Source:  Wiggins, G., & McTighe, J. Understanding by Design. Merrill Prentice Hall: 1998.
 For further information about Backward Design refer to http://www.ubdexchange.org/  

Title: _Fractional Parts_ Subject/Course: __Math ________

Topic: _Fraction Operations   Grade(s): ___6th___   Designer(s): _Devane/Rogers__
 

Stage 1 – Desired Results

Established Goal(s)

M6N1. Students will understand the meaning of the four arithmetic operations as related to positive rational numbers and will use these concepts to solve problems.

c. Determine the greatest common factor (GCF) and the least common multiple (LCM) for a set of numbers.

d. Add and subtract fractions and mixed numbers with unlike denominators.

g. Solve problems involving fractions, decimals, and percents.

(enter goals here)

Understanding(s) Students will understand that...

1.  Fractions must have common denominators before they can be added.

2.  Common multiples can help students rename fractions with common denominators.

3.  Unit fractions always represent one part of the whole.

4. Students will understand that all powers of two have a common multiple of two.

5. Fractions can be represented as a decimal value.



 



 

Essential Question(s)

1.     How are fractions with different denominators added?

2.  How can we find the least common multiple of a set of numbers?

3.  How do we determine successive powers of 2?

4.  How do we convert fractions to decimals?

 

Students will know...

…what is the least common multiple of a set of numbers.

…what are powers of two.

 

 

K

Students will be able to...

…add fractions with unlike denominators.

…convert fractions to decimals.

…recognize a pattern.

S

 

Stage 2 – Assessment Evidence

 

Performance Task(s)

Find the sum:


 

(Student will add the first seven fractions in this series.)

 

T

Key Criteria

Students should know that 2 to the seventh is the least common multiple of the denominators of the first seven powers of two.  The first seven fractions must be written as a fraction with a denominator of two to the seventh power.

   

Other Evidence

Solution:

1/2 = 64/128

1/2^2 = 32/128

1/2^3 = 16/128

1/2^4 = 8/128

1/2^5 = 4/128

1/2^6 = 2/128

1/2^7 = 1/128

The total is 127/128.

 

 

Stage 3 – Learning Plan

Learning Activities

 

1) Students will add a series of simple unit fractions to determine a sum.

1/2 + 1/4 = ?

1/2 + 1/4 + 1/8 = ?

1/2 + 1/4 + 1/8 + 1/16 = ?

1/2 + 1/4 + 1/8 + 1/16 + 1/32 = ?

 

2)  Change the solutions of each fractional sum above to a decimal value rounded to the nearest ten-thousandth?  What do you notice?

 

3)  Suppose the shading in the square below continues forever in the same pattern. What fractional part of the square is shaded? Can you change these fractions to decimal values rounded to the nearest ten thousandth?  What do you notice?  (Assume that all horizontal lines are parallel, all vertical lines are parallel, and all angles are right angles.)

 
 
Extensions:
 
What would happen if the square was cut into thirds, one third shaded in,  and then an additional third was cut into thirds and so on?  Can you draw an accurate example and follow through with the previous steps?  What fractional part of the square is shaded?