Chapter 11 - Building Strategies for Whole-Number Computation (Toward…
Chapter 11 - Building Strategies for Whole-Number Computation
Toward Computational Fluency
manipulative or drawings along with counting to directly represent meaning of an operation students use a tool to help them think, such as grouping counters
not the standard algorithm but doesn't involve the use of physical materials or counting by ones. strategies based on place value, operations or relationship between addition and subtraction. Students can write down middle steps to help work through problems
making sense of the procedure as a process. Algorithms should have certainty, reliability, transparency, efficiency & generalizability. Understanding how algorithms work and when they are the best choice is central to development of
Techniques for fading direct modeling include recording verbal explanations, ask if they can do it mentally, have students write a numeric record of how they solved with physical models
Avoid immediately giving answers, encourage student dialog, clarify previous knowledge, curiousity, right & wrong ideas, strategic questioning, show samples of work
Grade 4 - Addition & subtraction with whole numbers
Grade 5 - multiplication with multi digit whole numbers Grade 6 - Division of multi digit whole numbers
Development of Invented Strategies in Addition & Subtraction
Models to support invented strategies
there are three; split strategy, jump strategy & shortcut strategy.
Adding Multi digit Numbers
there are various ways for students to solve three digit numbers. Add tens, ones then combine, add on tens then add on ones, move some to make a ten & use a nice number & compensate
Subtraction as "Think-Addition"
students who know the think-addition strategy for basic facts can use with multi digit. Successful for students with disabilities. Numbers in these problems encourage use of multiple strategies emphasizing place value
harder to do mentally. Usually is the first strategy students think of who have been taught standard algorithm but emphasize think-addition whenever possible
Empty number line to incorporate a jump strategy. More flexible than regular number line because it can go anywhere, no tick lines and spaces between.
Bar diagrams - work well for subtraction comparison model and part-part-whole. The
involves flexible adjustment of numbers.
Standard Algorithms for Addition & Subtraction
Standard algorithms for multi digit addition & subtraction are different than nearly every invented method. Starting with right most digits and being digit orientated, trading or regrouping. Students must know the value of numbers as well as understand the process
Invented Strategies for Multiplication
story problems help to understand the problem. We want to move them away from repeated addition.
Multiplication by a One-digit multipler
Let students model problems in ways that make sense to them.
Multiplication of Multidigit Numbers
make sure to discuss how to say "forty-eight hundred" as student may just tack on zeros without understanding why. students will need to see peers work to understand more efficient ways of multiplying
students not comfortable decomposing numbers into parts will see numbers in sets of single groups, based on repeated addition
same as standard algorithm but students always begin with largest values
students change problem into an easier one then make an adjustment to the problem. Strategy can't be used for all numbers but valuable for mental math & estimation
Standard Algorithms for Multiplication
Standard algorithm is one of the most difficult for students who haven't explored.
Begin with Models
the area model gives students a drawing to understand and support their spatial understanding and reasoning
don't focus on individual digits but instead on the concept of multiplying tens times tens is hundreds
students use a grid to organize their thinking along place-value columns
Invented Strategies for Divison
might start with multiples of 10. equally good for one-digit divisors as two-divisors
gives students a sense that problem can be solved in different ways and with different starting points. Pose a variety of starting points to nudge student's thinking
Standard Algorithms for Division
Begin with Models
long division starts with left-hand or biggest piece & is taught with the partition/ fair share model
Round up to the nearest 10 to underestimate the amount of groups. This approach works for students learning division for the first time and reduces the mental strain
Partition or Fair Share Model
students don't understand how 4 goes into the 5 of 583 initially. We have to set them up with this thinking, usually with a story problem. (5 cartons, 8 boxes, 3 bars)
Develop the Written Record
Record the number of pieces shared in all
Record the number of pieces remaining
Trade for smaller pieces
Teaching Computational Estimation
goal of teaching is to be able to flexibly & quickly produce an approx. result that will work for the situation & be reasonable
Computational Estimation Strategies
mental calculations are more complex because they require deep knowledge of how numbers work.
Use real examples
Use estimation language
use a context
accept a range of estimates
don't reward or emphasize a student that is closest
focus on flexible methods, not answers
ask for information but no answer