Number Talks with Fraction

This post is based on the last class session for this semester. We went out with a bang. We tackled NT with fractions. We actually started our talk about fractions the previous class session. Today was more of a review for the final. After much practice order fractions without determining alike denominators and operations with fractions using models, my PST (Pre-service Teachers) engaged in a NT involving fractions. The following is a conversation with adding fractions:

 1/4 +1/2

Teacher: How did you get 3/4/?

PST 1: I know that 1/2 is equal to 1/4 + 1/4which is 2/4. So I used 1 /4  + 1/4+ 1/4. Therefore  it equals to 3/4. 

PST 2: I just found that common denominator which is fourths. Since the first fraction already has fourth, I don’t have to change that fraction. But the second fraction is ½. The second fraction has to be changed to the new fractions that is equivalent. The equivalent fraction is 2/4. So 1/4+ 2/4 is 3/4 . 

Both student used equivalency to add these fractions. PST 2 decomposed 1/2 into the 1/4s in order to add fraction. She used her prior knowledge on how those two fractions are related. This knowledge was foster by using tools such as the “Wonder Wall”, fraction tiles, and creating models. All of these things were used in class during the previous and that day’s class sessions. PST 2 used her ability to solve the fraction with equivalency like PST 1; however, she reverted to what was comfortable. 

 The next problem for this cluster deals with adding two fractions below.

1/8+ 1/2

Teacher: How did you get 5/8?

PST 3: I thought of in terms of finding common denominators. I used 16ths. But I had to make equivalent fractions by multiplying 2 by both numerator and denominator. This gave me 2/16. The second fraction I multiplied 8 to the numerator and denominator. This gave me 8/16. So 2/6 + 8/16 is 10/16 or simplified to 5/8 .

PST 4: I decomposed 1/2 into fourths first. I know 1/2 equals to 1/4 +1/4. Then I further broken down 1/4 into eights. 1/4 is 1/8 + 1/8. So I have another 1/4. That one is 1/8 + 1/8  too. So. I/8 (from the original problem  ) plus 1/8 + 1/8 + 1/8+ 1/8 (from the 1/2 decomposing to 2/4 to 4/8) equal to 5/8.

Again both PST 3 and 4  added fractions using equivalency. Both know they need alike fraction to add them.  The second approach involved decomposing the fraction in order to add. It was noted in class that one of the Common Core Standards involves decomposing fractions into its basic units.  . "Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model.  [4-NF3b] (Alabama State Department of Education, 2015)  PST 4 completed a double decomposition in order to get a like fraction and to justify her thinking.

PST study fractional reasoning using uncommon methods. These methods are listed in Common Core State Standards. We use fraction models to perform operations with fractions. Such models include addressing the three different models-set, area, and linear. For this NT, student used equivalency to add fractions. Using models helped them to visualize and conceptualize the importance of equivalency. Hopefully, they will use these method during their careers as teachers.