Download How Much Does a Kilogram Cost?

Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Patterns, Relationships, and Algebraic Thinking
Activity:
How Much Does a Kilogram Cost?
TEKS:
(5.5) Patterns, relationships, and algebraic thinking. The student
makes generalizations based on observed patterns and relationships.
The student is expected to:
(A) describe the relationship between sets of data in graphic
organizers such as lists, tables, charts, and diagrams;
(5.6) Patterns, relationships, and algebraic thinking. The student
describes relationships mathematically.
The student is expected to select from and use diagrams and
equations such as y = 5 + 3 to represent meaningful problem
situations.
(5.3) Number, operation, and quantitative reasoning. The student
adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to:
(B) use multiplication to solve problems involving whole numbers
(no more than three digits times two digits without technology);
(5.14) Underlying processes and mathematical tools. The student
applies Grade 5 mathematics to solve problems connected to everyday
experiences and activities in and outside of school.
The student is expected to:
(A) identify the mathematics in everyday situations;
(B) solve problems that incorporate understanding the problem,
making a plan, carrying out the plan, and evaluating the solution
for reasonableness;
(C) select or develop an appropriate problem-solving plan or
strategy, including drawing a picture, looking for a pattern,
systematic guessing and checking, acting it out, making a table,
working a simpler problem, or working backwards to solve a
problem; and
(D) use tools such as real objects, manipulatives, and technology to
solve problems.
(5.15) Underlying processes and mathematical tools. The student
communicates about Grade 5 mathematics using informal language.
The student is expected to:
(A) explain and record observations using objects, words, pictures,
numbers, and technology; and
(B) relate informal language to mathematical language and
symbols.
Patterns, Relationships, and Algebraic Thinking
How Much Does a Kilogram Cost?
Grade 5
Page 1
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
(5.16) Underlying processes and mathematical tools. The student
uses logical reasoning.
The student is expected to:
(A) make generalizations from patterns or sets of examples and
nonexamples; and
(B) justify why an answer is reasonable and explain the solution
process.
Note: Portions of this lesson address TEKS at other grade levels as well;
however, the intent of the lesson fits most appropriately at the grade level
indicated.
Overview:
Students will engage in the problem-solving process to discover patterns
and to make generalizations about how many nickels and how many
pennies it takes to equal the mass of one kilogram. Students will work
with partners to choose a problem-solving strategy, determine a solution,
and then share their approach with the class.
Materials:
Math Curse by Jon Scieszka
Mr. Newton’s Problem – Handout/Transparency 1
How Much Does a Kilogram Cost? – Handout/Transparency 2
Possible Solution Strategies – Handout/Transparency 3
Mr. Newton Strikes Again – Handout/Transparency 4
Blank paper
Large chart paper or blank transparencies
Markers
Calculators (optional)
Grouping:
Introductory activity – whole group
Problem-solving process – students will have a working partner
Time:
2 class periods
Lesson:
1.
2.
Procedures
Share the book, Math Curse, by Jon
Scieszka.
Notes
You may want to share only the
introduction and a few problem
pages that exemplify the problemsolving process along with some
problem solving strategies.
Review the 4-step problem-solving process
with students.
(5.14 B) Understand the problem,
make a plan, carry out the plan,
and evaluate the solution for
reasonableness.
Patterns, Relationships, and Algebraic Thinking
How Much Does a Kilogram Cost?
Grade 5
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Mathematics TEKS Refinement 2006 – K-5
3.
Procedures
Illustrate some of the problem-solving
strategies that might have been employed by
the main character in the book.
Ask students to share additional problemsolving strategies (5.14 C).
Record and post these strategies on chart
paper, if not already posted in the room.
4.
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Notes
Sample illustrations:
p. 6: Gallons/quarts/pints – make
diagram like Gallon Guy or a table
p. 7: Bus problem – write an
equation
p. 8: Birthdays – use a chart or
list
pp. 15-16: Find a pattern, etc.
Use the last page to introduce the science
teacher, Mr. Newton, and read his statement,
“You know, you can think of almost
everything as a science experiment!”
Tell students that Mr. Newton is setting up a
science experiment, and he needs our help
with a problem. They will need to rely on
their knowledge of the 4-step problemsolving process and one or more problemsolving strategies to solve it.
5.
Share Mr. Newton’s Problem
(Handout/Transparency 1) with the class.
Read for comprehension.
A method that can be used to help
students comprehend the problem
situation is:
a. Have students read the problem
chorally. You will set the pace
and then fade out so that you can
key in on the students’ oral
reading behaviors. Make note of
any difficulties (slowing down,
voices dropping out, vocabulary,
mispronunciations, etc.).
b. Remove the problem, and ask
students to tell you what they
remember about the problem.
Restate each student’s
contribution. It is likely that
students will have only partial or
incorrect recall of the details in the
problem. Emphasize the need for
rereading any problem.
Patterns, Relationships, and Algebraic Thinking
How Much Does a Kilogram Cost?
Grade 5
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Mathematics TEKS Refinement 2006 – K-5
Procedures
Tarleton State University
Notes
c. Show the problem again, and
repeat step a.
d. Remove the problem again, and
elicit details from the students,
restating each student’s
contribution.
e. Finally, present the problem
again and clarify anything that
might have caused difficulty during
the reading of the problem.
6.
Ask guiding questions to check for
understanding and summarization of the
problem.
Have students summarize what
Mr. Newton wants to know.
(number of nickels needed to
equal the mass of one kilogram;
number of pennies needed to
equal the mass of 1 kilogram)
Ask students what information in
the problem will help them to find
out.
(nickels have a mass of 5 grams;
pennies have a mass of 2.5
grams)
At this point, it is important that
students understand that we are
not interested in the value of the
coins. We are focusing on their
mass.
7.
Use the Think/Pair/Share method to have
students determine which strategies might
be used to solve the problem.
Patterns, Relationships, and Algebraic Thinking
How Much Does a Kilogram Cost?
The “Think/Pair/ Share” method
allows all students an opportunity
to communicate. The teacher
asks the question and allows
everyone 2 minutes of quiet “think”
time. Then students get with their
working partner and share their
answers. Finally the teacher gives
several pairs the opportunity to
share their thoughts with the entire
class.
Grade 5
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Mathematics TEKS Refinement 2006 – K-5
8.
9.
10.
Procedures
Elicit the strategies that students think might
be useful.
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Notes
The following are some of the
strategies that might be used with
this problem:
• make a table
• look for a pattern
• draw a picture or diagram
• guess and check
• formulate an equation
Have students work with their partners to find For those students who arrive at a
a solution to the problem.
satisfactory solution early, have
them demonstrate or justify their
solution through a different
strategy.
Monitor students’ progress, giving as little
guidance as is necessary.
The teacher should determine if
and when to allow the use of
calculators. It may be helpful to
allow their use when students are
solving for the number of pennies.
Since this problem involves a
multiplicative relationship, and
since 5th grade TEKS do not
require students to multiply or
divide with decimals, it may be
beneficial to provide some
students with calculators once
they have exhibited a reasonable
solution for the number of nickels
needed to equal the mass of 1
kilogram.
Some students may use
proportional reasoning to
determine that twice as many
pennies as nickels would be equal
the mass of 1 kilogram since the
mass of two pennies equals the
mass of one nickel.
11.
Have students transfer their findings to a
transparency or chart paper. Each pair
should be ready to share their solutions with
the group.
Patterns, Relationships, and Algebraic Thinking
How Much Does a Kilogram Cost?
Ask pairs of students to share
their strategies and solutions with
the whole group. Have students
with similar strategies contribute
any further ideas or thoughts. Call
Grade 5
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Mathematics TEKS Refinement 2006 – K-5
Procedures
12.
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Notes
on student pairs to share until all
strategies have been exhausted.
If different incorrect solutions are
presented, allow the students to
process their own work and look
for explanations for the
discrepancies.
Discuss the interrelatedness of the various
strategies that have been presented. Allow
students to make these connections as much
as possible. See Possible Solution
Strategies (Handout/Transparency 3).
Some possible points to bring out during the
discussion are as follows:
What does the table help us to see?
A pattern
What does a pattern allow us to do?
Formulate a rule or equation
If you used guess and check, what results
match data in the table or match the
equation? Which ones do not?
Those that fell within the range of acceptable
outputs without going over
How does your drawing or diagram represent
the data in the table or the solution equation?
Answers will vary depending on the picture
or diagram.
13.
When students have satisfactorily arrived at
the answers to Mr. Newton’s first two
questions, present questions 3 and 4 to the
students on Handout/Transparency 2.
14.
Pose the problem question:
So, how much does a kilogram cost?
A kilogram of nickels cost $10.00 and a
kilogram of pennies cost $4.00.
Patterns, Relationships, and Algebraic Thinking
How Much Does a Kilogram Cost?
Now that students know the
number of nickels (200) and
pennies (400) needed to equal the
mass of one kilogram, they can
employ any strategy to calculate
the values.
Grade 5
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Mathematics TEKS Refinement 2006 – K-5
Procedures
Have students share their strategies for
arriving at these solutions.
15.
Present the challenge questions to the
students on Handout/Transparency 2.
Answer:
Mr. Newton will need 5 rolls of nickels or 8
rolls of pennies to make a kilogram.
16.
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Notes
You might tell students that
besides knowing how much a
kilogram of each coin costs, Mr.
Newton is also interested in how
many rolls of each coin would be
needed.
Have students share their strategies for
arriving at their solutions.
Homework:
Have students complete Mr. Newton Strikes Again -Handout/Transparency 4.
Assessment:
One ton is equal to 2000 pounds. Measure your weight to the nearest
10 pounds. Based on this weight, about how many of you would it
take to equal a “ton of kids?” Use words and pictures, diagrams, or
tables to show how you arrived at your answer.
Extensions:
1. Have students gather items from home that approximate the mass
of one kilogram. Ask students to bring their items and objects to
class. Let them use a balance to determine how closely they were
able to approximate the mass of one kilogram.
2. Have students find other items/objects of uniform mass (washers,
screws, nails, milliliters of water, binder clips, etc.) and determine
how many it will take to approximate the mass of one kilogram. Let
students use balances to determine the mass of each item. Then,
students can calculate how many are needed to approximate one
kilogram.
Resources:
Literature Connection: Counting on Frank by Rod Clement.
Frank’s owner, the narrator of this book, might be called a measuring
maniac. Students will enjoy his wacky way of counting and his
methods for measuring the world around him.
Modifications: This activity might be configured as a problem set for the week.
Day 1: Read Math Curse and introduce the problem. Read the
problem for comprehension and summarize the important
information.
Day 2: Have students generate possible strategies and select a
Patterns, Relationships, and Algebraic Thinking
How Much Does a Kilogram Cost?
Grade 5
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Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
suitable strategy for solving the initial problem.
Day 3: Have students share their strategies and solutions with the
group. Connect the various strategies and representations
shared by the students.
Day 4: Have students solve questions 3 and 4. Let students share
strategies and solutions.
Day 5: Present the Challenge questions. Have students share
strategies and solutions.
Patterns, Relationships, and Algebraic Thinking
How Much Does a Kilogram Cost?
Grade 5
Page 8
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Mr. Newton’s Problem
Mr. Newton, the science teacher, needs some kilogram
masses for use in an upcoming science activity with the 4th grade
students. Rather than order standard kilogram masses from a
science supply store, he has decided to use coins because of
their small size and uniform mass. Nickels have a mass of about
5 grams each, and pennies have a mass of approximately 2.5
grams each. Mr. Newton needs to know the following:
1. How many nickels does it take to equal the mass of 1
kilogram?
2. How many pennies does it take to equal the mass of 1
kilogram?
Handout/Transparency 1
Patterns, Relationships, and Algebraic Thinking
How Much Does a Kilogram Cost?
Grade 5
Page 9
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
How Much Does a Kilogram Cost?
Continued:
3. How much would a kilogram of nickels cost?
4. How much would a kilogram of pennies cost?
Challenge:
5. If nickels come in rolls of $2.00, how many rolls of nickels
would Mr. Newton need to equal the mass of 1 kilogram?
6. If pennies come in rolls of $0.50, how many rolls of pennies
would Mr. Newton need to equal the mass of 1 kilogram?
Handout/Transparency 2
Patterns, Relationships, and Algebraic Thinking
How Much Does a Kilogram Cost?
Grade 5
Page 10
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Possible Solution Strategies
1. Make a table.
Nickels Process Grams
(N)
(g)
Pennies
(P)
Process Grams
(g)
1
1x5
5
1
1 x 2.5
2.5
2
2x5
10
2
2 x 2.5
5
5
5x5
25
5
5 x 2.5
12.5
10
10 x 5
50
10
10 x 2.5
25
100
100 x 5
500
100
100 x 2.5
250
200
200 x 5
1000
200
200 x 2.5
500
N
Nx5
g
400
400 x 2.5
1000
P
P x 2.5
g
2. Write an equation (possible responses).
1000 ÷ 5 = 200
5 x ____ = 1000
1000 ÷ 2.5 = 400
2.5 x ____ = 1000
3. Guess and Check (possible guesses).
10 nickels would = 50 grams
20 nickels would = 100 grams
100 nickels would = 500 grams
200 nickels would = 1000 grams
2 pennies would = 5 grams
4 pennies would = 10 grams
40 pennies would = 100 grams
400 pennies would = 1000 grams
Handout/Transparency 3
Patterns, Relationships, and Algebraic Thinking
How Much Does a Kilogram Cost?
Grade 5
Page 11
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Mr. Newton Strikes Again
Now you know that 200 nickels or 400 pennies are approximately
equal to the mass of one kilogram. Mr. Newton has some more
questions for you to answer. Use your knowledge of patterns to
help you answer his questions.
1. If 1 kilogram equals 1,000 grams, how many grams are
equal to 2 kilograms? _____________
How many nickels would be needed to equal the mass of 2
kilograms? ______________________
How many pennies would be needed to equal the mass of 2
kilograms? ______________________
2. If 1,000 grams equals one kilogram, how many grams would
equal 1 kilogram? _____________
2
How many nickels would be needed to equal the mass of
1
2
kilogram? ______________________
How many pennies would be needed to equal the mass of
1
2
kilogram? ______________________
3. Mr. Newton has a mass of pennies equal to 250 grams.
What fractional part of a kilogram is 250 grams?
_________________________________
How many pennies would be equal in mass to 250 grams?
_____________________________
How many nickels would be equal in mass to 250 grams?
_____________________________
Handout/Transparency 4
Patterns, Relationships, and Algebraic Thinking
How Much Does a Kilogram Cost?
Grade 5
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