Learning Objectives
- Translate verbal relationships into mathematical relationships.
- Find mathematical relationships that match given data.
Chapter 33 Mathematical Relationships
Section 33.1 Key Words are (Mostly) Evil
Students tend to dislike word problems. This is probably because word problems ask students to do something more difficult than pure calculations. Word problems ask students to understand information and then translate it into a form that allows them to use their mathematical toolbox.
In order to help students with this transition, some teachers started to introduce rule-based patterns into reading word problems. The problem is that this has led to students thinking incorrectly about the process of thinking through word problems. This approach is sometimes called "key words" and you can find lists of "key words" that are categorized by operation.
The problem with this is that these lists are (mostly) evil. There is a grain of truth in them, but it sells students short of an actual understanding of how to think through word problems. It causes students to read word problems in ways that are sometimes incorrect and does not advance the cause of mathematical thinking.
Activity 33.1. The Problem with Key Words.
Here is a classic example: "Bob has 5 rocks. Alice has 3 more rocks than Bob. How many rocks does Alice have?" The common thought process for people is that "more" tells us we need to add, so Alice has 8 rocks. The answer is correct, but the reasoning is wrong. But to understand why the reasoning is wrong, we need to look at a slightly different problem.
Try it!
Alice has 5 rocks. Alice has 3 more rocks than Bob. How many rocks does Bob have? Explain your reasoning.
Solution.
Since Alice has 3 more rocks than Bob, Bob has a smaller number of rocks compared to Alice. So if Alice has 5 rocks, we need to subtract to get the number of rocks that Bob has, which means Bob has 2 rocks.
In the example, the mathematical operation that was required was subtraction, but "more" means to add (if you believe in the rule-based key word approach). And so this leads us to the question of what’s really happening in the word problem.
Activity 33.2. Mathematical Relationships.
In most word problems, the words do not directly describe mathematical operations that you need to perform. They are not instructions. Rather, they are representations of mathematical equations using words. In the example above, the relationship is "Alice has more rocks than Bob." We can translate this into the following:
\begin{equation*}
\begin{array}{c}
a = \text{The number of rocks Alice has} \\
b = \text{The number of rocks Bob has} \\ \\
a = b + 3
\end{array}
\end{equation*}
There are some key points to remember. The first is that the variables used are all defined. This is an important element of communication. If you introduce a symbol, you need to be clear what the symbol means. Furthermore, it’s not acceptable to say that \(a\) represents "Alice." Variables are symbols that represent a quantity or a mathematical expression. Alice is a person, not a number. So it is good to get into the habit of being accurate in describing your variables.
Once we have the relationship established, we can then move to plug in the variables. If we are told that "Bob has 5 rocks" then we can set \(b = 5\) and solve:
\begin{equation*}
\begin{aligned}
a \amp = b + 3 \\
5 \amp = b + 3 \amp \eqnspacer \amp \text{Substitute $a = 5$} \\
b \amp = 2 \amp \amp \text{Subtract $3$ from both sides}
\end{aligned}
\end{equation*}
Or we are told that "Alice has rocks" then we can set \(a = 5\) and solve:
\begin{equation*}
\begin{aligned}
a \amp = b + 3 \\
5 \amp = b + 3 \amp \eqnspacer \amp \text{Substitute $a = 5$} \\
b \amp = 2 \amp \amp \text{Subtract $3$ from both sides}
\end{aligned}
\end{equation*}
The actual decision to add or subtract is based on the relationship that is established by the words, not through identifying a specific word as if it were an instruction. This distinction once again highlights mathematical thinking. You’re not looking for a specific rule or algorithm to follow, you’re looking to understand a relationship so that you can apply the appropriate tool to solve your problem.
If you are unsure about the mathematical relationship you’ve created, it can always be checked by plugging in a couple numbers. There is clarity in examining specific values that can sometimes be lost when looking at symbols. For example, when reading \(a = b + 3\) some students will think that \(b\) is the bigger value "because 3 is being added to it." But this is not true, and the clearest way to see it is to just pick a value of \(b\) and see what the formula actually gives as the result.
Try it!
Annika is 4 inches taller than Carlos. Write an equation that describes this relationship.
Solution.
\begin{equation*}
\begin{array}{c}
a = \text{Annika's height in inches} \\
c = \text{Carlos' height in inches} \\ \\
a = c + 4
\end{array}
\end{equation*}
Activity 33.3. Solving a Full Word Problem.
Once a mathematical relationship is established, it then becomes the foundation for solving a word problem based on that relationship.
Try it!
Annika is 70 inches tall. She is also 4 inches taller than Carlos. How tall is Carlos?
Solution.
\begin{equation*}
\begin{aligned}
a \amp = c + 4 \\
70 \amp = c + 4 \amp \amp \text{Substitute $a = 70$} \\
c \amp = 66 \amp \amp \text{Subtract $4$ from both sides}
\end{aligned}
\end{equation*}
Students shy away from word problems because of the extra step of effort that’s required to translate the words into mathematical symbols. However, that extra step is the step that covers the gap between "real life" and mathematics. Making that connection is an important piece of the puzzle for mathematical thinking.
In addition to having mathematical relationships described by words, there are situations where the relationship is described by a chart of values.
\begin{equation*}
\begin{array}{c|c}
x \amp y \\ \hline
1 \amp 2 \\
2 \amp 4 \\
3 \amp 6
\end{array}
\end{equation*}
After some thought, the pattern that you will likely see is that \(y = 2x\text{.}\) And you can check this by plugging in values.
It turns out that there are all sorts of formulas that would match those values. If you were to use the equation \(y = -\frac{1}{6} x^3 + x^2 + \frac{1}{6} x + 1\text{,}\) you would find that it also works! However, for these problems we are going to stick with the "obvious" answer and not worry about more complicated relationships.
Activity 33.4. Finding a Mathematical Relationship.
When trying to find a formula that matches a given table, there is no magic formula. And while there is a computational method if you can guess the "form" of the equation, it’s better to develop intuition. The relationships that you will be given to work with are going to be relatively simple, and you should be able to come up with them by trying to think logically and in an organized manner.
Try it!
Find a formula that relates and based on the following chart.
\begin{equation*}
\begin{array}{c|c}
x \amp y \\ \hline
1 \amp 9 \\
4 \amp 6 \\
8 \amp 2
\end{array}
\end{equation*}
Solution.
\begin{equation*}
x + y = 10
\end{equation*}
Section 33.2 Worksheets
PDF Version of these Worksheets
1
external/worksheets/33-Worksheets.pdfWorksheet Worksheet 1
1.
Matt and Reuben have 14 toys between the two of them. Write an equation that describes this relationship.
2.
Matt and Reuben have 14 toys between the two of them. Matt has 5 toys. How many toys does Reuben have?
3.
Reuben has 4 more toys than Matt. Write an equation that describes this relationship.
4.
Reuben has 4 more toys than Matt. Together, they have 14 toys between them. How many toys does each one have?
Worksheet Worksheet 2
1.
A box of cans of soda contains 12 cans. Write a relationship between the number of boxes and the number of cans of soda.
2.
A store has a sale on boxes of cans of soda. However, the store limit is 8 boxes per customer. If each box of soda contains 12 cans, how many cans of soda can be bought by a single customer during this sale?
3.
Dr. Pepper is a math professor. Every semester, she buys each of her students a can of her favorite soda. She has 30 students in her class. If a box of cans of soda contains 12 cans, how many boxes does she need for her class?
4.
In the previous question, there were two possible answers, depending on how you saw the problem. You might have had a fractional box of sodas, or you might have decided a fractional box of soda doesn’t make sense and rounded up to an integer number of boxes. Is it possible that both answers are correct, or is one answer correct and the other incorrect?
Worksheet Worksheet 3
1.
Find a formula that relates the variables in each chart. (Note: Each chart is separate.)
2.
Find a formula that relates the variables in each chart. (Note: Each chart is separate.)
3.
Find a formula that relates the variables in each chart. (Note: Each chart is separate.)
4.
Find a formula that relates the variables in each chart. (Note: Each chart is separate.)
Worksheet Worksheet 4
1.
A gym membership at Gym A costs $60 as an initiation fee plus $20 per month. Write a formula that relates the gym costs and the length of the membership.
2.
A gym membership at Gym B costs $100 as an initiation fee plus $10 per month. Write a formula that relates the gym costs and the length of the membership.
3.
Determine the price of months of membership at each gym. Which one is the better deal if you are a member for less than 4 months? Which is the better deal if you are a member for more than 4 months?
4.
Suppose you were thinking about starting an exercise routine, and that you were deciding between Gym A and Gym B. You know that you want to give it at least three months of effort, but you’re not sure if you’ll go beyond that. Which gym would you choose? Explain the logic of your decision.
Worksheet Worksheet 5
1.
Write a word problem that uses the phrase "more than" but that you cannot just add the numbers in the problem to get the correct answer. Then solve it.
2.
Write a word problem that uses the phrase "fewer than" but that you cannot just subtract the numbers in the problem to get the correct answer. Then solve it.
3.
Write a word problem that uses multiplication in the mathematical relationship between the variables. Then solve it.
Section 33.3 Deliberate Practice: Basic Word Problems
Algebra is a skill, which means it requires practice to become proficient. But it will take more than rote repetition to get there. Deliberate practice is the thoughtful repetition of a task. For each of these sections, you will be given a list of specific skills or ideas to focus on as you practice thinking through the problems.
Focus on these skills:
- Define your variables.
- Write the equation or equations that represent the given mathematical relationship or relationships.
- Present your work legibly.
Worksheet Worksheet
Instructions: Answer the question.
1.
Kai and Ama are math tutors. Kai has 12 clients, and Ama 5 has more clients that Kai. How many clients does Ama have?
2.
Kai and Ama are math tutors. Kai has 7 clients, which is 3 fewer than Ama. How many clients does Ama have?
3.
Kai and Ama are math tutors. Kai has 7 clients, and Ama has four fewer clients than Kai. How many clients do they have in total?
4.
Martin and Sylvia have each have marbles in their pockets. Sylvia has 7 more marbles than Martin, and Martin has 9 marbles. How many marbles does Sylvia have?
5.
Martin and Sylvia have each have marbles in their pockets. Sylvia has 5 fewer marbles than Martin, and Martin has 8 marbles. How many marbles does Sylvia have?
6.
Martin and Sylvia have each have marbles in their pockets. Together, they have 22 marbles. Sylvia has 2 more marbles than Martin. How many marbles do they each have?
7.
Leonard and Carlos are having a pickle-eating contest. Carlos ate three fewer pickles than Leonard. Carlos managed to eat five pickles. How many pickles did they eat in total?
8.
Leonard and Carlos are having a pickle-eating contest. Carlos ate three more pickles than Leonard. In total, they ate 11 pickles. How many pickles did they each eat?
9.
A food box program costs $30 per month if you pay as you go, but if you pay for a year’s worth of boxes up front, they will give you two months for free. What is your average monthly discount for paying in advance?
10.
A food box program costs $30 per month if you pay as you go, but if you pay for a six months at a time, you get two points towards their bonus program. If you get ten bonus points, they will give you a free box. How long will it take to earn a free box?
Section 33.4 Closing Ideas
This is the first section in the final branch of this book. The ideas are about trying to bridge the gap between formal mathematics and applications. In this first section, we focused primarily on relationships between variables because this is the core value of mathematical reasoning. Translating quantitative relationships into mathematical equations is the key skill that employers want when they talk about wanting employees with "math ]skills." They don’t need people to just perform calculations, since they have computers and calculators that can do that for them. They need people who can translate the needs of the business into a mathematical language that they can then use to make decisions. The problem of the two gyms in the worksheets is an example. There often isn’t just a single answer. It depends on non-mathematical features of the situation that can’t always be easily quantified.
And that’s an important takeaway for thinking about applications of mathematics. It’s not always the pursuit of "the" answer, but rather it’s a tool to help you think about the situation and provide quantitative information to help with the decision-making process.
