Home » Math

# Category Archives: Math

## Fractions in Action

Have you ever seen one of these? It’s a hands-on way to teach fractions.

All you need is construction paper, scissors, a stapler, and markers.

Staple five pieces of construction paper together and then slice them up to make a flip book. At this point, hand the book over to your kid. Let him figure out how to cut and label the pages to make a fraction flip book.

I did this project with fourth graders this afternoon, and it took them about thirty minutes to make their first book. Then I passed out more paper and it took them fifteen minutes to make a second book, using 1/3, 1/6, 1/9, and 1/12. The second set of fractions was more difficult, but by then they had mastered the whole concept behind the activity.

If you try this at home, don’t be afraid to let your child struggle. Tape is okay! That means learning happened. 😉

## Fifth Grade Math Triangle Challenge

Here’s an especially tricky problem from 5th grade geometry. Everyone knows that the area of a triangle is 1/2 (b * h). But with this particular triangle, what qualifies as “the height” is difficult to see. At least it was for me the first time I looked at it.

I don’t know–maybe you’ll look at this problem and say “Duh, Jenny.” But for me, the scalene triangle was strange looking.

When I first looked at this I saw that it would be easy to solve with the Pythagorean Theorem. But Houghton Mifflin Math Expressions hadn’t covered that yet. So there was another even easier way to solve this problem that wasn’t jumping out to my me or my son.

Can you figure out what it is?

From Houghton Mifflin Math Expressions Grade 5 Chapter Two

Figuring out the perimeter of the red triangle is easy. That’s 17 + 9 + 10 = 36 cm. But what about the area?

First, I’ll show the way that ends up being the most complicated: using the Pythagorean Theorem.

Using a squared + b squared = c squared, find out the area of the yellow triangle.

Now that you know b = 6, this lets you figure out that the length of the rectangle is 15 cm. That lets you figure out the area of the whole rectangle.

Use the Pythagorean Theorem to find the area of the yellow triangle.

Then you figure out the area of the green triangle, and subtract green and yellow from the area of the rectangle, finally find your answer.

This is a perfect example of how being algorithm dependent can screw up your number sense. I was so sure the Pythagorean Theorem was the way to go, I initially missed seeing the easier solution.

Another look at the original problem.

Triangles can always become parallelograms, which can be easier to deal with.

Here’s the “Duh!” moment. Now it’s super easy to see that 8 cm = the height of the parallelogram which means it also = the height of the triangle.

1/2 the area of the parallelogram is the area of your triangle.

Now after all of that, let’s look at the original problem and try a third method to solve this problem, using the formula 1/2 (b*h). This is arguably the easiest method.

1/2 (9 * 8) = 36 sq cm.

Okay, so why didn’t I use the formula to begin with? When my son first looked at this, why didn’t I say “Dude, plug in the formula 1/2 (b * h),”?

Because that’s not what good math teachers do. Math is more than memorizing and applying formulas. Math is about experimenting, visualizing, internalizing and sometimes struggling until you reach a higher level of understanding.

This is an example of a problem that is simple yet confusing. Those are the best types! I’ve gone through college level calculus and I still looked at the picture and couldn’t viscerally understand why 8 cm was the height of the triangle. Neither could anyone in my family. (My husband, btw, is a lot smarter in math than me!)

So we played with this problem. We turned it inside out. Now, it makes sense. Along the way, we got to do a lot of cool math.

## Putting the sense in Number Sense

The best teaching happens when you make a lesson visual, spatial and auditory. That’s why I love teaching kids number sense with a math balance.

Utilizing a math balance in a whole-class setting of twenty-seven kids would be tricky, but at home with one child it’s easy. The  balance we own came from Right Start and costs \$25.

5 does not equal 8. It’s so easy to see.

Just like it’s easy to figure out that there are many number combinations that equal 5.

In fact, we spent a full ten minutes just figuring out the number 5!

In pedagogy, we call this “Constructivism”.  It means learning a new concept through your own experimentation and discovery. Giving children the full Constructivist experience isn’t always possible, but a math balance makes it a lot easier.

## New ways to learn math (that have been around a long time)

Exploring rotational symmetry with cookie cutters

Here’s my “I Brake for Moms” column about Common Core Math in today’s Daily Herald:

http://www.heraldnet.com/article/20140601/BLOG5205/140609980/Common-Cores-math-method-is-different-not-evil-

If you’re looking for ways to support your children’s math development at home, here are some of my favorite activities:

Rotational symmetry

How to simplify fractions

Hands-on Algebra for third grades

Fifth grade Algebra and candy

Part-Whole-Circle math

Marshmallow math

Helping kids understand place value

Square numbers with crackers

Math without worksheets

Math for two and three year olds

Reducing fractions with peanut butter

Multiplication Memory

Using a math balance to develop number sense

Area of triangles with geoboards

Area of polygons on geoboards

Dreambox math review

For more of my thoughts on math, click here. If you are interested in learning more about my educational and professional background, click here to read the snobby stuff.

## Math with a Hello Kitty stencil

For some reason a Hello Kitty stencil makes this more fun.

Do you have any stencils or stickers laying around? Those are the types of things that breed in our playroom.

The other day my four-year-old found her Hello Kitty stencil. I told her, “Let’s do math with Hello Kitty,” and she was instantly intrigued.

Working on the number 9.

For this activity we worked on building the number nine with stickers. This was also a chance to work on the commutative property; 4+5= 9 means that 5+4 = 9.

So unleash your random art junk! I bet “Math with Bob the Builder” would be fun too.

P.S. Can you see the spot on my camera? (I’m a pretty clueless photographer.) Ugh!

## Life of Fred Pre-Algebra

Okay Public School families, this is where I introduce you to something from the Homeschooling world that can be useful for families like ours. Have you ever heard of Life of Fred by Stanley F. Schmidt, Ph.D?

The “Life of Fred” series is a very unusual way to teach math, science, history, literature, and plain old common sense, in an integrated format. Half chapter book, half graphic novel, half textbook (hmmmm… those fractions don’t add up); the Life of Fred books teach through story and humor. The hero of the books, Fred, is a five year-old math professor at Kittens University, somewhere in Kansas.

I’ve been a bit harsh on these books in the past because Life of Fred uses a lot of algorithms early on, especially in the Fractions book. I’m much more of a Constructivist teacher, so too many algorithms, too early, makes me nervous. I also don’t think Life of Fred would be good for “mathy ” 1st-3rd graders who were emergent readers. The reading would hold back their math, which would be very frustrating.

But! If you were to take the opposite type of child, let’s say a kid who loved words, stories, pictures and funny jokes but who wasn’t very interested in math, then Life of Fred is absolutely perfect. I was just talking with a teacher friend last week who was dealing with that exact situation. “Have you heard of Life of Fred?” I asked her. “It’s just what you need.”

Another important thing to note about Life of Fred is that it is a spiraling curriculum. This means that Dr. Schmidt introduces a concept and then circles back to it later on. So if your child doesn’t quite understand something in chapter one, don’t worry, it will be reviewed again later.

Right now my son is reading Life of Fred: Pre-algebra 1 with Biology. In terms of Algebra, there’s nothing harder than Hands On Equations or Continental Math League. But there is a lot of other stuff, like fractions, decimals, and conversion factors. Kids with strong fifth grade math skills would do fine with this book.

I’ve been reading Life of Fred : Pre-Algebra 2 with Economics  for fun. I’ve also got Hot X: Algebra Exposed! by Danica McKellar on my reading list. It doesn’t hurt for a mom to brush up on math she learned 25 years ago, right?  Luckily, Life of Fred makes that pretty fun.

Math on a plate.

Today my 4-year-old and I got out Cooking Is Cool: Heat-Free Recipes for Kids to Cook by Marianne Dambra. I am a little disappointed in the book because it doesn’t have step-by-step pictures for children to follow and it uses a lot of food coloring. But the full color illustrations of each recipe are nice.

Here’s our version of Butterfly Salad. The actual recipe called for dyed cottage cheese, which I thought was gross, so we used grapes instead.

Ingredients:

• lettuce leaf
• pineapple rings
• cottage cheese
• grapes or berries
• celery stalk (the body)
• 1 olive (the head)
• a carrot or bell pepper (the antennae)

Math Skills Involved:

• counting
• fractions (cut the pineapple rings in half)
• comparisons (more cottage cheese, less cottage cheese)
• ordinal numbers (first you do this, second you do that, etc.)
• symmetry (the goal is to make the wings look the same)

This recipe took about twenty minutes to make. My daughter and I both had a lot of fun!

## Just say no to busy work!

Don’t buy this at Costco!

I had very low expectations for this latest Leap Frog purchase and I wasn’t disappointed.

Leap Frog’s Complete Kindergarten Learning Kit (Math, Printing, Language Building, Early Reading) (Grade K) was selling for \$20 at Costco. (It’s \$89 on Amazon!!!)

I didn’t want to buy it but my preschooler made me.

Okay, that’s not totally true. I was curious. The former Kindergarten teacher in me was begging to see what was in that box.

Save yourself \$20 and just look at my picture:

Nothing in this box is bad exactly, it’s just that I don’t believe workbooks and flashcards are the answer.

Sometimes you’ll get kids like my daughter who “want” to do workbooks. Okay, fine. Whatever. We can get out the Leap Frog workbooks for fun.

But edutainment is different than education.

There are a hundred more meaningful things you could do with your emergent reader that would be more meaningful. Here’s roadmap of examples.

With flashcards, if you are going to use them selectively (as I sometimes do), they shouldn’t be confusing. Take a look at this:

Sooooo many numbers and soooooo confusing!

That’s a little bit better.

One thing the kit came with that I thought was pretty good were these dot cards:

These are awesome, but you could make them at home for free.

Here are some fun ideas for preschool math.

Final thoughts? Maybe the next time you are at Costco, you can save \$20!

P.S. Leap Frog does have four products that I highly recommend:

Sadly all of the other Leap Frog products I have purchased haven’t been as good.

## Rotational Symmetry with Cookie Cutters

Exploring rotational symmetry with cookie cutters.

Bruce had homework regarding rotational symmetry and it totally confused me because I’m really bad at visual-spatial things.

What is rotational symmetry? That means a shape that can be rotated less than 360 degrees and still look the same.  More info right here.

For spatially challenged people like me (you should see me parallel park!), rotational symmetry can be hard to picture. Hands-on learning can help.

A long time ago, I blogged about using flour and cookie cutters to learn about flips and turns. Guess what? That idea also works for rotational symmetry too!

First draw the X and Y axis in the flour. Then start experimenting.

This butterfly does NOT have rotational symmetry.

Neither does the shamrock. It does not look the same until a 360 turn.

A circle definitely has rotational symmetry. But this one’s pretty obvious.

A little bit of masking tape and this becomes more interesting.

At 180 degrees, this shape has rotational symmetry.

A word to the wise: this activity is messy! It’s the perfect example of something that would be really hard to do with thirty fifth graders in a classroom, but doable with your child at home.

Just be sure to have a vacuum ready!

## Teach how to simplify fractions in a hands-on way

Full disclaimer! This activity is a billion times more fun with candy and peanut butter. Unfortunately, we used plastic tiles today.

Happy Martin Luther King Jr. Day!

Free time at home gave us the chance to do some fraction review, to support what’s coming up next in my son’s Houghton Mifflin Math Expressions book.

This is one of those teaching activities that will either seems silly to you (if you had a strong childhood math experience), or else will be a huge light-bulb moment (if your childhood math was based on drill, kill and memorization.)

Are you ready to see what you think?

Here’s one way to make the abstract concept of simplifying and unsimplifying fractions, hands on and concrete.

Use tiles, candy, crackers, or other manipulatives to make fractions come to life. In this example, 8 out of 12 tiles are yellow.

“Peanut butter” them together, and all of a sudden, 8/12 becomes 2/3.

After you do it with manipulatives, then introduce the formula. Divide the numerator and denominator by 4, and you turn 8/12 into 2/3.

Here’s another one:

This time, we are going to unsimplify the fraction.

Now we can see that six out of nine tiles being red is the same thing as two thirds.

Okay, so plastic tiles makes fractions a little bit more fun. But if we were working with M&M’s, this would be awesome!

## Basher Books for Grown Ups

My son Bruce(8.5) loves Basher Books so much that he reads them over and over again. I still don’t exactly understand the appeal, but appreciate how much he’s learned. If Trivial Pursuit ever becomes popular again, I want Bruce on my team.

Today at the bookstore I came across The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics by Clifford Pickford. It’s the closest thing I’ve ever seen to a Basher Book for adults.

I am very sensitive to violating copyrights, so I’m not going to share an interior picture, but this book has topics like “St. Petersburg Paradox” on the left-hand page, and then a really cool picture on the right. Sound familiar?

I bought this book for myself, but I’m pretty sure Bruce will read it too, especially if I leave it on the kitchen table next to his cereal.

FYI: My version is leather bound and cost \$20 at Barnes & Noble. The paperback version on Amazon is a lot cheaper.

## Singapore vs. Right Start, an Afterschooler’s Dilemma

Afterschooling with Singapore Math?

The other day I was at the teacher store buying Christmas presents (yes, I’m weird), when I  decided to take a look at the Singapore math section.

Long time readers will remember that I’ve blogged about Singapore math before. A couple of years ago, I used the fourth grade Singapore Standards book to help supplement my son Bruce’s math work afterschool.

From our Right Start kit.

But with my daughter Jenna(4), we’ve been working through Right Start Level A, because I love Right Start.

I love the manipulatives…I love the constructivist approach…I love that handwriting doesn’t have to get in the way of progress… Dr. Joan Cotter is my hero!

The downside of Right Start is that parents have to set up a lot of stuff. You can’t just open a workbook and hand your kid a pencil.

Right Start Level A also seems to stretch out into 1st grade territory. So right now, Jenna’s on lesson 30 (out of 77), and we’re pretty much treading water.  I need to wait a bit for her to developmentally catch up and be ready to continue.  Some kids would be able to move faster. Some kids would need to move slower. But Jenna’s only four, and there’s no rush.

So while I was at the teachers store I picked up a copy of the Singapore Math textbook A for Kindergartners. It’s colorful (some would argue cartoonish), and really engaging for a little girl like Jenna, who loves to do “homework”. She breezed through half of the book in a few days, of her own accord, and then polished off a lot more over the weekend.

Now we’re at the point where Jenna really needs to learn to write the number 5 before she can finish up book A, and move on to book B. (Darn, that handwriting!)

Anyone familiar with Singapore can probably guess what we have not done this past week, which has allowed Jenna to breeze through those pages so quickly.

We haven’t been following all of the instructions that involve collecting toys to count, measuring objects in the house, or discussing potatoes as a food source.

That’s the real danger of using Singapore. It’s easy to skip all of the important, hands-on stuff, and just have your kid do workbook pages.

That doesn’t mean that I think Singapore is bad, I just think that parents need to be careful.

In our situation, I’m fine with Jenna using it as a fun workbook so that she can have “homework” like her brother. That’s because she’s been doing so many hands-on activities from Right Start.

But if Singapore was the only way I was supplementing math afterschool (or in this case before school), I would purchase the teacher’s guide and be a lot more careful.

Teaching is different than watching your kid to workbook pages, –even if you are drawing out dots for the number 5!

## What’s happening to Math Expressions?

Right now our school district uses the 2009 version of Houghton Mifflin Math Expressions. As math programs go, I like it. (Full review here.) Are there things I would like to change about Math Expressions? Yes. Do I think it’s horrible? No.

But here’s the thing. Washington State is fully implementing the Common Core State Standards Initiative in 2014. Knowing how these things work, I’m guessing that 2009 Math Expressions will be considered outdated, especially since Houghton Mifflin has come out with a new version of Math Expressions that is Common Core aligned.

It makes me feel sorry for our school district. Those poor people at the district office, trying to figure out how to find \$1 million dollars to buy new books that hopefully won’t be outdated in four years… Yikes!

In the meantime, the 2009 Math Expressions books are really cheap. I ordered volume one of the 5th grade textbook for under ten dollars.

Why did I do this? It’s nice to have the book at home so that I can see what my son Bruce is learning in school. Sometimes 5 minutes of “Mom Math” can really make a difference.

There are also a lot of things in the textbook meant to be cut out, like game pieces and little flashcards. This can’t happen in an classroom environment where the books need to be saved for future years…or something.

Vocabulary words can be underlined, important numbers can be circled, concepts can be pretaught or reviewed as needed. This was ten dollars well spent!

Having the textbook at home makes me wish we could all think bigger and be better about collaborating. School Districts, Teachers, Parents, Students; if everyone was literally on the same page, we could make good things happen.

But that would take two things that are constantly in short supply when it comes to education: trust and money.

If a school district is struggling to buy new math books that become obsolete almost as soon as they are printed, how could it ever afford to buy a second set for kids to use at home (maybe) with their parents?

That was a rhetorical question. Homeschools, feel free to give yourselves a smug little pat on the back right about now.

P.S. If you are looking for a Khan Academy alternative. Houghton Mifflin also has some sort of free online-tutorial that I haven’t checked out yet called Go Math Academy.

## Mini-Pumpkin Math

There’s math in here. I know it!

Jenna(4) has been working on taking a large quantity of objects and organizing them to make the objects easier to quantify.  (See here for more info.)

Today she practice that skill using mini pumpkins.

How many pumpkins are there?

One solution would be to line them up and count.

Or you could make “five flowers” with pumpkins.

Organizing makes it easier to visualize.

Can you see in one glance what the answer is?

## Measuring with apples

How tall are you in apples?

Full confession: I totally stole this idea from Jen-tilla the Mum. Her pictures are cuter than mine too, because she show’s her kids’ faces.  All you get to see are my ugly purple tights.  (They looked a lot cuter when I was wearing boots.  You’ve got to trust me on this one.)

Mommy gets measured.

Measuring with apples is a classic example of non-standard measurement.

Non-standard measurement is how Kindergarten teachers used to introduce the concept of measurement to young children before standardized testing made everything serious.  Now you’ve got to hope an activity like this is in the textbook, or that your child’s teacher is willing to be brave.

But don’t let the juicy yumminess of apples fool you.  There is a lot of learning in this activity:

• Counting
• Estimating
• Measuring
• Investigating
• Pattern work (optional)

P.S. Don’t forget to wash your math manipulatives before they go back into the fruit bowl!