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Fifth Grade Math Triangle Challenge

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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

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.

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.

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.

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

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 finding your answer.

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.

Another look at the original problem.

Okay, so everyone knows that the area of a triangle is 1/2 b * h. But with this particular triangle, that's tricky to see.

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

Now it's super easy to see that 8 cm = the height of the triangle, right?

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.

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.

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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.

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

Exploring rotational symmetry with cookie cutters

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

Cookie cutter geometry

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.

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.

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!

What’s happening to Math Expressions?

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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.


Math Expressions: Student Activity Book Softcover, Volume 1 Level 5 2009 (Math Expressions 2009 – 2012)

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.

Math through Osmosis

Try to ignore this while eating breakfast. I dare you!

There’s nowhere to hide from learning at our house!

Here’s a quick trick to help promote learning at home. If your kid is learning a new vocabulary term, post the definition somewhere he’ll see it during breakfast.

In this example, I’m including some Greek words for numbers that will be helpful when learning about decimeters.

Btw, there’s already been a debate at our house over whether or not I spelled these words correctly. In my defense, I copied the Greek straight from the math textbook. So If I spelled something wrong, blame Math Expressions.  🙂

Mini-Pumpkin Math

There's math in here.  I know it!

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?

How many pumpkins are there?

One solution would be to line them up and count.

One solution would be to line them up and count.

Or you could make "five flowers" with pumpkins.

Or you could make “five flowers” with pumpkins.

Organizing mates it easier to visualize. Twenty Six!

Organizing makes it easier to visualize.

Can you see in one glance what the answer is?

Part-Whole Circle Math

Part-Whole Circle Math

Part-Whole Circle Math

Right now Jenna(4) and I are doing a new Part-Whole Circle problem every day.  Focusing on word problems helps teach reading and math at the same time.  It also compliments what Jenna is doing in Right Start Level A.

But you don’t need a formal math program to try this idea with your preschooler at home.  Here’s how it works.

First we read and discussed the word problem.  I had Jenna write a "B" and a "G" in the part circles, for boys and girls.

First we read and discuss the word problem. Jenna writes a “B” and a “G” in the part circles, for boys and girls.

Next Jenna moves three markers to the boy circle.

Next Jenna moves three markers to the boy circle.

Jenna moves the remaining  pieces to the girl circle.  Now the problem is solved.

Jenna moves the remaining pieces to the girl circle. Now the problem is solved.

Here are some other problems we will be working this week:

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For more ideas for math with your preschooler, please click here.

Learning to add with a math balance

A math balance is a fun way for kids to develop number sense.

A math balance is a fun way for kids to develop number sense.

My daughter Jenna is 4, and is in the middle of completing Right Start Level A.

Here’s some fun we had this morning “playing math”, using our math balance.  This isn’t from a specific Level A lesson; it’s just the two of us making things up. But this will give you the idea of how you can use a math balance to help kids develop number sense and computation skills.

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First, your child puts weights on the numbers you are adding.

First, your child puts weights on the numbers she is adding.

Next, your child counts on fingers to get her answer.

Next, your child counts on fingers to get her answer.

Finally, check your answer on the balance.

Finally, she checks her answer on the balance.

Fun, right? Our balance is from our Right Start kit, but you can also purchase a Number Balance on Amazon.

Preschool Math Game

“Make Me Dinner”

That’s what I’m calling this latest math game my daughter Jenna(3.5) and I have been playing lately.  This game is deceptively simple. It works on loads of math skills, through play.

Learning Objectives

  • quantity visualization
  • counting
  • matching

Materials

  • Divided plates
  • counters (we use square inch tiles)

Directions

  1. Mom makes the dinner on one plate
  2. The kid copies it.
  3. 3 rounds is plenty.  Don’t push it!

P.S.  If you’re interested in more math ideas for preschoolers, please click here.

Marshmallow Math

Yes, this looks disgusting.

My three year old daughter Jenna felt like she was missing out on the fun after her brother Bruce got to build atoms with marshmallows this weekend.  So I got out the food dye and made up an activity for the preschool set.

It would have been a lot easier to used colored marshmallows to begin with, but we didn’t have any.  The up side of the DIY version is that blue marshmallows look gross and nobody wanted to eat them.

Think of the calories saved!

The first thing we did was build the number 5 using blue and white marshmallows.  This is very similar to what we do with Right Start Level A and the abacus.

We also built shapes.  Jenna is learning about triangles, squares and rhombuses.

Hands-on, fun, easy and meaningful.  That’s how I like to do math with three year olds.

Stuffed Animal Math

How many times does your preschooler drag out her stuffed animals?  Probably a lot!

Here’s how I turned doggy-time into math-time.

First I brought out the craft sticks. 

There are five doggies and two random Beanie Babies.  So we laid out tally sticks for how many animals there were: 7.

Then we showed the number 7 with our hands.

We also made the number 7 on the abacus.

Then my daughter brought out all of the animal friends!

Jenna doesn’t understand the number 15 yet, but she does understand 5 and 5 and 5.

So that’s how many stuffed animals she has.  5 and 5 and 5.

“Let’s Play Math” review

If you are familiar with my blog than you already know that I am passionate about teaching math from a Constructivist perspective.

Teaching math from a Constructivist perspective means enabling children to develop their own meaningful strategies for solving problems, instead of just blindly teaching traditional algorithms. It also means giving children time, space, and materials to explore mathematical concepts and create their own understanding.

Let’s Play Math: How Homeschooling Families Can Learn Math Together, and Enjoy It! (Homeschool Math Manuals) by Denise Gaskins, is a beautiful book that explains the “why” and “how” of teaching math from a Constructivist perspective.

I loved it!

This book is well researched, well annotated, and includes loads of activities that you can try with kids K-12 at home. While reading the book, I found myself remembering a lot of things I had forgotten from my teacher-training in Constructivist math.

The only weird thing, was that the author never actually uses the word “Constructivist”.

I’m still not exactly sure why that is. But clearly, I’m a former public school teacher bringing my own public school jargon with me. The author, Denise Gakins, approaches the topic from the world of homeschooling. So there you go…

The other comments I have about the book Let’s Play Math are not criticisms, they are only observations.

This is more of me and my public school background chiming in.

There were many instances where the author mentions public schools and textbooks teaching math from a very traditional “drill and kill” sort of way. This is definitely not trueand true, depending on the school and district.

Dale Seymour Math Investigations for example, is a solid, Constructivist program used by many public schools. It gets horribly bashed by the homeschooling mother of a certain blog I will not name, which is really unfair. You cannot judge the whole Investigations program by looking at the homework workbook. The real learning in Investigations happens in the classroom, on the floor, with kids playing and exploring math.

My last school, did not use textbooks at all.

Conversely, Saxon Math is a homeschooling program that is the total opposite of Constructivism. Drill, kill, “carry the one” etc. I could just as easily be prejudiced and suspect that many homeschooling moms don’t understand how Constructivist math is being taught in public school classrooms, and that they have a knee-jerk-back-to-basics reaction that leads them to Saxon.

My other comment is about algorithms, and when to teach them. I’m a proponent of not teaching traditional algorithms until fourth or fifth grade, once a child has a multitude of other stratgeties for solving problems.

My experience is that when you teach a child an algorithm too early, they will cling to the algorithm like a life raft, and thinking will stop.

It has been tricky to not let my own public school son learn algorithms until fourth grade math, but I’ve been able to accomplish this through a whole lot of Afterschooling.

I’m not sure if Denise Gaskins would agree with my approach about when to teach algorithms or not. I suspect she would, but I’m not sure. I think it would have been beneficial to slip an entire chapter on algorithms, somewhere in the middle of her book.

Lastly, I have to say that there were so many parts of this book that I highlighted that I really gave my Kindle a workout!

There is a whole section that I’m going to come back to this summer, to keep my kids busy. But was especially useful to me at this moment, were the talking points for helping kids solve problems on their own. Yes, I at one point learned all of talking points, but I really needed the refresher.

My son’s school does Continental Mathematics League, and those problems are really hard. I’m going to print up all of the talking points and post them in our kitchen so that my husband and I will have a list of questions to prompt our son’s thinking. Here are some examples from the book:

  • What do I want?
  • What can I do?
  • Does it make sense?
  • Can you draw a picture?
  • Can you act the problem out?
  • Can you make a chart?
  • Do you see a pattern?
  • Can you try the problem with smaller numbers?
  • Can you work backwards?
  • Can you try something and see if it works?

It doesn’t matter what grade your child is at, those questions are good places to start.

That’s just a teeny, tiny sample of all of insights Let’s Play Math has to offer.

I can’t wait until the author publishes Let’s Play Algebra!

“Let’s Play Math”

If you are familiar with my blog than you already know that I am passionate about teaching math from a Constructivist perspective.

Teaching math from a Constructivist perspective means enabling children to develop their own meaningful strategies for solving problems, instead of just blindly teaching traditional algorithms. It also means giving children time, space, and materials to explore mathematical concepts and create their own understanding.

Let’s Play Math: How Homeschooling Families Can Learn Math Together, and Enjoy It! (Homeschool Math Manuals) by Denise Gaskins, is a beautiful book that explains the “why” and “how” of teaching math from a Constructivist perspective. It is well researched, well annotated, and includes loads of activities that you can try with kids K-12 at home. While reading the book, I found myself remembering a lot of things I had forgotten from my teacher-training in Constructivist math.

The only weird thing, was that the author never actually uses the word “Constructivist”.

I’m still not exactly sure why that is. But clearly, I’m a former public school teacher bringing my own public school jargon with me. The author, Denise Gakins, approaches the topic from the world of homeschooling. So there you go…

The other comments I have about the book Let’s Play Math are not criticisms, they are only observations.

This is more of me and my public school background chiming in.

There were many instances where the author mentions public schools and textbooks teaching math from a very traditional “drill and kill” sort of way. This is definitely not trueand true, depending on the school and district.

Dale Seymour Math Investigations for example, is a solid, Constructivist program used by many public schools. It gets horribly bashed by the homeschooling mother of a certain blog I will not name, which is really unfair. You cannot judge the whole Investigations program by looking at the homework workbook. The real learning in Investigations happens in the classroom, on the floor, with kids playing and exploring math.

My last school, did not use textbooks at all.

Conversely, Saxon Math is a homeschooling program that is the total opposite of Constructivism. Drill, kill, “carry the one” etc. I could just as easily be prejudiced and suspect that many homeschooling moms don’t understand how Constructivist math is being taught in public school classrooms, and that they have a knee-jerk-back-to-basics reaction that leads them to Saxon.

My other comment is about algorithms, and when to teach them. I’m a proponent of not teaching traditional algorithms until fourth or fifth grade, once a child has a multitude of other stratgeties for solving problems.

My experience is that when you teach a child an algorithm too early, they will cling to the algorithm like a life raft, and thinking will stop.

It has been tricky to not let my own public school son learn algorithms until fourth grade math, but I’ve been able to accomplish this through a whole lot of Afterschooling.

I’m not sure if Denise Gaskins would agree with my approach about when to teach algorithms or not. I suspect she would, but I’m not sure. I think it would have been beneficial to slip an entire chapter on algorithms, somewhere in the middle of her book.

Lastly, I have to say that there were so many parts of this book that I highlighted that I really gave my Kindle a workout!

There is a whole section that I’m going to come back to this summer, to keep my kids busy. But was especially useful to me at this moment, were the talking points for helping kids solve problems on their own. Yes, I at one point learned all of talking points, but I really needed the refresher.

My son’s school does Continental Mathematics League, and those problems are really hard. I’m going to print up all of the talking points and post them in our kitchen so that my husband and I will have a list of questions to prompt our son’s thinking. Here are some examples from the book:

  • What do I want?
  • What can I do?
  • Does it make sense?
  • Can you draw a picture?
  • Can you act the problem out?
  • Can you make a chart?
  • Do you see a pattern?
  • Can you try the problem with smaller numbers?
  • Can you work backwards?
  • Can you try something and see if it works?

It doesn’t matter what grade your child is at, those questions are good places to start.

That’s just a teeny, tiny sample of all of insights Let’s Play Math has to offer. I can’t wait until the author publishes Let’s Play Algebra.

XtraMath

Tip of the Week to share with parents

Have you heard of XtraMath?

XtraMath.org is a free online math program that assesses, monitors, quizzes and drills your child on his math facts.  It takes just five minutes a day.

XtraMath also sends you weekly updates about your child’s progress. All you need is a computer hooked up to the Internet.

Once again, that website is: https://www.xtramath.org/

Math for Preschoolers

In the above picture you see my daughter Jenna learning the greater than/less than symbol, at age two. She thought she was playing “Hungry Guy”, but really she was learning a first and second grade skill.

I believe that children as young as two and three can do real math. The trick is to teach them mathematical concepts in a way that makes sense to them.

Play-based math will get results!

(Click here for some free activities to try.)

Jenna is three and a half now, and I felt like she was ready for something more formal. So three weeks ago, we begin using Right Start Mathematics Level A. I have no affiliation whatsoever with Right Start. I’m just a die-hard Joan Cotter fan.

We are on lesson 7 now, and I am thrilled. All of the activities are easy to set up, play-based, and conceptually very deep.

Jenna does between 5 -10 minutes of math every day, but only if she wants to.

Here is a sample of what we have done so far:

We made triangles and quadrilaterals out of craft sticks.

We learned about comparison words like long and longest.

We did ordering work of longest to shortest. (This is the before picture.)

We began to explore the abacus.

This is exciting. This is fun. This is easy.

The teacher in me wishes every child in America was benefiting from Dr. Joan Cotter’s wisdom. The mom in me wishes I had know about Right Start when Bruce was this age!