Barometric pressure is a measure of how much mass of air, i.e. how many air molecules, exist above the point of measurement, all the way up to the top of the atmosphere. Therefore, barometric pressure decreases with elevation. Any given volume of air has density. The density of air can vary from place to place and from time to time. The difference in barometric pressure between observation sites at different elevations is a measure of the density of air in a column of air between those two elevations.
Cold air is denser than warm air.
Drawing on hands-on observations to explain an aspect of a natural system. Building a chain of reasoning from cause to effect. Building a chain of reasoning from observation to interpretation. Instructions for this activity are contained in many middle school science books. In this video, a mercury barometer is carried up a mountainside, and the mercury is seen to fall as the climber ascends.