chapter 13 More Fractions! (Some important facts and ideas (Some other…
Chapter 13 More Fractions!
Addition and Subtraction
Set the expectation that students will use a variety of methods and that they will vary widely with the fractions encountered in problems.
In the real world kids first discover they must use fractions in measurements problems (usually).
Ergo, use contextual strategies!
Most initial problems shouldn’t use a denominator greater than 12.
Models- There are area, length, and set models that are useful for illustrating fractions. Best to being in with are and linear models.
Rectangles and number lines are useful things to begin with for measuring fractions.
Area models- circles have been found to be effective models for adding and subtracting fractions. Circles allow students to develop mental images of the size of different pieces.
Length models- Purportedly the best model, particularly in races or a sub sandwich.
Number line- excellent for adding and subtracting. Easily connected with the process of a ruler.
Estimation- a thinking tool. Should be highlighted as students build meaning for addition and subtraction with fractions.
Estimation required that students bring a sense of a fractions size to the process. They do this by the size of the fractions within the problems and estimating the size of the overall answer.
Without estimation kids may not understand how the answer relates to the problem.
The idea that the numerator counts and the denominator tells what is counted makes addition and subtraction of like fractions are the same as adding and subtracting while numbers.
When working on adding with like denominators, it is important to be sure that students are focusing on the key ideas: the units are the same, so they can be combined.
Begin adding and subtracting fractions with unlike denominators by having a task where only one denominator needs to be changed to a common denominator. For example, let student use fraction pieces and a context about the quantity of pizza eaten. The key question to ask is “How can we change this problem into one in which the parts are the same sized units?”
The main idea is so that students will see the connections and that seemingly unlike fractions lend themselves to making the same problem.
Fractions greater than One
Include mixed numbers in all of your activities activities with addition and subtraction and let students solve these problems in ways that make sense to them.
First use familiar concepts- iteration and partitioning.
There should be a four step progression of division. It begins in the 5th grade.
2 A fraction divided by a whole number.
3 A whole number divided by a fraction.
4 A fraction divided by a fraction.
1 A whole number divided by a whole number.
Two meanings of division-
Measurement interpretation- useful because students can draw illustrations to show the measures. Also used to develop an algorithm got dividing fractions. Have kids explore it.
To help students understand the meanings of division, ask them to describe what their equations are asking.
Partitioning is helpful with whole number division.
You are asking “How much share for each friend?”
Once they have done whole number division they need practice dividing any fraction with a whole number, using contexts.
Two algorithms of division.
Common denominator algorithm.
Algorithm is: To divide fractions, first identify common denominators, and then divide numerators.
Relies on the measurement/repeated subtraction concept of division. Uses estimation and fraction circles
Let students figure out patterns. When they do, bring those into discussions.
Invert and multiply algorithm
The most common, and probably most misunderstood, algorithm.
Partitioning and sharing examples effectively demonstrate this algorithm.
Multiply by the denominator and divide by the numerator.
The fractions don’t have to be inverted. Simply multiply by the denominator and divide by the numerator.
Kids need to know what the unit is.
You also need to know what the unit is to understand remainders.
Iterating and partitioning are foundational to multiplying fractions.
Use real world contexts to assist.
When working with whole numbers we should say “3 sets of 5.” The first factor tells how much of the second factor we have.
5th graders should think of these topics by thinking of multiplication of fractions from a perspective of scaling or resizing
First teach multiplication of a fraction by a whole number, then multiplying whole number by a fraction.
Set models, such as with counters, are an effective model for finding parts of a whole.
Area models, such as rectangle, are effective visual tools for illustrating and making generalizations.
Relative size of unit fractions.
After several weeks and using multiple representatives students will begin seeing patterns. Presumably the standard algorithm of multiplication will result. Make inquiries that press students to tell how the computation of connects to the illustration.
Have problems that include both fractions and fractions with a whole number. This way students will have opportunity to think about the impact of multiplying a number less than 1 by a number greater than 1.
Some important facts and ideas
Fractional success, particularly with computation, is closely tied to success with Algebra 1.
They must understand the procedures and why they work.
Having a deep understanding of fractions takes time.
Most errors are made on misunderstandings, not miscalculations.
In the grades
In 3rd grade students should be able to: add equal sized units to understand that a fraction a/b is the quantity that formed by a parts of size 1/b. This is foundational knowledge for addition of fractions.
Grade 4 should be able to add and subtract fractions with like denominators and multiply fractions with whole numbers.
Grade 5 should be able to add and subtract fractions fluently with like and in like denominators, multiply fractions, divide unit fractions by whole numbers, and divide whole numbers by unit fractions.
Grade 6 should be able to divide fractions by fractions.
If you focus too much on procedures, students will not understand what they are doing even while they carry out the procedure later. They will be set up for failure later on.
Strategically buildings their understanding about fraction operations from their knowledge of operations align with those for rational numbers.
4 steps for teaching fractions
1 Use contexts
2 Use a variety of methods
3 Include estimation and invented methods
4 Address misconceptions.
Some other important things to do and know
Give kids a large access to and variety of ways to solve fraction computation problems.
Mental and invented strategy can be applied and a standard algorithm developed over time.
Give solutions that make sense to the learners.
Problems don’t have to elaborate, they simply must matter to the students.
Explore each operation with a variety of models. Get kids to defend their solutions use a multiplicity of strategies.
Include estimation and invented methods as the foundation for strategy development.
Address common misconceptions.
Students can build on their knowledge of equivalence and invented strategies to develop a meaningful grasp of the common-denominator.
Have a public discussion that includes common misconceptions and an approach is in/correct.
Explicitly discuss common misconceptions with your students, especially with fraction operations because students incorrectly generalize rules from whole-number operations.
Give students time with problems and don’t rush them to use rules.
These develop side by side with the visual models and situations