5.3.4 Match Expressions and Tables with Situations

Multiple Representations Verbal Scenario

Here are verbal descriptions of two situations, followed by tables and expressions that could help answer one of the questions in the situations.

A food company currently has 5 convenience stores. It is considering two plans for expanding its chain of stores.

Plan A: Open 20 new stores each year.

Plan B: Double the number of stores each year.

How many stores will there be after 5 years under each plan?

Match each representation (a table or an expression) with one situation. Be prepared to explain how the table or expression answers the question.

1. $5+20+20+20+20+20$
2. $5·32$
3. $5·2·2·2·2·2$

4. years	0	1	2	3	4	5
   number of stores	5	10	20	40	80	160

5. $5+100$

6. years	0	1	2	3	4	5
   number of stores	5	25	45	65	85	105

To determine which options go with which scenario, you first need to analyze each scenario.

Let’s look at Plan A.

• Plan A is to open 20 new stores each year for a food company that currently has 5 convenience stores. In both scenarios, the initial value or starting point is 5 stores, which is why in each table the first value is $(0,5)$. If they are opening 20 additional stores each year, they are adding 20 stores to the total from the previous year. So in year 1 there would be 25 total stores, in year 2 there would be 45 total stores, and so on. These values are represented in option 6, so option 6 would represent Plan A.
• The mathematical expressions contain either multiplication or addition. Plan A is ADDING 20 stores a year, so year 5 could be represented as option 1 because it would be the initial number of stores plus 20 per year for 5 years, or $5+20+20+20+20+20$.
• Option 5 is also added, and represents Plan A since $5+20+20+20+20+20$ is the same as $5+100$.

Let’s look at Plan B.

• Plan B would also start at 5, but it says to double the number of stores each year. Double means to MULTIPLY by 2, so it would be $5·2·2·2·2·2$, which is option 3.
• Since $5·2·2·2·2·2$ can be simplified to $5·32$, option 2 also represents Plan B.
• Option 4 is the other table option, and as mentioned before, each table should have an initial value of 5 stores. Since Plan B is to double the number of stores, at the end of year 1 there should be 10 stores. When you check option 4, you can see that each term is double the term before.

Therefore, Plan A is options 1, 5, and 6, and Plan B is options 2, 3, and 4.