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Teaching Fractions and Decimals
Transcript of Teaching Fractions and Decimals
CCSS.Math.Content.3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
• CCSS.Math.Content.3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
o CCSS.Math.Content.3.NF.A.2a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
o CCSS.Math.Content.3.NF.A.2b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
• CCSS.Math.Content.3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
o CCSS.Math.Content.3.NF.A.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
o CCSS.Math.Content.3.NF.A.3b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
o CCSS.Math.Content.3.NF.A.3c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
o CCSS.Math.Content.3.NF.A.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
What the Standards say...
Grade 3- Develop understanding of fractions as numbers
Grade 4 -Extend understanding of fraction equivalence and ordering
-Build fractions from unit fractions
-Understand decimal notation for fractions and compare decimal fractions
Grade 5 -Use equivalent fractions as a strategy to add and subtract fractions
-Apply and extend previous understandings of multiplication and division
To effectively teach fractions, students must build a conceptual understanding of them. Often, fractions are introduced as concepts that are hard for students to contextualize. When first working with fractions, you should use examples that students would really use in real life examples (ex: 1/2, 1/3, 1/4). Rather than focusing on memorizing the definitions of denominator, numerator, and the rules for working with fractions, teachers need to make students understand why fractions work the way that they do. Before introducing any vocabulary, students should be provided with a plethora of visuals and manipulatives of how the numbers of fractions relate to their meaning and are applied to the real world. NCTM encourages the act of "tinkering", or students physically using objects to compare, model, and create better spacial awareness. Methods to visually show what fractions are include: equal parts, parts of a whole, divide and shade, and only after all of these ways, writing fractions as numbers.
Two Videos I Found Especially Helpful
This video is a lecture that demonstrates an effective way to introduce the concept of fractions and equivalent fractions using cuisenaire rods. Since it is a lecture the video is a little longer, but I found that she had very useful tips for teaching.
This video demonstrates how a classroom used pattern blocks to understand fractions. Pattern blocks are a good tool to use when learning about fractions because you can change the way that students see a whole. A whole may not necessarily be one thing, so by using pattern blocks kids can recognize that a whole is the total of all of the parts.
If you do not have the resources to provide students with the manipulatives previously mentioned, teachers can easily come up with creative new ways to help students understand fractions with a hands-on approach. Folding paper into parts of equal sizes can really come in handy as an easy way for students to conceptualize how as the denominator increases, the size of the pieces decreases. Another way that teachers can work around having materials is having students make their own rods by cutting up graph paper and coloring it in. This way, students are physically making their own examples and are more likely to understand.
Another hands-on way to get students thinking about fractions is through food or cooking. This way, you can make the topic of fractions not only more interesting, but also interdisciplinary and applicable to real life.
How to use a number line to understand fractions
By dividing the same number line into segments of different sizes, students can easily see which fractions are equivalent by looking to see which are the same size, or where they line up on the number line.
The number line also makes it clear to students that the larger your denominator gets, the smaller your "pieces" get.
Because in years past fractions were discussed mainly in terms of circles, pies, and pizzas, some students may have trouble connecting the ideas they have already learned back to the number line. This video comes up with a creative way to address this problem.
The wording that we use when we teach is very important. In order to maximize understanding of concepts, teachers should present a class with a word and ask them if they can think of any synonyms. Ex: "We are looking to find a fraction that is the same as this one, does anyone know of another word we could use for same?" This type of probing makes the students aware of the vocabulary that they are using, so that they can use it correctly.
Teachers must also keep in mind the words that they use to make sure they are telling the students correct information. The way that we word concepts can help students understand concepts that they once may not have. For example, many students call the denominator of a fraction the "bottom number". From this wording, it is easy to understand why students might be confused as to why we need common denominators in order to add, etc. Instead, we should refer to the denominator as the "name of the number" or as "how many pieces make a whole".
Ex: 1 apple + 1 apple = 2 apples ; 1 apple + 1 banana ≠ 2 banapples
• CCSS.Math.Content.4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
• CCSS.Math.Content.4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
• CCSS.Math.Content.4.NF.B.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
o CCSS.Math.Content.4.NF.B.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
o CCSS.Math.Content.4.NF.B.3b Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model.
o CCSS.Math.Content.4.NF.B.3c Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
o CCSS.Math.Content.4.NF.B.3d Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
• CCSS.Math.Content.4.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
o CCSS.Math.Content.4.NF.B.4a Understand a fraction a/b as a multiple of 1/b.
o CCSS.Math.Content.4.NF.B.4b Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.
o CCSS.Math.Content.4.NF.B.4c Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem.
• CCSS.Math.Content.4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.2
• CCSS.Math.Content.4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100.
• CCSS.Math.Content.4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
Comparing Fractions with unlike denominators
Oftentimes students will be able to spit out equivalent fractions, but have trouble ordering fractions. This is usually because they are using tricks or memorizing rules, rather than understanding what gives a fraction it's value.
Only after students are 100% comfortable with the concept of what fractions are and WHY they work the way they work, using pictures and ideas, do we introduce "rules" or patterns to help students. At this point, we can teach students that we can find equivalent fractions by multiplying the numerator and the denominator by the same number. This works because one whole can be represented by any number over itself (ex: 4/4, 6/6, 20/20), so you are technically just multiplying the fraction by one.
For those classrooms where students have access to computers or iPads, this website has a ton of good manipulatives to work with online.
Adding and subtracting fractions with like denominators
Up to this point, we have worked hard to make sure that students understand what each part of a fraction means. Once students grasp this, they can start to add and subtract fractions. It is important to make sure that students are still using visual representatives at this point, so that they fully understand the reasoning for why tricks, such as "add only the numerator, keep the denominator the same" work. Addition and subtraction of fractions is a good place to make use of the number line.
I found these videos to be helpful
I liked these videos because they take it slow and go step by step. They also each addressed common mistakes that students make.
In this video, we see how to teach addition of fractions using an area model, or fraction bars.
In this video, we see how to teach subtraction of fractions using a number line.
Both of these visual aides can be used to represent both addition and subtraction of fractions easily. There are other videos on this site that show specifically how to do it.
When you have more than one whole, fractions become either improper fractions or mixed numbers.
Short Activities to Help Students with Improper Fractions vs. Mixed Numbers
Professional Development Project
I chose to do my project on the instruction of fractions in grades 3-5. Additionally, I wanted to practice reading, understanding, and teaching to the Common Core content standards. I chose fractions to do this with because they are often a difficult concept for students to grasp. I know that I struggled with fractions when I was younger, so I am looking for the opportunity to find ways to make this topic more clear for students.
Multiplying Fractions by Whole Numbers
This video connects the concept of repeated addition to multiplication. I liked this video because it showed different ways to approach the same problem, addressing both standard 4a and 4b. It also made the concepts easy to follow by breaking down the problem step by step and using models.
How We Should Approach Word Problems
Word problems are an important part of lessons because they 1. relate concepts to real world situations and 2. force students to find entry points and create solution pathways to problems they may not have seen before. Because they deal with new situations, word problems allow students to demonstrate their deep understanding of overall concepts.
Applying Previous Knowledge to Word Problems
Here's a joke reflecting how students usually approach word problems...
Creating the Most Beneficial Word Problem
Ideally, word problems should be about a situation that students would have to deal with in real life. Additionally, they should be something that students have never seen before, nor do they know how to solve right off the bat. This forces students to think about the concepts behind what they are trying to figure out. Lastly, word problems should include extra information so that students must weed out what is important and what is not.
We have already learned how to find equivalent fractions, but how do we extend this concept to accommodate for our base ten system and changing fractions to decimals?
Easy- with the knowledge that we already have, it is safe to say that in order to create equivalent fractions, we have to multiply the numerator and denominator by the same
(This is the case because 3x10= 30 and 10x10= 100, then 30+27 = 57)
But how do we change that to a decimal? We use a Place Value Chart.
Because the fraction 3/10 is read as "three tenths", we place a 3 in the tenths column on the place value chart. 3/10 = 0.3
If we had a mixed number, say 5 6/100 ("five and six one-hundredths"), we would place 5 in the ones column, a decimal, a 0 as a place holder in the tenths spot, and then a 6 in the hundredths spot. 5 6/100 = 5.06
The video below shows the steps behind comparing decimals. As pointed out in the standards, it is important to remember that we can only compare decimals that are part of the same whole (we know at this point that that means common denominators). In order to make decimals part of the same whole, students need to make sure that both decimals have the same amount of numbers (this can be done by multiplying/dividing by tens to create equivalent fractions Ex:
• CCSS.Math.Content.5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.
• CCSS.Math.Content.5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.
• CCSS.Math.Content.5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
• CCSS.Math.Content.5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
o CCSS.Math.Content.5.NF.B.4a Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.
o CCSS.Math.Content.5.NF.B.4b Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
• CCSS.Math.Content.5.NF.B.5 Interpret multiplication as scaling (resizing), by:
o CCSS.Math.Content.5.NF.B.5a Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
o CCSS.Math.Content.5.NF.B.5b Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.
• CCSS.Math.Content.5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
• CCSS.Math.Content.5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1
o CCSS.Math.Content.5.NF.B.7a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.
o CCSS.Math.Content.5.NF.B.7b Interpret division of a whole number by a unit fraction, and compute such quotients.
o CCSS.Math.Content.5.NF.B.7c Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem.
Other Ways of Recognizing Equivalent Fractions Using Models
Students can recognize fractions as equivalent when they are the same "size". By lining up fraction bars, students can see which fractions take up equal amounts of space. Students also must recognize that 1 whole may be written differently than that. Since we know that the numerator is the number of parts we have, and the denominator is how many parts equal a whole, if the two are the same, that is equivalent to 1. Ex: 4/4 = 1 ; 12/12 = 1
I feel like I learned a lot from this project and I really enjoyed researching all of the different aspects of teaching fractions to students. Although I do feel like I am now much more equipped to teach this topic in a classroom, I think it is important to recognize that the material included in this prezi is not even half of what I discovered through my research. What I value most out of what I learned are the resources that I discovered. Throughout this project I came across numerous websites that I now have bookmarked and can refer back to at any point, for any topic I may need help teaching in the future. I learned that the amount of teacher resources out there are essentially limitless, and there is always somewhere you can go to read about, discuss, or share with others the latest, and most effective, ways to teach any topic. I also found countless websites about other teacher-related topics (that may not have anything to do with this project) that I know that I will be able to use daily in the future. To list a very select few of the ones I used most for this project:
secc.sedl.org/common_core_videos ; learnzillion.com ; teachingchannel.org ; mathlearningcenter.org ; khanacademy.org ; insidemathematics.org ; illuminations.nctm.org ; and also pintrest and several other blogging websites where teachers can share their creative new ideas for activities in the classroom.
I found this website somewhere throughout my research and I'm convinced that it was an actual gift from god! It's Professor Wu's (Berkeley) guide to teaching each of the content standards related to fractions throughout the grade levels.
Adding/Subtracting Fractions with Unlike Denominators
Once students have a firm grasp in the concept of making equivalent fractions, the idea of adding or subtracting fractions with unlike denominators becomes simple. Students must first make the fractions equivalent, and then add or subtract as they normally would.
We can make decimals into fractions and then add them (steps we have seen how to do already). Or we can add/subtract them while they’re still in decimal form:
I liked this lesson video because it took into account several of the standards. Not only did it discuss how this lesson completes the content standards, but also the practice standards. It also showed several good classroom techniques no matter what lesson you are teaching.
Multiplying By Fractions
Looking at fractions as the division of a numerator by the denominator
A way of looking at a fraction is division of two numbers. In general: a/b=a÷b
This video uses area models to show the same concept:
Multiplying by Fractions
In fourth grade, students were introduced to multiplying a fraction by a whole number. We did this through the idea of repeated addition. Now, in fifth grade, we need to explore how multiplying a whole number or a fraction by a fraction works. In order to multiply fractions, we need to multiply the numerators, and multiply the denominators. Here's a rhyme to help:
Multiplication as scaling
Solving Real World Problems
Games to help with fractions
Worksheets, activities, lessons
As with any topic, it is important for students to be able to relate what they have learned back to real life situations. The following video is a creative way that students can see how fractions work in real life.
Just as with whole numbers, multiplication of fractions can be explained through scaling. 3x10= 3 times the amount of 10.
Similarly, 3 x (25,421 + 376) is 3 times the amount of (25,421 + 376). With fractions, the same thing happens:
I found an abundance of different resources to use in regards to the topic of fractions. The Illuminations website has tons of lesson plans, for example:
There are also several other sites that are great resources. Some examples include:
I liked how she told the students the objective of the lesson after they had already done the work: when you multiply fractions less than one you are taking a part of part.
Dividing Fractions is the last step of working with fractions in fifth grade. The following video shows how division of fractions works through understanding with models:
A trick that is often used for dividing fractions is "drop, change, flip" this video shows how trick works:
Using computer games to help make a topic more interesting is a great way to get kids to practice math without them even realizing how much work they're doing. Here are a few online games that I found:
Books are also a great way to get students talking about fractions. By using reading books about fractions to your class, you are also making the lesson interdisciplinary. Here are some of the books I liked:
-Fractions Are Parts of Things
-The Hershey's Milk Chocolate Fractions Book
At this point, it is also important to note (and discuss with your students) why fractions break previous rules for operations that they may know. For example, Fractions decrease, rather than increase, when they are multiplied.