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

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

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.

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

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.

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

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

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.

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

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.
problem_activity: Activity
The purpose of this activity is to bring together identities and the work students did earlier with rational equations. For each question, students first rewrite the fraction
\(\frac{2}{15}\)
using the given formula and then prove that the formula is an identity.
The whole-class discussion at the end of the activity should focus on how students showed that the formula is an identity. Monitor for groups who use different methods to share during the discussion. For example, they may identify different common denominators when adding together the fractions in the expression on the right side of the formula.
Launch
Arrange students in groups of 2. Tell students to complete each problem individually, and then compare their work with their partner. Partners should discuss any similarities and differences in their methods, and decide which method they prefer.
If students need some practice adding fractions with unlike denominators, begin the activity by displaying the equation
\(\frac78 = \frac12 + \frac13 + \frac{1}{24}\)
for all to see and ask students how they can figure out without using a calculator that the equation is true. After some quiet work time, select 1–2 students to share their thinking about why the equation is true. The purpose of looking at this equation before the main activity is to make sure students recall that adding fractions requires a common denominator.
Speaking: MLR8 Discussion Supports.
Use this routine to support students in producing statements justifying their identity formulas. Provide sentence frames for students to use when comparing their methods and discussing which method they prefer, such as “I prefer _____ because _____ .”, or “We both have _____ which represents _____ .”
Design Principle(s): Support sense-making
Student Facing
[IMG]
In Ancient Egypt, all non-unit fractions were represented as a sum of distinct unit fractions. For example,
\(\frac49\)
would have been written as
\(\frac13 + \frac19\)
(and not as
\(\frac19+\frac19+\frac19+\frac19\)
or any other form with the same unit fraction used more than once). Let’s look at some different ways we can rewrite
\(\frac{2}{15}\)
as the sum of distinct unit fractions.
Use the formula
\(\frac{2}{d}=\frac{1}{d}+\frac{1}{2d}+\frac{1}{3d}+\frac{1}{6d}\)
to rewrite the fraction
\(\frac{2}{15}\)
, then show that this formula is an identity.
Another way to rewrite fractions of the form
\(\frac{2}{d}\)
is given by the identity
\(\frac{2}{d} = \frac{1}{d} +\frac{1}{d+1} + \frac{1}{d(d+1)}\)
. Use it to re-write the fraction
\(\frac{2}{15}\)
, then show that it is an identity.
Student Response
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Student Facing
Are you ready for more?
For fractions of the form
\(\frac{2}{pq}\)
, that is, fractions with a denominator that is the product of two positive integers, the following formula can also be used:
\(\frac{2}{pq}=\frac{1}{pr} + \frac{1}{qr}\)
, where
\(r = \frac{p+q}{2}\)
. Use it to re-write the fraction
\(\frac{2}{45}\)
, then show that it is an identity.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may not be sure how to find a common denominator when the denominators all involve variables. Remind them that a common denominator is a multiple of all of the denominators. If needed, students can guess possible common denominators, and then test them by checking whether they're divisible by each of the denominators.
Activity Synthesis
The purpose of this discussion is for students to share how they proved the given formulas are identities. Ask 2–4 previously identified groups who used different methods to share how they each showed that the formula is an identity, recording their strategies for all to see.


correct_option_index: 9