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Didactic Unit on Algebra.
MATHEMATICS AS “DNL”
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)
DIDACTIC UNIT : ALGEBRA
LEARNING OBJETIVES
A-2
evaluate an open expression, with one variable, using the
following to replace the variable
a. integers
b. rational numbers
A-3
evaluate an open expression, with two variables, using the
following to replace the variables
a. fractions and decimals
b. integers
c. rational numbers
A-5
solve word problems by using
a. equations
A-6
solve equations, in one variable, using
a. undoing
A-8
solve for the variable in a formula, using inverse operations
A-9
understand and use functions as
a. ordered pairs
Didactic Unit on Algebra.
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)
A-10
graph ordered pairs of numbers and tables of values on the
coordinate plan
a. in any quadrant
A-12
create a table of coordinates and the accompanying graph for a
simple
linear
equation
A-13
determine if an ordered pair is a solution to a particular linear
equation
A-14
display a relation as a set of ordered pairs, a mapping diagram,
and
a
graph
A-15
understand and use correctly the terms
a. polynomial, like terms, monomial, binomial, trinomial,
degree, coefficient
A-16
use
manipulatives
to
represent
a
polynomial
A-17
generate equivalent expressions by collecting like terms
A-18
add
and
A-19
multiply
A-20
multiply
subtract
and
a
two
divide
polynomial
polynomials
two
by
monomials
a
monomial
Didactic Unit on Algebra.
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)
A-21
multiply
A-22
divide
a
a
binomial
by
polynomial
by
a
a
binomial
monomial
A-23
solve and verify equations, in one variable, with variables on both
sides
of
the
equal
sign
A-24
solve and verify equations, in one variable, containing fraction or
decimal coefficients
Instructional Notes
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Encourage a variety of problem-solving strategies. Give
students an opportunity to explain their strategy to other
members of a group or to the class as a whole. Solving
problems in a cooperative group situation is a valuable
learning experience for many students. Encourage students
to make up their own problems and to bring problems from
other aspects of their lives and share these with other
members of the class.
A-2 Evaluating open expressions using integers and rational
numbers is an extension of work done in previous grades.
Students should be familiar with operation symbols used in
algebra and with the standard order of operations. You may
wish to have students express number properties such as the
commutative, associative, and distributive properties using
variables.
E.g.:
a(b + c) = ab + ac (distributive)
a + b = b + a (commutative)
Didactic Unit on Algebra.
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)
(a + b) + c = a + (b + c) (associative)
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A-3 The concept of substitution is very crucial to success in
algebra. Many students have difficulty substituting one
mathematical expression for another. The phrase "another
name for" is sometimes useful instead of substitution.
Substitution is also used to check solutions to open
sentences.
Indicate to the students that frequently there is more than
one way to solve any problem or to illustrate a particular
concept.
A-5 Ask students to write, in their journals, hints they may
have for others to help them translate problems and puzzles
into algebraic equations and then to solve them.
A-6 Solving equations by undoing is often demonstrated
using a balance scale. This idea could then be extended by
drawing a diagram of a balance with alge-tiles on the pans.
Encourage students to see that the two sides must be kept
balanced. A reverse flowchart may be useful for some
students particularly when there is more than one operation.
One way of using alge-tiles is to place them on a sheet of
paper that has been divided in half by a vertical line. Keep
the tiles representing one side of the equation on one side of
the page and those representing the other on the other side.
E.g.: -2x + 5 = 9
Isolate the x-term and then isolate the x.
-2x+5 = 9
-2x = 9-5
-2x = 4
-x = 2
Didactic Unit on Algebra.
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)
-x=2
To make x positive, change the signs on both sides
x = -2
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Other ways of using alge-tiles may be found in booklets
describing their use.
Whatever method or methods you use, start with equations
using one operation. To help students remember the order of
undoing operations, you could use the analogy of a person
putting on shoes and socks. You normally put on your socks
first and then your shoes because that is the easiest. When
you are removing them you go in the reverse order -- shoes
off first and then socks. Similarly when undoing operations,
start by undoing the last operation that would be done.
Encourage students to keep track of any new terms they
encounter by entering them in a journal along with an
example or description.
A-8 Students frequently have difficulty mastering this
concept as it constitutes further abstract thinking.
Emphasize that the methods here are the same as those in
solving equations in one variable. The only difference is that
the results are often expressions rather than numbers.
The first example in A-8 could be used in a problem-solving
context where you are given the perimeter and the length
and asked to solve for the width.
A-9 A function is a rule that pairs each element of one set
with one and only one element of another set. E.g.: the
perimeter of a regular octagon is a function of the length of
a side. For every length of a side, there is a unique
perimeter.
The rule for a function can be defined in a number of ways
such as a table, arrow diagram, or equation. Not all of these
ways may be appropriate for one particular function.
In a function, each first element corresponds to only one
second element. Function terminology can be left until
secondary mathematics courses but students should be
Didactic Unit on Algebra.
MATHEMATICS AS “DNL”
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)
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familiar with a variety of ways of writing a function. A good
understanding of function concepts is essential to success in
mathematics in subsequent years. f(x) 8x is read "f is a
function that pairs x with 8x".
A-10 Several mathematics books give suggestions for
"games" that provide students with practice activities in
locating points on a plane.
A-11 Integrate this objective with transformations in the
Geometry/Measurement strand. Additional transformations
(rotations, reflections) could be included as enrichment.
A-12 Begin with situations in which y is alone on one side
of the equation. If students have a good understanding of the
concept of graphing given an equation, the idea could be
extended by combining it with objective A-9 and asking
students to graph equations such as 2x - 3y = 6.
A-13 A couple of methods could be used to accomplish this
objective. The expectation is that students could substitute
the appropriate values for x and y and determine whether the
equation is true or false.
A-14 Integrate the graphing of functions with the graphing
of data.
A-15 Students should be familiar with the terms but should
not be required to memorize formal definitions. Provide
opportunities for students to discuss mathematical concepts
and encourage proper usage of terms in those discussions.
They should be able to give examples of each.
A-17, 18 Other than collecting like terms, work with
polynomials is new to students in this course. The
introduction to polynomials and their operations with the
use of manipulatives such as alge-tiles is useful for most
students particularly those who are still at the concrete stage
of developing an understanding of concepts. When first
using alge-tiles, give students time to become familiar with
them. The coloured sides represent positive values and the
white sides represent negatives. Alge-tiles are particularly
useful for operations with polynomials.
Other suggestions for the use of alge-tiles can be found in a
variety of print materials.
Didactic Unit on Algebra.
MATHEMATICS AS “DNL”
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)
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A-19 Multiplication of monomials requires an
understanding of the rules for exponents when multiplying
powers of the same base and when taking the power of a
power, or power of a product. Encourage understanding
rather
than
memorization
of
the
rules.
E.g.: m4 means m x m x m x m and m² means m x m
So m4 × m² = (m x m x m x m) (m x m) = m6 i.e., (m4 + 2)
Similar expansions can be used to develop the other rules.
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Division of monomials requires the introduction of negative
integers and zero as exponents.
A-20 The distributive property is an important concept for
the multiplication of polynomials. Alge-tiles can be used for
students to do a variety of questions such as 3(x + 2) i.e., use
three groups of x + 2 and simplify the result.
A-21 One idea for using alge-tiles for multiplication of
binomials (especially those with negative coefficients) is to
use a coordinate grid. Place the tiles representing the first
binomial along the x-axis (positives on the right, negatives
on the left). Then place the second binomial along the y-axis
(positives above the horizontal axis and negatives below).
Complete the rectangle to obtain the product.
For (x - 3)(x+ 2), see the example on the left.
Area models can be used for multiplication of polynomials.
E.g.: (a + 3)(a - 9)
(a+3)(a-9) = a2-9a+3a-27 =a2-6a-27
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A pupil or the teacher modelled the process of factoring x2 +
4x+ 4 by using algebra tiles and forming a square with them.
What are the factors of x2+4x+4?

Use this method to factor x2 + 5x + 6
Didactic Unit on Algebra.
MATHEMATICS AS “DNL”
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)
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Although factoring is not an objective at this course, you
may wish to introduce it as the opposite of division. Use
alge-tiles and the area model to help students understand the
process.
A-22 Since division is the same as multiplying by the
reciprocal
A-23 There are a variety of appropriate ways of introducing
solving equations with variables on both sides of the equal
sign. One method is an extension of the balancing idea. For
example,
3n
+
1
=
2n
+
3
Take 1 chip from each side. Then
3n + 1 - 1
= 2n + 3 - 1
3n + 0
= 2n + 2
3n
= 2n + 2
Take 2 boxes from each side. then,
3n - 2n
= 2n + 2 - 2n
n
= 0+2
n
= 2
Check by
substitution
3(2) + 1
6+1
7

= 2(2) + 3
= 4+3
= 7
Encourage students to discover that there is not one first
correct step but there is a most efficient first step.
Didactic Unit on Algebra.
MATHEMATICS AS “DNL”
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)
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A-24 Students are often reluctant to solve equations with
fractional coefficients. Emphasize that solving these
equations is the same as any others. The same methods can
be used -- guess and check, cover-up, or undoing.

You can also eliminate the fractions by multiplying both
sides of the equation by the least common denominator and
then solving the resulting equation.
EXAMPLES/ACTIVITIES
A-2
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3x² - 8x + 7 when x = -5
-6t + 4t when t = 8
2x + 6x² - 7 when x = -1

2a² - 3a + 1 when a = - 1½
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Evaluate:
Complete the table below:
x -3x+4
-3
0
3
5
Didactic Unit on Algebra.
MATHEMATICS AS “DNL”
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)
The Acme Car Shop charges its customers according to the
formula C = 45 + 35.60h where C is the repair cost in dollars and
h
is
the
number
of
hours
of
repair
work.
The Classic Car Repair Shop uses the formula C = 30 + 41.70h.
Nancy has 5 hours of repair work to be done to her car. How
much less would she pay by taking her car to the less expensive
shop?
Bill paid 169.60€ for repair work at the Acme Car Shop. How
many hours of repair work was he charged for?
Write an expression or equation to represent each situation.
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The cost to rent a VCR is a $25 deposit, plus $10 for each
day. How much will it cost to rent a VCR for 4 days? ... For
10 days? ... For d days?
Bruce bought some licorice. It cost $3.75 for the first
kilogram and $3.25 for each additional kilogram. How
much would he pay for 3 kg? 10 kg? m kg?
A-3
Evaluate:
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3ab - 6b when a = 2/3 and b = 5/6
-j + 7k - 3j when j = -2 and k = 3
x-3 + y³ when x = 2 and y = -2
Compact disks cost $14 for the first one and $13 for each
additional one. If you buy M compact disks and spend D dollars,
write an equation that represents the relationship between M and
D.
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ab + 3a when a = -¾ and b = 2/3
A-5
Blocks are used to make a staircase as shown below. How many
blocks are required if the base is 12 blocks long? (Ans: 78)
Didactic Unit on Algebra.
MATHEMATICS AS “DNL”
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)
Lisa has 1 more baseball card than 4 times the number Mary has.
If Lisa has 19 more cards than Mary, how many cards does Mary
have?
The sides of a triangle are consecutive integers. If the perimeter of
the triangle is 55 cm, how long is each side? (Write an equation
and solve.)
A string measuring 50 cm in length is cut into three pieces. One
piece is twice as long as the shortest piece and the other piece is
10 cm longer than the shortest piece. Find the length of each piece
of string.
Dennis has 25€ and can save 2.80€ per day. Jeena has18€ and can
save 3.70€ per day. Who will be the first to be able to buy a 72 €
tennis racquet?
A-6
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Solve:
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1.5x = 4.5
2m - 16 = -12
The equation 5x = 4 + 3x has been modelled with algebra tiles.
Explain how you can use the tiles to justify an algebraic solution
process.
Use algebra tiles to justify an algebraic solution to 3x-7 = - 2x+8.
A-8
P = 2(l + w). Solve for w.
Explain how
P = 2l + 2w
Didactic Unit on Algebra.
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)
P-2l = 2w
w = P-2l / 2
Given that density is mass divided by volume, explain why
volume is mass divided by density.
A-9
Write the function x 3 - 2x as a set of ordered pairs if the values
for x are -2, -1, 0, 1.
A-10
Locate each of the following points on the coordinate plane. Join
the points in order and calculate the area of the resulting figure.
A(5,5), B(-7,5), C(-13,-5), D(-1,-5)
One corner of a rectangle is at (5,2). The centre is at (-1, -1). Find
the coordinates of the other three corners.
In which quadrant is each of the following points located:
A(-3,1), B(-7,5), C(-13,-5), D(-1,-5)
A-12
Create a table of values for the following equation and graph your
results: y = 3x + 2
Stamps can be purchased individually or in packets of 5, 10, 15,
or 25 stamps. Prepare a table showing the number of stamps
purchased and the cost. Plot the ordered pairs from the table to
show the relationship. Find the rule that describes it.
A-13
Is the ordered pair (-1, 2) a solution of the equation
5x - 3y = -11? Why or why not?
A-14
Didactic Unit on Algebra.
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)
X2-3x Find the ordered pairs and draw a graph for this relation if
+1
the values for x are -2, -1, 0, 1.
A-15
What is the degree of
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4x³ y²
9x³ - 8x + 12
Give an example of a binomial of degree 2.
What is the numerical coefficient of -6a4b?
What is the constant term in the equation 4x - 3 = 2y?
A-16
Use alge-tiles to show the expression 2x² - 3x + 5.
Explain how the algebra tiles given below can be used to justify
an algebraic process for simplifying: (4x² - 3x + 5) + (4x - 2).
A-17
Simplify: 3a - 8b + 7a - 15b + 10
C represents the number of compact disks and C+C+4+2C=56.
Using this information, write a problem.
Write an expression for the perimeter of a triangle:
A-18
Explain how the algebra tiles given below can be used to justify
an algebraic process for simplifying: (4x² - 3x + 2) - (3 + x - 4x² ) .
Add:
3x² - 5x + 7
-4x² + 8x +2
Subtract: (2x² - x - 3) - (x² + 7x - 4)
Didactic Unit on Algebra.
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)
A-19

Multiply:

(3mn)(-m² n4)
(-2s² t4)³
Justin used algebra tiles and an area model to explain the
multiplication 2x(3y). He set up the model by drawing a frame
with dimensions 2x and 3y.
Show how he filled the area model in to get the product.
A-20
Multiply: 3x² y³ (4x³ - 8xy + 7y² )
Explain why the area model with algebra tiles can justify the
product: 2x(x-2)
A-21
Multiply: (3a - 2)(4a + 7)
Find the product of -2x-3 and 3x+4.
Use an area model with algebra tiles to explain your algebraic
solution to the product (4x+1)(x+2).
A-22 Divide (4x2+3x-5): (x-2)
A-23
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5a - 7 = 2a + 4
Solve and check: 2(4x-5) = (-2x+6)A-24
Solve and check:

¾a = 15
Didactic Unit on Algebra.
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)

0.03x + 0.05(x - 1) = 2.45
Explain the relationship between
Manipulatives/Resources
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Linking cubes
Pattern blocks
Coloured counters
Alge-tiles
Print material to support the use of alge-tiles

Teacher's Notes:
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Balance scale
Alge-tiles
A-5 Ask students to write, in their journals, hints they may
have for others to help them translate problems and puzzles
into algebraic equations and then to solve them.
Printed materials on alge-tiles.
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Didactic Unit on Algebra.
MATHEMATICS AS “DNL”
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)
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Teacher's Notes:
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Graph paper
Overhead grid
Student journals
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Teacher's Notes:
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Graph paper
Alge-tiles
Print material to support the alge-tiles resource.
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Teacher's Notes:
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Alge-tiles
Rules for Exponents
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Product of powers
am an = a m + n
Power of a power
(am)n = amn
Power of a product
(xyz)n = xny nzn
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Teacher's Notes:
Didactic Unit on Algebra.
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Author: ANTONIO LUDEÑA LÓPEZ. IES VEGA DEL ARGOS (CEHEGIN)

Alge-tiles
ASSESSMENT CRITERIA
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Use the real numbers when solving problems.
Use approximations of irrational numbers.
Use the calculator to work with numbers.
Distinguish the different sorts of rational numbers.
Use powers and square roots when solving problems.
Simplify and compare equivalent root expressions.
Find the real roots of a polynomial.
Factorise polynomial.
Use algebraic tools when solving problems related to
equations.
 Solve quadratic equations and problems related to them.
 Solve inequalities: linear and quadratic inequalities.