In this video we’re going to summarize
the Math One Lesson title Cafeteria Actions and Reactions. This is a develop
understanding task, the purpose of this lesson is for students to develop
strategies for creating solving and explaining each step when solving an
equation. Elvira is interested in collecting data about how many students
use each of the tables during each lunch period, she has recorded some data on
post-it notes to analyze later. Here are the notes that she’s recorded. Elvira is
wondering how many students were sitting at the front table when she wrote her
first note, unfortunately she’s not sure what order the middle three post-it
notes were recorded in since they got stuck together in random order. She’s
wondering if it matters. We need to determine how many students were
originally sitting at the front table. I want to approach this problem first with
a diagram and then the video will take you through how you could set up an
equation but let’s just start with a diagram to make sure we really
understand what this problem is asking, and how to model it. We start at the end
because this is the only place where we know exactly how many students were
sitting at the table . So let’s start with a diagram. So here I have the lunch table
and then I put 12 students around it so I’m going to write down that there are
12 students at this table, and let’s look at the sticky note what happened right
before that. The students at the front table separated into three equal sized
groups and then two groups left early leaving only 1/3 of the students at the
table. That means that a lot of students left and the remaining were 12 so
two-thirds left one-third stayed so that means we need three groups of this 12 to
be sitting at the table. So we have a group of 12 and then another group of 12
and then the third group of 12 so that’s a total of 36. All right let’s look to
see what happened right before that so at this point in time there were 36.
So this says 4 more students have just taken seats with the students at the
front table so if we are working backwards that means that there would be 4 less so that
means there are 32 total of 32 students. Each of the students at the
front table has been joined by a friend doubling the number of students at the
table, all right so then that would be 16 because if it’s doubling that means
we’ll just have to divide it into two so every other student wasn’t sitting there.
So these are the students that join and so our answer to this question would be
16. So let’s just go back through and make sure that this makes sense when
we’re doing this forward since we were having to undo everything to work
backwards. So some students were sitting at the front table said this there’s 16,
each of the students the front table has been joined by a friend doubling the
number of students at the table which means there would be 32 because
everybody had a friend join them. Then after that 4 more students have taken
seat so we’re adding 4, so that would be this 4 right here that have now just sat
down. Alright now that we have a total of 36 says the students at the front table
separated into 3 equal sized groups so there’s our three equal groups, and then
two of the groups left, leaving one-third of the students at the table, and that means
that there are total of 12 students there at the end of lunch. All right so
let’s look to see how we can represent this algebraically with an equation and
symbols. The first thing we’re going to want to do is define our variable. So X
needs to represent the students that were originally sitting at the front
table, so students originally sitting at the front table that is what X
represents, so a number of students. Alright whenever I want to write my equation, I’m
going to start at the beginning. So some students we’re sitting at the front
table, that is going to be our X. Each of the students at the front table has been
joined by a friend doubling the number of students at the table, so if they’re
doubling that’s multiplying by two, I have 2x, then for more students have
just taken seats at the front table so we’re adding 4 so we have 2x plus 4
and then we had to divide that by 3 because 1/3 is remaining,
since 2/3 left 1/3 is remaining so we want 1/3 of the number of students that
we have so that is going to be 2x plus 4, what we had before, divided by 3, and that
equals 12. Let’s take a look at how to solve this algebraically, when we first
found the answer we had to work backwards and that’s the exact same
thing that we do when we solve an equation we work backwards. So the first
thing when we were working backwards is we had to multiply by 3, so we’ll do the
same thing we’ll multiply by 3 on both sides of the equation. So that gave us
our 36 students at the table. After that we had to subtract 4 and that’s what we will do here, subtract 4 on both sides of the equation The next thing we
did was we divided by 2 which again we’ll do here divide by 2 and our
answer was 16. The only thing that I did here is I rearranged these two sticky
notes, so let’s do this problem the same way we did the last one let’s start with
a diagram and then we’ll set up our equation. Again as the lunch period ends
there are 12 students seated at the front table, before that they had split
into three equal groups so we have a total of 36 still right here, let me record
how many that we have, there were 12 here and 36 here. Now the difference in this
problem the last one is that this is when the students have doubled so they
doubled it to get to 36 so we have to divide by 2 to get the amount that we
had before, a total of 18, and then 4 more students so we have to take those 4 off,
subtract those 4 out and we have 14. So as you can see 14 is different than the
16 we got with a different order, so does it matter which order this the
notes were recorded in, yes it matters which order the notes were recorded in.
We rearranged them and we got a different answer.
So let’s take a quick look at setting up the equation, so again we have to
identify what our variables are going to represent, so define what the variables
representing so X is going to represent how many students were originally
sitting at the front table. So X is going to be right here. Now this time we added 4 first as we went from 14 to 18, so we have X plus 4.
Now everybody has a friend joining them doubling the amount so 2 times the
X plus 4, and after that we divided by 3 and we had 12. So when we solve this
equation we started with multiplying by 3, so 2 times the quantity of X plus 4
equals 36, and then divide by 2 to get the 18, and then subtract 4 subtracting 4 on
both sides of the equation to get 14. So here are 3 different
equations that could be written based on a particular sequence of notes, examine
each equation and list the order of the five notes that is represented by each
equation, find the solution for each equation. So this right here should look
familiar and this last one right here should look familiar because that was
the first example the original way that the question was ordered. So let’s take a
quick look at this one that we haven’t looked at yet, when trying to decide what
the order is if you start with the X and then look to see what was done to the X
and kind of work your way out, so here some students are sitting at the front
of the table and then they were separated into their three groups
leaving 1/3 of the students at the tables its divided by three first, and
then four students sat down and then everybody was joined by a friend, and
then at the end of the lunch period there were 12 remaining. So to solve this
equation I want to again work backwards, so I start with what happened last, so
this doubling so my first step is going to be 2 divide by 2 on both sides
of my equation, I’m going to represent it like this. That leaves me over here with
X divided by 3 plus 4 equals 6. My next step is going to be to subtract 4 on
both sides of my equation so I have X divided by 3 equals 2. And then my last
step here I’m going to multiply by 3 let me use parentheses and put a 3 for
multiplication, put the parentheses around this one and put a 3
for multiplication that leaves me with X over here on the left and 6 2 times 3
being 6 so there are 6 students at the front of the table. A quick summary
remember whenever you’re working with variables you need to define your
variable, never lose track of what X represents it will help you through the
solving process. Also when you’re setting up your equation remember we were
working forwards when we were setting up the equation here, so X divided by 3 the
first thing that happened was the dividing by 3, and then the plus 4, and
then the multiply by 2 to get to the answer. However when we were solving it
we had to work backwards to undo each of these steps and use the opposite
operations. Alright thank you for watching