standard_description: Use the structure of an expression to identify ways to rewrite it. For example, see x^(4) – y^(4 )as (x^(2))^(2 )– (y^(2))^(2), thus recognizing it as a difference of squares that can be factored as (x^(2 )– y^(2))(x^(2 )+ y^(2)).
problem_activity: Optional activity
This activity extends the work with rectangles and fractions to continued fractions. Continued fractions are not a part of grade-level work, but they can be reasoned about and rewritten using grade-level skills for operating on fractions (MP8). In particular, the insight that
\(\frac{1}{\frac{a}{b}}=\frac{b}{a}\)
(a special case of invert and multiply) is helpful.
In this activity, students consolidate their understanding about how the greatest common factor of the numerator and denominator of a fraction can help them write an equivalent fraction whose numerator and denominator have greatest common factor 1—sometimes called “lowest terms.”
Launch
Arrange students in groups of 2. Students work alone and compare their work with a partner.
Consider a quick warm-up like this:
Write a fraction that is equal to each expression:
\(3+\frac15\)
\(\frac{1}{3+\frac15}\)
\(2+\frac{1}{3+\frac15}\)
Representation: Develop Language and Symbols.
Eliminate barriers and provide concrete manipulatives to connect symbols to concrete objects or values. For example, provide students with access to the four large rectangles prepared on graph paper.
Supports accessibility for: Visual-spatial processing; Fine-motor skills
Student Facing
Accurately draw a 37-by-16 rectangle. (Use graph paper, if possible.)
In your rectangle, draw a line segment that decomposes the rectangle into a new rectangle and a square that is as large as possible. Continue until your original rectangle has been entirely decomposed into squares.
How many squares of each size are there?
What are the dimensions of the last square you drew?
What does this have to do with
\(2+\frac{1}{3+\frac15}\)
?
Consider a 52-by-15 rectangle.
In your rectangle, draw a line segment that decomposes the rectangle into a new rectangle and a square that is as large as possible. Continue until your original rectangle has been entirely decomposed into squares.
Write a fraction equal to this expression:
\(3+\frac{1}{2+\frac17}\)
.
Notice some connections between the rectangle and the fraction.
What is the greatest common factor of 52 and 15?
Consider a 98-by-21 rectangle.
In your rectangle, draw a line segment that decomposes the rectangle into a new rectangle and a square that is as large as possible. Continue until your original rectangle has been entirely decomposed into squares.
Write a fraction equal to this expression:
\(4+\frac{1}{1+\frac{7}{14}}\)
.
Notice some connections between the rectangle and the fraction.
What is the greatest common factor of 98 and 21?
Consider a 121-by-38 rectangle.
Use the decomposition-into-squares process to write a continued fraction for
\(\frac{121}{38}\)
. Verify that it works.
What is the greatest common factor of 121 and 38?
Student Response
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