Purpose: To give students some experience with methods of making estimations for real life situations, showing that estimations can be based on some research and will come closer to the true number than a "wild" guess when mathematical methods are used.

Objectives: Students will develop the understanding of how to form a ratio to figure out the variable in question.  Students will understand that estimations can be more than guesses and can be based on evidence and mathematical reasoning.

Materials: Paper bag for each group of 4 students, fill each with a few cups of pinto beans, each group receives a small bag with about a cup of white beans, each group receives a small scooper cup.

Opening: As part of their responsibilities, fish biologists monitor fish populations. One may want to know, for example, how many striped bass there are in the San Francisco Bay. Since this number changes throughout the year as fish move in and out of the bay to spawn, biologists need a fairly quick and inexpensive way to gather current data.  The following investigation is a simulation of the process used by Fish and Game scientists to estimate the number of fish in a lake. Each group has a "lake" (a paper sack) full of "fish" (pinto beans), and a "net" (small cup).  We will be estimating the total number of fish in the lake by drawing samples and using ratios.

Activities and Procedures:

Introduction/ Instruction:

1. Before making any statistical measurements, each group member will guess how many fish are in the lake. Record all of your group’s guesses.

2.
a) Use your group’s net to take out a whole scoop of fish. This is your first sample.

b) Count the number of fish you netted in your first sample. To "tag" the netted fish, replace each one with a white bean. Record the number of tagged fish as the total number of tagged fish:________.

c) Put all the tagged fish back into the lake. (Return the replaced fish to the empty ziplock bag). Be careful not to let any of the fish jump out on to the floor. Gently shake the bag to thoroughly mix all the fish in the lake. Try not to bruise them.

d) Use your net to take out another scoop of fish. (This is your second sample.) Count the number of tagged fish in the sample and record this number as number of tagged fish in sample:________. Also count and record the total number of fish in sample:_______.

e) We want to estimate the total number of fish in the lake. You now have three pieces of information with which to do this: the total number of tagged fish in the lake, the number of tagged fish in the sample and the total number of fish in the sample. Use this information to write two equivalent ratios, one having the unknown X to represent the estimate of the total number of fish in the lake.

f) Use the equivalent ratios equation (the number you came up with for X in part (e)) to re-estimate the total number of fish in the lake______. (One group estimate)

g) Return your second sample of fish to the lake, gently mix the fish, and take another sample. Repeat the counting procedure of part (d) and the use of ratios in parts (e) and (f) to get another estimate of the lake’s fish population.

3. It is important to get an accurate count of the fish population, but each time you collect a sample it costs the taxpayers $500 for your time and equipment. So far your two samples have cost $1500. If you feel your estimate at this point is accurate, record it and the total cost of the sampling on the class chart. If you think you should try another sample for better accuracy, follow the same steps as before. Draw as many samples as you feel you need, but remember each sample costs $500.

4. When you are satisfied with your group’s estimate, count the fish in your lake to find the actual population!________(actual population)

a) Compare the actual population to your original guesses and to your estimate.

b) Use your population estimate and the actual population to calculate the percent error of your estimate. (The difference of your estimated population and the actual population divided by the actual population).

5. Record your group’s data on the class chart.

Standards Met by number according to NCTM

1. Mathematics as Problem Solving – Students will formulate problems from situations outside of mathematics. This is accomplished because we have a problem (how many fish in the lake) that uses mathematical skills to solve the problem. The math concepts of estimation and ratios are formulated from a situation that is not involved with mathematics, instead it originates in the biological sciences. Students will also acquire confidence in using mathematics meaningfully through this demonstration, hopefully realizing that math is valuable for gaining information and not just pencil paper computations.

3. Mathematics as Reasoning – Students will understand and apply reasoning processes with special attention to reasoning with proportions. Students achieve this through taking sample estimations and then evaluating the information they gain from the samples with the original guessing estimation they made about the number of fish in the lake. They can change or support their original "guesstimation" using the new sampling estimation, reasoning based on the information gained in the sample collecting.
4. Mathematical Connections – Students are exploring problems and describing results using algebraic representations in the form of equivalent ratios. They are also applying mathematical thinking and modeling to solve problems in the science discipline. This lesson will hopefully bring about a valuing of the role of mathematics in our culture and society by seeing it used in a profession that students can relate to (Fish and Game as opposed to engineer).
5. Number and Number Relationships – Students will understand, represent, and use number in equivalent fraction forms in a real world situation. They will also further develop their number sense for whole numbers by estimating the amount of fish and checking their estimation. The lesson aims at teaching students to understand and apply ratios, proportions and percentages in one of many situations.

7. Computation and Estimation – The nature of this lesson deals with computation and estimation. Students will be using computation, estimation and proportions to solve a problem.

9. Algebra – The lesson requires students to apply algebraic methods to solve a real-world problem. It also develops the understanding of the concepts of variable and equation by representing the situation in an equation with an unknown variable.

There are other standards that are touched upon that are not mentioned here.

Algebra and Functions

2. Use a letter to represent and unknown number; write and evaluate simple algebraic expressions in one variable by substitution. (5^th Grade standard)

2. Interpret and use ratios in different contexts to show the relative sizes of two quantities using the notation a/b. Our lesson explores one context.
3. Use proportions to solve problems. Use of cross-multiplication, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.

3. Estimate unknown quantities graphically (they will visually estimate) and solve for them by using logical reasoning and arithmetic and algebraic techniques.

5. Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language, they will also show their work symbolically.

This math lesson is done in groups. In order to address the issues of diversity, the groups should be carefully chosen, incorporating a generous mix of abilities. Diverse students can learn through the modeling of others and peer tutoring.

Students can also be led in a discussion about who would be doing the job of counting fish for the Fish and Game department, focusing mainly on the gender of the workers.

In addition, the visual and hands-on work makes this activity more concrete for ESL students. It may pose a bit of a problem with the beans representing fish, but with the use of pictures this could probably be overcome.

For those not quite up to the more advanced math skills, they could contribute to the group by counting beans. Counting is an important part of this activity; it would be valuable to the group as a whole and valuable to the individual's self-esteem and sense of belonging. This also lends to the higher students pulling along the lower students. The lower performing students do not feel put on the spot, but instead can listen to the other students and watch how they do the math.

Assessment: This is done through observation of the group work and discussions happening while the students are working on the project. The teacher can ask questions of the students aimed at assessing their understanding of the concepts of ratios, estimation, and percentages. Another way would be to have the students write up a proposal at step 3 telling how many more samples they are going to take and justifying this. (Remember, at step three we recognize that it is costing money each time we take a sample. Students should begin to understand that the more samples they take the more accurate the count will be, and this should then bring in two pieces of information making the decision to take more samples a bit more complex.)

Assessment of estimation could also continue after the lesson by having the students make up other situations where the taking of a sample, in order to learn about the whole population, would be a simpler and easier way to go. Students can show their understanding by setting up a way to solve the problem, and then solving it. They would probably need to use some type of counting system, like with the beans and the fish. Exploring batting averages, or other real life situations that require the use of ratios, could also assess their understanding of ratios and percentages.