The Two Interpretations of Division Problems

There are two interpretation of division: the "How many groups?" interpretation and the "how many in each group?" interpretation. With the "How many groups?" interpretation of division, A ÷ B means the number of Bs that are in A––that is, the number of groups when A objects are divided into groups with B objects in each group. With the "How many in each group?" interpretation of division, A ÷ B means the number of objects in each group when A objects are divided equally among B groups. Watch the video below to review how to solve some simple divsion problems using picture representations and keep this process in mind as you read this blog post.

How Many Groups?

As shown below, the division problem 8 ÷ 2 can be translated as a "How Many Groups?" problem by writing an equivalent multiplication problem. In this case, "8 ÷ 2 = ?" can be rewritten as "? x 2 = 8". In this format, the multiplicand (? in our case) is considered to be the number of groups that are solution is asking for. Our multiplier (2 in this case) is the amount of cups in one popcorn ball. Finally the product (8) is the amount of cups that the number of groups we are looking for will produce. In simpler terms, our problem reads, "What number times 2 equals 8?"

Once you have figured out how to rewrite division expressions as multiplication expressions using whole numbers, the same methodology follows for fractions.

There are a few ways in which you can solve these types of problems including the use of tables, strip diagrams, double number lines, and/or annotated equations. All of these ways are listed in the chart below.

Practice Problem: In your math journals, write a “How Many Groups?” word problem for ⅔ ÷ ¾ and solve the problem with the aid of a strip diagram, table, and a double number line. Be sure to show your work.

How Many in One Group?

The second interpretation of division lies in the "How Many in One Group?" problems. For example, take the problem 6 ÷ 2. In order to demonstrate a "How Many in One Group" problem, we will say that 2 identical containers hold exactly 6 cups of popcorn. Our task is to determine how many cups of popcorn are contained within 1 of those containers. Therefore, we will rewrite the expression 6 ÷ 2 as a multiplication expression. As a result, our new expression will be 2 x ? = 6. The multiplicand (2) represents how many groups we have in our problem in terms of containers. The multiplier, or ?, is how many cups of popcorn are contained in each group or container. The product, or 6, is how many cups are held in 2 containers. In simpler terms, our equation reads as "2 times what equals 6?"

Notice the difference in how this equation reads compared to the "How Many Groups?" problem from before.

Much like the first interpretation, once you have determined how to rewrite the division expressions as multiplication expression for whole numbers, the same methodology is true for fractions (as shown below).

Practice Problem: In your math journals, write a "How Many in One Group?" word problem for 5 ÷ ⅓ with the aid of a strip diagram, table, and a double number line. Be sure to show your work.

The Distributive Property

The distributive property of multiplication over additions says that for all numbers, A, B, and C,

A x (B + C) = A x B + A x C

In the expression A x (B + C), we are multiplying A times the quantity of B + C. While the distributive property is often merely accepted as a working strategy with all numbers, there are ways explain why it works. This is perhaps best done by using arrays of objects.

For example, let’s use the expression 4 x (8 + 3).

There are two possible ways to show this expression by way of arrays. Since the quantity of 8 + 3 can be computed, we can find the sum of the two numbers and rewrite our expression as 4 x (11). In order to maintain our focus that the 11 in our expression is a result of adding 8 and 3, we could color code our objects in the array.

Using the definition of multiplication, we know that 4 x (11) can be thought of as 4 groups with 11 in each group. This is show in the array below. Notice that of the 11 circles that make up each horizontal row, 8 are red and 3 are black. This is what I meant by color coding our array to maintain that we added 8 + 3 to get our 11 in the expression.

If we count up all of our dots, we can see that our expression 4 x (11) is equivalent to 44, or:

4 x (11) = 44.

The second way to represent the expression 4 x (8 + 3) in the form of an array is as follows:

Instead of computing 8 + 3 and then multiplying that by 4, we distributed the 4 amongst the quantity of 8 + 3. In other words, we multiplied each digit within the parentheses by 8. As a result, we can now claim that 4 x (8 + 3) = (4 x 8) + (4 x 3).

Keep in mind that this lesson is about the distributive property. So, if you were asked to use the distributive property to solve 4 x (8 + 3), you would use our second method of solving it to do so.

Watch the following video from Khan Academy to get an interactive lesson of what we’ve already covered in this post.

Once you've watch the video, try the following problems and record your work in your math journals.

If you need any supplemental help with the distributive property, here are some games and supplemental graphics that may help.

Henry Anker Flash Game

Real World Example

Make sure to write down any questions you still have in your math journals so that we can cover them tomorrow in class.

Adding Fractions

MCC4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

• a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

Now that we know how how to compare fractions, we can work on adding fractions. For the purpose of this post, I will assume that the students I am working with with understand the concept of adding and subtracting whole numbers, equivalent fractions, and how to find the least common multiple of two numbers.

For a practical example of how adding fractions may play a part in your life, let's work with pizza. Let's say that as a class, we decide to have a pizza party. After the party, there are partial boxes of pizza left. I say that Ralph can take the left overs home as long as the remaining pizza is less that one whole pizza. Since there are multiple boxes filled with different number of slices of pizza, we must add fractions in order to determine whether the remaining pieces are less than one whole pizza.

Example 1: Adding fractions with the same denominator

We'll say that there are two pizza boxes remaining after the class party. It is important to note that each pizza was cut into 8 equal slices originally. One box of pizza has 3 pieces left in the box. The other pizza box has 1 piece left in it. The first pizza box can be described to have 3 of the 8 pieces left, or 3/8 of the whole pizza. The second pizza box can be described to have 1 of the 8 pieces left, or 1/8 of the whole pizza. In order to find out if the remaining pieces make up less than one whole pizza, we add our two fractions. Since our denominators are the same in this case, we can add our numerators and put the sum over our denominator of 8. 

So:

As you can see, 3/8 + 1/8 = 4/8 or 1/2 of a pizza. Since 1/2 of a pizza is less than 1 whole pizza, Ralph will be able to take the leftover pizza home in this situation.

Using our example as a reference, do the following problems and record your work in your math journals:

Example 2: Adding Fractions with unlike denominators

For this example, we'll say that there are two pizza boxes containing a portion of a whole pizza in each box. However, this time, one pizza was cut into 8 slices and the other was cut into 6 slices. Out of the pizza cut into 8 slices, 3 slices are left. Since 3 slices, of 8 slices remain, we can determine that 3/8 of the pizza is left. Of the pizza cut into 6 slices, 4 slices are left. Therefore, we can determine that 4/6 of the second pizza is left. In order to determine if the two fractions is less than 1, we must find the sum of the fractions.

Unlike the first example, these fractions do NOT share a common denominator. Before we add the two fraction, we need to find the least common multiple of our denominators (8 and 6). Using our previous knowledge of least common multiples, we already know that the LCM of 8 and 6 is 24. This means that our two fractions need to be translated as units composed of 1/24 parts. 

In the case of our fraction 3/8, in order to get an equivalent fraction consisting of 1/24 parts, we must multiply the numerator and denominator by 3.

As you can see, our 3/8 fraction can now be represented as 9/24.

In the case of our 4/6 fraction, in order to get an equivalent fraction consisting of 1/24 parts, we must multiply the numerator and denominator by 4.

As you can see, our 4/6 fraction can now be represented as 16/24.

Now that our two fractions have a common denominator, we can add the numerators and put the sum over our denominator of 24.

So:

As you can see, 9/24 + 16/24 is equivalent to 25/24. Since 25/24 is greater than 1, Ralph would not be able to take home the leftover pizza in this situation.

Do the following problems and record your work in your math journals:

After you've read the blog, watched the videos, and worked the practice problems, feel free to play FRUIT SHOOT FRACTION ADDITION for extra practice.

Comparing Fractions

Common Core Georgia Performance Standard MCC.4.NF.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.

When given any two fractions and asked to compare, it is sometimes difficult to determine which fraction is greater than another fraction. This is especially the case when the two fractions don't share a common denominator. For the purposes of this post, I am going to assume the students that I am working with are familiar with the concept of a common denominator and how to find one.

However, given two fractions without common denominators, how can we compare the two?

Visual Model Method

One way to compare two fractions would be to draw a picture and then make a comparison based upon your drawing. This method is not always the best route to take when working with paper and pencil. However, a computer program that gives you exact measurements can show this method with great ease. To work with this method, follow this link to an activity and work the problems listed in the google document.

Common Denominator Method

The second way to compare two fractions is to find a common denominator and then compare the numerators. A common denominator can alway be found by multiplying the denominators of both fractions. For example, to compare 1/2 and 3/8, we can give the two fractions the common denominator 2 x 8 = 24 in order to compare them. In order to represent 1/2 in units of twenty-fourths, you must multiply the numerator and denominator by 12. Thus, showing that 1/2 is equivalent to 12/24. Then, you must represent 3/8 in units of twenty-fourths. To do so, you must multiply the numerator and denominator by 3. Thus, showing that 3/8 is equivalent to 9/24. Since 9 units of 1/24 is smaller than 12 units of 1/24, then one can assume that 12/24 is greater than 9/24 (12/24 > 9/24). Follow this link to an activity containing a further explanation and problems to solve using this method.

Benchmark Method

The final way to compare fractions is to compare each fraction to a particular "benchmark" fraction. Some examples of good benchmark fractions are 1/4, 1/3, 1/2, and 1. In order to use this method, it is perhaps best to use the given fractions as units of length so that you can say that the fraction that is the closest distance to the benchmark (without surpassing the benchmark) is the greater of the two. Follow this link to view a video and work with this method.

So there you have it...three different ways to compare fractions. If you're a student who'd like more practice or an instructor who's looking for more supplemental activities for your kids, the following games should suit your needs:

Balloon Pop Fractions

Tug Team

Fractions Dolphin Racing

WELCOME!

Throughout this semester, this blog will serve as a means to share my thoughts, ideas, and potential techniques in regards to the lessons covered in my MATH 7020 class at the University of Georgia. Specifically, this blog will focus on various ways to "flip the classroom" concerning specific lessons covered by my professor (Dr. Sybilla Beckmann-Kazez). 

As a self-proclaimed computer-whiz, I am particularly interested in placing a heavy emphasis on technology as a tool to "flip" the aforementioned lessons. It is my hope that these posts will serve as a lesson plan of sorts and show how, exactly, I would go about "flipping a classroom" that covers the same material that Dr. Beckmann's course covers.

Prior to showing you how I would flip the classroom, I should discuss what "flipping the classroom" really means.

As shown in the MADDrawProductions video above, "flipping the classroom" is a technique in which teacher's can have their students cover the required curriculum without sacrificing the individualization and one-to-one time 

with their students. The traditional teaching model has teachers spending the class period standing in front of the class lecturing and having the students perform assignments outside of class. The "flipped" model does what it's name suggests and turns that model on it's head. Instead of surrendering the vital time with the teacher in order to hear the lecture, the students are assigned to watch, read, and investigate the content material outside of class. This allows precious class time to be spent actually working on assignments in which a teacher may be of assistance. 

Another important characteristic of the "flipped" model is that it allows children to learn the material at a pace that is beneficial to them. For students who struggle with the material, the ability to have more one-to-one time with the teacher in class may allow them to finally grasp it. On the opposite end of the spectrum, a student who may find an assignment relatively easy and boring may benefit from this model by moving ahead to more difficult tasks or serve as a peer tutor to another classmate.

Although I am a big proponent of the "flipped" classroom model, there are some foreseeable shortcomings that a teacher may face. Some of these shortcomings include:

• An increased work-load on teachers in the "pre-planning" phase of their year
• Requires a high degree of trust between students and teachers
• May become interrupted by the need to teach for standardized testing
• Relies on an availability of technology for the students wether it be at home, in class, or public access to a computer lab

Despite these shortcomings and those not listed, I am adamant that this model best suits how I would approach teaching as a profession.


Here's a cool infographic taken from Knewton's website pertaining to the "flipped" classroom model:

Created by Knewton and Column Five Media