Algebra
Starters
Algebra, Junior High Workshop Series
Algebra Starters /21
(2005)
22/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
DEVELOPING ALGEBRAIC THINKING
“Algebraic thinking encompasses the set of understandings that are needed to interpret the world
by translating information or events into the language of mathematics in order to explain and
predict phenomena. Also, algebraic thinking leads to the abstract thinking required for success
in textbook algebra. To develop true understanding, students must work with problem situations
that arise throughout the strands of mathematics and in various contexts that are familiar or make
sense to them. While students are solving engaging, meaningful problems, we teachers must
focus the students’ attention on the concepts vital for success in formal algebra.” Emphasizing
algebraic thinking does not require a new curriculum as must as it requires a shift in
emphasis and teaching methods.
“By viewing algebra as a strand in the curriculum from pre-kindergarten on, teachers can help
students build a solid foundation of understanding and experience as a preparation for the more
sophisticated work of algebra.” Principles and Standards for School Mathematics (37).
Attention to algebraic thinking should be incorporated into all strands of the mathematics
curriculum and should begin in the primary grades. Algebraic thinking can be developed by
providing students a sequence of learning experiences that build on previous understanding and
prepare the foundation for formal algebra. It should be developed in a systematic manner,
similar to number sense. If this is done, students will view algebra as a natural extension of their
previous mathematical learning.
From Lessons for Algebraic Thinking (pp. ix, x, xii) by Ann Lawrence and Charlie Hennessy. Math Solutions
Publications: Sausalito, CA.
WHY STARTERS?
Goal: This set of activities is designed as “starters” or openers. Some will require some
explanation first time out, but the intent is to develop quick and easy warm-ups that focus the
students at the start of mathematics class.
WHAT IS A STARTER?
To fit the category of a “starter,” activities will share several important attributes.
They will:
 appear simple enough that every student can give a response
 appear thought provoking, stimulating to the imagination, mysterious, puzzling and evoke
imaging
 lend themselves to a variety of approaches to solutions
 lend themselves to oral discussion and mental computations more than pen and pencil
calculations
 connect to important mathematical concepts
 expose some of the basic common knowledge/misconceptions that students bring
 focus on making sense and understanding, not remembering rules and procedures.
Algebra, Junior High Workshop Series
Algebra Starters /23
(2005)
LINKS TO THE NUMBER SENSE WORKSHOP
The first workshop in the series of the Junior High Mathematics Workshops provided by Alberta
Education was “Developing Conceptual Understanding of Number.” It was set up as a series of
activities that could be used as warm-ups to get class started. The pre-algebra section of that
material is included at the end of the Algebra Starters section to remind teachers they already
have a valuable resource.
INSTRUCTIONS
1. Avoid starter questions that only allow one solution or one way to find the solution.
2. Avoid “tricks.” The idea is to convince students that mathematics can make sense.
3. Avoid the temptation to over challenge—starters are meant to be accessible to all.
4. Avoid starters that become a “race” or competitive. Starters are meant to allow everyone
time to participate.
STRATEGIES FOR ELICITING RESPONSES FROM STUDENTS
1. “Are you sure? How do you know? Convince me. Can you tell us what you were
thinking, how you were thinking about that?”
These questions can all help encourage students to talk more.
2. One way to draw students out is to accept their answer, then ask if another student can
explain their thinking. Often as another student starts to try to explain, the first student
will interrupt to clarify or confirm.
3. Establish a signal system that keeps students from blurting out before others have had
time to think. Thumbs up if you think you have an answer, catch my eye and tap your
nose, stand up and sit back down if you are ready.
4. Make it a habit to always ask, “Did anyone think about it a different way?”
5. When students say “I just knew,” they may be serious. For some of us 8 × 7 = 56 is a
known fact. But if you want to encourage some strategizing, you might simply ask:
“Does anyone have a way that would help someone figure it out? I don’t know 8 × 7 but I
think 8 × 5 + 8 × 2.”
6. Encourage students to paraphrase each other. “Can someone repeat what ___ just said?”
7. “I know what you are saying makes sense, but I am still not clear. Can someone else help
me out? Explain what ____ is saying.”
8. When no answers are forthcoming, tell students a strategy that another class used.
“Someone in Mr. Smith’s class said…” What do you think? Does it make sense?”
REMEMBER, STARTERS ARE INTENDED TO GENERATE DISCUSSION AND
INTEREST. THEY ARE NOT MEANT AS WORKSHEETS.
24/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
LINKS TO RESEARCH
In a three-year study aimed at changing teachers’ practice, Good, Grouwer and Ebmeier (1983)
found that by starting every class with a short session of “mental mathematics” raised student
confidence and achievement.
Starting class with visual, spatial and mental mathematics activities increases brain activity.
Students are more alert for a longer time.
Starting with openers focuses students’ attention. It signals that mathematics class is starting and
it just might be interesting.
Starting class this way builds community. A social norm is reinforced that suggests everyone is
worth listening to and everyone has something to contribute. Communication is a critical process
in learning.
When teachers focus on listening to student thinking, they gain insights that can be used to
refine, adapt and improve their use of instructional strategies.
When mathematics class starts this way, students and teachers are reminded that mathematics is
about thinking, reasoning, justifying and making sense.
Algebra, Junior High Workshop Series
Algebra Starters /25
(2005)
To fit the category of a “starter,” activities will share several
important attributes.
They will:

appear simple enough that every student can give a response

appear thought provoking, stimulating to the imagination,
mysterious, puzzling and evoke imaging

lend themselves to a variety of approaches to solutions

lend themselves to oral discussion and mental computations
more than pen and pencil calculations

connect to important mathematical concepts

expose some of the basic common knowledge/misconceptions
that students bring

focus on making sense and understanding, not remembering
rules and procedures
26/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
In a three-year study aimed at changing teachers’ practice,
Good, Grouwer and Ebmeier (1983) found that starting every
class with a short session of “mental mathematics” raised
student confidence and achievement.
Starting class with visual, spatial and mental mathematics
activities increases brain activity. Students are more alert for
a longer time.
Starting with openers focuses students’ attention. It signals
that mathematics class is starting and it just might be
interesting.
Starting class this way builds community. A social norm is
reinforced that suggests everyone is worth listening to and
everyone has something to contribute. Communication is a
critical process in learning.
When teachers focus on listening to student thinking, they
gain insights that can be used to refine, adapt and improve
their use of instructional strategies.
When mathematics class starts this way, students and
teachers are reminded that mathematics is about thinking,
reasoning, justifying and making sense.
Algebra, Junior High Workshop Series
Algebra Starters /27
(2005)
Starter 1
n + 5 = 12
x + y = 10
How are these two equations the same? How are they different?
Possible Responses
They both have variables.
They both have numbers.
They are both adding.
They both have equal signs.
One uses n and the other uses x, y.
One has an answer and the other has many answers.
Teacher Prompt
What is the value of n in this equation?
What could the value of x be? Of y?
What could they not be?
As students offer pairs of numbers, prompt deeper thinking by asking:
Can you use fractions? Decimals? Integers?
Starter 2
n + 5 = 12
x + y = 10
Work with a partner to create a word problem for each equation.
Teacher Prompt
Encourage students to use specific contexts like measurement, money or weights.
Create a problem that might come from geometry.
28/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
n + 5 = 12
x + y = 10
How are these two equations the same?
How are they different?
3 + 5 * y = 40
5y + 3 = 40
How are these equations the same?
How are these equations different?
Put each equation into a word problem.
Is one easier to think about than the other?
Does it affect the answer you get?
Does it change the way you think?
Algebra, Junior High Workshop Series
Algebra Starters /29
(2005)
Starter 3
3 + 5 * y = 40
5y + 3 = 40
Possible Responses
They use the same letters, numbers and operations.
They both multiply but one uses a sign and the other doesn’t.
Both equal 40.
In the first one, if you add 3 + 5 you get 8 times y.
In the second one, you multiply y times 5 and then add 3 on.
What about order of operations?
I bought a pen for three dollars and 5 CDs. I spent $40. How much did each CD cost?
I bought 5 CDs and a pen. The pen cost $3. How much did each CD cost?
Starter 4
If you lived in Europe during the 1700s, you might have seen any of these equations in the work
of mathematicians.
4 × 5 = 20
4  5 = 20
4 * 5 = 20
4(5) = 20
Invite willing participants to come to the board and demonstrate one way to write 6 times 8.
Solutions
6 × 8 = 48
6  8 = 48
6 * 8 = 48
6 (8) = 48
(and of course you can reverse them all) :
8 × 6 = 48 8  6 = 48 8 * 6 = 48 8 (6) = 48
How many different ways could they have written 9 times an unknown number?
Because there are so many possible ways, mathematicians have agreed on a standard notation.
Nine times x is 9x. Four times y is 4y.
30/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
If you lived in Europe during the 1700s, you might
have seen any of these equations in the work of
mathematicians.
4 x 5 = 20
4  5 = 20
4 * 5 = 20
4(5) = 20
How many different ways could they have written
9 times an unknown number?
Algebra, Junior High Workshop Series
Algebra Starters /31
(2005)
Starter 5
Goal: This warm-up is simply to remind students of the variety of notations that can be used to
demonstrate multiplication. Be on guard for misconceptions as students explain what they think
the multiplication is and how they might believe it can be rewritten. With each item you may
want to remind students that by convention some of their choices, while possible, will never
occur. However, the goal is to bring their attention to how algebraic notation might differ from
whole number notation.
Possible Solutions
a) 2(3) 23 2*3
(2 is multiplied by 3 or 3 is multiplied by 2)
b)
5 times b, 5(b) but the standard convention is 5b
(b is being multiplied by 5 or 5 times and you could say 5 is being multiplied b times)
c)
7  8 = 56 (7 times 8, 7 is multiplied by 8 or 8 is multiplied by 7)
7(8)
7*8
7×8
d)
4(5) (4 times 5, 4 is multiplied by 5 or 5 is multiplied by 4)
4(5)
4*5
4×5
e)
2(3 + 5) = (2 times 3 + 5, 3 + 5 or 8 is multiplied by 2 or doubled)
2 × (3 + 5)
2 * (3 + 5)
The brackets make it unnecessary to use another symbol, but you could use one.
You could also write (2 × 3) + (2 × 5)
f)
3y/2 =(3 times y divided by 2 or divided in half)
You could write it 3  y ÷ 2
(3y) ÷ 2
g)
(9 – y)6 = (Subtract whatever y is from 9 and then multiply by 6)
However you rewrite this one, you need to be careful that you do the 9–y first.
9–y/6
(9–y) ÷ 6
h)
4p (5 + 4) = (Something (p) times 4 and then multiply that by the answer you get when you
add 5 and 4. Multiply p four times then use the answer to multiply with the answer to 5 + 4.
You could write 4  p (5 + 4) for example, or you could write 4 * p (9). The way it is on
the page is the most efficient and the most “conventional” but do students understand it?
32/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
These equations and expressions all show multiplication:
a)
2×3=6
b)
5b
c)
7  8 = 56
d)
4(5)
e)
2(3+5) =
f)
3y/2 =
g)
(9 – y) 6 =
h)
4p (5 + 4) =
Identify what is being multiplied in each equation.
Rewrite the multiplication a different way.
Algebra, Junior High Workshop Series
Algebra Starters /33
(2005)
Starter 6 (No handout required)
The teacher reads aloud each section of the story below. After each section, ask students what a
good note or diagram would be to capture the main idea. You may need to have a discussion
about what the piece is actually saying.
The use of symbols began with the Egyptians. Through the ages, mathematicians have chosen
and used their own preferred symbols to describe ideas in mathematics.
(Stop: What is the key idea?)
Once the printing press was invented, books could be printed and published for a wider audience.
It became necessary to agree on some notations so that readers could clearly understand.
(Stop: What is the key idea?)
Rene Descartes wrote the first printed text to introduce coordinate geometry.
Descartes particularly liked to use a, b, c for known values and x, y, z for unknown values.
(Talk your thinking aloud to students: Coordinate geometry, that is where you use x,y pairs and
plot them on a graph. Draw a four quadrant grid on the board and label the x,y.)
Now ask students to help with the note to use for this section.
Possible Prompts
What letters did Descartes like to use?
What do we mean by known values? Unknown values?
What are values?
In 1637 as the typesetter was setting the print, he found he was running short of the last letters of
the alphabet. The French language has lots of words that use y and z. He asked Descartes if it
mattered which letters were used in the equations and Descartes replied that it did not. So the
printer chose to use x most often, as he had many x’s compared to y’s and z’s.
(Stop: What is the key idea?)
34/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
Because Descartes used a raised dot for multiplication, there was no shortage of x’s. The raised
dot is a fairly universal sign for multiplication. Use of the × (“timsing sign”) is not popular
among mathematicians.
(Stop: What is the key idea? Descartes used a raised dot? What does that mean? If students
are not sure demonstrate 4·5 instead of 4 × 5. Why do you think he did that?)
Descartes was one of many who used the dot. It was introduced as a symbol for multiplication by
G. W. Leibniz.
In a letter to John Bernoulli, July 29, 1698, Leibniz wrote: “I do not like X as a symbol for
multiplication, as it is easily confounded with x; ... often I simply relate two quantities by an
interposed dot and indicate multiplication by ZC · LM. Hence, in designating ratio I use not one
point but two points, which I use at the same time for division.”
(Stop: What is the key idea? So why is it that the raised dot gets used? I wonder why textbooks
continue to use the × for multiplication if mathematicians do not.)
If students enjoy this read aloud, there are many other stories about
famous mathematicians available. Have students surf the Internet to
find more.
Algebra, Junior High Workshop Series
Algebra Starters /35
(2005)
Starter 7
This starter might be best done in pairs so students have someone to talk with.
Give each student a copy of the overhead so they have the equations in front of them, but no
pencils. The question is: “Which is easiest in your head and which would you want a pencil for?”
See Horne (attached article at end of section) for explanation of the value of this starter.
Starter 8
Similar to Starter 7. Students see that division can be notated many ways but again in algebra we
tend to use specific conventions.
36/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
How many ways can you write a division question?
How about 8 divided by 4?
8÷4=2
8 2
4
48
8:4
4
8
How about 4 divided by 8?
x divided by 3?
4x + 3y divided by x?
Algebra, Junior High Workshop Series
Algebra Starters /37
(2005)
Which of these equations:
a) is easy to solve in your head?
b) could be solved in your head but requires extra
thinking?
c) would you prefer to use a pen and paper to solve?
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
2x + 5 = 9
3x – 4 = x + 2
4x + 3 = 12
5 = 2x + 1
3x – 8 = 5x + 2
6x – 5 = 3x + 2
3(x – 4) = x + 2
2(x + 5) = 9
5x – 2 = 9
(11x + 5)/3 – 4 + 2 × 3 = 11
38/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
Starter 9
Instructions: Place one set of 6 exercises on the overhead. Tell students in each set of
6 exercises you may use the calculator three times to solve a problem. You may not use paper
and pencil for any of them.
Which three would you choose to solve and why?
Give students time to look and think and talk with a partner if they wish.
Record their strategies on the board.
There are 6 sets of boxes. You can use a day for six days.
Algebra, Junior High Workshop Series
Algebra Starters /39
(2005)
1. 547 x 382 = ?
________
2. The product of 2341 and zero is
________
3. Multiply 815 and 9774.
________
4. 382 x 547 =
________
5. 9774 x 689 =
________
6. One times 6 million is
________
1. 357 x 842 = ?
________
2. 42 x 357 = ?
________
3. 600 x 849 = ?
________
4. 358 x 842 = ?
________
5. 849 x 600 = ?
________
6. 357 x 42 = ?
________
40/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
M = 8473
N = 672
1. M plus N
________
2. M times N
________
3. N plus M
________
4. Zero plus N
________
5. (8400 + 73) times (900 + 72)
________
6. (600 + 72) times M
________
A = 575, B = 75, C= 500
1. 7345 times A
________
2. (B + C) times 7345
________
3. A minus C
________
4. 7345 divided by B
________
5. 7345 x B + 7345 x C
________
6. (476 x 899) – (899 x 476)
________
Algebra, Junior High Workshop Series
Algebra Starters /41
(2005)
A = 599, B = 701
1. A plus B
________
2. A x B
________
3. B - A
________
4. B x A
________
5. 100 times A
________
6. A x 600 + A
________
M = 639, N = 4848
1. N divided by M
_________
2. M times N
_________
3. Zero divided by M
_________
4. One times M times N
_________
5. Ten times M
_________
6. One hundred times N
_________
42/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
Starter 12 Algebra Number Lines
Materials
 string or rope (optional)
 file cards with 0, 5 10, –5, –10, x, y, x + 3, y + 6, x –5, 3x
 other assorted expressions of your choosing
Instructions
1. Draw a number line across the front board, or use the string to stretch a number line across
the front of the room or lay it on the floor.
2. Place evenly spaced increments on the line.
3. Hold up the zero card and invite students to decide where they would like it placed.
4. Do the same with the 10, 5, –10, and –5 cards.
5. Now hold up the x and invite a student to come up and place it.
(It can go anywhere as it is simply an unknown.)
Once a student has decided a placement, ask the following questions of the class:



How do you know the placement is possible?
Is that location the only place where that value could go?
Do we know anything about the value of x?
Now hold up the x + 3 and ask a student to come up and place it.
Once a student has decided on a placement, ask the following questions of the class:



How do you know the placement is possible?
Is that location the only place where that value could go?
Do we know anything about the value of x + 3?
Remove the x and the x + 3 cards.
Let’s start fresh.
Hold up x – 5. Where will it go?



How do you know the placement is possible?
Is that location the only place where that value could go?
Do we know anything about the value of x?
Algebra, Junior High Workshop Series
Algebra Starters /43
(2005)
Hold up 3x. Where will it go?



How do you know the placement is possible?
Is that location the only place where that value could go?
Do we know anything about the value of x?
Variations and Extensions
Facilitator Note
Invite participants to create some ways to use this idea at their table groups.
Have them share their ideas with the larger group.
Always ask the three questions:



Is the placement possible?
Is that location the only place where that value could go?
Do we know anything about the value of x?
1. Allow students to create and place their own cards.
2. What if I place x at (–3) and the next person gets x + 3?
3. Where would you place expressions like: 2y  9 3x + 5
x/2
3y/3
Once students have done some whole class discussion with the number, you can use it as a
regular review to keep ideas fresh.
As students enter the class have some cards already placed on the number line and ask them to
explain if they make sense.
Record equations:
If I place x at 5, then someone can write on the board:
x =5
Whoever places x + 3 can finish the equation x +3 = 8.
So what would 2x + 3=
44/ Algebra Starters
(2005)
3x/5=
Algebra, Junior High Workshop Series
This article is included as it provides some explanation for why many of these starters have been
chosen. It also offers further suggestions for discussion starters.
Algebra revisited (Reprinted with permission from Marj Horne, Australian Catholic
University).
Poor concepts of the symbols used in algebra contribute to students’ difficulties. Some
concerns include the understanding of the addition sign, the equals sign and the variety
of meanings for the pro numeral x. Following a discussion of student understanding of
algebraic concepts, some activities are suggested that foster discussion around some of
the “big ideas” of algebra and have the potential to make the concepts of algebra
explicit.
Introduction
The mention of the word algebra often brings a negative reaction from the listener.
Many adults comment that mathematics was “okay” until they started algebra. It then
became hard and sometimes they add that they subsequently failed mathematics. I
have heard teachers comment that when the word algebra was mentioned it was like a
chilly wind blew through the classroom. The perception seems to be that algebra is
difficult.
Why is it that algebra causes so many difficulties for children learning mathematics?
Many children seem to “hit a brick wall” in their mathematics learning early in Years 7
and 8 and this is usually attributed to algebra. Recently there has been a lot of
international attention on early algebra being introduced in the first few years of school.
Changes in curriculum in many places have included algebraic development right from
the start of schooling and this should make a difference but these changes will take
some years to filter through and affect students at Years 7 and 8.
Understanding of the Operation Symbols
So what are some of the causes of these difficulties with learning algebra? One of the
difficulties is that although one aspect of algebra is generalized arithmetic the signs and
symbols in algebra are not exactly the same as they are understood by many students
in arithmetic. For example when some students see 5 + 7 they immediately recognize
the + as a sign to combine the two numbers and give the response 12. Once the
number is seen as 12, the original components such as the + sign are no longer visible
and a single number, 12, replaces the expression 5 + 7. In algebra, however, a + 7 is
different. The plus sign does not mean “combine the two parts to make a single number
in the same way” as it did for the arithmetic expression (Chalouh & Herscovics 1988).
a + 7 can be considered as a single object made by combining the two components a
and 7 but these components maintain their identity within the object.
Algebra, Junior High Workshop Series
Algebra Starters /45
(2005)
Many children try a variety of ways to combine the separate components. Most teachers
of Years 7 and 8 have seen expressions such as 4x + 3 simplified by students to 7x.
Some students will have learned the procedure for simplifying expressions and can use
it to simplify quite complex expressions but then add an extra step to write the final
expression as a single term, thus eliminating the plus sign. Students who leave the
expression as 4x + 3 without trying to combine the parts are said to have “acceptance of
lack of closure” (Collis 1975). This understanding is a critical part of algebraic
development.
Understanding of “=”
Another aspect of differences between symbol use in arithmetic and algebra is the =
sign. Freudenthal (1983) claimed four different categories of meaning for the equals
sign: the result of sum; quantitative sameness; a statement that something is true for all
values of the variable (identity); and a statement that assigns a value to a new variable.
A full understanding of the equals sign as it is used in algebra requires all of these
meanings. However, for many students = is the sign that indicates the need to do
something—an operator sign—or to move to the next step, or even as an indicator of
where to write the answer—a syntactical indicator (Carpenter and Levi 2000), so they
will record incorrectly using equals signs. I saw this demonstrated in a classroom
recently where students were solving a problem. The question concerned how many
legs there were with 2 lions, 4 cubs and 4 storks. After the sharing time at the end of the
lesson, the display shown in Figure 1 was on the blackboard.
24=8+44
= 8 + 16 = 24 + 4  2
= 24 + 8 = 32
Fig. 1. Record of solution to legs problem.
This misuse of the equals sign during the solution to a problem is also common among
secondary school students who use = as a sign to do the next step in solving an
equation. For example cosA = 0.5 = 60o. Other students will use the equals sign at the
start of a row, so it becomes the sign for the next line of a solution even if the task is
solving an equation as in Figure 2.
3x – 4 = 2x + 5
= 3x – 4 – 2x = 5
=x–4=5
=x=9
Fig. 2. Misplaced use of “=” during equation solving.
46/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
The meanings of these symbol components of arithmetic and algebra need to be made
explicit for the students. The new curriculum in Queensland addresses this by including
the algebraic structure as part of early understanding of mathematics and recognizing
that this structure underlies both arithmetic and algebra. In other places also, the
primary school curriculum has recognized the need for improved understanding of the
equals sign. However, for Year 7 and 8, students who have not had these experiences
and who are still operating with the idea that the equals sign is an indication of where to
write the answer, there needs to be discussion about these issues so that the reasons
for them using the symbols in a particular way are based on understanding rather than
“because the teacher tells me I have to set it out that way.”
Understanding of the Pronumeral
Yet another group of errors arise because of lack of explicit explanation for the different
uses of the pronumeral. Figure 3 shows a list of equations where pronumerals can be
considered to have different meanings.
x+2=5
x(x – 2) – 15 = 0
2x + 3x = 5x
4 x + 3y = 12
A=l×w
3x + 4 = 15
a(x + b) = ax + ab
y = 2x – 4
y = mx + c
Fig. 3. Equations with different possible interpretations of “x”
Sometimes the x represents a number that is known, and sometimes it represents an
unknown number. Sometimes it represents one number and sometimes many. On
some occasions the pronumeral is a variable, and on others, a constant. A more
complete list of possible meanings for the “x” is given here:
 a specific known number
 a specific unknown number
 more than one specific number
 any number
 (any object)
 a variable that may be dependent or independent
 a constant
 a quantity that can be measured
 a quantity that can be calculated
In the first of the equations in Figure 3, most students look at and know immediately that
x = 3. In this situation x is not an unknown. The equation is transparent. Many students
thus find it difficult to understand why the textbooks use complicated algorithms to
“solve” such equations. The methods of solution given make much more sense when
applied to the second of the equations as nearly all students would need a formal
method of solution. The third equation not only requires a method of solution but yields
more than one value for the pro numeral.
Algebra, Junior High Workshop Series
Algebra Starters /47
(2005)
The distributive law, which is the fourth equation, is an identity that is always true for all
possible values of the pro numerals involved and relates to Freundenthal’s third
category of meaning for the equals sign. The fifth equation was placed in brackets
because, while an identity, the x is not restricted to pro numerals or algebraic objects
made up of pro numerals, but indeed could be any object. This is the root of what has
become known as “fruit salad algebra,” based on using the letter to represent an object
often starting with that letter so 3a + 4a = 7a is read with the a being “apples” rather than
the desired understanding at this level of a representing a number.
The understanding of x that relates to functions and relations is as a variable, and is
represented in equations six to eight in Figure 3. It also relates to Freudenthal’s fourth
category for the understanding of the equals sign. As an independent variable, x does
not just represent any number but rather all numbers in the possible domain. For
students there is a difference between an expression written in the assignation form
such as equation 6 and the linear relation represented in equation 7. Equation 8 also
raises the idea of constants and variables. I remember being puzzled over this
distinction for years as a student in high school and at University.
In the final equation in Figure 3, the l and the w represent the length and width of a
rectangle and as such in the student’s eyes are know quantities because they are easily
measured. The A is different because it is not measured directly but is rather calculated.
This difference in understanding explains why students who otherwise can solve an
equation like 3x = 21, have difficulty finding the length when the area is 21 cm2 and the
width is 3 cm (Usiskin 1988).
Students often adopt one meaning for the pronumerals and do not attend to others. A
classic situation arose when a teacher was returning a test to a Year 8 student. The
student complained that he had been unfairly treated as the teacher had marked the
question wrong when it was correct. The teacher looked at the linear equation, for which
the student had the answer 39 and explained to the student that 14 was the correct
value as it made the equation correct. The student responded in frustration “but all last
year you told us that x could be any number and so what is wrong with 39?” The symbol
x has many different meanings that are rarely if ever made explicit and this can
contribute to students misunderstandings. These multiple meanings need to become
part of the classroom conversation.
One key aspect of algebra is its use in generalization. The student above has over
generalized the meaning of the x but on other occasions we want students to
generalize. Algebra has often been described as generalized arithmetic, and part of it is
the abstraction from specifics in arithmetic to general underlying structures.
Difficulties in this abstraction process often occur because of students focusing on
inappropriate generalizations and interpretations as well as obstructions caused by
semantics and alternative approaches to semantics deduced from the “concrete”
situation. For example, given a simple one- or two-step linear equation in early algebra,
students will often solve it by a guess and check method in spite of the teacher
presenting a different approach. This is reinforced by success in the problems in the
Years 7 and 8 textbooks, and becomes entrenched but does not lead to further
understanding and allow transfer to more difficult situations.
48/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
Similarly in arithmetic, the equality symbol is often seen as a signal to perform
operations, but this is a limited conception and causes an obstacle in algebra. Left to
their own devices without direction, students are unlikely to develop the semantics of
algebra as we know them because the types of experiences they have are limited and
often lead to alternative representations that do not then relate to other situations.
Another example of students developing entrenched but non-productive understanding
is with students using a backtracking method. They might record it happily as 5x + 3 = 8
= 5 = 1 and all students involved at the time understand what this means, but it is a
misuse or different meaning of the symbols and will limit future development. This
means clear guidance is needed to assist the students to construct knowledge and use
mathematical language and sign systems that are compatible with the language and
sign systems of others.
Backtracking causes a further obstacle. Students are often shown how they can solve
fairly complex equations with one occurrence of the variable on the left hand side of an
equation and a single number on the right hand side. They practise this skill and
become adept at using it. This often leads to a strong reluctance to relinquish it when in
the following years they meet equations for which backtracking cannot be used, and
thus handicaps their further development in algebra.
Algebra Sense
Students need to develop a sense of algebra. What do I mean by algebra sense?
Algebra sense is an understanding of the objects of algebra and the different
representations as well as the ability to sense the form of the result of a particular
process (Horne and Maurer 1998). It is most important to visualize the nature and form
of the solution and to move readily between the representations or mathematical sign
systems, rather than the ability to work with the objects to produce the required
solutions—although of course producing solutions is also necessary in developing
algebra. In many ways, algebra sense corresponds to number sense, though algebraic
experiences are not as much part of the students’ world as numerical experiences.
A critical part of developing algebra sense is encouraging discussion in which the use of
language and student explanation can assist students in their developing
understanding. The few activities below are designed to allow all students to participate
in developing mental algebraic skills and more particularly to make sense of algebra.
The key part of these activities is the ensuing discussion in which the issues can be
made explicit and the big ideas of algebra be raised. Part of the focus is on some of
these key principles of algebra. For example, the first activity focuses on the
approaches students use in solving equations. The idea is to enable the students to
share their ideas about how to solve equations. The ensuing discussion should also
raise the issue of when different methods are useful and efficient and the difference
between “arithmetic” linear equations which can be solved using backtracking type
methods and “algebraic” equations which have more than one occurrence of the x.
Filloy and Sutherland (1989) call this separation between what they see as arithmetic
and algebraic, the didactic cut.
Algebra, Junior High Workshop Series
Algebra Starters /49
(2005)
Activity 1
Which of these equations:
 is easy to solve in your head?
 could be solved in your head but requires extra thinking?
 would you prefer to use a pen and paper to solve?
2x + 5 = 9
3x – 4 = x + 2
4x + 3 = 12
5 = 2x + 1
3x – 8 = 5x + 2
6x – 5 = 3x + 2
3(x – 4) = x + 2
2(x + 5) = 9
5x – 2 = 9
(11x + 5)/3 – 4 + 2 × 3 = 11
Another question to ask then is what different methods could be used to solve these
equations and which is the most efficient method for each question? We know many
students use guess and test even though teachers have often tried to insist on the
students setting out their equations solutions by using a balance method. For this
activity the key focus for the students could be which methods are most suitable for
which equations. Instead of the question above about doing the problems in their heads,
the question might be “for which of these equations could you use A Guess and test; B
Backtracking; C the balance method; or D other?”
The activity can also involve group discussion before the whole class has a sharing
time. Allow the students some time of individual work to decide on their answers then
have them share their strategies in groups of three or four. The question did not
actually ask for the solutions to the problems, but during the discussion about the
strategies the solutions will arise. Following a time of group discussion, the key
approaches can be discussed with the whole class with the students also suggesting
how to decide on the best method to use each time. The other key point that will arise is
that there is not one best method. While the balance method always works and is often
the taught algorithm, it is not the most efficient method for an equation like 20/(2x + 3) =
4 or 32/(3x + 1) = 4. The focus of this question was solution of equations. Activity 2 also
focuses on solving equations.
50/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
Activity 2
Write down five different equations that have a solution of
x = 3.5.
The approaches that students use to do this task can be shared with the class. To elicit
a variety of answers from the students, criteria can be added such as at least one of the
equations has to have an x on each side of the equals sign.
The activities used can be from any aspect of algebra. The critical aspect is that they
are fairly open and encourage the students to share and discuss meaning. Activity 3 is
an open task that focuses on equivalent expressions and raises the whole issue of
simplification.
Activity 3
Ask the students to write down three different expressions equivalent to 2x + 3. Collect
verbal answers from all students (the teacher acting just as scribe), arranging them in
up to five different groups on the board as students give their answers to you. The
answers should be recorded on the board with no corrections. It is up to the students to
discuss any discrepancies. The groups might be:
• those which change the order of terms or insert symbols, e.g., 3 + 2x, x × 2 + 3
• those in which the number term is changed, e.g., 2x + 6 – 3, 2x + 1 + 10/5
• those in which the coefficient of x is altered or a series of x terms are added or
subtracted, e.g., 8x/4 + 3; 2x + 3 + x
• those which are a combination of the last two groups
• a miscellaneous group which may include changes to the x, e.g., x2 + x + 3.
If too many answers are coming in for any of the first three groups, ask them to try to
change some other aspect of the expression. The students also will need to check that
they agree with each recorded expression. When answers have been collected from the
whole class, the students can explain why you have grouped them in the way you have
by explaining the common aspects of each group and the differences between them. Of
course with older students’ expressions can be with different powers. There should be
class discussion about how we know the expressions are equivalent and students
should try to explain how they arrived at their answers. Another way to do this is to put
up the expression and focus the nature of the student answers by specific questions
while still leaving them partly open.
For example:
Write down an expression with no 3 in it.
Write down an expression with no 2 in it.
Write down an expression with a – sign.
Write down an expression that begins with a negative number.
Write down an expression with a fraction in it.
Write down an expression with a b in it.
Algebra, Junior High Workshop Series
Algebra Starters /51
(2005)
One of the early rules students suggest is often to change the order, so the negative
raises that question. Students often think a – b is the same as b – a. Rather than
immediately correcting the students, suggest that the order does not matter, following
up by using the same task but with the starting point 2x – 6, or some other similar
expression. As part of the discussion one of the questions becomes, “How do you know
when two expressions are equivalent?” Another key issue to raise in the discussion is
which of the expressions is simplest. For many students x + x + 1 + 1 + 1 is the simplest
as it shows the basic meaning.
Activity 4
Write down an ordered pair that satisfies the equation 2x + 3y = 6.
An important part of all these activities is the discussion that ensues. Students should
explain how they arrived at their answers and discuss the relative ease of using different
types of numbers and approaches.
Try it again with y = x2 + 3.
Did strategies change for this problem and if so why?
Concluding Comments
These activities and the associated discussions are an attempt to engender in students
a sense of algebra. Estimation and number sense are acknowledged as critical to our
teaching. An important part of the introduction of ordinary calculators in schools is the
corresponding emphasis on estimation skills as students develop the number sense
necessary in tandem with calculator skills. Symbolic manipulators (CAS, computer/
calculator algebra systems) are to algebra as ordinary calculators are to number,
although there is one important difference. Students are continually meeting number
and measurement in a variety of ways in the world around them and in their out-ofschool experiences. A corresponding algebraic world experience is not as accessible.
Algebra provides a language, notation and procedures that enable problems from the
world to be more easily and efficiently solved. The rarity of this experience in everyday
life means we must be extra careful to include experiences that can support the
development of algebraic estimation skills and assist in the development of algebra
sense. Our approach to teaching algebra has to allow for a variety of approaches.
Efficient mental methods are not always the same as written algorithms and change
more with the components of the question rather than with the nature of the question.
Number sense plays an important part in this. How will the corresponding algebra sense
be developed? We will need to change our teaching programs to include approaches
that will build algebra sense.
Reprinted with permission from Marj Home, Australian Catholic University.
52/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
References
Carpenter, T. P., and Levi, L. Developing Conceptions of Algebraic Reasoning in the Primary
Grades. Wisconsin Center for Educational Research. 2000.
http://www.wcer.wisc.edu/ncisla/publications/reports/RR-002.pdf. Accessed Sept. 8, 2004.
Chalouh, L, and Herscovics, N. “Teaching Algebraic Expressions in a Meaningful Way.” The
Ideas of Algebra K–12. Reston, Virginia: NCTM, 1988, pp. 33–42.
Collis, K. The Development of Formal Reasoning. Newcastle, Australia: University of
Newcastle, 1975.
Filloy, E., and Sutherland, R. “Designing Curricula for Teaching and Learning Algebra.”
International Handbook of Mathematics Education Vol. 1, 1996, pp. 139–160.
Freudenthal, H. Didactical Phenomenology of Mathematical Structures. Dordrecht, Holland:
Reidel Publication, 1983.
Horne, M., and Maurer, A. “A New Angle on Algebra.” Exploring All Angles. Brunswick:
Mathematical Association of Victoria, 1998, pp. 194–200.
Kieran, K., and Chalouh, L. “Prealgebra: The Transition from Arithmetic to Algebra.” Research
Ideas for the Classroom: Middle Grades Mathematics. New York: Macmillan, 1992,
pp. 179–198.
Usiskin, Z. “Conceptions of School Algebra and Uses of Variables.” The Ideas of Algebra K–12.
Reston, Virginia: NCTM, 1988, pp. 8–19.
Algebra, Junior High Workshop Series
Algebra Starters /53
(2005)
Pre-Algebra Patterns 1
Vocabulary
Notes
Answers

1.
a)
b)
c)
7
256
each number is double the number in
the previous step.
2.
a)
b)
8
10
3.
a)
b)
bat
ret, rot, rut or rat
For 3b), “rit” has not
been included
because it is not an
English word. “Rat”
has been included
because there is no
indication that an “a”
cannot be replaced
with another “a”.
54/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series
Pre-Algebra Patterns 1
1. The following chart reveals a number pattern.
Step
Number
1
1
2
2
3
4
4
8
5
16
6
32
7
64
8
128
a) What is the first step where the number is greater than
50?
b) If the pattern continues, what is the number at step 9?
c) Describe the pattern in the “number” row of the chart.
2. The following diagrams show a pattern in the number of
seats arranged around an increasing number of small
tables.
What is the value of the missing number in each of the
following charts?
a)
b)
Number of
Small
Tables
Number of
Seats
1
2
4
6
Number of
Small
Tables
1
2
3
4
3
Number of
Seats
4
6
8
3. Consider the following chart:
top
tap
a)
b)
lip
lap
men
man
bet
A
B
rat
Give a word to replace A in the chart.
Give two possible words to replace B in the chart.
Algebra, Junior High Workshop Series
Algebra Starters /55
(2005)
Pre-Algebra Patterns 1
Vocabulary
Notes
Answers

1.
Possible Answers:
 Place the object in term 3 on a row of 7
blocks
 Add 7 blocks underneath the object in
term 3
 …
2.
a)
b)
3:30
BONG BING BING BING
3.
a)
Possible Answers:
 the number of cats in the school
yard after 3 ran away.
 …
Possible Answers:
 the number of cats caught by the
pound if half the cats escaped.
 …
It is important that
students always
indicate what the
variable represents.
b)
56/ Algebra Starters
(2005)
4.
Let a be Carole’s age. Then her mother’s
age is 2a + 10.
5.
a)
b)
9
Let the term number be t. Then, the
number of squares is 2t -1.
Algebra, Junior High Workshop Series
Pre-Algebra Patterns 2
1.
Describe how to build the fourth shape in the following
series:
1
2.
2
3
A clock goes BONG for every hour and BING for every 15
minutes.
For example BONG BONG BING represents 2:15.
a) What time is it if the clock goes BONG BONG BONG
BING BING?
b) What would you hear at 1:45?
3.
If c represents the number of cats in the school yard, what
situation could each of the following represent?
a) c  3
c
b)
2
4.
When you double Carole’s age and add 10, you get her
mother’s age. Write a mathematical expression that
shows the mother’s age. Tell what your variable
represents.
5.
The following chart reveals the number of squares in a
pattern.
Term Number
1
2
3
4
Number of Squares
1
3
5
7
a) How many squares would exist for term number 5?
b) Write a mathematical expression that would allow you
to determine the number of squares for any term.
Algebra, Junior High Workshop Series
Algebra Starters /57
(2005)
Pre-Algebra Patterns 2
Vocabulary
Notes
Answers

1.
For 1b), it is acceptable
if students come up with
a name that only
satisfies the two patterns
they identified in part a).
a)
b)
2.
a)
b)
c)
58/ Algebra Starters
(2005)
Possible Answers:
 pattern is alphabetical in order
 the number of letters in each
name increases by 1 each time
 the names alternate girl then boy
then girl, etc.
Eleanor, Emmalou, Eveline or
Ellymae are suitable answers
because they satisfy all three
conditions in part a. (ie. girl’s name
starting with “E” and having 7 letters)
Term
1
2
3
4
Number
Number
of Small
3
5
7
9
Squares
You could find the number of small
squares in term 7 by:
 Drawing all the figures and
counting squares for figure 7
 Extending the chart in part a).
 Creating an algebraic expression
as in part c) and substituting 7 for
the variable
 …
Total small squares is 2t + 1 where
t = term number.
Algebra, Junior High Workshop Series
Pre-Algebra Patterns 3
1.
There are several possible patterns in the
following list of names:
Ann, Brad, Carol, Daniel, _____ , ______
a) Describe two of the patterns you found.
b) What is a possible fifth term in the list of
names? Why?
2.
1
2
3
a) Construct a chart showing the term number
and the increasing number of small squares
in the pattern above.
b) How could you find the number of small
squares in the seventh term?
c) For the pattern shown above, write an
algebraic expression showing the total
number of small squares where t = the term
number.
Algebra, Junior High Workshop Series
Algebra Starters /59
(2005)
Pre-Algebra Patterns 3
Vocabulary
 rule
 sequence
 regular die
Notes
Answers

1.
a)
b)
2.
a)
For #3, have students
discuss the patterns they
found to justify their
picture for term 4.
Some possible patterns:
 Filled in Boxes:
9, 21, 33, …
 Size of figures: 3 x 3;
5 x 5; 7 x 7; so the
next one is 9 x 9. All
figures have the
perimeter and the
diagonals filled in.
 Filled in Boxes: 32 – 02;
52 – 22; 72 – 42; so the
next one is 92 – 62.
 For #4, have dice
available for students to
confirm that the sum of
the opposite sides is
always 7.
b)
Term
Number
Number of
Small
Squares
3.
1
2
3
4
5
3
4
5
6
7
Possible Answer:
33 + 12 = 45
squares should be
shaded. All
squares on the
diagonal of the
large square are
shaded.
4.
60/ Algebra Starters
(2005)
11 triangles
The number of triangles is one more
than the term number.
a)
b)
7
Sum of the hidden numbers is
6 + 7 + 7 = 20 assuming you can see
the numbers on 4 sides of each die.
[6 is the number on the bottom of the
top die]
Algebra, Junior High Workshop Series
Pre-Algebra Patterns 4
1. The number of triangles in a pattern is shown in the
following chart:
Term Number
1
2
3
4
Number of Triangles
2
3
4
5
a)
b)
Find the number of triangles for term number 10.
Give a rule describing the number of triangles for any
term.
2. The number of small squares is increasing in the following
pattern:
Term 1
a)
b)
Term 2
Term 3
Draw a picture showing Term 4 in the pattern.
Make a table showing the number of small squares
for each of the first 5 terms.
3. Build or draw the fourth term in the following sequence.
Justify your answer.
#1
#2
#3
4. Three dice are stacked on top of each other. The number
on the top of the highest die is 1.
a) What is the sum of the numbers on opposite
sides of a regular die?
b) What is the sum of the hidden numbers for your
stack of 3 dice? Explain.
Algebra, Junior High Workshop Series
Algebra Starters /61
(2005)
Pre-Algebra Patterns 4
Vocabulary Write a mathematical expression for each
1.
of the following:
a) a number, n, increased by 3.
b) The number of birds, t, in a nest after 2 flew away.
c) Your brother’s age if he is twice your age
decreased by 1.
Notes
Answers
Complete the following
the rule “the number
For2.#3, students
1.
a)chart
n +using
3
should discuss
the is one more b)
– 2 the term number”.
of people
than ttriple
relationship
2n – 1 3where n = your4 age
Term
Numberof the 1
2 c)
term number to “x”
Number
of
and the relationship
2.
People
of the number of
Term
1
2
3
4
triangles to “y”.
Number
 For #4, this is a
Number of
4
7
10
13
review for Set H.
People
 Points on the x-axis
have a y-coordinate
The following
chart
the
number of triangles
in
of 3.
0. Similarly,
points
y
3. reveals
a)
11
triangles
on the y-axis
have an
an increasing
pattern. b)
x-coordinate of 0.
Term Number
Number of
Triangles
y
4. d) Q
a)
b)
2
5
3
2
81
4
x
Term Number
y
a)
(0, -3)
How many triangles
4? R
b) occur
(1, 4) for
andterm
(1, -4)number
Q
c)
Sand
= (8,y1)is the number of
If x is the term number
R
S1
P1
Q1
2
4.
triangles,
draw a graph showing the pattern in the
S
P
S
x
table.
P
4.
1
Number of
Triangles

a)
3
b)
62/ Algebra Starters
(2005)
c)
x
d) ofP1all
=(2,points
-1) ; Q1on
= (2,
-7)y-axis
; R1 = (8,and
-7) ;
Find the coordinates
the
units
below the x-axis. S1 = (8, -1) [See sketch on the left]
R1
Find the coordinates of all points 4 units from the
x-axis and 1 unit right of the y-axis.
Algebra, Junior High Workshop Series
PQRS is a square with P =
,Q=
, and
R=
. Find S.
Pre-Algebra Patterns 5
1.
Write a mathematical expression for each of the following:
a) a number, n, increased by 3.
b) The number of birds, t, in a nest after 2 flew away.
c) Your brother’s age if he is twice your age decreased
by 1.
2.
Complete the following chart using the rule “the number of
people is one more than triple the term number”.
Term Number
Number of
People
3.
2
3
4
The following chart reveals the number of triangles in an
increasing pattern.
Term Number
Number of
Triangles
4.
1
1
2
3
2
5
8
4
a)
b)
How many triangles occur for term number 4?
If x is the term number and y is the number of
triangles, draw a graph showing the pattern in the
table.
a)
Find the coordinates of all points on the y-axis and 3
units below the x-axis.
Find the coordinates of all points 4 units from the
x-axis and 1 unit right of the y-axis.
PQRS is a square with P = 2,1 , Q = 2,7 , and
R = 8,7. Find S.
If PQRS is reflected in the x-axis to get square
P1Q1R1S1, find the coordinates of P1,Q1,R1 and S1.
b)
c)
d)
Algebra, Junior High Workshop Series
Algebra Starters /63
(2005)
Pre-Algebra Patterns 5
Vocabulary
 generate
Notes
Answers

1.
a)
b)
21 triangles
The number of triangles is 1 more
than 2 times the term number.
2.
a)
b)
18 small squares
n 2  2 where n is the term number.
3.
Possible Answer:
For #3, this is exactly the
same question as #3 on
8I-4.
 For #3, have students
discuss the patterns they
found to justify their
picture for term 4.
Some possible patterns:
 Filled in Boxes:
9, 21, 33, …
 Size of figures: 3 x 3;
5 x 5; 7 x 7; so the
next one is 9 x 9. All
figures have the
perimeter and the
diagonals filled in.
 Filled in Boxes: 32 – 02;
52 – 22; 72 – 42; so the
next one is 92 – 62.
 For #4, have dice
available for students to
confirm that the sum of
the opposite sides is
always 7.
64/ Algebra Starters
(2005)
33 + 12 = 45
squares should
be shaded. All
squares on the
diagonal of the
large square are
shaded.
4.
a)
b)
34 ( 6 + 7 + 7 + 7 + 7)
71 (1 + 14 + 14 + 14 + 14 + 14)
Algebra, Junior High Workshop Series
Pre-Algebra Patterns 6
1. The number of triangles in a pattern is shown in the
following chart:
Term Number
Number of Triangles
a)
b)
1
3
2
5
3
7
4
9
Find the number of triangles for term number 10.
Give a rule describing the number of triangles for any
term.
2. The number of small squares is increasing in the following
pattern:
Term 1
a)
b)
Term 2
Term 3
How many small squares are needed to generate
Term 4?
Give an expression describing the number of small
squares for any term.
3. Build or draw the fourth term in the following sequence.
Justify your answer.
#1
#2
#3
4. Five dice are stacked on top of each other.
The number on the very top is 1.
a)
b)
What is the sum of the hidden numbers?
What is the sum of all the numbers showing?
Algebra, Junior High Workshop Series
Algebra Starters /65
(2005)
Pre-Algebra Patterns 6
Vocabulary
 quadrant
Notes
Answers

1.
For #2, the algebraic
expression is 4(n + 1)
where n is the term
number.
a)
b)
c)
t–5
c + 12 where c is the number of
candies before you add the dozen.
n
1
 10 or n  10 where n is your
2
2
age
2.
Term
Number
Number of
Small
Squares
3. b)
1
2
3
8
12
16
… 8
36
Number of
Triangles
y
1
1
x
Term Number
3.
a)
b)
y = 4x – 3
See sketch on the left.
4.
a)
b)
c)
(0,3) and (0,-3)
(1,4); (1, -4); (-1,4); (-1, -4)
y
Q 6 R
6
P 6 S
d)
66/ Algebra Starters
(2005)
x
R = (8,7) and
S = (8,1)
P2 = (-2, -1); Q2 = (-2, -7);
R2 = (-8, -7) and S2 = (-8,-1)
Algebra, Junior High Workshop Series
Pre-Algebra Patterns 7
1.
Write a mathematical expression for each of the following:
a) a number, t, decreased by 5.
b) the number of candies in a bowl after you add a
dozen.
c) your brother’s age if he is ten years older than half
your age.
2.
Complete the following chart using the rule “the number of
small squares is the product of 4 and the sum of the term
number and 1”.
Term Number
Number of Small
Squares
3.
2
…
3
8
The following chart reveals the number of triangles in an
increasing pattern.
Term Number
Number of Triangles
a)
b)
4.
1
a)
b)
c)
d)
1
1
2
5
3
9
4
13
If x is the term number, write a mathematical
expression for the number of triangles.
If x is the term number and y is the number of
triangles, draw a graph showing the pattern in the
chart.
Find the coordinates for all points on the y-axis and
3 units from the x-axis.
Find the coordinates for all points 4 units from the
x-axis and 1 unit from the y-axis.
PQRS is a square with P = 2,1 and Q = 2,7 . Find
R and S if they are in the same quadrant as P and Q.
If PQRS is reflected in the x-axis and the new square
is then reflected in the y-axis to get square P2Q2R2S2,
find the coordinates of P2, Q2, R2 and S2.
Algebra, Junior High Workshop Series
Algebra Starters /67
(2005)
68/ Algebra Starters
(2005)
Algebra, Junior High Workshop Series