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
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