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This interactive applet covers **everything about 3d Grade’s Fractions** that is included in the **Common Core Standards**. This is the **visual fraction model** that the Standards suggest to use in the classroom to help students develop an understanding of this topic faster and easier.

Description of the teaching flow:

A vertical slider in the control area guides you through 3 modes:

► The **Definition **mode, where visuals like a bar, a circle and a set of identical objects are used to set the concept of the fraction.

► The **Number Line** mode, where the previous conclusions are transferred to the number line and the fractions take their position on it. This extends to also include fractions >1.

► The **Comparison** mode where two bars are presented and you can partition them in different ways to explore when the represented fractions are equivalent, when they are different, which one is greater and why etc.

**Definition** mode:

► Drag the horizontal slider to partition the bar. Start a discussion:

◍ How many parts is the bar divided into?

◍ What is their size?

◍ What notation is reasonable to use for each one of them and how to call them?

Press the little cross in the box to get the answer and see that each one of the equal parts is 1/(number of equal parts) and is called “unit fraction”.

► Drag the orange pointer at the top of the bar to start coloring the parts. Question to start a discussion:

◍ How many parts – unit fractions have been colored?

◍ Can you extend the notation used for the “unit fraction” to name these fractions too?

Press the little cross in the box to get the answer and see that the colored part is the sum of x unit fractions and is written as x/number of equal parts defining the “fraction”.

► Drag the horizontal slider at the bottom to show that fractions can be built using different shapes (circle in this case) as long as these are partitioned into equal parts and using set of objects as long as these are identical.

Move to the **Number Line** mode:

► Drag the slider to partition the bar. Discuss:

◍ What is happening to the number line right below the bar?

► Drag the orange pointer to select a certain number of parts.

◍ Again what is happening to the number line? (Observe how the red thick segment on the number line moves along with the pointer and “selects” the corresponding parts)

◍ What fraction corresponds to the endpoint of the red thick segment?

◍ How do the numerator and denominator of the fraction compare?

Press the little cross below the endpoint of the segment to reveal it.

► Drag the orange pointer to select all parts that the bar is divided into.

◍ What fraction corresponds to the endpoint of the red thick segment?

◍ How do the numerator and denominator of the fraction compare?

Press the little cross below the endpoint of the segment to reveal it.

◍ Can you come up with a rule on what the fractions with equal numerator and denominator are equal to?

► Click in the checkbox to change the setting and show numbers >1. Discuss:

◍ How many units are now shown on the number line?

► Drag the slider to partition all three units on the number line.

► Drag the orange pointer to select a certain number of parts. Extend the selection beyond the number 1 on the number line.

◍ How many unit fractions have now been selected?

◍ Can you extend the previous notation used to represent fractions <1 to this case too where the endpoint of the red segment is beyond the number 1 on the number line?

◍ How do the numerator and denominator of this fraction compare?

Move to the Comparison mode.

► Partition the two bars in a way that the number of parts of one bar is a multiple of the number of parts of the other bar (e.g 5 and 10). Drag the orange and green pointers to select parts in both bars of equal total length (e.g. 3 and 6). Question to discuss:

◍ What are the two fractions that are represented?

◍ How do the total lengths of the selected parts in both bars compare?

◍ What is a reasonable way to call these fractions? (equivalent)

Press the little cross in the two boxes to reveal the fractions.

◍ What is the relationship between the numerators and between the denominators?

◍ Can you come up with a rule on when two fractions are equivalent?

► Partition the two bars in equal number of parts (e.g. 12 and 12). Drag the orange and green pointers to select different parts in the two bars (e.g. 5 and 8). Question to discuss:

◍ What are the two fractions that are represented?

◍ How do the total lengths of the selected parts in both bars compare?

◍ How do the two fractions compare?

Press the little cross in the two boxes to reveal the fractions

◍ What is the relationship between the numerators and between the denominators?

◍ Can you make a rule on how two fractions with equal denominators compare to each other?

► Partition the two bars in different number of parts (e.g. 9 and 14). Drag the orange and green pointers to select the same number of parts in the two bars (e.g. 6). Question to discuss:

◍ What are the two fractions that are represented?

◍ How do the total lengths of selected parts in both bars compare?

◍ How do the two fractions compare?

Press the little cross in the two boxes to reveal the fractions.

◍ What is the relationship between the numerators and between the denominators?

◍ Can you make a rule on how two fractions with equal numerators compare to each other?

**With these easy, intriguing and ready-to-use GeoGebra applets, you will enrich your regular classroom practice and enhance student understanding of math concepts. With the help of a smartboard or a projector, you can demonstrate for the entire class or let students explore independently on their computers. No knowledge of how to use GeoGebra is required, you just use an app. No internet connection is needed.**

To be able to see and play with the applet you will need to download and install the free software GeoGebra in your computer or tablet if you haven’t done this before. The process is very simple and quick: The link is

https://www.geogebra.org/download

from which you can download GeoGebra Classic 5 (this is the version that I also use).

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