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Understanding Linear Equations and Proportions

Explore linear equations, direct variation, proportions, and inverse functions. Learn to solve proportions and apply ratio concepts with real-life problems. Practice using linear equations to represent sequences and solve equations.

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Understanding Linear Equations and Proportions

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  1. Overview of Chapter 2 • In this chapter students use linear equations to represent a sequence of calculations and solve those equations by undoing (working backwards) • In chapter 3 students will use the linear equations to model linear growth, graphs, and extend their solution techniques to include balancing.

  2. Lesson 2.1 and 2.2 • Review previous work with proportions and introduce students to the idea of undoing to solve a proportion. • Lesson 2.3 • Develops the idea of deriving linear expressions from measurement conversions with an emphasis on dimensional analysis • Lesson 2.4 • Introduces direct variation as an alternative to solving proportions • Students create scatter plots of real data, draw a line through the data points, and find an equation y=kx to fit the data.

  3. Lesson 2.5 • Students are introduced to the topic of an inverse function • Lesson 2.6 • Through an activity students explore a real life problem on gears that relates direct and indirect variations. • Lesson 2.7 • Students practice the rules for order of operations by analyzing how the steps in linear expressions describe by a number trick undo each other to end up with the same number. • Lesson 2.8 • Students write linear equations to represent sequences of steps and solve those equations by undoing.

  4. Proportions Rename fractions as decimal numbers Write ratios and proportions that express relationships in data Solve proportions by multiplying to undo division Solve proportions by inverting both ratios Solve problems using proportions Review skills in working with percents

  5. Proportions • When you say “I got 20 out of 24 questions correct on the last quiz,” what ratio are you describing? • Picture this ratio using colored cubes. • What other names can you give this ratio? • How do the cubes help you see those equivalent ratios? • What is the equivalent decimal for this ratio?

  6. Write 20:24 in a fraction. How do you change this ratio to a decimal? • What are some other ways to write this ratio?

  7. Using the Calculator to convert ratios • Calculator Note 0A shows • How decimal numbers can be converted to fractions • How fractions or ratios can be changed decimal numbers

  8. What is a proportion? • A proportion is an equation stating that two ratios are equal. • Check to see if the following ratios are proportions by finding the equivalent decimal for each.

  9. Think about it • The variable M stands for an unknown number. • Replace the variable M to make the following statement true.

  10. Multiply and Conquer p. 97

  11. Multiply and Conquer Step 1: Multiply both sides of the proportion by 19. Why can you do this? What does M equal?

  12. Multiply and Conquer Step 2: For each equation, choose a number to multiply both ratios by to solve the proportion for the unknown number. Then multiply and divide to find the missing value.

  13. Step 3: Check that each proportion in Step 2 is true by replacing the variable with your answer. Step 4: In each equation in Step 2, the variables are in the numerator. Write a brief explanation of one way to solve a proportion when one of the numerators is a variable.

  14. Step 5: The proportions you solved in Step 2 have been changed by switching the numerators and denominators. That is, the ratio on each side has been inverted. (You may recall that inverted fractions, like and are called reciprocals.) Do the solutions from Step 2 also make these new proportions true?

  15. How can you use what you just discovered to help you solve a proportion that has the variable in the denominator, such as Why does this work? Solve the equation.

  16. Step 7 There are many ways to solve proportions. Here are three student papers each answering the question “13 is 65% of what number?” What are the steps each student followed? What other methods can you use to solve proportions?

  17. Applying what you’ve learned • Jennifer estimates that two out of every three students will attend the class party. She knows there are 750 students in her class. Set up and solve a proportion to help her estimate how many people will attend. Students who will attend Students who will attend Students who are invited Students who are invited

  18. Applying what you’ve learned • After the party, Jennifer found out that 70% of the class attended. How many students attended? • 70% is one way to write a ratio. What ratio will it equal? • Solve the ratio:

  19. 70% of the class attended. • 750 people were in the class. • How can we find out how many people attended the party? 70% 750

  20. 750

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