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Chapter 8 - Exploring Number & Operation Sense (Developing…
Chapter 8 - Exploring Number & Operation Sense
Developing Addition & Subtraction Operation Sense
Addition & Subtraction Problem Structures
Students with disabilities should be taught underlying structures so they can identify important characteristics of the situations
Problem types
Join (add to) initial/start amount, change amount/added, result amount/after change takes place (largest)
**
easiest
Separate (take from) Start is largest
Part-Part-Whole (put together/take apart) two parts combine into one
Compare(two quantities) third amount doesn't exist but is the difference
Change problems (join and separate)
Part-Part-Whole problems
Compare Problems
Problems are described in terms of
structure and interpertation
not as addition and subtraction
Teaching Addition & Subtraction
Word Problems need to be given in meaningful contexts
Focus class time on a few deep discussion questions instead of many brief questions
Have students correct other "students" work in order to practice going through the problem and thinking of the process
Use real world data for questions that link to student interests
Developing Multiplication & Division Operation Sense
Four classes of structures
equal groups
comparison
area
combinations
Equal Group
one number or factor counts how many sets, groups or parts of equal size are involved. Other factor tells size of each group and final factor is the whole or total of all the parts
Comparison
There are 2 different sets or groups. There are 3 possibilities for the unknown, the product, the group size and the number of groups
Array and Area
Model for an equal-group situation shown as a rectangle. Number of groups by number of items in that group. Area is different because of the unit attached to it
Combination
More complex of multiplication. Counting the number of possible pairings that can be made between 2 or more sets. Important in determining probablities
Teaching Multiplication & Division
Often taught separately but its important to combine soon after for students to see the inverse relationship
Hurdle is understanding that a group is a single entity but that it is also a group of containing a certain number of objects
Create problems that relate with students interest. They will develop more meaningful and spontaneous thinking when they are interested
Leave extra space on the paper for students to try multiple strategies and for them to explain their thinking
Representations of multiplication/division may look different but we can't be rigid in their meaning.
Properties of Multiplication & Division
Commutative & Associative Properties
the order of the factors doesn't matter. Have students use arrays to see their work can vary
Zero & Identity Properties
these are hard concepts for students to grasp. Encourage exploration of these by making arrays or finding patterns
Distributive Property
can split one of the factors into smaller factors then multiply all of those by the original leftover to get the same answer. visualize this thinking for students
Why not Division by 0?
students are used to hearing you just can't do it but why? explain that there is no answer because 0 is undefined, it just doesn't make sense
Strategies for Solving Contextual Problems
Analyzing Contextual Problems
think about the answer before solving the problem
ELL's need time to think about the vocab and context of the question along with the math that needs to occur
Work a simpler problem to practice the strategy needed to solve the more challenging one
Avoid the key word strategy!!
Students will ignore important parts of the problem because we are teaching them to focus on only key words. Key words are often misleading in quesitons
Multistep Word Problems
Give students a one-step problem and have them solve it. Then have them create a second problem for the rest of the class to solve.
Make a hidden question. Give different groups different questions. have them solve the first question, write a second question then write a combined question getting rid of the hidden question.
Give multistep problems to students and have them analyze and find the hidden question throughout the problem.