Insights Into Algebra 1: Teaching for Learning
Properties Lesson Plan 2: Curses and Re-Curses! It’s Happening Again.
In this lesson, students learn about basic recursion by exploring patterns in the data they generate from two simple probability-based experiments.
One 50-minute period
Students will be able to:
- Write a Now-Next equation to describe a mathematical sequence.
- Use different equations to determine the type of model needed for a given situation.
Principles and Standards for School Mathematics, National Council of Teachers of Mathematics (NCTM), 2000.
NCTM Algebra Standard for Grades 6-8
NCTM Algebra Standard for Grades 9-12
Teachers will need the following:
- Chalkboard or overhead projector
Students will need the following:
- A coin
- Two dice of different colors
- Math journals
- Centimeter graph paper
Teachers Activities and Assignment
1. Introduce the day’s lesson by asking students how to get out of jail in the board game Monopoly® if they don’t have a “Get Out Of Jail Free” card. Likely, at least one student will know that a player is allowed to roll three times to try to get doubles. If the player doesn’t roll doubles in three tries, the player pays $50 to get out of jail.
2. Explain that the class will investigate the question “How long will you typically have to wait until a chance event occurs?”
1.Explain to students the idea of waiting-time distributions. (A waiting-time distribution is a description, either verbal or numerical, of how long (e.g., how many coin tosses, how many rolls of the dice, how many days) it will take before a particular event occurs. Tell students that they are to conduct an experiment to discover how many times they must flip a coin before it lands on “heads.” Ask students to predict the number of tosses it will generally take before landing on heads.
2.Have each student complete 10 trials, recording the number of tosses in each trial before heads was obtained.
3.Compile the results from the entire class, and use the data to create a bar graph. One possible way to create the bar graph is to give each student 10 Post It® notes, one for each trial they conducted. Draw the horizontal axis shown below on the chalkboard or on a large piece of paper and have each student put a sticky note in the appropriate columns. Note that approximately half of the trials will yield heads on the first flip. Consequently, the first column will be very tall. You may wish to divide the Post It® notes in half and apply them vertically so that they take up less space.
Number of Coin Tosses to Get Heads
4.Explain that the trials they conducted give experimental data. Ask students, “What should happen, in theory?” Students should realize that heads will occur on the first toss about half the time, because the probability of heads for any given toss is 1/2.
5.Lead them through the “square model” using the file CoinTossSquare. Share and discuss this representation with your students, either by displaying the PowerPoint file page by page or by drawing the same representation step by step on the chalkboard or overhead projector. This activity gives students a visual representation of why the probability of heads on any given toss is 1/2 the probability of the previous toss. (The probability of heads on the second toss is also 1/2, but if heads occurs on the second toss, then tails must have occurred on the first toss; otherwise, there would be no need to toss the coin a second time. Consequently, the probability of heads on the second toss is one-half of 1/2, or 1/4, and this pattern continues.)
6.Have students generate a table showing the number of tosses and the probability of heads on that toss. Have a student volunteer draw the same table on the chalkboard or overhead projector.
7.Ask students to describe this situation with a Now-Next equation. As stated above, the probability of heads on any given toss is 1/2 the probability of the previous toss, so Next = 1/2 × Now or Next = Now ÷ 2, Start = 1 (the value of toss number 0), or Start = 1/2 (the value of the first toss).
8.Have students verify that the Now-Next equations that they suggest do indeed fit the pattern. Discuss the value of the start. Because most students are used to starting with the y-intercept, they may choose a start value of 1, which would be the value of the 0 term if it existed. A start value of 1/2 is also acceptable because it is the value of the first term. The important thing is to get students to clearly communicate their thinking, and which term they are using for the starting value.
The purpose of steps 9-12 is to help students determine the type of function they would use to model a given situation. They should be able to recognize whether they have a linear model, a polynomial model of a certain degree, or an exponential model.
9.Have students find the differences between terms of the sequence. Then, have them find the second differences, the third differences, and so on. Students should notice that the pattern keeps repeating; that is, the difference between terms is continually 1/2, 1/4, 1/8, and so on, no matter how many times the difference is taken. (This means that an exponential function can be used to model the data.)
10.Have students complete a table of values for y = 3x + 4, for x = 0 to 6. Then, have students determine the differences. Students should notice that the difference between terms is always 3, so this sequence reaches a finite difference in one step.
Using the values in the table, have students generate a Now-Next equation (Next = Now + 3, Start = 4).
11.Have students consider the finite differences for x2, x3, and x4. Respectively, they reach finite differences in the second, third, and fourth steps. Students can use finite differences to determine if the pattern fits a polynomial model of any given degree. If a common difference is never reached – that is, if a pattern is obtained that makes it obvious that a finite difference will not be reached – students must look for a different model.
12.Have the students again consider the bar graph from the coin toss experiment. Students should see that it might be a decay model. Have students generate an equation of the form y = abx. Remind students that a is the start value and b is the growth or decay rate.
13.Have students talk about what y and x represent in the context of the coin toss. Students will typically believe that the start value is a = 1/2, based on the table.
14.Have students talk with partners about the value of b. Students might generate the equation y = 1/2 × (1/2)x. But this doesn’t work; in theory, the equation y = 1 × (1/2)x is needed. (An interesting discussion can result from the question of why the constant a is not 1/2 if time permits, you may wish to pursue this discussion.)
15.Tell students that now that they’ve modeled the coin toss data with recursive and explicit equations, it’s time to consider a more complex scenario: How long you will have to wait in jail in a Monopoly® game if you must wait until you roll doubles.
16.As with the coin toss experiment, have each student complete 10 trials, recording the number of rolls before they get doubles. Again, compile all of the data into a bar graph. Ask students, “How is this graph different from the graph for coin tosses?” Students should notice that the columns are not as tall. Ask students, “Why is this graph different from the graph for coin tosses?” Students should realize that the probability of heads was 1/2, but the probability for doubles is much less. (In fact, P(doubles) = 1/6, which will be shown later in the lesson.)
Using centimeter grid paper, show students how to determine the probability of getting doubles. Create a 6 by 6 grid. Number six squares (1-6) along the horizontal axis to represent the possible outcomes for one die, and number six squares (1-6) along the vertical axis to represent the outcomes for the other. The squares along the diagonal, then, represent rolls of doubles. Of a possible 36 different outcomes, six are doubles, so the probability of doubles on a given roll is 1/6. (A possible grid is shown below. Note that numbers along the top are in red and numbers along the side are in blue, indicating that two different color dice were used. The numbers in the squares represent the pairs as indicated by the colors. The doubles occur along the diagonal.)
17.Using a representation similar to the square model used for the coin toss, divide a square into sixths and color in one-sixth to represent the probability of rolling doubles on exactly the first roll. Ask students to work in groups to determine the probability of doubles on exactly the second roll. Have students share their results and how they obtained them. Then, students should discuss in their groups the probability of getting doubles on exactly the third and fourth rolls.
18.With the class, create a table that numerically expresses each result.
19.Have students gather in groups to brainstorm ways in which they could generate this result without drawing the square model. Students should recognize that they can write a Now-Next equation that describes the probability of rolling doubles on a given turn. Because the probability is equal to 5/6ths of the previous probability, the Now-Next equation would be Next = 5/6 × Now.The start value is 1/6, which is the value of the first toss. (Note that an explicit function for this situation is y = 1/6 × (5/6)x – 1.)
1.In groups, students should take five minutes to make a list of the things they learned today.
2.Invite students to share items from their lists. Use this opportunity to bring out the key ideas and rectify any misunderstandings. Work with the students to create a class list to record in their math journals.
Related Standardized Test Questions
The questions below dealing with recursion have been selected from various state and national assessments. Although the lesson above may not fully equip students to answer all such test questions successfully, students who participate in active lessons like this one will eventually develop the conceptual understanding needed to succeed on these and other state assessment questions.
- Taken from the Michigan High School Test (Spring 2002):
During a visit to his physician, John received a single dose of medication that decays or loses its effectiveness at a rate of 20% per hour. The amounts of medication that remain in his body at the end of each hour for the first 3 hours are represented by the sequence 150, 120, 96 mg. Approximately how many milligrams of medication remained in his body after the first 6 hours?
C. 50 (correct answer)
- Taken from the Kentucky Core Content Test (Spring 1999):Triangle Patterns
The diagrams above show the first three steps of a process used to make a triangle pattern. In Step 1, the large triangle has an area equal to 1 [square] unit. In Step 2, the large triangle is divided into 4 congruent triangles, and the middle triangle is removed.
A. What is the area of the shaded region in Step 2?
Solution: 3/4 square units
In Step 3, the dividing process continues. Each shaded triangles in Step 2 is divided into 4 congruent triangles. Each middle triangle is removed, and the shaded parts remain. The dividing process continues until the total shaded area is less than 1/2 square units.
B. How many steps does it take until the shaded area is less than 1/2 square unit? (Give a complete explanation to justify your answer.)
Solution: 4 steps, because the area for each step equals 1, 3/4, 9/16, and 27/64. A recursive formula for this sequence is Next = Now × 3/4, Start = 1.
- Taken from New York Regents High School Examination, August 2002:
A used car was purchased in July 1999 for $11,900. If the car depreciates 13% of its value each year, what is the value of the car, to the nearest hundred dollars, in July 2002?
Solution: $7,800. A recursive formula for this is Next = Now – Now × 0.13 or Next = Now × 0.87, Start = 11,900. The answer is the fourth term.
- Taken from the Mississippi Subject Area Practice Test, Mathematics:
The terms below belong to a famous mathematical sequence.
1, 1, 2, 3, 5, ___, 13, 21, 34, …
Which term comes between 5 and 13?
B. 8 (correct answer)
Student Work: Factoring Assignment
I consider this a good summary. The student has captured all of the core concepts. The first three entries in the summary are typical “warm-up” items. By this, I mean that they specifically reiterate information developed in the lesson. The remainder of the summary is where the student’s conceptual understanding emerges. The student has listed seven characteristics relating to mathematical models that he can generalize beyond the day’s lesson. This is always the aim of good instruction.
Looking at this summary and the day as a whole, I can see that I need to spend some time helping students nail down the starting value in the recursive model. I think this student also needs some assistance rewriting the first item in the summary. It is unclear from the writing whether he is referring to the cumulative probability or the probability of a single event. I believe he is referring to the cumulative probability. It is my experience that students who are ambiguous in their writing have some ambiguities in their understanding.
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.