options: Use properties of operations to generate equivalent expressions.

Solve systems of equations.

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

Work with addition and subtraction equations.

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

Create equations that describe numbers or relationships

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

Write and interpret numerical expressions.

Use polynomial identities to solve problems.

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

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

Add and subtract within 20.

Represent and solve equations and inequalities graphically.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Write expressions in equivalent forms to solve problems.

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

Work with radicals and integer exponents.

Gain familiarity with factors and multiples.

Generate and analyze patterns.

Represent and analyze quantitative relationships between dependent and independent variables.

Multiply and divide within 100.

Analyze patterns and relationships.

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

Represent and solve problems involving addition and subtraction.

Understand the relationship between zeros and factors of polynomials.

Solve equations and inequalities in one variable.

Use the four operations with whole numbers to solve problems.

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

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

Interpret the structure of expressions.

Perform arithmetic operations on polynomials.

Reason about and solve one-variable equations and inequalities.

Represent and solve problems involving multiplication and division.

Rewrite rational expressions.
problem_activity: Warm-up
The purpose of an Estimation warm-up is to practice the skill of estimating a reasonable answer based on experience and known information, and also help students develop a deeper understanding of the meaning of standard units of measure. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it (MP3). Asking yourself, “Does this make sense?” is a component of making sense of problems (MP1), and making an estimate or a range of reasonable answers with incomplete information is part of modeling with mathematics (MP4).
Launch
Display the image for all to see. Ask students to silently think of a number they are sure is too low, a number they are sure is too high, and a number that is about right, and write these down. Then, write a short explanation for the reasoning behind their estimate.
Student Facing
[IMG]
The book in the center is 276 pages long. What is the mean number of pages of the 3 books?
Record an estimate that is:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "too low"}, {"_attributes": {"scope": "col"}, "_value": "about right"}, {"_attributes": {"scope": "col"}, "_value": "too high"}]}]}], "tbody": [{"tr": [{"td": [{}, {}, {}]}]}]}]}
Explain your reasoning.
Student Response
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Activity Synthesis
Select a few students to share their estimate and their reasoning. If a student is reluctant to commit to an estimate, ask for a range of values. Display these for all to see in an ordered list or on a number line. Add the least and greatest estimate to the display by asking, “Is anyone’s estimate less than
\(\underline{\hspace{.5in}}\)
? Is anyone’s estimate greater than
\(\underline{\hspace{.5in}}\)
?” If time allows, ask students, “Based on this discussion, does anyone want to revise their estimate?”
Then, reveal the actual value and add it to the display. The books are 830, 276, and 32 pages long, so the average number of pages in the books is about 379 pages. Ask students how accurate their estimates were, as a class.
“Is the actual value inside the range of values?”
“Is 379 near the middle of the estimates?”
“How variable are the estimates?”
“What are the sources of errors?”
Consider developing a method to record a snapshot of the estimates and actual value so students can track their progress as estimators over time.


correct_option_index: 26