The geoboard is the perfect manipulative for a student to demonstrate understanding of this standard. Esmeralda shows her octopus polygon and proves that the area is 11 square units.

Galilea proves the area of her "X" by showing the four hidden right triangles inside each of the 4 "arms" of her polygon. This was a popular design, so several were able to compare solutions to prove that the area is 10 square units.

Although Johnny made the same design, he proved the area by finding the

*outside*area of 6 square units and subtracted it from 16 to get a polygon area of 10 square units.

Another fun activity I like to do with this is ask the children to find a shape that will trick the teacher. Thrilled with the challenge, several tried to create difficult shapes. Soon, they become so good at solving the difficult puzzles that they don't need me to intervene.

Sometimes students are confused about the size of the square units. Whenever they see any square, they think it is a square unit, forgetting that all of the units must be the same size and measured with the same unit value. Ivette redoes her polygons to show 1 (square for sq. unit) + 2 sq. units + 2 sq. units + 1 sq. unit = 6 (square units).

Carl shows how he determined the area of the yellow quadrilateral by drawing a square around the top right-hand portion of his polygon.

He knows the area of the square is 4 square units, so he can subtract the 1 red square to get 3 square units. In addition, he can see that one-half of the top 2 squares is 1 square unit, so he can use this information to determine that the yellow area inside this square is also 1 square unit. Since his design is symmetrical, the area for the rest of the polygon is easy to figure out.

Esmeralda shows how she combined half squares to count whole square units.

Rigoberto and Angel are both very proud of their shapes. I love the critical thinking skills involved here!

Although this is now a 6th grade standard, my 4th graders love this activity and everyone feels successful.

Carl and his group prove his solution to the class, giving everyone a chance to demonstrate his or her understanding.

I also taught the students how to prove the area by subtracting the outside or

*negative*space from the total geoboard area. In this way, the children are able to prove their area in two different ways.

Galilea and Leslie present their solution for Leslie's fancy "Spiked Heel" design.

## Don't have geoboards?

#### Here's a great virtual geoboard!

Can you help Cesar and Alejandra prove the area of this quadrilateral as 5 square units?They prove the area by finding a hidden right triangle on the bottom right, and use the idea of a 4-unit square to determine the top portion of this shape.

Geoboards are an excellent way to explore fractional parts of the one-whole square on the geoboard. The children also enjoy creating quilt or garden designs.

Alejandra, a third grader, illustrated her geoboard garden. You can also see the Communication Guide I created to help my ELD students write down their mathematical ideas and strategies.

How many of these shapes can you solve and prove?

There are a number of great interactive websites where the children can create interesting shapes and click the button to compare their answers.

Great geoboard outfit, Andrew!

Interactive Geoboard

I am truly proud of these math geniuses! :) All of them!!

Angel and Gabriela explain how they solved Angel's "Chinese Fighter Fish" polygon.

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