make sense of quantities 22. how do quantities fit into the problem 23
TRANSCRIPT
![Page 1: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/1.jpg)
Make sense of quantities
1
![Page 2: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/2.jpg)
How do quantities fit into the problem
2
![Page 3: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/3.jpg)
Abstract and Recontextualize
3
![Page 4: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/4.jpg)
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
4
![Page 5: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/5.jpg)
5th Walls With Windows
5
![Page 6: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/6.jpg)
Draw a picture
6
![Page 7: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/7.jpg)
Use a table
7
![Page 8: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/8.jpg)
Build on previous knowledge or calculations
8
![Page 9: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/9.jpg)
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
9
![Page 10: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/10.jpg)
Singing at the Ballpark- 4th21 students need to get to the ballpark.Each car will carry one adult and up to 4 students.
5 ¼ cars please? 10
![Page 11: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/11.jpg)
Interpret results into context to check for reasonableness
11
![Page 12: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/12.jpg)
Check for Reasonableness
12
![Page 13: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/13.jpg)
Check for Reasonableness
You can’t leave someone behind!!!! 13
![Page 14: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/14.jpg)
Solve a design problem – 5th
14
![Page 15: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/15.jpg)
Real Life Situation
15
![Page 16: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/16.jpg)
Write an equation to model
16
![Page 17: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/17.jpg)
Map relationships between Quantities
17
![Page 18: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/18.jpg)
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
18
![Page 19: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/19.jpg)
Strategy - Slope
19
![Page 20: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/20.jpg)
Tools - Calculator
20
![Page 21: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/21.jpg)
Tools Calculator – Strategy Convert to decimals
21
![Page 22: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/22.jpg)
Mathematical Practice:
Attend to precision. Communicate precisely with clear definitions State meaning of symbols Calculate accurately, efficiently, and use
appropriate level of precision
22
![Page 23: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/23.jpg)
Have a Clear Definition
• Teacher comments on 2-4 tests: Why do the tasks use the word angle when our textbooks use “vertex”?
23
![Page 24: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/24.jpg)
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?
24
![Page 25: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/25.jpg)
Cognitive Demand
• How many dimensions are there?
25
![Page 26: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/26.jpg)
Dimensions?
26
![Page 27: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/27.jpg)
Dimensions?
27
![Page 28: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/28.jpg)
Diagonal
28
![Page 29: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/29.jpg)
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
29
![Page 30: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/30.jpg)
5th- Box of Cubes- Structure
30
![Page 31: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/31.jpg)
Making Sense of Structure
Seven most important words to transform education: How did youFigure that out? 31
![Page 32: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/32.jpg)
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.
32
![Page 33: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/33.jpg)
Mathematical Practice:
Look for and express regularity in repeated reasoning. Look for repeated calculations in both general
methods and for short cuts
33
![Page 34: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/34.jpg)
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.
34
![Page 35: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/35.jpg)
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
35
![Page 36: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/36.jpg)
Resources
• SVMIMAC.org website• Inside Mathematics.org directly or through
link in NCSM
36
![Page 37: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/37.jpg)
TOOLS BY SUBJECTAlgebra & FunctionsAlgebraic Properties & RepresentationsData AnalysisFunctions & RelationsGeometry & MeasurementMathematical Reasoning & ProofsNumber OperationsNumber PropertiesPatterns, Functions & AlgebraProbabilityStatistics
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
37
![Page 38: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/38.jpg)
38
![Page 39: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/39.jpg)
39
![Page 40: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/40.jpg)
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
![Page 41: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/41.jpg)
Butterfly and Moth Collection
How much longer was the longest wingspanthan the shortest?
![Page 42: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/42.jpg)
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
![Page 43: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/43.jpg)
Knowing Fractions
![Page 44: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/44.jpg)
Knowing Fractions
![Page 45: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/45.jpg)
![Page 46: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/46.jpg)
![Page 47: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/47.jpg)
![Page 48: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/48.jpg)
![Page 49: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/49.jpg)
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.
![Page 50: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/50.jpg)
Purpose of Resources
![Page 51: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/51.jpg)
![Page 52: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/52.jpg)
![Page 53: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/53.jpg)
![Page 54: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/54.jpg)
![Page 55: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/55.jpg)
![Page 56: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/56.jpg)
![Page 57: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/57.jpg)
![Page 58: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/58.jpg)
Developing Algebraic Thinking
Describe T pattern 5.
![Page 59: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/59.jpg)
![Page 60: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/60.jpg)
Lattice Fence – 6th
Define the pattern to explore relationships.
![Page 61: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/61.jpg)
![Page 62: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/62.jpg)
Aussie Fir Tree - Algebra
![Page 63: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/63.jpg)
![Page 64: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/64.jpg)
Proportional Reasoning
![Page 65: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/65.jpg)
![Page 66: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/66.jpg)
Rate Concentrate – 6th
![Page 67: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/67.jpg)
![Page 68: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/68.jpg)
Using the 8th Grade Test
![Page 69: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/69.jpg)
8th Grade - Tool
• Understand the 8th grade Mathematics Common Core Course and the deep rich mathematical expectations for students
• Includes rich Algebra Strand but also works to expand and deepen understanding and facility with other strands
• Use as Placement Test or Summative Test - Could your 7th graders pass this test? Could your Algebra and Geometry students meet standard on test?
• Facilitate Course Discussion with Staff and Parents – Is this the mathematics we want students to be able to know and do?
![Page 70: Make sense of quantities 22. How do quantities fit into the problem 23](https://reader035.vdocument.in/reader035/viewer/2022081602/5514bd5e55034640138b56c9/html5/thumbnails/70.jpg)
Implementing Formative Assessment Lessons- Forming Professional
Learning Communities
– How do we create classroom culture?– How do we facilitate “staff” doing mathematics
together to understand purpose of the lesson, understand the mathematics before giving it to students, work as a learning community to discuss techniques, purpose of using student work, types of comments to put on student papers, how to have a plenary discussion? What are the important mathematical ideas to be drawn from students? What teacher moves help hold all students accountable for their own learning?
– Making time for lessons?