BJU Press Blog

math comprehension

Dividing Fractions a Simpler Way

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dividing fractions printable
The twelve-year-old’s T-shirt reminded me of my own homeschool experience with fractions and made me chuckle with delight. The shirt said, “Did you know that 5 out of 4 people struggle with fractions?”dividing fractions materials

When I was in the upper elementary grades, my mom switched us from a homeschool math program that used a hands-on approach to a program that was trendy among homeschool families at the time. It was during that phase of my math education that fractions became increasingly difficult. I had to multiply and divide with fractions, and it was all so confusing. Why is it that when I multiply, I get a smaller number, but when I divide, I get a larger number?

I think the hands-on approach would have helped me understand the basic arithmetic of fractions. If your child is struggling with dividing fractions, try the following hands-on approach to see if it helps. (I found this activity in BJU Press’s Math 6 textbook, which has lots more hands-on activities for teaching fractions and other challenging math concepts.)

dividing fractions circlesInstructions

  • Print out these circles and have your child cut along the outer solid lines. (Note: You can also use two apples instead.)
  • We’re going to work through the problem 2 ÷ ¾, but that’s rather abstract, so let’s use a real-world problem.

“We want to make apple tarts. Each tart requires ¾ of an apple. We have two apples. How many tarts can we make?”

dividing fractions cutting

  • Start with the two complete circles. These represent the number of apples we have.
  • Ask: “How do we need to cut the apples to make the tarts?” Since we’re going to need to separate out ¾ of an apple, the answer is “into quarters” or “into fourths.”
  • Now cut the whole circles into fourths.
  • Ask your child: “How many fourths are needed for each tart?”
  • Now have your child separate the fourths into piles of three.
  • Ask: “How many piles of three do you have?” Answer: 2
  • Ask: “How many quarters are left?” Answer: 2

Expressing the Answer

This might be tricky, but we want to express the exact number of tarts we can make. We could answer, “Two tarts,” but we would have two fourths or ½ an apple left over.” If we want to make a tart with the remaining apple fourths, how much of a tart could we make?  We would say, “Two and two-thirds tarts.” This confused me as a child. Why would I use thirds when I started with fourths? Walk your child through these questions to help with understanding:
dividing fractions coloring

  • Ask: “How many fourths do we need to make a whole tart?” Answer: 3
  • Ask: “How many fourths do we have left?” Answer: 2
  • Ask: “If it takes three to make a whole tart, and we have two of the three to make a whole one, how would we write that as a fraction?” Help your child see that the bottom number of the fraction tells us how many parts make a whole, and the top number tells us how many parts of that whole we have. So the whole tart has three parts (apple fourths), and we have two of those three parts or ⅔.
  • So we can make 2⅔  apple tarts with 2 apples.
  • Have your child color each group of three fourths and the two remaining fourths a different color so they can see which group of fourths will go into each tart.
  • Help your child write out how this would look as a math problem and solution: 2 ÷ ¾ = 2⅔.

dividing fractions completeWhy Hands-On Works

Working through this activity helped me understand some math that always challenged me as a child and into adulthood. Using hands-on activities helps many of us understand math. That’s because math is an expression of physical realities. When we see, touch, and talk about those physical objects that math is representing, the math makes more sense.

So if your child is struggling with fractions, try this and some other hands-on activities.

 

4 Steps for Teaching Math Comprehension

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comprehension with math manipulatives
When most of us think back to our experiences with math, we think of endless speed drills and a driving need to get the right answer. As a child, you may have wondered how memorizing formulas was going to help you understand math, and now it’s the same with your children.

You may feel like teaching your children to understand math is some kind of grand undertaking or incredible journey. Looking at it that way can be overwhelming. Where do you begin? How do you keep going? Maybe there’s a different way of achieving math comprehension?

As adults, we often remind ourselves to take everything one step at a time, and that’s probably some of the best advice, but what are the first steps of an approach that teaches your children to comprehend math rather than just memorizing meaningless facts?

1. Connecting numbers to tangible ideas

In and of themselves, numbers are abstract and nebulous. In order to ground number concepts in the real world for your children, you can use manipulatives. The manipulatives can be objects you use every day or the specially designed manipulatives found in the BJU Press elementary math curriculum. This physical understanding of numbers—or number sense—brings math comprehension into your children’s grasp.

2. Building a foundation for future learning

This step is two-fold. Once your children have made strong connections between the abstract concept of numbers and real objects, they’ll need to visualize the processes they’re using so that they remember that numbers mean something. In order to make sure that your children are visualizing math, try asking them questions that encourage them to think about their processes and how math works with real objects.

3. Expanding math comprehension

For math comprehension to grow deeper, it’s important that your children understand base ten math. You and I use base ten every time we break numbers into columns of thousands, hundreds, tens, ones, and so on. Making sure your children understand the concept of base ten math will make grasping the more difficult math processes easier for them.

4. Reaching real-world application

Learning comes full circle to the real purpose of math—a tool to assist us in wisely stewarding God’s creation to serve others—when it begins to reach real-world situations. The skills your children learned during carefully prepared lessons and activities will come into play naturally in real-world situations, such as grocery shopping. These life skills develop as a result of how well they understand math, not because of how well they do speed drills or what grade they make on a test.

Notice how each step after the first relies heavily on connecting numbers back to the real world in some way? As important as all the subsequent steps are, the first is still the key to really comprehending math.

As an ancient Chinese philosopher once wrote, “A journey of a thousand miles begins with a single step.” Once you make strong connections between abstract numbers and tangible objects, the other steps will begin to feel more natural.

Building Long-Term Math Mastery

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BJU Press Math
Have you ever stopped and asked yourself, “Why does one plus one equal two?” Most people who find themselves faced with this math question will probably say something along the lines of “it just does.” But if you thought about it, you could probably come up with a more articulate answer. For example, if you have one marker, then buy another one, and count how many markers you have, you will have two. That’s why one plus one equals two.

What did I have to do in order to answer the question? For most adults, connecting number problems to familiar objects is automatic. That’s because, as children, we learned number sense by working with objects that we could count and move around. You know that manipulatives are key to developing math comprehension. But are there other keys to math comprehension that are essential for long-term mastery?

Yes. Our Math 2 student worktext constantly asks your child questions about the processes he used and how he found his answer.

Students must learn to think about and explain the processes and formulas they use in order for math to have any lasting purpose for them.

For example, in high school, I was at the top of my class in math, but not because I had any idea of what all those numbers meant. It was simply because I could put numbers in the right places in a formula. In fact, I always loved the quadratic equation, and I still remember x equals negative b plus or minus the square root of b-squared minus 4ac all over 2a. But when I got to geometric proofs, I was hopelessly lost. My high school math textbooks never reminded me that the formulas and numbers I loved playing with actually meant something. It was like putting together a puzzle with edge pieces but no middle.

A child’s ability to solve equations doesn’t mean he gets it.

So how do you ensure that your child will be able to make sense of math as he moves into more and more complex ideas?

  • Ask him to explain his process. What would the equation he’s solving look like in real life?
  • Ask him for observations about the concepts he’s learning. Does he see any new patterns? Do other math concepts apply to the new one? Does the new math concept apply to old ones?
  • Incorporate common objects as manipulatives. As he begins learning about area and surface area, can he find the surface area of a tissue box? How would that information be useful to the person designing the box or putting it together?

Look through the thought bubbles in Math 2 student worktext for more ideas of questions to ask your child.

When math has meaning, it also has purpose. If you can create a strong connection between your child and the meanings of the numbers and formulas, you might also connect him to a future career.