Analysis: Multiplying Fractions Versus Whole Number Multiplication
Question: Do we always get a greater product on multiplying two numbers?
Observe the following multiplication problems.
6 × 4 = 24
Case 4: When one of the multiplicands is a fraction greater than 1
The product is will be greater than the other fraction
8/3 × 2/5= 16/15
6/10 × 9/5= 54/50= 27/25
9/6 × 1/7= 9/42= 3/14
Case 5: When both of the multiplicands are fractions greater than 1
The product will be greater than both the fractions.
8/3 × 3/2 = 4
6/4 × 5/2= 30/8
Shade It In Multiply Fractions With Area Models
Fractions and area models take center stage in this math worksheet that shows students how to easily visualize and multiply fractions. Step-by-step instructions and two practice models get students comfortable with the process before they are asked to draw their own area models to solve the equations. Area models are a useful visual tool to help your fifth-grade student confidently tackle fraction multiplication.
Multiplying Fractions With Whole Numbers
When I introduce multiplying fractions, I start out with multiplying fractions by whole numbers. Its important to review basic fractions and ensure that students know how to represent these. Then, when multiplying a fraction by a whole number, they will repeat the fraction model as many times as the whole number.
My students receive an information page with the steps to create a model for multiplying fractions by whole numbers. After a mini-lesson and interactive examples on a smartboard, theyre given an outline of the visual models and practice problems to try on their own. They fill in the models to match the problem and find their answers.
Next, I challenge my students to create models on their own. Theyre given additional problems and the space to draw their models. The difficult part of this is organizing them neatly, so using a ruler is important a great opportunity for them to use appropriate mathematical tools.
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Why Fraction Visual Models Matter
Research shows that fraction visual models are incredibly effective for teaching fraction concepts.
Conceptual understanding means knowing the meaning of a numerators and a denominators. As well as understanding what actually happens when we perform fraction operations. Knowing why the algorithms work, as opposed to just getting answers.
Without conceptual understanding, students cant use what they know to solve word problems, or apply their understanding in real-world situations.
Many students learn to calculate with fractions, but most never developing conceptual understanding. Some educators believe that a procedural approach is good enough I didnt learn about fraction visual models in school, and I turned out fine!
But a number of studies highlight the disappointing results of the old school approach. For one, American students consistently perform poorly on international math assessments. Many of us continue to struggle with fractions through collegeand into adulthood.
In addition to their instructional benefits, fraction visual models also make effective assessments. As in my Secret Whiteboard example, when students draw a fraction visual model, you can instantly tell whether or not they have understand the concept. This isnt always clear when theyre just calculating.
Fraction Visual Models: What Every Teacher Should Know
Its no secret that fractions can be a great hurdle for many students. And textbooks and online platforms now use fraction visual models to help overcome that hurdle.
So why do so many students still struggle? If fraction visual models are so effective, shouldnt fraction frustration be a distant memory by now?
The answer is that its complicated. Sure, visual representations can offer tremendous benefit to our students. And not just for fractions. But theyre not a magic cure all.
In order for visual models to serve their purpose, we need to choose the right model, at the right time. And we need to see models as a vehicle to understanding, not just another item in the curriculum.
After making every possible mistake in my own classroom, Ive figured out what works and what doesnt. So if youre not sure how to teach your fraction visual models, dont fret.
Just follow the five tips below, and check out the recommended resources, to make fraction visual models a breeze for you and your students.
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Fractions Activities For Your Classroom
- Lesson Plans and Classroom Activities
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But many textbooks make fraction visual models more complex and confusing than they have to be. And there is not enough emphasis on having students draw models instead, they mostly just interpret the models printed in the books.
As a result, they dont get the full benefits of using visual models. And teachers dont get to see whats really going on in their students heads.
Leaving The Denominator Unchanged
Students can make the mistake of forgetting to multiply equal denominators. This is likely due to the fact you dont have to touch equal denominators in fraction addition.
For example, they might see × and incorrectly answer instead of ².
How to help your students
In the practice guide, expert panelists suggest explaining the conceptual basis of fraction multiplication using unit fractions .
In particular, teachers can show that the problem ½ × ½ is actually asking what ½ of ½ is, which implies that the product must be smaller than either fraction being multiplied.
Verbalizing this misconception is helpful, but visualizing it is especially effective. Enter the fraction wall!
Fraction walls are a brilliant way to help students see what, in this case, an abstract one half of one half looks like.
Have a chat with your superstar this weekend! Look at the Fractions Wall together.. ask them about our key vocabulary… NUMERATOR, DENOMINATOR, MIXED NUMBER, EQUIVALENT… how many can they explain to you?! #rudstonmaths
For this activity, split your students into groups of four. Next, theyll divide themselves in teams of two, one being Player A and the other Player B.
Give each group a deck of shuffled cards .
As in the picture above, each student will draw a numerator card and a denominator card .
Depending on your schedule, you can assign the whole deck or give students a timer to complete as many as they can.
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First Step To Dividing Fractions: Modelling
Just like multiplication, once students gain expertise in modeling division problems for fractions, then the procedural method to find the answer is a cakewalk. Therefore, for the division of fractions as well, its suggested to use the approach from visual to non-visual method. For students, such an approach is more plausible as it lays a strong conceptual foundation.
Lets see a few examples.
How To Multiply A Fraction By A Fraction Using A Model
In this video, well show you the multisensory way of multiplying a fraction by a fraction using a model.
Youll see how to:
- Draw a model of the fraction
- Know which one to start with
- How to identify the area that needs shading
We offer all online math services featuring the multisensory math method which you can learn about here: madeformath.com/services
Okay, my name is Adrianne from Math for Middles. Im a multisensory math tutor, and today I wanted to show you how to do multiplication of fractions by a fraction using the model method. Now if you have a kiddo thats in the public school system or any school thats using the National Council for Math Standards or NCTM, theyre going to be expected to know how to do modeling with fractions.
I wanted to talk to you about this. The thing that gets really confusing for a lot of kids is this multiplication of fractions when were using a model, its confusing because theyre not sure whats really going on. Theyre just using numbers. Its super easy. I multiply across the numerator and then I multiply across the denominator. But when were trying to model it, it can be hard because were taking a part of a part, and that is confusing. And thats more like division than it is multiplication.
Did you like our math video? Make sure to send in your questions and well answer it the multisensory way. Dont forget to subscribe, share, and like our videos.
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Key To Excel Practice More
Understanding fractions and their operations become a lot easier once children gain expertise in visualizing the problem and know what needs to be calculated. But to gain such confidence, they need to practice a lot of problems. Its necessary that they model the problem and then solve them.You can refer to these worksheets on SplashLearn which are easily downloadable and printable, to help solidify your kids understanding of multiplying fractions.
How To Draw Models For Dividing Fractions
To divide a whole number with a fraction make the whole number into a fraction by putting it over a denominator of 1. Big idea we can use visual models to help us represent and solve problems involving dividing a whole number by a fraction.
Dividing Fractions Ccss 6 Ns 1 Apply And Extend Previous
3 Ways To Divide A Whole Number By A Fraction Wikihow
How To Draw A Model For Dividing Fractions Fresh The Best By Images
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What Is An Area Model
An area model is a useful tool you can use to model certain fraction concepts. An area model is a square that you divide into equal-sized rectangles to represent a fraction. Here are some math concepts you can model with fraction sticks and area models:
A prevalent theme in the Grade 4 Common Core standards is understanding equivalent fractions, or, more precisely, the notion that a fraction remains the same when you multiply the numerator and denominator by a non-zero whole number. Understanding equivalent fractions is important when comparing and ordering fractions, adding and subtracting fractions with unlike denominators, and reducing fractions to their lowest term. So, lets talk about finding equivalent fractions!
This is a great time for students to experiment informally with fraction sticks. They can choose a fraction, such as 2/3, and see what combinations of other fractions are equivalent, such as 8/12. After various opportunities to experiment informally with fraction sticks and write down their observations, they will be ready to learn a more formal rule: when you multiply the numerator and denominator by the same non-zero number, you will obtain an equivalent fraction. If your students are ready to be challenged with the symbolic form, you can explain:
If we were to demonstrate 2/3 = 8/12 fact using an area model, first divide the area of the unit square into three rectangles. Shade in 2/3 of the area of the unit square.
Dividing Fractions In Real Life: Introduction
We come across situations in our day-to-day lives where we apply the concept of fraction division. Fraction division can be made interesting and the concept can be fully instilled in our kids minds if they are challenged to solve scenarios from real life.
This would help the kids to visualize and make sense of fraction division.
Example: Suppose you have 3 apples, each cut in half. To how many people can you distribute these 3 apples, if each gets half a piece?
Lets picture this situation & solve it. Also, an apple a day will definitely keep the math blues away!
The problem can be further extended by modifying the fraction.
For example, what if you decide to give each person a quarter piece? How many people can be served now?
Or what if you decide to give each person three quarters? How many people can be served now?
Comprehending a problem and then modeling it is a crucial step in any problem-solving scenario of fraction division. Encourage students to try visualizing the scenario and then look for its solution.
Draw and Solve!
A group of friends bought a pizza. They shared the pizza equally and finished it.
Try identifying how many friends were there if each of them got
|one-eighth of the pizza|
|So, 1 ÷ 2/8 = 4|
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How to draw model math. Just think of addition and subtraction as directions on the number line. We also have tons of graph paper for you to print and use. A complete unit for teachers.
We can use the number line as a model to help us visualize adding and subtracting of signed integers. Classkick is a free app that shows teachers in real-time exactly what students are doing and who needs help so they can provide instant feedback. The worksheets can be made in html or PDF format — both are easy to print.
Predict how addition or subtraction of a proton neutron or electron will change the element the charge and the mass. The printable worksheets contain skills based on writing multiplication sentences describing models completing equations and drawing models. Provides an outstanding series of printable graphing worksheets.
Use the number of protons neutrons and electrons to draw a model of the atom identify the element and determine the mass and charge. Read the scenario carefully share the objects evenly draw the correct number of specified objects in each group to complete the worksheets. Model to build understanding of bar models.
While solving math word problems in worksheets ask your 2nd grade child to draw bar models to understand which operations should be used and how the problem should be solved. Bar Model Pictorial models to divide numbers is an interesting strategy and most helpful in solving word problems. You can also customize them using a generator.
Tip #: Start With The Meaning Of Fractions
The essence of understanding fractions comes down to numerators and denominators.
The denominator cuts a whole into equal parts. The numerator counts those parts. Students first learn to understand fractions as parts of a whole.
Later, students are introduced to the idea of fractions as division. The fraction ¼ is the same as 1 divided by 4. Students should be able to model both meanings of fractions before moving on to fraction operations.
But many students are expected to work with models of fraction operations or equivalence before theyve mastered the basics.
Whether youre teaching 2nd grade or 22nd grade, do yourself a favor and take a minute to review numerators and denominators with your students.
Its also important for students to understand simpler visual models before using advanced ones. The typical progression for modeling fractions starts with circular and extends to rectangular area models. Next, students can move on to more complex representations, like number lines and bar models.
Skipping a step in the progression is a common cause of confusion. Number lines and bar models are still representative models. But theyre more abstract than the circular and rectangular models, which show a fractions size in two-dimensions. I see this issue a lot with middle school students who struggle with bar models most never became fluent with arrays, area models, and number lines.
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Major Errors Students Make With Fraction Multiplication
Though some students will quickly grasp your fraction multiplication lessons, others can struggle with these new concepts.
The earlier teachers catch these misconceptions, the sooner students can learn from and correct their errors.
According to the What Works Clearinghouse Institute of Education Sciences practice guide, Developing Effective Fractions Instruction for Kindergarten Through 8th Grade, these are some of the most common misconceptions in regards to learning how to multiply fractions.
Multiply Fractions With Visual Models
One of the more challenging skills in teaching fractions is using visual models to show mathematical thinking. In 5th grade, this is an element in both decimal and fraction operations. I like to introduce these operations with the visual models before moving to the standard algorithm of any skill. Let me show you how I use visual models to multiply fractions.
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Model Improper Fractions And Mixed Numbers
In an earlier example, you had eight equal fifth pieces. You used five of them to make one whole, and you had three fifths left over. Let us use fraction notation to show what happened. You had eight pieces, each of them one fifth, \frac, so altogether you had eight fifths, which we can write as \frac. The fraction \frac is one whole, 1, plus three fifths, \frac, or 1\frac, which is read as one and three-fifths.The number 1\frac is called a mixed number. A mixed number consists of a whole number and a fraction.
The Simple Way To Multiply Fractions
There are three simple steps your students need to follow when learning how to multiply fractions:
And before your keen students ask, the answer is yes unlike adding fractions, you can multiply two fractions with different denominators.
Lets go through an example together!
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First Step To Multiplying Fractions: Modelling
Using visual models as a learning aid makes the process of both teaching and learning more effective, fun, and interactive. It is a student-centered approach and helps kids in visualizing key mathematical concepts, which further helps them to gain a deep conceptual understanding at a root level.
For the teachers, visual models can trigger discussions of mathematical ideas and relations with previously known concepts. It helps them have a better understanding of students thought processes about the concept.
For that reason, teaching fraction multiplication initially through models and then proceeding towards standard procedures should be the ideal approach.
To begin with, Fraction Multiplication can take up 3 different forms.
Level 1A: Whole × Fraction
Level 1B: Fraction × Whole
Level 2: Fraction × Fraction
Please note: We will not be taking up mixed numbers as a separate case because these are also fractions written in a different form. To help your kids learn about mixed numbers and improper fractions through fun games, you can sign up here!
Level 1A: Whole × Fraction
Example 1: Tim used ¼ of the pumpkin to make one pumpkin pie.
She made 3 pies. Lets find out how many pumpkins she used in total.
As she used one-fourth of the pumpkin 3 times, its multiplication expression would be 3 × ¼
So, Tim used three-fourth of the pumpkin to make 3 pies.
Example 2: If Tim had to make 5 pies, how many pumpkins would she need?
5 ×¼ =5/4
Level 1B: Fraction × Whole
Example: ¼ × 8