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Make sense of quantities. How do quantities fit into the problem. Abstract and Recontextualize. Mathematical Practice:. Construct viable arguments and critique the reasoning of others. Build logical progression of statements to explore conjectures Recognize and use counterexamples

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Make sense of quantities

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Make sense of quantities

Make sense of quantities

How do quantities fit into the problem

How do quantities fit into the problem

Abstract and recontextualize

Abstract and Recontextualize

Mathematical practice

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 th walls with windows

5th Walls With Windows

Draw a picture

Draw a picture

Use a table

Use a table

Build on previous knowledge or calculations

Build on previous knowledge or calculations

Mathematical practice1

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

Singing at the ballpark 4th

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?

Interpret results into context to check for reasonableness

Interpret results into context to check for reasonableness

Check for reasonableness

Check for Reasonableness

Check for reasonableness1

Check for Reasonableness

You can’t leave someone behind!!!!

Solve a design problem 5th

Solve a design problem – 5th

Real life situation

Real Life Situation

Write an equation to model

Write an equation to model

Map relationships between quantities

Map relationships between Quantities

Mathematical practice2

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

Strategy slope

Strategy - Slope

Tools calculator

Tools - Calculator

Tools calculator strategy convert to decimals

Tools Calculator – Strategy Convert to decimals

Mathematical practice3

Mathematical Practice:

Attend to precision.

  • Communicate precisely with clear definitions

  • State meaning of symbols

  • Calculate accurately, efficiently, and use appropriate level of precision

Have a clear definition

Have a Clear Definition

Teacher comments on 2-4 tests: Why do the tasks use the word angle when our textbooks use “vertex”?

Words are tools for thinking

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?

Cognitive demand

Cognitive Demand

  • How many dimensions are there?







Mathematical practice4

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

5 th box of cubes structure

5th- Box of Cubes- Structure

Making sense of structure

Making Sense of Structure

Seven most important words to transform education: How did you

Figure that out?

7 th grade freezing in fargo

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.

Mathematical practice5

Mathematical Practice:

Look for and express regularity in repeated reasoning.

  • Look for repeated calculations in both general methods and for short cuts

6 th freezing in fargo

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.

Tools for practices and standards

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



  • website

  • Inside directly or through link in NCSM

Make sense of quantities

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

Practices require content

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

Butterfly and moth collection

Butterfly and Moth Collection

How much longer was the longest wingspan

than the shortest?

Research suggests

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

Knowing fractions

Knowing Fractions

Knowing fractions1

Knowing Fractions

Preparing for geometry

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.

Purpose of resources

Purpose of Resources

Developing algebraic thinking

Developing Algebraic Thinking

Describe T pattern 5.

Lattice fence 6th

Lattice Fence – 6th

Define the pattern to explore relationships.

Aussie fir tree algebra

Aussie Fir Tree - Algebra

Proportional reasoning

Proportional Reasoning

Rate concentrate 6th

Rate Concentrate – 6th

Using the 8 th grade test

Using the 8th Grade Test

8 th grade tool

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?

Implementing formative assessment lessons forming professional learning communities

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?

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