*Contributor: Marlene Vogel. Lesson ID: 10794*

Is your family spending grocery and utility money the most efficient way? By going to the grocery and using your toilet, you'll learn how to choose the right rates in math.

categories

subject

Math

learning style

Kinesthetic, Visual

personality style

Otter, Golden Retriever

Grade Level

Middle School (6-8)

Lesson Type

Dig Deeper

**The Big Shake**

A group of people has organized The Big Shake. The plan is to have citizens from New York to Los Angeles line up and pass a handshake from east to west.

- How would you estimate the number of people it will take for The Big Shake to be successful?

Here's how you can figure out your estimation for The Big Shake:

- Have several classmates, friends, or family members form a line. Count how many people are in the line and measure its length.
- Start a handshake at one end of the line. Use a stopwatch to time how long it takes the handshake to go from one end of the line to the other.
- Use your results from Step 1 to estimate the number of people per foot. Explain why this number is a
*rate*. - The road distance from Los Angeles to New York is 2,825 miles, or almost 15,000,000 feet.
- Approximately how many people need to be in the line?

- Use your results from Step 2 to estimate the speed of the handshake.
- How did you choose your units for this rate?

- Estimate when the handshake would have to start so it ends on midnight in Los Angeles. Explain how you found your answer.
- What rates did you use to help solve this problem?

The focus of this lesson is how to choose the appropriate rate and unit when solving a math problem, so let's solve this problem at the end of the *Get It!* section!

**What is a rate?**

According to Jenny Eather at A Maths Dictionary for Kids, rate is when you compare two measurements with different units.

It is important to know the appropriate rate to use in a situation.

For example, it is more appropriate to use *miles-per-hour* instead of *feet-per-minute* when talking about how fast a car is moving. However, when you are discussing the movement of a snail, it is better to use the rate of inches-per-minute instead of miles-per-hour.

- Do you know why?

We use miles-per-hour when discussing the movement of a car because cars can move long distances in a short period of time. We use inches-per-minute when discussing a snail because snails move slowly and it would take a long time for us to measure how far a snail moved in an hour.

**Sample rate problem**

Below is a sample problem that will help you learn how to choose the correct rate when solving a math problem:

Suppose you want to examine your family's electricity and water use. You plan to find out how efficiently your family uses these resources, and what changes you might make to increase efficiency. Name rates that will be helpful in carrying out your plan.

- Your first step is to make a list of the important information from the problem:
- You are examining your family's electricity use.
- You are examining your family's water use.
- Choose rates that will help you describe how much electricity and water your family uses.

Think about what rates to use when examining your family's energy use.

You are going to keep track of how much electricity your family uses.

- One way to choose the appropriate rate for electricity is to think about what objects in your home use electricity to work.
- You do not need to choose every object that uses electricity, just a few. For example, lamps use electricity.
- Light bulbs are labeled by how many watts they use. Knowing this information, you could calculate how much energy a specific lamp in your home uses by calculating the watts-per-hour.
- The rate would be
*watts per hour*.

Normally, you could measure water usage by gallons. However, it is difficult to measure how much water your family uses in your home by gallons as your plumbing is hidden in the house and underground.

- What you could do is measure your family's water usage by identifying the objects in your home that use water. Focus on your toilet and your shower.
- To measure how much water is used by your toilet in your home, you could measure how many times it is flushed in an hour. The rate would be the number of times the toilet is flushed per hour.

- What about your shower? What rate could you use to measure the amount of water used by your shower?

Discuss this with your teacher.

- Did you say minutes per shower?

Excellent! A good way to measure how much water is being used during a shower is to time how long the shower is running.

**What is a reciprocal rate?**

An important point to know about rates is that they can be reciprocal. Simply put, a *reciprocal* is the opposite of the original rate. For example, as you have read earlier, car speed is measured in miles-per-hour. A reciprocal for that rate is hours per mile.

**Sample reciprocal rate problem**

Here is a sample problem to help you see how reciprocal rates work:

Stephen recycles 25 pounds of newspapers, cans, and bottles per week. Give a reciprocal rate that has the same meaning.

- The original rate is pounds per week.
- The reciprocal rate is weeks per pound.

You can solve the problem by writing the rates as *fractions*.

Pounds per week would be written as the following fraction:

25 pounds | |

1 week |

The reciprocal rate is written as:

1 week | |

25 pounds |

Knowing how to write a rate in the reciprocal form will help you solve problems later on.

To find a unit rate, you divide the numerator in the fraction by the denominator.

For example, imagine that you are filling up a 10-gallon container with water from a hose.

- You timed how long it took for the container to fill to the 2-gallon mark.
- You found that it took 1 minute to fill the container with 2 gallons of water.
- Your rate would be minutes per gallon.
- Write the rate in fraction form:

1 minute | |

2 gallons |

Now you can figure out how long it takes for 1 gallon of the container to be filled with water.

- To do this, divide the 1 in the numerator by the 2 in the denominator.
- Your answer will be 0.5.
- Don't forget to add your unit.

Your final answer will be 0.5 minute per gallon. This is your *unit rate*.

Now, you can complete the problem at the beginning of the lesson. Access *The Big Shake Answer Key*, found in **Downloadable Resources** in the right-hand sidebar, to check your answers when you are done:

In the *Got It? *section, you will find activities to help you practice the skill of choosing the appropriate rate.

We help prepare learners for a future that cannot yet be defined. They must be ready for change, willing to learn and able to think critically. Elephango is designed to create lifelong learners who are ready for that rapidly changing future.

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