Jamee Childs
MAE
5318
Fall 2006
Description
of the Unit
This unit is a hands-on math unit for students in grades 4 and 5. The lessons locus on patterns and algebraic situations. The lessons included in this unit plan are organized to follow the Learning Cycle.
Prior
Knowledge (K-3rd grade)
Below is a summary of what
students are expected to know by the time they enter 4th grade.
Standard 1: The student describes, analyzes, and
generalizes a wide variety of patterns, relations, and functions.
By 4th grade, students have
learned to identify, describe, and recognize patterns using a wide variety of
materials and attributes. Students should also be able to predict and extend
patterns as well as identify missing parts in patterns. Standard 2: The
student uses expressions, equations, inequalities, graphs, and formulas to
represent and interpret situations.. Upon entering
4th grade, students are expected to know that symbols can be used to
represent missing or unknown quantities, solve addition and subtraction
sentences where an unknown number is represented by a geometric
shape._ Students also begin using concrete objects to solve
number sentences with equalities and inequalities.
Standards
Addressed
NCTM Standards
The student will understand patterns, relations, and
functions
· Describe, extend, and make generalizations about geometric and numeric patterns,
·
Represent and analyze patterns
and functions, using words, tables, and graphs. The student will represent and analyze mathematical
situations and structures using algebraic symbols
·
Identify such properties as
commutativity, associativity, and distributivity
and use them to compute with whole
numbers;
·
Represent the idea
of a variable as an unknown quantity using a letter or a symbol;
·
Express
mathematical relationships using equations.
The student will use mathematical models to represent
and understand quantitative relationships
·
Model problem
situations with objects and use representations such as graphs, tables, and
equations to draw conclusions.
The student will analyze change in various
contexts
·
Investigate
how a change in one variable relates to a change in a second
variable;
·
Identify and
describe situations with constant or varying rates of change and compare them.
MA.D.1.2 The
student describes, analyzes, and generalizes a wide variety of patterns,
relations, and functions.
1.
describe a wide variety of patterns and
relationships through models, such as manipulatives, tables, graphs, rules using algebraic
symbols.
2.
generalizes a pattern, relation,
or function to explain how a change in one quantity results in a change
in another.
MA.D.2.2 The
student uses expressions, equations, inequalities, graphs, and formulas to
represent and interpret situations.
1.
represents a given simple
problem situation using diagrams, models, and symbolic expressions translated
from verbal phrases, or verbal phrases translated from symbolic
expressions, etc.
2. uses informal methods, such as physical models and graphs, to solve real-world problems involving equations and inequalities.
Sequence
of Lessons
Patterns
Invitation1
1. What's Next? (NCTM Illuminations):
Students begin their study of growing patterns by making linear
patterns with pattern block shapes using several pattern cores. They extend a
partner's pattern and find the missing element in a pattern. (See http://illuminations.nctm.org/LessonDetail.aspx?ID=L302)
Exploration
2.
Snake
Patterns (PBS Mathline): The teacher
presents students with rules for a growth pattern that involves red and black
rings in imaginary snakes. Students use rules to produce the first several
stages of the pattern. They then work in small groups to find mathematical
patterns in their results. Then students attempt to use those patterns to
predict how subsequent stages would look. Finally, students answer questions
that allow the teacher to assess their understanding of the mathematics
underlying the patterns. (See http://www.pbs.org/teachers/mathline/lessonplans/atmp/snake/snake_procedure.shtm.)
Concept
Introduction
3. Patterns on Charts (NCTM Illuminations): Students find, record, and analyze patterns on hundred and multiplication charts. They also use an online calculator to generate patterns and then record them on a chart. (See http://illuminations.nctm.org/LessonDetail.aspx?ID=L303)
4.
Growing
Patterns (NCTM Illuminations): Students use numbers to make growing
patterns. They create, analyze, and describe growing patterns and then record
them. They also analyze a special growing pattern called Pascal's Triangle. (See http://illuminations.nctm.org/LessonDetail.aspx?ID=L304)
5.
Watch
Them Grow (NCTM Navigations): Students will construct growing
patterns using pattern blocks and isosceles right triangles; record patterns
numerically in a table; state rules for extending the patterns. (See Watch Them
Grow, NCTM)
6.
Tiling
a Patio (NCTM Navigations): Students observe patterns and
relationships; make conjectures about patterns and test those
conjectures; discuss, verbalize, generalize, and represent patterns and
relationships. This activity is set in a problem-solving scenario whereby
students are asked to determine how many tiles will be in the next-largest
patio. (See Tiling a Patio,
NCTM.)
7.
Ups
and Downs of Patterns (NCTM Navigations): Students will identify and analyze
situations with constant or varying rates of change. (The
Ups and Downs of Patterns, NCTM.)
Application
8. Peddling Petals (PBS Mathline): Using the setting of a flower-making fundraiser, students explore a variety of patterns. They examine patterns in a flower design made of triangles surrounding a square. They complete a table listing the number of triangles needed to complete various flower designs. Using Excel, students also make tables showing the cost of buying different cut-outs of flowers. (See http://www.pbs.org/teachers/mathline/lessonplans/pdf/esmp/peddlingpetals.pdf.)
Algebraic Situations
Invitation
9.
Triangle
Rule (NCTM Navigations): Students investigate the perimeters of figures composed of equilateral triangles
arranged in a row; describe (verbally and symbolically) the "rule", or
function, that will produce the perimeter for any given arrangement of triangles; and make a
connection to the idea of function when describing the rule for a pattern
with numbers. (See Triangle-Rule
Machine, NCTM.)
Exploration
10.
Function
Machine (Virtual Manipulatives): Students use an online manipulative to explore the function concept
through the "machine" metaphor. The domain elements (input) are dragged
into the machine, which then goes through some (unseen) process and spits out
the range element corresponding to the input. The results are then displayed in
tabular form. (See http://nlvm.usu.edu/en/nav/frames_asid_191_g_3_t_2.html.)
Concept
Introduction
11.
Variable
Machine (NCTM Navigations): Students explore the idea of variable
as a symbol that can stand for any member
of a set of numbers; substitute numbers for variables (letters) to
discover unknown values. (See The
Variable Machine, NCTM.)
12. Catch of the Day (NCTM Navigations): Students will work with variables as they determine the number of each kind of fish caught; record algebraically the statements of the results of their "catch." See Catch of the Day, NCTM.)
13.
Algebra
Scales (NCTM Navigations): Students will determine if expressions
constitute an equation (balanced scale) or
an inequality (unbalanced scale); understand that quantities on both sides of an
equation must be equal; use logical thinking to find a replacement set to
solve equations. If needed, students can use the Hands-on-Equations kit as a
resource. (See Algebra Scales,
NCTM.)
14.
I Spy
Patterns (NCTM Navigations): Students will partition the given array
into different parts; translate visual
patterns into numerical expressions; explore how equivalent numerical
expressions represent the commutative and associative properties of
operations. (See I Spy
Patterns, NCTM.)
15.
Building
Houses (NCTM Navigations): Students will verbalize the numerical
relationship in each problem; translate each relation into an algebraic
equation. (See Building
Houses, NCTM.)
16.
That's
Odd (NCTM Navigations): Students will observe various patterns in an
array; represent observed visual patterns as numerical patterns; and represent a
numerical pattern as a functional relationship. (See That’s Odd,
NCTM.)
Application
17. Squares Cubed (NCTM Navigations): Students will investigate the functional relationships between the length of the sides of a square and its perimeter and area; describe the functional relationship between the lengths of the sides of squares and (a) their perimeters as the sides increase in length, (b) their areas as the sides increase in length; and describe the functional relationship between the lengths of the sides of cubes and their volumes as the sides increase in length. See Squares Cubed, NCTM.)
What’s Next?
Below is a description of the expectations for students in
grades 6-8.
Standard 1: The student describes, analyzes, and
generalizes a wide variety of patterns, relations, and functions. Students
describe, predict, and create numerical and geometric patterns through models.
They are able to use those models to develop algebraic expressions, equations,
and formulas. Students also predict outcomes based on a generalization of a
pattern or relationship and function rules. Finally, students create and interpret tables, graphs, equations, and verbal
descriptions to explain cause-and-effect
relationships.
Standard 2: The student uses expressions, equations, inequalities, graphs, and formulas to represent and interpret situations. Students represent and solve real-world problems graphically, with algebraic expressions, equations, and inequalities.. They solve linear equations and graph solutions to equations and inequalities on a number line and/or coordinate plane. Students use algebraic problem-solving strategies to solve real-world problems involving linear equations and inequalities. They begin by using concrete materials to solve equations and inequalities. They then move to solving single- and multi-step linear equations and inequalities that represent real-world situations.
Bibliography
Cuevas, Gilbert J. and Yeatts, Karol.
(2001). Navigating through algebra in grades 3-5.
Kaput, Jim. (unknown). Algebraic thinking math project: snake patterns-s-s-s. Retrieved October 3, 2006, from http://www.pbs.org/teachers/mathline/lessonplans/atmp/snake/snake_procedure.shtm.
Harelson, Tracey. Using strip models with algebra. Handouts
from NCTM Regional Workshop attended October 2005. PBS Mathline (unknown). The elementary school math
project: peddling petals. Retrieved
November 5, 2006, from http://www.pbs.org/teachersource/mathline/lessonplans/pdf/esmp/peddlingpetals.pdf
Virtual Manipulatives: http://nlvm.usu.edu/en/nav/vlibrary.html.