Dimensional Analysis in Math: Master Problem Solving with Metrics

2. Dimensional Analysis Why do it? Kat Woodring

3. Benefits for students • Consistent problem solving approach • Reduces errors in algebra • Reinforces unit conversion • Simplifies computation • Improves understanding of math applications • Multiple ways to solve the same problem

4. Benefits for teachers • Successful problem solving strategy for advanced or special needs students • Vertically aligns with strategies for Chemistry and Physics • Improves Math scores • Easy to assess and grade

5. 5 Steps of Problem Solving • Identify what you are asked. • Write down what is given or known. • Look for relationships between knowns and unknowns (use charts, equations). • Rearrange the equation to solve for the unknown. • Do the computations, cancel the units, check for reasonable answers.

6. Teaching Opportunities with Metric System • Beginning of year • Review math operations • Assess student abilities • Re-teach English and SI system • Teach unit abbreviations • Provide esteem with easy problems • Gradually increase complexity

7. 5 Steps of Dimensional AnalysisUsing the Metric Conversion • Start with what value is known, proceed to the unknown. • Draw the dimensional lines (count the “jumps”). • Insert the unit relationships. • Cancel the units. • Do the math, include units in answer.

8. Lesson Sequence • English to English conversions. • Metric to Metric conversions. • English to Metric conversions. • Metric to English conversions. • Complex conversions • Word problems

9. Write the KNOWN, identify the UNKNOWN. • EX. How many quarts is 9.3 cups? 9.3 cups = ? quarts

10. 9.3 cups = ? quarts Draw the dimensional “jumps”. 9.3 cups x * Use charts or tables to find relationships

11. Insert relationship so units cancel. quart 1 9.3 cups x 4 cups *units of known in denominator (bottom) first *** units of unknowns in numerator (top

12. 1 quart 9.3 cups x 4 cups Cancel units

13. 1 quart 9.3 cups x 4 cups Do Math • Follow order of operations! • Multiply values in numerator • If necessary multiply values in denominator • Divide.

14. 9.3 = 4 Do the Math 1 quart 9.3 x 1 9.3 cups x = 4 cups 1 x 4 = 2.325 s

15. Calculator /No Calculator? • Design problems to practice both. • Show how memory function can speed up calculations • Modify for special needs students

16. Sig. Fig./Sci. Not.? • Allow rounded values at first. • Try NOT to introduce too many rules • Apply these rules LATER or leave SOMETHING for Chem teachers!

17. Show ALL Work • Don’t allow shortcuts • Use proper abbreviations • Box answers and units are part of answer • Give partial credit for each step • Later, allow step reduction • If answer is correct, full credit but full point loss

18. Vocabulary • KNOWN • UNKNOWN • CONVERSION FACTOR • UNITS

19. Write the KNOWN, identify the UNKNOWN. • EX. How many km2 is 802 mm2 ? 802 mm2 = km2?

20. 802 mm2 = km2? Draw the # of dimensional “jumps” 802 mm2 x x x x x x

21. 802 mm2 = km2? Insert Relationships cm2 dm2 m2 dkm2 hm2 km2 802 mm2 x x x x x x mm2 cm2 dm2 m2 dkm2 hm2

22. cm2 dm2 m2 dkm2 hm2 km2 802 mm2 x x x x x x mm2 cm2 dm2 m2 dkm2 hm2 Cancel Units *Units leftover SHOULD be units of UNKNOWN

23. cm2 dm2 m2 dkm2 hm2 km2 x x x x x x mm2 cm2 dm2 m2 dkm2 hm2 Cancel Units (1)2 (1)2 (1)2 (1)2 (1)2 (1)2 802 mm2 (10)2 (10)2 (10)2 (10)2 (10)2 (10)2 *Units leftover SHOULD be units of UNKNOWN

24. cm2 dm2 m2 dkm2 hm2 km2 x x x x x x mm2 cm2 dm2 m2 dkm2 hm2 Do the Math… (1)2 (1)2 (1)2 (1)2 (1)2 (1)2 802 mm2 (10)2 (10)2 (10)2 (10)2 (10)2 (10)2 *What kind of calculator is BEST?

25. Differences from other math approaches • Solve for variables in equation first, then substitute values • Open ended application • No memorized short-cuts • No memorized formulas • Reference tables, conversion factors encouraged

26. Outcomes • Usescience • Thinkscientifically • Communicatetechnical ideas • Teachall students • Bescience conscious not science phobic