MMD#5

Using the Sketchpad tool shapes1.gsp, create each of the following designs and four more of your own.

 Technologies Needed:

• The Geometers Sketchpad
•  Geometers Sketchpad file: http://math-ed.com/shapes1.gsp   [You must first download this file, save it to your computer, then open it with The Geometers Sketchpad. ]

For additional resources and ideas, visit my website math-ed.com 

Copyright David A. Thomas 2010  All Rights Reserved

MMD#4

How is the area of a triangle related to its sides and perimeter?
Is there a triangle of greatest area for a given triangle perimenter?

Technologies Needed:

• The Geometers Sketchpad
• Directed Activity: Perimeter vs. Area
• 2_MaxTriangle.gsp  [You must first download this file, save it to your computer, then open it with The Geometers Sketchpad. ]

For additional resources and ideas, visit my website math-ed.com 

Copyright David A. Thomas 2010  All Rights Reserved

MMD#3: Triangle Angle Sum

In every triangle, the angle measures add up to 180 degrees. Why is that?
Want to know.

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Dear Want to know,
For starters, let's consider a demonstration based on paper folding.

1. Starting with a sheet of unlined paper, cut out a triangle ABC, making side AB about 5 inches in length.
2. Fold side AB on itself so that the crease passes through vertex C.  This crease, labeled CD, is perpendicular to side AB.
3. Position vertex C on top of point D and crease the paper to form EF. 
4. Repeat Step 2 to create perpendiculars to AB through points E and F.
5. Using all three creases, fold the three vertices of triangle ABC together so that they converge at point D.  Together, these three angles precisely cover a 180 degree angle GDH.  Therefore the sum of  the three angles is 180 degrees.
6. Repeat this entire process with several other triangles.
7. How would you summarize your findings?  
8. This demonstration, while meant to be persuasive, is not a proof.  If you'd like to see that, let me know. 

For additional resources and ideas, visit my website math-ed.com 

Copyright David A. Thomas 2010  All Rights Reserved

MMD#2: Balancing Point

Dear Math for Mom & Dad,
      My child likes to make things out of paper, tape, and cardboard.  How can I use that interest to foster her interest in geometry?
Geometry Mom

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Dear Geometry Mom,

Use this activity to find an interior point of a cardboard triangle on which the entire triangle will balance.

Begin by finding the midpoint of each side of an acute triangle.  If you have a paper template of the triangle, you could fold each side of the triangle on itself, endpoint to endpoint, then identify the midpoint by creasing the paper.  When the paper template is placed on top of the cardboard triangle, use the creases to mark the midpoint of each side.  Alternatively, you could measure the length of each side of the cardboard triangle, divide by 2, then find the midpoints by measurement.  Better yet, do it both ways and discuss the merits and limitations of each approach.

Next, draw segments called medians from each vertex to the midpoint of the opposite side.  As seen in the figure below, the medians of a triangle intersect at a point called the centroid.  This point is also called the center of gravity of the triangle, the point where a cut-out version of the triangle will balance on the point of a pen or pencil.

 

Try this same approach with a right triangle.  With an obtuse triangle.  Does it always work?

To find the centroid of a quadrilateral, connect the midpoints of opposite sides.  These bimedians intersect at the centroid.  Try this with several quadrilaterals.  Does it always work?  

For additional resources and ideas, visit my website math-ed.com 

Copyright David A. Thomas 2010  All Rights Reserved

MMD#1: Fraction Addition

Dear Math for Mom & Dad,

     My son is struggling with fraction addition.  Rules by themselves make no sense to him.  Any suggestions?

Parent of a 4^th grader

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Dear Parent of a 4^th grader,

Area models provide a nice context for thinking about the meaning of fraction operations.  We’ll use that approach to model the sum 1/2 + 1/3.  First, take a rectangular sheet of paper and fold it in half the long way (i.e., a Hog Dog fold).  Then open the paper and fold it in thirds the other direction (i.e., a Letter fold).  This divides the sheet into 6 smaller rectangles, each of which has an area 1/6^th of the entire sheet (See Step 1).  Next, shade half of the sheet horizontally (See Step 2) and a third of the sheet vertically (See Step 3).  Finally, count the shaded rectangles, taking double shading into account.  You may also rearrange the shaded cells as seen in Figure 4 to avoid double shading.  Since 5 out of 6 rectangles are shaded, 1/2 + 1/3 = 5/6.  Alternatively, rectangles may be drawn on large cell graph paper and shaded to illustrate fraction operations.  

Use this approach to model the following sums.  Let me know how it goes!

·         1/4  + 1/3

·         1/3 + 3/8

For additional resources and ideas, visit my website math-ed.com 

Copyright David A. Thomas 2010  All Rights Reserved

It's about the kids

To many parents, helping their children with mathematics is more of an ordeal than an opportunity.  This blog seeks to build a foundation for an authentic mathematical dialogue between parents and children.