6.7: Practical Applications

By understanding these practical applications, students can see the value of mathematical reasoning beyond the classroom and develop the skills they need to succeed in a variety of fields.

These examples illustrate how mathematical reasoning can be applied to real-world situations, helping students understand its practical significance in everyday life.

What type of reasoning skill does each situation use?

Exercise \(\PageIndex{1}\)

First Grade Situations:

1. A student is sorting toys into two boxes. The teacher says, "Put the cars in the blue box and the dolls in the red box."
2. At a grocery store, there are two checkout lines. One line has three people waiting, and the other line has five people waiting. A student observes, "If I join the line with three people, I will wait less than if I join the line with five people."
3. A child notices that every time they add one more block to their tower, it gets taller. They think, "If I keep adding blocks, my tower will keep getting taller."
4. A student is counting apples. They start with 3 apples and their friend gives them 2 more. The student thinks, "If I start with 3 apples and get 2 more, I will have 5 apples."

Answers

Solutions:

1. The student uses the connective "and" to follow the instructions correctly, putting cars in the blue box and dolls in the red box.
2. The student uses deductive reasoning to understand that joining the shorter line will likely result in less waiting time.
3. The child uses inductive reasoning to understand that adding more blocks increases the height of the tower.
4. The student uses a direct proof by adding the numbers directly: 3 + 2 = 5, showing they will indeed have 5 apples.

Exercise \(\PageIndex{2}\)

Second Grade Situations:

1. A student observes that every time they skip count by 2s, they get even numbers. They conclude, "If I skip count by 2s, I will always get even numbers."
2. A child is packing lunch and wants to make sandwiches using bread slices. They know they have two slices of bread for each sandwich and six slices of bread in total. The child thinks, "If I make three sandwiches, I will use all the bread."
3. A child is making a sandwich. Their parent says, "You can use peanut butter or jelly, but not both."
4. A teacher shows students a collection of shapes. There are 4 triangles and 4 squares. The teacher asks, "Do we have more than 5 shapes in total?"

Answers

Solutions:

1. The student uses inductive reasoning to generalize that skip counting by 2s always results in even numbers.
2. The child uses deductive reasoning to determine that making three sandwiches will use all six slices of bread.
3. The child uses the connective "or" to understand that they can choose either peanut butter or jelly for their sandwich, but not both.
4. The students use a visual proof by counting each shape. They see there are 8 shapes (4 triangles + 4 squares) which is more than 5, proving the statement visually.

Exercise \(\PageIndex{3}\)

Third Grade Situations:

1. A group of friends notices that every time they share a pizza equally, each person gets a fair share. They think, "If we divide any pizza equally, everyone will get the same amount."
2. Students are solving a puzzle where they need to find numbers that are both even and greater than 10.
3. A group of friends wants to share 12 cookies equally. One friend says, "If we each take two cookies at a time, we will finish all the cookies."
4. A student observes that every time they add an even number to another even number, the result is always even. They check with 2 + 2 = 4, 4 + 6 = 10, and 8 + 12 = 20. They conclude, "Any time I add two even numbers, the result will always be even."

Answers

Solutions:

1. The friends use inductive reasoning to generalize that dividing any pizza equally among them will result in everyone getting the same amount.
2. The students use the connective "and" to identify numbers like 12, 14, and 16 that meet both criteria.
3. The friends use deductive reasoning to understand that taking two cookies each time will result in them finishing all 12 cookies.
4. The student uses inductive reasoning to generalize that adding any two even numbers results in an even number based on specific examples.

Exercise \(\PageIndex{4}\)

Fourth Grade Situations:

1. A family is planning a road trip. They know that their car gets 30 miles per gallon of gas. The family says, "If we drive 300 miles, we will need 10 gallons of gas.".
2. A student notices that every time they multiply a number by 10, the digit in the ones place becomes 0. They conclude, "If I multiply any number by 10, the ones place digit will always be 0."
3. A student is baking cookies and the recipe says, "Add 2 cups of sugar if you want them sweet, or 1 cup of sugar if you want them less sweet."
4. A student is solving a problem where they need to determine if the sum of two odd numbers is always even. They check with 1 + 3 = 4 and 5 + 7 = 12. They conclude, "If I add two odd numbers, the result is always even."

Answers

Solutions:

1. The family uses deductive reasoning based on their car's mileage to calculate the amount of gas needed for a specific distance.
2. The student uses inductive reasoning to generalize that multiplying any number by 10 always results in the ones place digit being 0.
3. The student uses the connective "or" to decide how much sugar to add based on their preference for sweetness.
4. A student is solving a problem where they need to determine if the sum of two odd numbers is always even. They check with 1 + 3 = 4 and 5 + 7 = 12. They conclude, "If I add two odd numbers, the result is always even."

Exercise \(\PageIndex{5}\)

Fifth Grade Situations:

1. In a math class, the teacher says, "If a number is divisible by 4, then it is also divisible by 2."
2. A farmer has a field that is 60 meters long and 40 meters wide. The farmer thinks, "If I plant crops in rows that are 1 meter apart, I can fit 40 rows in the field."
3. A family goes grocery shopping and notices that every time they buy 1 gallon of milk, it costs $3. They think, "If we buy any number of gallons of milk, it will cost $3 per gallon."
4. A student is given a problem to prove that if a number is divisible by 6, then it is divisible by 3. They assume the opposite: "If a number is not divisible by 3, it is not divisible by 6." They check with 6 (divisible by both), 12 (divisible by both), and 14 (not divisible by 6, not divisible by 3). They conclude, "Since 14 is not divisible by 6 and not divisible by 3, our assumption holds."

Answers

Solutions:

1. The student uses the connective "if...then" to understand that any number divisible by 4 (like 8, 12, 16) will also be divisible by 2.
2. The farmer uses deductive reasoning to calculate the number of rows that can fit in the field based on its dimensions and the spacing between rows.
3. The family uses inductive reasoning to generalize that the cost of milk is $3 per gallon, regardless of the quantity purchased.
4. The student uses an indirect proof (contrapositive) to show that if a number is divisible by 6, it must be divisible by 3.

Exercise \(\PageIndex{6}\)

Sixth Grade Situations:

1. A baker observes that every time they double the recipe, they need to use twice the amount of each ingredient. They conclude, "If I double any recipe, I will need to use twice the amount of each ingredient."
2. A student is planning a trip and calculates the budget. They think, "I will spend $50 if I go to the amusement park, or $30 if I go to the zoo."
3. The family uses inductive reasoning to generalize that the cost of milk is $3 per gallon, regardless of the quantity purchased.
4. A student is asked to prove that the area of a rectangle is the same when calculated in different ways. They are given a rectangle with length 5 units and width 3 units. The teacher says, "Draw a rectangle with these dimensions and count the squares inside."

Answers

Solutions:

1. The baker uses inductive reasoning to generalize that doubling any recipe requires using twice the amount of each ingredient.
2. A student is planning a trip and calculates the budget. They think, "I will spend $50 if I go to the amusement park, or $30 if I go to the zoo."
3. The baker uses deductive reasoning to determine that making 5 cakes will use all 15 cups of flour.
4. The student draws the rectangle and counts 15 squares inside (5 rows of 3 squares each). This visual proof shows that the area calculation (length × width) matches the count of squares inside the rectangle.

Exercises

Fallacies in Common Language

For each of the following statements, name the type of logical fallacy being used.

1. If you don’t want to drive from Boston to New York, then you will have to take the train.
2. Every time I go to Dodger Stadium, the Dodgers win. I should go there for every game.
3. New England Patriots quarterback Tom Brady likes his footballs slightly underinflated. The “Cheatriots” have a history of bending or breaking the rules, so Brady must have told the equipment manager to make sure that the footballs were underinflated.
4. What you are doing is clearly illegal because it’s against the law.
5. The county supervisor voted against the new education tax. He must not believe in education.
6. “Apples a day keeps doctors away.” No one has said apples are bad, so this old saying must be true.
7. Wine has to be good for your health because… I mean, can you imagine a life without wine?
8. Studies show that slightly overweight senior citizens live longer than underweight ones. The extra weight must make people live longer.
9. Whenever our smoke detector beeps, my kids eat cereal for dinner. The loud beeping sound must make them want to eat cereal for some reason.
10. There is a scientist who works at a really good university, and he says there is no strong evidence for climate change, especially global warming. Some politicians also question climate change. So I don’t really believe it.
11. My neighbor cheats on his tax returns. I don’t believe anything he says.

A: “Don’t fight over small things. Just let them go.”

B: “What exactly are ‘small things’? How do I know what is small and what isn’t?”

A: “Well, small things are things you really don’t want to fight over.”

Propositions and Logic

13. List the set of integers that satisfy the following statement: A positive multiple of 5 and not a multiple of 2
14. List the set of integers that satisfy the following statement: Greater than 12 and less than or equal to 18
15. List the set of integers that satisfy the following statement: Even number less than 10 or odd number between 12 and 10
16. You qualify for a special discount if you are either

• a full-time student in the state of California or
• at least 18 and your income is less than $20,000 a year.

For each person below, determine if the person qualifies for this discount. If more information is needed, indicate that.

1. A 17-year-old full-time student at a California community college with no job
2. A 28-year-old man earning $50,000 a year
3. A 60-year-old grandmother who does not work and does not go to school
4. A boy in first grade
5. A mother making $18,000 a year and not enrolled in any college
6. A 22-year-old full-time student at Arizona State University with no job
7. A 18-year-old earning exactly $20,000 a year while attending UCLA part-time

17. Write the negation: Everyone failed the quiz today.
18. Write the negation: Someone in the car needs to use the restroom.
19. How can you prove this statement wrong?

“Everyone who ate at that restaurant got sick.”

20. How can you prove this statement wrong?

“There was someone who ate at that restaurant and got sick.”

21. How can you prove this statement wrong?

“There is no baseball player who can excel at both pitching and hitting. Everyone must choose one or the other.”

22. How can you prove this statement wrong?

“Every student must have an ID number before registering for classes.”

Analyzing Arguments

39. Determine whether each of the following is an inductive or deductive argument:
    1. The new medicine works. We tried it on 100 patients, and all of them were cured.
    2. Every student has an ID number. Sandra is a student, so she has an ID number.
    3. Every angle of a rectangle is 90 degrees. A soccer field (pitch) is rectangular, so every corner is 90 degrees.
    4. Everything that goes up comes down. I throw a ball up. It must come down.
    5. Every time it rains, my grass grows fast. Rain speeds up the growth of grass.
    6. Sports makes a person strong. My daughter plays basketball. She will be strong.
40. Analyze the validity of the argument.

Everyone who gets a degree in science will get a good job. I got a science degree. Therefore, I will get a good job.

41. Analyze the validity of the argument.

If someone turns off the switch, the lights will not be on. The lights are off. Therefore, someone must have turned off the switch.

42. Suppose the statement: “If today is Dec. 25, then the store is closed.”
    1. “Today is Dec. 25. Thus, the store is closed.” Which property was used? Is this valid?
    2. “The store is not closed. So today is not Dec. 25.” Which property was used? Is this valid?
    3. “The store is closed. Therefore, today must be Dec. 25.” Which property was used? Is this valid?
    4. “Today is not Dec. 25. Thus, the store is open.” Which property was used? Is this valid?

For the following questions, use a Venn diagram or a truth table to determine the validity.

43. If a person is on this reality show, they must be self-absorbed. Laura is not self-absorbed. Therefore, Laura cannot be on this reality show.
44. If you are a triathlete, then you have outstanding endurance. LeBron James is not a triathlete. Therefore, LeBron does not have outstanding endurance.
45. Jamie must scrub the toilets or hose down the garbage cans. Jamie refuses to scrub the toilets. Therefore, Jamie will hose down the garbage cans.
46. Some of these kids are rude. Jimmy is one of these kids. Therefore, Jimmy is rude!
47. Every student brought a pencil or a pen. Marcie brought a pencil. Therefore, Marcie did not bring a pen.
48. If a creature is a chimpanzee, then it is a primate. If a creature is a primate, then it is a mammal. Bobo is a mammal. Therefore, Bobo is a chimpanzee.
49. Every cripsee is a domwow. Mekep is not a domwow. Therefore, Mekep is not a cripsee. (This sentence has a lot of made-up words, but it is still possible to check for the validity of the argument. This is a good practice for abstract thinking.)
50. Whoever dephels a kipoc will be bopied. I did not dephel any kipoc. Therefore, I will not be bopied. (Again, you do not need to know the meaning of each word to do this exercise.)