Unit Title: The Overland Trail
Primary Unit Author: Dr. Gene Fite
Edited provide and support documentation:
Matthew Hillmer
Time Frame: Four 75 - 90 minute sessions plus one 75 - 90 minute session for summative assessment
Description: Students will work in pairs to solve a series of complex open-ended problems. These problems develop the foundation for a conceptual understanding of algebraic expressions and associated computational skills including the order of operations and applications of positive and negative numbers. I will follow these open-ended problems with a more traditional introduction to the abstract symbolism associated with algebraic expressions along with basic traditional procedures for dealing with expressions will be introduced.
Standard 2 - Algebraic Concepts
The student uses algebraic concepts (e.g., patterns, functions, equations, inequalities, systems, graphs, and matrices) in a variety of situations.
Benchmark 2:VARIABLES, EQUATIONS, AND INEQUALTIES: The student uses variables, symbols, real numbers, and algebraic expressions to solve equations and inequalities in a variety of situations.
Indicator
9.2.2.2 uses symbols, variables, expressions, inequalities, equations and simple systems of equations to represent problem situations that involve variable quantities
Objective: Upon completion of this learning activity and when asked to diagram, demonstrate, or respond either orally or in written form, the student will:
Upon completion of this learning activity and when asked to diagram, demonstrate, or respond either orally or in written form, the student will:
a) Clearly state the problem within a real-world application of mathematics
b) Give a detailed description of how the problem was solved
c) State the solution to the problem as clearly as possible including how symbols, variables, expressions, and equations were used in the solution to the problem.
d) Create extensions or variations of the problem (not applicable to the summative assessment)
Conduct a self-assessment of their solution and problem solving strategies
Assessment
Summative
Description: Upon completion of learning activities, students will be given an open-ended problem requiring them to apply the concepts developed in the learning activities. Their task will be to solve the problem and use the “Standard Write-Up” to convey their knowledge and understanding of symbols, variables, and algebraic expressions. Students should be given approximately five minutes to present and defend their answer to members of a group of four students using a “rally robin” format.
Task prompts:
Summative Problem
This assessment involves the use of algebraic variables, expressions, summary phrases and equations.
You will be given specific algebraic expressions and will be asked to write, summary phrases for them; you also will be given specific summary phrases and asked to write algebraic expressions. You will then be asked state the expressions and summary phrases as algebraic equations.
Reminder: The summary phrase for FC is “the number of children on the wagon train” and not “the number of families in a wagon train times the number of children in a family.”
The symbols below are the same as those used in “Ox Expressions.” Though no specific numerical values are assigned here, you should assume that each symbol represents a single number.
Symbol Meaning
F the number of Families in a wagon train
M the number of Men in a family
W the number of Women in a family
C the number of Children in a family
V the number of wagon (Vehicles) per family
T the number wagon Trains in one year
Y the number of pairs (Yokes) of oxen per wagon
A the number of oxen (Animals) per yoke
P the weight of one ox (in Pounds)
L the Load for one wagon (in pounds)
G the amount of Grass eaten by one ox in one day (pounds)
H the gallons of water (H2O) consumed by one ox in one day
D the number of Days on the trail
B the gallons of water (Beverage) consumed by one person in one day.
Questions:
1. Write a summary phrase for the expression W + M + C
2. Write an algebraic expression for the water consumed in a day by a family.
3. Write a summary phrase for the expression D(H + B)
4. Write an algebraic expression for the number of people in a wagon train
5. Write a summary phrase for the expression FM
6. Write an algebraic expression for the amount of water consumed by an ox on the trip.
7. Does the expression WL have a meaning? If so, what is it.
8. Make up a meaningful algebraic expression of your own and give a summary phrase for it.
9. Discuss how an expressions and it corresponding summary phrase could be use to create an algebraic equation. Convert your answer to #8 into an algebraic equation using your expression and summary phrase.
10. Compare and contrast the meaning of an algebraic expression and an algebraic equation.
Time Frame (75 to 90 minutes)
· Students work individually to develop a solution to this problem
· Students use the Standard Write-Up to individually express their solution to the problem.
The “Standard Write-up” involves:
I. Problem Statement: State the problem clearly in your own words. The problem statement should be clear enough that someone unfamiliar with the problem could understand what it is being requested.
II. Process: A detailed description should be given to explain what was done in an attempt to solve the problem. The student should use his or her notes as a reminder and should include things that didn’t work out or that seemed like a waste of time. This part of the write-up should be done even if the student didn’t solve the problem.
III. Solution: The student should state the solution as clearly as possible and explain how it is known that the solution is correct and complete. If only a partial solution was obtained, it should be presented. If only a generalization of the problem was completed, the student should, at the very least, give generalized results.
The explanation should be written in a way that will be convincing to someone else, even someone who initially disagrees with the given answer.
IV. Extensions (not applicable to the summative assessment): The student should invent some extensions or variations to the problem. That is, write down some related problems. They can be easier, harder, or about the same level of difficulty as the original problem. These do not have to be solved by the student.
V. Evaluation: The student should discuss his or her personal reaction to the problem. For example,
· Was the problem educationally worthwhile? What was learned from it?
· How could the problem be changed to make it better?
· Was the problem enjoyable to work
· Was the problem too hard or too easy?
Performance Standard:
A ninth grade student scoring at the proficient level demonstrates a high level of knowledge and comprehensive understanding. The student can apply basic facts and various mathematical concepts and procedures to make inferences and formulate or solve challenging problems. Sometimes the student may have difficulty in solving complex problems without minimal assistance. The student can accurately and confidently perform specified whole number computation without the aid of a calculator. The student can usually successfully solve problems from situations the student may or may not have seen before
On a rubric scale of 1 to 5 the following criteria assess student mastery.
An average score of 3 is satisfactory performance
1 Unsatisfactory
2 Basic
3 Satisfactory
4 Proficient
5 Advanced
Rubric for Scoring Student Work
/ 5-Advanced/Distinguished / 3 - Satisfactory / 1-UnsatisfactoryProblem Statement / The problem statement is clear enough for someone to understand what is being requested / The problem statement is present but confusing / The problem statement is unacceptable
Process / The description is detailed and tells what did and did not work / Details are missing in the description / The description does not make it clear what was done, what worked, and what did not work.
Solution / The solution is stated clearly with a convincing explanation for why the solution is correct and complete / Details are missing in the statement of the solution. The explanation lacks convincing evidence / The solution is ambiguous or unclear and the explanation is insufficient
Extensions(Not Applicable to the Summative Assessment) / Extension problem(s) are provided that are challenging and relevant / Extension problem(s) are provided but lack the rigor and relevance of the original problem / Extension problems are not provided or if the are, the are not challenging or relevant.
Evaluations / All four evaluation problems are thoroughly answered / The four evaluation problems are only partially answered / Only parts of the evaluation problems are answered
Assessment
Formative
Description: Formative assessments are embedded in the learning activities of this lesson and may take the form of informal teacher observation to graded work done during class time on outside of class as homework.
Materials/Resources:
Period maps of the continental US with the westward trails marked will be provided to the students. This allows them to study the historical migration of families along the Overland Trail.
Learning Activity:
The Learning Activity #1:
TEACHER NOTE: Read aloud this passage describing the Overland Trail experience. Follow this with learning activity #1, “To Kearny By Equation”
Passage and material are taken from the IMP- Year 1 Program textbook
The Overland Trail
(From Women’s Diaries of the Westward Journey)
Between 1840 and 1870, a quarter of million Americans crossed the continental United States, some twenty-four hundred miles of it, in one of the great migrations of modern times. They went West to claim free land in the Oregon and California Territories, and they went West to strike it rich by mining gold and sliver. Men and women knew they were engaged in nothing less than extending American possession of the continent from ocean to ocean…The westward movement was a major transplanting of young families. All the kinfolk who could be gathered assembled to make that hazardous passage together…
The emigrants came from Missouri, Illinois, Iowa, and Indiana, and some all the way from New York and New Hampshire. Most of them had moved to “free land” at least once before, and their parents and grandparents before them had similarly made several removals during their lifetime. These were a class of “peasant proprietors.” They had owned land before and would own land again. They were young and consumed with boundless confidence, believing the better life tomorrow could be won by the hard work of today…
The journey started in the towns along the Missouri River between St. Joseph and Council bluffs. These settlements came to be known as the “jumping-off places.” In the winter months emigrants gathered to join wagon parties and to wait for the arrival of kin. It was an audacious journey through territory that was virtually unknown.
Guidebooks promised that the adventure would take no more than three to four months time – a mere summer’s vacation. But the guidebooks were wrong. Often there was no on in the wagon train who really knew what the roads would bring, or if there were any roads at all. Starting when the mud of the roads began to harden in mid-April, the emigrants would discover that the overland passage took every ounce of ingenuity and tenacity they possessed. For many, it would mean six to eight months of grueling travel, in a wagon with no springs under a canvas that heated up to 110 degrees by midday, through drenching rains and summer storms. It would mean swimming cattle across rivers and living for months at a time in tents.
Overland Trail Families
(From the diary of Catherine Haun)
Ada Millington was twelve when her family set out for California. It was a large family; her father, who had five children by his first marriage, was traveling in two wagons with his second wife and their six children. The youngest was a year and a half old. In addition, there were five young hired hands. And there was Mrs. Millington’s sister and brother-in-law in a third wagon with their children, and their hired hands. And there was the brother-in-law’s sister, stepfather and mother in a forth wagon. The party seemed large and secure.
Our own party consisted of six men and two women and three wagons. Mr. Haun, my brother Derrick, Mr. Bowen, three young men to act as drivers, a woman cook and myself… four regulation “prairie schooners” drawn by sets of four oxen and filled with suitable supplies, with two pack mules following on behind was the equipment of the Kenna family. There were two men, two women, a lad of fifteen years, a daughter thirteen and their half brother six weeks of age. This baby was our mascot and the youngest member of the company…
One family by the name of Lemore consisted of man, wife and two little girls. They had only a large express wagon drawn by four mules and meager but well chosen, supply of food and feed. A tent was strapped to one side of the wagon, a roll of bedding to the other side, baggage, bundles, pots, pans and bags of horse feed hung on behind; the effect was really grotesque…