Numbers and Operations Fractions: Grades 3-5

1. Numbers and Operations Fractions: Grades 3-5 Jessica KuiperChantal Skinner Michael Suzuki

2. Introduction to Learning progressions According to Week Two’s Google slideshow, learning progressions are “Narrative documents describing the progression of a topic across a number of grade levels.” In other words, learning progressions describe the “skills and understandings that students are expected to have at each stage on their path to mastery.” (ETS 2019)

3. How does this Help our students’ learning experience? • Each standard builds upon the last standard. • It helps the learning process be simple and practical for students to easily demonstrate their skills by encouraging instruction to build on prior knowledge. • Students can receive differentiated instruction that tells the teacher what they know, what they can do, and how to design future lessons Learning progressions reflect student abilities and achievement levels, therefore providing teachers and students the building blocks to help them receive valuable and differentiated instruction.

4. Plan curricula better to meet learning targets • Teachers can gain information about their students’ progress that can better help them to differentiate instruction • Learning progression assessments show students precursor skills • Teachers understand each student’s level of expertise on an individual basis How learning progressions help our teachers teach

5. How do learning progressions align with the standards • Common Core State Standards tell the teacher what standards need to be met at which grade level. • Learning progressions help teachers understand how to get students to the level of thinking required for standard achievement based on their brain and expertise development.

6. What Does It Mean To Unpack a Standard? As discussed in Chapter 2 of Teaching Outside the Box, it is important to make material relevant to students. In order for students to understand the Common Core Standards, teachers must make the academic language engaging for students. In other words, teachers are encouraged to “unpack” standards, pulling out important verbs and nouns from the standard, and incorporating student’s prior knowledge on the subject.

7. Unpacking 3rd Grade Common Core Standards Target: Develop understanding of fractions as numbers. Standard: CCSS.MATH.CONTENT.3.NF.A.1 Objective: Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. The first step is to break the objective into nouns and verbs: Nouns Verbs Fractions UnderstandQuantity Formed1 part PartitionedWholeEqualParts (A/B)

8. Connecting Prior Knowledge The next step in unpacking a standard is to pull from prior knowledge students should be familiar with in 3rd grade. Standard: CCSS.MATH.CONTENT.3.NF.A.1 For this example, students should be aware that numbers can be broken down into smaller units. Students should also have a basic understanding of number sense, and that numbers convey meaning.

9. Using Multiple Means of Representation Recent studies show that students learn best through different exposures to new material. This can be broken down into three categories: auditory, kinesthetic, and visual learners. For this standard, learners would benefit from a variety of different learning aids when presented with new information. Visual: PowerPoints, Prezi PresentationKinesthetic: fraction tiles, physically breaking down items into smaller unitsAuditory: creating a phrase students can remember, periodically re-stating important information

10. UDL Model Along with means of representation, it is important to give students the opportunity to express themselves whichever way they see fit. For this standard, students can express their knowledge in variety of ways: PowerPoint presentations, a poster, a written paper, a creation from digital media, a verbal presentation, etc. Students should feel comfortable to share out in the classroom, and these strategies contribute to a more inclusive learning environment

11. Number and operations-fractions 3.NF.A.1 manipulatives • Students can reason with physical models and peers to “make meaning of experiences” (Dickenson & Coddington, 2019) • Virtual manipulatives • Area models • Fraction strips • Number lines • Visual representations • With repeated reasoning, students learn to make “generalizations as they build understanding of the meaning of common fractions” (Gojak & Miles, 2016, p. 114)

12. Number and operations-fractions 3.NF.A.1 Misconceptions According to Gojak and Miles (2016), students need multiple opportunities with “concrete representations” (p. 116). A building block, or sign that student needs more modeling and guidance is if they have one of these perceptions: • Given the same size whole, the smaller the denominator, the smaller the piece. • Fraction pieces must be the same shape and size. • Student considers the number of pieces in the whole but does not understand they must be the same size. • Student writes the fraction as a part to part relationship rather than part to whole. 1/3

13. Math Learning Progressions for all Math

14. lEARNING PROGRESSION FRACTIONS GRADES 3-5

15. Math Progression Grade 3-5 Fractions Click on the hyperlinked image to watch a video pertaining to math learning progression for fractions within grades 3-5 and how Focus, Rigor, and coherence pertain to developing instruction.

16. Meaning of a Unit Fraction Click on the hyperlinked image to learn about the building block of fractions and unit fractions. In addition, other topics include representing rectangles and halfs, and thirds. Other topics include partitioning shapes, partitioning number lines,

17. Comparing fractions Click on the hyperlinked image to learn about misconceptions while comparing fractions, comparing fractions using the denominator and numerator, and using benchmark numbers.

18. aDDITION OF FRACTIONS Click on the hyperlinked image to learn about how foundations skills in grade one and two impact grade 3-5 learning in math. In addition, key skills including finding the common denominator are discussed.

19. Multiplying fractions Part I Click on the hyperlinked image to learn about multiplication across grade levels. Topics include multiplication scaling, multiplying by a fraction greater than or less than 1, and multiplication on the number line,

20. Multiplying fractions part II Click on the hyperlinked video to learn more about how a fundamental understanding of multiplying fractions progresses into the later grade levels.

21. Dividing fractions Click on the hyperlinked video to gain an better understanding of how multiplication is necessary for dividing fractions. In addition, a conceptual understanding of how numbers become larger when dividing by numbers less than 1 is addressed. A common misconception is that division mean a number gets smaller. Numbers get smaller when they are divided by whole numbers but get bigger when divided by fractional components

22. Visual Models of Multi-div progression grade 3-5

23. Visual Model Learning Progression grade 3-5

24. Visual Models of Multi-div progression grade 3-5

25. Building blocks for proceeding with standards According to the learning progression for this domain, students should be successful with whole numbers on the number line, subdividing the distance from 0 to 1, and can reason and justify the location of unit fractions by folding strips or on the number line before proceeding to Standard 3.NF.A.2. (Gojak & Miles, 2016) By understanding how the learning progressions work, teachers can better pinpoint the building block that a student struggles with, therefore providing them with differentiated instruction and reteaching of the subject.

26. References Coddington, L. & Dickenson, P. Teaching Outside the Box. Kendall Hunt. Retrieved 11 January 2019. Common Core Standards. http://www.corestandards.org/Math/Content/3/NF/. Retrieved 11 January 2019. ETS. 2019. What are learning progressions and why are they important? Retrieved 14 January 2019 from https://news.ets.org/stories/what-are-learning-progressions-and-why-are-they-important/ Popham, J. . 2019. ASCD: All About Accountability/ the lowdown on learning progressions. Retrieved 14 january 2019 From http://www.ascd.org/publications/educational-leadership/apr07/vol64/num07/The-Lowdown-on-Learning-Progressions.aspx Tomlinson, C.A., Understanding differntiated instruction. ASCD.