options: Understand the relationship between zeros and factors of polynomials.

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

Multiply and divide within 100.

Add and subtract within 20.

Represent and solve equations and inequalities graphically.

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

Create equations that describe numbers or relationships

Gain familiarity with factors and multiples.

Represent and solve problems involving addition and subtraction.

Use polynomial identities to solve problems.

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

Rewrite rational expressions.

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

Work with radicals and integer exponents.

Perform arithmetic operations on polynomials.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Use properties of operations to generate equivalent expressions.

Solve systems of equations.

Represent and solve problems involving multiplication and division.

Reason about and solve one-variable equations and inequalities.

Write and interpret numerical expressions.

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

Represent and analyze quantitative relationships between dependent and independent variables.

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

Solve equations and inequalities in one variable.

Interpret the structure of expressions.

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

Analyze patterns and relationships.

Work with addition and subtraction equations.

Use the four operations with whole numbers to solve problems.

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

Write expressions in equivalent forms to solve problems.

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

Generate and analyze patterns.
problem_activity: Activity
In this activity, students determine the length of a “diagonal” line segment on a grid. Students can give an exact value for the length of the line segment by finding the area of a square and writing the side length using square root notation. The goal of this activity is for students to connect values expressed using square roots with values expressed in decimal form—a form they are more familiar with.
Monitor for students who draw a tilted square for the first problem during the first two minutes of work time. Then monitor for students who use the following strategies to find the length of the segment:
drawing a square and finding the area
using tracing paper
using a compass to make a circle
Launch
For this activity, it is best if students do not have access to a calculator with a square root button. If student calculators do have a square root button that students are familiar with, tell students that their explanations about their answers to the second problem need to dig deeper than pressing a button. In later lessons, however, they will be able to use it.
Before students begin, remind students that “exact length” means it can’t just be an approximation, so if it is not a rational number, we should write it with square root notation. For example, a square with area 17 has a side length of exactly
\(\sqrt{17}\)
, which is a little larger than 4, since
\(4^2=16\)
.
Begin by displaying the diagram for all to see. Ask students how this diagram is similar and how it is different from the diagram in the warm-up. Then 2–3 minutes after students begin working, pause the class and select a previously identified student who drew a square on the grid to share what they did and why. Give 2–3 minutes work time to finish the problems followed by a whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge. Provide students with access to tracing paper, straight edges, and compasses to support information processing in estimating the length of the line segment.
Supports accessibility for: Visual-spatial processing; Organization
Student Facing
[IMG]
Estimate the length of the line segment to the nearest tenth of a unit (each grid square is 1 square unit).
Find the exact length of the segment.
Student Response
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Activity Synthesis
Select students to present in this sequence:
Someone who drew a square and used the area to find the exact side length.
Someone who used tracing paper. This is essentially like the number line as a ruler. Ask students what this tells us about the exact value we found with the square. (
\(\sqrt{10}\)
is about 3.1.)
Someone who used a compass to find the approximate side length. This is a more formal geometric construction, but it is just another way to use the number line as a ruler.
Speaking, Listening: MLR8 Discussion Supports.
As students share their methods for finding the exact length of the line segment, press for details in students’ reasoning by asking how they know the figure they drew in the coordinate plane is a square. Listen for and amplify the language students use to describe the important features of the square (e.g., opposite sides are equal, opposite sides are parallel, each angle is
\(90^\circ\)
). Then ask students to explain why the side length of the tilted square must be
\(\sqrt{10}\)
. This will support rich and inclusive discussion about strategies for finding the exact length of a line segment in the coordinate plane.
Design Principle(s): Support sense-making


correct_option_index: 13