Justifying Area

Our recent emphasis in the Math Workshop has been with estimating and justifying the area of irregular shapes.

One tool that we have used to justify the area formulas for rectangles and right triangles has been the Geoboard.

Formula for Rectangles (Area = Base x Height) Formula for Triangles (Area = 1/2 Base x Height)



Consider the irregular polygon shown on this Geoboard template.

What is the area of this pentagon?

During Math Workshop, we have approached the task of finding solutions to problems similar to this by decomposing the figure into smaller rectangles and right triangles so that we can use the area formulas for the smaller parts and then put those smaller totals together to find a justified total area.




The green rectangular area in this pentagon has a base of 3 units and a height of 2 units, so the area of this rectangle is 6 square units.

   Area of a Rectangle = Base x Height, so 6 = 3 x 2.

   (We also reference the area formula as length x width.






The right scalene triangle (brown) shown on the grid is exactly HALF of 2 square units (the dotted lines are shown to illustrate this idea). Therefore, the area of the triangle is 1 square unit.

Area of a Triangle = 1/2 Base x Height

The base of the triangle is 1 unit and the height of the triangle is 2 units.

(1 x 2) ÷ 2 = 1

A right scalene triangle is exactly half of a rectangle.

 
The right isosceles triangle (blue) shown on the grid is exactly HALF of 1 square unit (the dotted lines are shown to illustrate this idea). Therefore, the area of this triangle is 1/2 square unit.

Again, Area of a Triangle = 1/2 Base x Height

The base of this triangle is 1 unit and the height of the triangle is 1 unit. (1 x 1) ÷ 2 = 1/2

A right isosceles triangle is exactly half of a square.


All decomposed parts are then combined to identify the total area of the figure.

The area of the pentagon is 7 1/2 square units!!

Students, you can have fun with virtual Geoboards by clicking the link below, which will take you to the Virtual Library of Math Manipulatives. (A parent may have to install Java in order for you to utilize this site.)

Click Here: VIRTUAL GEOBOARD FUN

Also, if you would like to practice with transformations, you can click the links below for some more virtual fun!

Virtual TRANSLATIONS (Slides)


Virtual REFLECTIONS (Flips)


Virtual ROTATIONS (Turns)






If you utilize these online resources, leave a comment to let your teachers know so you can earn some extra Behavior Bucks!

Erosion and Deposition

EROSION & DEPOSITION (teach) from Moira Whitehouse

This slideshow is very helpful in emphasizing many of the main points we have been talking about in Science this week. Enjoy!

The Amazing Rock Cycle

Huge boulders, tiny grains of sand, silt, clay ~ all are made of rocks. Even though rocks are non-living, they can change in form both above and below the Earth's surface. In Science, we have been learning about how scientists classify rocks according to the way in which they were formed and we know that there are three main "kinds" of rocks: igneous, sedimentary, and metamorphic.

 

After using hand lenses to closely observe various properties of rocks (perusing through our classroom's three rock collections), we then turned our attention to two awesome virtual labs (Gizmos): Classifying Rocks and The Rock Cycle. We also researched our textbook for important information and also had fun with a Snickers candy bar correlation- be sure to discuss this at home! :-)

 

Here are some highlights of what we have learned:

 

• Igneous rocks form from other rocks that have MELTED. Melted rock below the Earth's surface is "magma" and melted rock that reaches the Earth's surface is called "lava".
• Mineral crystals form when lava cools IF they have time to form. Quickly cooling lava may have NO crystals- the longer it takes for molten rock to cool, the larger the crystals may form.
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• Sedimentary rocks form when layers of sediments settle on top of one another and harden. Sediments are made up of soil, shells, bits of rock, and dead plant and animal matter It is in these rocks that FOSSILS are found!  (Be sure to discuss our bread and gummy bear layering demonstration at home.) :-)
• Water, wind, ice, and gravity move the sediments that form these rocks. Over time, these particles settle in layers and press older layers together, which hardens them and forms rocks.
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• Metamorphic rocks form from high temperatures DEEP inside the Earth. These rocks are also under a lot of pressure from the weight of the rocks that are above them. These "forces" can cause rocks to change their form into metamorphic rocks. The heat and pressure can even cause the mineral crystals in the rock to change! Chemicals in some rocks also cause other minerals to form. (Most rocks are made up of different combinations of minerals.)
• It is amazing that metamorphic rocks can form from sedimentary, igneous, AND other metamorphic rocks! Look at the diagram of the rock cycle at the beginning of this post to see just how interconnected the three types of rocks are.
• Students, what has been your favorite experience in class in learning about the Rock Cycle? Leave a comment to earn Behavior Bucks!
• Happy Science Sleuthing,