math

Math Automaticity: The Secret Key to Math Success

October 16, 2018 by Jenna

$math automaticity with M&Ms$
If you’ve read any educational articles about teaching math, you may have come across the term automaticity. It may seem like a scary word, but it’s not. It simply refers to the ability to do something without thinking about it, or automatically. In math, it most often refers to the ability to use basic math facts without thinking about them. So when solving 5 + 5, you don’t have to count up the fingers on your left hand and the fingers on your right hand to make 10. You just know that 5 + 5 equals 10. Math automaticity isn’t scary at all, right?

But how do you bring your children to the point where they are demonstrating math automaticity? As with most subjects, learning math is a process. Working their way through the process guides them towards automaticity.

Start Early to Develop Number Sense

Most children begin developing number sense early—before kindergarten. The BJU Press K3, K4, and K5 materials use songs, number games, and other activities to help children develop an early sense of math. Having your children participate in board games and dice games while they’re still very young also helps them develop number sense. These early stages are crucial in your child’s development. Studies show that early math and number skills (even more than reading skills) predict how well a child will perform academically later on.

Add Manipulatives

Manipulatives play a key role in math learning by helping students to visualize throughout the process. They help solidify children’s number sense and give them tools to connect abstract math concepts to the real world and creation. That’s why all the elementary level BJU Press math courses include manipulative packets. Many of the high school level math courses even offer suggested manipulatives, like this candy sorting activity from Algebra 2.

Review and Review Again

Once your children have a strong number sense and manipulatives to work with, they will need to practice their math skills. Simply understanding a math concept and how it works doesn’t mean that they have automaticity for that math concept. They need to practice the same concepts in new situations to create familiar problem-solving paths in their minds.

For example, when I was a kid, I struggled with knowing my right from my left. But I knew that I was right handed. I also knew that the hand that makes the L was my left hand. Whenever I needed to figure out right and left, I thought about which side my dominant hand was and which side made the L. I did that so often as a child that the thought process made mental ruts. I don’t go through that whole process now because I don’t have to. That kind of repetition is what makes the review books in elementary math so important. They provide additional practice opportunities of the day’s lesson as well as a spiral review of previous lessons. Each time your children encounter a familiar problem, they’re digging those mental ruts deeper and coming to solutions faster.

Unless you’re a math person, you may be wondering why it’s important for your children to be so proficient in math that they can do it all in their heads. In fact, many parents often think of math as less important than reading. But by going through this process of developing math automaticity in your children, you’re also enabling them to go further and do more with math. And the further they go in math, the more developed their logic and critical thinking skills become. In the end, you may not be teaching the next Einstein, but you definitely want to train up children who think logically and critically when faced with difficult decisions and complicated variables.

Building Understanding with Negative Numbers

June 19, 2018 by Ben

$negative numbers activity$
By the time I got to junior high, I was really confused about negative numbers. At first it was simple, but concepts like multiplying negatives and absolute value taxed my understanding. Of course, I joined in on the chorus sung by every math student who doesn’t understand what he or she is supposed to be learning: “When am I ever going to use this?” Once I reached high school, I realized that negative numbers play a significant role in science. Today, negative numbers play a major role in my problem solving in home budgeting and in business.

So how can we set up children for success when they start using negative numbers so they can solve ever more complex problems? The answer is exposure, development, and mastery. We need to expose children to significant concepts well before they need them to problem solve. Next we need to develop their understanding on the path to mastery. Then, children will be ready for success later on.

Exposing a New Concept

I’ve learned that with negative numbers you need to expose your child to the concept in the elementary grades. A good time to begin is in the second half of fourth grade. At this point in their math journey, children are ready to explore negative values but not quite ready to perform mathematical operations with those numbers.

If you live in a northern state, your child may have been exposed to negative numbers through real-life winter temperature. In the South, we’re never sure what to do when the mercury drops below 20 degrees Fahrenheit, so my daughters will need some other exposure to real-world negative numbers.

Interacting with Negative Numbers

So how do we introduce children to negative numbers? As with all math concepts, we need to make observations with our children as we do hands-on activities—but how? What makes negatives difficult for children initially is that you can’t look at or touch negative three apples.

The best way to interact with negative numbers is with a number line. Try the following questioning strategy for introducing your child to negative numbers.

Materials

You can use this printable, which includes the following:

Number line (from negative 9 to positive 9)
Markers to cut out

(BJU Press has also created a fourth-sixth grade manipulative kit that helps children learn this and other math concepts.)

Activity

Give your child the number line and the markers.
Ask your child to place the blue marker above the number 5. This will be a point of reference.
Ask your child to place the red marker above the number 7. Ask, “Which of the two numbers is larger?” and “Is it farther to the left or farther to the right?” Point out that the larger number is to the right and the smaller is to the left.
Direct your child to take the red marker from 7 and put it above the number 1.
Ask, “Now which number is the larger?” and “Is it farther to the left or to the right?”
Have your child move the red marker above number negative 1.
Ask, “What do we call that first number to the left of zero?” Your child might say, “Minus 1.” Explain that it is called negative one and that any time that negative symbol appears before a stand-alone number, it shows the number is less than zero.

Seeing the Math

This is a critical point because your child is going to compare two negative numbers. In your child’s experience, 7 has always been more than 1. But when both are to the left of zero (negative), the 1 is larger. This may seem obvious to an adult, but to a child it is counterintuitive. Allow your child to be “wrong” without correcting. You can set up the manipulatives so your child can see the difference.

Leaving the red marker on the negative 1, ask your child to move the blue marker to negative 7. Ask, “Which number do you think is larger, negative 1 or negative 7? Remember when we compared 1 and 5, which number was larger? Was it farther to the left or farther to the right?”
Now ask your child to compare negative 1 and negative 7 again. Say, “Remember if the number is the farther to the right, it’s the larger.” Then ask, “Which number is larger, negative 1 or negative 7.”
If your child is struggling with how this can be true, ask, “What would you prefer, to owe me one dollar or seven dollars?”
Explain that a negative number is like owing money when you don’t have any money at all. It is less than zero. So the bigger the numeral, the smaller or less the value.
Try comparing several more numbers.

Developing Understanding

This is a basic introduction to negative numbers. In the next few days, have your child try these exercises to develop his or her understanding:

Give your child a number line with negative 10, negative 5, 0, and 10 marked. Have your child fill in the rest of the numbers.
Create greater-than and less-than exercises for your child, using negative numbers.
Give your child a set of four or five numbers (including both negative and positive) to arrange from smallest to largest.
If your child is familiar with coordinate graphs, try extending the x to the left beyond the zero and the y down below the zero.

Try a variety of these activities over the next week or two. Never take more than twenty minutes or so, and make sure you are keeping the time engaging. You will want to revisit negative numbers again in fifth grade. Once you review what you’ve already covered, you can begin to introduce adding with negative numbers.

If you introduce, review, and develop a mathematical concept in the elementary years, your child will be prepared to be successful in junior high and high school math.

Embracing the Unfamiliar in Math and Handwriting

May 24, 2018 by Megan

I have a confession to make. When I started homeschooling last year, I was scared. I was scared of messing up my children’s education. I was scared that my carefully ordered life was about to be turned upside-down. And I was scared of two subjects in particular—math and handwriting.

As I read through the Math 2 and Handwriting 2 Teacher’s Editions in preparation for the beginning of our homeschool year, I began to get really nervous. I knew, of course, how to write in cursive and how to add and subtract. But this teaching methodology, this style of writing—all of it was new. I couldn’t help thinking to myself this is not how I learned it.

My memory of second grade isn’t perfect, but I remember drilling math facts, the long homework papers filled with math problems, and the stressful timed tests. I didn’t learn cursive writing until third grade, and my uppercase Ts, Gs, and Fs looked nothing like the letters that I saw on BJU Press’s handwriting chart.

Maybe you find yourself having similar thoughts as you evaluate the BJU Press math or handwriting curriculum. Like me, you might want to avoid the unfamiliar in favor of the familiar. Here are a couple of things to keep in mind that I hope will be a help to you.

BJU Press math is different because it emphasizes understanding.

I grew up with a “drill and kill” math curriculum that heavily emphasized the memorization of facts, formulas, principles, and so on. And I did fine . . . until I had to actually apply the math. Since I had no real depth of understanding, I struggled to apply what I learned.

Without ignoring the need for learning math facts, BJU Press aims to develop a depth of math understanding that leads to application from the very beginning. I personally learned so much as I was teaching second-grade math—I went from a mindset of “I didn’t learn it this way” to “I wish I had learned it this way.”

BJU Press handwriting is different because it’s based on how kids naturally write.

Many young children struggle with writing. I know I did. In first grade, I even had a tutor to help me with my handwriting. Straight lines and round circles aren’t easy for small hands to form. That’s why BJU Press developed the PreCursive handwriting style. It’s simpler for little hands. Plus, it makes the transition to cursive a breeze.

It took me a couple weeks to retrain my own pen strokes, so kids who didn’t start out learning BJU Press PreCursive from the very beginning might need a little time to get used to the new handwriting style. But once they learn it, it will be easy for them to produce neat writing.

BJU Press math and handwriting have been a big blessing to our homeschool. I no longer consider them odd—I think they’re awesome.

Check out our math and handwriting products for yourself, and see what a difference “different” can make.

Learning Place Value with a Hands-on Activity

May 17, 2018 by Ben

$place value cover$
As a homeschool dad, I found it easy to fall into the trap of believing that my young child understood a number because she could say it and count to it. When my second daughter was in kindergarten, we enjoyed skip counting to a trillion by 100 billions. She felt like she grasped the number 1 trillion. But I was overlooking place value and the role it plays in deep understanding of foundational math concepts.

The concept of place value is crucial not only in gigantic numbers but also in more relatable numbers. It’s so easy for a six-year-old to recognize and count to a number such as 23 but not understand what the 2 and the 3 mean in practical, real-life experience.

One of the best ways to develop your child’s understanding of place value is through less-than and greater-than activities. Doing hands-on exercises develops number sense that will enable your child to solve real-world problems.

Preparing

Cut out twenty small squares of paper and ten strips of paper (ten squares each). [You can make strips and squares using the first page of the printable, or use the counting cubes from the Math 1 Student Manipulatives packet.]
Label three sheets of paper, the first as “Ones,” the second as “Tens,” and the third as “Answers,” or use pages 2–4 of the printable.

Maneuvering

Give your child the following instructions orally, and guide him in carrying them out.

Position the Tens paper and the Ones paper side by side with the Tens on the left. Make the number 32 on the top half of the Tens paper and the Ones paper.
Skip count the tens aloud (i.e., count 10, 20, 30).
Count the ones aloud (i.e., count 1, 2).
Write the number on the answer sheet (i.e., 32).
Now make the number 23 using the bottom half of the Tens paper and the Ones paper.
Skip count the tens and count the ones.
Write the number immediately below the 32 on the answer sheet (Example 1).

Observing

This is the key point in the activity. You’re ready to ask your first-grade child to make some comparisons based on what he’s observing with the manipulatives.

Ask, “Which group is bigger, the one on the top (32) or the one on the bottom (23)?”
Follow up with, “Which one has more ones?” and “Which one has more tens?”
Finally ask, “Which place should you look at to determine which number is bigger (greater), the ones place or the tens place?” Have your child respond by circling the correct digit on the answer sheet.

Repeating

Remove the strips and squares from the Tens and Ones pages. Go through the process of building 19 and 45 but without the counting steps.
Tell the child to write the numbers side by side with a box in between (Example 2).
Ask which number is larger. The child should indicate his answer in the box by drawing the < symbol in the box.
Now have your child pick two numbers (greater than 10 and less than 50) to repeat the activity.
Have your child guess which number is larger before building the two numbers.
Guide him in writing the numbers with a box in between and then drawing the correct symbol in the box (Example 3).

Practicing

Write out five or six less-than or greater-than problems for your child to practice what he just learned. Encourage him to make each number on the Tens and Ones pages before writing in the greater-than or less-than sign.

Try to spend about twenty-five minutes on this activity before moving on to something else.

Do the the activity again on another day with different numbers to continue developing your child’s number sense and understanding of the critical role that place value plays in determining how large a number is.

Help Your Second Grader with Division

April 19, 2018 by Ben

$hands-on division activity$
As your homeschool child becomes proficient in multiplication, it’s time to introduce division. Typically, children become comfortable with division in their third-grade year of math. But the end of second grade is a good time to introduce the concept.

Since math models God’s creational order, it helps to introduce division with physical objects. Using objects from God’s world to touch, play with, and observe, we can help children develop a number sense that fosters deep understanding.

Guided Hands-on Discovery of Division

Here’s a fun hands-on activity to introduce division in a non-pressure way. All you need is an empty egg carton with the lid removed and twelve small objects. I used jelly beans, but dried beans, beads, or any other small household items will work.

Remember, the goal is to develop number sense that leads to deep understanding. We don’t want to overwhelm our children.

Have your child place all twelve jelly beans in the egg carton, putting two in a compartment until all the jelly beans are used.
Ask, “How many jelly beans did we start with?”
Ask, “How many compartments are in use?” To find the answer, he can count out the six compartments that have two jelly beans.
Ask “How many jelly beans are in each of these compartments? It’s the same number in each one.”
Dump out all the jelly beans and focus on the compartments.
Direct your child, “Fill four compartments so that each compartment has the same number of beans.” Make sure your child feels free to make mistakes at this stage. Have him keep adjusting the number of beans until four compartment hold three beans each.
Say, “This time fill three compartments so each compartment has the same number of beans. Were we able to still use all of our beans? How many are in each compartment?”
Finally, ask, “How many compartments would we use if we put six jelly beans in each compartment?”
After your child makes a prediction, allow him try it out.
Once your child becomes comfortable with the process, explain that he has done division! Division is simply making equal groups or sets out of a larger number. So we could say that 12 beans divided into 6 sets equals 2 beans in each set. Or 12 beans with 3 beans in each set makes 4 sets.
Now write out the equations that your child has already worked out with beans: 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3, and 12 ÷ 6 = 2.
As you write out each equation, have your child recreate it with the jelly beans. This will help your child visualize what the equations mean.

Sustained Learning

This activity shouldn’t take more than twenty minutes. Over the next couple of days, review with your child, and let him practice independently. Keep the egg carton and jelly beans handy, so your child can work out the problem hands-on each time. After he works a problem out with the jelly beans, he can write down his discovery.

Remember, you’re not trying to guide your child to achieve mastery of division right now. You’re introducing it ahead of third grade, when your child will spend more time developing skill with division. So after a week, move onto another math concept your child has more experience with.

My wife and I have found that these interactive math activities ignite a joy of learning for our children. It turns play into guided discovery and develops a strong number sense in our children. Try this activity out toward the end of your child’s second-grade year.