Transcript Algebra for ALL - Indiana University

Algebra for ALL
Indiana Council of Teachers
of Mathematics
October 12, 2003
Donna McLeish
[email protected]
Email for a copy of this presentation
The FEAR of
Algebra
New Content in Math Textbooks
100%
90%
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Algebra for ALL Must Begin
in Kindergarten…for ALL
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Identify, sort, and classify objects by size,
number, and other attributes.
Identify objects that do not belong to a
particular group.
Identify, copy, and make simple patterns
with numbers and shapes.
And continue every year…
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The Indiana Academic Standards-Mathematics
specify year-by-year content standards and
indicators for:
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Number Sense
Computation
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Algebra and Functions
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Geometry
Measurement
Data Analysis and Probability
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Grade 1
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Write and solve number sentences from
problem situations involving addition and
subtraction.
Create word problems that match given
number sentences involving addition and
subtraction.
Recognize and use the relationship
between addition and subtraction.
Create and extend number patterns using
addition.
Grade 2
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Relate problem situations to number sentences
involving addition and subtraction.
Use the commutative and associative rules for
addition to simplify mental calculations and to
check results.
Recognize and extend a linear pattern by its
rules.
Create, describe, and extend number patterns
using addition and subtraction.
Grade 3
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Represent relationships of quantities in the form of a
numeric expression or equation.
Solve problems involving numeric equations.
Choose appropriate symbols for operations and relations
to make a number sentence true.
Understand and use the commutative and associative rules of
multiplication.
Create, describe, and extend number patterns using
multiplication.
Solve simple problems involving a functional relationship
between two quantities.
Plot and label whole numbers on a number line up to 10.
Grade 4
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Use letters, boxes, or other symbols to represent any number in simple expressions,
equations, or inequalities (i.e., demonstrate an understanding of and the use of the
concept of a variable).
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Use and interpret formulas to answer questions about quantities and their
relationships.
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Understand that multiplication and division are performed before addition and
subtraction in expressions without parentheses.
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Understand that an equation such as y = 3x + 5 is a rule for finding a second
number when a first number is given.
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Continue number patterns using multiplication and division.
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Recognize and apply the relationships between addition and multiplication,
between subtraction and division, and the inverse relationship between
multiplication and division to solve problems.
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Relate problem situations to number sentences involving multiplication and
division.
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Plot and label whole numbers on a number line up to 100. Estimate positions on
the number line.
Grade 5
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Use a variable to represent an unknown number.
Write simple algebraic expressions in one or two variables and evaluate
them by substitution.
Use the distributive property in numerical expressions and
equations.
Identify and graph ordered pairs of positive numbers.
Find ordered pairs (positive numbers only) that fit a linear
equation, graph the ordered pairs, and draw the line they
determine.
Understand that the length of a horizontal line segment on a coordinate
plane equals the difference between the x-coordinates and that the
length of a vertical line segment on a coordinate plane equals the
difference between the y-coordinates.
Use information taken from a graph or equation to answer
questions about a problem situation.
Grade 6
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Write and solve one-step linear equations and inequalities in one variable
and check the answers.
Write and use formulas with up to three variables to solve problems.
Interpret and evaluate mathematical expressions that use grouping
symbols such as parentheses.
Use parentheses to indicate which operation to perform first when writing
expressions containing more than two terms and different operations.
Use variables in expressions describing geometric quantities.
Apply the correct order of operations and the properties of real numbers (e.g.,
identity, inverse, commutative, associative, and distributive properties) to
evaluate numerical expressions. Justify each step in the process.
Identify and graph ordered pairs in the four quadrants of the coordinate
plane.
Solve problems involving linear functions with integer values. Write the
equation and graph the resulting ordered pairs of integers on a grid.
Investigate how a change in one variable relates to a change in a second
variable.
Grade 7
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Use variables and appropriate operations to write an expression, a formula, an
equation, or an inequality that represents a verbal description.
Write and solve two-step linear equations and inequalities in one variable and check the
answers.
Use correct algebraic terminology, such as variable, equation, term, coefficient,
inequality, expression, and constant.
Evaluate numerical expressions and simplify algebraic expressions by applying the
correct order of operations and the properties of rational numbers (e.g., identity, inverse,
commutative, associative, and distributive properties). Justify each step in the process.
Solve an equation or formula with two variables for a particular variable.
Define slope as vertical change per unit of horizontal change and recognize that a
straight line has constant slope or rate of change.
Find the slope of a line from its graph.
Draw the graph of a line given the slope and one point on the line, or two points on the
line.
Identify functions as linear or nonlinear and examine their characteristics in tables,
graphs, and equations.
Identify and describe situation with constant or varying rates of change and know that a
constant rate of change describes a linear function.
Grade 8
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Write and solve linear equations and inequalities in one variable, interpret the solution
or solutions in their context, and verify the reasonableness of the results.
Solve systems of two linear equations using the substitution method and identify
approximate solutions graphically.
Interpret positive integer powers as repeated multiplication and negative integer
powers as repeated division or multiplication by the multiplicative inverse.
Use the correct order of operations to find the values of algebraic expressions involving
powers.
Identify and graph linear functions and identify lines with positive and negative slope.
Find the slope of a linear function given the equation and write the equation of a line given
the slope and any point on the line.
Demonstrate an understanding of rate as a measure of one quantity with respect to
another quantity.
Demonstrate an understanding of the relationship among tables, equations, verbal
expressions, and graphs of linear functions.
Represent simple quadratic functions using verbal descriptions, tables, graphs, and
formulas and translate among these representations.
Graph functions of the form y=nx2 and y=nx3 and describe the similarities and differences in
the graphs.
So… How do we teach it?
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Textbook exercises – not fun
Workbook pages – not fun
Worksheets – not fun
Games – FUN – But do they teach
algebra topics?
Let’s try a few …
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Kindergarten – “What’s My Rule?” Fishing
Grade 1 – Guess the Rule; Magic Bag
Grade 2 – “What’s My Attribute Rule?”
Grade 3 – “What’s My Rule?” Function Machine
Grade 4 – Credits/Debits
Grade 5 – Top It: Positive/Negative Numbers
Grade 6 – “What’s My Rule?” Formulas
Grade 7 – Solution Search
Now ask yourself …
Did you have fun…
Did you learn more about…
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attributes
patterns
positive and negative numbers
equations
inequalities
Did you learn some algebra???
If you can answer “Yes”
then you can say
I learned some
algebra and it
was painless!
Let’s give our students the
same opportunity.
Life isn’t all fun and games, and
neither is the learning of algebra.
But fun and games can and should
be an important part of how we
teach algebra at every grade level.
Resource
Teacher’s Guide to Games
Grades K-6
from
Everyday Mathematics
Wright Group
The McGraw-Hill Companies
(wrightgroup.com)
What’s My Rule? Fishing
K181
Object: To identify a sorting rule
Directions
1.
The teacher “fishes” for a few students using some obvious
attribute. The teacher fishes out a few people wearing brown,
for example, without explaining what he is fishing for.
2.
The teacher asks “What sort of a fish am I going to catch next?”
or “What’s my rule?”
3.
Students guess until someone says “Students wearing brown.”
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The teacher asks “Who are the students NOT in my net?”
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Students identify those students.
6.
Repeat steps 1-5 using other obvious attributes
Magic Bag
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Object: To Identify the Rule
Directions:
1.
The teacher tells the class that they are going to play a guessing
game using a magic bag. The teacher puts 1 stick in the bag,
says a magic word, and takes 2 sticks out of the bag. Next, the
teacher puts 2 sticks in the bag and pulls out 3 sticks.
2.
The teacher repeats the activity several times by always taking
out 1 more stick than is put into the bag. The class describes a
rule for the magic bag’s action (add 1, 1 more, next number…).
3.
Teacher (or student volunteer) repeats the activity for other
rules. The rest of the class guesses the rule.
What’s My Attribute Rule?
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2-180
Object: To be the first player to identify a rule used to sort a
set of attribute blocks.
Directions:
Players label one sheet of paper: These fit the rule.
Players label another sheet: These do NOT fit the rule.
Players take turns. Each player rolls the die once The player with the lowest
number is the first “Rule Maker.”
The Rule Maker mixes the Attribute Rule cards and then stacks them
facedown.
The Rule Maker turns over the top card but does not show it to the other
players or tell them what the rule is.
The Rule Maker chooses 3 or 4 attribute blocks that fit the rule on the card,
then places the blocks on the sheet labeled These fit the rule.
(continued on next page)
7.
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The Rule Maker chooses 3 or 4 attribute blocks that do NOT fit the
rule on the card, then places the blocks on the sheet labeled
These do NOT fit the rule.
The other players are the “Guessers.” The Guessers take turns.
Each one chooses a block that he or she thinks might fit the rule.
The Rule Maker tells each Guesser Yes or No. The Guesser puts the
block on the correct sheet. The Guesser suggests what the rule
might be. The Rule Maker tells the Guesser if his or her rule is
correct.
The Guessers continue until someone figures out the rule. That
player is the winner and becomes the Rule Maker for the next
round.