Division: Laying the Groundwork, Part 3

Manipulatives make learning fun.
Manipulatives make learning fun.

 Laying the Groundwork for Teaching Division, Part3

In Part 1 and Part 2 of Laying the Groundwork for Teaching Division, I discussed ways parents can begin laying the groundwork for the future teaching of division to their children.

Teaching division may seem like a complicated task but using manipulatives makes the skill easy to teach and to acquire.

For our purposes in this post, we will use recycled ice cream sticks. Family members could assist in recycling so that you have a useful amount.

Ask extended family members to save items that can be used as manipulatives to teach mathematical skills.
Ask extended family members to save items that can be used as manipulatives to teach mathematical skills.

(Another alternative would be to use tongue depressors. Just ask your doctor for a handful next time you have an appointment. If you tell him or her the purpose for your request, you may find yourself with a generous handful free of charge.)

Let’s Review Place Value First

Before we continue let’s take a moment to discuss the place value of numbers. This is just for the parents’ information.

The number 2 is a one-digit number which is made up of 2 one’s.

The number 23 is a two-digit number which is made up of 2 tens and 3 ones.

The number 234 is a three-digit number which is made up of 2 hundreds, 3 tens, and 4 ones.

In the number 2, the digit 2 is in the one’s place.

In the number 23, the digit 2 is in the ten’s place.

In the number 234, the digit 2 is in the hundred’s place.

In other words, a 3-digit number has 3 digits, one which represents the hundred’s place, one which represents the ten’s place, and one which represents the one’s place.

In the number 2, there is only one digit. This tells us we have 2 ones.

In the number 23, there are two digits. This tells us we have 2 tens and three ones.

In the number 234, there are three digits. This tells us we have 2 hundreds, 3 tens, and 4 ones.

Prepare Your Manipulatives
Choose your colors.

Back to our ice cream sticks. Choose a color you wish to use to represent the hundred’s place. For our demonstration we will use purple.

Choose another color to represent the ten’s place. For our demonstration we will use red.

Now choose another color to represent the one’s place. For our demonstration we will use yellow.

Use markers, paints, crayons, etc., per your color choices

Use markers, paints, or crayons according to your color choice to dye your assortment of ice cream sticks.  If your budget allows, colored ice cream sticks may be purchased inexpensively at various craft stores.

Teaching the Skill of Division Using Your Manipulatives

Now that you have your own manipulatives, let’s see how we can use them to continue laying the groundwork for teaching the concepts of division. You will find it much easier than you thought.

Ice cream sticks may be used to represent numbers.
Ice cream sticks may be used to represent numbers.
To represent the number 234, we will use 2 purple sticks, 3 red sticks, and 4 yellow sticks (per our color choices).

(To make this task even easier, you may decide to only use even numbers for all digits, especially if your child/student is very young.)

234/2 = ____
The example problem we will be solving is 234 divided by 2.
  • Lay the sticks out in proper order to represent the number 234.
Place value represented by manipulatives
Representing place value
Explain to your child what the colors represent.
  • Explain to your child what the colors represent. Always use the same colors to represent the same place value.
  • Tell your child to “pretend” he or she has two friends. The number 234 can represent marbles, cookies, building blocks, or whatever your child likes to play with.
  • Tell your child to divide the 234 between his or her two pretend friends equally.
Direct them through the process.

Draw attention to the 2 “hundred’s” sticks (purple). Share these two sticks equally.  They will place one stick in each pretend friend’s pile.

The “hundreds” divided.

Then direct them to divide the three “ten’s” sticks (red). They will place one stick in each pretend friend’s pile. But what will they do with the third stick? No, they cannot break it.

Problem represented thus far

Show them that they can solve this by trading the “ten” stick for ten “ones” sticks.

Now show them to place the ten “ones” sticks with the 4 “ones” sticks. How many “one” sticks (yellow) do they have now?

Ask them to divide these 14 “ones” sticks between the two “pretend” friends. 7 “ones” sticks should be placed in each pretend friend’s pile.

7 yellow sticks go to each friend
Explain the answer to the problem.

234 has now been divided into two equal but separate groups. The answer to the problem is the amount in one of the groups.

  • So, if we share 234 building blocks equally between two friends, each friend will have 117 blocks.

Therefore, 234 divided by 2 is 117.

234 divided equally between 2 friends.
234 divided equally between two friends.

This is an example of the division of a three-digit number by a one-digit number.

Where to start
  • Begin with the division of a one-digit number by a one-digit number, continue with the division of two-digit numbers by a one-digit number, and then the division of three-digit numbers by a one-digit number.
  • Go at your child’s pace.

Remember this is to be a time of enjoyment. Don’t get stressed if your child struggles at first. Eager learners will pick up on what you are requesting as you continue along.

I do hope these posts have been helpful. Parents can do a tremendous job of helping their children enjoy learning these somewhat difficult tasks.

Just take a few minutes each day to lay the groundwork for division and your child will find this task conquerable.

You may leave any questions in the comment box below.

Copyright 2017 by Peggy Clark

 

 

Laying the Groundwork for Division, Part 2

Laying the Groundwork for Teaching Division, Part 2

The process of division may seem intimidating to students and parents. However, it is not as complicated as many think.

Parents can make the passage to proficiency of this skill easier by laying the groundwork early in their child’s life.

In Part 1 of this post, two ways were discussed that parents may employ.

In this post we will discover another easy and inexpensive way to further aid in helping children develop the skills necessary to achieve mastery.

That way is through a simple technique that employs the use of manipulatives.

Manipulatives make learning fun.
Manipulatives make learning fun.

Yes, manipulatives may be costly to purchase but they can be readily made by recycling items already found in most households. One only need the imagination to come up with a variety of useful tools made from simple everyday objects that are usually thrown in the garbage.

For our purposes in this post, we will use recycled ice cream sticks.

Enlist the aid of extended family members in recycling so that you have a useful amount.

Make this a fun game. Your child will be learning concepts of division without realizing it.

Do this discovery activity.

Pile a number of ice cream sticks on the middle of a table or other flat surface. Ask a variety of questions that will engage thinking skills.

Ask extended family members to save items that can be used as manipulatives to teach mathematical skills.
Ask extended family members to save items that can be used as manipulatives to teach mathematical skills.

Change the amount of sticks. Then continue asking questions for that amount.

Adjust questions according to your child’s skill level.

Let your child discover the answers.

Whether your little one can count past ten or not is not important. Mathematical concepts such as one-to-one correspondence is being developed.

If your child is older, he or she should be able to give you a numerical answer through this discovery activity.

Some sample activities when working with one child:

Divide the pile of sticks so that you and I have the same number of sticks. How many sticks do you have?  How many sticks do I have?

Put all the sticks back in the middle of the table.

Now pretend that there are 3 people here. Divide the sticks into three equal piles. (Give the child time to complete the activity.) How many would each of us have?

Put 10 of the sticks in the middle of the table.

Divide the sticks into two equal piles.

Put 8 sticks in the middle of the table.

Divide the sticks into four equal piles or groups.  How many sticks are in each pile or group?

Use mathematical terms as you see your child progressing.

As your child progresses in understanding, begin to add in mathematical terms.

In the last question above, the word groups was added to the instruction. This is a simple but nonthreatening way for your child to begin understanding mathematical terms. Do this in a gentle way. When the child has grasped the understanding of the new term, use that term instead.

For instance, when you perceive that your child has grasped the understanding of what you are requesting when you use the word group, drop the word pile. “How many sticks are in each group?”

Don’t rush.  Just add in a new term as you see your child progressing.

Another example from above is the use of the term equal. The term equal should eventually replace the words same number.

Do a few minutes of discovery activities each day.

Adding just a few minutes of this activity to your child’s playtime each day will go far in laying the groundwork for what will not be daunting, but will actually become a welcome task of teaching the process of division.

More to come in Part 3.

Copyright 2017 by Peggy Clark.

Division: A Daunting Task

Laying the Groundwork for Teaching Division

Manipulatives Make Learning Fun Photo
Manipulatives Make Division Easy and Fun
Division is that part of math class that seems daunting to teachers and students alike.

I remember watching my older brother do long division. His homework pages had problems that seemed so long they looked as if they covered half of his paper. I assumed those problems must have been very important.  Doing that type of math looked so grown-up. It was a process that I wanted to be able to conquer. I couldn’t wait to get to that level of math.

Division is not so complicated as many think. It just needs to be taught correctly. Part of that comes with laying the proper groundwork.

Long before division class comes around, children should already be engaging in the concepts of dividing.

Those “official” concepts begin with kindergarten and continue level by level. But division has already been introduced to the child even before that “first” day of formal schooling begins.

For example, children see division at work around the supper table as they share the meal. A family of four shares four pieces of chicken equally. The pumpkin pie is divided into six or eight pieces.  A gallon of milk is poured into glasses. Rolls are shared round the table.  The last one may be divided in two parts to share between eager siblings.

So how can parents begin that formal groundwork for division?

Parents can add to what the child is observing day by day by simply adopting the usage of mathematical terms.

“Please share with your brother,” may be changed to “Please divide the roll between the two of you.”

Look for opportunities to use mathematical terms when appropriate.

Divide these sandwiches so we have two (or four) pieces each.

There’s only four cookies left. Share them equally with your sister.

Divide the candy bar equally between yourself and your brother.

Divide the last of the milk between yourself and Dad.

Divide the potato tots so that each of you get the same amount.”

Children will be alert to the methods their siblings are using because they want to be sure they are getting their “fair share.” This unknowingly draws their attention to the division process.

Another way parents can lay the groundwork is to allow their children to measure.

Mother can let the children measure the ingredients for her favorite recipes. Dad can let the children measure items for his next building or repair project.

We tend to think of measuring as adding to something that we are making; however, we also measure because we are about to subtract from something. We are taking a cup of milk from the gallon.  We are sawing a foot of lumber from six foot of lumber.

Measuring is an indirect way of preparing the child for future concepts concerning division.

This may seem odd, but division is really just a fast way to subtract the same number over and over.

A third way parents can lay the groundwork for division is to actually divide groups of items – not on paper, but with manipulatives.

I’ll discuss an easy and inexpensive way for parents to do this in my next blog post.

Copyright 2017 by Peggy Clark

 

 

 

 

Be an Effective Teacher Without the Frustration

Sometimes we have a struggler who just doesn't seem to get the lesson we're trying to present no matter how many times we have presented it. Frustration may try to overtake us but we must not allow it to take us captive to its destructive vice.

Instead we must find another way to present the lesson that is geared toward our child's learning style and learning ability.

A greater time may need to be spent on teaching the concept. Let's face it. Do we always grasp how to do something the first time we see or hear the instructions? We must remember to teach effectively, not hurriedly. 

Use manipulatives whenever possible. Manipulatives are objects, drawings, charts, number lines, or other tools that a child may touch, handle, and manipulate. For example, instead of trying to describe how to do an addition equation, picture the process first.

Draw the addition problem on the page. 
9 + 8 = ? 
Draw a set of 9 triangles and a set of 8 triangles. 
Then circle both sets to show that you are going to group them together. Let the child give the answer.

Next use objects to show the equation process. 
Place 9 objects in one group. Place 8 objects in another group. Then pull the objects into one group. Let the child give the answer.

Then have the child use the objects to show the equation process using the same equation. If he or she does this correctly, then give the child another equation. 

7 + 5 = ?
Let the child use the objects to show the equation process. 7 objects should be placed in one group. 5 objects should be placed in another group. Allow the child to explain what he or she is doing so you can see if they are processing the problem correctly. Let the child give the answer to the equation. 

When you are satisfied that the process has been understood correctly, go back to paper and do several equations together if necessary. 

Stay focused on the goal of the lesson. Teach the lesson effectively without frustration. An effective teacher will seem to overteach at times but the goal of understanding will be achieved at a higher rate. And is understanding not what we're trying to achieve?


For precept must be upon precept, precept upon precept; line upon line, line upon line; here a little, and there a little. Isaiah 28:10

Jelly Bean Math

Let’s have a fun math class today with our young child. Be prepared by purchasing a bag of jelly beans.

Parent to child:

“Let’s take a bag of jelly beans and put them on the table.”

“Let’s sort them by color.” You might work with the child or let him or her do this particular task on their own.

Continue with the following instructions.

“Now let’s compare the groups (sets). Which group has the most jelly beans? Which group has less jelly beans?

Which group has more — the yellow group or the red group? Which group has less — the red group or the green group?”

Put all the jelly beans back together in one group on the table. Give the child 2 jelly beans.

“How many jelly beans do you have?” Give the child 1 more jelly bean. “How many jelly beans do you have now?”

Continue the conversation by giving jelly beans and having the child give you jelly beans. At times, ask the child which of you have more jelly beans or less jelly beans.

“Separate (divide) the jelly beans into 2 equal groups.” Let the child do this on his or her own. Take note of how the child does this. Some children may separate by putting 1 in each of 2 groups one at a time. Or the child may place several beans in one pile; then several in another pile. Or they may line them up side by side in a line. This observation will help you to see how the child is processing this instruction. If they get the right answer, do not attempt to correct them or show them another way at this time. They will learn various ways as they continue to work with manipulatives.

In the simple exercises above, children are recognizing, classifying, categorizing, arranging, separating, adding, dividing, sorting, identifying, more than, less than, associating, etc.

There are so many mathematical skills than can be caught with manipulatives that are as simple and inexpensive as a bag of jelly beans. So think outside the box. Every math concept or principle does not have to be taught with pencil and paper. Sometimes it helps to just have a little fun and learn at the same time.

Enjoy your jelly beans.

Struggling with Addition, Part II

Strugglers who find addition to be a chore may find it a fun adventure when given the opportunity to learn with manipulatives. The use of these hands-on objects leads to purposeful learning in a stress-free manner. Allowing children time to “play” with such objects increases their imagination, stimulates brain activity, develops motor skills, and lays a foundation for learning mathematical concepts.

Cubes, dice, building blocks, building logs, buttons, beads on a string, ice cream sticks, pinto beans, even army men — what do these have in common? Each of them are wonderful tools that can be manipulated in order to accomplish learning.

The use of such tools is an inexpensive way to build upon the knowledge that a child already possesses. During play with various objects, a child learns how to build up and take down, gather together and take away, group together and take from, and sort by his or her preference. At the beginning stages, however, a child does not know the official mathematical label that we would use for such manipulation, but through play the groundwork is being laid upon which we purposefully begin to teach the foundational concepts and principles that lead to the understanding of the processes of addition, subtraction, multiplication, division, etc.

The introduction of handmade or purchased rectangular rods that are proportionally sized lead to even greater learning. Several companies have manufactured rectangular rods that are a great addition to your arsenal of manipulatives. However, knowing the limitations of household budgets, poster board or card stock can be used to create a representation of these rods, howbeit, they are not three-dimensional.

Place value is traditionally taught in some schools through the use of an abacus. Some students struggle with the use of the abacus because not all students have their own to manipulate. It is easier for a child to understand the concept when they have their own manipulatives to handle and engage in learning.

An inexpensive way to teach children place value is through the use of dried beans and ice cream sticks. Single beans are used for counting by ones, sorting into groups of equal amounts, etc. Ten beans will fit nicely onto an ice cream stick.

Using proper glue, allow the child to make his or her own “ten” sticks so he or she will realize that there are ten and only ten beans on each stick. Ten of the “ten” sticks can be placed side by side to create “hundred” sticks — just glue 2 sticks across the back to hold them together. A variety of mathematical concepts can be taught using these sticks.

So how can we use manipulatives to teach addition? By now, you probably have a variety of ways rambling through your mind, but we will give some direction in a later post. For now, begin to let your mind wander back to your childhood days when you played with blocks and logs. Yes, learning can be fun — at least some of the time.

Struggling with Addition

One of a child’s greatest needs is to memorize the addition facts. This task comes easy for many; but some struggle and fall behind in their math studies because of this one single necessity.

What can parents do when they find their child struggling in this area?

Immediate intervention is needed. More complicated math utilizing the process of addition should be delayed until this situation is remedied.

First, process why he or she may be struggling. Were foundational steps skipped or overlooked? Does the child understand what the individual numbers represent? For instance, does the child understand that the picture 8 represents a collection of 8 objects or 8 sets of objects?

When teaching numbers, sometimes this knowledge is easily overlooked. A child may be trying to memorize images of number figures without any understanding of what those images represent. For example, the picture 8 plus the picture 8 equals the picture 16 instead of 8 objects added to 8 more objects equals 16 objects total. Some children need more time to process this information.

The use of manipulatives will aid understanding. Yes, I know some teachers frown on the use of objects to help a student learn their facts. They insist on memorization. Yet, when students are struggling with a particular math problem, such teachers model the addition by the use of their fingers. This is a bad practice and greatly hinders the child. Children following this teaching model quickly learn to rely on fingers.

The use of manipulatives diverts the child away from the use of fingers. Each object is representative of 1 and when added together those 1’s become a collection of more than 1. No matter the learning style of the individual, this hands-on approach to addition greatly increases the struggling child’s ability to understand and memorize the facts.

Just what are manipulatives and what are some ways that they can be used to teach math? We’ll discuss this in a later post. Right now, since you have read this far, you may have a struggler about which you are concerned. Hopefully, the next post will be of benefit to you.