Teaching Fractional Arithmetic Visually

I was recently asked to teach fractions to a group of kids in my daughter's second grade class. I think most kids are quick to learn the concept of a fraction, and as a consequence, they get bored with the endless drills of coloring exercises. Many stop paying attention, which leads to problems later on. Trying to figure out how to do things differently, I recalled an afternoon two years ago when I taught my daughter how to add and subtract fractions. I figured she that if she understood the concepts of a whole and a part of a whole, she would be able to tackle basic fractional arithmetic. I used visual representations, and she quickly learned how to add and subtract of homogeneous and heterogeneous fractions. Given my prior success, I decided to try this in my daughter's classroom.

Step 1:
Explain that fractions are the part of a whole that you are talking about. Give a ton of examples. This is a critical step before moving on. Kids must understand what a fraction is and how to read and write them using the standard notation.

Step 2: Review how to represent fractions as pictures and vice versa. For example,


Use a few more examples. Teach the kid how to draw squares divided into thirds, fifths, tenths, and a few other common denominators.

Step 3: Explain what it means to add fractions. Say that it means adding parts of a whole. Explain that if we add parts of a whole, it is easier to add pieces of the same size. Give the example of adding 1/2 and 1/4. 1/2 and 1/4 have different sizes. So, how do you tell how much of the whole you have when you put together 1/2 and 1/4? The best thing to do here is to use four blocks of the same size. Manipulate two of the four blocks as 1/2 of a whole. Give the kid enough time to realize that 1/4 is 1/2 of 1/2.

Step 4: Once the child understands that cutting a square into pieces of the same size is the key to adding fractions (i.e. making the fractions homogeneous), proceed with a few exercises such as the following. Draw the two fractions we used before.


Ask your child to figure out how to divide both fractions into pieces of the same size. Tell him that he is not allowed to erase lines already drawn. Tell him that he is allowed to draw new lines, but that the goal is both fractions to divided into pieces of the same size. Clearly, the answer here is to subdivide the 1/2 horizontally as follows:


Now (this is critical), make sure the child understands that how much is shaded in the square on the left does not change simply because we drew another line. It should be clear by now that 1/2 and 2/4 are the same fraction. Make sure this is clearly understood before proceeding.

Step 5: It is now time to learn how to convert heterogeneous to homogeneous fractions. I would suggest easy cases first, followed by slightly more complicated ones. Let's start with the following two fractions. Always draw one fraction using vertical lines and the other using horizontal lines.


The answer should be

By drawing vertical lines in one fraction and horizontal in the other, it becomes clear how to draw new lines to divide both pictures into pieces of the same size.

Let's try one more example.


The answer now is


Make sure your child understands that 3/5 is equal to 6/10. Likewise, make sure it is understood that 1/2 = 5/10.

Step 6: It is now time to introduce the pictorial representation of improper fractions. Ask your kid to draw the following fractions: 3/2, 5/4, 4/2, 5/2, etc. The answers follow:

Do as many examples as necessary until the child is proficient at drawing improper fractions.

Step 7: It is now time to bring it all together to add and subtract fractions visually. Tell the child to draw each fraction in a problem. Ask the child to complete the problem visually. Finally, ask the child to convert the drawing representing the answer to a written fraction.

As you probably realize, you can teach reduction to lowest terms visually as well.

I hope this blog entry helps you introduce fractions faster and earlier than is typically done in schools. It took only 40 minutes to teach a group of second graders in my daughter's school how to add and subtract homogeneous and heterogeneous fractions. All they knew before I taught them was what a fraction is and how to write them down in standard notation.

A Sample Problem Set for Second Grade

Most of my posts are about issues surrounding gifted education. Some of them are about dealing with the complications of raising a girl gifted in areas typically dominated by men. I figure it was time to give a sample problem set.

I wrote the problem set attached to the bottom of this post for my daughter's second grade class. I help the teacher once a week by splitting the classroom into three groups. I write problems sets to challenge the "gifted" group. I work with the "average" kids to make sure they are proficient on the topics mandated by the State of California -- essentially preparing them for the CST exams by reinforcing what is taught in class. Finally, I tutor the bottom third of the class to help it understand the basic concepts of arithmetic. There are clear cognitive differences between the three groups, and this makes my job quite difficult -- what works for one third of the class does not work for the other two.

The point of the including the problem set below is to give an example of how to teach the basic ideas of proof construction to kids in early elementary school. Notice the structure of the problem set. I try to emulate the way college math books are structured:

1. Definitions
2. Example and computational exercises
3. Proof construction as a vehicle to learning math and deepen understanding

I am guided by three principles. First, computation is important. Second, learning how to derive new ideas from definitions and first principles is central to the philosophy of mathematics. Finally, there is no substitute for learning by discovering, and proof construction is the door to the wonderful world of mathematical discovery. I firmly believe even 6 and 7 year olds should be taught how to construct logical arguments.


Sample Problem Set for Second Grade Statistics and Set Theory
Prerequisites: Addition, subtraction, an informal understanding of the concept of a set, and Venn diagrams. We do not assume any knowledge of multiplication, division, or fractions. We assume students do not know anything about negative numbers.

Definition: The range of a set of numbers is the distance between the biggest and the smallest.

Exercise 1: Find the range of the set of even numbers between 0 and 100. Assume that 0 and 100 are in the set.

Exercise 2: Define a set of numbers using the following four properties:

1. Every number in the set is bigger than or equal to 30 and smaller than or equal to 60.
2. You can get every number in this set by counting by tens.
3. You can get every number in this set by counting by fives.
4. You can get every number in this set by counting by twenties.

Find the set of numbers defined by the above properties. Compute the range of the set.

Exercise 3: Which one of the following two sets has the bigger range?

• Set 1: The set of even numbers between 10 and 20, including 10 and 20.
• Set 2: The set of odd numbers between 10 and 22.

Exercise 4: What is the range of the set of numbers equal to their doubles?

Exercise 5: What is the set of numbers not equal to themselves:? What is the range of this set?

Definition: The mode of a set is defined as the element that appears most often. A set may have no mode, one mode, or more than one mode. We say that a set is bi-modal if it has exactly two modes.

Exercise 6: Find the mode of {100, 99, 50, 3, 2, 1, 60 ,1 ,85}

Exercise 7: Find the mode of {Pablo, Paulina, Alex, Kolane, Pablo, Paulina, 2, 3, 10, 1}

Exercise 8: Find the mode of {2, 3, 4, 5, 6, 7, 8}

Definition: The median of a set is the number for which half the elements in the set are smaller and half bigger than the mode. Sometimes, the mode is part of the set. Sometimes, the mode is not.

Exercise 9: Find the median of {1,2,3}.

Example 1: The median of the set in the exercise 9 was a member of the set. However, the median is often not a member of the set. For example, 3 is median of the {2,4}. There are only two numbers in the set {2,4}. We pick the number right in between these two. That number is 3. Half the elements in {2,4} are smaller than 3 and half bigger than 3.

Exercise 10: Find the median of {100, 1, 80, 52, 48, 10, 12, 15}

Exercise 11: If you throw away the smallest and the largest numbers in a set, does the median change?

Exercise 12: Assume the following things:

1. There are twenty students in the classroom.
2. Every student likes either apples, pears, or both.
3. 15 students like apples.
4. 10 students like pears.

How many students like both apples and pears? Prove your answer using a Venn diagram.

Exercise 13: Assume the following three things:

1. There are twenty students in the classroom.
2. A student likes only one type of fruit.
3. 15 students like apples.
4. 10 students like pears.
   

How many students like neither apples nor pears? Prove your answer using a Venn diagram.

Definition: The mean of a set of numbers is one that repeated as many times as there are elements in the set gives you the same number as adding all the numbers in the set.

Example 2: Given the set {2,3,4}, computer the mean.

Solutions: First, we add all the numbers in the set. 2+3+4 = 9. We have to find a number that added to itself three times equals 9. That number is 3, because 3+3+3 = 9. Hence, 3 is the mean of {2,3,4}.

Example 3: Compute the mean of {1,2,3,4,5,6,7}.

Solution: 1+2+3+4+5+6+7=28. What number can I add to itself 7 times to get 28? You can find out by trial and error, but the answer is 4 because 4+4+4+4+4+4+4=28. Hence, the mean of the set is 4.

Example 4: Show that the mean of {2,3,4,5,6,7,8}=35 is bigger than or equal to 2 without computing the sum of the numbers?

Solutions: If the mean were smaller than 2, the first choice would be 1. This would mean that adding 1 to itself seven times would equal the sum of the elements in the set {2,3,4,5,6,7,8}. However, 1 is smaller than every element in the set. Hence, adding seven ones cannot equal the sum of the elements in the set. This implies that the mean must be larger than or equal to 2, the smallest element in the set.

Example 5: Show that the mean of {2,3,4,5,6,7,8}=35 is smaller than or equal to 8 without computing the sum of the numbers?

Solutions: If the mean were bigger than 8, the first choice would be 9. This would mean that adding 9 to itself seven times would equal the sum of the elements in the set {2,3,4,5,6,7,8}. However, 9 is larger than every element in the set. Hence, adding seven 9s must be larger than then sum of the elements in the set. This means that 9 must be smaller than or equal to 8, the largest element in the set.

Exercise 14: Can the mean of a set be smaller than the smallest number in the set?

Hint: Look at the prior two examples. This problem is solved the same way. Assume that the mean is smaller than the smallest number in the set. Compare it to every number in the set. What do you see?

Exercise 15: Can the mean of a set be bigger than the largest number in the set?

Hint: Assume that the mean is bigger than the biggest number in the set. Compare it to every number in the set. What do you see?

Exercise 16: Show that the mean of any three numbers is both

• larger than or equal to smallest element of the set
• smaller than or equal to the largest element of the set

Hint: Use what you did in exercises 14 and 15.

Exercise 17: Give an example of a set for which the mean and the median are NOT equal.