teaching fractions with learning progressions
DESCRIPTION
Kristin Frang, Mathematics Consultant MAISD Regional Math & Science Center Laura Wilson, Teacher Leader Vicksburg Community Schools. Teaching Fractions with Learning Progressions. Series Goals. Explore a framework that balances conceptual understanding, procedural fluency and application. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/1.jpg)
TEACHING FRACTIONS WITH LEARNING PROGRESSIONS
Kristin Frang, Mathematics ConsultantMAISD Regional Math & Science Center
Laura Wilson, Teacher LeaderVicksburg Community Schools
![Page 2: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/2.jpg)
SERIES GOALS Explore a framework that balances conceptual
understanding, procedural fluency and application.
Identify critical components to understanding fractions and instructional strategies that support their development.
Understand students’ development of core concepts and ways of reasoning about fractions
![Page 3: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/3.jpg)
YOUR QUESTIONS
How do I get in so much higher order thinking?MAISD CCSSM wiki
Overwhelming, but every journey begins with a step.
![Page 4: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/4.jpg)
TODAY’S AGENDA Review last session & homework Major Emphasis in CCSS SBAC Performance Task Organizing Instruction Levels 0-2 Student Reasoning
LUNCH Levels 3-4 Student Reasoning Matching Instruction to Student Reasoning
![Page 5: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/5.jpg)
MIND STREAMINGFind a partner at your table: Partner A talks for 1 min about the topic Partner B listens and encourages Partner A
The role reverses: Partner B talks for 1 min about the topic Partner A listens and encourages Partner B
![Page 6: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/6.jpg)
![Page 7: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/7.jpg)
Equal Partitioning
![Page 8: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/8.jpg)
Iterating
![Page 9: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/9.jpg)
Unit Fractions
![Page 10: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/10.jpg)
Fraction Comparison
![Page 11: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/11.jpg)
Big Ideas of Fraction
Concepts
![Page 12: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/12.jpg)
Chapter 1 Thoughts
![Page 13: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/13.jpg)
It is difficult for students to move from working with whole numbers to working with fractions because a fractional quantity is described with two numbers, not one, and understanding fraction requires one to explicitly comprehend a relationship between two quantities – the whole and its parts.Battista
By focusing on relationships, children come to see fractions as connected to other things they know about number and operation Empson & Levi
![Page 14: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/14.jpg)
CCSS CRITICAL AREA
3rd grade – page 21
4th grade – page 27
5th grade- page 33
![Page 15: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/15.jpg)
CLUSTER HEADINGSCluster headings function like topic sentences in a paragraph in that they state the point of, and lend additional meaning to, the individual content standards that follow…Materials do not simply treat the Standards as a sum of individual content standards and individual practice standards.
K-8 Publishers’ Criteria for the CCSS for Mathematics
![Page 16: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/16.jpg)
Connections at a single grade level can be used to improve focus, by tightly linking secondary topics in the major work of the grade. K-8 Publishers’ Criteria
for the CCSS for Mathematics
![Page 17: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/17.jpg)
Focus on Major Work: In any single grade, students and teachers using the materials as designed spend the large majority of their time, approximately 3/4 , on the major work of each grade level… In addition, major work should especially predominate in the first half of the year
K-8 Publishers’ Criteria for the CCSS for Mathematics
![Page 18: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/18.jpg)
![Page 19: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/19.jpg)
NUMBER WORK (10 – 15 min)
Develops
• number sense without story context
• communication• connections• algebraic reasoning
Allows• multiple entry points / answers• access for wide range of skills &
abilities
Can be• single problem• series of problems
![Page 20: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/20.jpg)
NUMBER WORK EXAMPLEDifferences of (Purpose: differences)
Ask students to write pairs of numbers with a given difference (e.g. difference of 7) for a predetermined number of minutes
Share responses and repeat several times. (e.g. 32/39, 54/61, 1009/1016)
Variation: Use decimals and fractional differences
![Page 21: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/21.jpg)
NUMBER WORK EXAMPLES
![Page 22: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/22.jpg)
Number work is …
• commonly done whole class
• beginning of math block
• can also be tied into ‘calendar’ routines for younger students
• work was done independently, but chosen specifically for individual students.
![Page 23: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/23.jpg)
Balanced Mathematics
Kristin Frang, MAISD Regional Mathematics & Science CenterAdapted from Cathy Seely, Balance is Basic and Larry Ainsworth, Five Easy Steps to a Balanced Math Program
Formative Assessment Skills, Facts, Procedures
Conceptual
UnderstandingProblemSolving
Flexible Set of Thinking& Reasoning
![Page 24: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/24.jpg)
![Page 25: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/25.jpg)
Instruction
Problem Solving (30 – 45 min)
Number Work or Inspecting
Equations (10 – 15
min)
Practice
Fluency & Maintenance
(10 – 15 min)
![Page 26: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/26.jpg)
PROBLEM SOLVING All content domains
Large & small group settings
May serve multiple purposes, such as: Introducing new conceptsChallenging students to develop and apply
effective strategiesProviding a context for improving skills
![Page 27: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/27.jpg)
PROBLEM SOLVING
Utilizes problems that: Emerge from the students’ environment
Can be found in the school’s curricular materials
Allow for multiple solution strategies
Lead to justification and generalization
![Page 28: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/28.jpg)
INSPECTING EQUATIONS Focus is on learning about how the equal sign expresses equality relationships.
Students:
Explore equality & meaning of equal sign
Analyze conventional symbols of #, operations & relationships
Look for & use #
relationships.
Use computation to confirm equality of
relationships.
![Page 29: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/29.jpg)
INSPECTING EQUATIONS EXAMPLE
Can students accurately solve equations with repeated variables?
g + g + 4 = 16h + h – 3 = 11a + a + a = 15
Do students take advantage of familiar number relationships?
25 + 47 + 75 = y98 + 69 + 2 = y
![Page 30: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/30.jpg)
FACT FLUENCY
Illuminates a child’s sense of number relationships and the thinking strategies he/she uses to calculate single-digit computation
Identify size of numbers a student
can compute mentally
Defines numbers sizes to use in other blocks
Defines numbers sizes to work on for individual fluency &
maintenance
![Page 31: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/31.jpg)
3-2-1 REFLECTION 3 things you want to remember from our
time this morning
Highlight 2 things you want to focus on implementing first
Choose 1 and create a plan to help you implement in your current unit
![Page 32: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/32.jpg)
Problem solving should be the central focus of the mathematics curriculum. Problem solving is not a distinct topic, but a process that should permeate the entire program and provide the context in which concepts and skills can be learned
![Page 33: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/33.jpg)
ACCESSIBLE COGNITIVE JUMPS
![Page 34: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/34.jpg)
LEVELS OF SOPHISTICATION IN STUDENT REASONING
![Page 35: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/35.jpg)
LEVEL 0 (SUBLEVEL 0.1 & 0.2)Tell what fraction is shaded.
Response: 1
![Page 36: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/36.jpg)
LEVEL 0 (SUBLEVEL 0.1 & 0.2)
Level 0.1 Level 0.2
Three people want to share the pizza below equally. Show how much each person gets.
![Page 37: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/37.jpg)
LEVEL 1Student recognizes only familiar pictures of fractions
![Page 38: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/38.jpg)
Shape A looks like a half. The rest are not halves
LEVEL 1
![Page 39: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/39.jpg)
I think A,B,C and D look like ¾ . E and F don’t look like ¾.
LEVEL 1
![Page 40: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/40.jpg)
LEVEL 2 Count all parts, count shaded parts
Don’t explicitly focus on whole Difficult time finding fractions of sets
Don’t use iteration/partitioning into equal parts
Can make comparisons by drawing pictures
![Page 41: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/41.jpg)
A, B and F are halves because one part is shaded and one part is not.
LEVEL 2
![Page 42: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/42.jpg)
LEVEL 2 Count all parts, count shaded parts
Don’t explicitly focus on whole Difficult time finding fractions of sets
Don’t use iteration/partitioning into equal parts
Can make comparisons by drawing pictures
![Page 43: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/43.jpg)
A, B, C and D are ¾ because they all have 4 parts and 3 are shaded. E and F are not ¾ because they have to many pieces.
LEVEL 2
![Page 44: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/44.jpg)
LEVEL 1 VS. LEVEL 2
![Page 45: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/45.jpg)
LEVEL 3 Partition into equal parts only when whole
is explicitly specified
Difficulty find fractions of sets, improper fractions and performing arithmetic operations because whole is more difficult to maintain
Reasoning is restricted to shape, do not reason about quantities
Use splitting and iterating
![Page 46: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/46.jpg)
• A is one-half because there are 2 equal pieces and 1 is shaded. Same thing on B.
• C, move square in bottom row to empty spot so the gray & white parts are the same (see red arrow)
• D, move the bottom shaded squares to the left to make one half.
• E, move all shaded squares to the left.
• F is not one half because the gray and white parts aren’t equal.
LEVEL 3
![Page 47: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/47.jpg)
LEVEL 3 Partition into equal parts only when whole
is explicitly specified
Difficulty find fractions of sets, improper fractions and performing arithmetic operations because whole is more difficult to maintain
Reasoning is restricted to shape, do not reason about quantities
Use splitting and iterating
![Page 48: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/48.jpg)
• B and D because they both have 4 equal parts and 3 shaded.
• The parts aren’t equal in A and C.
• E is ¾ because you can divide it so that 3 out of 4 equal parts are shaded.
• F is not because there are 6 squares shaded.
LEVEL 3
![Page 49: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/49.jpg)
![Page 50: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/50.jpg)
LEVEL 3 Partition into equal parts only when whole
is explicitly specified
Difficulty find fractions of sets, improper fractions and performing arithmetic operations because whole is more difficult to maintain
Reasoning is restricted to shape, do not reason about quantities
Use splitting and iterating
![Page 51: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/51.jpg)
Read pages 26-27 and add to your mind-maps
LEVEL 3Other Characteristics & Illustrations
![Page 52: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/52.jpg)
LEVEL 4 (SUBLEVEL 4.1 & 4.2)
Students have created a new link between number & shape
Flexibly go back and forth between actions on shape and actions on number
Generalize understanding to multiple contexts
Keep track and maintain the whole
When students reason about the numerical quantities
![Page 53: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/53.jpg)
A,B,C,D and E are each one-half because 8 squares are shaded and 8 are not shaded.
LEVEL 4 (SUBLEVEL 4.1 & 4.2)
![Page 54: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/54.jpg)
• B and D because they both have 4 equal parts and 3 shaded.
• The parts aren’t equal in A and C.
• E is ¾ because you can divide it so that 3 out of 4 equal parts are shaded.
• F is not because there are 6 squares shaded.
LEVEL 3
![Page 55: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/55.jpg)
• B and D because they both have 4 equal parts and 3 shaded.
• The parts aren’t equal in A and C.
• E is ¾ because you can divide it so that 3 out of 4 equal parts are shaded.
• F is not because there are 6 squares shaded.
LEVEL 3 VS 4
![Page 56: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/56.jpg)
• E: each row has 6 squares, so the rows are equal; and 3 of the 4 rows are shaded.
• F: 3 groups of 2 squares shaded, and 1 group of 2 squares unshaded (see circled groups above)
LEVEL 4
![Page 57: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/57.jpg)
![Page 58: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/58.jpg)
Read pages 33 and add to your mind-maps
LEVEL 4Other Characteristics & Illustrations
![Page 59: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/59.jpg)
LEVELS OF SOPHISTICATION IN STUDENT REASONING
![Page 60: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/60.jpg)
FRACTIONS
AND
OPERATIONS
![Page 61: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/61.jpg)
![Page 62: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/62.jpg)
ACCESSIBLE COGNITIVE JUMPS
[I]f algorithms are taught to early in students’ development of reasoning about fractions, students cannot understand the algorithms conceptually, so they learn them by rote…Students who learn computational algorithms for fractions before achieving Level 5 reasoning will learn them rotely. Battista, pg.10
![Page 63: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/63.jpg)
Level 0.1Help students develop understanding of partitioning Focus on equal
partitions not naming fractions
TEACHING STUDENTS AT LEVEL 0.1 & 0.2
Level 0.2Help students develop initial understanding of fractions Link Unit fractions to equal
partitions
Ask questions such as “How many thirds are in a whole?
Introduce Non-Unit Fractions
Ask questions such as “How many thirds are in two-thirds?
![Page 64: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/64.jpg)
NUMBER LINESThe focus on fractional parts is an important beginning, but number sense with fractions demands more – it requires that students have some intuitive feel for fractions.
John Van de Walle
![Page 65: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/65.jpg)
PARTITIONING WITH LENGTH MODELS
Give students number lines partitioned such that only some of the partitions are showing.
Nicole walks 1 mile to school everyday. Using the number line below, how far has Nicole gone? How do you know?
![Page 66: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/66.jpg)
FRACTION NUMBER SENSE
• More and less than ½ or 1
• Distance from ½ or 1
• Same # of parts,
• Different size parts
• More of same-size parts
![Page 67: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/67.jpg)
CLOSE TO
![Page 68: Teaching Fractions with Learning Progressions](https://reader035.vdocuments.net/reader035/viewer/2022062305/568166df550346895ddb1293/html5/thumbnails/68.jpg)
HOMEWORK
1. Read Chapter 2, pages 12-40
2. Choose several assessment tasks from pages 105-111 to try with students.
Record Student Strategies Remember you goal is to learn about your students
thinking
At session 3 on April 9th, we will explore operating with fractions.