# Tutor profile: Miguel P.

## Questions

### Subject: Pre-Calculus

Use the Pythagorean theorem to prove that $$sin^2(x)+cos^2(x)=1$$.

This question will not challenge students' understanding of the Pythagorean theorem as much as their understanding of the unit circle. To illustrate it, they will draw the unit circle and plot an angle x, sin(x), and cos(x). Then, they will just need to apply the Pythagorean theorem to prove the equality.

### Subject: Pre-Algebra

a) Solve $$7-x<10$$. b) Solve $$3\le7-x$$. c) Solve $$3\le7-x<10$$.

Many students struggle with inequalities; especially with double inequalities. Although compound inequalities might be an Algebra 1 question depending on where students are or their levels, the concept of solving inequalities is the same whether you have one or two. We can solve the problem one side at a time (following the same steps on both sides), or work on both inequalities at the same time. Students will be asked to solve both simple inequalities first and then predict what the answer to the compound inequality would be by making sense of their answers to parts a) and b), allowing them to make connections between problems with one inequality and problems with two. Then, they will solve the compound inequality and notice that they used the same steps for solving all three problems. Step by step, a solution to the compound inequality would be: $$3\le7-x<10$$ $$-7 +3\le-7+7-x<-7+10$$ $$-4\le -x<3$$ Multiply everything by -1: $$4\ge x>-3$$ The solution interval for the problem can then be rewritten as: $$x\in(-3,4]$$

### Subject: Statistics

Exploring data with CODAP (free online data exploration tool available at https://codap.concord.org/): 1. First, open a new document on CODAP and drag the Energy.csv file (provided by me) onto the workspace to import the data. 2. Use the tools necessary to answer the following questions: a. Which is the most used source of energy throughout the year? And the least? b. How common is the use of renewable energies compared to non-renewable? Is there a trend throughout the years? c. At what times of the year are the different types of energy more prominent? Can you explain the phenomena you observe? d. Pose one question of your own for these data and find an answer with CODAP.

CODAP is an educational tool that makes data exploration very easy and intuitive. The focus of the question is to encourage students to think about how to approach data exploration and not just apply formulas to calculate statistics. In this case, I would provide a file with data on energy generation in the US. Students will then have to think about what variables they would like to represent on graphs and what statistical tools to use to answer the questions posed. Based on the data in the spreadsheet, the answers would look like this: a. This question is vague on purpose, and students will have to make a decision as to what year to look at (data available from 2000-2021) or what methods to use to look at an overall usage for the two decades. Different answers will be given by students but the most common will be that coal is the most used source of energy. b. Non-renewable energies are still used more than renewable. However, there is a trend that indicates a decrease in non-renewable energy use and an increase in renewable energy use. Students can look at trends by plotting the appropriate variables and then the least-squares line, which CODAP plots accompanied by its formula, giving students the slope that helps them report their findings. c. Again, based on the data, students will make sense of the numbers. Looking at the amount of solar energy, for example, produced throughout the year, students will find out that it peaks in the summer months. They will theorize about their findings for each type of energy (nuclear, fuels, solar, wind, hydroelectric...). d. Students now pose their own questions so they can experience something similar to the research process. What would you like to know? => Use the data and your knowledge of statistics to find the answer to your question. => Share your newly acquired knowledge.

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