"Trick or Treat" Math Fun!

Halloween is in the air and it's time for some math fun!  Our kids went trick-or-treating last night and brought home tons of candy.  So for a "Fun Friday" math activity, we decided to use our sweet treats to practice math skills.

Preschool:  Candy Monster

Our 4 year old has been working on early numeracy skills: number recognition, counting, one-to-one correspondence and the idea of "adding one more."  To begin our activity, I wrote the numbers one through ten on index cards and placed them across the floor.  Then, I told him to dump out all of his candy.  This was lots of fun!  I told him he was the candy monster and it was important to organize his candy.  He was to place the correct number of candies on each index card.  Check out his work!

From here, we talked about different ways to make each number on the card by rearranging the candy.  Also, we talked about how you add one more piece of candy to get to the next number card.  We concluded the lesson by counting the total pieces of candy as we cleaned up our mess.  Of course, no candy lesson would be complete without letting our candy monster choose his favorite treat!  Bon Appetit!

Elementary Math: Halloween Candy Activity

Our second, third and fifth graders have been working on classifying data, fractions, and percents.  This activity began by having them categorize a "pumpkin" of Halloween candy.  From here, they created a tally chart of each type and displayed the results in a bar graph using Create-A-Graph.  They also determined the fraction and percent of each type of candy.  The spreadsheet option on Google Docs was used to record their data.  Finally, they used Skitch to capture an image of the information.

This activity was great!  In addition to the concepts mentioned, we reviewed various ways to display data and when to use each type of graph.  We talked about independent and dependent variables.  Our second grader was introduced to fractions and our third grader practiced reducing fractions to lowest terms.  again, we concluded our lesson with a sweet treat and a big smile!

Happy Halloween!

Long Division Made Simple!

As a math teacher and homeschooling mom of four, I'm not sure which is more fun to teach: making change, subtraction with borrowing, fractions or long division.  I say this with a sarcastic smile on my face.  So, when beginning the unit on long division with our second grader, I set out to find a way to teach the concept instead of a mindless algorithm.  I came up with the idea to relate division to Legos.  Before I go any further, there are two things I must clarify.  First, I know Legos is not a "real word."  I should technically use the more appropriate term, Lego playing bricks.  Secondly, this lesson has much more to do with division than with "Lego playing bricks."  I was hoping to grab our son's attention by using his absolute favorite topic of conversation.  I know it is a dirty trick, but sometimes necessary! 

Lego Dilemma:
You have been playing Legos all afternoon.  Your mom walks in and wants you to pick up everything, but throws in an added twist.  She places three containers on the floor and ask you to place equal amounts of Legos in each. Lucky you have been counting these Legos and know for a fact there are 763 Lego pieces scattered about the floor.  How many would you place in each container?

Method 1

Drop one Lego at a time into each of the three containers until they are all gone.  This might take a while!  One, two, three...

Method 2:

You have three containers and 763 Lego bricks.  So, you could subtract groups of three until you reach zero.

 

You can see this method works well for small values, but what about our problem? We have only subtracted off 6 groups of three and we have a long way to go!  This will take forever!

Method 3:

Let's think about something new: long division.  It's really not long, just a new way of thinking!

You want to divide 763 by 3.  In other words, you need to determine how many equal groups of three we can make out of 763.

Let's begin by expressing the problem in the traditional way.  We see that our dividend is 763 and our divisor is 3.

We want to first look at the hundreds place.  In the value 763, there are 7 hundreds.  If you were to split those into three equal groups, how many would go into each group?  Think: What number times three makes a number close to seven?

From the diagram, you can see that 2 hundreds would go into each group for a total of 6 hundreds and there would be one left over.  To show this traditionally, you would have:

Now, we are finished with the hundreds place.  We have 1 hundred leftover or 10 tens.  Right? 

This relates back to borrowing when subtracting.  Moving forward!  These 10 tens are now added to the 6 tens that are present.  Very quickly, you can see there are 16 tens.  We now have:

Just as we did previously, we will see how to divide these 16 tens into three equal groups.  Think: How many times will 3 go into 16 or, even better, 3 times what number makes something close to 16?

From the diagram, you can see that five tens can go into each group.  So, 3 will go into 16 five times because five times three is fifteen.  You can easily see there will be one ten left over.  Now, if we continue our division problem, you will have:

 Now that we are finished with the tens place, we must address the ones place.  Our 1 ten breaks down into 10 ones and we add that to the 3 ones already there.  This is illustrated below:

It is easy to see we have 13 ones.

Hang in there!  We're almost finished! Now, we want to see how to split these 13 ones into three equal groups.  Think: What number times three gives a number close to thirteen?

Again, it is easy to see from the diagram that we can split the 13 ones into three equal groups of four with a 1 one leftover.  Unfortunately for this last, little one it is the remainder.  There are no more places to consider and no other place for it to go.  So, our final problem looks like this:

Oftentimes, we write the quotient as 254 R.1 to show there is a remainder of one.

Solution:

Basically, we have discussed three different methods to address the dilemma of how many Legos to place in each of the three containers so there is an equal number in each.  Each method is perfectly correct!  There is no wrong way, just faster ones.  Obviously, Method 3, will provide the quickest answer to the problem.  We see you would need to place 254 Legos in each container and give the one remaining to either your little brother or sister.  Of course, the better solution would be to keep your Legos picked up so your mom doesn't have to put you in this situation.

Moral:

Keep your room clean and math is everywhere!

Just SCRATCHing the Surface of Fractions

Fractions are always fun, right?  I can hear the sighs and cringes even as I mention the word fractions.  For some reason, even children who enjoy math are not fond of this concept.  I wanted to share one of our adventures into the mysterious realm of fractions.

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The Problem:

Ben had 50 more coins than Joan.  After Joan had given 29 of her coins to Ben, she had 1/3 as many coins as Ben.  How many coins did Ben have at first?

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This problem came from the Singapore Math Intensive Level 5A and was included after studying operations with fractions.  In typical fashion, our daughter completely ignores the accompanying diagram and sets out to solve this problem armed with her new math skills.  She quickly found herself at an impasse and extremely frustrated.

I approached her and tried to see into her thinking strategies.  Then, we realized the diagram! 

 

She knew Ben's amount was equal to Joan's amount plus fifty.  Also, that after Joan gave Ben 29 coins, her amount was equal to 1/3 of Ben's.  Good start!  Then, the trouble began.  She kept adding 29 to 50 to get 79 and tried to divide this value by 3.  Really close, but not quite!  She kept running in to the problem that 79 is not divisible by 3.  I asked her what she was trying to do with the amount of  "79."  She thought if she could divide this by three it would represent the amount in each rectangle for Ben.  Still, close but not exactly correct.

After struggling with this for a while, she decided to put it away and move on to other problems.  As a high school math teacher, I kept wanting to set up two equations with two unknowns and just solve it for her.  I knew, though, she needed to keep struggling with it and work it out.  So, she went to her dad and they wrote a Scratch program to solve it.  Not exactly, the approach I would have taken, but it provided a different type of learning experience. 

For those of you who are unfamiliar with Scratch, it is a computer programming language developed by the Life-Long Kindergarten group out of MIT University.  Our daughter has been using this for a couple of years to animating her drawings.  She has been wanting to write a program that served a different purpose.  So, she and her dad decided to tackle the dreaded "fractions problem".  To quote my husband directly, he refers to their process as a "brute force" method of solving.  They created a screencast to explain what they did and posted it to Vimeo.  I also need to add that Abby was unimpressed with the names and decided to change them to something more creative like Percy and Ashley Kit.  Check it out!

Brute Force Solving in Scratch (screen cast) from Rhett Allain on Vimeo.

Actually, they did a slightly different problem in that they are trying to determine how many coins Joan has.  Nevertheless, this was and is a really cool learning experience for our daughter. 

Back to the problem...

From the program you see that Percy must have had 133 coins in the beginning and Joan would have had 83. Now, I still thought it was extremely important for her to know how to solve this problem without technology.  So, after some time, she and I revisited the problem.  This time, we decided to pay attention to the given diagram.

After a break, she realized her mistake.  She had forgotten to add the 29 to Ben's total.  So, really, we know that 2/3 of Ben's total could be found by adding 29+50+29.  This equals 108.  If you divide 108 by 2, it becomes apparent that each rectangle represents 54 coins.  So, after receiving the 29 coins from Joan, Ben had (54*3) or 162 coins.  To find the amount he had in the beginning, you simply subtract 29 from this total.  So, 162-29 = 133.  Therefore, Ben had 133 coins in the beginning.  How many did Joan have?  Well, you would subtract 50 from Ben's total.  So, 133-50 = 83.  This matches what Abby and her dad found using Scratch. 

I have to say I had to fight the urge to just do things for her.  That would be easier!  I feel it is so important to allow students the opportunity to wrestle with concepts regardless of the discipline.  Ultimately, if you do this, the student takes ownership of the problem and gets on the track towards developing genuine understanding.

All in all, this was a valuable learning experience for everyone.  Here's what we discovered:

• Technology is fabulous and really cool!
• In order to use technology effectively, you still need a good foundation of the concept being studied.
• Never ignore the diagrams!
• Sometimes it helps to get a fresh start.  Put it away for a while and revisit the problem later.

Our approach to education is multi-dimensional: exploratory, creative, hands-on, and integrated with technology. The ultimate goal is to create an environment that fosters inquiry, creativity and a genuine love of learning.  I feel this experience has provided our daughter the opportunity to "think outside of the box" while having a meaningful learning experience.  I think this is a problem she will not soon forget.

P.S.  It's okay to have a "P.S." on a blog post, right?  I really am excited that our children have such a wonderful dad who is extremely involved in their education.  Not many kids have a dad who is a physics professor!  It has really opened many doors to them and given them many wonderful opportunities.  I can't wait to share what we have been doing with circuits!  Anyway, please check out his blog if you are interested in physics and how things work in general.  Lots of cool stuff at Dot Physics!