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

Use the four operations with whole numbers to solve problems.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Analyze patterns and relationships.

Add and subtract within 20.

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

Work with radicals and integer exponents.

Use polynomial identities to solve problems.

Interpret the structure of expressions.

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

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

Solve equations and inequalities in one variable.

Use properties of operations to generate equivalent expressions.

Represent and solve problems involving addition and subtraction.

Represent and solve equations and inequalities graphically.

Work with addition and subtraction equations.

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

Understand the relationship between zeros and factors of polynomials.

Create equations that describe numbers or relationships

Gain familiarity with factors and multiples.

Rewrite rational expressions.

Write expressions in equivalent forms to solve problems.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Represent and analyze quantitative relationships between dependent and independent variables.

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

Perform arithmetic operations on polynomials.

Represent and solve problems involving multiplication and division.

Reason about and solve one-variable equations and inequalities.

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

Solve systems of equations.

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

Generate and analyze patterns.

Write and interpret numerical expressions.

Multiply and divide within 100.
problem_activity: Activity
This activity is an opportunity for students to practice rewriting expressions using the distributive property. It is a step up from the same type of work in grade 6 because arithmetic with signed numbers is required.
The row with
\(6a-2b\)
is designed to allow students to figure out how to factor by reasoning based on structure they already understand, instead of learning how to factor based on a procedure that a teacher demonstrates first. The rows with
\(k(4 - 17)\)
and
\(10a-13a\)
are designed to prepare students for combining like terms in the next lesson.
Launch
Draw students’ attention to the organizers that appear above the table, and tell them that these correspond to the first three rows in the table. Let them know that they are encouraged to draw more organizers like this for other rows as needed.
Arrange students in groups of 2. Instruct them to take turns writing an equivalent expression for each row. One partner writes the equivalent expression and explains their reasoning while the other listens. If the partner disagrees, they work to resolve the discrepancy before moving to the next row.
Action and Expression: Internalize Executive Functions.
Provide students with blank diagrams to organize their work for the factored and expanded expressions. Include example diagrams labeled “expanded” and “factored.”
Supports accessibility for: Language; Organization
Student Facing
In each row, write the equivalent expression. If you get stuck, use a diagram to organize your work. The first row is provided as an example. Diagrams are provided for the first three rows.
[IMG]
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "factored"}, {"_attributes": {"scope": "col"}, "_value": "expanded"}]}]}], "tbody": [{"tr": [{"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42054"}, "_value": "\\(\\text- 3(5 - 2y)\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42097"}, "_value": "\\(\\text- 15 + 6y\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42137"}, "_value": "\\(5(a-6)\\)"}]}, {}]}, {"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "4097"}, "_value": "\\(6a-2b\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42180"}, "_value": "\\(\\text- 4(2w-5z)\\)"}]}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42214"}, "_value": "\\(\\text- (2x-3y)\\)"}]}, {}]}, {"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42247"}, "_value": "\\(20x-10y+15z\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42283"}, "_value": "\\(k(4-17)\\)"}]}, {}]}, {"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "4113"}, "_value": "\\(10a-13a\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42333"}, "_value": "\\(\\text- 2x(3y-z)\\)"}]}, {}]}, {"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42375"}, "_value": "\\(ab-bc-3bd\\)"}]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "42413"}, "_value": "\\(\\text- x(3y-z+4w)\\)"}]}, {}]}]}]}]}
Student Response
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Student Facing
Are you ready for more?
Expand to create an equivalent expression that uses the fewest number of terms:
\(\left(\left(\left(\left(x\strut+1\right)\frac12\right)+1\right)\frac12\right)+1\)
. If we wrote a new expression following the same pattern so that there were 20 sets of parentheses, how could it be expanded into an equivalent expression that uses the fewest number of terms?
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
If students are unsure how to proceed, remind them of tools and understandings they have seen recently that would be helpful. For example, “Draw an organizer and think about how the organizer represents terms in the expression.” Also, “Rewrite subtraction as adding the opposite.”
Activity Synthesis
Much of the discussion will take place in small groups. Display the correct equivalent expressions and work to resolve any discrepancies. Expanding the term
\(\text-(2x-3y)\)
may require particular care. One way to interpret it is to rewrite as
\(\text-1 \boldcdot (2x-3y).\)
If any confusion about handling subtraction arises, encourage students to employ the strategy of rewriting subtraction as adding the opposite.
To wrap up the activity, ask:
“Which rows did you and your partner disagree about? How did you resolve the disagreement?”
“Which rows are you the most unsure about?”
“Describe a process or procedure for taking a factored expression and writing its corresponding expanded expression.”
“Describe a process or procedure for taking an expanded expression and writing its corresponding factored expression.”
Writing: MLR3 Clarify, Critique, Correct.
Present an incorrect response for one of the expanded expressions in the table. For example, “An equivalent expression for
\(6a − 2b\)
is
\(4(a−b)\)
because
\(6−2\)
is 4 and
\((a−b)\)
is left on its own.” Prompt students to clarify any of the language and reasoning in the incorrect response and then to identify the error(s). Invite students to work with a partner to write a correct response. This helps students evaluate and improve on the written mathematical arguments of others.
Design Principle(s): Maximize meta-awareness


correct_option_index: 12