Make sense of quantities

1. Make sense of quantities

2. How do quantities fit into the problem

3. Abstract and Recontextualize

4. Mathematical Practice: Construct viable arguments and critique the reasoning of others. • Build logical progression of statements to explore conjectures • Recognize and use counterexamples • Justify their conclusions and respond to other’s ideas using drawings, diagrams, actions

5. 5th Walls With Windows

6. Draw a picture

7. Use a table

8. Build on previous knowledge or calculations

9. Mathematical Practice: Model with mathematics. • Solve problems in everyday life • Write equation to describe a situation • Solve a design problem • Make assumptions and approximations to simplify a complicated situation • Interpret results to see if they make sense in terms of the situation

10. Singing at the Ballpark- 4th 21 students need to get to the ballpark. Each car will carry one adult and up to 4 students. 5 ¼ cars please?

11. Interpret results into context to check for reasonableness

12. Check for Reasonableness

13. Check for Reasonableness You can’t leave someone behind!!!!

14. Solve a design problem – 5th

15. Real Life Situation

16. Write an equation to model

17. Map relationships between Quantities

18. Mathematical Practice: Use appropriate tools strategically. • Use paper pencil, concrete models, ruler, protractor, calculator, spreadsheet, etc. • Tools might also include choosing an appropriate mathematical strategy

19. Strategy - Slope

20. Tools - Calculator

21. Tools Calculator – Strategy Convert to decimals

22. Mathematical Practice: Attend to precision. • Communicate precisely with clear definitions • State meaning of symbols • Calculate accurately, efficiently, and use appropriate level of precision

23. Have a Clear Definition Teacher comments on 2-4 tests: Why do the tasks use the word angle when our textbooks use “vertex”?

24. Words are tools for thinking Not having words to use limits the mathematics we can think about. – Harold Asturias, Lawrence Hall of Science Teacher comments: Why did you use the word “dimensions” on the 5th grade task box of cubes? Why didn’t you just ask for length, width, and height?

25. Cognitive Demand • How many dimensions are there?

26. Dimensions?

27. Dimensions?

28. Diagonal

29. Mathematical Practice: Look for and make use of structure. • Discern pattern or structure • See complicated things, such as algebraic expressions as single objects or as composed of several objects

30. 5th- Box of Cubes- Structure

31. Making Sense of Structure Seven most important words to transform education: How did you Figure that out?

32. 7th Grade Freezing in Fargo • How many times colder is Wednesday Feb. 25th than Tuesday Feb. 3rd? Almost 40% of the students in the sample subtracted.

33. Mathematical Practice: Look for and express regularity in repeated reasoning. • Look for repeated calculations in both general methods and for short cuts

34. 6th Freezing In Fargo • Which week (Sunday through Saturday) recorded the average lowest temperature? A student noticing that all the averages are divide by 7 days should realize that comparing totals will yield some comparative results without needing to divide.

35. Tools For Practices and Standards • Use MARS Tasks • Define the meaning of the standards and practices • Raise expectations for teachers about what students are capable of accomplishing • Help teachers anticipate misconceptions so that they can be surfaced and addressed in class discussion and re-engagement lessons

36. Resources • SVMIMAC.org website • Inside Mathematics.org directly or through link in NCSM

37. 3rd GradeCore IdeasRecognize and use characteristics, properties, and relationships of two-dimensional geometric shapes and apply appropriate techniques to determine measurements.Choose appropriate units and tools for particular tasks and use these units and tools to estimate and measure (length, weight, temperature, time, and capacity).Identify and compare attributes of two-dimensional shapes and develop vocabulary to describe the attributes.Calculate perimeter and area and be able to distinguish between the two measures. (Area may be measured by covering a figure with squares.)Use visualization, spatial reasoning, and geometric modeling to solve problems.Recognize geometric ideas and relationships and apply them to problems.MARS TasksLooking Glass LandTaskRubricCore Mathematical Ideas and ChallengesQuestions for Teacher ReflectionDiscussion of Successful Examples of Student WorkDiscussion of Student MisconceptionsGraph and Analysis of the MARS Task DataSummary of Student Understandings and MisunderstandingsImplications for Instruction TOOLS BY SUBJECTAlgebra & FunctionsAlgebraic Properties & RepresentationsData AnalysisFunctions & RelationsGeometry & MeasurementMathematical Reasoning & ProofsNumber OperationsNumber PropertiesPatterns, Functions & AlgebraProbabilityStatistics

40. Practices Require Content • Looking at and Understanding Number System • Using Place Value Strategies to Make Sense of and Solve Problems • Understanding Number Line as a basic mathematical concept and tool

41. Butterfly and Moth Collection How much longer was the longest wingspan than the shortest?

42. Research Suggests: • Number lines help students understand fractions as a “single number” instead of two – unique point or location on the line • Number line concepts and reading fractions can be introduced through rulers, clocks, scales • Number lines help students develop the ability to generalize about number and operations

43. Knowing Fractions

44. Knowing Fractions

49. Preparing for Geometry • To do the type of work needed to be successful in geometry, students need to have a variety of experiences at earlier grades. • Ideas build over time.

50. Purpose of Resources