Here is a math problem in honor of one of our family’s favorite movies…
Han Solo was doing much-needed maintenance on the Millennium Falcon. He spent 3/5 of his money upgrading the hyperspace motivator. He spent 3/4 of the remainder to install a new blaster cannon. If he spent 450 credits altogether, how much money did he have left?
Stop and think about how you would solve it before reading further.
For children, learning always begins with play. This is how they wrap their minds around new ideas and make them their own.
“There should be no element of slavery in learning. Enforced exercise does no harm to the body, but enforced learning will not stay in the mind. So avoid compulsion, and let your children’s lessons take the form of play.”
—Plato, The Republic
If we want our children to enjoy learning math, our first job is to establish an attitude of playfulness.
This is especially important for anyone working with a discouraged child or a child who is afraid of math. The best way to help a discouraged child is to put away the workbook. Try something different, fun, and challenging.
Far too many people find themselves suddenly, unexpectedly homeschooling their children. This prompts me to consider what advice I might offer after more than three decades of teaching kids at home.
Through my decades of homeschooling five kids, we lived by two rules:
Do math. Do reading.
As long as we hit those two topics each day, I knew the kids would be fine. Do some sort of mathematical game or activity. Read something from that big stack of books we collected at the library.
Conquer the basics of math and reading, then everything else will fall into place.
One of the most common questions I get from parents who want to help their children enjoy math is, “Where do we start?”
My favorite answer: “Play games!”
And as the world slowly recovers from the pandemic crisis, it’s even more important for families to play together. So my publisher agreed to make my ebook Let’s Play Math Sampler: 10 Family-Favorite Games for Learning Math Through Play free for the duration.
If you haven’t seen the meme going around, this is a palindrome week because the dates (written American style and with the year shortened to ’19) are the same when reversed.
Here’s a math puzzle for palindrome week — or any time you want to play with math:
What do you think: Will all numbers eventually turn into palindromes?
You can find all sorts of hundred charts on my Free Math Printable Files page.
Read about the history of palindromes on Nrich Math’s Palindromes page.
Find out more about the Palindromic Number Conjecture in Mark Chubb’s article An Unsolved Problem your Students Should Attempt.
Or play with Manan Shah’s advanced palindromic number questions.
Did you know that numbers can be polite? In math, a polite number is any number we can write as the sum of two or more consecutive positive whole numbers.
(Consecutive means numbers that come one right after another in the counting sequence.)
For example, five is a polite number, because we can write it as the sum of two consecutive numbers:
5 = 2 + 3
Nine is a doubly polite number, because we can write it two ways:
9 = 4 + 5
9 = 2 + 3 + 4
And fifteen is an amazingly polite number. We can write fifteen as the sum of consecutive numbers in three ways:
15 = 7 + 8
15 = 4 + 5 + 6
15 = 1 + 2 + 3 + 4 + 5
How many other polite numbers can you find?
Are all numbers polite?
Or can you find an impolite number?
Can you make a collection of polite and impolite numbers? Find as many as you can.
How many different ways can you write each polite number as a sum of consecutive numbers?
What do you notice about your collection of polite and impolite numbers?
Can you think of a way to organize your collection so you can look for patterns?
Make a conjecture about polite or impolite numbers. A conjecture is a statement that you think might be true.
For example, you might make a conjecture that “All odd numbers are…” — How would you finish that sentence?
Make another conjecture.
And another.
Can you make at least five conjectures about polite and impolite numbers?
What is your favorite conjecture? Does thinking about it make you wonder about numbers?
Can you think of any way to test your conjectures, to know whether they will always be true or not?
This is how mathematics works. Mathematicians play with numbers, shapes, or ideas and explore how those relate to other ideas.
After collecting a set of interesting things, they think about ways to organize them, so they can look for patterns and connections. They make conjectures and try to imagine ways to test them.
And mathematicians compare their ideas with each other. In real life, math is a very social game.
So play with polite and impolite numbers. Compare your conjectures with a friend.
Share your ideas in the comments section below.
And check out the list of student conjectures at the Ramblings of a Math Mom blog.
CREDITS: Numbers photo (top) by James Cridland via Flickr (CC BY 2.0). I first saw this activity at Dave Marain’s Math Notations blog, and it’s also available as a cute printable Nrich poster. For a detailed analysis, check out Wai Yan Pong’s “Sums of Consecutive Integers” article.
This week, I’m working on graphics for my upcoming book 70+ Things to Do with a Hundred Chart. I had fun with this complex fraction image.
It looks a bit cluttered. Possible tweak: Remove the brackets and instead use a thicker dividing line to show the thirds.
While I’m thinking about that, would you like a sneak peek at an activity from the book?
You don’t need a set of worksheets or lesson plans to learn math. All you need is an inquiring mind and something interesting to think about.
Play. Discuss. Notice. Wonder.
Enjoy.
Here’s how you can play complex fractions with your kids…
Print a few blank 120 charts and turn them sideways, so each chart has ten rows with twelve squares in each row.
Cut out the rows to make fraction strips with twelve squares on each strip.
Color a different set of squares on each strip. On some strips, arrange the colored squares all together at one end. On other strips, mix them around.
If we count each strip as one whole thing, what fraction of its squares are colored?
Match the strips that represent the same fraction.
On some of the strips, there will be more than one way to name the fraction. For example, if six squares are colored, we can call that 6/12 or 2/4 or 1/2 of the strip. These alternate names are easiest to see when the colored squares are all at one end of the strip, because you can fold the strip to show the halves or fourths.
How many different fraction names can you find for each set of colored squares?
We could also call the strip with six colored squares “1 1/2 thirds” of the whole strip. Can you show by folding why that name makes sense?
Or we could call the strip with five colored squares “2 1/2 sixths.”
When we have a fraction within a fraction like this, we call it a complex fraction, because it is more complicated than a common (or simple) fraction.
Another way to say it: Complex fractions have other fractions inside them.
A complex fraction is like a puzzle, challenging us to find its secret identity — the common fraction that names the same amount of stuff.
For example, how much is 3 1/3 fourths? One fourth would be three of the twelve squares on a fraction strip. So three fourths would be three sets of those three squares, or nine squares. Then we need to add one-third of the final fourth, which is one of the remaining three squares. So 3 1/3 fourths must be ten squares in all.
3 1/3 fourths = 10/12 = 5/6
How many complex fractions can you find in your set of fraction strips?
Can you figure out how much a one-and-a-halfth would be?
That is one piece, of such a size that it takes one and one-half pieces to make a complete fraction strip.
A one-and-a-halfth is a very useful fraction and was a favorite of the ancient Egyptian scribes, who used it to solve all sorts of practical math problems.
How about a one-and-a-thirdth? How many of those pieces make a whole strip? What common fraction names the same amount of stuff?
Or how much would a two-thirdth be? In that case, it only takes two-thirds of a piece to make a complete strip. So the whole piece must be greater than one. A two-thirdth’s secret identity is a mixed number. Can you unmask it?
Make up some challenge fraction mysteries of your own.
I’m still working on the graphics for my hundred chart book. Here’s the latest version of the complex fraction strips.
I like this one much better.
What do you think?
CREDITS: The slogan “Make Math Your Own” comes from Maria Droujkova, founder and director of the Natural Math website. Maria likes to say: “Make math your own, to make your own math!”
70+ Things to Do with a Hundred Chart is now available from Tabletop Academy Press.
This fall, my homeschool co-op math class will play with math journaling.
But my earlier dot-grid notebooks were designed for adults. Too thick, too many pages. And the half-cm dot grid made lines too narrow for young writers.
So I created a new series of paperback dot-grid journals for my elementary and middle school students.
I hope you enjoy them, too!
Click here for more information
So, what can your kids do with a math journal?
Here are a few ideas:
I’m sure we’ll use several of these activities in my homeschool co-op math class this fall.
Learning math requires more than mastering number facts and memorizing rules. At its heart, math is a way of thinking.
So more than anything else, we need to teach our kids to think mathematically — to make sense of math problems and persevere in figuring them out.
Help your children learn to see with mathematical eyes, noticing and wondering about math problems.
Whenever your children need to learn a new idea in math, or whenever they get stuck on a tough homework problem, that’s a good time to step back and make sense of the math.
Kids can write their noticings and wonderings in the math journal. Or you can act as the scribe, writing down (without comment) everything child says.
For more tips on teaching students to brainstorm about math, check out these online resources from The Math Forum:
Problem-solving is a habit of mind that you and your children can learn and grow in. Help your kids practice slowing down and taking the time to fully understand a problem situation.
Almost anything your child notices or wonders can lead to a math experiment.
For example, one day my daughter played an online math game…
A math journal can be like a science lab book. Not the pre-digested, fill-in-the-blank lab books that some curricula provide. But the real lab books that scientists write to keep track of their data, and what they’ve tried so far, and what went wrong, and what finally worked.
Here are a few open-ended math experiments you might try:
Create math art. Check out my math doodling collection on Pinterest and my Dot Grid Doodling blog post. Can you draw an impossible shape?
I’d love to hear your favorite math explorations or journaling tips!
Please share in the comments section below.
* * *
P.S.: Do you have a blog? If you’d like to feature a math journal review and giveaway, I’ll provide the prize. Send a message through my contact form or leave a comment below, and we’ll work out the details.
This blog is reader-supported.
If you’d like to help fund the blog on an on-going basis, then please head to my Patreon page.
If you liked this post, and want to show your one-time appreciation, the place to do that is PayPal: paypal.me/DeniseGaskinsMath. If you go that route, please include your email address in the notes section, so I can say thank you.
Which I am going to say right now. Thank you!
“Math Journals for Elementary and Middle School” copyright © 2018 by Denise Gaskins. Photos of children © original artists / Pixabay.
“As we go through each lesson, it seems like my daughter has a good handle on the concepts, but when we get to the test she forgets everything. When I ask her about it, she shrugs and says, ‘I don’t know.’ What do you do when your child completely loses what she has learned?”
Forgetting is the human brain’s natural defense mechanism. It keeps us from being overwhelmed by the abundance of sensory data that bombards us each moment of every day.
Our children’s minds will never work like a computer that can store a program and recall it flawlessly months later.
Sometimes, for my children, a gentle reminder is enough to drag the forgotten concept back out of the dust-bunnies of memory.
Other times, I find that they answer “I don’t know” out of habit, because it’s easier than thinking about the question. And because they’d prefer to be doing something else.
I’ve been working on my next Playful Math Singles book, based on the popular Things to Do with a Hundred Chart post.
My hundred chart list began many years ago as seven ideas for playing with numbers. Over the years, it grew to its current 30+ activities.
Now, in preparing the new book, my list has become a monster. I’ve collected almost 70 ways to play with numbers, shapes, and logic from preschool to middle school. Just yesterday I added activities for fraction and decimal multiplication, and also tips for naming complex fractions. Wow!
Gonna have to edit that cover file…
In the “Advanced Patterns” chapter, I have a section on math debates. The point of a math debate isn’t that one answer is “right” while the other is “wrong.” You can choose either side of the question — the important thing is how well you support your argument.
Here’s activity #69 in the current book draft.
When you add fractions, you face a problem that most people never consider. Namely, you have to decide exactly what you are talking about.
For instance, what is one-tenth plus one-tenth?
Well, you might say that:
+
= 
But, you might also say that:
+
= 
That is, you started off counting on two independent charts. But when you put them together, you ended up with a double chart. Two hundred squares in all. Which made each row in the final set worth 
So what happens if you see this question on a math test:
If you write the answer “
Is that fair? Why, or why not?
CREDITS: Feature photo (above) by Thor/geishaboy500 via Flickr (CC BY 2.0). “One is one … or is it?” video by Christopher Danielson via TED-Ed. This math debate was suggested by Marilyn Burns’s blog post Can 1/3 + 1/3 = 2/6? It seemed so!
Denise Gaskins' Let's Play Math 