Lesson 1: Piggy Banks and Pools

My little sister, Savannah, is three years old. She has a piggy bank that she wants to fill. She started with 5 pennies, and each day when I come home from school, she is excited when I give her 3 pennies that are left over from my lunch money. What is the equation for the number of pennies in the piggy bank on day n?

• p(n) = 5 + 3n

• p(n) = 3 + 5n

• p(n) = 5n + 3

• p(n) = 3n + 5

Answer explanation

Savannah starts with 5 pennies and receives 3 more each day. The total number of pennies after n days is given by the equation p(n) = 5 + 3n, which simplifies to p(n) = 3n + 5, making this the correct choice.

Tags

CCSS.8.F.B.4

CCSS.HSF.LE.A.2

Our family has a small pool for relaxing in the summer that holds 1,500 gallons of water. I decided to fill the pool for the summer. I was getting bored just standing there watching water flow and began to think about a mathematical model for the time it takes to fill the pool while I was waiting. I checked the flow on the hose and found that it was filling the pool at a rate of 2 gallons every minute. When I had 5 gallons of water in the pool, I started the timer. Use a mathematical model to determine how long it will take to fill the pool.

• 745 minutes

• 750 minutes

• 745.5 minutes

• 740 minutes

Answer explanation

The pool holds 1,500 gallons. Starting with 5 gallons, 1,495 gallons remain. At 2 gallons per minute, it takes 1,495 / 2 = 747.5 minutes. Since the timer started after 5 gallons, the total time is 745 minutes.

Tags

CCSS.7.EE.B.4A

I'm more sophisticated than my little sister, so I save my money in a bank account that pays me 3% interest on the money in the account at the end of each month. (If I take my money out before the end of the month, I don’t earn any interest for the month.) I started the account with $50 that I got for my birthday. What will be the amount in the account after 1 month?

• $51.50

• $50.50

• $53.00

• $52.00

Answer explanation

To find the amount after 1 month, calculate 3% of $50: 0.03 * 50 = $1.50. Add this interest to the initial amount: $50 + $1.50 = $51.50. Therefore, the correct answer is $51.50.

Tags

CCSS.HSF.BF.A.2

At the end of the summer, I decided to drain the 1,500-gallon swimming pool. I noticed that it drains faster when there is more water in the pool. That was interesting to me, so I decided to measure the rate at which it drains. I found that 3% was draining out of the pool every minute. What is the equation that models the gallons of water in the pool at t minutes?

• A) W(t) = 1500 * (0.97)^t

• B) W(t) = 1500 - 0.03t

• C) W(t) = 1500 * (1.03)^t

• D) W(t) = 1500 * (0.03)^t

Answer explanation

The pool drains 3% per minute, meaning 97% remains. The equation W(t) = 1500 * (0.97)^t models the gallons left after t minutes, as it represents exponential decay of the water volume.

Tags

CCSS.HSF.LE.A.2

Compare problems 1 and 3. What similarities do you see? What differences do you notice?

• Both problems involve algebraic equations.

• Both problems require solving for x.

• Problem 1 is a word problem, while problem 3 is a numerical problem.

• Problem 1 and problem 3 have different mathematical operations.

Answer explanation

The correct choice highlights that problem 1 and problem 3 involve different mathematical operations, indicating a key difference in their approaches. This distinction is crucial for understanding how to solve each problem effectively.

Tags

CCSS.7.EE.B.4A

Compare problems 1 and 2. What similarities do you see? What differences do you notice?

• Both problems involve mathematical calculations.

• Both problems require logical reasoning.

• Both problems have the same solution.

• Both problems are unrelated.

Answer explanation

Both problems involve mathematical calculations, highlighting a key similarity. However, they may differ in context or complexity, but the shared element of math is significant in their comparison.

Tags

CCSS.1.OA.B.3

Compare problems 3 and 4. What similarities do you see? What differences do you notice?

• Both problems involve algebraic equations.

• Problem 3 is about geometry, while problem 4 is about algebra.

• Both problems require solving for x.

• Problem 3 is a word problem, while problem 4 is a numerical problem.

Answer explanation

Both problems involve algebraic equations, indicating they share a common mathematical foundation. However, they differ in context, with problem 3 being a word problem and problem 4 a numerical problem.

Tags

CCSS.7.EE.B.4A

Use your model in problem 4 to find when the pool will be empty. Justify your answer.

• The pool will be empty in 5 hours.

• The pool will be empty in 10 hours.

• The pool will be empty in 15 hours.

• The pool will be empty in 20 hours.

Answer explanation

The model indicates that the pool empties at a rate that results in it being empty in 10 hours. This is the only option that aligns with the calculations made in problem 4, confirming it as the correct answer.

What is a continuous function?

• A function that is defined only at discrete points.

• A function that has no breaks, jumps, or holes in its graph.

• A function that is not defined for all real numbers.

• A function that is always increasing.

Answer explanation

A continuous function is defined as one that has no breaks, jumps, or holes in its graph, ensuring that it can be drawn without lifting the pencil. This distinguishes it from functions that are discontinuous.

Tags

CCSS.HSF-IF.C.7B

What is a discrete function?

• A function with a continuous domain

• A function with a discrete domain

• A function with no domain

• A function with a variable domain

Answer explanation

A discrete function is defined by a discrete domain, meaning it takes specific, separate values rather than a continuous range. Thus, the correct choice is 'A function with a discrete domain'.