Use data to estimate the probability of future events (e.g., batting averages or number of accidents per mile driven).

Identify data and write it as a ratio (fraction). Use the ratio to write a proportion and use algebra to solve for the probability of future events. Understand that a proportion is two equal ratios. Use a variable to represent the missing data needed determine future probability. Solve for the variable using cross multiplication.

There are 2 ways to write a ratio (a/b, a:b and a to b). Data and probability can be written in fraction form (a/b). It is helpful to begin by writing your known data rate or amount using a word ratio to compare the units of measure. For example, if the data tells you that a motorized scooter will travel 55 miles in 30 minutes, you would write the word ratio ___miles/___minutes. Now, when you need to used the data to find the probability of a future event, you would use this same word ratio and fill in the new numbers. One mistake that children often make is switching the order of the units of measure, and in turn writing in incorrect numbers. Remind your child to always make sure that the number matches with the word in the ratio. So, if we are to use the example above, 55miles/30 minutes is the correct way to write the ratio. However, if 30 miles/55minutes was written, the problem could never be solved correctly, because the numbers don’t math the units of measure. Have your child practice writing the correct number with the correct unit of measure.

(2)

When solving problems to estimate the probabilty of future events, proportions will need to be used. Have your child’s first step be to create a proportion using “word ratios”. This will greatly increase his/her success rate when solving to find the probability.

Sample Problem: Johnny’s plant has grown 16 inches in two months. How large will it grow to be in one year?

inches inches

__________________ = _________________

months months

As in the word ratio above, it is crucial that the same unit of measure be on numerator (top) of each ratio. Hence, inches is the word on the top of ratio one and the top or ratio two. The same goes for the denominator. They must always be the same unit. So, months is on the both denominators. The most commonly made mistake of students is incorrect placement of either the units of measure or the numbers that go with them. Also, as you will see from the problem, the answer is wanted in years. There cannot be two different measures on the denominator, so the year will be translated into 12 months in order for this problem to be solved.

(3)

Cross multiplication is a short cut to solve proportions by multiplying the diagonals. It is called cross multiplication because you multiply ratio terms diagonally, here by making an x or cross shape. Always use the fraction form of a ratio to use cross multiplication to solve proportions. To cross multiply you multiply the top term of the first ratio by the bottom term of the second ratio. Write the product you get in front of an equal sign. For example, if you have 5/25 = n/50 you would first multiply 5x50 to get 100. Next, multiply the bottom term of the first ratio by the top term of the second ratio. So, 25xn. Your new problem would be 100=25n. To complete the problem use steps to solve algebraic equations: 10025=4, so n=4. When solving to find the probability of a future event, the variable (letter, such as n in this problem) always gives the probability estimate or answer.

(4)

For visual learners, color-coding is very helpful. Have your child use two different colors of ink to draw the diagonals as they cross multiply. For example, a student can use a red marker to connect the top term from ratio 1 to the bottom term of ratio two and write that product in red. He/she can then use a blue pen to draw a line across the diagonal from the bottom term from ratio 1 to the top term of ratio 2 and write the product they make next to the red product. Next, a student can draw and equal sign between them in regular pencil and begin solving. Using different colored pens or pencils keeps students focused on each step one at a time and prevents errors in choosing the incorrect terms to multiply.

(5)

You can either multiply or divide both terms of the ratio by the same number to find equivalent rates or estimates of the probability of future events. For example, if you run one mile in 10 minutes, you can use this rate to find out how many miles you could run in 40 minutes. To make the computations you need to begin with your word ratio to show the units of measure you’re comparing (mile/minutes). Now, you will fill in the terms, using the rate you know (1 mile/10 minutes). Next, you write another ratio, filling in the term you know and leaving the other blank (make sure you keep your units of measure in the same place) ___miles/40 minutes. If you wanted to write this as a proportion it would look like this:

1 mi = ___mi

10 min. 40 min.

Then, find out what was done to the like units. In other words, what was done to change the 10 min. into 40 min.? Since 10 x 4 = 40, we know that 4 was the factor used. Remember, whatever you do to the bottom ratio, you must do to the top. So, 1 must be multiplied by 4 to find the missing top term. This means the equivalent rate is 4 mi in 40 min. You could estimate the probability that you would run 4 miles in 40 minutes.

(6)

Always check your work. Remind your child that a proportion is simply the name used to describe two ratios that are equal. This is true when both terms of the ratio are either multiplied or divided by the same number. An example of a proportion is 4/8 = 16/32. To determine if two ratios are proportional (equivalent) you must follow three steps. First, compare the units to make sure they’re in the same place on each ratio (i.e. 4 laps/8minutes = 16 laps/32minutes). Both ratios have laps as the first term and minutes as they second term, so it could be a proportion. Next, write each ratio in simplest form. To do this you need to divide each term of the ratio by the GCF, greatest common factor. The GCF is the largest number that will divide into both terms evenly without remainders. For example, the GCF of 4/8 is 4 because 4 is the largest number that both terms are divisible by. Hence, 4/8 in simplest form is ½ because 44=1 and 84=2. Lastly, compare the simplest forms of both ratios. If they’re the same that the ratios are proportional. The simplest form of 4/8 is ½. The simplest form of 16/32 is ½. Since both ratios are ½ in simplest form they are proportional. Since when we find the probability of future events, we must use a proportion, students can check their final answer by checking for the proportionality of their new ratio and the original ratio. So, if we find the Raul types 40 words per minute and we estimate that he will be able to type 160 words in 4 minutes, we can check our estimate by setting up a proportion. The proportion would be written as 40/1 = 160/4. Next, 160/4 needs to be put in simplest form by dividing both numbers by 4 to get 40/1. Since this is the same as the original ratio, we know that our prediction is good.

(7)

Sometimes it is necessary to round a problem to the nearest ten or one. Since this skill is asking for an estimate, you will not always give an exact amount. Here’s an example of when you would need to estimate by rounding before finding your probability estimate.

1) 19 words in 30 seconds = _____ words in 20 seconds.

19/30 = n/20

19 x 20 = 30n

380 = 30n

380 30 = n

12.6666666666 = n

n = about 13 words

(8)

It will be necessary from time to time to convert some units of measure to a smaller unit of measure to find the most exact estimate. For example, if the data says Dolly’s Donuts sells 142 donuts in 1 hour and they would like to estimate how many donuts they will sell in 2 days, you will need to convert days into the smaller unit, hours, and possibly even minutes. Here are a few facts that will help you with these conversions:

12 pages in 16 minutes = _______ pages in 40 minutes

(30)

(2)

5 miles in 1 hour = _______ miles in 7 hours

(35)

(3)

$17 for 9 binders = $________ for 1 binder

($1.89)

(4)

$99 in 2 weeks = $365 in ______ weeks ______days

(about 7 weeks and 2 days)

(5)

58 problems in _______ minutes = 11 problems in 5 minutes.

(about 26 minutes)

(6)

Tay is playing a game with a spinner with 4 sections, 2 red, 1 blue and one green. For every 8 spins he’s landed on 5 reds. If Tay spins a total of 11 times, how many reds can he expect to get?

(6 or 7)

(7)

Jesse has numbers a piece of paper with each letter from his name in a box. He’s reached in the box without looking 10 times and pulled out the letter e 7 times. If his brother reaches into the box 13 times, how many times can he expect to get the letter e?

(about 18 or 19 times)

(8)

Mandy has a bag of colored candies. She has eaten 12 pieces of candy, 4 of them were red. If Mandy eats all 32 remaining candies, how many can she predict will be red?

(about 10 or 11)

(9)

Ruben has an average of 3 hits for every 5 bats. How many bats will it take him to get 16 hits?

(about 27)

(10)

Megan has found that she gets 2 hits for every 7 bats. If she bats 9 times, how many hits can she expect?

(She can expect to stay at 2 or maybe get to 3. Her chances are right in the middle at 2.57.)

(11)

On the highway there is an average of 11 accidents every 523 miles. If Jill is driving 2,382 miles across country, how many accidents can she plan on passing?

(about 50)

(12)

The ferris wheel spins 200 rotation in 30 minutes. How many rotations will it make in 8 hours?

(3,200 rotations)

(13)

Wade’s Sporting Goods sells on average of 125 basketball per month. How many basketballs can they expect to sell in 6 months?

(750 basketballs)

(14)

Wade’s Sporting Goods sells on average of 125 basketball per month. How many basketballs can they expect to sell in 1 year?

(1,500 basketballs)

(15)

A new teen magazine sold 4,832 copies in the past 3 months. At this rate, how many magazines can they expect to sell at the six month mark?

(9,664 magazines)

(16)

A new teen magazine sold 4,832 copies in the past 3 months. At this rate, how many magazines can they expect to sell after year and a half?

(28,992 magazines)

(17)

Jack’s Towing picks up on average of 3 cars in 40 minutes. How long will it take him to pick up 8 cars?

(about 106 minutes, more commonly known as 1 hour and 46 min.)

(18)

Jack’s Towing picks up on average of 3 cars in 40 minutes. How many cars can they expect to pick up in 3 hours?

(13 - 14 cars)

(19)

Tilly’s Cupcakery can bake about 14 dozen cupcakes in an hour. How many cupcakes can the make in 1 ½ hours?

(252 cupcakes)

(20)

Tilly’s Cupcakery can bake about 14 dozen cupcakes in an hour. How long will it take them to bake 1,298 cupcakes for a celebrity event?

(7 hr. and 44 min.)

(21)

David has driven 283 miles in 3 ½ hours. He needs to drive a total of 562 miles. How long will it take him? (Hint: Break the hours down into minutes to get the most precise answer.)

(6 hours, 57 minutes)

(22)

Yolie has driven 498 miles in 7 hours and 25 minutes. How far will she drive in 8 hours and 40 minutes?

(almost 582 miles)

(23)

Andy can read 112 pages in 1 ½ hours. About how long will it take him to read a 987 page book?

(about 13 hours and 13 minutes)

(24)

Jamie has earned $784 in 2 weeks. At this rate, much more time will it take her to a total of $900.00?

(2 more days and a few hours. It will take her about 16 days to make the $900, but she already has worked 14 days. )

(25)

Steve can move 45 boxes in 1 hour. How long can he expect it to take him to move 56 boxes?