Tying the Knot Concept Lesson
2009-2010
Unit 2: Algebra 1
LESSON OVERVIEW
Overview: Students work in small groups tying knots in a length of rope. They measure the length of the knotted rope after each new
knot and represent this information in a table, a graph, a written description, and an algebraic formula. They draw a line of best fit
through their data points and determine the slope and y-intercept of this line, and interpret the physical meaning of this slope and yintercept.
CA Standards Addressed:
6.0 Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the
region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).
7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the
point-slope formula.
Mathematical Goals of the Lesson: Develop and analyze a linear function to approximate and describe the real-world relationship
between the length of a rope and the number of knots tied in the rope.
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Funded by the James Irvine Foundation
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Access Strategies: Throughout the document you will see icons calling out use of the access strategies for English Learners, Standard English Learners, and
Students With Disabilities.
Access Strategy
Cooperative and Communal Learning
Environments
Icon
Instructional Conversations
Academic Language Development
Advanced Graphic Organizers
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Description
Supportive learning environments that motivate students to engage
more with learning and that promote language acquisition through
meaningful interactions and positive learning experiences to achieve
an instructional goal. Working collaboratively in small groups, students
learn faster and more efficiently, have greater retention of concepts,
and feel positive about their learning.
Discussion-based lessons carried out with the assistance of more
competent others who help students arrive at a deeper understanding
of academic content. ICs provide opportunities for students to use
language in interactions that promote analysis, reflection, and critical
thinking. These classroom interactions create opportunities for
students’ conceptual and linguistic development by making
connections between academic content, students’ prior knowledge,
and cultural experiences
The teaching of specialized language, vocabulary, grammar,
structures, patterns, and features that occur with high frequency in
academic texts and discourse. ALD builds on the conceptual
knowledge and vocabulary students bring from their home and
community environments. Academic language proficiency is a
prerequisite skill that aids comprehension and prepares students to
effectively communicate in different academic areas.
Visual tools and representations of information that show the structure
of concepts and the relationships between ideas to support critical
thinking processes. Their effective use promotes active learning that
helps students construct knowledge, organize thinking, visualize
abstract concepts, and gain a clearer understanding of instructional
material.
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Academic Language Goals of the Lesson:
Assumption of Prior Knowledge:
Graph points on a coordinate plane
Academic Language:
Slope
y-intercept
x-intercept
linear
table
graph
algebraic formula
Materials:
Concept Task
Four Corners Representation Sheet
(Table, Graph, Verbal, Symbolic)
Rope: 1 piece for each small group (about
3 to 4 feet in length). There should be
ropes of at least two different thicknesses.
Tape measures
Connections to the LAUSD Algebra 1, Unit 2, Instructional Guide
Understand, Analyze, and Graph
Linear Equations
6.0, 7.0, 8.0
•
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Evaluate the slope of a line
Use rate of change to solve
problems
• Write and graph different forms of
linear functions
• Identify characteristics of parallel
and perpendicular lines
Derive the equation of a line:
• Given the slope and a point on a line
• Given 2 points on the line
• Parallel or perpendicular to a given
line through a given point
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Key:
Suggested teacher questions are shown in bold print.
Questions and strategies that support ELLs are underlined identified by an asterisk*.
Possible student responses are shown in italics
Phase
Students work in groups as they
investigate
Students create and compare graph
that model the problem
SET UP PHASE: Setting Up the Mathematical Task—Part 1
INTRODUCING THE TASK
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The students should be working in small groups of three for this activity. (One student to tie the knots, one to measure the
resulting length of the rope, and a third to record the data in a table)
After placing the materials at each table and grouping the students, have a student read the first paragraph. Invite other
students to tell you what they feel is important information contained in the paragraph. Have another student restate the
paragraph in their own words to ensure that the whole class is clear about the relationship they are about to investigate.
Continue in this manner, reading paragraphs a) and b), making sure that the whole class is clear about how they are to
initially undertake this task. Tell the groups that one student will tie the knots, one will measure the rope, and the other will
record the data in the table. Have them decide who will take on each of these tasks and then have them begin.
Clarify any confusions students may have but do not suggest specific values for their investigation.
Note: For the Share, Discuss, and Analyze phase, it will be helpful to compare graphs from different groups. To that end
you could either: have students graph their results on a transparency (Using the same scale and axes) – if access to an
overhead projector is available, OR have a representative from each selected group copy their graph onto a single large
“class graph” at the front of the room.
To assist ELLs’ participation in the class discussion*:
• Allow time for students to first talk in small groups (pairs) and then have the groups report to the whole class.*
• Reinforce appropriate language as students communicate their ideas (e.g. re-voice a student’s contribution in
complete, grammatically correct language). Ask students if you have captured what they said*.
• Create work groups that are heterogeneous according to language proficiency*
• Model appropriate mathematical language, emphasizing vocabulary used in appropriate context.*
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Students create multiple
representations of a function
Students explain their reasoning
Phase
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EXPLORE PHASE: Supporting Students’ Exploration of the Task
STRUCTURE
SMALL GROUP WORK (Gathering Data – parts (a) and (b))
Students work in their small groups to make measurements of the rope as they tie an increasing number of knots in it.
As students are working, circulate around the room.
o Be persistent in asking questions related to the mathematical ideas, exploration strategies, and connections
between representations.
o Be persistent in asking students to explain their thinking and reasoning.
o Be persistent in asking students to explain, in their own words, what other students have said.
o Be persistent in asking students to use appropriate mathematical language.
What do I do if students have difficulty getting started?
Ask questions to the group such as; “What happens to the rope when you tie a knot in it?”, and “What things are changing
and which are staying the same?”, and “How could you represent this information in a table?”
What do I do if students finish early?
If students have completed making measurements then have them move to graphing their data and developing a written
description and an algebraic formula to represent the data.
SMALL GROUP WORK (Analyzing Data – parts (b) and (c) and additional questions)
Once students have made their measurements and entered these into table, the next task is to get each group to fill out
the rest of the four corners representation sheet (Table, Graph, Verbal, and Symbolic).
Have a student read part (c) and make sure the class understands they are to graph the data points and then draw the line
of best fit.
As students are working, circulate around the room.
o Be persistent in asking questions related to the mathematical ideas, exploration strategies, and connections
between representations.
o Be persistent in asking students to explain their thinking and reasoning.
o Be persistent in asking students to explain, in their own words, what other students have said.
o Be persistent in asking students to use appropriate mathematical language.
What do I do if students have difficulty getting started?
Ask questions such as “How long is the rope after you tied two knots in it? Where would you find that point on the graph?”,
and “Can you show me where you could draw a line that would go through or very near most of your points?”
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Critical Understanding
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The verbal description of this function is a precursor to being able to write the function symbolically. So you should be
looking for descriptions such as, “The rope was initially 105cm long and then for each knot we tied in the rope, it got
shorter by about 8cm”. If students’ descriptions are not as developed then ask advancing questions such as:
“What happens to the rope when you tie a knot in it?”
“What do you notice about the rope after you’ve tied lots of knots in it?”
“Do you notice any patterns in your table?”
“How long was the rope to begin with?”
Once the students have a written description ask them to analyze their line of best fit and find an equation that describes
it. If students are having difficulty doing this ask such questions as:
“When there are zero knots, what is the length of your rope?”
“When there is one knot, what is the length of your rope?”
“When there are two knots, what is the length of your rope?”
“What quantities are changing in the relationship?”
“Could you use variables to represent these quantities?”
“What does your verbal description say?”
As students are developing their algebraic equation, continue to circulate, and ask questions such as:
“How does your verbal description relate to your algebraic equation?”
“Show me where the length of the rope with no knots in it is represented in your equation.”
“Show me where the amount that the rope gets shorter each time you tie a knot is represented in your equation.”
Through teacher questioning,
students are encouraged to make
the connection between a
written/verbal description of their
function and the equivalent
algebraic equation
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Use re-voicing and questioning to
develop terms in context
EXPLORE PHASE: Supporting Students’ Exploration of the Task
STRUCTURE (continued)
Phase
Additional Questions
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What are the slope and y-intercept of this line (of best fit)?
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Interpret the meaning of the slope and y-intercept with regard to the rope and the knots.
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Predict the length of a rope with 10 knots
(Note: it is possible that the rope will not be able to have 10 knots tied in it – it may be too short, in which case this may be
the perfect opportunity to discuss that we are dealing here with a mathematical model which approximately describes
reality. So even though it is possible to put the number 10 into an equation, the output may be nonsensical depending on
the circumstances. A good question to ask should this arise is, “Why can I put 10 into your equation but not tie 10 knots in
your rope?”)
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(Extension) What is the maximum number of knots that you can tie with your piece of rope?
Students may assume that this question is answered by finding the x-intercept of the line of best fit. However, further
experimentation should reveal that the amount of rope available to tie a new knot not only decreases additionally by the
length of the knot. Hence if each time you tie a knot the rope length decreases by 5cm, if each knot is 2cm long then the
amount of rope available in which to tie a new knot has decreased by 7cm. (Questions to ask to stimulate students’
thinking about this would be, “Once I’ve tied a knot, where can I tie a new knot?” AND “If I have 5 knots tied in a rope next
to each other, how much rope do I have left in which to tie new knots?”)
MONITORING STUDENTS’ RESPONSES
• As you circulate, attend to students’ mathematical thinking and to their conjectures, in order to identify those responses
that will be shared during the Share, Discuss, and Analyze Phases.
If the ropes used by the class are of a variety of thicknesses (at least two) then their graphs will have different slopes, and
if the ropes were of different lengths to begin with, then they will have different y-intercepts. As you circulate, look for a few
groups whose work would collectively express these differences. Chose these groups’ work as the basis for the Share,
Discuss, and Analyze phase.
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Phase
Possible Solutions
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A possible solution for the table could be:
Possible Questions
Ask questions such as:
“How does your verbal description
relate to your algebraic
equation?”
Misconceptions/
Errors
Students may well
have problems
with measuring
the rope.
“Show me where the amount that
the rope gets shorter each time
you tie a knot is represented in
your equation.”
A possible solution for the graph could be:
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Explain in your own words what
_______(another student) said.*
Funded by the James Irvine Foundation
Show me how did you
get the data for your
table?
What did you measure
to get the length of the
rope after tying 1 knot?
“Show me where the length of the
rope with no knots in it is
represented in your equation.”
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Questions to Address
Misconceptions/Errors
They will also
need time to
discuss that the
points they
generate may not
be exactly in a
straight line, and
thus the line of
best fit (drawn by
eye) may not go
through all of the
points.
Are the changes in the
length of your rope the
same when you tied
each knot?
Why do you think these
might be different?
Did you round your
measurements up or
down?
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A possible solution for the verbal description
could be:
Students may well
experience
difficulty writing a
description of the
function.
How long was your rope
to start with?
What happened when
you tied one knot? Two
knots?
Do you see any patterns
in your data?
A possible solution for the algebraic formula
could be:
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How could you describe
the patterns you see?
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Sharing, Discussing, and Analyzing
Orchestrating the mathematical discussion: a possible Sequence for sharing student work, Key Questions to achieve the goals
of the lesson, and possible Student Responses that demonstrate understanding.
Revisiting the Mathematical Goals of the Lesson:
The purpose of this sharing/discussion is to make explicit the conclusions the exploration.
Phase
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Sequencing of Student
Work
Order of group
presentations:
A) Display two groups’
work that have
significantly different yintercepts but similar
slopes. Allow students a
few minutes to observe
the graphs before
asking questions.
Rationale and
Mathematical Ideas
Observing the work from
different groups who have used
ropes of different lengths and
thickness allows the students to
see how changes in the realworld have an effect on the
mathematical models (i.e., the
graphs and the equations) and
reinforces the relationship
between original rope length
and y-intercept, and rope
thickness and slope.
B) Next display two
groups’ work that have
significantly different
slopes. Allow students a
few minutes to observe
the graphs before
asking questions.
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Students discuss their conclusions
and share their thinking
Possible Questions and Student Responses
“What do these graphs tell us about the relationship
between the number of knots and the length of the rope?”
“As the number of knots increases the length of the rope gets
shorter”
“What similarities do you observe between the graphs?”
A: “They both have the same downwards slope”
B: “They both have the same y-intercept”
“What differences do you observe between the graphs?”
A: “They have different y-intercepts”
B: “One has a steeper slope than the other”
“What does the y-intercept of the graph tell us?”
“How long the rope was to begin with”
“What does the slope of the graph tell us?
“How much rope it takes to tie a knot”
“Why do the graphs have different y-intercepts?”
“Because the ropes are different lengths to begin with”
“Why do the graphs have different slopes?”
“Because the thicker rope uses more length to tie a knot”
“Where else (i.e., where in the other representations) do you
see these differences?”
“In the equation I can also see the different coefficients
representing the different slopes, and the different constants
representing the different y-intercepts”
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