## Advent Math Activity Calendars

Once again, some of my favorite websites offer a seasonal selection of activities to encourage your children’s (and your own!) mathematical creativity, one for each day in the run-up to Christmas.

Including an especially tough Advent meta-puzzle for truly determined problem-solvers…

Click the images below to visit the corresponding December Math Calendar pages.

### For Primary Students

Easier activities for elementary and middle school.

A problem or game that uses dice for each day in the run up to Christmas.

When you get to the Nrich website, click a number to go to that day’s math.

### For Secondary Students

Activities for middle and high school.

Most of the tasks require dotty paper or circles which you can find on the Nrich Printable Resources page.

When you get to the Nrich website, click a number to go to that day’s math.

### For Teens and Adults

“Because sometimes, in the midst of all the family fun, it’s good to put your headphones in and retreat into a word of your own, this year’s advent calendar brings you some of our favourite Plus podcasts. From the secrets of the Universe to the maths of football stadiums, there should be something there for everyone. Happy listening and happy Christmas from the Plus team!”

When you get to the +Plus Magazine website, you can tell which links are live because they jump to a larger size when you tap or mouse over the picture.

One link becomes live each day — so come back tomorrow and discover something new!

### Christmas Meta-Puzzle

Or try your hand at the biggest mathematical mystery of them all — and save Christmas!

“It’s nearly Christmas and something terrible has happened: one of Santa’s five helpers — Jo Ranger, Fred Metcalfe, Kip Urples, Meg Reeny, and Bob Luey — has stolen all the presents during the North Pole’s annual Sevenstival. You need to find the culprit before Christmas is ruined for everyone.”

If you solve all the clues and enter the answer on Christmas day, you may win a present for yourself, too.

### Still More Mathy Fun

Colleen Young has collected several more holiday math calendars — enough to keep your kids playing and learning well into the New Year. Take a look!

“Peanuts Christmas Panorama” photo [top] by Kevin Dooley via Flicker. (CCBY2.0)

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## Playing Complex Fractions with Your Kids

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?

### Make Your Own Math

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…

### Start with Fraction Strips

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?

### Look for Complex Fractions

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?

### Challenge Puzzles

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.

### Update…

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 part of my Playful Math Singles series. Coming soon to your favorite online bookstore…

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## Math Journals for Elementary and Middle School

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!

## Math Journaling Prompts

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.

### 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?

### Or Try Some Math Doodles

Create math art. Check out my math doodling collection on Pinterest and my Dot Grid Doodling blog post. Can you draw an impossible shape?

## How Would YOU Use a Math Journal?

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.

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## FAQ: Forgetting What They Learned

“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.

And still other times, I find out they didn’t understand the topic as well as I thought they did when we went through it before.

No matter how we adults try to explain the concepts, some kids want to be answer-getters. They don’t want to do the hard work of thinking a concept through until it makes a connection in their minds. They want to memorize a few steps and crank through the lesson to get it over with.

In all these cases, what helps me the most is conversation.

My children and I always talk about our math. I ask questions like “What do you think? What do you remember? Can you explain the question to me? What are they asking for?”

And, whether the child’s answer is right or wrong, I practice my poker face. Trying not to give anything away, I ask, “How did you figure it out? Can you think of a way to confirm your answer?”

### Talking Math with Your Kids

Not sure how to talk about math with your children?

If you have preschool and elementary-age kids, read Christopher Danielson’s inspiring book and blog:

For middle school and older students, check out Fawn Nguyen’s wonderful collection of Math Talks. Be sure to read the “Teachers” page for tips and talking points:

“You don’t need special skills to do this. If you can read with your kids, then you can talk math with them. You can support and encourage their developing mathematical minds.”

— Christopher Danielson
Talking Math with Your Kids

### Playful Ways to Learn or Review Math

Games are a great way to practice math. Check out these (free!) math games for all ages:

And if you have elementary-age children, here are a few grade-level tips to help them learn (and remember) math concepts:

Credits: Girl in field photo by SOURCE Hydration Systems and Sandals technology via Flickr. (CC BY 2.0) Nigerian classroom photo by Doug Linstedt and young girl studying by pan xiaozhen on Unsplash.

This post is an excerpt from my book Let’s Play Math: How Families Can Learn Math Together—and Enjoy It, as are many of the articles in my Let’s Play Math FAQ series.

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## Math Debate: Adding Fractions

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.

### 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?

Well, you might say that:

$\frac{1}{10}$  of one hundred chart
+ $\frac{1}{10}$  of the same chart
= $\frac{2}{10}$  of that hundred chart

But, you might also say that:

$\frac{1}{10}$  of one chart
+ $\frac{1}{10}$  of another chart
= $\frac{2}{20}$  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 $\frac{1}{20}$  of the whole pair of charts.

So what happens if you see this question on a math test:

$\frac{1}{10}$  + $\frac{1}{10}$  = ?

If you write the answer “$\frac{2}{20}$”, you know the teacher will mark it wrong.

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!

Want to help your kids learn math? Claim your free 24-page problem-solving booklet, and sign up to hear about new books, revisions, and sales or other promotions.

## A Beautiful Puzzle

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:

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