Category Archives: Geometry

Inspiration to Innovation

DJD has begun his fourth grade innovation project.  He is designing light fixtures.  He began to codify his design language.

Color: The way light reflect off objects and is perceived by the human eye.

Warm Bronze and Textured Nickel

Shapes: Two or three-dimensional areas with a boundary. Shapes encourage us to have specific moods or emotions and emphasize specific areas of interest.

Equilateral triangle-based pyramids

Reflective: The ability for object to be projected back from a different angle.

Opaque: The ability of a medium to allow light to appear indirectly and less intensely.

Textured and Tinted

Tessellation: Tiling of a surface using one or more geometric shapes with no overlap and no gaps.

Mathematical, Interlocking, Patterns of patterns,
Dramatic, Historic, Repeatable, New yet familiar, Randomness

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Filed under DJD, Geometry

Angles work continues

BW is having a difficult time with the concept of angles and we have continued to work on angles rather extensively. (He is fascinated by the sides and their lengths and will create an obtuse angle with two sticks only to end up with an acute angle when he chooses the stick color he likes and it is a short stick.  He wants to insist he still has an obtuse angle.

The usual Casa angle works won’t keep him attentive until his eyes are trained to objectively look at the angles, so I’ve begun asking him to figure out the “laws” of closed objects and angles.  The other children keep wandering by.  Occasionally they’ll pull out some sticks to think through BW’s questions at their level.

I write three questions about angles for him to answer each day:

1.  Can you make me a triangle with two obtuse angles?

2. Can you make me a triangle with one right angle?

3.  Can you make me a triangle with two right angles?

When he comes back to report his findings, I ask more questions:

1. How many sides does it take to close a figure with all obtuse angles?

2.  Can an equilateral triangle have a right angle, too?  (Ohhh – all the angles in an equilateral triangle are what type.  Always?)

3.  So what happens if you make a closed figure using the right angles you discovered?  (Squares are part of a larger grouping of four sided objects.  Do you remember their name?  Quadrilaterals. Do all quadrilaterals have right angles?)

Those questions will lead to more questions to appear on his list tomorrow.

His “Laws of Angles” for yesterday were:

1.  All equilateral triangles have acute angles.

2. A pentagon is the smallest figure that can have all obtuse angles.

3. A square and a rectangle both have all right angles.


Filed under BW, Closed Figures, Geometry, MMcC, Students

Geometry Doodling

The kids asked that I post this video.  We did it one afternoon.  We will hit more on the series of infinates but meanwhile you need to try this.

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Filed under Art, Closed Figures, Geometry

Circumference to Area of a Circle – Montessori Style

Trying to explain something as cool as geometry done in the Montessori style is hard to do, but I’m diving in head first.

Prior to this lesson we had discussed the nature of a circle and reviewed its parts.

1.  We rediscussed the formula for finding the area of a decagon.  A=p x a/2

2.  We worked out the problem for the geometric cabinet’s decagon. We talked carefully about the perimeter.  (It was great that AR was observing the lesson; the older guys were better at explaining it carefully with a younger listener.)

3.  I handed the older guys the circle and said how will we measure its area?  The same way they said.  AV was the first to realize there is not apothem in the language governing the circle. JV made the transition to radius.  Then everyone thought it was good. BR and DW could tell by my face that this was going to get interesting. BR began looking around – on high alert. I handed LR the circle and asked him to find the perimeter.  JV picked up the ruler. Difficulty ensued.

4.  AR sidled up next to me and reminded me about the Sir Cumference book.  Wouldn’t a string work?  I said that it would but I had another way too.  I took a ruler and drew a line on the chalkboard.  I took the circle and made a mark with the chalk on the circle.  I drew a beginning line on the chalk line and began to roll the circle until it came around to the chalk line again; I marked this. How big is the perimeter of the circle?  BR snatched the ruler and measured.  This was perplexing because it was larger than they thought it should be.  So I did it again.  Same distance.  Then we discussed the word for perimeter is only for polygons.  Circles have their own language.  It is a circumference.

5.  I then said, “hmmmmm, I wonder something.”  I took the circle and laid it down at the staring point.  I marked the diameter of the circle.  I repeated myself three times.  But there was a little left over. We tried a different sized circle beginning with measuring the circumference.  Same little bit left over.

6.  I took a compass up and measured the little bit and then wondered home many time the compass measurement would fit into one of the diameters of the circle.  7 times.  Ahhhhh. so the left over it 1/7th of the diameter of the circle.  So the whole circumference is 3 and 1/7 diameters.  Ohhhhh.  About this point AV begins to put the pieces together.  He gets very still.  I make it an improper fraction 22/7.  He blurts out.  That’s pi.

7.  I divide it out and see that 1/7 is .142…..  The kids really were having a hard time believing that pi was not magically derived, but rather was an observation by the Greeks.  They named the universal thing peripharein; the first letter is Pi.  This is why we call it pi.

8.  Of course now we HAD to measure things and find their circumference.  So off we went.

BR and DW worked together.

That was all for that day.

The next day.

9.  We began again.  All we had found in our hunt for the area of a circle had been how to find the circumference.  We need to find the rest.

10.  We wrote down the decagon’s formula again to guide us.  Then we drew a circle and began to replace the terms for the decagon with circle language.  A = c x r/2.  JV mentioned that this was not the formula he had seen for the area of the circle.  I agreed.  But we all agreed that it was the right way of saying it.

11. HHHHmmmmmm.  So we decided that we could replace the “c” with the parts of the formula.  A= pi x d x r/2.  Hmmmm.

12.  Could we combine things now because diameter and radius were related?  Hmmm – yes.  D = 2r.  Ok our new formula says: A = pi x 2r x r/2.  After some math review here we realized that we couldn’t make it 3r.  I was a little disappointed that AV didn’t realize the 2’s could cancel each other out.

13.  I made that transition.  Now A = pi x r x r.  Ohhhh.  BR, whose been doing quite a bit of cubing work, saw the square immediately.  Now it says pi x r squared.

14.  Off they went to find their papers from the previous day and now find the area.

PS: There is another layer to the proof.  I did not do this proof.  If anyone has photos of the layout, I would appreciate seeing them.

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Filed under AR, AV, BR, Closed Figures, DW, Geometry, JV, LR

AR is into Montessori Math

AR is working through the checkerboard.  She didn’t begin checkerboard until the middle of this year and she is wanting to abstract in many ways too soon, however she works until she gets it right.  She sits down everyday to do dynamic multiplication (carrying)  on the checkerboard.

Here she is counting 5 taken 6 times.  It appears she has picked-up the 2 that she had carried from the previous step in the problem and is going to just add it in right there.  I’m not sure it works for me, but I’m not too twitchy about it.  I think I’ll observe it  and see if it messes her up in the longer term.

After this work she asked for a lesson on geometry.  I invited her to observe one I was doing with the older guys.  It was on circumference.  She made the connection with the Sir Cumference and the Dragon of Pi. She had read this before Christmas break when she was interested in three dimensional objects.

She still wanted more. So as the others worked out their new formula for finding the circumference of the circle, she and I began making the elementary equivalency works.  BW joined us as a product tester.  We’ve made all the square pieces.  Triangle equivalency to go.

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Filed under AR, Checkerboard, Closed Figures, Equivalency, Geometry, Mathematics

Math in the Montessori Mind

We are having technical difficulties with the camera. Bear with the blurry photos.

The upper kids have been hanging out in the world of geometry for quite some time.  They’ve been proofing various geometry proofs – from triangle equivalencies to Pythagorean theorems to the reasons for the area formulas (most recently).  So today we began shifting back into the realm of pure math.

We began cube root lessons and had a lovely Montessori moment.  We beat our heads into the wall with the layout of the visual formula.

Why are certain rectangular prisms on certain levels?

Name three things you notice about the layout.

What do we know about math rules for addition?  

What do we know about math rules for multiplication?

Eventually LR and AV helped out with some concepts and began touching the materials.  Even in this time when the ability to abstract is becoming easier, touching and manipulating the materials helped them understand some of the concepts.

They began to have moments of clarity that began to involve manipulating the math concepts.  What if we put this here? Is it the same?

What happens if it is “mathematically wrong” but “geometrically correct?”

How much volume do the black rectangular prisms take up.

Are these like the “wings” on the square rooting work?

At lunch the other moms wanted a bit of info,  Ms. JW needed to see.  After we began to discuss the problem at hand, we worked backwards to easier concepts and lighted on squaring and square rooting.

Ms. JW got down the peg board and began square rooting at 625.


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Filed under AV, Binomial & Trinomial Cube - Cube rooting, BR, Geometry, JV, LR, Mathematics, Peg Board - squaring/rooting, Triangle Theorms

First day back – two feet into the mire

Today we began organized classes again.  Due to limited heat in my home, we’ve set-up shop at AR, BR, and LR’s home.

DR researching her sedimentary rock paper in the living room.

DW researching her sedimentary rock paper in the living room.

JV researching his sedimentary rock project – Salt.

The sewing table.  We will be felting for the next few weeks.  This book is amazing.

The Geography and Letter work shelf and comfy chair.

BR researching his geology in front of the geology, geometry, and math shelf.

Supply shelf with LR’s Forgotten English calendar.

AV working on Art History and AR making a book for her sedimentary project.

We began lessons in the basics of multiplication, geometry proofs – triangle to rectangle, rhetoric devices (today hyperbole), tessellation, and citation of sources in work.


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Filed under AR, Art, AV, BR, BW, DW, Geology, Geometry, JV, LR, Mathematics, Operations, Science, Triangle Theorms, Writing