Scientific Notation

Think back to your days in geometry. Do you remember the two-column proofs and the joy that would come with writing "by definition" as the reason to support a statement? There are just some things in mathematics that exist by declaration. By definition, the square root of negative one is represented with i. Why? Because the Italian mathematician Rafael Bombelli said so.

There is no "But why?" behind scientific notation either. It is a definition created to ease the cumbersome nature of incredibly large or microscopically small numbers. Scientific notation is a method of writing a number as a decimal multiplied by a power of ten. The definition restricts the decimal to a number greater than or equal to 1 and less than 10. It's still the same number...just written in a specified format.

Have you used place value and powers of ten to rewrite numbers according to the scientific notation format? Base ten blocks can model the decomposition of a number and serve as a beneficial launching pad for writing large numbers in scientific notation.

Scientific Notation with Base Ten Blocks

Once students grasp the definition, continue with the "Sort This!" activity to rewrite numbers from different forms including standard notation, product or quotient form, as a power of ten, or scientific notation. The "I have... Who has..." activity provides a quick informal assessment of converting numbers. Also the "RallyCoach" and "Stations" activities require students to solve problems involving numbers written in scientific notation.

Scientific Notation Resources

This activity set highlights Common Core State Standards 8.EE.A.3 and 8.EE.A.4 included in MATH-8 and Accelerated MATH-7.

The Distributive Property

How do you teach the distributive property? When planning lessons, be sure you are maintaining perspective. Ask: What have your students already experienced with the content? This is particularly critical when students have gaps and require support to bridge that gap. You will need to have the perspective of how the related content has developed over the years/courses so that you can replicate that progression for struggling students via individualized learning. Ask: What leaps will your students make in upcoming courses that stem from what you do with the content now? It is important to identify future learning so that you can ensure adequate depth in your course. Let's review the progression of the distributive property...

3.OA.B.5

In Grade 3, students use the distributive property as a strategy for multiplying. This is the concept builder for the distributive property. If students were unsuccessful when working strictly with numbers, then their efforts in applications with variables will be a stretch at best. Or worse yet...what if students memorized multiplication facts and never made sense of the concept through the distributive property?!?

6.EE.B.5

In Grade 6, students use the distributive property to produce equivalent expressions. Notice the example includes using the distributive property to toggle between factored form (the result from division) and expanded form (the result from multiplication).

7.EE.A.1

In Grade 7, students continue with the same concepts learned in Grade 6 with an extension to using rational numbers. Should 7th graders be solving equations that require using the distributive property? Continue reading.

8.EE.C.7b

In Grade 8 (and Accelerated MATH-7), students will use the distributive property to solve linear equations with rational coefficients. Notice the standard says "whose solutions REQUIRE expanding expressions using the distributive property..." in order to solve the linear equation.

Differentiate the necessity of the distributive property.

Perhaps MATH-7 students should be asked to solve equations that are similar to the equation in the left column above; however, this begs the question "How do we hope students solve that type of equation?". We also hope MATH-8 students recognize the need for the distributive property before they start the process of solving equations that are similar to the equation in the right column above.

This discussion highlights the Common Core State Standards 7.EE.A.1 and 8.EE.C.7b included in MATH-7, MATH-8 and Accelerated MATH-7.

Solving Systems of Equations

If MATH-8 teachers ever doubted that they teach algebra, diving into the Common Core State Standard 8.EE.C.8 would certainly end the debate. Students are required to solve and analyze pairs of simultaneous linear equations. The depth of this standard involves understanding the meaning of solution(s) of a system, solving a system graphically and algebraically, and using this knowledge to solve real-world problems that lead to a system of two linear equations.

The prerequisite for this content includes solving one-variable equations in Grade 7 and previously in Grade 8 as outlined in this Solutions of Linear Equations post. The tiling activity towards the end of the post would be a great review of types of solutions to set the stage for systems.

In a recent Genius Hour, we focused on how to teach 8th grade students to solve a system of linear equations algebraically. While it may be common to ask student to solve each equation for the variable "y" and set the two resulting expressions equal to solve for the variable "x", this process begs the question "Why?". Are we helping our students make connections so that they understand the origin of the algebraic method and know this process is valid? Do they understand the concept behind solving algebraically?

We looked at an introduction activity that extends from the "Balance This!" Activity in the Solutions of Linear Equations post. The visual of the balanced scale was used to discuss solutions and the adjustments necessary if the conditions are represented with two different scales. In this activity, students reason with a balance of shapes and a given value to determine the value of each shape.

Solving Systems of Linear Equations

We started with a school store scenario in which two different "store specials" were offered. One special involved a purchase of three items from the top shelf and two items from the bottom shelf for a total of $16. The second special involved a purchase of two items from the top shelf and three items from the bottom shelf for a total of $14.

• What are some possible values that keep the firsts scale balanced? How many different combinations can you find? How many combinations are possible?

• What about the second scale? How many total combinations are possible?

• What if you consider both scales simultaneously...meaning the possible values that keep the first scale balanced must also be the same values that keep the second scale balance? How many different combinations can you find? How many combinations are possible?

How much do the items on each shelf cost? Which shelf has the more expensive items? Justify your reasoning. Many students will use trial and error to begin solving the puzzle, but beware of students who verbalize a strategy that is actually substitution! The activity opens several paths, one of which is solving numerically and then making sense of how that cumbersome numeric approach connects to a systematic algebraic approach. The activity extends to all types of solutions and coordinates with several support activities that were reviewed.


These activities highlight the Common Core State Standard 8.EE.C.8 that is included in MATH-8 and Accelerated Algebra 1.