Lesson 1: Using Mathematical Models/Proportions
In this lesson, students learn that an equation works like a scale, with the equal sign serving as the balancing point. They manipulate these equations in order to isolate variables and find the variables’ values. Additionally, students learn that scale keeps a map’s distances in proportion to real-world areas depicted on a map.
STANDARDS (CCSS and NCTM):
- Grade 6: Constant speed, unit rates, dependent variable, solving equations (CCSS 6.RP.A.3.b, 6.EE.B.7)
- Grade 7: Proportional relationships, solving proportional equations scale drawings (CCSS 7.RP.A.2.c, 7.EE.B.4.a, 7.G.A.1)
- Grades 6–8: (CCSS MP1, 2, 4, 5 & 6); NCTM Algebra
Students will be able to
- Describe how equations work like scales, with equal signs serving as balancing points that mean “is the same as” rather than “the answer to the problem is.”
- Solve for variables in equations by performing the same inverse operations on both sides of the equation, similar to balancing weights on a scale.
- Recognize that scale keeps a map’s distances in proportion to real-world distances depicted on the map.
- Use a map's scale to calculate distances.
- Identify variables in, use, and solve for variables in the constant speed formula d = r • t.
TIME REQUIRED: 40–50 minutes, depending on class needs; additional time for worksheets
- Balance scale with weights (drawings on the board will also suffice)
- For each pair of students, small paper tiles with “x” written on each tile and counters (about 6 “x” tiles and 30 or 40 counters per pair)
- 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)
INTRODUCTION TO NEW MATERIAL:
1. Show students a balancing scale without weights on either arm. Ask: Why is the scale in balance? (because each side has the same weight, zero grams). 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 you operated on only one side of the scale? What if you took 2 grams off one side? (The scale would become unbalanced.) 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 so that a weight of 5 grams is on each side.
3. Ask: What would happen if you tripled both sides? What amount would you have on each side of the scale? (15 grams on each side) Demonstrate that the scale is still balanced. Then ask: How would you get each side of the scale to return to having just 5 grams of weight? (Students might answer that you can subtract 10 grams from each side or divide each side by 3.) Use this question as an opportunity to discuss and demonstrate inverse operations. Divide the weight on each side by 3, showing that inverse operations, multiplying by 3 and then dividing by 3, "cancel each other out."
4. Indicate that equations work like the balancing scale, with the equal sign acting as the fulcrum, or balancing point. (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. Distribute the “x” tiles and counters to pairs of students. Display a simple equation, 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. Have students model the equation that you displayed with their “x” tiles and counters by treating half of their desks as each side of the scale. For instance, have them put one “x” tile and 7 counters on the left sides of their desks and 9 counters on the right sides of their desks. Ask: What can you do to the left-hand side of the scale to isolate x, or get x alone? (take away 7 counters) If you take away 7 counters from the left side, what must you do to keep the equation balanced? (take away 7 counters from the right side)
6. Have students practice modeling and manipulating these types of equations (and two-step equations with whole, positive coefficients before the x) with “x” tiles and counters. As students practice, ask them to take note of what operations they see in the equation and which operations they have to perform in order to isolate the variable. This may help them develop the concept of why using inverse operations can help solve equations.
7. Give students additional practice problems in which to solve for the variable. But this time, have them model with “x” tiles and counters as well as showing work underneath the written equations. For instance, if students are to find the value of x in 3x + 4 = 19, have them first write “–4” underneath each side of the equation (yielding 3x = 15) and then “÷ 3” underneath each side of that answer (yielding x = 5). Modeling as they show work may help them move from concrete to abstract concepts. This may also help them more fully grasp why they should use addition and subtraction as inverse operations before using multiplication and division. As students seem more and more comfortable with solving for the variable, feel free to take away their “x” tiles and counters and challenge them to solve the equations without concrete models. Tell students to check that their answer makes sense by substituting x with the value of the variable and ensuring that the sides of the equation are equal.
8. If students seem ready for a more complex equation, display 2x + 4 = 6x – 8. Ask students to solve and explain. Some students may choose to add 8 to each side, and some may choose to subtract 4 from each side. Some pairs may then subtract 2x from each side, while others may subtract 6x from each side. Have pairs with different solution strategies present their work for the class so that students can see that x has the same value for each of those viable solution strategies.
9. Mention that keeping equations in balance applies to many real-life situations. Show students the map and ask them to explain how a map represents the region it depicts. (Students might answer 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 the scale on the map states that 1 inch = 200 miles, and you measure 2.5 inches, ask: What is the real distance represented by this map distance? Show students how to set up a proportion, such as 1 in./200 mi. = 2.5 in./x mi. or 1 in./2.5 in. = 200 mi./x mi. Emphasize that, just as they had to keep equations balanced, they must keep proportions balanced. If they operate on one part of a ratio, they must complete the same operation on the other. After demonstrating one example, have volunteers participate in using proportions to find real distances of other distances on the map.
10. Use distances calculated. Ask: How long would it take to travel between the two points? (Students should indicate that the time depends on how fast one is traveling.) Write “Distance = Rate • Time” on the board and explain what each term means. Indicate that this formula represents the relationship between distance, rate, and time. Write the formula as d = r • t and show how to manipulate the formula, e.g., r = d/t and t = d/r. Show that it is possible to determine one of the terms if the two others are known, e.g., one can calculate rate if time and distance are known. For the distances calculated in step 9, practice calculating rate, distance, or time given values for the two other variables. When doing this, make an appropriate assumption depending on the means of travel (e.g., a driving speed of 55 mph or flying speed of 600 mph). Have students practice a number of calculations based on the map distances.
INDEPENDENT PRACTICE11. 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)