Tree Growth Problem 7.EE - Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

Tree Growth Problem Several years ago, I planted a 3-foot tree that has grown the same amount each year. At the end of the third year, the tree was 1/5 taller than it was at the end of the second year. The tree is now 18 feet tall. How many years ago did I plant the tree?

Reasoning Abstractly and Quantitatively 1. Explain how the chart below models the following statement: ‘ Several years ago, I planted a 3-foot tree that has grown the same amount each year.’ Original Height End of 1st Year End of 2nd Year End of nth Year 3 feet (3 + x) ft. (3 + 2x) ft. (3 + nx) ft. X is the rate at which the tree is growing per year. 3 + 2x represents the height of the tree at the end of the 2nd year, where 3 is the original height and 2x is the additional growth for 2 years. The expression nx represents the total growth for n years. What does x represent? What does 3 + 2x represent? What does nx represent?

Reasoning Abstractly and Quantitatively 2. Explain how the equation below models the following statement: ‘At the end of the third year, the tree was 1/5 taller than it was at the end of the second year.’ Original Height End of 1st Year End of 2nd Year End of nth Year 3 feet (3 + x) ft. (3 + 2x) ft. (3 + nx) ft. 3+3x represents the height of the tree at the end of the 3rd year. 1/5(3+2x) is the additional height the tree had grown from the end of the 2nd year to the end of the 3rd year. X = 1 foot/year. What does 3+3x represent? What does 1/5 (3+2x) represent? What is the value of x?

Reasoning Abstractly and Quantitatively 3. Explain how the equation below models the following statement: ‘ The tree is now 18 feet tall. How many years ago did I plant the tree?’ Original Height End of 1st Year End of 2nd Year End of nth Year 3 feet (3 + x) ft. (3 + 2x) ft. (3 + nx) ft. N represents the number of years since the tree was planted. 3+nx represents the the height of the tree at the end of the nth year. Since x = 1 ft/year, the value on n is 15 years. What does n represent? What does 3+nx represent? What is the value of n?