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BNM2 Task 4: Understanding and Teaching Fractions, Decimals, or Percentages Western Governors University August 31, 2021

2 A. 1. First Standard: Fourth Grade  CCSS.MATH.CONTENT.4.NF.B.3.B: 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 (Common Core Standards, 2021). Second Standard: Fifth Grade  CCSS.MATH.CONTENT.5.NF.A.1: Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators (Common Core Standards, 2021). Third Standard: Sixth Grade  CCSS.MATH.CONTENT.6.NS.A.1: Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem (Common Core Standards, 2021). 2. Sample Problem: Fourth Grade  CCSS.MATH.CONTENT.4.NF.B.3.B – Decompose the following fractions into smaller fraction equations with the same denominator. Draw a picture of the fraction to show your work. o 4 5

3 o 2 7 8 Sample Problem: Fifth Grade  CCSS.MATH.CONTENT.5.NF.A.1 – Find the common denominator to solve the equations. o 3 8 + 1 2 o 4 5 - 2 3 Sample Problem: Sixth Grade  CCSS.MATH.CONTENT.6.NS.A.1– Solve the following word problem: How many 2 3 cup servings are in a 7 3 cup of sugar? 3. First Sample Problem Solution: Fourth Grade  To solve the problem, the student will first need to read the instructions and see that they are decomposing or breaking down the fractions into smaller fraction equations. To do so, they will need to look at the numerator first, as the denominator will be staying the same. For the first problem 4/5, the students could choose to do the equation 2/5 + 2/5 = 4/5, to which they would use the visual: + = showing how they broke down the fraction into smaller pieces. For the second problem 2 7/8, the students will need to break the whole number apart as well. An example answer could be 1 + 1 + 3/8 + 4/8 = 2 7/8. This answer

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4 shows the student breaking apart the whole number as well as decomposing the fraction into two smaller pieces. The visual for this answer would be: + + + = Second Sample Problem Solution: Fifth Grade  First, the student will need to identify the LCD, or least common denominator in this problem. After going through a multiplication table, the student will find that the LCD for 3/8 and 1/2 is 8. To get 1/2 to have a denominator of 8, they must multiply both the top and bottom by 4 to get the fraction 4/8. The student will then add together the fractions to get the answer to the equation which is 7/8. For the second problem, the student will need to find the LCD of 4/5 and 2/3. Since 5 and 3 share 15, the student will need to multiply both fractions to get the LCD of 15. They will need to multiply both numerator and denominator of 4/5 by 3, and multiply both numerator and denominator of 2/3 by 5. They will then have the equation 12/15 – 10/15. The students will then subtract the fractions now that there is a similar denominator and arrive at the answer of 2/15. Third Sample Problem Solution: Sixth Grade  The student will first need to identify what the question is asking. They will realize that they will need to divide the cups of sugar by the serving size. This will give them the equation 7/3  2/3. To solve this equation, the students will need to turn the division problem into a multiplication problem by multiplying by the reciprocal. This will turn 7/3  2/3 into 7/3 x 3/2. Now the students will multiply the fractions straight across, giving them 7 x3/ 3x2. After solving the

5 multiplication for the numerator and denominator, the student is left with 24/6. This can be simplified into the answer 4. 4. Fraction, Decimal, and Percentage Understanding Across Grade Levels  Fourth Grade: The standard chosen for students in the fourth grade helps them build their understanding of fractions as it helps them identify how to decompose and build fractions. This builds upon their prior knowledge of adding and subtracting friendly numbers to solve an equation that they have learned in previous grades, as well as comparing fractions with the same denominator as they did in the third grade. Per the standard, the students will be able to show that they can decompose a fraction into a sum of fractions with the same denominator in more than one way as well as using a visual to justify their work. In the example problem, the students will need to decompose the fractions into simple fraction equations that have the same denominator. This standard and example problem chosen will build a foundational understanding of fractions as well as adding fractions that the students will need to know as they encounter more advanced fraction equations in the upper grades.  Fifth Grade: The standard chosen helps the students build on their prior understanding of fractions as it helps them add and subtract fractions with differing denominators. In the fourth grade the students learned how to decompose fractions into friendlier options for the equation, and then justify the equation with a visual. This prior knowledge is built on with this standard and example problem, as they will be adding and subtracting fractions with like

6 denominators after they solve for the least common denominator, or LCD. The standard chosen will help the students continue to build on their understanding of fractions by having the students write out the fractions, finding the LCD, and then adding or subtracting the resulting fractions to get the answer. The students will use the example problem as practice of adding and subtracting fractions with unlike denominators, which will continue to build on their foundational understanding of fractions. This will help students as they go through the upper grade levels as they will need to know how to find the LCD of the fraction equation to solve more complex algebraic expressions.  Sixth Grade: The chosen standard for the students will help them build their understanding of fractions and identify the relationships between the integers in the given problem. The students have the prior knowledge of working with fractions in the previous grades, specifically decomposing fractions in the fourth grade as well as multiplying fractions in the fifth grade. Having this previous practice from the lower grades, the students can build on this foundational understanding to grasp the concepts of dividing fractions in the standard. They are solving division of fraction word problems in the example problem to do higher order concepts such as multiplying the fraction by its reciprocal which is part of their given standard. The example problems will help build the student’s understanding of dividing integers, which is an essential foundational skill to have throughout their mathematical education as they move to higher grades and harder concepts involving fractions.

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7 B. 1. In the video “Understanding Fractions Through Real World Tasks” I noticed the teacher Ms. Franco was a facilitator as she did not operate under the traditional construct of teaching. Rather, she guided the students towards learning using such strategies like activating prior knowledge and using real life examples in her lesson. This allowed the students to learn through building on previous knowledge, dialogue, and picking apart ideas to form their own thoughts about them. Through this hands-on teaching opportunity with the watermelons, Ms. Franco’s students are able to see equality in fractions and allowing them to physically see how we can make equal fractions with different denominators. 2. The teacher in this lesson, Ms. Franco, designed her lesson using the “Three-Phase Lesson Structure” (The Teaching Channel, 0:17) that activates prior knowledge, establishes clear expectations, and makes sure that the students understand the problem before the lesson. During the lesson, Ms. Franco takes notice of her students’ mathematical thinking and provides appropriate support and worthwhile extensions, such as at 2.55 where she asks the student about his fraction compared to his tape diagram. By checking for understanding, she can formatively assess that her student understands the problem and is ready to move on to part three of the structure (The Teaching Channel, 2:55). For the end of the lesson, Ms. Franco promotes a community of learners by listening actively to their thinking without evaluation, and then summarizes the main idea so she can identify future problems. 3. At the end of the lesson, Ms. Franco had the students turn and talk to their peers about what they did during the lesson, how they “attacked” the task, and how they justified

8 their answer. By discussing what they learned in their own words and explaining it to their fellow students, they are able to solidify the summarization of the lesson and reinforce what they learned during the lesson. By walking around and listening to the students’ summaries, Ms. Franco is able to check for any misconceptions and errors to identify future problems. Yet as she stated in the video, the summarization was successful and showed that they grasped the lesson (The Teaching Channel, 5:20). 4. One way that Ms. Franco formatively assessed her students through this lesson was by activating their prior knowledge through questioning by asking them what they knew about fractions at the beginning of the lesson (The Teaching Channel, 0:23). Through this strategy, Ms. Franco was able to see how much her students knew about fractions, and how she would need to scaffold the lesson to get the students to the level she needed them to be at the end of the lesson. Additionally, Ms. Franco used observation with open questioning to formatively assess her students. (The Teaching Channel 3:35). By observing her students, she can check for misconceptions and see where she needs to assist the student with scaffolding questions. a. By activating prior knowledge, the teacher can see what information the students have gain from either previous schooling or life experiences. It’s important to know what the students already know about the lesson before starting, so the teacher can understand why the students may struggle and help fill in the learning gaps for the students to be successful. b. By observing the students and asking open ended questions to encourage discussion, the teacher can see and hear the students’ thought process in

9 action. This provides the teacher with important information about the student’s progress with the lesson, and to check for students struggling with misconceptions. The open-ended questioning by the teacher during the observations allows the students to explain in their own words their understanding of the lesson, which can help the teacher check for misconceptions or errors that need to be addressed. 5. An activity that would be helpful in clarifying the need for common denominators would be pizza party flashcards. The teacher passes out two cards with differing amounts of pizza slices (fractions) to each group. The groups need to find the common denominator and see which pizza would let everyone have the bigger slice. For example, one group may get the fraction 1/2 and another card with 1/4. The common misconception would be that the bigger denominator means a bigger slice. The students would need to go through the steps of finding the least common denominator using a multiplication chart manipulative and seeing which fraction would let them have more pizza. C. 1. An instructional strategy that I will use in my fractions lesson is the concrete- representational-abstract method, commonly known as the CRA method. The CRA method is broken up into three stages, with the first stage being concrete. During the concrete stage, the students are given a physical manipulative to help them during the lesson. In the next stage, known as representational, the students turn those physical manipulatives into a drawing to represent the same concepts. Finally, during the abstract stage, the students use the drawings that they made in the previous stage to help them to write the equation and solve the problem (Pennsylvania Department of Education. 2020).

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10 a. The CRA method can help deepen the students understanding of fractions, decimals, and percentages as “After receiving CRA instruction, students significantly improved their conceptual understanding of equivalent fractions. While engaging in a post-test, students would fall back on the representational drawings to guide their way through a problem” (Pennsylvania Department of Education. 2020). The students will be able to use their resources from the first two stages as a way to process and replay the steps in their head when coming to the abstract stage, which will deepen their understanding of equivalent fractions when moving forward in the lesson. The student will be able to understand fraction comparisons with the help of the concrete-representational stages which will help them succeed in the abstract phase. References: Common Core Standards. 2021. Common Core State Standards for Mathematics. Web. Retrieved from: http://www.corestandards.org/wp-content/uploads/Math_Standards1.pdf . Pennsylvania Department of Education. 2020. Concrete-Representational-Abstract: Instructional Sequence for Mathematics . Web. Retrieved from: https://www.pattan.net/getmedia/9059e5f0-7edc-4391-8c8e-ebaf8c3c95d6/ CRA_Methods0117 Teaching Channel. 2020. Understanding Fractions Through Real World Tasks . Web. Retrieved from: https://library.teachingchannel.org/landing- page? mediaid=bf55qvbv&playerid=Cogatb0c .