Hands-on Activity: Solving with Seesaws
Educational Standards :
Pre-Req Knowledge (Return to Contents)
A basic understanding of solving equations with one unknown.
Learning Objectives (Return to Contents)
After this activity, students should be able to:
Materials List (Return to Contents)
(Note: The seesaw for this activity may be constructed entirely with parts from hardware stores, or a carpenter-machinist might be helpful in its construction. Or, any seesaw can be used for the activity, as long as the LEGO MINDSTORMS NXT sensors can be attached to a moveable beam to demonstrate balance.)
The classroom seesaw set-up needs:
Note that single licenses and site licenses are available; site licenses may make sense for schools with high use.
Introduction/Motivation (Return to Contents)
Two-step equations may not look pleasant on paper. However, learning how to solve them is important for solving many real-world problems that come up in math and science. Two-step equations even come up in daily life. Figuring out how many boxes of cereal you can get with $15 while taking into account coupons or discounts is one example. Calculating if a cardboard box with a weight limit can hold smaller boxes of cans along with other loose cans is another example. The list goes on and on.
A simple example of a two-step equation is shown in Figure 1. Such an equation is composed of variable and constant terms, which can appear on either side of the equation. (Write on the board the Figure 1 equation, along with its corresponding terminology. Solve step-by-step, and you ultimately obtain n = 12.) Solving an equation like this involves moving terms from one side to another, keeping the equation "balanced."
A good physical example of the use of two-step equations is the seesaw or a physical balance. It is a simple machine that embodies an equation. (Draw a diagram of a seesaw under the equation, positioning the fulcrum under the equal sign). We do not have an equation if there is no balance. In this way, it helps to physically view what you need to do to solve an equation. For example, the Figure 1 equation can be used as follows: you have 27 loose bottles of water on one side of a seesaw, and 3 loose bottles on the other, which are placed alongside 2 full bags of bottles.
Engineers use equations all the time when designing structures or machines. For example, you may have seen a construction site where workers, architects and engineers work together to build the large skyscrapers found in big cities.
Most skyscrapers are made of basic structures called beams. Beams are simple structures whose lengths are much longer than their widths and heights. These structures are used in buildings because they are shaped to be stiff enough to resist a lot of bending. They are made of wood, metal or a mix of materials. Many beams are placed inside buildings so that they work together to keep a building standing still.
Civil engineers are the type of engineers who design bridges, road, buildings and other structures for our daily use. Believe it or not, the designs behind buildings involve equations, especially when balancing the weight of the entire building. If not balance, a building may tilt or not be able to withstand the amount of weight (load) it was designed to hold. That is why civil engineers check and re-check their work to make sure the design functions, especially for everyone's safety. Engineers go as far as to re-check every single beam in the entire building design to ensure that it remains "balanced" when supporting weight. Keeping that in mind, let's try another example (see Beam Example).
With the equipment we have today, we're going to use the help of sensors, which are electronic devices that measure the changes in a system as it is being used. Sensors can be used to measure temperature, distances, the level of light and darkness in a room, weight and other data. Sensors have been used inside buildings and machines to make sure that they are maintained and working correctly. In our activity today, we will use this balance system to help us see if the equations we solve by hand (and by using the seesaw) are "balanced."
Vocabulary/Definitions (Return to Contents)
Procedure (Return to Contents)
Before the Activity
With the Students
Attachments (Return to Contents)
Safety Issues (Return to Contents)
Troubleshooting Tips (Return to Contents)
After each weight transfer, level the seesaw by hand and then let it balance by itself, so that the ultrasonic sensors can read the correct distance from the floor.
Make sure that no nearby objects (such as tables, desks or your own arms, legs and feet) are too close to the sensors, since they may obstruct the sensor's readings.
Assessment (Return to Contents)
Two-Step Equations: Have students complete the Seesaw Pre-Assessment to assess their prior knowledge of solving two-step equations.
Real-World Conceptual Identification: Before the activity, briefly identify and discuss two-step equations, as well as some real-world examples. Ask students how two-step equations are involved in the design of buildings and large-scale structures. Does the design of bridges and skyscrapers involve solving equations? (Answers: Yes! The design of bridges and skycrapers involve solving equations based on forces, moments, length and width dimensions, area, volume and many other concepts. In the process of planning these structures, engineers solve many equations to make sure everything will fit together correctly and the structure will support itself and not fall down. These equations involve two-step equations and some equations that take even more than two steps to solve!) Make sure that students realize that structures and buildings, much like equations, need to be balanced in terms of weight.
Activity Embedded Assessment
Analogies and Real-World Examples: Evaluate students on the following criteria: cooperation in problem solving, and step-by-step reasoning using a physical analogy of the problem at hand. Since they are working in groups, all students in the group must participate and cooperate with each other. It is also important for each group to demonstrate a logical progression from each step to the next in order to understand how to solve basic two-step equations. Ask students: What other kinds of existing structures require such mathematical analysis? (Possible answers: Factory machinery, or objects in their own homes.) For example: What needs to be considered to build a kitchen table? What objects and forces does a kitchen table need to support? (Possible answers: The weight of plates and bowls of food, bags of groceries, a vase of flowers, a person leaning on it.) Ask: Does the structure or object need to be balanced? Can you help make sure it is balanced using mathematical equations? (Answer: Yes, by using mathematical equations, you can determine how much force from the load on the table is distributed to each leg or support of the table. If you were designing the table, you could test the material you wanted to use to make sure the legs or supports of the table could hold the weight of a Thanksgiving dinner or a seven-layer wedding cake, for example.)
Class Discussion: Re-iterate the importance of mathematics in building design. To give a general idea of the amount of work engineers undergo to assess building designs, ask students how many equations engineers may have to solve when dealing with the design of a skyscraper. Are beams the only structures inside of buildings that can be designed using equations? (Answer: No, many other objects within a structure can be designed using equations. A table is a one example that we already mentioned. We focused on beams because they play a major supporting role in large structures. Many equations must be solved to make sure each beam can support the load expected to be applied to it. Larger structures usually means more beams... and more equations to solve!) As an additional inquiry, ask students: How might how sensors play roles in building design and assessment? Do you think that buildings can use sensors? How and why? (Answer: Sensors can be used to measure the force being applied to a particular beam based on its load. Also, as in our activity today, sensors can be used to collect all sorts of data, such as to make sure things are level. That's important for how things look, but even more important for building stability. If a force causes a beam to lean or fall in one direction, a sensor can alert an engineer that this is happening before the beam leans or falls so far that other objects it is supporting fall down as well.)
Test: Have students complete the Seesaw Post-Assessment to demonstrate their understanding of solving two-step equations.
Activity Extensions (Return to Contents)
Create a force balance activity using a similar structure. Using the concepts of force and moment, cut several notches into the beam so that the distance from the seesaw center to each load on the beam becomes another parameter to consider for beam balance.
Supporting Program (Return to Contents)Applying Mechatronics to Promote Science (AMPS) GK-12 Program, Polytechnic Institute of New York University
Acknowledgements (Return to Contents)
This activity was developed under the Applying Mechatronics to Promote Science (AMPS) Program funded by National Science Foundation GK-12 grant no. 0741714. However, these contents do not necessarily represent the policies of the NSF and you should not assume endorsement by the federal government.