Lesson 1: Using Mathematical Models/Proportions
In this lesson, students learn how an equation works like a scale with the equal sign serving as the balancing point, as well as how scale keeps a map's distances in proportion to real-world areas depicted on a map.
- Students will understand that an equation works like a scale with the equal sign serving as the balancing point; that the equal sign means "is the same as" rather than "the answer to the problem is"; that, like balancing weight on a scale, what is done to one side of an equation must be done to the other.
- Students will understand how scale keeps a map's distances in proportion to real-world areas depicted in the map; use a map's scale and coordinates to calculate distance; identify the variables in the formula d = r · t; and manipulate the formula to solve for variables.
- Balance scale with weights (drawings on the board will also suffice)
- Map (e.g., world, state, etc.) with longitude/latitude and/or coordinates, and a key including scale
- Worksheet 1 Printable (PDF) — Content Connections: Ordered Pairs
- Bonus Worksheet 1 Printable (PDF) — Content Connections: Measurement
- Take-Home Activity 1 Printable (PDF) — Content Connections: Measurement
- Resource: Answer Key (PDF)
- Resource: Mini-Poster (PDF)
Time required: 30-40 minutes, depending on class review needs; additional time for worksheets
1. Show students a balancing scale without weights on either arm. Ask why the scale is in balance (because each side has the same weight). Ask what would happen if 5 grams were added to both sides. Demonstrate, reiterating that the scale is still balanced because the same amount was added to each side.
2. Ask what would happen if we only operated on one side of the scale. What if 2 grams were taken off one side? Demonstrate that the scale is out of balance because the action taken on one side wasn't repeated on the other. Return the 2 grams to balance the scale.
3. Ask what would happen if we tripled both sides. Demonstrate that the scale is still balanced. Then, to demonstrate inverse operations, divide the weight on each side by 3, showing that inverse operations "cancel each other out."
4. Indicate that equations work like the balancing scale, with the equal sign acting as the fulcrum. (Many students have the common misconception that the equal sign means "the answer to the problem is" rather than "is the same as.") Before proceeding, make sure students understand this distinction and the goal of keeping things in balance.
5. Write a simple equation on the board, e.g., x + 7 = 9. Indicate that the goal is, as with the scale, to keep the equation in balance. Indicate that we can solve for x if we isolate it on one side of the equation through manipulations that keep the equation in balance. Depending on class support needed, show how to solve for x using the scales metaphor, and keeping the equation in balance by taking the same action on both sides. Write -7 under the 7 on the left side of the equation and -7 under the 9 on the right side. Complete the subtraction, leaving x = 2. Repeat with other one-step equations, such as x - 4 = 7, 5x = 45, and x/6 = 7. Show how to check the accuracy of the calculation by incorporating the value of x into the original equation. Both sides will be equal.
6. Follow up with an equation requiring more than one manipulation, e.g., 2x + 4 = 8. Ask students to solve and explain, step by step, how they kept the equation in balance. For something more complex, try 2x + 4 = 6x - 8. Again, ask students to solve and explain. Show the step-by-step manipulation, first adding 8 to both sides, then subtracting 2x from both sides, leaving 12 = 4x. Show how dividing both sides by 4 keeps the equation in balance, leaving x = 3. Incorporate the value of x back into the original equation to show both sides being equal.
7. Mention that keeping an equation in balance applies to many real-life situations. Show students the map and ask them to explain how a map is a representation of the area it depicts. Answers should include that map distances are proportional to the real distances they represent. Demonstrate by measuring the distance between two map locations and using proportions to calculate the actual distance. For example, if 1 inch = 200 miles and you measured 2.5 inches, show how 1 in./200 mi. = 2.5 in./x mi. or 500 miles.
8. Using distances calculated, ask how long it would take to travel between the two points. Students should indicate that the time depends on how fast one is traveling. Make an appropriate assumption depending on the means of travel (e.g., a driving speed of 55 mph or flying speed of 600 mph). Divide distance by rate to determine trip length.
9. Write "Distance = Rate · Time" on the board and explain what each term means. Indicate that a formula is a representation of a relationship in the real world, just as a map is a representation of a real place. Show how it's possible to determine one of the terms if the other two are known, e.g., one could calculate rate if time and distance are known. Write the formula as d = r · t and show how to manipulate the formula, e.g., r = d/t and t = d/r.
10. Point out coordinates and/or longitude/latitude markings on the map. Determine whether students know how to express a location's coordinates in x,y format (and longitude/latitude format if desired). Make sure they can determine horizontal and vertical distance using coordinates.
11. Distribute Worksheet 1: The Case of the Doubtful Distance (PDF). Read the introduction as a class and review the "Additional Clues" (key facts) before students complete the worksheet. Make sure students are able to determine each route's distance using the coordinates and the map's scale. Review answers as a class. Worksheets Answer Key (PDF)
12. Students will build on these skills in Bonus Worksheet 1: The Case of the Perilous Planting (PDF) and Take-Home Activity 1: The Case of Sweet Proportions (PDF). If students require additional support, review worksheets as a class. Worksheets Answer Key (PDF)