# Bus, Train, and other problems for changing values

## Overview

Bus and train problems asking the number of passengers after several board and exit are interesting and challenging problems to which learners can relate. They can be used to introduce ways to represent problems, record value changes, and as an introduction to recursive expressions and formulas.

This page includes information to plan and use activities for passengers on a bus and train with introduction for formulas and suggestions to extensions of changing values in: a bank account, amount of snow on the ground over a month as it accumulates and melts, money box at a garage sale, super market or other business cash register with multiple sales. How the sales are reordered for purchases with multiple items. Height of stacks of chairs or cups as they are used and replenished.

## Concepts and outcomes

### Mathematical

#### Concepts

• Problems can be represented with words, pictures, symbols, numbers, tables, diagrams, charts, graphs, equations, formulas, models, simulations, animations, videos
• Different ways to communicate and represent problems have different advantages.
• Formula is an equation that can describe an event or actions.
• Sequence is an order in which numbers or things follow each other.

Mathematical concepts, misconception, and outcome knowledge base

#### Outcomes

• Identify value changes with addition and subtraction.
• Recognize problem solving strategies. Acting out a problem, working backwards.
• Reason, explain, and justify solutions with representations and symbols.

## Pedagogical notes

### Activities sequence

• Bus driver problem - strategy work backwards
• Train engineer problem - strategy work backwards
• Vehicle of your choice - bus or train problem act out. Use tables to record actions. Recursive formulas to solve problems,
• Bank account Amount of snow on the ground over a month Money box at a garage sale. Super market with multiple sales. How the sales are reordered for all items. Height of stacks of chairs. Cups. Find prices for individual items put on a supreme pizza or sub sandwich and calculate the total for 20 subs and 10 pizzas.

### Focus questions

• What is a sequence?
• How would a bus driver know the number of passengers on their bus?
• What strategies do you know to solve problems?
• How are problems represented?

## Activity discussion and teaching suggestions

### Activity 1 - Bus problem

Discussion: Define start. In this problem, start is defined. In some problems, like this, start may not be defined and could be assumed to be before the bus driver arrives. Therefore, a plausible answer would be: zero (0).

However, with the start being defined, a backward strategy could be used to start with the total on the bus at the, now point and undo events at the two stops. Think of a video clip running backwards or acting it out backwards.

• At stop 2. Eleven get on the bus. So they are included among the 34 passengers. Therefore, to undo what happened at stop 2, you have to subtract 11 passengers. See them backing off the bus and stopping at the bus stop to wait for the bus to arrive as the bus backs away?
34 passengers - 11 passengers = 23 passengers
• At stop 1. Thirteen people get off the bus so to undo that action, you need to add 13 passengers. See them backing onto the bus and sitting down?
Therefore, you have:
23 passengers + 13 passengers = 36 passengers
• All together:
34 passengers - 11 passengers + 13 passengers = 36 passengers
• Notice. Each equation is separate.

A second solution would be to use algebra.

• In that case, you could start with the unknown, the number of passengers on the bus at the start.
x = passengers on the bus at the start
• Then, take the thirteen passengers that got off and represent it in the equation as - 13.
• x - 13
• Then, represent the eleven passengers that got on as + 11.
• x - 13 + 11
• Next, set it equal to the thirty-four passengers left on the bus.
• x - 13 + 11 = 34
• Then, solve the equation for x and x = 36.

Why do people find this problem difficult?

A couple of reasons. First, some people want a starting number and when there isn't one, they don't know where to begin.

Some may pick a number, use a guess and check strategy, and alter their selected starting number by adding and subtracting in an attempt to arrive at the final number of people.

Others, may overcome there not being a starting number, go to the end, and try to work backwards. However, some of them will freeze when the problem says people get on the bus, because they don't believe they can increase the final number beyond its final value, since it is the final value. Others may not think they should subtract passengers since the problem says they get on the bus and that usually means addition.

Let's look at a mistaken equation and compare it to an accurate one.

• A person thinks, 34 passengers, and 11 got on and 13 got off. They would conclude:
• (34 + 11) - 13 = start.
• Others think to start with an unknown, take 13 off and add 11 on. They might conclude:
• x - 13 + 11 = 34.
• If both are solved, the solutions are different.
• (34 + 11) - 13 = x; x = 32
• &
• x - 13 + 11 = 34; x = 36

Which should cause students to consider how this happened and explain each with a representation that will justify the difference.

### Activity 2 Train problem

Same as activity 1

### Activity 3 Vehicle problem

Slightly different than the previous bus and train problem.

Need to assume the vehicle is empty to start.

Then using recursive formulas.

• 0 + 6 = 6
• 6 + 5 - 2 = 9
• 9 + 3 = 12

If the something is written like the following: 0 + 6 = 6 + 5 - 2 = 9 + 3 = 12 or 6 + 5 - 2 = 9 + 3 = 12.

Then, attention needs to be made that the values on each sides of the equal sign aren't equal. If they are simplified, you have 6 = 9 = 12. Not equalities.

A driver who wants to record a running record might use a table with headers such as: Start, passengers getting on, Passengers getting off, Passengers aboard.

Have learners record the first four stops in the table.

Leaving Passengers on board Passenger boarding Passengers disembarking
Station 0
Stop 1 7 7
Stop 2 13 6
Stop 3 16 5 2
Stop 4 19 3

## Bus Logic Problem

Pretend you are a bus driver. You start your route:

### Start

There are some people on a bus.

### Stop 1:

13 people get off.

11 get on.

### Now:

There are 34 people on the bus.

How old is the bus driver?

Thought the first question would be how many people on the bus ...

Okay!

How many were on the bus at the start?

Hint: Use a, work the problem backwards or undo strategy.

## Train logic problem

Pretend you are a train engineer. You start your route:

### Start

There were some people on a train.

### Stop 1:

16 people get off.

19 get on.

### Now:

There are 68 people on the train.

When is the engineers birthday?

Thought the first question would be how old is the engineer ? ...

Okay!

How many were on the train at the start?

Hint: Use a, work the problem backwards or an undo strategy.

## Another vehicle problem

Pretend you are a ______________. You start your route:

### Stop 1

The first stop seven people get on.

### Stop 2

Six people get on.

### Stop 3

Five people get on and two off.

### Stop 4

Three people get on and no one gets off.

How many are on the bus after the fourth stop?

Hint:

Act it out.

If a train engineer or bus driver wanted to record information on how many passengers got on, got off, and were currently on the vehicle, what kind of table might they make?