Math Curriculum, 2013-2014

Highhill Education has started a new Lesson-Planning Linkup and this week's topic is Math Curriculum. It took me some time to find a math curriculum that I was comfortable with, and with the price of most of them, it was an expensive trial and error process. I have finally found a system that works for all of my children, and I am going to keep with it as long as it continues to work, so this year's math curriculum looks very much like last years. We are just moving up a level for each of them in their understanding and the topics that will be covered.

• Math on the Level
• Mathematics, A Way of Thinking
• Teaching Textbooks: Algebra I

Quentin began third grade last year, and so he has graduated out of Math Their Way, (Pre-K-2nd) and he started Mathematics, A Way of Thinking, (3rd-6th) which is the same book James used, just at a different place in the book. Quentin was just starting the book, and James is finishing it up. Both books were written by a husband and wife team of math teachers, Bob and Mary Baratta-Lorton. Both books are very hands-on and sequential. 

James will begin the school year doing the last activities from Mathematics, A Way of Thinking on Negative Numbers, and then we will begin Teaching Textbook's Algebra I, taking it VERY slowly, adding in lots of hands-on applications from Pinterest. I will also use materials from Math on the Level.

Sam was involved in some of my trial and error process of finding the right math curriculum for us. He began Algebra with Videotext Algebra, which looked very good in the beginning (I might even use some of the beginning activities with James), but then somewhere along the way, it became very confusing. Even Steven and I became confused about where the curriculum was going. Sam's grades began to go down, understandably. So, with all the great reviews we have heard from my blog friends, we decided to try Teaching Textbooks. It has worked for us. We decided to have him start from the beginning, which put him a half to three-quarters of a year behind at the start of this past year. Between that and the various things that came up this year to interfere with our regular schooling, Sam is still working on Alegbra I and will be this next school year.

Alex has topped out on his understanding of math concepts. We have worked on addition, subtraction, and multiplication for about eleven years, and each time we sit down at the table, we are at the same place. He can add fine, he can subtract once I finally get him to understand what I want him to do. He cannot multiply. I have tried a variety of hands-on activities over the years, and some special needs curriculums, to no avail. When it comes to special needs children, sometimes it comes to the point that you have to accept them for who they are and their abilities for what they are. In terms of meeting diploma requirements, we will continue to practice these concepts, but I am not stressed or frustrated any more at how to unlock any more potential in this area. We will have fun instead. We will play games, make things together and in general enjoy the time together.

More discussion on the sequence and use of our math curriculum, What To Teach, and When.

We will also be adding in some Living Math, where appropriate. Here is an example of some of the living math books we will use in conjunction with our history and science studies

• Joy of Mathematics, T. Pappas
• Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians
• Senefer, A Young Genuis in Old Egypt, B. Lumpkin
• How to count like a Martian, Glory St. John
• The Warlord series, Virginia Pilegard
• The Librarian Who Measured the Earth, Kathryn Lasky
• Science in Ancient Mesopotamia, Carol Moss
• Science in Ancient Egypt, Geraldine Woods
• Science in Ancient China, George Beshore
• Science in Ancient Greece, Kathlyn Gay

Links:

• Highhill Education Lesson Planning Link-up- Writing Curriculum

Next post will be about our Science Curriculum.

Highhill's Lesson Planning Link-Up ScheduleJuly 11 - Writing
July 18 - Math
July 25 - Science
August 1 - History
August 8 - Music
August 15 - Art & Handicrafts
August 22 - Geography
August 29 - Foreign Language
September 5 - Reading
September 12 - Organization your Classroom/Schedule

Fun with Pascal's Triangle

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We had a lot of fun today playing with Pascal's Triangle, invented by Blaise Pascal, a French Mathematician in 1653. First I started out giving them the following math problem to solve.

Imagine you're buying an ice cream cone. If there are 5 flavors, how many possible combinations are there? How many ways of having no flavors? How many ways of having 1 flavor? How many ways of having 2 flavors? How many ways of having 3 flavors? How many ways of having 4 flavors? And, how many ways of having 5 flavors?

This was a good word problem for my 10-year old to solve. While he was working it out, I cut out squares of colored paper and glued them on another sheet of paper to form a pyramid or triangle. Once he finished solving the problem, I read to him some about Pascal and then I had him fill in all the outside squares on our triangle with the numeral 1. I then asked him to fill in the rest of the squares by adding together the numerals in the two squares that joined above each square. We only had seven rows, including the top 1, before we ran out of room. If I do this again, I might get a larger piece of paper to work from.

Now I told him to count down five rows, not counting the top 1, and look at the numbers that ran across this row. Did he see any similarities to the answers he had for the problem he had worked on earlier?

They, of course, matched.

We counted down five rows because the possible flavors we had to work with in the problem was five. Pascal's triangle can be used to find the answers to any of such type problems with ease.

I then asked him to add up the numbers in each row.

What pattern do you see?

He began saying the numbers out loud, "2, 4, 8...they double each time!"

I then read to him about something called a Galton board. The board is made with nails arranged like Pascal's triangle. Marbles are then poured through it. The probability of an marble ended up in a particular column is easy to work out by looking at the numbers in Pascal's triangle. The final pattern, too, is in the shape of a bell curve.

Galton board, photo from Wikipedia

There are so many fascinating patterns to be found in Pascal's Triangle. What patterns do you see?

Why not build one yourself and see?

The 12 Days of Christmas Math

How many of each type of gift did she receive from her true love? (How many Partridges in Pear Trees? How many Turtle Doves?)
Which gift(s) did she receive the most of?
How many gifts in total did she receive?
Bonus Questions: How many birds did she receive? How many people?

"If you read a child a math book" (with apologies to Laura Numeroff)

If you read a child a math book...(like Mummy Math)

he will want to make Platonic Solids models...

with toothpicks and gumdrops...

like tetrahedrons...

and octahedrons...

and that might make him a little confused...

but he'll keep working at it and he will get it...

 and then he might want make polyhedra models...

and that will remind him of one time we counted shape faces and he'll want to do it again...

and that will make him want to count the vertices and edges...

and that will lead the oldest child to remember when we did this before and that will lead to

Euler's Polyhedron Formula,
V-E+F=2...

and that will lead to counting in positive and negative numbers...and then he'll want you to read another.

Living math books with these geometric concepts:

• Mummy Math by Cindy Neuschwander (grades 1-4)
• Sir Cumference and the Sword in the Cone by Cindy Neuschwander (Grades 3-5)
• Mathematicians Are People, Too: Stories From the Lives of Great Mathematicians, Luetta Reimer (ages 8 and up)
• Leonhard Euler and the Bernoullis: Mathematicians from Basel, M. B. W. Tent (6th grade and up)

What is a polygon? 

The first time we visited the topic of polygons, we used some paper polygons. The we made a chart, counting the edges and vertices with a marker. The older students could plug in Euler's formula. While talking about it we discovered that the same formula can be applied to 2-dimensional figures with a constant of 1 instead of 2. Makes one think, huh? 

This time we used toothpicks and gumdrops, which we have done before. Toothpicks and gumdrops are good to work with to make regular polygon because the equal sized toothpicks make them automatically have all their sides the same length. This time when we counted the edges and vertices, they were easy to count because the edges are the sides, denoted by how many toothpicks are used and vertices are where two sides meet up, denoted by how many gumdrops are used. This time, because we wanted to look closely at the Platonic solids, they measured the angles with a protractor and (with a little adjusting) correctly determined that three of the five Platonic Solids are made out of equalateral triangles. I thought it was interesting that Tiffini at Child's Play, when she did this lesson, also brought the children's attention to Polyhedral Dice and their shapes, which made counting the sides even easier as that is how the dice are named. We use those dice all  the time for our math games.

In Circles

Today we looked at circles. I thought that starting with the most favorite circle they know might be a good idea, so we started our math on the trampoline.

Since a circle is defined as a set of all points in a plane that are the same distance from a given point, I thought the best place to start would be to find the center of this circle.

We took yarn and measured from one point on the trampoline, straight across to the other side, and then we curved the yarn back on itself, folding it in half, to find the center point. We marked this with a big dot. We understood that this wasn't an exactly precise measurement, but it gave us an idea of where the center point was.

Then, using the diagram at the top of this post, we made sections on our trampoline circle. We discussed each term, drew the lines and then labeled them. They remembered some of the terms from when we had read this book a few weeks ago.

 Even if they didn't quite understand some of the different terms at first, as we kept talking about them, asking questions and using them, they caught on.

"Just as when we folded the yarn in half and found the center we can draw a line from the centre to any point along the circle. This is called the radius.

Is there just one radius?

No, there are many radii. Picture it like the spokes of a bicycle wheel.

A radius is also a line segment – it has two endpoints.

You could label the endpoints with letters, but let's not this time so we won't get confused with too much written on the trampoline.

Does the length of the radius change at all? No – they are all the same length.
Now draw a line from one point of the circle to another point on the circle. This is called a chord.

What is the longest chord we can make? Through the center.What is it called? Diameter.

Not only is it the longest chord, what else does it do? It divides the circle into two equal parts called a semi-circle.

You can also see by the way we folded the yarn in half to make the diameter turn into the radius that the diameter of a circle is twice the length of it’s radius.

Next we covered perimeter and circumference.

We talked about how the path around any geometric shape is called a perimeter.

Then I had them walk around the edge of the trampoline and told them the name for the perimeter of a circle is the circumference. It is the distance around the circle. 

How about a tangent? Yes, it is a straight line or plane that touches a curve at a point but does not intersect it at that point.

And a secant? A straight line that intersects a curve at two or more points. Is it the same as a chord? Yes, a chord is the bit of a secant that lies within the circle.

I showed them how to show that the line goes on; by putting arrows on the ends.

Now, for some fun.

Simon says, "Lay on the radius."

Simon says, "Put your elbow on the diameter."

Simon says, "Hop on one foot to the secant."

Simon says, "Run around a half circle."

Simon says, "Make a tangent with your body."

Simon says, "Put your foot on a chord and one hand on the center."

Now that they got some of the wiggles out of them, we went inside and made circles on construction paper and cut them out. The little boys traced around canning lids while Sam used a compass.

Speaking of using a compass, I have found this lovely book, Circles by Mindel and Harry Sitomer, to be excellent at teaching kids how to use a compass. Not only does it show you how, but it also gives you fun exercises using the compass to make art. It is out of print, but you may be able to find it at your library, or a used copy through Amazon.

 Back to  our circles, I made a chart on the white board listing circumference, diameter, radius and ratio. Along the top I listed their names.

We used some more yarn to measure the circumference of each of the circles. We listed these measurements on the chart. We then measured the diameter of each of the circles and had them figure out and then confirm by measuring each of the circle's radius. I told them how to write the fraction of c/d for each circle and wrote this on the ratio row. Then Sam determined roughly what the fraction equaled and they quickly saw the pattern that the circumference was always a little more than three times longer than the diameter. It was then I gave them the name for this ratio, Pi and showed them its symbol. I told them that the exact ratio, had our measurements been precise would always be 3.146.

I read them of the story of Archimedes from Mathematicians are People Too.

We then took our circles and found the center by folding them first one way and then again the other way (a 90 degree angle) and putting a dot where both the folds met. I then had them cut out angled sections.

We had fun calling each of them Pacman. Then we measured them with a protractor and I wrote down what their measurements were. If they were way off, I helped them measure them correctly.

We then added the measurements of the two pieces together and they were thrilled to find out they always equaled 360 degrees. We then went on to cutting them into three or four pieces and finding out the degrees of those pieces, adding them and always getting 360 degrees. By this time they were getting pretty good at guessing the degrees of the angles.

And what better thing is there to practice measuring angles than this?
By the way, did you know that 3/14 is National Pi Day?
You could have a little fun with it then.

sources and inspiration: 

Other lessons like this are at Jimmie's Collage, where she has her daughter make her own protractor and angle maker, at A Pilgrim's Heart and at Child's Play, where I got the idea for using the trampoline to teach math.

Pythagoras

We all remember learning A squared plus B squared equals C squared and for many of us it was just one more meaningless fact to memorize. We never really saw it in action and we didn't learn who first made this discovery. I don't want my kids to just memorize to forget. I want them to really understand and know math facts in some sort of context. During our study of Ancient Greece, I wanted us to learn about Pythagoras. We can contribute many math discoveries to him, but we don't know a whole lot more about him as a person, so the book What's Your Angle, Pythagoras? A Math Adventure by Julie Ellis and Phyllis Hornung is a purely fictionalized story of the young Pythagoras. It is, however, a wonderful introduction to looking at number patterns and being able to apply them to real problems. It covers topics like right triangles and their properties as well as his famous theorem.

And, yes, I know that the sketch that Sam made has an extra row on his 5 squared section...he might still be working on attention to detail, but he certainly knows what the theorem means and can picture what it means in his head. I am satisfied.

Sam copied a page of the book in his math journal because it was so clear and visually appealing. As a group, we played with right triangles and Math-U-See blocks. We also played with knotted ropes and right triangles, also pictured in the book. 

Tangrams: The Seven Magic Shapes

Three Pigs, One Wolf and the Seven Magic Shapes by Grace Maccarone is a wonderful introduction for younger students to tangrams. The Tangram is an ancient Chinese puzzle in which one is to form a specific shape given only in outline or silhouette using all seven pieces, which may not overlap. In this book, the seven shapes make different characters or objects that further the plot. The tangrams in this book are not a single silhouette; the pieces are set slightly apart so that the young readers can manipulate the tans (the individual shapes that make up the set of tangrams) and copy the pictures. After copying several different pictures from the book, Quentin felt confidant enough to make his own pictures. These can be traced onto paper and given to other people to solve. It is a lot of fun for young children to see others working on a puzzle they have created.

resources:

• Another resource about tangrams can be found at Love2Learn2day.