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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

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.


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.

Math Without Worksheets

One of the best things about teaching math from a Constructivist perspective is the emphasis on metacognition, or “thinking about thinking”. In a good Constructivist classroom the teacher should know, and be constantly assessing, the thinking strategies her students are using to solve problems. One of the ways I accomplished this when I was a teacher was through small group white-board activities.  I would group three to five students according to ability level, and let them create and solve word problems on the white-board for me, while I reordered their thoughts and strategies.

From a Constructivist perspective, listening to how children solve problems is where the true art in teaching comes. You need to know when to ask probing questions, when to lead them on to more efficient ideas, and when to pick apart what they are saying. The idea behind Constructivism is not to let kids wildly flounder, but to gently encourage them towards efficiency and accuracy.

My son Bruce(6.5) attends a school where the district uses Houghton Mifflin Math Expressions, which I like, but would not describe as a Constructivist program. Although there are occasionally times built into the curriculum where Bruce is asked to write out his own math problem as part of his homework, or explain his thinking in words and pictures, there doesn’t seem to be the same heavy focus on metacognition that there was in my own classroom.

Last night after dinner, I decided to do an experiment. How would Bruce handle a third grade Constructivist classroom? How would he do if he was in one of my white-board groups, from long ago? It turns out, not so well, at least not without more practice. And let me tell you, after realizing this type of activity was a struggle for him, we will be working on it at least once a week from now on!

Here is our first example. I asked Bruce to tell me a subtraction story problem about the number 263. That was the first bit of difficulty as you can see!  Then I asked him to solve the problem with numbers on the left side of the paper, and to keep his work neat, and organized within the box. Second bit of difficulty! I also wanted him to write down his final answer and circle it, but I forgot to articulate this very well, so that was “my bad”. Lastly, I asked Bruce to explain his thinking to me while I reordered his strategies. This is when he really started to flounder.

Okay, let’s try again! If you look at this next example, you can see a slight improvement. The story follows the word problem, his number work stays to the left, and he circled his final answer…but his explanation? On paper it looks okay, but I really had to tease this out of him. It did not come clearly or easily even with me doing the writing for him!

In a Constructivist classroom, by the end of fourth grade I would expect a learner to be able to see a number problem, write a story to go with it, and then write out their proof of how to solve it both in numbers and words. I would expect to see clear thinking, organized work, and neatness. None of that, is as easy as it might sound.

Sometimes, these types of Constructivist activities make certain parents turn vinaceous with annoyance and anger that comes from misunderstanding. I would never expect a child to have to write about every problem they solved. Nor do I think it is fair to hold children back mathematically because they might struggle with reading, writing or language. But doing proof work like this once a week in a small group setting is really helpful for everyone. Teachers get insight into how children are thinking, kids learn from the strategies of each other, and mathematical communication skills get practiced.

As an at-home activity, this has the added bonus of being completely free. I am going to continue working on this type of proof work with Bruce in an Afterschooling setting. I like Houghton Mifflin and I am extremely happy with Bruce’s teachers, but I want him to be successful at work like this too.

Staying Sane Over Christmas Break

Do you want to know what I’ve been waking up to every morning of vacation?  My six year old Bruce hovering and inch from my face saying “Wake up Mom; it’s time to do spelling.”  The first morning of vacation I didn’t move fast enough for him, and he said, “You’re asking for it—Freeze Out!” and yanked the covers off of me.  I couldn’t really object because that’s how I often resort to getting him up and ready for school each morning.  🙂

Two days into my Afterschooling Over Christmas Break project, he had read through eight books and I needed to add an extra poster board of options.  He’s also done four Hands On Equations lessons, finished third grade in Dreambox Math, and as result, has earned a lot of time playing Lego Ninjago on the computer.

There has also been a lot of piano playing over the past few days, since Bruce suddenly decided he wanted to learn how.  I’ll probably write more about this later because I have a lot to say about gifted children and intensity.  Suffice to say for now, yesterday I was basically chained in the living room trying to keep Jenna occupied while Bruce (entirely of his own accord)  spent about six hours at the piano learning 23 songs from his Primer Level Piano Book.  I’ve never seen anything like it. When I would try to get him to take a break from learning he would become irate.  So instead of waking up to my six year old asking me for a spelling lesson this morning, I woke up to piano playing instead.

Today was my husband’s first day of vacation, and as soon as we had breakfast I grabbed my canvas grocery bags, said “See ya later sucker!” and drove off in his new car!  (Okay, maybe I didn’t really say that, but there was some evil laughter involved.)  I finally got the chance to go to the teacher store by myself and look through all of the homeschool math curriculums I’ve been curious about.

Granted, twenty minutes of examination really can’t tell you everything you need to know about a program, but it seemed to me that the only similarity between Saxon and Math Expressions was the quantity of work they have children do.  Philosophically, they are quite different even though both programs are published by Houghton Mifflin.  Singapore Math Standard Edition seemed a lot more similar to Math Expressions, in terms of how they fall somewhere in the middle of the Back to Basics vs. Constructivist spectrum.  (Please correct me if I’m wrong about this!)  From what I could tell, both programs teach multiple strategies along with traditional borrowing and carrying methods. Where they differ, is that Singapore employs a lot more curriculum compaction, whereas Math Expressions has kids do page after page after page of work.  I went ahead and purchased a Singapore 4a textbook just because I wanted to examine it further, and see what Bruce thought about it.  Maybe it will keep him busy…


Third Grade Math — Transferring Answers

Bruce(6) came home from school yesterday super excited about his first day of third grade math.  It’s the first time he has said “School was great!” so far.  Mathematically, the Houghton Mifflin Math Expressions Volume 1 book is in the slightly easy to perfect level range for him.  But oh no!  It is a textbook and so he has to transfer his answers into a spiral notebook.

Uhg!  I am feeling like such an idiot right now.  Bruce has been doing all sorts of math up the wazoo this summer, but it never occurred to me to teach him about transferring his answers.  He did this a little bit with Life of Fred Fractions, but that was many months ago.  Since LOF was for fun, I didn’t push him to make things neat and tidy either.  I really should have known better and foreseen that transferring answers was coming down the pipeline because I use to teach third grade for four years.  Two of those years were with textbooks, but the other two we used math menus, with no transferring required.  Teaching Bruce to transfer answers just didn’t occur to me.

Bruce was really confused by having to transfer his answers, and had actually written in the book at school before he realized he wasn’t supposed to.  Eraser to the rescue!  I hope he gets the hang of this spiral notebook thing quickly without being discouraged, because the third grade math is so perfect for him.  I don’t want his six-year-old fine motor skills to drag down his mathematical progress either.  There is a huge dichotomy there in what his brain can do and what his hand can write.  I need to teach him how to grid out his answers on the page, and neatly write page numbers at the top ASAP.

For any of you homeschooling families who might be reading this, take note!  If your kid can cruise through a Singapore or Miquon book without any problem that’s great.  But could that same child transfer all of those answers in an organized way to a spiral notebook?

Horizons Math

(Don’t buy this!)

When Bruce turned four and was still going to Montessori, I decided to begin formal math instruction with him at home. He was (and is) so energetic and inquisitive, that I thought his behavior would improve if some of his energy was channeled into academic pursuits.  I looked around blindly for a curriculum to get him started on, and discovered the Horizons math program through Alpha Omega Publications.  There are two workbooks in each curriculum year, for a total of 180 lessons.  There is also a teachers guide to go with it, which I did not purchase, and I suspect a lot of people do not buy either.  (That may have been a big mistake!)  Bruce took the online placement test which scored him as being ready for the first grade.

The first grade curriculum was pretty good, and Bruce sailed through it in about six months.  It is a spiraling curriculum, so there is a lot of coming back at topics previously covered for review and practice.  I also liked that the workbook pages were very colorful, and had pictures.  The workbooks seemed to be very equation heavy, with not a lot of word problems, which is okay for first grade because that way reading skills do not hamper math progression.  ( I have since found out that the Teacher’s Guide includes a lot more word problems.)

The first grade workbook has a lot of drill-and-kill.  Often times Bruce would get tired of actually writing out the numbers (since he was only four), so I’d be the secretary and he would solve problems in his head and then tell me what to write.  Near the end of each book the lessons were getting too easy and repetitive, so we just crossed off big sections of them and skipped to the next page.

When Bruce started Kindergarten I bought him the second grade program thinking we would continue to chug along.  (He was also doing our school district’s second grade curriculum, Houghton Mifflin Math Expressions.)   I purchased the Horizons math, reading and spelling kits.  This was all a big mistake.  By the second grade book it was clear that the Horizons program was designed for back-to-basics home school families, which certainly doesn’t describe my teaching approach!

The second grade math program has a strong focus on traditional algorithms such as borrowing and carrying, to the point that pages are set up with little “carry the one” signs.  They have kids doing 4 digit addition and subtraction by the second or third month of second grade, which is only possible if you are mindlessly solving equations with algorithms but are not doing the deeper work of creating true number sense.  By contrast, Bruce has finished the Hougton Mifflin Math Expressions second grade program, is now half-way through with Right Start Level C, and is only now capable of solving 4 digit equations in his head, meaning he really understands how to do it.  Teaching kids to crank out algorithms is not teaching higher order mathematical thinking.

The Horizons Second Grade reader was truly bizarre.  It was an adaptation of Robinson Crusoe written at a second grade level and broken into 90 chapters, interspersed with excerpts from a second grade reader from the 1800s.  After about ten chapters, Bruce was bored out of his mind and refused to read any further.  It was the exact opposite of the high interest reading material necessary to inspire young children into becoming self motivated readers.

To be fair, I didn’t have the Horizons Math Teachers Addition, which the website clearly states in an integral part of the program.  But based on the workbooks, which have 2nd graders cranking out 4 digit subtraction with regrouping problems using traditional algorithms, Horizons did not seem to be a program I felt comfortable using for Bruce.  We have switched to Right Start, and have been much happier.

The Horizons reading program seemed to be from the standpoint of “If it was good enough for my great-great-great-grandpa, then it’s good enough for my son.”  Um… no in fact, it is not good enough  at all.  I just wish it hadn’t taken me almost $300 to figure that out.