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