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

People Problems and Word Problems {3-6}

Word problems are my exception to the not buying math curriculum rule. I do usually buy a book of word problems for the grade appropriate for the child. Evan-Moor makes a nice book with one word problem a day for about $17. I think word problems are so essential. Students may be skilled at additon and yet not understand in what situation that skill may be applied. They may have the technical ability to solve problems when the numbers are provided but be lost when asked to extract the same numbers from words. For their daily word problems, I sometimes ask if they can prove them, even if they get them right, so that I can see what method they use to get their answers.

source: Mathematics; a Way of Thinking, Bob Baratta-Lorton

Unstandardized Measurement (3-6)

I like to start out teaching measurement by helping them understand the kinds of problems that historically led to the development of standardized systems of measurement. Civilization felt the need for standard units of measurement because to be without such units was a constant source of frustration. Exposing them to these frustrations helps them to see the need for such standardized units of measure which can be satisfied later with problem solving.

People
I familiarize the boys with taking measurements by using lengths of string or yarn. This introduces them to the idea that some displays of measurements are more useful in answering questions that others. By letting them take control of how the measurements are displayed they learn some valuable concepts on their own, such as making sure they label and place them on a page.  I then let them come up with questions that this data might answer. They came up with:

• who has the largest or smallest of a particular measurement
• which measurements are larger than others, such as the neck was always larger than the wrist
• which measurements were the same?

I then ask them how they could display the data they collected to see the answers to these questions more easily.

Now that I am not always well enough to give assignments orally, I am writing them more in their journals for them to follow. Using only one body measurement; in this case, the wrist, I got him to get measurements from everyone in the family. This exercise showed James very quickly that it is hard to make sure that the measurements were consistent.  He developed a way of making sure that the wrist on each person was in the same place by placing the string between the wrist bone and the hand. It was the beginning of thinking of all the factors that must be considered if two people are to measure an object and get the same results.

"Students may find they know less about measuring at the end of this lesson than they thought they did at the start. The confusion they face now, however will eventually lead them to a fuller understanding of measurement."- Mathematics, A Way of Thinking 

Next, I had him take his own wrist measurement and measure his neck, his arm and his first finger. The purpose of this exercise is for him to find ratios of length that exist between the various parts of the body.  Most human bodies have similar proportions. Not every neck will be exactly two wrists around, but most do. James and Quentin will find the the ratios between their own body parts will be similar.

Objects

The next task is for him to use the body as a measuring tool to measure some object in the room.  The purpose of the assignment is to force him to examine the imprecise nature of their measuring devices.

"This exposes him to the kinds of problems that have historically lead to the development of standardized systems of measurement"- Mathematics, A Way of Thinking 

Using Unifix cubes to measure the string or yarn makes a nice transition before introducing standardized measurement.Use the Unifix cubes to measure your family's hand-spans, which then can be used for comparison and averaging.
What they have learned about unstandardized measurement will help them to more easily understand standardized measurement and the reasoning behind its development.

related posts:

• Comparing Body Measurements
• The Annunciation; Leonardo DaVinci and human proportions
• Tostada Cost Analysis

source:

•  Mathematics; a Way of Thinking, Bob Baratta-Lorton
• GEMS guide, Math On The Menu

Advanced Division {3-6}

With Beans, Cups and Bowls
We have worked with beans, cups and bowls as math manipulatives for a long while. We have used them to explore different base systems, place value, addition and subtraction, beginning multiplication, beginning division, and decimals. Now we use them for some advanced division problems. Sometimes I give them the problems and sometimes I will have them randomly make up problems using a die to roll out the numbers and sometimes I let them make up their own problems. Today I had him roll a 10-sided die to make up the problems. Here is the first one and how we worked through it using beans, cups and bowls. The first roll was for the beans, or ones column and he rolled a 6. The second roll was for the cups, or tens column, and he rolled a 0. The third roll was for the bowls, or hundreds column, and he rolled a 1. The fourth roll determines the number of groups he will be dividing the beans, cups and bowls into. In this case, he rolled a 4. We set up a little chart in his math notebook that holds this information. It looks like the above picture. He draws columns with four lines to them to represent the fact that we are dividing by four.

He sets up the board with the appropriate amount of beans; for this problem, 1 bowl (containing 10 cups with 10 beans in each cup), and 6 beans, equaling 106 beans in total.  For his own explorations, I give him ample time to make his own discoveries by trying to solve the problems in any way he wishes, but for my teaching sessions, I show him the method that I find is the easiest for me, which means starting with the bowls.

It is easy for him to see that the 1 bowl of 100 beans must be broken down into amounts of 10 in order to divide them, so he takes the cups out of the bowl and begins sorting them into four groups.

He notes at the top of the bowls column that 1 bowl cannot be divided without breaking it down by putting a zero above the bowls. He then breaks down the cups of ten into the four rows on the chart and sees that they break down into four groups, each containing two cups of ten and that he has two cups of ten left over, which he notes on the chart. It doesn't show it in this picture, but he also noted it  with a two above the tens column.

He then moves on to the remaining cups, or the tens. He cannot divide that by four, so he must take the  beans out of the cups and put them all in the ones column to be divided there.

He discovers that the two bean cups, once the beans have been taken out and the beans already in the beans column divides into four groups of six beans, with two beans remaining.

He writes the six above the ones column and puts the two for the two remaining beans as the top part of a fraction, with the bottom part as the divisor, or four, in this case, making 2/4. He didn't finish the re-drawing of the beans once they were put in the ones column because he didn't need to. He could see the answer without drawing it, so I did not insist on this for this problem. He may see the need for it in future problems, but he has to see the need for it himself for it to be useful to him. I also don't worry about simplifying the fraction at this point. That is to be done at another time.
Now that he has completed the process, he can begin making up his own problems and working them out, using the chart and board as much as he wants to.

Division with Chips

We started out our working with chips instead of beans by asking him to put five orange chips, four purple chips, three red chips. two blue chips and one green chip on the new trading board.
He has worked with this color scheme before and knows that each color is a multiple of ten of the chip to it's direct right.

We began by dividing this amount (54,321) by three. We began by starting at the ones/green column, making trades from the left, as needed. It wasn't too long before he discovered that this method did not work right.

So, we tried again, dividing the orange/ten-thousands column first into three groups. The leftover chips are converted to purples/thousands. This continues with purples, and all the other colors, moving towards the right. Finally the green/ones chips are divided, with any leftovers places to the right of the trading board, for the remainder fraction.

I also noted what he was doing on the board on the notebook page. I also noted what was happening on the board in the traditional form.

Now, he can roll dice to determine the number of chips to place in each column of the chip trading board, and the number of groups into which they are to be divided. He needs to learn how to draw a mini board on the notebook page with the appropriate number of rows beneath the chip trading board. He also can begin recording the problems in the traditional form, if he wishes, but is not required to at this time.

Long Division
There comes a point in which the problems become so large that the trading board becomes too cumbersome.
"...a time comes in both multiplication and division when the scope of a problem outstrips the practicality of using materials. In multiplication, this situation was alleviated by lattice multiplication. To supplement the students' capabilities in division  the next series of lessons presents an abstract system of dividing that serves the same purpose as the boxes in multiplication." -Bob Baratta-Lorton, Mathematics; a Way of Thinking

We began by his writing the numbers one through 10, putting a circle around each number. I then get him to go by five down the line, so by the 1, he puts 5 and by the 2, he puts 10 and so on.
So now we begin with a division problem, but I want it to be small enough so that we can check his problem with the chip trading boards. This should bridge the gap between them.
4321 divided by 5, and I write it in the traditional manner.
I get him to make a by 5's column as before and I get him to look for a number that could be subtracted from four. When he tells me there are none, I tell him to put a 0 over the 4.
What is the largest number in the fives column that we can take from 43?
40. Can I subtract 40 from 43?
Yes. What about the next number, 45?
No. Okay, we can't use that then. Let's use 40. The circled number next to 40 is what number?
8
Okay, so put the 8 above the 3 in the problem and put the 40 below the 43, and subtract.
Okay. We've used the four and the three. The next number we use is the two. We bring it down beside the three, like this.
Now we have 32. What is the largest number in the fives column we can take from 32?
We continue in this manner, writing the division problem in the usual way, and using the fives column to assist us.
When we get to the last number, which doesn't take any numbers from the fives column, we write it as a fractional remainder.

Advanced Division: Long Division Theory

Now, I get him to check his answer with the chip trading board, (just as I have done with his brother.)

I then give him more problems, with different column numbers to work with.

At some point, I ask him to teach me by telling me the steps, one-by-one as he does the problem.
At this point, he can create his own problems, either from whole cloth, or by rolling a die.
He also is asked to solve word problems...
We are having a party at our homeschool group, and we have been asked to bring peanuts to share. If there are 32 people who are coming to the party and the bag of peanuts we are bringing holds 125 peanuts, how many peanuts would each person get?...

source: Mathematics; a Way of Thinking, Bob Baratta-Lorton

related posts:

• Beginning Division
• Advanced Division: Long Division Theory

links:

• Math Monday Blog Hop at Love 2 Learn 2day

Advanced Multiplication (3-6)

Multiplication with Beans and Cups

For cups and beans, I begin with two cups and a stockpile of beans. I first tell them to start with putting 1 bean in each cup, and I begin recording while they count the beans. 2 (cups) x 1 (bean in each cup) = 2

Then I ask them to put 1 more bean in each cup and I record while they count.

2 x 2 = 4

We continue on like this for a few minutes.

Somewhere along the way, I just give them 1 instead of a set of two beans and ask them what do we do about this?At this point we talk about some numbers not coming out evenly, and then we add 1 more to make a set he can add to the cups, and continue on from there. We can add more beans or more cups as long as there is interest in this activity. In this way, young children can bridge the gap between addition and multiplication seamlessly.

Lattice Multiplication and Napier's Bones

As problems increase in size, the use of these materials becomes impractical. Traditionally an abstract system of long multiplication the distributive process has been taught.
"Research has shown, however, that the lattice system of multiplication allows students to compute multi-digit multiplication problems in significantly less time and with greater accuracy than is possible using the distributive method. "- A Comparison of Two Methods of Teaching Multidigit Multiplication [University of Tennessee, Frank George Hughes]

The lattice method came out of a calculating aid made by John Napier called "Napier's Rods."
"John Napier was a Scottish nobleman who loved mathematics. He invented logarithms, worked in spherical trigonometry and designed "Napier's rods," a mechanical calculating aid...These were an assortment of rods marked off with numbers. When these rods were arranged correctly, they could be used for multiplication and division...They were a sort of movable multiplication table -an early type of slide rule, which is what people used before pocket calculators. Because they were made of bone or ivory strips they were sometimes called "Napier's Bones."

-Mathematicians Are People, Too, Reimer and Reimer

To use them, you take out the rods that have at the top the first number you are working with.

For example if your problem is 298 x 7, you take out the 2 rod, the 9 rod and the 8 rod and lay them down on the table. You can see, if you go down to the 7th row of this set (because 7 is the second number you are working with), on the top of the slashes is 165. Write that down. These are your tens. Below the slashes are the numbers 436. Since these are your ones, you will write them under the 165, but you will indent out one column, to make the number above start in the tens column. Add these two numbers together, and you will get 2086, which is the answer to 298 x 7.


Let's try another problem; 31 x 24. This system is sometimes called the "Lattice System" because when you get to working with larger numbers, it becomes easier to make a grouping of boxes. You make a graph with the number of boxes across as there is in your first numeral; in this case two. You make the number of boxes down as you have numerals in your second numeral; in this case also two. So you have a graph with two boxes going across and two boxes going down for this problem.
Now put diagonal lines going from one corner of each box diagonally to the other corner. I extend my diagonal to make it clear where the answers go. The drawing of the boxes may seem complicated, but it is easy once you have done it a few times and kids find it easy to do on any blank piece of paper. Next, write the digits of your problem along the sides of your boxes. Now, get out your rods for the first two digits of your problem; in this case the 3 rod and the 1 rod. Go down the number of boxes according to the numerals along the side of your graph; in this case, 2 first. Copy down the boxes just as they are on the rods, the numerals above and below the slashes will correspond to the boxes and slashes you have in your graph; in this case 0/6 and 0/2. Continue this way with the next digit; in this case, 4. Now you have numerals all around your boxes. Ignore the numerals along the top and right sides now, and add only the numbers within the boxes on the diagonal. Starting at the bottom corner the one diagonal triangle box has the numeral 4, so I write 4 down below it. The next row of diagonal triangle boxes contain the numerals 2,0 and 2, which if you add them together equal 4, so I write 4 below them. The next diagonal row of numerals are 1,6 and 0, which equals 7, so I write 7 below it. The corner diagonal box contains 0, which I chose to just leave out since it won't affect the answer, but you could have your students write a 0 there just to be consistent and get in the habit of always writing down the numerals so as not to forget any by accident. It depends on how old they are and their understanding of the concepts.
The answer to this problem, reading the numerals from right to left is 744.

This method can be used for as large a problem as you want. Just make sure you draw the correct number of boxes according to the numerals that are in your problem. Here is the problem 123 x 12. We got out our 1, 2 and 3 rods and made our graph 3 boxes by 2 boxes and wrote the numerals around the top and right hand edges. We then copied what the first (1) and second (2) boxes of the bones had in them. We then added the numerals on a diagonal, getting the correct answer; 1,476.
You have to use a slightly different method of adding up the numbers if you get numbers that have carrying in them. For this problem, for example, 123 x 67, you will get the numerals from left to right, 7 1,14,11 and 7. If you get numerals over 9, you must carry them over into column addition. You write down the numeral(s) and then add the number of 0's after it that corresponds to the number of columns after it. Since 1 is in the first row, it gets no 0's after it. 14 is in the next row, and it has only 1 row after it, so you write down 14 with 1 0 behind it, or 140. The next row has 11 in it (pardon the odd looking numerals; my son made a mistake in his addition the first time and got 10, but then changed it to an 11) so you write 11 with two 0's behind it for the two columns, or 1100. The last row has a 7 in it and it has three 0's behind it for the three rows, or 7000. Add these numbers together and you will get 8,241; the correct answer. This is something that is complicated to explain but easy to use once you understand how it goes. All a child needs to be able to do is add two digit numbers.

 More about how it works here at Math is Good For You!

"The lattice method produces the same kind of understanding as the distributive method but is easier to teach, faster to use, and less prone to error. " -Mathematics...A Way of Thinking, Robert Baratta-Lorton

How to Make Napier's Bones

If you would like to make some Napier Rods, just get 9 wide craft sticks (like tongue depressors) and divide them into 9 fairly even sections. I just eyed mine; I did not measure them. If you are using these with children younger than 3rd or 4th grade, I would divide them into 10 sections and use the top section to put the number of the rod on the top. My Kindergartner can use these, but sometimes has difficulty reading the top number to identify which rod he is using. I ended up putting the number of the rod on the back for him. He chooses the rods by the numbers on the back and then turns them over to use them. My severely dyslexic son has trouble sometimes counting down the blocks correctly. Perhaps using different colors for the block divisions than for the numbers would help with this. (It would also be possible to make a thin column down one side to mark the rows with numbers.) Divide each of these sections with a diagonal line and copy the numbers above. The numbers are just the traditional multiplication tables. I used an Ultra-fine point Sharpie to write with, but it did bleed into the wood a little, making the numerals fuzzy. I am not sure if there would be a better writing instrument to use for this.

You can instead go to Mathwire and print and cut out these Napier's Bones. They can be used on their own or cut out and glued to large craft sticks.
"Students who have difficulty reasoning abstractly with numbers are frequently unable to grasp the numberical logic behind either the distributive or the lattice approach to long multiplication. Knowledge of why an abstract system of producing answers works is not as important as the knowledge that it does wok. For this reason, answers to the initial problems students work using a lattice method are checked against the materials, ususally chips on the chip trading boards."
-Mathematics...A Way of Thinking, Robert Baratta-Lorton

source: 

• Mathematics Their Way, Mary Baratta-Lorton