options: Use properties of operations to generate equivalent expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

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

Perform arithmetic operations on polynomials.

Use the four operations with whole numbers to solve problems.

Rewrite rational expressions.

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

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

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

Multiply and divide within 100.

Add and subtract within 20.

Represent and solve equations and inequalities graphically.

Reason about and solve one-variable equations and inequalities.

Generate and analyze patterns.

Gain familiarity with factors and multiples.

Work with radicals and integer exponents.

Analyze patterns and relationships.

Solve equations and inequalities in one variable.

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

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

Understand the relationship between zeros and factors of polynomials.

Represent and solve problems involving addition and subtraction.

Represent and solve problems involving multiplication and division.

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

Solve systems of equations.

Work with addition and subtraction equations.

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

Write and interpret numerical expressions.

Create equations that describe numbers or relationships

Write expressions in equivalent forms to solve problems.

Interpret the structure of expressions.

Use polynomial identities to solve problems.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Understand properties of multiplication and the relationship between multiplication and division.
problem_activity: Activity
In previous activities and lessons, students found the exact area of a square in order to find an approximation for the square root of an integer. In this activity, students start with a square root of an integer, and draw a square to verify that a given approximation of the square root is reasonable. This is the first time students have seen or drawn squares that do not have vertices at the intersection of grid lines, so it may take them a few minutes to make sense of the new orientation.
Launch
Give students access to four-function calculators. Display the diagram for all to see. Ask students what is the same and what is different about this diagram and diagrams they have seen in earlier activities. Arrange students in groups of 2. Give students 2–3 minutes of quiet work time followed by partner and whole-class discussions.
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer for data collection and organizing information.
Supports accessibility for: Language; Organization
Writing, Speaking, Listening: MLR1 Stronger and Clearer Each Time.
After students have had time to identify a point on the number line that is closer to
\(\sqrt{3}\)
, ask them to write a brief explanation of their reasoning. Give students time to meet with 2–3 partners, to share and get feedback on their writing. Display prompts that students can ask each other that will help students strengthen their ideas and clarify their language. For example, “How do you know that
\(\sqrt{3}\)
is between 1.5 and 2?” and “How did you determine the area of the square that you drew?” Students can borrow ideas and language from each partner to refine and clarify their original explanation. This will help students refine both their ideas and their verbal and written output.
Design Principles(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
Diego said that he thinks that
\(\sqrt{3}\approx 2.5\)
.
[IMG]
Use the square to explain why 2.5 is not a very good approximation for
\(\sqrt{3}\)
. Find a point on the number line that is closer to
\(\sqrt{3}\)
. Draw a new square on the axes and use it to explain how you know the point you plotted is a good approximation for
\(\sqrt{3}\)
.
Use the fact that
\(\sqrt{3}\)
is a solution to the equation
\(x^2 = 3\)
to find a decimal approximation of
\(\sqrt{3}\)
whose square is between 2.9 and 3.1.
Student Response
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Student Facing
Are you ready for more?
A farmer has a grassy patch of land enclosed by a fence in the shape of a square with a side length of 4 meters. To make it a suitable home for some animals, the farmer would like to carve out a smaller square to be filled with water, as in the figure.
What should the side length of the smaller square be so that half of the area is grass and half is water?
[IMG]
Student Response
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for free access to Extension Student Response.
Activity Synthesis
Invite one or two students to share their squares. Then tell students, “The square of a point on the number line can be visualized as the area of a literal square. This can help us estimate the value of a square root. Simply squaring the number can as well. Let’s check the squares of some numbers that are potential approximations of
\(\sqrt{3}\)
.”
Then ask students to suggest decimal approximations, and check together as a class by finding their squares. Students should be using each guess to make a better guess next. For example, if they try 1.5, then the square is 2.25, which is too low. This suggests trying bigger. Because we know that 2 is too big (because
\(2^2 = 4\)
, it should be somewhere in between 1.5 and 2.) For example, students might suggest this order:
\(1^2 = 1\)
and
\(2^2=4\)
\(1.5^2 = 2.25\)
\(1.8^2 = 3.24\)
\(1.7^2 = 2.89\)
\(1.72^2 = 2.9584\)
\(1.73^2 = 2.9929\)
So 1.73 is a pretty good approximation of
\(\sqrt{3}\)
.


correct_option_index: 15