1.  Modeling for Middle School Connecting Context with Math

2. Who am I?  Started teaching 7 years ago (Orosi, CA)  Intern = lowest on the totem pole  1 st year “Algebra for ALL” (8 th grade)  Math = NOT a priority  How do I reach these kids???  “Access” prior knowledge… Go WAY back  Teach Algebra by connecting it to primary math  “Math is so much easier now”

3. Now?  Started new position this year in Sanger, CA (8 th grade)  Personal goal = Improve every day so that I can help make math learning a positive experience for as many students as possible  Offer my learning and experiences – spark thought, creativity, collaboration  Not an expert (…yet!) – just a teacher who tries stuff!

4. What about YOU?  Name  From (Location)  Grade(s)/Role  Hope tolearn/do/accomplish inthis session?

5. What is Modeling?

6. Modeling in Middle School  The Common Core Standards focus heavily on pushing students towards abstract mathematical models (algebra) in middle school and examples with “un-friendly” (rational) numbers  Many students see algebraic models as something “new” – no connection  Our focus as educators should be on finding ways to connect concrete examples (that make sense to students) with the abstract mathematical representations of those models.

7. Concrete  Abstract  The problem is making the connection between each of these representations (they all mean the SAME thing)

8. Example: Adding Fractions Common Errors? Why do students make these errors? *Comprehension

9. Let’s Try: Adding Fractions

10. Questions Probing Comprehension What does ___ mean? Could you give an example? How could we show/represent/draw ___? How could you use your model/picture to demonstrate what happens? How can you prove/justify your answer using your model/picture? What if….? …other ideas???

11. Got Comprehension? NOW what?  CONNECT the understanding of concept with a mathematical representation:  “How can we represent our work with the model/picture just with math?”

12. Let’s go back: Adding Fractions

13. The REAL point of my time with you today…  Build comprehension through concrete and pictorial representations  Use these models to demonstrate and find the mathematical representation.  If students can “discover” the mathematical “short cut” (abstract) through a model they understand, it will make a lot more sense (and will carry over to more complex mathematics in the future)  CONTINUE ASKING the same questions – have students explain, justify, demonstrate, and connect the math to the original conceptual models

14. Something to Think About…  My personal belief: It isn’t so much about the models themselves as it is about students’ sense-making process.  If students have not used a particular strategy before, you will have to teach both the strategy itself, AND the math content relative to your grade level (and you thought time was an issue BEFORE!)  Sometimes it is more effective to leave it open ended and see what students come up with

15. Common modeling strategies  Area models: Nice connection with real life (area of a rectangle) and many different concepts that continue into higher levels of mathematics (multiplication, division, distribution, factoring, polynomials, estimating square roots, completing the square…)  Bar Models/Tape Diagrams: I like these best for word problems and representing numbers (fractions, percentages), but can be used for almost any type of problem. *Algebra tiles  *Number lines: Can be very helpful, but not innately understood by most students – have to build up understanding (this is already an abstraction in most situations)

16. Some resources for modeling strategies:  Grade level progression documents (a little more work, but show suggested strategies from lower grades that carry over): http://math.arizona.edu/~ime/progressions/ http://math.arizona.edu/~ime/progressions/  Tool for Bar Modeling with lots of video demos (apps that go along with this site are called “Thinking Blocks” but might be over-scaffolded for many students) http://www.mathplayground.com/ThinkingBlocks/thi nking_blocks_start.html http://www.mathplayground.com/ThinkingBlocks/thi nking_blocks_start.html  National Library of Digital Manipulatives: http://nlvm.usu.edu/en/nav/grade_g_3.html http://nlvm.usu.edu/en/nav/grade_g_3.html  Honestly… I google a lot… Use vetted resources (NCTM, illustrativemathematics, illuminations, etc.)

17. What concepts cause the most confusion for your students?  We will focus on models and strategies that support YOUR requests/needs

19. Connect with me! Mg.aoki@gmail.com @MarisaAoki

20. Shortcut to the Payoff: (what was my point)

22. 1 st #: Speaker was well-prepared and knowledgeable (0-3) 2 nd #: Speaker was engaging and an effective presenter (0-3) 3 rd #: Session matched title and description in program book (0-3)