The Playful Math Carnival is like a free online magazine devoted to learning, teaching, and playing around with math from preschool to high school. This month’s edition features articles from bloggers all across the internet.†
You’re sure to find something that will delight both you and your child.
By tradition, we start the carnival with a puzzle in honor of our 123rd edition. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.
†Or more, depending on how you count. And on whether I keep finding things to squeeze in under the looming deadline. But if there are more, then there are certainly 36. Right?
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’m sure we’ll use several of these activities in my homeschool co-op math class this fall.
Noticing and Wondering
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.
Puzzles Are Math Experiments
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:
Explore Shapes
Pick out a 3×3 set of dots. How many different shapes can you make by connecting those dots? Which shapes have symmetry? Which ones do you like the best?
What if you make shapes on isometric grid paper? How many different ways can you connect those dots?
Limit your investigation to a specific type of shape. How many different triangles can you make on a 3×3 set of dots? How many different quadrilaterals? What if you used a bigger set of dots?
Explore Angles
On your grid paper, let one dot “hold hands” with two others. How many different angles can you make? Can you figure out their degree without measuring?
Are there any angles you can’t make on your dot grid? If your paper extended forever, would there be any angles you couldn’t make?
Does it make a difference whether you try the angle experiments on square or isometric grid paper?
Explore Squares
How many different squares can you draw on your grid paper? (Don’t forget the squares that sit on a slant!) How can you be sure that they are perfectly square?
Number the rows and columns of dots. Can you find a pattern in the corner positions for your squares? If someone drew a secret square, what’s the minimum information you would need to duplicate it?
Does it make a difference whether you try the square experiments on square or isometric grid paper?
I’d love to hear your favorite math explorations or journaling tips!
Please share in the comments section below.
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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.
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My son can’t stand long division or fractions. We had a lesson on geometry, and he enjoyed that — especially the 3-D shapes. If we can just get past the basics, then we’ll have time for the things he finds interesting. But one workbook page takes so long, and I’m sick of the drama. Should we keep pushing through?
Those upper-elementary arithmetic topics are important. Foundational concepts. Your son needs to master them.
Eventually.
But the daily slog through page after page of workbook arithmetic can wear anyone down.
Many children find it easier to focus on math when it’s built into a game.
“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.
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.
Have a Math Debate: Adding Fractions
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?
1/10 of 100
Well, you might say that:
of one hundred chart
+ of the same chart
= of that hundred chart
But, you might also say that:
of one chart
+ of another chart
= of the pair of charts
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 of the whole pair of charts.
So what happens if you see this question on a math test:
+ = ?
If you write the answer “”, you know the teacher will mark it wrong.
Today we have a guest post — an exclusive tale by Sasha Fradkin and Allison Bishop, authors of the new math storybook Funville Adventures. Enjoy!
Funville Adventures is a math-inspired fantasy that introduces children to the concept of functions, which are personified as magical beings with powers.
Each power corresponds to a transformation such as doubling in size, rotating, copying, or changing color. Some Funvillians have siblings with opposite powers that can reverse the effects and return an object to its original state, but other powers cannot be reversed.
In this way, kids are introduced to the mathematical concepts of invertible and non-invertible functions, domains, ranges, and even functionals, all without mathematical terminology.
We know about Funville because two siblings, Emmy and Leo, were magically transported there after they went down an abandoned slide.
When they came back, Emmy and Leo shared their adventures with their friends and also brought back the following manuscript written by their new friend Blake.
But my favorite way to celebrate any new year is by playing the Year Game. It’s a prime opportunity for players of all ages to fulfill the two most popular New Year’s Resolutions: spending more time with family and friends, and getting more exercise.
So grab a partner, slip into your workout clothes, and pump up those mental muscles!
For many years mathematicians, scientists, engineers and others interested in mathematics have played “year games” via e-mail and in newsgroups. We don’t always know whether it is possible to write expressions for all the numbers from 1 to 100 using only the digits in the current year, but it is fun to try to see how many you can find. This year may prove to be a challenge.
Use the digits in the year 2018 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.
You must use all four digits. You may not use any other numbers.
Solutions that keep the year digits in 2-0-1-8 order are preferred, but not required.
You may use a decimal point to create numbers such as .2, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.
You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.
My Special Variations on the Rules
You MAY use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal. But students and teachers beware: you can’t submit answers with repeating decimals to Math Forum.
You MAY use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. I’m including these because Math Forum allows them, but I personally try to avoid the beasts. I feel much more creative when I can wrangle a solution without invoking them.
This lovely puzzle (for upper-elementary and beyond) is from Nikolay Bogdanov-Belsky’s 1895 painting “Mental Calculation. In Public School of S. A. Rachinsky.” Pat Ballew posted it on his blog On This Day in Math, in honor of the 365th day of the year.
I love the expressions on the boys’ faces. So many different ways to manifest hard thinking!
Here’s the question:
No calculator allowed. But you can talk it over with a friend, as the boys on the right are doing.
You can even use scratch paper, if you like.
Thinking About Square Numbers
And if you’d like a hint, you can figure out square numbers using this trick. Think of a square number made from rows of pennies.
Can you see how to make the next-bigger square?
Any square number, plus one more row and one more column, plus a penny for the corner, makes the next-bigger square.
So if you know that ten squared is one hundred, then:
… and so onward to your answer. If the Russian schoolboys could figure it out, then you can, too!
Update
Simon Gregg (@Simon_Gregg) added this wonderful related puzzle for the new year:
You want your child to succeed in math because it opens so many doors in the future.
But kids have a short-term perspective. They don’t really care about the future. They care about getting through tonight’s homework and moving on to something more interesting.
So how can you help your child learn math?
When kids face a difficult math problem, their attitude can make all the difference. Not so much their “I hate homework!” attitude, but their mathematical worldview.
Does your child see math as answer-getting? Or as problem-solving?
Answer-getting asks “What is the answer?”, decides whether it is right, and then goes on to the next question.
Problem-solving asks “Why do you say that?” and listens for the explanation.
Problem-solving is not really interested in “right” or “wrong”—it cares more about “makes sense” or “needs justification.”
Homeschool Memories
In our quarter-century-plus of homeschooling, my children and I worked our way through a lot of math problems. But often, we didn’t bother to take the calculation all the way to the end.
Why didn’t I care whether my kids found the answer?
Because the thing that intrigued me about math was the web of interrelated ideas we discovered along the way:
How can we recognize this type of problem?
What other problems are related to it, and how can they help us understand this one? Or can this problem help us figure out those others?
What could we do if we had never seen a problem like this one before? How would we reason it out?
Why does the formula work? Where did it come from, and how is it related to basic principles?
What is the easiest or most efficient way to manipulative the numbers? Does this help us see more of the patterns and connections within our number system?
Is there another way to approach the problem? How many different ways can we think of? Which way do we like best, and why?
What Do You think?
How did you learn math? Did your school experience focus on answer-getting or problem-solving?
How can we help our children learn to think their way through math problems?
I’d love to hear from you! Please share your opinions in the Comments section below.
CREDITS: “Math Phobia” photo by Jimmie (blog post title added) via Flickr (CC BY 2.0). Phil Daro video by SERP Media (the Strategic Education Research Partnership) via Vimeo.
I’d love to hear your favorite math explorations or journaling tips! 
of one hundred chart
of that hundred chart
of the pair of charts
of the whole pair of charts.
Denise Gaskins' Let's Play Math 