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Groupable Models for Place Value
Models that help students understand the concepts of ones, tens and hundreds through methods that include several objects that can be sectioned off into groups of ten. Models can include cups of beans, sticks, or cubes.
Pre-Grouped Models for Place Value
Models that help students understand the concepts of ones, tens and hundreds through methods that include objects that are previously created in groups of ten. For example, strips and squares, base ten blocks, and little ten frame cards all come in pre-grouped sections making children put the pieces together themselves, rather than counting out objects in groupable models.
Used by students who no longer need to understand how ten units makes "a ten" or by some students who need to return to place value- concepts as they struggle with more advanced computations. These models do not show the model for a ten as physically ten times larger than the one. Models include money or a bead frame with different colors rep[resenting different place values.
3 ways of counting sets of objects
- Direct modeling (Count by ones-> Use of base ten models)
- Student-Invented Strategies (Supported by written recordings-> Mental methods when appropriate)
- Traditional Algorithms (Use base-ten materials to model the steps. Prove that it produces a correct answer)
What are equivalent representations?
Strategies that give the equivalent representations of numbers. It also allows students to create their own representations.
When students are able to flexibly and quickly produce an approximate result for a computation that will be adequate for the situation.
The three developmental phases towards computational fluency
- Direct modeling
- Invented strategies
- Traditional algorithms
3 differences of invented strategies vs. traditional algorithms
- I.S.s are number oriented rather than digit oriented. Ex. an invented strategy for 68x7 begins 7x60 is 420 and 56 more is 476. The first product is 7 times 60, not the digit 6 like it would be in T.A.
- I.S.s are left-handed rather than right-handed. They begins with the largest numbers, those represented by the leftmost digits. For 26x47, invented strategies will begin with 20x40 is 800 providing some sense of the size of the eventual answer in just one step. T.A. begins with 7x6 is 42, hiding the result until the end.
- I.S.are flexible rather than "one right way." I.S. tend to change with numbers involved in order to make the computation easier. The T.A. suggests using the same tool on all problems.
The benefits of invented strategies
- Students make fewer errors
- Less reteaching is required
- Students develop number sense
- Invented strategies are the basis for mental computation and estimation
- Flexible methods are often faster than the traditional algorithms
- Algorithm invention is itself a significantly important precess of "doing mathematics."
What is computational fluency
Having efficient, flexible and accurate methods for computing.
Strategies for computational estimation
- Front-End- focus on the leading or leftmost digits in numbers, ignoring the rest. After and estimate is made on the basis if only these front-end digits, an adjustment can be made by noticing how much has been ignored.
- Rounding- changing the numbers in the problem ti others that are easy to compute mentally
- Compatible Numbers- when numbers ca be adjusted slightly to produce groups with benchmark values, making finding an estimate easier.
Place value mats
A worksheet that helps students organize their number count in order to better understand the place value when adding and subtracting.
base-10 block area model and 4 partial products
- models with sets of ten
- circular disks
- meter stick
- blank number lines
- ??? four partial products (be able to name the four partial products using both the physical model and Base 10 language and the symbols and be able to state the dimensions of each rectangle that represents each partial product - the 3 columns)???
The content of a lesson taught ABOUT problem solving
problem solving strategies
The content of a lesson taught THROUGH problem solving
the 5 Math Content Strands – Number & Operations, Algebra, Geometry, Measurement, Data Analysis & Probability
The three things that mathematics teachers should assess
- Concepts & Procedures
- Mathematical Processes
The 4 DOK Levels
- Level 1: Recall
- Level 2: Skill/Concept
- Level 3: Strategic Thinking
- Level 4: Extended thinking
the structure of the Mathematics Framework
5 Competencies across grade levels with objectives “under” each competency