standard_description: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
problem_activity: Activity
The purpose of this activity is to extend the definition of exponents to include negative exponents. The meaning of zero exponents is also revisited. When students use tables to explore expressions with negative exponents, they are noticing and making use of structure (MP7).
Launch
Arrange students in groups of 2. Tell students to work on their own and then share their reasoning with their partner after each has had a chance to complete the problem. Follow with a whole-class discussion.
Student Facing
Complete the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "exponential form"}, {"_attributes": {"scope": "col"}, "_value": "number form"}, {"_attributes": {"scope": "col"}, "_value": "calculations"}]}]}], "tbody": [{"tr": [{"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "1145"}, "_value": "\\(2^5\\)"}]}, {}, {}]}, {"td": [{}, {}, {"_value": "16"}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "25037"}, "_value": "\\(\\frac{2^4}{2}=2^{4-1}=2^3\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "1932"}, "_value": "\\(2^3\\)"}]}, {}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "25045"}, "_value": "\\(\\frac{2^3}{2}=2^{3-1}=2^2\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "1131"}, "_value": "\\(2^2\\)"}]}, {"_value": "4"}, {}]}, {"td": [{}, {}, {"_value": "2"}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "25058"}, "_value": "\\(4 \\boldcdot \\frac12=2\\)"}]}]}, {"td": [{}, {}, {"_value": "1"}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "failed-conversion"}, "_value": "\\(2 \u00a0\\boldcdot \\frac12=1\\)"}]}]}, {"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "8478"}, "_value": "\\(2^{\\text-1}\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "497"}, "_value": "\\(\\frac{1}{2}\\)"}]}, {}]}, {"td": [{}, {}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "451"}, "_value": "\\(\\frac{1}{4}\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "failed-conversion"}, "_value": "\\(\\frac12 \u00a0\\boldcdot \\frac12 = \\frac14\\)"}]}]}, {"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "25113"}, "_value": "\\(2^{\\text-3}\\)"}]}, {}, {}]}, {"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "25133"}, "_value": "\\(2^{\\text-4}\\)"}]}, {}, {}]}, {"td": [{}, {}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "2239"}, "_value": "\\(\\frac{1}{32}\\)"}]}, {}]}]}]}]}
Student Response
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Activity Synthesis
Display the completed table, and resolve any questions students have. Questions for discussion:
“How would you make sense of
\(2^{\text-6}\)
?” (I know
\(2^{\text-5}=\frac{1}{32}\)
, and I know that
\(2^{\text-6}\)
would be the result of multiplying by another
\(\frac12\)
. So,
\(2^{\text-6} = \frac{1}{64}\)
.
“How could we figure out the value of
\(7^{\text-2}\)
?” (Possible response: We could rewrite it as
\(\frac{7^0}{7^2}\)
, and then
\(\frac{1}{49}\)
.


aligns: no