Multiplying/Factoring 2
Tic Tac Times
I.
Content:
Tic Tac Times is an algebra game that combines mathematical skills with a competitive strategy. It is designed to be a
highly motivational skill practicing exercise that involves the problem-solving strategy of working backward. In the
previous lesson students have been introduced to the concept of multiplying monomials and binomials. This game will
give them A LOT more skill practice with those concepts, without it seeming like plain skills practice. Of course, this is
only the first day of the game.
After playing and enjoying it, students will create a cheat sheet, beginning to find which factors match which answers.
Then, from there, we will look for patterns in both the factors and the answers – for example, some answers only have
two terms while some have three – Why is that? What do their factors look like? Students will be discovering the ideas
of monomials, binomials and polynomials on their own as well as quadratic terms, linear terms and constants.
The fun doesn’t stop there though. The students will be looking for more patterns. For example, some of the answers
have two addition signs between the terms while some have two subtraction signs and some have one of both – why is
that? What do their factors look like? With this students will be discovering the ideas of factoring on their own. For
example, if both of the signs in the answer are addition signs, both of the factors must have addition signs between their
terms.
Great fun, great practice, great discussion, great discovery! All of this from just a little game of Tic Tac Times.
II.
Learning Goal(s):
Students will know and be able to:
- Identify and define the difference between a monomial, binomial and polynomial
- Multiply monomials and binomials
- Correct their own mistakes in multiplying monomials and binomials as they play
- Match two factors to their binomial/polynomial answer
- Identify and define the quadratic, linear and constant terms in a monomial, binomial and polynomial
- See patterns between the factors and the binomial/polynomial answer, such as:
o When multiplying two binomials they add to the middle (linear) term and multiply to the last (constant)
term
o If there are two addition signs in the polynomial answer, the factors must have addition signs between
both their terms
o If there are two subtraction signs in the polynomial answer, the factors must have two different signs
between their terms and the greater will be negative
o If there is an addition sign first and a subtraction sign second in the polynomial answer, the factors must
have two different signs between their terms and the greater will be positive
o If there is a subtraction sign first and an addition sign second in the polynomial answer, the factors must
have subtraction signs between both their terms
III.
Rationale:
Tic Tac Times is just what I said in my content section – great fun, great practice, great discussion and great discovery!
First and foremost, it gives the kids additional practice multiplying monomials and binomials without them really
thinking it’s practice. Setting it up as a game instead of simply a worksheet makes it more fun and they get more out of
it. After simply the playing of the game and all that practice, we are able to pull patterns from it, introduce new
vocabulary and set the stage for factoring. The students need to learn the rules for factoring, however, instead of me
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telling them that: if there are two addition signs in the polynomial answer, the factors must have addition signs between
both their terms; they will discover it on their own. I will be introducing the vocabulary though. Some of it they may
have heard already, such as, constant term, linear term and maybe even monomial (but I doubt it). These vocabulary
terms are imperative moving forward though so introducing them now is setting them up for the future.
IV.
Assessment:
During this lesson, assessment is going to happen in a variety of ways since there are multiple days and multiple parts.
The lesson will begin with a POD that focuses on multiplying binomials. It will be an easy way to assess where the
students are when we begin. Then, during the game, students will learn very quickly whether or not they actually know
how to multiply monomials and binomials. As I am circulating, it will also become very apparent to me who is struggling
and who is excelling. The next part of the lesson, during day two, involves creating a cheat sheet. This will be done in
partners but each partnership will be responsible for sharing out certain answers. If a partnership can come up with
their assigned answers and present them to the class, both they and I can see how they are doing with the material. The
finding patterns part also focuses on groups sharing out what they find. Lastly, after our whole class discussion on
patterns, I will give the students an exit slip. The exit slip will have both algebraic problems to do out and also questions
they need to respond to (3 things you learned from Tic Tac Times, 2 things you liked/didn’t like – if you didn’t like
something you need to explain how it can be improved, 1 question you still have about the material).
V.
Personalization:
During this lesson students will be grouped homogeneously. Due to the competitive nature of the game I want to make
sure that students are adequately challenged but also comfortable and somewhat evenly matched. As we move onto
the cheat sheet part and finding patterns, the students will stay in their same groups but there will be lots of sharing out
and discussing. It is easily accessible. Those parts focus on “what do you notice”. Any group or student can answer that
regardless of ability level. I have actually found that some of my students that tend to struggle more are grasping these
concepts and feeling confident in their ability with them.
As with the previous lesson, all types of problem solving methods are encouraged. For the actual game of Tic Tac Times
the algebra tiles are required as game pieces, therefore, the students have to have them out on their desks within reach
of being used if needed. This way, any student that wants the tiles will already have them there and will not have to ask
or go up and get them. I know that a lot of the students that would prefer to use the tiles would not ask for them or go
up and get them. In addition, I will also provide scrap paper so that the students can work out their answers using the
box method or double distribution if they choose. All methods are encouraged. The game, however, also lets students
find and correct their own mistakes in what they have been making.
Some students will be confused by the directions of the game. To prevent this, we will go over the instructions as a class
before beginning play. I know the instruction sheet is very wordy and some students are intimidated by that. During
game play students will have a scrap sheet of paper that they are required to turn into me at the end of class, showing
that they were doing something during the period, this prevents students from slacking off and only pretending to play.
There will be a time limit on finding patterns in the pairs before we share out. I do not want some groups to be sitting
around waiting and I do not want other groups to be stuck looking for patterns endlessly. Groups will share out patterns
they find, teaching other students about them. During the time when students are looking for patterns in their pairs, I
will be circulating and if groups are stuck asking questions like “Well, why does this answer only have two terms while
this one has three?” and “Why does this answer have two addition signs while this one has two subtraction signs?” to
get them going. However, I think most groups will have no trouble noticing patterns on their own, they’ve proven to be
good at those assignments. Lastly, I know the vocabulary is a lot for them. But this will only be the introduction to it, we
will have a lot more practice with it in the coming lessons (notes, crosswords, concept maps, etc).
VI.
Activity description and agenda:
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Grouping: homogenous pairs to encourage competition and challenge the students
Materials: Game boards/directions for each pair, algebra tiles
Day 1
0-7
7-13
Students enter the class and sit with
their assigned partners. They begin to
work on the POD that is on the board.
They write their answers and show
any work on the index card provided.
They may use the algebra tiles, box
method or double distribution to
solve the problems.
Students will raise their hands and
volunteer awesome things about the
example on the board. If a student
volunteers an answer without raising
his/her hand I will ignore it. Students
need to get in the habit of raising their
hands. They will pick out all of the
awesome things about the problem.
After we have picked all the awesome
things, they will identify the mistake.
13-20
Students will volunteer as I ask to read
each section of instructions. They will
ask any clarifying questions if need be.
20-55
Students will play the game in their
pairs. They will do out any work they
need to on their scrap piece of paper.
They will utilize the algebra tiles if
they need to. If someone wins, they
play again.
55-60
Clean up! Each pair needs to turn in
their game board and work for the
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(x+2)(x+1)
(x-1)(x+2)
(x-1)(x-2)
The POD will be projected on the
board. I will hand out the algebra tiles
for students to use. When students
start to turn in their index cards I will
sort them into yes/no piles based on
the answer to the last question.
“My favorite no” – I will explain to
students that I have sorted their
answers only looking at the last
question and picked my favorite
WRONG answer. But before we talk
about why it’s wrong, we’re going to
pick out all the awesome things about
it – because it does have a lot of
awesome things, that’s why I picked
it! I will open it up for students to pick
the awesome things…
I will capitalize on the fact that the
student SHOWED work! YES! So even
though this person made one tiny
mistake, they showed the work for us
to find it! Which means that in the
future, I will be able to tell where they
went wrong and address it as well as
give them partial credit.
I will pass out the game boards. We
will go over the instructions. I will ask
for volunteers to read each section.
We will stop after each section to
summarize and I will demonstrate on
the ELMO with my own game board
and algebra tiles what it should look
like.
I will be circulating around the room
to see how students are doing. At the
very beginning I will make sure that
each group has started and is playing
before answering any other questions.
Once each group is started, I do not
foresee many more questions.
I will stop when there is about 5
minutes left to the end of class. I will
Day 2
0-20
25-30
30-40
40-55
55-60
Day 3
0-7
7-13
day. All of the algebra tiles need to be
put back into their backs and returned
to the box.
instruct students to put their algebra
tiles away. They should have 20 little
squares, 8 rectangles and 4 big
squares. They will hand in their work
together and their game board.
Students will play the game in their
groups to get more practice. Some of
them will hopefully begin to develop
strategies as they see which factors
lead to which answers.
Students will listen as I explain the
instructions.
I will circulate around the room as
students are playing answering any
questions, directing students back on
task if need be.
Students will look for the factors that
match the answers on their game
board. They will write them under the
polynomials.
Each group will present out on their
findings. They will show which factors
they got for each answer and any
methods they found for finding the
factors.
Clean up! Students will put away the
algebra tiles and make sure their
names are on their game boards and
work for the day before passing them
in.
Students enter the class and sit with
their assigned partners. They begin to
work on the POD that is on the board.
They write their answers and show
any work on the index card provided.
They may use the algebra tiles (to
some extent, they’re going to run
out), box method or double
distribution to solve the problems.
Students will raise their hands and
volunteer awesome things about the
example on the board. If a student
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I will call everyone back together and
explain what we are moving on to. We
are going to make a cheat sheet to
playing. This means under each
answer on the board, you are going to
fill in the factors you need to get it.
I will assign different groups different
rows of the board to figure out.
I will circulate around the room as
students are doing this. Some groups
may need more help than others
finding those factors.
I will call groups up to present but this
part is them presenting their findings.
As per usual, students should not be
talking while other groups are
presenting. This rule will need to be
enforced; they are still not doing it.
After each group has presented, I will
commence the clean up process.
Again, there should be 20 small
squares, 8 rectangles and 4 big
squares in the algebra tile bags.
(x+10)(x+8)
(x-10)(x+8)
(x-10)(x-8)
The POD will be projected on the
board. I will hand out the algebra tiles
for students to use. When students
start to turn in their index cards I will
sort them into yes/no piles based on
the answer to the last question.
“My favorite no” – I will explain to
students that I have sorted their
answers only looking at the last
volunteers an answer without raising
his/her hand I will ignore it. Students
need to get in the habit of raising their
hands. They will pick out all of the
awesome things about the problem.
After we have picked all the awesome
things, they will identify the mistake.
13-15
15-18
18-25
25-50
Students will listen as I give
instructions.
Students will come up with as many
patterns as they can in 3 minutes.
Each group will share a pattern they
noticed.
Students will look for the patterns I’m
asking for if they haven’t already
noticed them on their own.
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question and picked my favorite
WRONG answer. But before we talk
about why it’s wrong, we’re going to
pick out all the awesome things about
it – because it does have a lot of
awesome things, that’s why I picked
it! I will open it up for students to pick
the awesome things…
I will capitalize on the fact that the
student SHOWED work! YES! So even
though this person made one tiny
mistake, they showed the work for us
to find it! Which means that in the
future, I will be able to tell where they
went wrong and address it as well as
give them partial credit.
Looking back at our game boards for
Tic Tac Times. We’re going to look for
patterns now – both in our factors and
answers and them together. These
patterns can be different ways to
categorize your game board answers
or things you notice about certain
factors combined together, etc.
Students have 3 minutes to come up
with as many patterns as they can.
Anyone who comes up with more
patterns than I can write down in that
amount of time, gets a point for the
day. Any pair that comes up with a
cool pattern I haven’t noticed, gets a
point for the day. At the end of 3
minutes we will each be sharing a
pattern so you better have enough
that you can share even if other
groups have stolen some of yours.
I will be writing down my own
patterns.
I will be calling on groups. I will share
my own pattern. As groups are
sharing out I will be capitalizing on
vocabulary – for example, two terms =
binomial, one term = monomial, 3
terms = trinomial
We will get into discussing factoring.
How do you tell if the factors are
positive, negative, etc? Look at the
signs in the answer. What patterns do
we notice?
50-60
VII.
Students will complete the exit slip
individually.
Exit slip:
(x-3)(x+4)
x2 -5x + 6 = (______)(_______)
3 things you learned from Tic Tac
Times
2 things you liked/didn’t like –
remember, if you didn’t like it you
need to give a way it could’ve been
improved
1 (or more) question you still have
about the material
List the Massachusetts Learning Standards this lesson addresses.
A.SSE.1A - Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such
as terms, factors, and coefficients.
A.SSE.2 - Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus
recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
A.SSE.3 – Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity
represented by the expression.
A.SSE.3A - Factor a quadratic expression to reveal the zeros of the function it defines.
A.APR.1 - Perform arithmetic operations on polynomials. Understand that polynomials form a system analogous to the
integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and
multiply polynomials.
A.REI.4B - Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the
quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic
formula gives complex solutions1 and write them as a ± bi for real numbers a and b.
VIII.
Resources:
http://www.pleacher.com/mp/mlessons/algebra/tictac.html
IX.
Reflection
As far as Day 1, it was awesome for the first 55 minutes. Things went exactly as I had planned. The kids got really into
the “My Favorite No”. I actually picked the same mistake in both classes. When multiplying a negative times a negative,
Amber and Kenisha had left it as a negative when it should have been positive. But they both solved it using the box
method and showed all their work so they were great examples to use. I think it really got the point across. It’s not all
about the answer, sometimes it’s about showing work too. AND everyone makes mistakes. I harped that point
immensely. They all know that every time I make a scavenger hunt or something to that effect that they all get extra
points for finding my mistakes… and there always are mistakes. Commence Lilly singing Hannah Montana, which I totally
approve of in this case (because “everybody makes mistakes... nobody’s perfect!”). Moving on, the kids played the game
and got the practice I wanted them to. In 9A, Josh, Nate and Juan had not completed the previous worksheet and
demonstrated no knowledge of the concepts so I had pulled them aside to be their own group to finish said worksheet.
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They did much better during this class. They asked for help when they needed it. I feel like I spent most of the class
helping them but by the end they really seemed to get it so it was worth it. However, when it was time to wrap up,
chaos ensued, unacceptable. As a result, after everything was collected and put away we had a nice chat about their
behavior. I told them that they were awesome for the first 55 minutes of class but that as soon as I said “Okay, we’re
about done for today. I need everyone to …” they checked out and chaos ensued. Just because I said class was almost
over, does not mean class is over. Until I dismiss them from the room they need to act like the students I know they are
in my classroom. I even went as far as reminding them that while I had been treating them like mature high schoolers, I
could go back to treating them like middle schoolers at any point if I needed to. This meant that if they did have an awful
clean up like they just did, I could make them do it again. Or if they were talking over me as I was trying to give
instructions, the entire class could go get their lunches and come up and eat with me in silence. If I need to start making
those changes because that’s what their behavior is showing me then I told them I would. Of course, I would really like
not to have to do those things. But it is coming to that point. I am a broken record. I keep reinforcing the idea that when
someone else is talking you are not talking, you are listening, whether it is me or a peer. There are other ones but that
one has been most prominent lately. They all loved my analogy of a broken record but hopefully they also got the idea
behind it. It’s beginning to annoy me that I have to say it so much but I will continue to say it and enforce it until they get
it and do it.
Unfortunately, Day 2 of this LAP, I did not actually teach myself. I was incredibly sick so I left lesson plans for Shannon to
do with the kids. Of course, just to be safe I left her with more than they would have gotten through. But I’d rather be
over planned than under planned. They got through the plans for Day 2. She said a lot of the students were still
struggling with the idea that the factors add to get the linear term and multiply to get the constant at the end. That is
something that will need to be solidified during the next day. In addition, she said she assigned a row of the game board
to Izzy and Camilo and when it was their turn to present they had nothing to show for it. She had the rest of the groups
present and came back to them, still nothing. Finally they pulled it together and did a good job presenting; they just
really needed to be pushed. She also gave them much less time to find the factors for the corresponding answers; I had
originally planned for 10 minutes but she had them do it in 3. I guess I do need to cut down on time limits that I give
them sometimes. However, something like this I would have thought would have taken them longer. If I were them, it
would have taken me longer. I think. It’s hard to think about myself without all the knowledge I currently have.
Regardless, they can do things in less time than I currently give them credit for. It’s something to keep in mind and adapt
for in the future.
The third and final day went completely differently in the two classes. In 9A, they came up with awesome patterns with
little to no guidance. Each pair presented one and then there were even more after that. We barely got to the
vocabulary I wanted to cover. However, we had a lot of really good discussions. Also, they did awesome on the exit slip
in their POD books. There were a few questions about factoring but other than that they did awesome on the
multiplying binomials and made good attempts at factoring. After class, Shannon recommended that I give 9B a bit more
structure when looking for patterns. In the past she said that she has drawn on her own cheat sheet with different colors
and asked the students to find the patterns associated with the colors. I decided to try it. It did seem to make things
more accessible for them. However, it also mean that they were more focused on finding the pattern for each color than
finding the pattern between the factors and the answers. For example, the green color all had factors with minus signs
in them. That’s a great thing for them to notice but then how does that affect the answers. I had to prompt them there
more than I had to prompt 9A. There were pros and cons to doing it both ways. In addition, in 9B we got to a lot more
vocabulary and they came up with some awesome connections for monomial, binomial, trinomial and polynomial. We
talked about other words they knew in their lives with those prefixes and Steven, before I even asked, was the first to
point out that monomial sounded like monochrome which meant one color so mono must mean one (SO HAPPY!). It
was a good segway into the notes tomorrow. I introduced the terms briefly today and we’ll solidify them along with the
rules to multiplying binomials in the next lesson.
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