Our Goal

"Understanding Multiplication of Exponents"
To create an instructional sequence (a.k.a. lesson plan) featuring:
- different models to present the concept
- select problems to facilitate student understanding
- explanations to bring about student understanding
- common student misconceptions
- ways to address misconceptions
- connections to real world experiences
- connections among mathematical ideas

Lesson Plan for Junior-High School Exponents

Lesson Plan for Junior-High School Exponents
A concept-map of multiplication of exponents

Student Misconceptions

Common mistakes that students make ..
Multiplying base and exponent     
     Incorrect:    23 ≠ 2 x 3 
     Correct:    22 x 2 x 2 = 8   

Exponents "distribute"

     Incorrect:  (2 + 4)3 ≠ 23 + 43
     Correct:    (2 + 4)3 = 63 = 6 x 6 x 6 =216

Anything to the power of "zero" is? ...

     Incorrect:  70 ≠ 0 
     Correct:    70 = 1  
     Proof:       70 = 7(m-m) = 7÷ 7m = 1 (anything divided by itself is just "1") 

Scientific Notation! A negative on an exponent and a negative on a number ... uhh? Are they the same? 
     Correct:  0.00036 = 3.6 × 10-4 
                 -0.00036 = -3.6 × 10-4                  
                 36,000 = 3.6 × 104
                 -36,000 = -3.6 × 104
A negative on an exponent and a negative on a number are equal 
     Incorrect:   -42 ≠ 16 
     Correct:    -42 = -16
     Correct:    -42 = -(4)= -16 
     Correct:    (-4)2 = 16 




How to fix this? 
A common misconception that can occur with instances 3 is that students believe that since
(3 x 2)² = 3² x 2² then (3 + 2)² = 3² + 2², or the exponent rules when multiplying numbers with exponents is applicable to all operations.
Perhaps further examples can be used to illustrate this point:
i.e. (pq)³ = (pq) x (pq) x (pq) = (p x p x p) x (q x q x q) = p³ x q³
(p + q)³ = (p + q) x (p + q) x (p + q) = use of distributive property to solve.

i.e. (3 x 4)³ = (3 x 4) x (3 x 4) x (3 x 4) = (3 x 3 x 3) x (4 x 4 x 4) = 3³ x 4³, which is same as 12³   but
i.e. (3 + 4)³ = (3 + 4) x (3 + 4) x (3 +4) = 7³, which is not the same as 3³ + 4³

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