Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Irrational Numbers
- Number Line
- Rational Numbers
- Real Numbers
Purpose and Instructional Strategies
In grade 7, students expressed rational numbers using terminating and repeating decimals. In grade 8, students define irrational numbers and determine their approximate location on a number line. In Algebra 1, students will expand their knowledge of the real number system to understand that complex numbers exist from their work with quadratics.
- Instruction includes using and creating a graphic organizer to show the relationship between the subsets of the real number system.
- Once students understand that (1) every rational number has a decimal representation that either terminates or repeats, and (2) every terminating or repeating decimal is a rational number, they can reason that on the number line, irrational numbers must have decimal representations that neither terminate nor repeat.
- Students sometimes overgeneralize that all square roots are rational numbers concluding that irrational numbers are unusual and rare. Instruction includes a variety of examples of irrational numbers. Irrational numbers are much more plentiful than rational numbers, in the sense that they are denser in the real number line.
- Instruction includes the understanding that the square root of a whole number is either another whole number or is irrational. When the result is another whole number, the original whole number is a perfect square. This fact is particularly relevant when the Pythagorean Theorem is applied to find a missing side length of a triangle whose other two side lengths are whole numbers.
- Instruction includes the understanding that the cube root of a whole number is either another whole number or is irrational. When the result is another whole number, the original whole number is a perfect cube.
- Instruction includes the understanding that adding or subtracting a rational number and an irrational number produces an irrational number. The same is true of multiplication or division unless the rational number is 0.
- Students should develop estimating skills when working with square roots without the use of a calculator. One strategy is to use benchmark square roots to determine an approximate value.
- For example, to find an approximation of , first 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.
Common Misconceptions or Errors
- Students may incorrectly believe that pi (π) is a rational number since they have only been introduced to a decimal approximation and a fraction approximation. To address this misconception, instruction includes looking further at the decimal representation of pi (π) so that students will notice that a pattern will not emerge. In fact, a pattern never emerges, therefore, pi is irrational.
- Students may incorrectly think that the number line only has the numbers that are labeled.
- Students may incorrectly think a numerical expression that includes addition or subtraction cannot be placed on a number line.
- For example, 2 + can be placed on a number line at approximately 3.73.
Strategies to Support Tiered Instruction
- Teacher provides opportunities to look at the decimal representation of pi (π) and comparing decimal representations of rational numbers and irrational numbers.
- Teachers provide opportunities for practice with performing decimal expansion of rational and irrational numbers to check for a pattern.
- Teacher models how to estimate a numerical expression with addition or subtraction and locates a place on the number line.
- o For example, show a representation of pi (π) and compare it to a decimal representation of . Students should identify that is a rational number and repeats 0.3333, and that is pi (π = 3.1415…) and doesn’t repeat.
- For example, a first approximation of 3 + , students could approximate as 2.25 since 5 is between the perfect squares of 4 and 9, but closer to 4. Therefore would be in between 2 and 3, but closer to 2. So, a reasonable guess for can be 2.25 and therefore a reasonable estimate for 3 + would be 3 + 2.25 which equals 5.25. If more accuracy is required, students should understand that a calculator is needed.
- Instruction includes looking further at the decimal representation of pi (π) so that students will notice that a pattern will not emerge. In fact, a pattern never emerges, therefore, pi is irrational.
Instructional Task 1 (MTR.1.1)
Part A. Provide an example of a rational number and explain how you can determine that it is rational.
Part B. Choose which number(s) below are irrational and explain how you can determine that they are irrational.
e) −4 +
Instructional Task 2 (MTR.4.1)
Part A. Use the number lines below to estimate the value of
. Explain why you put the points where you did.
Part B. Plot 1 +
on a number line. Explain your process with a partner.
Instructional Item 1
Plot the following numbers on a number line showing their approximate location to the nearest hundredth.
a. π − 2Instructional Item 2
b. − ( π)2
d. 2 +
Is 0.12345 a rational or irrational number? Explain your answer.
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.