Integer Exponents

Middle school students generally enjoy working with patterns and find much success with number patterns. So discovering patterns with positive and negative exponents is a great way to conceptually teach integer exponents.

Integer Exponents Discovery Activity

The general laws of exponents can be developed from the number patterns discovered in the Integer Exponents RoundRobin activity and record sheet. And the discussion of base 10 will set the stage for scientific notation.

Integer Exponents Quiz-Quiz-Trade

Integer Exponents Quiz-Quiz-Trade

Integer Exponents Fan-N-Pick

This standard (CCSS 8.EE.A.1) will also bridge understanding to Algebra 1 as students explore geometric sequences and exponential functions.

Solutions of Linear Equations

Once students are fluent with the order of operations, they are ready to solve linear equations. When teaching the order of operations, I emphasize that we are simplifying numerical expressions. So when I provide the total value of a numerical expression BUT make one of the values in your expression a mystery, then my students have an equation to solve. And the process of solving for the mystery value requires students to use the order of operations in reverse.

2x + 5 = 17

What happened to the mystery value? It was doubled (or multiplied by 2) and then 5 was added for a total of 17. What happened last? The 5 was added to result in a total of 17. So take away (or subtract) 5. Now the total is 12...which is the result when I double the mystery value. So divide the total by 2 and you have revealed the mystery value of 6.

In middle school, students are moving from concrete thinking to abstract thinking...some a little more quickly than others. But our goal is to help all students find success dealing with abstract concepts. Solving linear equations may require the use of manipulatives to support concrete thinkers as they conceptualize this abstract concept.

And then when you add the possibility of multiple mysteries value that would reveal the same total result or no mystery value at all that would yield the total, the abstract becomes more abstract. Sarah over at Everybody is a Genius posted an excellent approach to Solving Special Case Equations. I love the introductory "puzzle" approach with balanced scales. I altered the format to a Showdown activity to provide structure for the discussion with the record sheet.

Balance This! Showdown Activity

Balance This! Record Sheet

Sarah also shared the notes her students created for their Interactive Notebooks to summarize their learning. I created the foldable in PowerPoint so it could be sent to the print shop if you prefer.

Types of Solutions Foldable (Student)

Types of Solutions Foldable (Teacher)

And this tiling activity is a great activity to informally assess student understanding of solution types.

Types of Solutions Tiling Activity

These activities highlight the Common Core State Standard 8.EE.C.7a included in MATH-8 and Accelerated MATH-7.

Decimal Form of a Rational Number

The progression of understanding rational numbers bridges between MATH-7 and MATH-8. (Note: This content is merged in Accelerated MATH-7.) And in the midst of discussing process teaching and concept teaching, we have highlighted the following standards.

7.NS.A.2d
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.

8.NS.A.1
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Hop over to the MATH-7 tab to see the Decimal Form of Rational Numbers post to jump start the conceptual understanding of these standards.

First ensure that students understand the classification of rational and irrational numbers. You could engage students in a sorting activity based on a few provided examples and counterexamples. (If you use all cards in the set, this activity will set the stage for Algebra 1 teachers to discuss the results of rational and irrational number operations.)

Sorting Activity

And then you could spin off of the MATH-7 activity of why some rational numbers terminate and others repeat to continue towards the concept of rewriting a repeating decimal as its equivalent fraction.

Algebraic Discovery

Would it be easier to teach the process for converting a repeating decimal to its equivalent fraction? Which benefits the students today, next month, next year? Process teaching or concept teaching?

Be sure to reflect on other related posts:

CCSS Call for Rigor

Process vs. Concept Teaching

Irrational Approximations

The Common Core State Standards are filled with number sense. Students engaged in the Common Core aligned curriculum will have a deep understanding of rational and irrational numbers...including a rational approximation of an irrational number (CCSS 8.NS.A.2).

How would you find the rational approximation of this number?

Ummm...without a calculator. ;)

Here's how we approach this one in my classroom...Well. Let's start with square roots that are rational. We know the square root of 20 is somewhere between the square root of 16 and the square root of 25...so something between 4 and 5. Can you be more specific? Hmmm...the list begins to form...
root 16 = 4
root 17 = ?
root 18 = ?
root 19 = ?
root 20 = ?
......................about 4.5
root 21 = ?
root 22 = ?
root 23 = ?
root 24 = ?
root 25 = 5

Since we have 9 numbers to cover an increase of 1 unit, each increase will be a little more than 0.1. Why? Because 1/9 > 1/10. So we can approximate that root 20 is approximately 4.4. {Ish! At this point we discuss if the "little more" is "enough more" to justify rounding.}

Find Someone Who

But what about simplest radical form?!? We've searched high and low. No sign of "simplest radical form" in our copy of the CCSS document. (If you find it...please put me out of my misery!)

Could we teach our students to simplify radicals and then in turn use that to find the rational approximation of an irrational number? In simplest radical form, the square root of 20 is exactly 2 times the square root of 5. Hmmm...the square root of 5 is some number between the square root of 4 and the square root of 9...between 2 and 3.

root 4 = 2
root 5 = ?
root 6 = ?
..................about 2.5
root 7 = ?
root 8 = ?
root 9 = 3

Since we have 5 numbers to cover an increase of 1 unit, each increase will be more than 0.1. Why? Because 1/5 > 1/10 (some students will automatically go to the fact that 1/5 = 0.2). So we can approximate that root 5 is approximately 2.2 and therefore 2(2.2) = 4.4.

Bottom line. We reach a valid approximation either way. I think it's important for our students to discover both paths...and even more important for them to critique which approach they prefer and under what circumstances they would select that particular method. I'm ready for a viable argument. How about you?!?

Cooperative practice would be crucial for students to have perspective to support their argument.

Quiz-Quiz-Trade or Inside-Outside Circle

This topic has a natural connection to Pythagorean Theorem application (CCSS 8.G.B.7). When students are determining the length of an unknown side in a right triangle, they should be able to state their solution as both an exact and approximate number. And how do we teach students to solve using the Pythagorean Theorem when they don't know the first thing about a quadratic equation? That will be for another post...after we finish investigating rational and irrational numbers...stay tuned!