1. CCGPS Mathematics Fourth Grade Update Webinar Unit 3: Fraction Equivalents October 14, 2013
Update presentations are the result of collaboration between members of 2012 and 2013 Unit Review and Revision Teams and classroom teachers
Microphone and speakers can be configured by going to: Tools – Audio – Audio setup wizard
Turtle Toms-
Elementary Mathematics Specialist
These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.

2. Thank you!
Jenise Sexton and Graham Fletcher to the rescue!

3. Fraction Standards Progression by Grade Level
1.G.3. Partition circles and rectangles into two and four equal shares…
2.G.3 Partition circles and rectangles into two, three or four equal shares…
3.NF.1Understand unit fractions as part of a whole (paraphrased)
3.NF.2 Understand a fraction as a number on the number line…
3.NF.3 Explain equivalence of fractions in special cases…
(Jenise) Take a moment to read this slide. These standards show from where you students come with experiences with fractions. The first goal in the development of fractions should be to help students construct the idea of fractional parts of the whole, the parts that result when the whole or unit has been partitioned into equal-sized portions or equal shares. This is done in the primary grades as you can see above.

4. Progression Continued…
3.NF.3 Explain equivalence of fractions in special cases…
4.NF.1 Explain why a fraction is equivalent to a fraction by using visual fraction models…
4.NF.2 Compare two fractions with different numerators and denominators…
5.NF.1 Add and subtract fractions with unlike denominators by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference…
(Jenise) Looking deeper into the 3rd grade standards the emphasis is on the use of visual models such as area models and number lines to explore the idea of equivalent fractions. This continued into fourth grade’s experience with fraction equivalents. Research has suggested that effectively using physical models in fraction tasks is important. The use of physical models lead to the use of mental models and this builds students’ understanding of fractions. Properly used tools can help students clarify ideas that are often confused in a purely symbolic form. You’ll find that textbooks rarely encourage manipulatives and when they do they tend to only use area models.

5. Fraction Progression By Concept
The Meaning of Unit Fractions
Equivalent Fractions
Comparing Fractions
Addition of Fractions
Graham

6. Unit 3 and Fractional Understanding
Area/Region Models
Length/Measurement Models
Here are two types of models for fractions discussed in the Van de Walle text and seen in unit 3. According to Van de Walle, length models should used more often in instruction because they are very important in the development of student understanding of fractions. Length models are introduced with the first task within unit 3. This task will help support the eventual use of the number line to think about fractional problems.

7. New Tasks Red Rectangles Pattern Block Puzzles Revisited
(Jenise) Red Rectangles actually involve set models. This task provides another opportunity for students to find patterns within equivalent fractions and development an understanding of what makes fractions equivalent. Since students will work with a fraction of a set, proportional reasoning comes into play. However, proportional reasoning is not the purpose of the task. Pattern Block Puzzles Revisited address area models and number lines as part of the culminating task. The number line is a more advanced length model and a powerful tool for thinking about fractions. Researchers suggest number lines be heavily empahsized in the teaching of fractions.

8. Conceptual Focus on Equivalence
How do you know 4/6 = 2/3? Try to think of at least two different explanations.

9. Do You Find Yourself Here?
They are the same because you can simplify 4/6 and get 2/3.
If you have a set of 6 items and you take 4 of them, that would be 4/6. but you can put the 6 items into 3 groups, and the 4 items would then be 2 groups of the 3 groups. That’s means it’s 2/3.
If you start with 2/3, you can multiply the numerator and the denominator by 2, and that will give you 4/6, so they are equal.
If you had a square cut into 3 parts and you shaded 2, that’s 2/3 shaded. If you cut all 3 of these parts in half, that would be 6 parts with 4 parts shaded, or 4/6.
The above responses show two different types of thinking, conceptual and procedural. Conceptual: Two fractions are equivalent if they are representations for the same amount or quantity, if they are the same number. Procedural: To get an equivalent fraction you have to use multiplication or division.

10. Students “get” simplifying when trying to identify equivalent fractions….
Bill McCallum says… “It is possible to over-emphasis the importance of reducing fractions …. There is no mathematical reason why fractions must be written in reduced form, although it may be convenient to do so in simple cases.”
The term simplifying fractions is purposefully omitted from the standards and here is what Bill McCallum and author of the Common Core says about simplifying….
And you might be asking what about the dreaded “test”…well my response to that is that, equal parts are equal whether or not they are simplified.

11. What’s My Point?
Most of us were taught procedures with fractions that were not based on fraction sense.
Emphasize number sense and meaning of fractions
Physical models and context
Extensive use of number lines
Allow students time to understand equivalence
Use benchmark fractions and estimation
Procedures are easy to follow but hard to understand and transfer

12. Misconceptions… Numerator and denominator are separate values
Fraction such as 1/5 is smaller than a fraction such as 1/10 because 5 is less than 10.
According to Van de Walle.
Finding fraction values on a number line or ruler can help students see fractions as single values. Many visual models and contexts that show parts of the same whole are essential to student understanding. Just being told the rule the bigger the denominator, the smaller the fraction without providing concrete examples and reasons may lead to overgeneralization.

13. What do you believe about learning?
Choose one- don’t tell what you believe, just remember for reference later.

14. With which statements do you agree?
My intelligence is something very basic about me which I can’t change very much.
I can learn new things, but I can’t really change how intelligent I am.
No matter how much intelligence I have, I can always change it quite a bit.
I can always substantially change how intelligent I am.
Again, choose those with which you agree, and remember them for later reference.
Research has shown that the brain is like a muscle- the more you exercise it, the stronger it becomes. Every time you try hard and learn something new, your brain forms new connections over time, and makes you smarter. Intellectual development is not the natural unfolding of a fixed intellegence, but rather the formation of new connections brought about through effort and learning.
Understanding that you have a growth mindset unleashes motivation due to the realization that your intellectual growth is largely in your hands.
quote from article about student taught about mindset- “Students were riveted by this information. The idea that their intellectual growth was largely in their hands fascinated them. In fact, even the most disruptive students suddenly sat still and took notice, with the most unruly boy of the lot looking up at us and saying, “you mean I don’t have to be dumb?”

15. What do you believe about learning?
The fixed and growth mindsets create two different psychological worlds. In the fixed mindset, students care most about how they will be judged: smart or not smart. Students reject opportunities to learn if they might make mistakes. When they do make mistakes or reveal deficiencies, they try to hide them rather than correct them. They are also afraid of effort, because effort makes them feel dumb. They believe if you have the ability, you shouldn’t need effort. This causes many bright students to stop working in school when the curriculum becomes challenging.
The student with a fixed mindset also doesn’t recover well from setbacks. When they hit a setback, they decrease effort and consider cheating. The fixed mindset doesn’t offer viable ways to improve so they seek tasks that will prove their intelligence and avoid ones that might not and the desire to learn takes a backseat. (all from Dweck’s “Perils and Promises of Praise” which I’ve pushed out)

16. What do you believe about learning?
Students with a growth mindset realize that when they make a mistake or exhibit a deficiency, they can correct it. They understand that their brains are growing all the time, especially when they examine mistakes. For them, effort is a positive thing, because it causes their intelligence to grow. In the face of failure, they increase their efforts and look for new learning strategies. They are determined to understand, because they believe they can. Not worrying about how smart they appear, they take on challenges and stick to them.

17. What do you believe about learning?
Alfred Binet, the inventor of the IQ test, had a strong growth mindset. Far from intending to measure fixed intelligence, he meant his tool to be used to identify students who weren’t benefitting from their current curriculum so that other ways to learn could be devised.

18. Want more? http://mindsetonline.com/whatisit/about/
So, as a teacher, we also have a mindset, not just about our students, but about ourselves. We either believe we can learn from mistakes, and so are vulnerable to making them, or we hide them and avoid growth opportunities. This also affects how we interact with students, and how our interactions can change and influence what they believe about their ability to learn. If you don’t read anything else, read the article pushed out- “The Perils and Promises of Praise”

19. Graphic from- http://www.rti4success.org/
Why RTI?
Screening- identification of who may be at risk for poor learning outcomes
Progress Monitoring
Progress monitoring is used to assess students' academic performance, to quantify a student rate of improvement or responsiveness to instruction, and to evaluate the effectiveness of instruction.
Multi-level Prevention System
The Multi-level Prevention System includes three levels of intensity or prevention. The primary prevention level includes high quality core instruction. The secondary level includes evidence-based intervention(s) of moderate intensity. The tertiary prevention level includes individualized intervention(s) of increased intensity for students who show minimal response to secondary prevention. At all levels, attention should be on fidelity of implementation, with consideration for cultural and linguistic responsiveness and recognition of student strengths.
Data-Based Decision Making
Data analysis and decision making occur at all levels of RTI implementation and all levels of instruction. Teams use screening and progress monitoring data to make decisions about instruction, movement within the multi-level prevention system, and disability identification (in accordance with state law).
Graphic from-

20. Why RTI? What does it mean to be mathematically fluent?
What prevents kids from becoming mathematically fluent?
How can we support mathematical fluency development?
Don’t get confused- this isn’t just fact or algorithmic fluency. To be truly mathematically fluent is to be numerate.
So- what does it mean to be numerate?
Numeracy is defined as the ability to reason and to apply simple numerical concepts. Basic numeracy skills consist of comprehending fundamental mathematics like addition, subtraction, multiplication, and division. For example, if one can understand simple mathematical equations such as, = 4, then one would be considered possessing at least basic numeric knowledge. Substantial aspects of numeracy also include number sense, operation sense, computation, measurement, geometry, probability and statistics. A numerically literate person can manage and respond to the mathematical demands of life. By contrast, the lack of numeracy or innumeracy can have a negative impact if the skills are absent. Numeracy has an influence on career professions, literacy, and risk perception towards health decisions.
(from wikipedia!)

21. GOT RTI?
These are the standards assessed using gloss and ikan- what do you notice?
All are about developing understanding of strategies.

22. GET STRATEGIES.
These strategies make all the other standards useful. If you’ve got the strategies, you’ve got the keys to the rest of the mathematics.
What does this have to do with mindset? Well, mathematics isn’t something you are either good at or not. Your ability isn’t fixed. You can learn these ideas and ways to understand number which are really useful in all computation, and which help your mathematical intelligence grow. So, you need a way to identify which strategies students have, and which need development, so they can grow as mathematicians.

23. The What? GloSS Global Strategy Stage IKAN
Individual Knowledge Assessment of Number

24. Why?

25. GET STRATEGIES.

26. GET STRATEGIES.

27. GET STRATEGIES.

28. GET STRATEGIES.

29. Want more?
Push prezi

30. Feedback http://ccgpsmathematicsk-5.wikispaces.com/
Turtle Toms-
Elementary Mathematics Specialist

31. Teachers,
Look- a gosling!

32. Turtle Toms Program Specialist (K-5) tgunn@doe.k12.ga.us
Thank You! Please visit to share your feedback, ask questions, and share your ideas and resources! Please visit to join the K-5 Mathematics listserve. Follow on Twitter!
Turtle Toms Program Specialist (K-5)
These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.