Tuesday, May 24, 2011 Fraction Division - third example

Dear Alexander Morning Math Families:


For this example, if I laid out three of the fraction “half” pie pieces, then say, “I want to share three halves of a pizza it with four friends – what do I do?” the children respond, “cut each half in four equal pieces.”

So we took out four little green “skittles” (shaped to resemble people, with little round heads) then found the four fraction pie pieces that together equaled each of the halves. Altogether, we “exchanged” the three halves for 12 smaller pieces.

The Montessori pie pieces have the value printed on them, and the child can assemble them into a whole to “check” that (for example) 8 of the eighths pieces will equal one whole.

We shared the pieces equally with each of the four skittles, and found that each skittle (person) got 3/8 of a whole pie.

So I invited the students to LOOK at the materials as they worked with them, and consider how to write statements that are OBSERVATIONS of what they see, then to copy my diagram.

When a child learns to DRAW the picture of the materials, we see that some “process” the example more deeply than if they did not.

They actually traced the fraction pieces, OR simply drew the picture as a copy of my own, and labeled the diagram.

I also wrote the number sentence that was true:  3/2 ÷ 4 = 3/8.

Here’s the part I did NOT explain to the children:

In school, “years ago”, we adults (as students) might have been taught to “set it up and solve it” like this, using the algorithm:

Step one:

3    ÷      4   = ?

2

Step two:  “invert the divisor and multiply across”

3    x    1   =    3

2          4        8

When we carry out step two, in fact we are saying, “if we take one fourth of three halves, we get three eighths (of a whole).

However, the FIRST number sentence, stating the division, is the correct description of the operation in the word problem.

The SECOND number sentence simply shows the relationship of the reciprocal, same as the “facts family” for a multiplication and division fact.

The discovery that this is true, led to mathematicians using the “step two” process to solve such fraction division process.

This method is still taught today, but how many students can understand and explain the concept?

That’s all for now!

Morning Math – Fraction Division (May 24th) Second Example

Dear Alexander Morning Math Families:


For this example, if I laid out two of the fraction pie pieces of ½, then say, “I want to share two halves of a pizza it with six friends – what do I do?” the children respond, “cut each half in three equal pieces.”

So we took out six little green “skittles” (shaped to resemble people, with little round heads) then found the three fraction pie pieces that together equal each of the halves.

We shared the pieces equally with each of the six skittles, and found that each skittle (person) got 1/6 of a whole pie.

So I invited the students to LOOK at the materials as they worked with them, and consider how to write statements that are OBSERVATIONS of what they see, then to copy my diagram.

When a child learns to DRAW the picture of the materials, we see that some “process” the example more deeply than if they did not.

They actually traced the fraction pieces, OR simply drew the picture as a copy of my own, and labeled the diagram.

I also wrote the number sentence that was true:  2/2 ÷ 6 = 1/6.

Morning Math – Fraction Division (May 24th)

Dear Alexander Morning Math Families:


 Today we started with short group lesson on division of fractions. Today we used the Montessori “round”, “circle” or “pie” shaped fraction material.

A smaller group of students stayed after the lesson to work with materials and “diagram” the lesson. Others worked independently.

When we “diagram” a lesson, it means that we draw a picture and label it, to help reinforce the concept we studied.

The lesson started by taking out the materials to demonstrate simple story problems: sharing pieces of pizza with small groups of friends.

I emphasized again to the children to that when we write number sentences, it can describe a “situation”, or be a statement of some thing we OBSERVE.

For example, if I lay out the fraction pie piece of ½, then say, “I want to share it with three friends – what do I do?” the children respond, “cut it in three equal pieces.”

So we took out three little green “skittles” (shaped to resemble people, with little round heads) then found the three fraction pie pieces that together equal ½.

We shared them equally with the skittles, and found that each skittle (person) got 1/6 of a whole pie.

So I invited the students to LOOK at the materials as they worked with them, and consider how to write statements that are OBSERVATIONS of what they see, then to copy my diagram.

When a child learns to DRAW the picture of the materials, we see that some “process” the example more deeply than if they did not.

They actually traced the fraction pieces, OR simply drew the picture as a copy of my own, and labeled the diagram.

I also wrote the number sentence that was true:  1/2 ÷ 3 = 1/6.

Please see the following posts with the other two examples – we created three examples of fraction division in all.

They did a great job!

Please note that these pictures may not be perfectly proportioned or to scale…

These are the drawings I made on pieces of paper, which the children copied.

The plan here is to reinforce CONCEPT. Many adults and children understand basic fractions concepts, BUT as we explore advanced fractions work, such as fraction multiplication, many adults and children stop understanding the concepts and simply learn the rules of the algorithm, like (for division)  “invert and multiply across”…

…but then we have to remember when to apply which rule, when what is best is to learn the concept that resulted in the rule.

Please, never get frustrated with the child for not “getting” or “remembering” a concept… it takes TIME, PRACTICE, and mental, developmental GROWTH to fully understand concepts. I didn’t really “get” some of these concepts until I had completed my Montessori training and taught the material for a number of years!

I plan to continue to add this reminder at the end of the blog.

Please note:

The separate blog for general school information and updates is now:




www.alexandermontessorisunblog.blogspot



The morning math blog page is:

http://alexandermontessorimorningmathsunblog.blogspot.com/

I suggest you bookmark these addresses for ease of access.

We have placed restrictions on them so that for the moment they would not be searchable via Google, for example.


Please don't expect a "mandatory" daily posting at either site. I will try to post at least weekly, and to inform families by Alertnow e-mail when I make the "weekly" post.

Thanks!
James McGhee II, Headmaster

May 23rd bonus post - Delano's largest fraction diagram (edited)

Notice: Above left he started with 5 x 10 = 50 (2 wholes).
                                               5     5     25
Then (upper right) he expanded the drawing to be 8/5 "over" and 10/5 "up", equaling 80/5, which we reduced to 3 and 1/5.


In this work, children practice multiplication facts as they develop an understanding of fractions.


Well. that's all for today!

Morning Math - Fractions (May 23rd)

Dear Alexander Morning Math Families:


A short post today:

We started with short group lesson on fractions. This reviewed and then expanded on Friday’s lesson. Most of the students stayed after the lesson to work with materials and “diagram” the lesson. Others worked independently

When we “diagram” a lesson, it means that we draw a picture and label it, to help reinforce the concept we studied.

The lesson started by reviewing the definition of numerator and denominator.

I also wanted the children to consider that when we write number sentences, it can describe a “situation”, or be a statement of some thing we OBSERVE.

For example, if I lay out the fraction squares that represent 3/3 + 3/3 , and on the other side of an equal sign I lay out the fraction squares that represent 16/8…

Then I state that 3/3 + 3/3 = 16/8, it would be TRUE, and that is what I would observe with my EYES, but without the picture before me, the EXPECTED answer is 6/3, and it would be true, too.

The amounts are EQUAL, but the “expectation” is that I should ADD 3/3 + 3/3.

So I invited the students to LOOK at the materials as they work with them, and consider how to write statements that are OBSERVATIONS of what they see, like a scientist observing a creature in the wild.

The children that stayed in my lesson each started with a different fraction square.

They actually traced the fraction pieces, drew and labeled the diagrams “over”.

Then they each chose a different way of measuring the fraction “up”, by drawing horizontal lines and making the whole into smaller rectangles of equal parts.

As they did so, I wrote the dimensions on the board for them to copy for their unique fraction.

I also wrote the number sentence that was true: the “over” ‘times” the “up” equaled a whole with a certain number of pieces.

I also reminded them that this is HOW the rule for “multiplying across” was discovered… by describing pictures of fractions!

They did a great job!

Please note that these pictures may not be perfectly proportioned or to scale…

These are the drawings I made on the white board, which the children copied.

Take special note of Delano’s largest picture, which I added to the NEXT post: he chose to make a picture LARGER than one whole (8/5 x 10/5 = 80/25), then he and I worked to “reduce” the fraction into its “lowest terms” (3 and 1/5).

The plan here is to reinforce CONCEPT. Many adults and children understand basic fractions concepts, BUT as we explore advanced fractions work, such as fraction multiplication, many adults and children stop understanding the concepts and simply learn the rules of the algorithm, like “multiply across”…

…but then we have to remember when to apply which rule, when what is best is to learn the concept that resulted in the rule.

Please, never get frustrated with the child for not “getting” or “remembering” a concept… it takes TIME, PRACTICE, and mental, developmental GROWTH to fully understand concepts. I didn’t really “get” some of these concepts until I had completed my Montessori training and taught the material for a number of years!

I plan to continue to add this reminder at the end of the blog.

Please note:

The separate blog for general school information and updates is now:




www.alexandermontessorisunblog.blogspot



The morning math blog page is:

http://alexandermontessorimorningmathsunblog.blogspot.com/

I suggest you bookmark these addresses for ease of access.

We have placed restrictions on them so that for the moment they would not be searchable via Google, for example.


Please don't expect a "mandatory" daily posting at either site. I will try to post at least weekly, and to inform families by Alertnow e-mail when I make the "weekly" post.

Thanks!
James McGhee II, Headmaster

Morning Math - Fractions (May 20th)

Dear Alexander Morning Math Families:


 Today we started with short group lesson on fractions. Then most of the students worked independently while a smaller group stayed with me to “diagram” the lesson.

I have created a pdf file that shows details of the lesson. I am having difficulty uploading to the blog (probably due to file size) but I worked hard on it, and it's worth seeing, so please e-mail me at jrm2@alexandermontessori.com and I'll send you the jpg file.

Below you will find the text, but the pictures don't "show" as they do in the jpg file... and the pictures help you understand how your child sees the lesson, with the Montessori materials. So I strongly recommend you request the jpg file.

Thanks!

Morning Math - Fractions (May 20th)

Dear Alexander Morning Math Families:


Today we started with short group lesson on fractions. Then most of the students worked independently while a smaller group stayed with me to “diagram” the lesson.

When we “diagram” a lesson, it means that we draw a picture and label it, to help reinforce the concept we studied.

The lesson started by reviewing how fractions can look in “square” form.

We often see fractions represented by cutting a circle into halves or quarters. In the Montessori classroom we have materials that show fractions represented as circles, but also triangles, rectangles, and squares.

Here is an example of what these materials look like:

I took group photos of the students holding the fraction pieces, and I will post them when I obtain permission from the families of the students.

We emphasized how we SAY the fractions and how we WRITE the fractions in words and in numbers.

We emphasize that in each case, the size of the whole is the same, BUT the NAME of the fraction piece come from the number of EQUAL parts in which the whole happens to be divided.

“four fourths”      4

                           4

“three thirds”       3

                           3

Then we emphasized that the TOP number is called the “numerator”. “Numerator” has the same three letters as the words “numero” and “number”.

I explain that as a “study and learning tip” we can help our memory by connecting related ideas.

So the “numerator” tells us the number of fraction pieces we have.

Then we emphasized that the BOTTOM number is called the “denominator”. “Denominator” has the same three letters as the word “nom”, which means “name” in French.

I show again that as a “study and learning tip” we can help our memory by connecting related ideas.

I point out the “D” in denominator is the same letter as the “D” in “down”, and the denominator is “DOWN below the line.”

So the “denominator” tells us the NAME of fraction pieces we have.

The NAME of the fraction piece will come from the number of EQUAL parts in which the whole happens to be divided.

         4   four      

         4   fourths

I then explained that just like we can measure the dimensions, “how far over” and “how far up” a rectangle is shaped, we can do the same with a fraction picture.

3

3   →

4

4   →

I also showed that using a clear plastic “overlay”, we can make a fraction picture that shows more pieces, and represents fraction multiplication.

In the picture below, we have diagrammed twelve twelfths as made by     3    x    4    =    12

        3          4          12

     

We then showed a similar picture, but this time with

4    x   3     =    12

4         3           12

→         ↑

                    

 

 

                                                 

So note here that the 1/12 piece is the same SIZE, but it simply turned or “rotated” – one is wider but shorter, the other is taller but more narrow.

The children that stayed in my lesson actually traced the fraction pieces, drew and labeled the diagrams.

They did a great job!

Please note that these pictures may not be perfectly proportioned or to scale in these pictures, BUT the materials DO represent good proportions and scale when the children and I work with them.

The plan here is to reinforce CONCEPT. Many adults and children understand basic fractions concepts, BUT as we explore advanced fractions work, such as fraction multiplication, many adults and children stop understanding the concepts and simply learn the rules of the algorithm, like “multiply across”…

…but then we have to remember when to apply which rule, when what is best is to learn the concept that resulted in the rule.

Please, never get frustrated with the child for not “getting” or “remembering” a concept… it takes TIME, PRACTICE, and mental, developmental GROWTH to fully understand concepts. I didn’t really “get” some of these concepts until I had completed my Montessori training and taught the material for a number of years!

I plan to continue to add this reminder at the end of the blog.

Please note:

The separate blog for general school information and updates is now:




www.alexandermontessorisunblog.blogspot



The morning math blog page is:

http://alexandermontessorimorningmathsunblog.blogspot.com/

I suggest you bookmark these addresses for ease of access.

We have placed restrictions on them so that for the moment they would not be searchable via Google, for example.


Please don't expect a "mandatory" daily posting at either site. I will try to post at least weekly, and to inform families by Alertnow e-mail when I make the "weekly" post.

Thanks!
James McGhee II, Headmaster

Latest Morning Math post - May 19, 2011

Dear Alexander Morning Math Families:


Dr. Bill Lukes, Ed.D. was the substitute instructor for me today. He is my “go to” teacher when I have another commitment in the morning. Thanks, Dr. Lukes!

Dr. Lukes, his wife Mrs. Robbie Lukes, and Mrs. Lessie Fleissfrescher all provide ESOL and supplementary educational services to Alexander School students.

Please note:

The separate blog for general school information and updates is now:




http://alexandermontessorisunblog.blogspot.com

The address "where you are now" for morning math info, is:

http://alexandermontessorimorningmathsunblog.blogspot.com


I suggest you bookmark these addresses for ease of access.

We have placed restrictions on them so that for the moment they would not be searchable via Google, for example.


I have transfered all previous Morning Math posts to this blog - where you are now.




Please don't expect a "mandatory" daily posting at either site. I will try to post at least weekly, and to inform families by Alertnow e-mail when I make the "weekly" post.


Thanks!
James McGhee II, Headmaster

May 18, 2011 Review of "Go To" Lessons

11-5-18_blog_morning math.doc    

Dear Alexander Elementary Morning Math Families:

This morning we started with a review of the “Go To” lessons.

I have told the children that a “Go To” lesson is a lesson they can choose to do, if:

• they are unsure of what to do

• they are short on time and want to practice something to make good use of time

• they do not have their work plan or folder to help start them on their next activity.

This morning I reviewed lessons A (Fact Families), B (Fact Lists), and a “version” of lesson D (Building Cubes with Small Plastic Units) which we call “prism building”, because we build not only cubes, but various regular three-dimensional rectangular prisms.

Below you will find a summary of each kind of activity.

These can also be done at home to reinforce skills and concepts. HOWEVER, please be POSITIVE if and when encouraging a child to practice. “It builds your skills, it builds your character – THAT’s what practice does”, not “you have to do this JUST BECAUSE.”

“GO TO” LESSONS

The four “Go To” Lessons taught so far are:

A, Fact Families:

B, Fact Lists

C, Addition, Subtraction, Multiplication, and Division with the Stamp Game

D, Building Cubes (and othercrectangular prisms) with Small Plastic Units and Counting Bars

A, Fact Families:

The student chooses any fact, and writes the four associated facts.

The principles we have taught are:

1, “In every fact, four facts are hiding”

2,         “The OPPOSITE of Addition is Subtraction”

“THE OPPOSITE of Multiplication is Division”

3, There is a PATTERN for Facts Families. The OPPOSITE of a fact is the “partner fact”

We have not introduced the “letters substituted for numbers” as you se below, but they are provided for your information:

A + B = C       C - B = A

Then we simply reverse the addends

B + A = C      C – A = B

So a child may write a “fact family” in horizontal or vertical notation.

Example - Horizontal notation:

7 + 6 = 13                  6 + 7 = 13             13 – 6 = 7           13 – 7 = 6

Example - Vertical notation:    

  

	6 + 7  13		13 -   7     6
	7 + 6  13		13 -   6     7

 

Once a child writes the pattern on a paper or 3 x 5 card, PRACTICING involves READING it aloud OVER AND OVER – at least six TIMES – yes, this is rote learning, and repetition DOES aid memory!

THEN the child tries to repeat the facts family aloud without looking at the paper or card.

Another method is that the child writes the first fact (whichever it is) – for example:

	13 -   6     7

 

And then reads the fact “13 minus 6 equals 7”, then repeats the other related facts out loud:  “13 minus 7 is six, 6 plus 7 is 13, 7 plus six is 13.”

Then we can try an oral quiz without looking at the fact:

“13 minus 6 is?”  “7”   6” plus 7 is?”   “13”

Remember, some students have an AUDITORY MEMORY strength, some have a VISUAL MEMORY strength. Others may have a SPATIAL  / “kinesthetic” (movement related) strength.

Repeating aloud may help foor all, but ESPECIALLY for those with AUDITORY MEMORY strength. Still, the VISUAL image may aid the memory of this child.

Likewise, other students may have a VISUAL MEMORY strength. So SEEING the facts goes top this child’s strength. Still, HEARING the facts as the child repeats it MAY aid the memory of this child.

Other students with a SPATIAL  / “kinesthetic” (movement related) strength may benefit from manipulating counters and counting bars as they practice.

Every person has areas of natural strength, and other areas that CAN BE stringer with practice. So, we build confidence and skill by focusing on the has areas of natural strength.

THEN, we build MORE confidence and CHARACTER by working on areas that are NOT as strong, helping the child realize, “When I put forth the effort, I CAN DO IT. Error is my friend and teacher, when I have the appropriate attitude and I am willing to learn from my mistakes. The greatest discoveries were often made by people who made mistakes but kept trying until they succeeded.”

“Good, better, best – I never let it rest, ‘til my good is better and my better’s best!”

4, There is a PATTERN for Facts Families. The OPPOSITE of a fact is the “partner fact”

A x B = C       C ÷ B = A

Then we simply reverse the factors

B x A = C      C ÷ A = B

	6 + 7  13			13 -   7     6
	7 + 6  13		13 -   6     7

 

B, Fact Lists:

The principle in Fact Lists is that the child can write or copy a list of facts, noticing the pattern in the list. For example:

9 + 1 = 10

9 + 2 = 11

9 + 3 = 12

9 + 4 = 13

9 + 5 = 14

9 + 6 = 15

9 + 7 = 16

9 + 8 = 17

9 + 9 = 18

9 + 10 = 19

If the child does not notice the pattern, we can lead her to note it: “Can you see the end number on the answer? It is ONE LESS than the number we add to nine, because NINE is ONE LESS than TEN.”

Then we invite the child to write the OPPOSITE of each fact, next to that fact:

9 + 1 = 10                  10 – 1 = 9

9 + 2 = 11                  11 – 2 = 9

9 + 3 = 12                  12 – 3 = 9

9 + 4 = 13                  13 – 4 = 9

9 + 5 = 14                  14 – 5 = 9

9 + 6 = 15                  15 – 6 = 9

9 + 7 = 16                  16 – 7 = 9

9 + 8 = 17                  17 – 8 = 9

9 + 9 = 18                  18 - 9 = 9

9 + 10 = 19                19 – 10 = 9

Then we invite the child to read the fact list “across”, from left to write, so each fact and it’s opposite are named:

“9 + 1 = 10                 10 – 1 = 9”

“9 + 2 = 11                 11 – 2 = 9”… etc.

Then the child can try to COVER the OPPOSITE fact and NAME it without looking, then she uncovers the fact to see if she did so correctly.

In writing fact lists for multiplication and division, the same process is followed. However, to expedite the process, we can give the child a Multiplication Chart and allow him/her to copy it, THEN write the OPPOSITE division fact alongside each multiplication fact:

9 x 1 = 9                       9  ÷ 1 = 9

9 x 2 = 18                  18 ÷ 2 = 9

9 x 3 = 27                  27 ÷ 3 = 9   etc…

Then the child reads the list as described for the addition List, above.

Then the child can try to COVER the OPPOSITE fact and NAME it without looking, then she uncovers the fact to see if she did so correctly.

C, Addition, Subtraction, Multiplication, and Division with the Montessori Stamp Game (this we can only do at school, UNLESS you have similar material at home)

For Addition and Subtraction: Students can make up their own problems of 4 digits added to or subtracted from 4 digits, to practice regrouping (or problems without regrouping.)

For Multiplication and Division: Students can make up their own problems of 4 digits multiplied or divided by ONE digit, to practice regrouping (or problems without regrouping.)

When the student has completed the operation and recorded it, the student MUST check the operation by inputting the result into the OPPOSITE operation –

CHECKING Addition by Subtraction, or Subtraction by Addition

CHECKING Multiplication by Division, or Division by Multiplication

This may be done with a calculator but must ALSO be recorded on the student’s paper by hand using pencil, indication the regrouping and showing proper place value.

D, Building Cubes and prisms with Small Plastic Units (this we can only do at school, UNLESS you have similar material at home)

This lesson is more challenging and many of the newer of the students don’t get the point of it yet… If they build a cube or prism, some of them are not yet understanding how to write the dimensions or how to calculate the total.

This lesson has to be reviewed before we encourage students to do it as  “Go To” lesson.

I only invite the newer students to do this lesson under my supervision. Many of the “veteran” students can practice the entire activity properly with little or no coaching.

However, if they need the help, I TELL them the cube or prism to build, or ASK them “What cube or prism are you building?”

To be “expert”, they need to be able to say ALL of the following:

“It will have a BASE of (for example) FOUR and it will be FOUR units tall.”

“It will be 4 over, 4 up, and 4 back.”

“It will be by 4 by 4 by 4.”

OR

“It will be “over” FOUR (four units wide)

It will be THREE units tall.”

It will be FIVE units “back”

“It will be 4 over, 3 up, and 5 back.”

“It will be by 4 by 4 by 4.”

Then we have the student WRITE (or copy after we write) all three statements BEFORE S/HE starts building.

Then after the prism is built, we “deconstruct” it, group the bars from largest to smallest (like a similar Montessori Math exercise with bead bars), then we collect the products, then add them.

I will try to upload an video later that shows thye process. Meanwhile, you can visit the class to see how it is done.

Well, that’s all for today! Thanks for visiting the blog!

James McGhee II