Math Misconceptions and Considerations 8.EE.1 8.EE.2 8.EE.3 8.EE.4
3 + 4 = 34 Look closely at errors in students’ work (formative assessment) to help you reflect and make instructional decisions to suit all students’ needs.
One area of difficulty in algebra is simplifying expressions with exponents. Exponents are used to convey repeated multiplication. Students seem to struggle to understand the rules of exponents. Students first, need to understand that a m = a ⋅ a ⋅ a ⋅ a ⋅⋅⋅⋅a (m factors of a). If they don’t get this, they will struggle to understand and apply the rules of exponents.
CONSIDERATIONS:
Without this understanding, students will likely make the mistake of multiplying the exponents instead of adding them when working with the Product of Powers rule. If the rule is taught in isolation, without conceptual development of its function, students are less likely to remember it and more likely to make mistakes. The same applies to the Powers of a Power rule. Students need practice working with concrete problems before being introduced to more abstract ones.
There are many more rules of exponents. All of which should be introduced conceptually FIRST, going back to the basic definition of exponentiation. Once students have an understanding of the basic concept the rule can be defined and committed to memory.
“The exponent tells how many zeros to place behind the number or in front of the number.� This statement is true in the case of 6 x 104 because the digit six is followed by four zeros (60,000). But is the statement also true with 6 x 10-4? If we add four zeros before six we get 00006, but where do we place the decimal point? .00006 is not correct; this is 6 x 10-5. Tricks may work for some cases but may not work for all and students can become easily confused. When algorithms are presented without conceptual understanding, they are more easily forgotten, do not build number sense, and can lead to errors. These algorithms should be developed after the concepts have been mastered.
MISCONCEPTION:
WHAT TO DO: Students should have experience writing numbers in a variety of ways as the number times a power of ten. By exploring various expressions for the same number students should see the evolving pattern of multiplying or dividing by ten. Writing numbers in scientific notation can then be shown as moving the decimal to the right or left.
Square roots are the opposite operation of applying exponents. You can undo a power with a radical and vice versa. For example, 22 = 4 and 4 = 2 . You can take any counting number, square it, and end up with a nice neat number. But it doesn’t always work going backwards. For example, 3 ≈ 1.732050808 ; you may round this to 1.7 but it is still not a nice neat number. A common misconception is that the answer for a square root is negative and positive. While (+2) and (-2) is squared to get four, the square root is only the positive option. This can be confusing when solving the equation x2 = 4, because you are trying to find all possible values that might be squared to equal four. However, when simplifying the expression 4 , the radical symbol, , tells us to only report the “principal” square root, which is the positive result. So while 4 has two square roots, 2 and 2, the expression 4 equals +2.
MISCONCEPTION:
WHAT TO DO:
Giving students visuals showing the difference between square roots and cube roots helps build their understanding. Show students pictures of two perfect squares or small perfect cubes and have them find the missing perfect square. Square Roots:
8 ?
6 36
8
64 49
6
6 x 6 = 36 62 = 36
?
? x ? = 49 x2 = 49
8 x 8 = 64 82 = 64
Cube Roots:
5 4
3
27
125
? 3
3 x 3 x 3 = 27 33 = 27
5
4
3 4 4x4x4=? 43 = ?
5 5 x 5 x 5 = 125 53 = 125
Students can be challenged to find solutions to equations such as x3 = 216. This will lead students to understand and 3 are numbers and not just an exercise to be done. For example, 3 8 is just another way of writing 2.
The following are things to consider when performing operations with numbers in scientific notation. ADDITION AND SUBTRACTION Students have had experiences with decomposing numbers since Kindergarten (K.OA.3). Students then expanded this knowledge as they learned to apply the distributive property to express a sum of two whole numbers with a common factor in 6th grade (6.NS.4). This understanding will help them tie meaning to adding and subtracting numbers written in scientific notation.
It’s important for students to make connections between whole number operations with scientific notation operations. The example to the left shows two ways to write the same expression. The 1st method distributes the 102 to decompose the expression. The 2nd method composes the 3.1 and 4.7 together. The goal is for students to recognize that both answers are equal to each other AND the exponents in the decomposed scientific notation form are the same. When we add or subtract numbers in scientific notation form the exponents must be the same.
Students can solve this problem using two different methods. The first method shown demonstrates converting scientific notation to standard form, performing the operation and converting it back to scientific notation. The benefit with this method is that students are more comfortable with whole numbers so they can use this method to check their work as they practice the second method. In the second method, students leave the numbers in scientific notation. This method ties to the distributive property example shown above. The key is to remember that it is necessary to distribute out the common factor‌ which means the exponents must be the same in the problem. It doesn’t matter which exponent you change. The example to the left demonstrates converting both exponents to three, which worked fine but required one additional step at the end to convert the answer to scientific notation.
MULTIPLICATION The laws of exponents apply when multiplying or dividing numbers written in scientific notation.
DIVISION