illustrated math

I think it was the word illustrate that got Sprite. She sincerely thought this was what the question wanted her to do. And I guess on one level, she actually did follow the directions.

I asked her why a math test would want her to draw faces on the numbers. She agreed, “Yeah, I thought it was strange, but I did what they asked me to do. I illustrated it.”

Yet another reason to take these tests every now and then.

In our math lessons from livingmath.net, we’re up to the contributions of India and Arabia in the Middle ages, focusing on the number zero and positive and negative numbers.

To make integers more coherent for Sprite, I hunted for some illustrations.  Of course, there is the number line, but here are some real life parallels to positive and negative numbers which we started with:

• thermometer
• timeline
• money — assets and debts
• floors in a building,  above and below the ground floor

Here’s the timeline parallel. I titled this photo Jesus is Zero. Isn’t that mathematically perfect? The printable came from Montessori for Everyone.

Zero is an essential number both in place value and on the number line. I love the part in the poem “Zero” in Math Talk that says, “I’m the origin on the number line.” What an amazing parallel. Numbers, both positive and negative are valued in their relationship to zero. Time itself is measured in relation to the life of Christ. So in that way, Christ is Zero — the center, the origin, the base and foundation.

There is no way we’d get these insights from Singapore Math textbook. Lovin’ living math!

For the thermometer application, we used page 13 from the FREE ebook –Fast Ideas for Busy Teachers, Grade 4.

We read the first three chapters of The Journey of Al & Gebra to the Land of Algebra. This book is a bit hokey. Okay, it’s really hokey. The story is terribly strained to get in the math concepts. BUT, it’s making Sprite laugh at the silliness in it. And the concepts and vocabulary (integers, absolute value) are getting through. So even though I wouldn’t say it qualifies as a true living book, I’m leaving it in our curriculum for now. (The reading for this title is scheduled in our livingmath.net lesson plans.)

We used this Less Than Zero lesson plan that includes a printable game board pictured above plus a link to a very fun online math game that Sprite enjoyed. (Sprite just loves it when I say, “Your math activity is an online game today.”)

Sprite made a notebooking page about the zero in which she told about its history, its importance, and some of its unique characteristics.

Education World has another good game idea for learning about the number line. I adapted the game into a life-sized version.

I asked Sprite to create a life-sized number line so that she could serve as the playing piece. That got her attention! She was filled with anticipation about the game.

I pulled out two numbers from my prepared stack and had Sprite physically move out the problem. The first number was her starting point. Then she had to add the next number.

I saved each combination and had Sprite write down the math problems and answers after she had walked them out across the numberline.

To help her grasp subtracting integers, I told a story which she had to act out on the numberline.

“You had $7 when we went out. You found a totebag that cost $13 and asked me to loan you the extra money to cover it. Now how much money do you have? (Owe $6 / -6.) Then you did the dishes, and I subtracted -$2. How much do you have now?”

And so on with Sprite going into debt and working back out of it in various humorous situations. The sillier the better. It hides the fact that you’re actually learning math.

Here is a notebooking page to go along with this study –Positive and Negative Numbers Notebooking Page (82). Later this week, I’ll have Sprite record what she learned on it.

To be clear, the activities I’ve listed here were not done in on day or even one week. Actually, this study covered about two weeks worth of math instruction although we were working on other concepts as well (parallel and perpendicular lines, to be exact). I have found that with Sprite and math, Charlotte Mason’s advice is perfect — small doses with repetition work much more effectively than long lessons.

Related posts:

1. Living Math with Factors, Multiples, and Primes
2. Living Math with Angles
3. Living Math is not Quick or Neat

Taking a break from our intense multiplication study, we’ve moved on to a new topic — angles.  With living math, I do use our Singapore textbook. I browse through it to find another topic to study. (grin) Actually, I do have Sprite do a workbook exercise now and then.

Angles was one such topic that we hadn’t touched on this year, and it seemed easy enough to learn in a hands-on approach. (This is the face she wanted on the blog.)

Above and below you can see our most important tools:

1. a home made protractor in 360°
2. an angle maker
3. a store bought protractor

I find that in planning living math activities, I often have to reverse engineer the workbook exercises. I look at the problems and ask myself, “How can I convert this skill or concept into a hands-on or real problem solving activity?” And that’s where these activities came from.

First of all, I knew that the entire concept of degrees was important. Sprite needs to understand what an angle is and that it can be big or little and that the size is measured in degrees.

[The technical terms such as acute and obtuse are not vital at this point, so I didn't use them.  Vocabulary can come later after the concepts are cemented. However, we had already learned about right angles (90°) with Pythagoras, so it was easy to talk about smaller than and larger than a right angle.]

So I had Sprite make a 360° protractor by simply marking degrees on a large circle. (Got to sneak in some x9 facts too — 90°, 180°, 270°, and 360°.)

The angle maker is simply two strips of cardstock connected at one end with a paper fastener. Using these two tools together, we arranged the angle maker in various positions and measured the resulting angle, approximately, with our protractor. We did this for a couple of lessons so that Sprite would get a feel for the sizes of angles — a 20° angle is tiny and a 280° angle is big. A 100 ° angle is a tad larger than a right angle, and so on.

Then come the bits of paper. (It seems we can’t do a math activity without loads of cut up paper.) I used some scrap paper to make circles. I made sure to mark the center with a pen so Sprite could cut out angled sections. Some circles were cut in two pieces, others in three.

Then she measured. First I asked her to estimate visually the degree of angle. Her skill at estimating got better and better. Then I had her use the homemade protractor and then the official one.

I had her label each angle with its measurement. And then we verified the rule that each circle has 360° by adding up the total of the angles. It worked each time!  Of course. But letting a child discover rules rather than dictating them is a halmark of living math.

It all made its way onto a notebooking page.

Next approach — triangles.

I gave Sprite some triangles and asked her what she thought we’d find when we measured the angles and totaled them. She wasn’t sure, but she felt certain it wouldn’t be 360° like the circles.

Oops… she ended up with several triangles whose angles totaled over 180°. She somehow sensed that 181° and 182° must be wrong and that the triangles that were measured at 180° must be correct. It actually was easy enough to go back and remeasure and find her small errors. It ended up that some of our “right angles” were not actually 90°.

We corrected the measurements on the backsides, and put a few sample triangles onto a notebooking page.

Just to be clear, these angle lessons were spread over about a two week period. Interspersed in with these hands-on activities were some workbook exercises and at the end I used this BBC worksheet as an overall wrap up.

Related posts:

1. Living Math is not Quick or Neat
2. Living Math with Factors, Multiples, and Primes
3. Living Math Update – Fractions

We’re still drilling multiplication facts. (It seems this is going to take a while.) I’m constantly saying out of the blue, “Count by eights!” or “What’s 9 x 7?” It’s as much work for me to remember to review it as it is for Sprite to learn the facts.

This is a book I bought from Amazon –Math Games to Master Basic Skills: Multiplication & Division.  Sure, I could find or make my own bingo charts and times tables cards, but it’s so easy to have it all in a clean format, ready to copy and use.  (I wrote a review for the book at Amazon, so I won’t go into that here.)

This latest lesson was X7 and X9 facts.

• I gave Sprite all the products and had her divide them into two stacks — multiples of 7 and multiples of 9.
• Then I had her order the products and skip count, reading them outloud. She came up with her own variation — reading in a low voice for the smaller numbers, and progressively raising her voice as the numbers got larger. It was quite silly, but anything that makes math fun is okay with me.
• I asked her why 63 could be found in both rows. She nailed it!
• Then I randomly pointed at products and had her tell me the math problem. We discussed the whole 9 x 8 is the same as 8 x 9 issue.
• To top it all off, we played TWO board games from the NC Math site. I laminated them so we could use Vis-a-vis pens to mark our spots instead of using paper markers. Sprite always loves using those pens, and I like to use lots of variety to keep math fun.

More of our quest to master the multiplication facts! This time we’re working on times nine.

(It seems to me that being able to sprawl on the floor to work on a project is a huge benefit of homeschooling! Do you use the floor a lot like we do?)

Basically my strategy for x9 was to help Sprite see the pattern in the products by actually placing the digits in their correct spots. I’ve shown her this before, but somehow it didn’t seem to help her solve the x9 facts. Maybe she needs to discover it on her own. That’s far superior to my telling her.

So I set up the situation for her to discover the patterns in the x9 facts. I cut out ten green squares for the problems, and then twenty smaller cards with 2 each of the numbers 0-9. (Sprite did the writing,  of course.)

It amazes me that in all my years of schooling, I don’t remember any teacher telling me about the trick for the x9 facts. And I wasn’t sharp enough to see it myself, I guess.

After the hands-on activity, she did describe some patterns! And she seemed to “own” them better than when I’d simply pointed them out. I could see her face light up with discovery and with wonder at the perfect symmetry of the line of products. Initially she’d been skeptical that the stack of numbers was all she needed to solve the x9 facts.

Then I gave her an option. She could either put the math facts into a minibook of her choosing as she’d done with the x8 facts, or she could write down the x9 pattern on a notebooking page.

I was pleasantly surprised that she chose to write rather than the rote activity. It was quite difficult to write the rule, and we had to work together to get it written clearly. But she did it. And day two, she still remembered her hack for solving the x9 facts.

Related posts:

1. Learning x8 Facts

I’ve got a math confession to make. My ten year old, fifth grade daughter does not know her multiplication facts. And that may be partly my own fault. I never pressed her to memorize them. I kept hoping that it would click and she would just know them. But there’s no clicking yet. In fact, even some of the more “easy” math facts still give her problems.

I thought it was like so many other things — potty training and  reading, for example. The child has got to be ready or the teaching just won’t “take.” I still think that, and I’m not stressed about the times tables, but her lack of fluency is having a negative impact on our math studies. Sprite gets so bogged down in 7×6 that she forgets the broader point of the word problem we’re working on.

When Sprite was tested in the spring, her computation (called numerical operations) score was a full two grades lower than her math reasoning. I felt good about it, actually, since her computation score was right on grade level. I know better than to hold back on the advanced math reasoning just because arithmetic skills are lagging. We just do both side by side, stretching her mind with tough math ideas as we drill the basics.

So with all that said, one of my goals for math right now is the multiplication facts. She’s got to master these! (Prayers, ideas, links, and advice are all welcome!) I’ve got to come up with some sort of system with lots and lots of review, something like Spelling Power for multiplication.

This week we worked on x8 and x9, some of the tougher ones. We played a game from the wonderful, free materials at Public Schools of NC Mathematics Instructional Resources. There’s a x8 game in this section –

Resources for Mathematics: Grade Four (2003 SCS), Week by Week Essentials, Weeks 1-4 .

(It still amazes me how playing a game makes Sprite excited to do math. She never says, “Aww.. this is just a sneaky way to get me to practice my x8 facts!”)

Also, I gave her a x8 math facts chart for reference and told her she had to make her own minibook reference. Sprite chose lots of small matchbooks inside a single fold.

That day, she could call out all the x8 math facts. Victory! Two days later, most of it was forgotten. Sigh… It’s going to take a lot more than a single lesson to master these facts.

Related posts:

1. Learning x9 Facts

Our latest arithmetic topic has been factors, multiples, and primes. We used our Singapore math textbook and workbook as a supplement to these activities which formed the meat of the lessons. It wasn’t a very linear study. We skipped around somewhat, revisiting topics from different angles to really cement the understanding.

Paper and Pencil

Identifying multiple patterns I  printed eleven multiplication charts and Sprite labeled each one with a number from 2-12 at the top. Then she highlighted the multiples of that number all over the grid. It was very interesting to see the patterns that emerged. Multiple Matrix is an online version of this activity. Our page is actually from the Living Math book, but you can use any multiplication chart just as easily.

multiples Venn diagram

Venn Diagrams for multiples

I love graphic organizers mainly because Sprite and I are so visual. This triple Venn diagram was a great critical thinking application of the arithmetic we were studying. It was one of the last activities we did.

primes

Prime Numbers

For primes, we used a hundred chart and  The Sieve of Eratosthenes to find all the primes (up to 100).

Printable Games

1. Factor ‘Em In: Exploring Factors and Multiples
   This PDF is an entire lesson plan about multiples. My favorite part is the last page — a game called Multiple Mania. All you need is the printed page, 10 markers per person and a die.
2. 97 Prime board game (or the same board game in a PDF file)
3. Factor Finding Game (pictured below) Another simple game that requires only the printed board game, two dice, and 10 markers per person.

factor finding game

Here are two more that I found but didn’t actually use.

Factor Game (PDF)
A printable factors game plus extensive lesson plan for teaching primes, multiples, and factors.
Factor Game & Lesson Plan
Click on the PDF lesson link for complete directions, game board, and charts for analyzing how to play the game.

Since I published the blog post, I bought a reproducible game book from Amazon called 25 Super Cool Math Board Games (Grades 3-6). It has a few games that deal with factors and multiples as well as a variety of other basic math skills game. It’s a worthy addition to a math resource bookshelf.

Audio Helps

factors & multiples notebooking page

Factors and Multiples Song This PDF has games and worksheets for learning factors and multiples. The best part is the song on the last page sung to the tune of “I’ve Been Working on the Railroad.”

I find that songs are a great way for Sprite to learn, and this song really helprd her remember the differences between factors and multiples. Plus it’s just fun to be silly and sing about math! The song was really helpful in making this notebooking page about factors and multiples.

Online Games

Related posts:

1. Living Math with Angles
2. Living Math is not Quick or Neat
3. Living Math with Positive and Negative Numbers

At the HSS forum a mom explained how she took the math manipulatives down from the bedroom closet shelf. She sorted the goodies into baskets, shelves and drawers, all accessible to the children. Wonderful! The fun math tools aren’t helping anyone on a bedroom shelf.

Sprite and her FREE manipulatives

That started me thinking about the hands-on things we have. Are they really FREE? Does Sprite KNOW they are free to be used at any time? I thought the answer was an unequivocal yes. But then I remember just a few days ago she asked me where the balance scale was. I was surprised that she didn’t know it was on the shelf with the math manipulatives. Granted, that shelf is a bit crowded. Maybe too crowded to invite exploration. I mean, if you’re afraid that things will fall and crash to the floor, it inhibits your desire to poke around.

When I gave her the balance scale, I was quick to put in one of those mom phrases like, “Be careful with it!” Now, why did I say that? What is it about our educational resources that makes us treat them like fine china? We ONLY pull them out a few days each school year. Then we painstakingly put them away once the lesson is done. Why not leave them out all day for exploratory play? Why aren’t they being used more than just a few days a year?

To be honest, I fear that the pieces will get lost. And then I won’t have a perfect tool for homeschool. How ridiculous is that? I am trying to keep it perfect so it can sit on the shelf for the next 20 minute lesson that specifically addresses that theme (which may be months from now). I’ve got a nice collection of math manipulatives, but what good are they really doing us, sitting on the shelf?

Set the math manipulatives FREE! I declare this coming school year Free the Manipulatives Year. Set out the pattern blocks, the dice set, the balance scale, and the counting bears. Make sure little hands can reach them; make sure little minds know they are welcome to do so. Blur the boundaries between math and play. ( I realize that moms of toddlers probably can’t set their manipulatives free quite yet. But be looking forward to that day, and when it comes — liberate the tools!)

I reorganized our shelf that holds the science and math tools. Doesn’t it look better? Neat is so much more inviting, I think, than cluttered. (How do shelves got so cluttered so quickly?) Here are the before and after photos.

And I came up with a very thrifty way to store our graduated cylinders. I used a tissue box plus a lot of paper towel tubes and a couple of toothpaste boxes.  (We keep things like that on hand for crafty projects.) I hot-glued the tubes to each other and to the tissue box so that it’s all quite sturdy. The cylinders go into the tubes upside down. Now they are neat, protected from breaking, AND (most importantly) accessible for chemistry experimentation. The science tools are FREE too!

Graduated Cylinder Holder from Recycled Materials

Platonic polyhedra sounds so much more sophisticated than 3D shapes, doesn’t it? Actually, in our math history lessons, we found that the 5 now called “Platonic” solids were actually identified by the Pythagoreans (Pythagoras and his students) before Plato’s name became attached to them. Of course, we were not content to see them in a book, we had to construct them. That is only logical to me; otherwise how can you truly understand these figures?

We used this wonderful resource for printable templates:  Platonic Solids Mobile. I printed the nets (as flattened 3D shapes are called)  onto cardstock, then cut and scored them. Scoring makes a huge difference for a mathematical craft like this; it enables you to fold precisely. Just place a ruler along the line to score, and press with the rounded edge of a paperclip. It took patience and white glue to get them to hold together.

Before stringing the mobile, Sprite had to fill out this comparison chart (I created) for analyzing the features of the five figures.  She found it helpful to mark the faces, vertices (tips), and edges to keep track.

Sprite used the diagrams from page 2 of this book excerpt to create a notebooking page which we referred to quite a lot since the names of these figures don’t come too easily. But they are mighty fun to say!

1. icosahedron
2. tetrahedron
3. hexahedron (also known as a cube)
4. dodecahedron
5. octahedron

Putting the mobile together was another mathematical experience. Sprite had to discover which way to shift the strings to balance the mobile.

When we found the perfect arrangement, we taped the strings to the straws.

And here it is, ready for the cats to attack. (That is what happened, you know.)

This whole project was a tangent that I took off on.  Our scheduled reading, the chapter on Pythagoras in Mathematicians are People, Too, is what inspired me. So this was actually a four day activity that was a footnote to our (supposed to be) two week long Pythagoras study. Yeesh!

Living math is so messy! Not just in glue and squares of paper, either. I like things to be linear, but I’m finding that living math is not a neat path from A to B to C and so on. Instead it  jumps ahead to QRS and then circles back to pick up DE and F. So I’m adjusting to this new style and trying to enjoy where it takes us.

We do have an outline, but when we find a mathematical concept we want to veer off track towards, we go ahead. It’s all math, and it doesn’t bother me anymore that the tangents we choose are not in the textbook or not “grade appropriate.”

Later when we study Euler, another mathematician, we’ll revisit these shapes with this chart — Euler’s Formula with Platonic Solids.

I’ve written a special review of Living Math (from Livingmath.net) over at The Curriculum Choice. Instead of reposting it here, I’ll simply direct you over there. (Click the logo.)

We’ve been using Livingmath.net’s curriculum for about 12 weeks now. It’s been a worthwhile change although it has certainly necessitated some adjustments on my part. Go read the rest….

Related posts:

1. Opal Wheeler Biographies Curriculum Review
2. Living Math with Angles
3. Living Math is not Quick or Neat