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Solving Equations With Tape Diagrams
In this podcast we’re going to explore how to represent and solve equations with
tape diagrams. A tape diagram is a drawing that looks like a segment of tape and is
used to illustrate number relationships. It is also commonly known as a strip
diagram, bar model, fraction strip, or length model. The problems we’re going to
explore most directly tie to standard 6EEB7. First, lets begin with a problem. Justin
had 50 baseball cards. He gave his best friend half of the cards and then gave 10 to
his brother. How many cards does he have left? You may be surprised to learn that
this problem is similar to problems students are exposed to as early as common
core grade 3. Now students will not have done algebra representations with
variables at that point, but they do write number sentences and they do represent
with visual models including tape or strip diagrams. So let’s solve this problem with
problem solving. So we would first want to take our tape diagram or strip diagram
and divide it into two pieces because he’s going to take his baseball cards and give
half to his friend. So the entire tape diagram represents 50 cards, and then he’s
going to keep half and give half to his friend. Among the half he keeps, he then gives
10 to his brother. So here we have our representations. So the far left is the half he
gave to his friend then the 10 he gave to his brother and then this would represent
how much he has remaining. So if half of the 50 would be 25, he then gives 10 to his
friend, and we then could see we could subtract 50-25 then subtract 10 to get 15
cards left, so Justin would have 15 cards remaining. So, again, students have had
exposure to problems like this for the past 3 years. Let’s look at how we can then
translate that to equations with variables and representations with variables. So
let’s start with some one step equations. Let’s think about our tape diagram and
how we would represent x+5=12. Well first, we would want to take our tape
diagram and split it into 2 parts; and in this problem we want to think what the
entire strip or piece of tape would represent, and in this case it would represent our
total of 12. So then the other 2 component parts would be x+5, so this would be a
representation for x+5=12. What happens if we look at x-5=12? Well we still would
want two parts. 12 is no longer our total, in fact, x is now our total, and then we are
breaking that up into 5 and 12. Let’s look at a multiplication example, 3x=12. Well
we have our tape diagram, we know that 3 segments of x equals 12, so we want 3
components here of equal length and then that means our total would be 12, so each
of these sections would represent a section of x. Division is very similar. Again,
we’re going to divide that into 3 pieces. This time x is our total because we’re
dividing x into 3 pieces and each of those pieces reps 12 units so that’s how you can
simply represent our one-step equations using tape diagrams. Let’s look at some
actual problems that are tagged for 6EEB7. So the task we have here: Sierra walks
her dog, Pepper, twice a day. Her evening walk is twice as far as her morning walk.
At the end of the week she tells her mom, “I walked Pepper for 28 miles this week!”
How long is her morning walk? This was adapted from an illustrated math task; you
can see at the bottom is the direct link for that task. I’m going to show you some
representations that may be used. There are a lot of strategies for solving this
problem; we’re going to look at just a few. Again, this is one is specifically tagged for
6EEB7. So a student may choose to divide this up into 7 days for the 7-day week
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product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License
and then each day representing the morning walk and the evening walk knowing
that the evening walk is twice as long as the morning walk. So they could see the
blue represents the one day and they may say, “Oh if there are 7 days of 28 miles
then that means one day would represent 4 miles. Here is a separate similar
representation that may be used. Again, showing the 7 days, showing the morning
and evening walks, or they could choose to do this. They could say, “Well each day
they have to walk four miles so let me just focus in on one day with the morning
walk and two equal parts to represent the evening walk. So let’s say we isolate
down to just the one day and the four miles they could also represent with a tape
diagram here so doing a little bit of arithmetic to realize that for each day she’s
walking a total of 4 miles. Now we have a tape diagram to represent here’s the
morning walk. The evening walk is twice as long, so we have 3 equal parts 2 of
which represents evening one represents the morning, 4 miles total. So now we can
start to assign a variable x to each of these equal segments and then to write an
equation we can simply write that 3x=4 and then when you solve that you know x
equals one and a third. So that means Sierra walks one and a third miles for her
morning walk. So summing up, tape diagrams provide a nice visual model for
representing and solving problems. They’re wonderful because they provide
flexible options for solving some problems and different representations can be
accepted as long as it’s a correct representation of the problem, and it also helps to
build a students conceptual understanding of what is happening with the variables
and the quantities and the problem. So thank you for watching and I hope you enjoy
the tape diagrams.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this
product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License