Getting Started |
Misconception/Error The student attempts to convert the numbers to scientific notation or the formÂ Â but does so incorrectly. |
Examples of Student Work at this Level The studentÂ writes 39.9 xÂ .
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The studentÂ writes 0.529 xÂ Â and says, â€śI put the decimal in front of the first number.â€ť
The student counts the zeros to determine the exponent before estimating to a single digit number and says, â€śThe exponent is 11 because I counted 11 zeros after the nine.â€ť
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The student writes 5.29 xÂ Â and says, â€śThe exponent is -11 because there are 11 zeroes and they are on the left side of the number.â€ť
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Questions Eliciting Thinking Were you trying to convertÂ these numbers to scientific notation? What do you think is the difference between writing these numbers in scientific notation and estimating them by writing them in the formÂ aÂ xÂ Â whereÂ aÂ is a is a single digit number? What do you think is meant by a single digit number?
How did you determine where to put your decimal? How did you determine what exponent to write?
What would that number be if you estimate it using a single digit integer?
How do you know if the exponent will be positive or negative? |
Instructional Implications Provide direct instruction on estimating very large and very small numbers in the form ofÂ aÂ xÂ . Further explain that theÂ aÂ inÂ aÂ xÂ Â represents a single digit number. Demonstrate to the student that it is much easier to compare numbers when they are written in the formÂ .
Model how to estimate the extremely large and extremely small values (e.g., 421,372,201 is about 400,000,000 and 0.00000271 is about 0.000003). Next, demonstrate how to write the estimate in the formÂ aÂ xÂ . Emphasize the placement of the decimal point when writing large or small estimates in the form ofÂ aÂ xÂ . Provide the student with opportunities to practice estimating and writing estimates in the form ofÂ aÂ xÂ .
Address any issues with writing numbers in scientific notation. Provide instruction on the use of whole number exponents when writing powers of 10. Help the student understand the connection between place values and powers of 10. Have the student compare the standard forms of 3.99 xÂ Â and 4 xÂ . Point out that the exponent does not refer to the number of zeroes but to the number of place values.
Once the student can determine the (positive) exponent for large values, provide explicit instruction on how to determine the (negative) exponent for small values. |
Making Progress |
Misconception/Error The student rewrites the numbers in scientific notation rather than estimates them. |
Examples of Student Work at this Level The student writes 3.99 xÂ Â for the first question instead of 4 xÂ .
The student writes 5.29 xÂ Â for the second question instead of 5 xÂ .
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Questions Eliciting Thinking Were you trying to convert these numbers to scientific notation? What do you think is the difference between writing these numbers in scientific notation and estimating them by writing them in the formÂ aÂ xÂ Â whereÂ aÂ is a single digit number? What do you think is meant by a single digit number?
What would 3.99 be if you estimate it using a single digit integer?
What would 5.29 be if you estimate it using a single digit integer?
If you changed 3.99 to 4, would the exponent change? Why or why not?
If you changed 5.29 to 5, would the exponent change? Why or why not? |
Instructional Implications Make explicit the difference between scientific notation and estimation written in the formÂ aÂ xÂ . Model how to estimate extremely large and extremely small values (e.g., 421,000,000 is about 400,000,000 and 0.00000271 is about 0.000003). Next, demonstrate how to write the estimate in the formÂ Â
Have the student try to write his or her (scientific notation) answer in the formÂ Â Provide additional opportunities for the student to demonstrateÂ understanding of estimating extreme values and writing them in the formÂ aÂ xÂ . |
Got It |
Misconception/Error The student provides complete and correct responses to all components of the task.
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Examples of Student Work at this Level The student writes 4 xÂ Â for the first estimate.
The student writes 5 xÂ Â for the second estimate. |
Questions Eliciting Thinking How did you determine whether the exponent was positive or negative? Another student wrote 4 xÂ Â as the answer. What do you think that student did wrong? |
Instructional Implications Challenge the student to double 4 xÂ . Notice whether the student doubles the four or doubles the exponent (or both).
Show the student how to compare numbers in the form ofÂ aÂ xÂ . Consider using MFAS taskÂ Compare Numbers. |