options: Interpret the structure of expressions.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Understand the connections between proportional relationships, lines, and linear equations.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Understand and apply properties of operations and the relationship between addition and subtraction.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Multiply and divide within 100.

Understand solving equations as a process of reasoning and explain the reasoning.

Work with equal groups of objects to gain foundations for multiplication.

Create equations that describe numbers or relationships

Use polynomial identities to solve problems.

Reason about and solve one-variable equations and inequalities.

Rewrite rational expressions.

Use properties of operations to generate equivalent expressions.

Solve equations and inequalities in one variable.

Represent and analyze quantitative relationships between dependent and independent variables.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Generate and analyze patterns.

Work with addition and subtraction equations.

Work with radicals and integer exponents.

Use the four operations with whole numbers to solve problems.

Solve systems of equations.

Understand the relationship between zeros and factors of polynomials.

Understand properties of multiplication and the relationship between multiplication and division.

Analyze patterns and relationships.

Write and interpret numerical expressions.

Represent and solve problems involving addition and subtraction.

Represent and solve equations and inequalities graphically.

Write expressions in equivalent forms to solve problems.

Add and subtract within 20.

Perform arithmetic operations on polynomials.

Represent and solve problems involving multiplication and division.

Gain familiarity with factors and multiples.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
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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for free access to Student Response.
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: 12