In this discovery oriented lesson, students will explore the use of non-standard units of measurement. They will convert linear measurements within the metric system and also convert measurements given in astronomical units (AU) into more familiar units, specifically meters and kilometers. The unit conversions will be completed with measurements that are expressed in scientific notation. Students will recall their prior knowledge of how to add and subtract numbers given in scientific notation. They will also use their knowledge of exponent rules to determine an efficient method for multiplying and dividing numbers expressed in scientific notation.
Learning Objectives: What will students know and be able to do as a result of this lesson?
Students will be able to:
Convert between non-standard units of measurement and more common or useful units, depending upon the distance is being measured.
Understand why units that measure astronomical distance are appropriate and most conveniently expressed in scientific notation.
Multiply and divide numbers that are expressed in scientific notation, and express the result in scientific notation.
Prior Knowledge: What prior knowledge should students have for this lesson?
Prior to the lessons, students should know how to:
Convert between standard and scientific notation
Take accurate measurements with a meter stick
Convert units between systems
Convert between metric units, based upon the standard prefixes
Use the laws of exponents to simplify multiplication and division of terms having the same base
Add or subtract numbers expressed in scientific notation
Convert a number such as 4.2 x 105 to scientific notation
Guiding Questions: What are the guiding questions for this lesson?
What process do we use to convert a measurement from one unit to another?
Astronomical distances are generally expressed in AU…how can we convert AU to different units?
How are units defined?
How can we efficiently multiply and divide numbers that are very large by using scientific notation?
Introduction: How will the teacher inform students of the intent of the lesson? How will students understand or develop an investigable question?
Review measurement systems with your students. Discuss the differences between different systems (Metric, customary units, etc.), their similarities, and how to convert between systems. Also ask students to describe how an accurate measurement is taken: what tools should be used, how many measurements should be taken, and acceptable error.
Next, ask students if they know what a "Smoot" is. Chances are students will be unfamiliar with this term. Tell students about the Smoot: a non-standard unit of measurement, equal to the height of Oliver Smoot, who was a student at MIT. He repeatedly lay down on a bridge so that his friends could measure it in unts of "Smoots." Ask students why or why not they think that a Smoot is a valid form of measurement. You can show the video "Reenact the Smoot Act of Bridge Measuring" uploaded by YouTube user Alison Leedham to demonstrate how this measurement system was developed.
Some students will say that it is not valid, because it is not a standard measurement that they are familiar with. Ask students what might be required to have a valid unit of measurement. Expected responses include the following: It must be repeatable, standard, and well defined. A Smoot is defined as the height of Oliver Smoot at the time the bridge was measured (1.7 meters). Does this fits all the criteria for a valid unit of measurement? (Yes). Although a Smoot is not a standard unit of measurement, it is a valid one nonetheless.
Investigate: What will the teacher do to give students an opportunity to develop, try, revise, and implement their own methods to gather data?
Give each student a piece of twine or string and a meter stick. Instruct each student to cut the twine to a length of 1.7 meters. They will use this twine to measure objects in smoots.
Pair students up, then distribute the attached "How Many Smoots?" worksheet to each pair. Instruct them to measure various objects in the classroom using their twine that is one smoot in length, and record the length of each object in smoots. If desired, you can also let students measure larger objects outside the classroom.
As they determine their measurements, ask questions such as:
"If an object is between 1 and 2 Smoots, how do you determine the measurement?" Finding by estimation what fraction of a Smoot is needed.
"How can we determine if our measurement is accurate enough? Do we need the same measure of accuracy if we're wrapping a present as when we're designing a bridge or a spacecraft?" The level of accuracy needed depends on how important or critical the measurement is. Taking a measurement where someone can be endangered by a less accurate measurement, or ones where vast distances must be travelled require greater degrees of accuracy.
"Can you suggest any uses for a non-standard measurement like this one? Or situations where it might be useful?" Non standard units of measurement could be used in situations where the idea will be more memorable! It is easier to remember how many Smoots something is because it's so strange. Two common examples of using non-standard units that students may know about are as follows.
If no customary measurement tool such as a tape measure or ruler is available, distance or length may be estimated by: 1) measuring a distance by walking it off and counting the number of paces or steps, and 2) measuring a length of rope, yarn, or fabric by extending it from your chin to your outstretched arm. This is often used to estimate a yard.
Analyze: How will the teacher help students determine a way to represent, analyze, and interpret the data they collect?
Ask students: "Do we have to take new measurements if we wanted to know how many smoots it takes to reach a new destination? For example, if I wanted to know how many smoots it took to reach the moon, would I have to get Oliver Smoot to lie end to end all the way to the moon?" At this point, students should bring up the idea of unit conversions. Lead in with simpler problems, such as "How do you determine the number of meters in 5 smoots?" and follow with "How do you determine the number of smoots in 5 meters?" These questions will reinforce the importance of how to correctly set up a proportion or equation and use units when converting between different units of measure. Then ask, "How do you find how many smoots are in 1 kilometer?"
After discussing these questions, the teacher should provide a clear and concise summary of the technique used in converting units. The teacher should reiterate and demonstrate the method, with labels on units clearly shown. Verify that students understand this process before moving on to the next part of the activity. Additional examples may be presented if needed, before setting students to work on the other example conversions.
Ask students: "Now, we're going to focus back on standard units. If I asked how many meters were in 10 kilometers, how can you determine the answer?" Students should be able use their knowledge of metric units and of scientific notation to complete the first 2 problems on the worksheet.
Many students will try to write the numbers in standard form, which will prove very time consuming. Ask those students if there is a more compact and efficient way to represent those numbers. As students begin to convert this numbers into scientific notation, they will be required to operate on them.
For examples #4 and 5, remind students that when adding numbers in scientific notation, they must pay close attention to the place values, so that they aren't adding hundreds and millions, etc. (i.e., 2.31 x 103 + 3.56 x 104 = 0.231 x 104 + 3.56 x 104 = 3.791 x 104). Remind students that they can add or subtract numbers that are given in scientific notation, but first they have to write both numbers so they have the same power of 10, or magnitude. Teachers should ask students if they think using the larger or the smaller exponent for the common power would be more convenient.
For problems 6 and 7 on the worksheet, students will be discovering how to multiply and divide numbers represented in scientific notation. Problems 5 and 7 are meant as practice; allow students to answer those problems using numbers in standard form, and then encourage them to make a rule for those operations in scientific notation. Then, they can test their hypothesis on the other problems in the sheet.
Provide scaffolding as students explore the processes and the concepts involved in converting the results into scientific notation by asking questions such as "Now that you've converted your result back to scientific notation, what has changed?", "How are the laws of exponents related to how we multiply and divide numbers in scientific notation?", and "If we performed multiplication/division rules for exponents individually, what should happen to the coefficients?" Ask students to refer to question 7 to help them formulate an answer. Give students a few minutes to think this problem through before asking for responses.
Closure: What will the teacher do to bring the lesson to a close? How will the students make sense of the investigation?
Have students discuss their results, giving some examples of objects and their smoot measurements. The teacher should review, explain, clarify and emphasize the main concepts used in the activity, including how units were converted, and how to perform operations on numbers in scientific notation.
Key concepts to elicit:
To convert units, you must ensure that the unit of measurement cancels out. For example, if you divided AU by smoots, your measurement would be in AU/smoot, but if you divide meters by smoots, your measurement is a number with no units.
When adding/subtracting numbers in scientific notation, you may convert to standard form first, or you may add/subtract the coefficients in the appropriate place value so long as you convert the numbers such that they are expressed using the same power of 10.
To multiply or divide numbers in scientific notation, you multiply/divide the coefficients and add/subtract the exponents separately.
With any scientific notation operations, after our operation, you must ensure the result is still in scientific notation. This means verifying that the coefficient is still greater than or equal to 1 and less than 10.
By the end of the lesson, students will be able to convert between units of measurement. They will also be able to add, subtract, multiply, and divide numbers given in scientific notation. Students will complete the "Astronomic Distances!" worksheet to practice skills.
The measurement of mastery will be if they can successfully calculate the sum, difference, product, and quotient of two numbers in scientific notation.
Students learning will be monitored by way of the first worksheet, "How Many Smoots?" The teacher will be able to get feedback on each student's level of understanding and provide adjustments to instruction and clarifications as needed in response to students' work.
Feedback to Students
Students should be provided feedback throughout the lesson as the teacher circulates and observes and listens to discussions. The teacher should facilitate discussion and probing questions to help with any misconceptions.
Accommodations & Recommendations
Students will work in groups for the majority of the lesson, scaffolding ESOL and special needs students.
Students who need extra time will be able to finish any portions of the task not completed in class as part of the homework.
Students who finish early can convert their calculated measurements to Smoots or their own version of Smoots: their height in meters.
To extend the ideas of the lesson, you can have students discuss why so many measurement systems have been moving towards theoretical units; units that are not defined in terms of any physical object, but in terms of constants such as the speed of light. There are pros and cons to these units which your students should be able to highlight.
Students could make up their own measurement systems based on their heights.
Students could create visual posters of each of the mentioned planets, galaxies, etc with scientific notation of distances labeled.
length of twine or string to be cut into Smoot lengths (1.7 meters)
For each pair:
1 meter stick
Some problems will require students to perform operations on the numbers in scientific notation, rather than just converting units. Students should not be allowed calculators for this portion of the lesson, as they need to determine how operating on numbers in scientific notation affects them.
Math Practices addressed in this lesson:
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.