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Clarifications
Clarification 1: Within this benchmark, it is not the expectation to work with the number e.Clarification 2: Within this benchmark, the expectation is to plot, order and compare square roots and cube roots.
Clarification 3: Within this benchmark, the expectation is to use symbols (<, > or =).
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Irrational Numbers
- Rational Numbers
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 7, students expressed rational numbers with terminating and repeating decimals. In grade 8, students define irrational numbers, recognizing and expressing them in various forms, and students compare rational numbers to irrational numbers. In Algebra 1, students will perform operations with radicals. In Geometry, students will extend their understanding of radical approximations to weighted averages on a number line.- Students should have the opportunity to draw number lines with appropriate scales to plot the numbers to provide an understanding of where the numbers are in relation to numbers that are greater than and less than the number to be plotted.
- Students locate and compare rational and irrational numbers on the number line. Additionally, students understand that the value of a square root or a cube root can be approximated between integers.
- For example, to find an approximation of , two methods are described below, each using the nearest perfect squares to the radicand:
- determine the perfect squares 28 is between, which would be 25 and 36. The square roots of 25 and 36 are 5 and 6, respectively, so we know that is between 5 and 6. Since 28 is closer to 25, an estimate of the square root would be closer to 5.
- since 28 is located of the distance from 25 to 36, the is approximately located 11 of the distance from to . So, this reasoning gives the approximation + , which is about 5.27. This method is particularly relevant when students determine weighted averages on a number line in Geometry.
- For example, to find an approximation of , two methods are described below, each using the nearest perfect squares to the radicand:
- Students also recognize that every positive number has both a positive and a negative square root. The negative square root of is written as −.
- Instruction includes the use of technology, including a calculator.
Common Misconceptions or Errors
- Students may not understand that square and cube roots can be plotted on a number line.
Strategies to Support Tiered Instruction
- Instruction includes providing students with examples of square and cube roots for them to place on a number line and facilitating a conversation on understanding the value of each square and cube root.
- Teacher provides opportunities to co-construct number lines with appropriate scales and plot approximate values of cube roots and square roots.
- For example, provide partially completed examples of non-perfect square roots and non-perfect cube roots by using perfect square roots and perfect cube roots as benchmark quantities.
- Teacher provides support in recognizing that every positive number has both a positive and negative square root.
- For example, show examples of how multiplying two negative numbers gives a positive number, and how the square root of a number can be both positive and negative.
- Teacher assists students in writing an inequality to represent written statements.
- For example:
- Monique has more books than Mary.
- Mark has 5 pencils and Barry has 8.
- Animal Kingdom has at least 100 different species of animals.
- For example:
Instructional Tasks
Instructional Task 1 (MTR.6.1)Below are irrational and rational numbers.
- Part A. Order the numbers from least to greatest by plotting on a number line.
- Part B. Identify which numbers are irrational.
- Part C. Write an inequality that compares a rational number and an irrational number from the list.
Instructional Items
Instructional Item 1Plot −3.42857… on the number line below and explain how you determined its location.
Instructional Item 2
Using the chart below, compare the irrational and rational numbers shown.
Instructional Item 3
Plot the following cube roots on a number line , and .
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Problem-Solving Tasks
MFAS Formative Assessments
Students are asked to plot the square root of eight on three number lines, scaled to progressively more precision.
Students are asked to estimate the value of several irrational numbers using a calculator and order them on a number line.
Students are asked to graph three different irrational numbers on number lines.
Students are asked to find and interpret lower and upper bounds of an irrational expression using a calculator.
Student Resources
Problem-Solving Tasks
When students plot irrational numbers on the number line, it helps reinforce the idea that they fit into a number system that includes the more familiar integer and rational numbers. This is a good time for teachers to start using the term "real number line" to emphasize the fact that the number system represented by the number line is the real numbers. When students begin to study complex numbers in high school, they will encounter numbers that are not on the real number line (and are, in fact, on a "number plane"). This task could be used for assessment, or if elaborated a bit, could be used in an instructional setting.
Type: Problem-Solving Task
Students are given a pair of numbers. They are asked to determine which is larger, and then justify their answer. The numbers involved are rational numbers and common irrational numbers, such p and square roots. This task can be used to either build or assess initial understandings related to rational approximations of irrational numbers.
Type: Problem-Solving Task
Parent Resources
Problem-Solving Tasks
When students plot irrational numbers on the number line, it helps reinforce the idea that they fit into a number system that includes the more familiar integer and rational numbers. This is a good time for teachers to start using the term "real number line" to emphasize the fact that the number system represented by the number line is the real numbers. When students begin to study complex numbers in high school, they will encounter numbers that are not on the real number line (and are, in fact, on a "number plane"). This task could be used for assessment, or if elaborated a bit, could be used in an instructional setting.
Type: Problem-Solving Task
Students are given a pair of numbers. They are asked to determine which is larger, and then justify their answer. The numbers involved are rational numbers and common irrational numbers, such p and square roots. This task can be used to either build or assess initial understandings related to rational approximations of irrational numbers.
Type: Problem-Solving Task
