作者：Jovana Stanivuk 4 年以前
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Teachers have identified the positive impacts of teaching fractions throughout the whole school year, rather than in one distinct unit. This is called punctuated, or chunked learning. Punctuated instruction allows for teachers to be to responsive to students' thinking and responses while working with fractions. This type of instruction also gives students the opportunity to connect fractions to other mathematical strands/concepts and begin to see the relationship between various math units rather than isolating concepts into distinct units/strands. Teachers who use this type of instruction with the teaching of fractions tend to narrow their instruction to key fraction concepts, which allows students to master the basic concepts and spend the time needed to deeply explore such concepts.
It is also stressed how critical it is that Teachers have a deep understanding of all of the main concepts of not only fractions, but of the main ideas of each math strand in the specific grade they are teaching. This allows Teachers to have a greater ability to connect main concepts from various strands, giving students a holistic educational experience.
The article also shared many great resources for teachers who may be struggling or may just want to learn some new instructional strategies for teaching fractions. One resource I found extremely useful was LearnTeachLead.ca/paying_attention_fractions. This document explores four documents to support Teachers, but 'Fractions Across the Stands and Grades: Simple Tasks' was very effective in my eyes.
Examples: 3/2, 125/28
Examples: 3 4/5, 12 1/29
Examples: 1/15, 1/29
Examples: 1/2, 21/26
Hi Jovana,
I think your subtopic related strongly to mine. I researched and summarized 'Fractions Across the Strands' and I read that it is important for Teachers to have a good foundational understanding of the main concepts of fractions in order to teach students properly and appropriately. I think having the knowledge about the main concepts of fractions not only makes you a better Teacher, but also makes you feel more competent in your ability to teach mathematical concepts.
Great ideas! Tara.
When students are ordering fractions like 10/11 and 12/13, they can use the notion that the gap between the numerator and denominator is 1 fractional unit in both. They might further reason that since thirteenths are smaller than elevenths, there is less missing from the whole partitioned into thirteenths, so 12/13 is closer to 1 and therefore is greater than 10/11.
Using Equivalent Fractions
Convert fractions to have the same numerator to be able to judge which fraction is bigger or smaller.
Example: “I know that 3 is less than half of 7, so 3/7 < 1/2. I also know that 5 is more than half of 9, so 5/9 > 1/2 . I know then that 5/9 > 3/7 .”
When students have a strong understanding of fractions they are able to use number sense and proportional reasoning to make comparisons.
Using a Set Model to Represent Equivalent Fractions
4/6
However, although it is true that 4/6= 8/12 numerically, this shows an equivalent ratio rather than an equivalent fraction, since the whole has been changed from 6 to 12.
Using a Number Line Model to Represent Equivalent Fractions
Hi Emma,
As a visual learner myself I really like the options of different models to represent the fractions. Most of these models are ones that I forgot about so I am happy to be reminded of them. I think it is a great idea to share each of these models with students so they can in turn work with whichever model helps them to understand fractions the best!
-Brooke
Holly - I particularly think that using a set model is helpful when learning fractions. I think this model is really useful for people who have a visual learning style to understand how fractions work. In the document, they use an example of 3 groups of fruit. It is easy to look at the picture and identify that one-third (1/3) of the fruit are pears. I think the set model closely relates to the "representation" aspect of the C.R.A. In addition, the set model makes me think of the coding activity we did yesterday where we saw the fractions represented with the coloured balls on the screen.
E.g. addition can only occur between quantities with like units
E.g. number lines, rectangles, fraction bars, number rods
Emma - I like that unit fractions can be used as a starting point when teaching about fractions. Using unit fractions as benchmarks for referencing is a good strategy to teach students, and using models and pictures to demonstrate these parts of a whole in a concrete way will help learners get a better understanding of this new concept.
E.g. the fraction 5/4 is read as five-fourths, NOT five over four, five out of four, five-quarters
Use benchmarks for referencing
Compose and decompose fractions into unit fractions
Count by unit fractions to familiarize with numerator and denominator
As teachers: avoid providing pre-partitioned figures
Jovana- I think the differentiation between part-part fractions and part-whole fractions is very interesting. Personally, in primary school we focused on part-whole fractions. Instinctively, when I think of fractions I think of parts of a whole or part of 1 so the fraction 3/2 would be 1 and a half rather than three oranges and two non-oranges and the total of the numerator and denominator is the total amount of fruit. Once we got into ratios, I learned to communicate part-part fractions as 3:2. It's important to be consistent when teaching math and to remember that students connect to various methods of explanation so introducing this method in primary could be very useful.
Three ways to think about Part-Part Relationships.
Set Model
In the set of pieces of fruit, the number of pieces of fruit that are apples is 2/6 the number of pieces that are not apples.
Area Model
In the rectangle, 7/2 as many regions are shaded as are unshaded.
Number Line Model
Number Lines show a part-part relationship in which the distance a flag is hoisted up a pole.
Set Models Representing Part-Whole Relationships
When looking at a set of objects, it is important to be explicit about what is considered a whole.
Example: A whole is a carton of 12 eggs.
Number Line Model Representing Part-Whole Relationships
A fraction on a number line, as shown, is another example of a part whole relationship.
Enlarging or shrinking a quantity by a factor
Illustrations can be used to simplify complex questions. For example, if half of a lasagna is left and dad eats two-thirds of the remainder, how much of the whole lasagna is left?
Illustration (CONCRETE)
Algorithm (ABSTRACT) 1/3 x 1/2 = 1/6
ENLARGING E.g., Walking one and a half times as far (1 1/2) on Tuesday as on Monday
SHRINKING E.g., one-third (1/3) of an object
Dividing the numerator by the denominator
The result is the decimal equivalent for the fraction
The context of the question suggests equal sharing
Students introduced to the concept through equal-share contexts
Important for teachers to be precise depending on the construct of the fraction
E.g. in 3/2, the numerator is the WHOLE, and the denominator is the PARTITION
Example
This can also be represented on a number line
This is only one way to divide up the brownies. Other options include giving one brownie to each person and partitioning the remaining two or partitioning the brownies equally into four parts.
