Getting Started |
Misconception/Error The student converts the numbers to standard notation in order to compare them. |
Examples of Student Work at this Level The student first converts all numbers to standard notation and then either correctly or incorrectly compares them.
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Questions Eliciting Thinking Do you think you could compare these numbers without converting them to standard notation?
Do you know how many times larger Â is than? Can you tell just by looking at the exponents?
What kind of a number is -5 x ? What kind of number is 3 x ? |
Instructional Implications If needed, provide the student with opportunities to order from least to greatest a set of numbers in standard notation and then ask the student to rewrite each number as a single digit times an integer power of 10. This will enable the student to observe the relationship between the ordered numbers in each form. Guide the student to reason about the size of numbers written in the form Â by inspecting a and n.
Give the student additional opportunities to write and compare negative numbers in the form , where n is an integer. |
Moving Forward |
Misconception/Error The student makes comparisons solely on the basis of the size of the exponent. |
Examples of Student Work at this Level For problem B the student says, â€śThe negative number does not matter; the exponent is bigger.â€ť
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The student justifies each comparison by saying, â€śBigger exponent.â€ť
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Questions Eliciting Thinking When you made your comparisons, did you look at the first number at all?
When looking at a number line, in which direction are the larger numbers? Where would these numbers be on a number line? Which number is further to the right? |
Instructional Implications Ask the student to convert the numbers in problems B and E to standard notation and then compare them. Make explicit the importance of considering the value of both parameters, (i.e., a and n, when comparing numbers in the form a x ). Give the student additional opportunities to write and compare negative numbers in the form a x , where n is an integer. |
Almost There |
Misconception/Error The student has a correct strategy for making comparisons but makes a minor error. |
Examples of Student Work at this Level The student reverses the direction of the greater than and less than symbols, but has explanations to support all correct answers.
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A justification given contradicts an answer.
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The student describes the number of zeros in the numbers when written in standard notation but does so incorrectly.
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Questions Eliciting Thinking Why did you use this symbol in your answer? What does it mean? How do you remember the difference between each of these symbols: < , > ? What words are associated with each one?
In this problem, I think the justification you gave does not support your answer. Can you check this one again?
If you converted 7 x Â to standard notation, what would it look like? How many of its digits are zero? |
Instructional Implications If the student reversed the inequality symbols, provide instruction on the meaning of the symbols, and then give the student additional opportunities to compare numbers using these symbols.
Have the student practice writing justifications for answers given. Pair the student with another Almost There or Got It student and ask the students to review the othersâ€™ justifications and ask questions as needed. Also consider pairing the Almost There student with a Getting Started or Moving Forward student to provide more opportunity to clearly explain his or her reasoning and to help the student who is struggling. |
Got It |
Misconception/Error The student provides complete and correct responses to all components of the task. |
Examples of Student Work at this Level The student uses an understanding of exponents and the relative sizes of positive and negative numbers to compare each pair of numbers. The student explains his or her decisions in a mathematically correct way.
Correct answers and possible explanations: (A) < ; the larger the exponent, the larger the number. (B) < ; the first is less than zero, the second is greater than zero; a negative number is always less than a positive number. (C) > ; the one with the negative exponent is smaller since it is a number between zero and one. (D) < ; the more negative the exponent, the smaller the number. (E) < ; the more negative the number, the smaller it is; when they are both negative numbers, the one closer to zero is bigger.
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Questions Eliciting Thinking Is it always the case that the larger the exponent, the larger the number? Can you think of an exception?
How is it that 6 xÂ is smaller than 4 x Â when six is greater than four?
Is it possible to say â€śalways look at the exponent when ordering numbers in scientific notation?' Why or why not? |
Instructional Implications Introduce the student to performing operations with numbers written in scientific notation. Consider using MFAS task Mixed Form Operations and other 8.EE.1.4 tasks.
Have the student practice converting numbers written with any number times a power of 10 into scientific notation, including numbers with decimals, (e.g., ask the student to write 173.4 x Â in scientific notation). |