Keeping the Meaning in Mathematics: The Craft of Word Problems

Classroom Activities

Lesson 1: Projectile Motion

To lead into the Projectile Motion lesson, I would have students practice evaluating expressions for given values of the variables. In particular, I want students to recall that the product of any number of factors is zero if any one of the factors is zero. This is a key concept behind factoring quadratic functions that my students sometimes lose sight of. A possible Warm-Up activity might be: Evaluate 18ab(c + d)(e - f) when

1. a = 1, b = 0, c = 4, d = -8, e = 100, f = 73
2. a = 4, b = -2, c = 0, d = 1, e = 7, f = 7
3. a = -3, b = 7, c = -6, d = 6, e = 2, f = 5
4. a = 2, b = 1, c = 2, d = 0, e = 3, f = 1

Before beginning the word problems, I would define the variables and describe the physics (height would increase linearly forever, except that gravity becomes a greater force over time because of t ² to pull the object back down to earth) behind the projectile motion formula h(t) = h ₀ + v ₀t + ½ at ². Next, I would demonstrate how to write the equation given the information in a problem. Students would then begin to work on the sports-related word problems in their assigned groups. Appendix B provides an assortment of problems, but I might give a more extensive list to students so that they can have some choice in which problems they do within each category. You can tweak the problems to fit the sports that most interest your own students; however, be cautious with your choice of parameters and units to ensure that they're realistic.

As groups reach Dimension 7A (solve for a specific height), be sure to check that they manipulate the equations so they equal zero (as described earlier) before applying any algebraic solution method. I would also be prepared for a class discussion to emphasize the need to set the equation equal to zero if many groups don't recognize it themselves.

Within 2 or 3 90-minute block periods, I would expect all students to complete, and be held accountable for, word problems from Dimension 1A through 9A. Because of the range of ability levels within most classrooms, I know not every group will work at the same pace, but there are additional problems available for those that are prepared to move on. Ideally, I would love for my serious athletes to apply the principles relating the horizontal and vertical components of velocity to their own sports to see how they might improve their game, but I think it will depend on time, interest and ability.

Lesson 2: Geometry

I expect this geometry lesson to last about 2 days on a 90-minute block schedule. I will review basic perimeter, area, surface area and volume formulas for a variety of 2- and 3-dimensional shapes in my Warm-Up activity for the quadratic geometry problem suite. Some of the questions are trivial, but some require multiple steps.

Part I. Find the area and perimeter of a) square with side length 15 cm, b) rectangle with length = 40 in, width = 24 in, c) isosceles right triangle with hypotenuse = 3 m, d) equilateral triangle with side length = 8 in, e) circle with radius = 6 cm. Part II. Find the volume and surface area of f) cylinder with radius = 2 in and height = 10 in, g) box with length = 70 mm, width = 60 mm, height = 130 mm, h) box with square bottom with area = 81 ft ², height = 20 ft. Part III. Given the perimeter of a rectangle = 18 cm and length = 4cm, find the width. Given the perimeter of a rectangle = 50 cm and width = x, find the length (in terms of x).

From previous experience, I expect my students to have trouble writing the equations for the geometry word problems, especially using the perimeter to write dimensions in terms of just one variable. Therefore, before assigning the word problem set, I will do one or two examples with the full class. Again, students will work in their groups so they will have support as they practice writing and solving quadratic equations. Hopefully, students will make some observations as they work through the geometry problems. I will let their observations and difficulties lead to full-class discussions.

Once all groups have completed the first five categories (the "faster" groups will get to surface area), I will have students find a partner (or triple) that is in the same career area. The assignment for the pairs is to write and solve a minimum of three word problems related to their career area. (Non-vocational students can create problems about anything of interest to them.) One problem should focus on perimeter, one on area, and the third on volume. To help them, I will talk about the baseboard molding of the classroom measuring the same as its perimeter (this would work for a student's bedroom, also). The tiles on the floor cover the area of the floor, and the air in the room, or cabinet space are measures of volume. By the way, I will save these student-generated problems as a source of future problems! I don't expect the students to create three quadratic problems, and that's OK; they need to recognize the difference between quadratic and linear equations.

Lesson 3: Dilations

One more day for geometry, but this one focuses on dilations. As a Warm-Up, and reinforcement, I would take a problem or two from the previous geometry problems and change the numbers. Continuing with the pairs from the same career area, I will hand out a set of problems related to an assortment of careers, and have students select 3-4 problems of their choice. Since I only wrote one or two problems per career area, they will have to do some unrelated ones, also. However, by doing multiple problems they should start to see the relationship between changes in dimensions (scale factor) and changes in area.

I have some general instructions and tips for this problem suite. First, pay attention to units! Second, compare (by ratio) the original dimensions to the new ones; record the ratio (aka, scale factor). Third, compare (by ratio) the original and new area; record the ratio. Fourth, compare the ratio of areas to the scale factor. In each problem, students are asked to predict new dimensions or area and compare predictions to calculated answers. So, fifth, reason why predictions are right or wrong. After doing several problems, I hope students will be making correct predictions because they've learned that area increases/decreases by the square of the scale factor. That is, when the area is doubled, the dimensions only increase by a factor of √2» 1.4, but when the dimensions are doubled, the area increases by a factor of 2 ² = 4!

The follow-up part of this lesson is for the pairs to write and solve another (quadratic this time) problem related to their career area and create a poster illustrating the problem. Next, I will have the partners split up and find new partners from a different career area. The new partners will each be an expert (good for self-esteem) and explain their problems to each other. Finally, everyone will solve his/her partner's problem. If time allows, I will also have pairs present the problems posed on the posters to the rest of the class. Again, I will keep the student-generated problems for future use since they know more about their career areas than I do. I can also use them to add to the problem set so future classes will have more choices.