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# Dimensional Analysis

Lesson 3 of 6

## Objective: SWBAT convert from one unit of measurement to another using conversion factors they create based on equivalence statements.

*60 minutes*

This lesson addresses the need for students to both conceptually understand that we can measure something using different sets of units as well as to be able to convert between units. Being able to successfully complete dimensional analysis lays the groundwork for future calculative skills used in stoichiometric analyses.

The activity itself involves two major concepts (taken from the Crosscutting Concepts and the Science and Engineering Practices) addressed in the Next Generation Science Standards: **Crosscutting Concept 3 (CCC 3) -- Scale, Proportion, and Quantity** and **Science and Engineering Practice 5 (SEP 5) -- Using Mathematics and Computational Thinking**.

Students engage in first taking equivalence statements (such as 1 hour = 60 minutes) and turning them into two reciprocal conversion factors. For example, 1 hour = 60 minutes will become the two conversion factors 1 hour/60 minutes and 60 minutes/1 hour. Students then use these conversion factors to convert between one set of units to another (i.e. How many minutes are in 2.5 hours?). Such calculations clearly address **SEP 5**. The ability to convert between units while understanding that though the units we are using to express that amount are different, the actual amount has not changed, addresses **CCC 3**.

Developing an understanding of dimensional analysis, particularly the ability to cancel like units when one appears in the numerator and the same in the denominator, leads to success later in stoichiometric calculations involving conversions between grams and moles, determining limiting reactants, and predicting amounts of products, to name a few.

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#### Warm-Up

*5 min*

While I take attendance, students do a warm-up activity in their composition Warm-Up/Reflection books. I use warm-ups to either probe for students' prior knowledge about the day's upcoming lesson or to have them bring to mind and review what they should have learned during the Flipped Classroom recording the night before. (To read more about Warm Up and Reflection Books and the Flipped Classroom please see the attached resources.) I have assigned a video from Hank Green's YouTube Crash Course in Chemistry series for students to watch outside of class to prepare for today's activity:

Today's Warm-Up: "**How can we convert from one set of units to another?**"

In this case, the warm-up is asking students to both apply knowledge from the flipped lesson and is also preparing them for today's activity during which they are expected to use equivalence ratios to develop and use conversion factors.

If time permits, I walk around with a self-inking stamp to stamp the completed warm-ups indicating participation, but not necessarily accuracy. On days when there is too much business keeping, I do not stamp. Students have been told that warm-ups are occasionally immediately checked and other times not. At the end of each unit, Warm-Up/Reflection Books are collected and spot-checked.

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I begin whole class discussion by asking, "*How many minutes are there in 2 hours?"* to which students respond quickly with "120 minutes."

I follow-up with, "*How do you know that?*" which generally causes pause among students. Students have difficulty articulating how they go about converting from one set of units to another, but they are capable of performing those conversions automatically with units such as time because they are familiar with those units.

As students think about *how* they made the conversion, they will usually express understanding that there are 60 minutes in one hour. I acknowledge that as true and proceed to show them that they have actually created a conversion factor in their brains, and used it to convert from hours to minutes. I write the equivalence statement 1 hour = 60 minutes on the white board, and go through showing that from that equivalence statement, we can make two conversion factors: 1 hour/60 minutes and 60 minutes/1 hour. We then choose the appropriate conversion factor in order to cancel out, in this case, the hours and end up with an answer in minutes.

The process of dimensional analysis that I explain is detailed in this video:

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#### Small Group Work

*40 min*

Before passing out the Small Group Activity - Dimensional Analysis handout packets (one per group) I ask students to move into cooperative learning groups. Depending on the activity, I will assign groups (either randomly or by choosing deliberately based on student capability) or allow students to self-select. This particular activity is slightly differentiated to address the differing needs of my general level students and honors level students (as one of my chemistry classes is a mixed combination class of both general and honors level). I allow students to self-select with the caveat that general students work in general chemistry groups and honors students work in honors chemistry groups. Once students have chosen groups and moved seats if necessary, I hand out one Small Group Activity - Dimensional Analysis packet to each group. Then, I hand out a Dimensional Analysis Conversions assignment sheet to each individual student. I point out that there is a section at the end that is designated as Honors level only, but general level chemistry students may do that section as well if they like. (Keys are available here for Dimensional Analysis Conversions Key Page 1 and Dimensional Analysis Conversions Key Page 2.)

Students work in groups of 3 - 4 to first create the conversion cards (using index cards cut into fourths with a large paper cutter; an alternative is to have the students themselves cut the index cards with scissors if no large paper cutter is available). Each group needs 11 mini-cards (or 3 index cards to cut into fourths). I pass around two small school boxes with the cards in them so that groups can take what they need and pass to the next group.

Groups follow along the steps listed on page 1 of their group handouts, asking questions of each other or myself as they are reading and questions arise. Once they understand the workings of conversions (from the flipped video explanation, from reading the handout, and/or from asking me), they can move on to the next step during which they take equivalence statements (such as 1 m = 1 cm) and develop two different conversion factors for each one. It is crucial that students can make conversion factors on their own from equivalence statements, as often they can look up that kind of information but conversion factors are not typically readily searchable but what we use in unit conversions.

The third page of the handout packet details how to make each conversion factor card using the mini-cards, and then how to use those cards to complete the individual dimensional analysis handouts.

I expect students to work in groups to understand dimensional analysis, to determine the conversion factors from the equivalence statements, and to construct the cards. I also expect each individual student to complete his/her own Dimensional Analysis Calculation handout. Students can collaborate and help each other with those calculations, but in the end, each student is held accountable for correctly completing the assignment.

As students are working, I walk around answering questions and double-checking that students are, in fact, showing their work as they work through the calculations. Final products should looks something like Dimensional Analysis Page 1 and Dimensional Analysis pg 2 Honors (these resources are sample student work).

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#### Wrap-Up

*5 min*

If students have completed their individual Dimensional Analysis Calculations handout, I have them turn their work in. Most of my students will not be done by the end of the period, in which case I allow them to complete the rest of the calculations as homework that is due the next day.

The only part of the assignment that I collect is the individual handouts. Students can keep the small group packet and/or conversion cards. These tools are specifically created for this activity alone, and I will not be coming back to them at a later date.

Students return desks that may have been moved to their original places. Then I ask students as a whole if there are still any trouble spots that they have questions about. I answer any questions that may be posed.

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