options: Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
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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correct_option_index: 4