Models, Models, Everywhere

Teacher Notes

A. Student Performance Objectives – EOC/TEKS Correlation

Objective 2: The student will graph problems involving real-world and mathematical situations.

(b)(1) Foundations for Functions. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways.

(D)The student represents relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.

Objective 8: The student will use problem-solving strategies to analyze, solve, and/or justify solutions to real-world and mathematical problems involving one-variable or two variable situations.

(b)(1) Foundations for Functions.

(E)The student interprets and makes inferences from functional relationships.

Objective 9: The student will use problem-solving strategies to analyze, solve, and/or justify solutions to real-world and mathematical problems involving probability, ratio and proportion, or graphical and tabular data.

(b)(1) Foundations for Functions.

(B)The student gathers and records data, or uses data sets, to determine functional relationships between quantities.

(b)(2) Foundations for Functions. The student uses the properties and attributes of functions.

(D)In solving problems, the student collects and organizes data, makes and interprets scatter plots, and models, predicts, and makes decisions and critical judgments.

(b)(3) Foundations for Functions. The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations.

Given situations, the student looks for patterns and represents generalizations algebraically.

  1. Critical Mathematics Explored In This Activity

The fundamental objective in this lesson is to get the student to realize the word “model” has the samedefinition in mathematics as it does in other real-world circumstances.

A model is a “representation of something.” A model can be a plan or design used to describe a concept or a concrete object. As designs, models are found in dress patterns, electronic schematics, and architectural plans. The connection that we want from this activity is that mathematical tables, graphs, and rules are “models” which represent real physical circumstances. We also want the student to begin creating verbal models by providing written descriptions of circumstances.

A secondary concept is the idea of scale. A physical model on a 1-to-1 scale will be made the actual size of the object. NASA uses a 1-to-1 scale routinely when they are practicing for a space mission or problem solving. The military also does this when they build a mock village for practicing operations. Scale models students will recognize are those for cars, airplanes, and ships. Dolls are also scale models. These models are all smaller than the original object. Models larger than the real thing include those of a DNA molecule or plant cell. The point here is that scale lets us handle or work with a representation of an object in a meaningful way when working with the actual object would be impossible.

There is some potential for confusion on the part of the students with the word scale. Used in the context of this lesson and in transformations, scale refers to a ratio that compares an original to a copy that may be congruent or similar. If the copy is a 1-to-1 copy, then the original and the copy will be the same size. A copy with a scale factor that is bigger than 1 will be bigger than the original. A copy with a scale factor between 0 and 1 will be smaller than the original. When used in the context of the axes of a graph, scale refers to the distance between the unit marks.

  1. How Students Will Encounter the Concepts

In this activity students are going to measure the height of a stack of blocks for different numbers of blocks. The goal is for students to be able to determine a relationship between a number of blocks in a stack and the height of the stack. They will use this relationship to calculate values of the variables in the relationship. Students will record data and represent it with a table, a graph, a verbal description, and with a mathematical rule. This activity brings together the concepts of a physical model – the stack of blocks – with the abstract model – the mathematical rule.

This investigation is a good opportunity to review the ideas of a variable, evaluating expressions, graphing, and solving simple equations. This latter occurs when students are asked to predict in reverse, i.e. if a stack of “x” blocks is “y” units tall, then given a particular height of a stack of blocks, predict how many blocks are in the stack.

All through the course we will use the notion of mathematical modeling. Our main intent is to have the student understand that the real world can be modeled using mathematics. We want through the course to repeatedly emphasize that when we are creating a table, sketching a graph, describing a relationship or writing an equation, we are using a mathematical model.

We want sentences like to have meaning to the students in a concrete way. There is certainly a place for study of the concepts of graphing, linear equations, etc. in the abstract, but the majority of your students are not going to be pure mathematicians. For algebra to have use to the majority, we must make use of its power to model.

The reason this lesson begins with blocks having a dimension that is 1 cm. is that we want the students to stack the blocks so that the 1 cm dimension is used as the height of each block. This gives the opportunity to have the students encounter the parent linear function. This function is also called the identity function. You won’t use these terms yet, but we want the experience with this pattern so that when we are ready to discuss linear functions we can say, “Remember…”

Part 2 of the lesson has the groups work abstractly with blocks of 3 cm. in height. This is background work for direct variation that will be covered after Foundations for Functions.

D.Equipment Needed and Lesson Setup

Gather a collection of various models. Include a model car, airplane, dress pattern, scale drawing of a floor plan, a science model of an atom or molecule, etc. Remember that this lesson is intended for the first or second day of school, and students should be working on it while the teacher is dealing with administrative detail.

Each group of students needs about six pattern blocks, tiles, or cubes that have one dimension that is 1 cm, such as pattern blocks (any shape) or centimeter cubes and a ruler with metric units.

Have the students work together in groups of 2 or 3. Ask the students to follow the instructions on the activity pages. Let the students do the work. If a group is able to proceed without teacher assistance, let them.

  1. Questions to Assess Learning

After you have finished with administrative detail, you should circulate among the groups to assess how they are progressing with the investigation. Some questions you should ask are these:

  • Is there any pattern to how the height of the stack increases as you add blocks?
  • What does an ordered pair in this activity describe?
  • (Select a point on the group’s graphs, the ask:) What is the significance of this point of the graph in relation to a stack of blocks?
  • How would you decide the height of a stack of blocks that is 37 blocks high? (Do not allow the student to get by with simply giving the correct height of the stack. Require an explanation as to how the height was chosen.)
  • Explain how I would determine how many blocks are in a stack that is 96 cm. high.
  • What do the variables in the formula stand for? (This could reference question 13, or any of the questions where a formula is used.)

After students have had sufficient time to build and measure the required stacks of blocks, a general discussion with the students (not a lecture) about models should be held. Ask them to name all the kinds of models they know about. Usually they will mention one of the "super models." This is a good answer since clothes designers use them to represent how clothes look when worn by a real person. Some will mention scale models like cars and airplanes. As each type of model is mentioned, bring out the sample you brought. If the discussion stalls, bring out another of your samples and ask them to mention others of a similar nature. Finally, mention to them that mathematics is used all over the world to represent things. Say to them "This is called mathematical modeling." Then tell them that they are exploring the way this is done by working with the stacks of blocks. Since the principal intent of this lesson is to have the student understand the concept of a mathematical model, please focus on repeatedly using the word “model” in your instruction throughout this lesson and the rest of the course.

In the group discussion relating to the actual experiment, use the following questions:

  • What formula did you write to represent the relationship between the number of blocks in a stack and the height of the stack? Explain why your formula makes sense as a model for each stack.
  • How is the formula for experiment 2 different from the formula for experiment 1?

Explain the reason for this difference.

  • Explain how the graphs for both parts of the exercise are similar; are different.
  • For this lesson, what does an ordered pair such as (9, 27) mean?
  • Suppose we had blocks that were 2 inches in height. What would be the height of a stack of 15 such blocks? What would be the mathematical model formula for the height of stacks of these blocks?
  • What if the blocks were ¼ of an inch tall? How would this information change the formula? Use the formula to predict the height in inches of stacks composed of 12 blocks, 20 blocks, 200 blocks. How many blocks would be in a 4-inch stack?
  • Which model, the graph, the table, or the formula, do you consider the most useful and why? (Accept all answers)

F. Answers and Notes for the Student Activity

  1. Height of Stack column: 0, 1, 2, 3, 4
  2. Answers will vary. Sample: (2, 2) Means A stack of 2 blocks is 2 cm high.
  3. Pattern of data points should fall along the line, discrete vs. continuous data (ex. connect dots or not?)
  4. 6 cm; 10 cm; 100 cm
  5. 2.5 cm
  6. 170; sample explanation: the number of blocks in the stack is the same as the height.
  7. Sample answer: I would multiply the number of blocks in the stack by the height of one block.
  8. 12
  9. 10
  10. 5
  11. Answers will vary.
  12. Sample answer: h = n, where h = height and n = number of blocks (or other symbols)
  13. Sample answer: n = 0: 0 = 0; n = 1: 1 = 1; etc.
  14. Height of Stack column: 0, 3, 6, 9, 12. The pattern of data points in the scatter plot should fall along the line.
  15. Sample answer: h = 3n, or other symbols.
  16. 18 cm; 30 cm; 300 cm
  17. 7.5 cm
  18. 50; sample explanation: I divided by 3 because the height is 3 times as large as the number of blocks.
  19. Sample answers:

Data sets: the numbers in the height column are multiples of the # of blocks; the larger the number of blocks, the larger the height of the stack.

Scatter plots: the points fall along a linear pattern; both graphs are increasing from left to right.

Formulas: the height is a multiple of the # of blocks; neither formula involves addition or subtraction.

  1. Suggested Homework Assignments

An activity called Reflect and Apply: Models. Models, Everywhere is available in the Models, Models folder.

  1. Possible Extensions of the Lesson

Any number of activities can be created to allow the student to develop the concept of modeling. Among these are:

  • The Hand Squeeze. Have the students determine the amount of time it takes for a hand squeeze to be passed along a group of students. 6 students hold hands. A timer holds a stopwatch. The timer starts the clock when the first student in the line says, “go” and squeezes the hand of the next person in the line. The last person in the line says stop upon receiving a hand squeeze. Record the data in a table built on this ordered pair (number of students in the line, time for a hand squeeze to travel through the whole group). Add 2 students to the line and repeat the experiment. Remind students to pass the squeeze at the same rate for each trial of the experiment. Record the data. Continue adding students until all of the class has participated. Use the data to determine the time needed to pass the squeeze to one student, and ask questions similar to those in the Models, Models activity.
  • We have provided an additional activity called Weight of the World. This activity has the students build spaghetti bridges and determine the amount of weight needed to break the bridges. The data is often not “clean,” and you have to develop averages based upon the experiences of the whole group to derive a fairly reasonable model. This extra level of complexity, and the larger number of variables that can influence the result create a problem if you want “clean” data. Many teachers get concerned about the “dirty” data and feel it distracts from the effort to help students understand how mathematics models a situation. You may want to talk with a science teacher about real-world data collection prior to using this activity. This activity is extremely valuable if you want to show that a model is often best created using a large number of data samples, rather than just four or five samples.
  • Ask students to create a list of things they know about or can discover that use mathematical models. You should create the list and post it in a conspicuous place. As students discover other applications of mathematical modeling, you should add the new discoveries to the list.
  • While not essential to this lesson, you have the option of using graphing calculators or computer programs to create scatter plots and to test the validity of a suggested mathematical formula model. The graphing capability is extremely useful for testing to see if a suggested formula produces a graph that “best fits” the data. These two uses of technology are an excellent way to introduce technology into your instruction.

Models, Models Everywhere!

Student Activity

In this activity you will:

  • Measure the height of stacks of blocks.
  • Organize you data in a table.
  • Represent the data you collect in a graph, and with a formula.
  • Make predictions based on your investigation.

EXPERIMENT #1:

1. Measure the height of stacks of blocks containing varying numbers of blocks. Record your data in the table below. Stack your blocks as illustrated.

# of Blocks
/ Height of Stack (cm)
0
1
2
3
4

Data in a table can be represented with a mathematical tool called an ordered pair, such as (15, 20). In your table, the numbers in the ordered pairs would represent (# of blocks, height of stack). Thus, if the ordered pair (15, 20) was in your table, it would mean “a stack of 15 blocks is 20 cm high.”

  1. Write two ordered pairs for data in your table, and explain what each ordered pair means.
Ordered Pair: ______Means:______

Ordered Pair: ______Means:______

  1. Create a scatterplot of your data on the grid below.

Your table of data and your scatterplot are both models of your experiment. Other people looking at your models will know the heights of the various stacks of blocks without having to do the experiment themselves.

4. Using information contained in either your table or your scatterplot, predict the height of a stack that contains:

6 blocks ______10 blocks ______100 blocks ______

5. Suppose we could cut a block in half and stack 2.5 of the original blocks. Predict the height of the stack. ______

6. What if you are told that a stack of blocks is 170 cm high? How many blocks would you expect to find in the stack? Explain how you got your answer.

Number of blocks: _____ Explain: ______

______

______

7. How would you predict the height of a stack of blocks if a block of a different height were used?

______

______

It takes a lot of space to display a table or a graph containing a large number of values in a model, so mathematicians have developed another kind of model called a formula. You have used formulas before when you found the area of a rectangle or the perimeter of a square. Find out what these formulas are and write them in the spaces below.

8. Area of a rectangle ______

9. Perimeter of a square ______

10. What would be the area of a rectangular piece of cloth having a length of 6 meters and a width of 2 meters? Area = ______m2

11. Suppose that a square shaped vegetable garden has a perimeter of 40 feet. The length of each side of the garden is ______ft.

12. If a rectangular garden has an area of 75 square feet and a length of 15 ft., the width of the garden is ______ft.

13. Give another example of a formula you have used and explain its purpose.

______

In the work above, you can see the power of a formula model to help you answer questions when only partial information is known. Another advantage of a formula is that the formula represents all possible cases for a particular situation. Formula models can be expressed in words as well as in algebraic form (using variables). For example, in the area of a rectangle model used above, we can say: “The area of a rectangle in square units is the length of the rectangle multiplied by the width of the rectangle.”

  1. Using symbols, write a formula model to describe your data for the height of a stack of blocks for any given number of blocks: ______
  1. Use the space below to show that your formula will make predictions that are close to the data you collected in your experiment. You do this by substituting the number of blocks in a stack into your formula and showing that the result matches the data in your table.

EXPERIMENT #2: