13
RULES
That Expire
Overgeneralizing commonly
accepted strategies, using imprecise
vocabulary, and relying on tips
and tricks that do not promote
conceptual mathematical
understanding can lead
to misunderstanding later
in students’ math careers.
By Karen S. Karp, Sarah B. Bush,
and Barbara J. Dougher ty
18
August 2014 • teaching children mathematics | Vol. 21, No. 1
Copyright © 2014 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
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FUSE/THINKSTOCK
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Vol. 21, No. 1 | teaching children mathematics • August 2014
19
3+5=□
□+2=7
8 =□ + 3
2 + 4 = □ + 5.
Stop for a moment to think about which of
these number sentences a student in your class
would solve first or find easiest. What might
they say about the others? In our work with
young children, we have found that students
feel comfortable solving the first equation
because it “looks right” and students can interpret the equal sign as find the answer. However,
students tend to hesitate at the remaining
number sentences because they have yet to
interpret and understand the equal sign as a
symbol indicating a relationship between two
quantities (or amounts) (Mann 2004).
In another scenario, an intermediate student
is presented with the problem 43.5 × 10.
Immediately, he responds, “That’s easy; it is
43.50 because my teacher said that when you
multiply any number times ten, you just add a
zero at the end.”
In both these situations, hints or repeated
practices have pointed students in directions
that are less than helpful. We suggest that
these students are experiencing rules
that expire. Many of these rules “expire”
when students expand their knowledge
of our number systems beyond whole
numbers and are forced to change their
perception of what can be included in
referring to a number. In this article,
we present what we believe are thirteen
pervasive rules that expire. We follow
up with a conversation about incorrect
use of mathematical language, and
we present alternatives to help
counteract common student
misunderstandings.
The Common Core
State Standards (CCSS)
for Mathematical Practice
advocate for students to
become problem solvers
who can reason, apply,
j u s t i f y, a n d e f f e c t i v e l y
20
August 2014 • teaching children mathematics | Vol. 21, No. 1
use appropriate mathematics vocabulary
to demonstrate their understanding of
mathematics concepts (CCSSI 2010). This,
in fact, is quite opposite of the classroom in
which the teacher does most of the talking and
students are encouraged to memorize facts,
“tricks,” and tips to make the mathematics
“easy.” The latter classroom can leave students
with a collection of explicit, yet arbitrary, rules
that do not link to reasoned judgment (Hersh
1997) but instead to learning without thought
(Boaler 2008). The purpose of this article is to
outline common rules and vocabulary that
teachers share and elementary school students
tend to overgeneralize—tips and tricks that do
not promote conceptual understanding, rules
that “expire” later in students’ mathematics
careers, or vocabulary that is not precise.
As a whole, this article aligns to Standard of
Mathematical Practice (SMP) 6: Attend to
precision, which states that mathematically
proficient students “…try to communicate
precisely to others. …use clear definitions …
and … carefully formulated explanations…”
(CCSSI 2010, p. 7). Additionally, we emphasize
two other mathematical practices: SMP 7: Look
for and make use of structure when we take
a look at properties of numbers; and SMP 2:
Reason abstractly and quantitatively when we
discuss rules about the meaning of the four
operations.
“Always” rules that are not
so “always”
In this section, we point out rules that seem
to hold true at the moment, given the content
the student is learning. However, students later
find that these rules are not always true; in fact,
these rules “expire.” Such experiences can be
frustrating and, in students’ minds, can further
the notion that mathematics is a mysterious
series of tricks and tips to memorize rather
than big concepts that relate to one another.
For each rule that expires, we do the following:
1. State the rule that teachers share with
students.
2. Explain the rule.
3. Discuss how students inappropriately overgeneralize it.
4. Provide counterexamples, noting when the
rule is not true.
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ALINA555/THINKSTOCK
I
magine the following scenario: A primary
teacher presents to her students the following set of number sentences:
5. State the “expiration date” or the point
when the rule begins to fall apart for many
learners. We give the expiration date in
terms of grade levels as well as CCSSM content standards in which the rule no longer
“always” works.
problem. Keywords can be informative but
must be used in conjunction with all other
words in the problem to grasp the full meaning.
Expiration date: Grade 3 (3.OA.8).
Thirteen rules that expire
Students might hear this phrase as they first
learn to subtract whole numbers. When students
are restricted to only the set of whole numbers,
subtracting a larger number from a smaller one
results in a negative number, an integer that is
not in the set of whole numbers, so this rule is
true. Later, when students encounter application or word problems involving contexts that
include integers, students learn that this “rule”
is not true for all problems. For example, a grocery store manager keeps the temperature of the
produce section at 4 degrees Celsius, but this is
22 degrees too hot for the frozen food section.
What must the temperature be in the frozen
food section? In this case, the answer is a negative number, (4º – 22º = –18º). Expiration date:
Grade 7 (7.NS.1).
1. When you multiply a number by ten,
just add a zero to the end of the number.
This “rule” is often taught when students are
learning to multiply a whole number times
ten. However, this directive is not true when
multiplying decimals (e.g., 0.25 × 10 = 2.5, not
0.250). Although this statement may reflect
a regular pattern that students identify with
whole numbers, it is not generalizable to other
types of numbers. Expiration date: Grade 5
(5.NBT.2).
2. Use keywords to solve word problems.
This approach is often taught throughout
the elementary grades for a variety of word
problems. Using keywords often encourages
students to strip numbers from the problem
and use them to perform a computation
outside of the problem context (Clement and
Bernhard 2005). Unfortunately, many keywords are common English words that can
be used in many different ways. Yet, a list of
keywords is often given so that word problems
can be translated into a symbolic, computational form. Students are sometimes told that
if they see the word altogether in the problem,
they should always add the given numbers.
If they see left in the problem, they should
always subtract the numbers. But reducing
the meaning of an entire problem to a simple
scan for key words has inherent challenges.
For example, consider this problem:
John had 14 marbles in his left pocket. He
had 37 marbles in his right pocket. How
many marbles did John have?
If students use keywords as suggested above,
they will subtract without realizing that the
problem context requires addition to solve.
Keywords become particularly troublesome
when students begin to explore multistep word
problems, because they must decide which
keywords work with which component of the
www.nctm.org
3. You cannot take a bigger number
from a smaller number.
4. Addition and multiplication make
numbers bigger.
When students begin learning about the
operations of addition and multiplication,
they are often given this rule as a means to
develop a generalization relative to operation
sense. However, the rule has multiple counterexamples. Addition with zero does not create a
sum larger than either addend. It is also untrue
when adding two negative numbers (e.g., –3 +
–2 = –5), because –5 is less than both addends.
In the case of the equation below, the product is
smaller than either factor.
MathType 1
1 1 1
× =
4 3 12
This is also the case when one of the factors
MathType
2 factor is
is a negative number and
the other
1
positive, such as –3 × 8= –24. Expiration
date:
8 ÷ 4 = 2 or 4 ÷ 8 =
Grade 5 (5.NF.4 and 5.NBT.7) and2 again at
Grade 7 (7.NS.1 and 7.NS.2).
1
2
3
4
5
MathType 3
5. Subtraction and division make
1 2 5
numbers smaller. ÷ =
4 5 heard
8 in grade 3: both
This rule is commonly
subtraction and division will result in an
MathType
answer that is smaller than
at least4one of the
1
Vol. 21, No. 21 | teaching children mathematics • August 2014
21
numbers in the computation. When numbers
are positive whole numbers, decimals, or fractions, subtracting will result in a number that
is smaller than at least one of the numbers
MathType 1However, if the
involved in the computation.
MathType 1
1 1 1two negative numbers,
subtraction involves
× =1 1 1
students may 4notice
(e.g., –5 –
×a contradiction
=
3 12
4 the
3 rule
12 is true if the num(– 8) = 3). In division,
bers are positive whole
numbers,
MathType
2 for example:
MathType 2
1
8 ÷ 4 = 2 or 4 ÷ 8 =
1
8 ÷ 4 = 2 or 4 2÷ 8 =
2
However, if the numbers you are dividing are
MathType
3
fractions, the quotient
may be larger:
MathType 3
1 2 5
÷ =
1 2 5
4 5 ÷
8 =
4 5 8
6
7
8
9
10
11
This is also the case when dividing two negaMathType 4
tive factors: (e.g., –9 ÷ –3 MathType
= 3). Expiration
dates:
4
1 again at Grade 7 (7.NS.1
Grade 6 (6.NS.1) and
1
2MathType 1
and 7.NS.2c).
2
1 1 1
× MathType
=1
MathType
6. You always
larger
number
5
4divide
3 12the
MathType 5
1
1
1
by the smaller
number.
× =
1
1
3 12
2 =1 students
1
This rule4may
be true÷ when
begin to
÷
42 = 2
2MathType
4
learn their basic facts for2 whole-number
diviMathType 2
1 contextusion and the computations
are
not
8 ÷ 41= 21or 4 ÷ 8 =
8 ÷ 4But,
4 ÷ 8 example,
= 2 or for
=
ally based.
if 2the problem
2 1
states that Kate has72 cookies
to
divide among
7
herself and two
friends,
then
the
MathTypeMathType
3
3 portion for
1 2 is5 2 ÷ 3. Similarly, it is possible to
each person
÷ = 1 2 5
4 5 8 in
÷ which
=
have a problem
one number might be
4 5 8
a fraction:
MathType 4
1
MathType
4
Jayne has of a pizza
and wants
to share it
2
1
with her brother. What
portion of the whole
pizza will each
get?25
MathType
1
÷2 =
1
In this case, the
is as
2 computation
4
MathType
5 follows:
1
7
1
1
÷2 =
2
4
Expiration date: Grade 5 (5.NF.3 and 5.NF.7).
1
7. Two negatives make
a positive.
7
Typically taught when students learn about
multiplication and division of integers, rule 7 is
to help them determine the sign of the product
or quotient. However, this rule does not always
hold true for addition and subtraction of integers, such as in –5 + (–3) = –8. Expiration date:
Grade 7 (7.NS.1).
22
August 2014 • teaching children mathematics | Vol. 21, No. 1
8. Multiply everything inside the
parentheses by the number outside
the parentheses.
As students are developing the foundational
skills linked to order of operations, they are often
told to first perform multiplication on the numbers (terms) within the parentheses. This holds
true only when the numbers or variables inside
the parentheses are being added or subtracted,
because the distributive property is being used,
for example, 3(5 + 4) = 3 × 5 + 3 × 4. The rule is
untrue when multiplication or division occurs
in the parentheses, for example, 2 (4 × 9) ≠
2 × 4 × 2 × 9. The 4 and the 9 are not two separate
terms, because they are not separated by a plus
or minus sign. This error may not emerge in
situations when students encounter terms that
do not involve the distributive property or when
students use the distributive property without
the element of terms. The confusion seems to be
an interaction between students’ partial understanding of terms and their partial understanding of the distributive property—which may not
be revealed unless both are present. Expiration
date: Grade 5 (5.OA.1).
9. Improper fractions should always be
written as a mixed number.
When students are first learning about fractions,
they are often taught to always change improper
fractions to mixed numbers, perhaps so they
can better visualize how many wholes and parts
the number represents. This rule can certainly
help students understand that positive mixed
numbers can represent a value greater than one
whole, but it can be troublesome when students
are working within a specific mathematical context or real-world situation that requires them
to use improper fractions. This frequently first
occurs when students begin using improper
fractions to compute and again when students
later learn about the slope of a line and must
represent the slope as the rise/run, which is
sometimes appropriately and usefully expressed
as an improper fraction. Expiration dates:
Grade 5 (5.NF.1) and again in Grade 7 (7.RP.2).
10. The number you say first in counting
is always less than the number that
comes next.
In the early development of number, students
are regularly encouraged to think that number
www.nctm.org
MathType 2
8 ÷ 4 = 2 or 4 ÷ 8 =
1
2
TABL E 1
MathType 3
1 2 language
5
Some commonly used
“expires ” and should be replaced with more appropriate alternatives.
÷ =
4
5
8
Expired mathematical language and suggested alternatives
What is stated
What should be stated
1
MathType 4
2 or
Using the words borrowing
carrying when subtracting or adding,
MathType 5
respectively
1
÷2 = 4
Use trading or regrouping to indicate the actual action of trading or
exchanging one place value unit for another unit.
Using the phrase ___ out 2of __ to
describe a fraction, for example, one
1
out of seven to describe
Use the fraction and the attribute. For example, say one-seventh of the
length of the string. The out of language often causes students to think
a part is being subtracted from the whole amount (Philipp, Cabral, and
Schappelle 2005).
Using the phrase reducing fractions
Use simplifying fractions. The language of reducing gives students the
incorrect impression that the fraction is getting smaller or being reduced
in size.
Asking how shapes are similar when
children are comparing a set of shapes
Ask, How are these shapes the same? How are the shapes different? Using
the word similar in these situations can eventually confuse students about
the mathematical meaning of similar, which will be introduced in middle
school and relates to geometric figures.
Reading the equal sign as makes, for
example, saying, Two plus two makes
four for 2 + 2 = 4
Read the equation 2 + 2 = 4 as Two plus two equals or is the same
as four. The language makes encourages the misconception that the
equal sign is an action or an operation rather than representative of
a relationship.
Indicating that a number divides
evenly into another number
Say that a number divides another number a whole number of times or
that it divides without a remainder.
Plugging a number into an expression
or equation
Use substitute values for an unknown.
Using top number and bottom number to describe the numerator and
denominator of a fraction, respectively
Students should see a fraction as one number, not two separate numbers.
Use the words numerator and denominator when discussing the different
parts of a fraction.
7
relationships are fixed. For example, the relationship between 3 and 8 is always the same. To
determine the relationship between two numbers, the numbers must implicitly represent a
count made by using the same unit. But when
units are different, these relationships change.
For example, three dozen eggs is more than eight
eggs, and three feet is more than eight inches.
Expiration date: Grade 2 (2.MD.2).
11. The longer the number, the larger
the number.
The length of a number, when working with
whole numbers that differ in the number of
digits, does indicate this relationship or magnitude. However, it is particularly troublesome
to apply this rule to decimals (e.g., thinking that
0.273 is larger than 0.6), a misconception noted
by Desmet, Grégoire, and Mussolin (2010).
Expiration date: Grade 4 (4.NF.7).
www.nctm.org
12. Please Excuse My Dear Aunt Sally.
This phrase is typically taught when students
begin solving numerical expressions involving multiple operations, with this mnemonic
serving as a way of remembering the order of
operations. Three issues arise with the application of this rule. First, students incorrectly
believe that they should always do multiplication before division, and addition before
subtraction, because of the order in which they
appear in the mnemonic PEMDAS (Linchevski
and Livneh 1999). Second, the order is not as
strict as students are led to believe. For example,
in the expression 32 – 4(2 + 7) + 8 ÷ 4, students
have options as to where they might start. In
this case, they may first simplify the 2 + 7 in the
grouping symbol, simplify 32, or divide before
doing any other computation—all without
affecting the outcome. Third, the P in PEMDAS
suggests that parentheses are first, rather than
Vol. 21, No. 1 | teaching children mathematics • August 2014
23
Other rules that expire
We invite Teaching Children Mathematics (TCM) readers to submit
additional instances of “rules that expire” or “expired language” that
this article does not address. If you would like to share an example,
please use the format of the article, stating the rule to avoid, a case of
how it expires, and when it expires in the Common Core State Standards
for Mathematics. If you submit an illustration of expired language,
include “What is stated” and “What should be stated” (see table 1).
Join us as we continue this conversation on TCM’s blog at
www.nctm.org/TCMblog/MathTasks or send your suggestions and
thoughts to [email protected]. We look forward to your input.
grouping symbols more generally, which would
include brackets, braces, square root symbols,
and the horizontal fraction bar. Expiration date:
Grade 6 (6.EE.2).
13. The equal sign means Find the
answer or Write the answer.
12
13
An equal sign is a relational symbol. It indicates that the two quantities on either side of
it represent the same amount. It is not a signal
prompting the answer through an announcement to “do something” (Falkner, Levi, and
Carpenter 1999; Kieran 1981). In an equation,
students may see an equal sign that expresses
the relationship but cannot be interpreted as
Find the answer. For example, in the equations
below, the equal sign provides no indication of
an answer. Expiration date: Grade 1 (1.OA.7).
6=□+4
3 + x = 5 + 2x
Expired language
In addition to helping students avoid the thirteen rules that expire, we must also pay close
attention to the mathematical language we use
as teachers and that we allow our students to
use. The language we use to discuss mathematics (see table 1) may carry with it connotations
that result in misconceptions or misuses by
students, many of which relate to the Thirteen
Rules That Expire listed above. Using accurate
and precise vocabulary (which aligns closely
with SMP 6) is an important part of developing
student understanding that supports student
learning and withstands the need for complexity
as students progress through the grades.
No expiration date
One characteristic of the Common Core State
Standards for Mathematics (CCSSM) (CCSSI
2010) is to have fewer, but deeper, more rigorous standards at each grade—and to have less
24
August 2014 • teaching children mathematics | Vol. 21, No. 1
overlap and greater coherence as students
progress from K–grade 12. We feel that by using
consistent, accurate rules and precise vocabulary in the elementary grades, teachers can play
a key role in building coherence as students
move from into the middle grades and beyond.
No one wants students to realize in the upper
elementary grades or in middle school that their
teachers taught “rules” that do not hold true.
With the implementation of CCSSM, now is
an ideal time to highlight common instructional
practices that teachers can tweak to better prepare students and allow them to have smoother
transitions moving from grade to grade. Additionally, with the implementation of CCSSM,
many teachers—even those teaching the same
grade as they had previously—are being required
to teach mathematics content that differs from
what they taught in the past. As teachers are
planning how to teach according to new standards, now is a critical point to think about the
rules that should or should not be taught and the
vocabulary that should or should not be used in
an effort to teach in ways that do not “expire.”
REF EREN C ES
Boaler, Jo. 2008. What’s Math Got to Do with
It? Helping Children Learn to Love their Most
Hated Subject—and Why It’s Important for
America. New York: Viking.
Clement, Lisa, and Jamal Bernhard. 2005. “A
Problem-Solving Alternative to Using Key
Words.” Mathematics Teaching in the Middle
School 10 (7): 360–65.
Common Core State Standards Initiative (CCSSI).
2010. Common Core State Standards for
Mathematics. Washington, DC: National
Governors Association Center for Best
Practices and the Council of Chief State
School Officers. http://www.corestandards
.org/wp-content/uploads/Math_Standards.pdf
Desmet, Laetitia, Jacques Grégoire, and
Christophe Mussolin. 2010. “Developmental
Changes in the Comparison of Decimal
Fractions.” Learning and Instruction 20 (6):
521–32. http://dx.doi.org/10.1016
/j.learninstruc.2009.07.004
Falkner, Karen P., Linda Levi, and Thomas P.
Carpenter. 1999. “Children’s Understanding
of Equality: A Foundation for Algebra.”
Teaching Children Mathematics 6 (February):
56–60.
www.nctm.org
Hersh, Rueben. 1997. What Is Mathematics,
Really? New York: Oxford University Press.
Kieran, Carolyn. 1981. “Concepts Associated
with the Equality Symbol.” Educational
Studies in Mathematics 12 (3): 317–26.
http://dx.doi.org/10.1007/BF00311062
Linchevski, Liora, and Drora Livneh. 1999.
“Structure Sense: The Relationship between
Algebraic and Numerical Contexts.” Educational Studies in Mathematics 40 (2): 173–96.
http://dx.doi.org/10.1023/A:1003606308064
Mann, Rebecca. 2004. “Balancing Act: The Truth
behind the Equals Sign.” Teaching Children
Mathematics 11 (September): 65–69.
Philipp, Randolph A., Candace Cabral, and
Bonnie P. Schappelle. 2005. IMAP CD-ROM:
Integrating Mathematics and Pedagogy to
Illustrate Children’s Reasoning. Computer
software. Upper Saddle River, NJ: Pearson
Education.
Karen S. Karp, [email protected],
a professor of math education at the
University of Louisville in Kentucky, is
a past member of the NCTM Board of
Directors and a former president of the
Association of Mathematics Teacher
Educators. Her current scholarly work
focuses on teaching math to students
with disabilities. Sarah B. Bush, sbush@
bellarmine.edu, an assistant professor
of math education at Bellarmine
University in Louisville, Kentucky, is a
former middle-grades math teacher who
is interested in relevant and engaging
middle-grades math activities. Barbara
J. Dougherty is the Richard Miller Endowed Chair for
Mathematics Education at the University of Missouri. She
is a past member of the NCTM Board of Directors and
is a co-author of conceptual assessments for progress
monitoring in algebra and an iPad® applet for K–grade 2
students to improve counting and computation skills.
NCTM’s Member Referral Program
Ma k in g C on n e c t i on s
Participating in NCTM’s Member Referral Program is fun, easy, and
rewarding. All you have to do is refer colleagues, prospective teachers, friends,
and others for membership. Then, as our numbers go up, watch your rewards
add up.
Learn more about the program, the gifts, and easy ways to encourage your
colleagues to join NCTM at www.nctm.org/referral. Help others learn of the
many benefits of an NCTM membership—Get started today!
Learn More
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Vol. 21, No. 1 | teaching children mathematics • August 2014
25
Engage Students in Learning:
Mathematical Practices
An NCTM Interactive Institute for Grades K-8
@fawnpnguyen
Atlanta, July 11-13, 2016
Fawn Nguyen
[email protected]
Twitter:
Blog: fawnnguyen.com
CCSS Grade 6: Understand ratio concepts and use ratio reasoning to solve problems.
CCSS.MATH.CONTENT.6.RP.A.1
CCSS.MATH.CONTENT.6.RP.A.3
CCSS.MATH.CONTENT.6.RP.A.3.B
Understand the concept of a ratio and use
ratio language to describe a ratio relationship
between two quantities. For example, "The
ratio of wings to beaks in the bird house at
the zoo was 2:1, because for every 2 wings
there was 1 beak." "For every vote candidate
A received, candidate C received nearly
three votes."
Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by
reasoning about tables of equivalent ratios,
tape diagrams, double number line diagrams,
or equations.
Solve unit rate problems including those
involving unit pricing and constant speed.For
example, if it took 7 hours to mow 4 lawns,
then at that rate, how many lawns could be
mowed in 35 hours? At what rate were lawns
being mowed?
CCSS.MATH.CONTENT.6.RP.A.2
CCSS.MATH.CONTENT.6.RP.A.3.A
CCSS.MATH.CONTENT.6.RP.A.3.C
Make tables of equivalent ratios relating
quantities with whole-number measurements,
find missing values in the tables, and plot the
pairs of values on the coordinate plane. Use
tables to compare ratios.
Find a percent of a quantity as a rate per 100
(e.g., 30% of a quantity means 30/100 times
the quantity); solve problems involving finding
the whole, given a part and the percent.
Understand the concept of a unit
rate a/b associated with a ratio
a:b with b ≠ 0, and use rate
language in the context of a ratio
relationship. For example, "This recipe
has a ratio of 3 cups of flour to 4 cups of
sugar, so there is 3/4 cup of flour for each
cup of sugar." "We paid $75 for 15
hamburgers, which is a rate of $5 per
hamburger."1
1
Expectations for unit rates in this grade are
limited to non-complex fractions.
CCSS.MATH.CONTENT.6.RP.A.3.D
Use ratio reasoning to convert measurement
units; manipulate and transform units
appropriately when multiplying or dividing
Engage Students in Learning:
Mathematical Practices
An NCTM Interactive Institute for Grades K-8
@fawnpnguyen
Atlanta, July 11-13, 2016
Fawn Nguyen
[email protected]
Twitter:
Blog: fawnnguyen.com
quantities.
CCSS Grade 7: Analyze proportional relationships and use them to solve real-world and mathematical problems.
CCSS.MATH.CONTENT.7.RP.A.1
Compute unit rates associated with ratios of
fractions, including ratios of lengths, areas
and other quantities measured in like or
different units. For example, if a person walks
1/2 mile in each 1/4 hour, compute the unit
rate as the complex fraction 1/2/1/4 miles per
hour, equivalently 2 miles per hour.
CCSS.MATH.CONTENT.7.RP.A.2
Recognize and represent proportional
relationships between quantities.
CCSS.MATH.CONTENT.7.RP.A.2.A
Decide whether two quantities are in a
proportional relationship, e.g., by testing for
equivalent ratios in a table or graphing on a
coordinate plane and observing whether the
graph is a straight line through the origin.
CCSS.MATH.CONTENT.7.RP.A.2.B
Identify the constant of proportionality (unit
rate) in tables, graphs, equations, diagrams,
and verbal descriptions of proportional
relationships.
CCSS.MATH.CONTENT.7.RP.A.2.C
Represent proportional relationships by
equations. For example, if total cost t is
proportional to the number n of items
purchased at a constant price p, the
relationship between the total cost and the
number of items can be expressed as t = pn.
CCSS.MATH.CONTENT.7.RP.A.2.D
Explain what a point (x, y) on the graph of a
proportional relationship means in terms of
the situation, with special attention to the
points (0, 0) and (1, r) where r is the unit rate.
CCSS.MATH.CONTENT.7.RP.A.3
Use proportional relationships to solve
multistep ratio and percent problems.
Examples: simple interest, tax, markups and
markdowns, gratuities and commissions,
Engage Students in Learning:
Mathematical Practices
An NCTM Interactive Institute for Grades K-8
@fawnpnguyen
Atlanta, July 11-13, 2016
Fawn Nguyen
[email protected]
Twitter:
Blog: fawnnguyen.com
fees, percent increase and decrease, percent
error.
Engage Students in Learning: Fawn Nguyen Mathematical Practices [email protected] An NCTM Interactive Institute for Grades K­8 Twitter: @fawnpnguyen Atlanta, July 11­13, 2016 Blog: fawnnguyen.com Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning Grades 6­8 (source: ​NCTM​) Essential Understanding 1. Reasoning with ratios involves attending to and coordinating two quantities. 2. A ratio is a multiplicative comparison of two quantities, or it is a joining of two quantities in a composed unit. Question How does ratio reasoning differ from other types of reasoning? What is a ratio? 3. Forming a ratio as a measure of a real­world attribute involves isolating that What is a ratio as a attribute from other attributes and measure of an attribute in understanding the effect of changing each a real­world situation? quantity on the attribute of interest. 4. A number of mathematical connections link ratios and fractions. How are ratios related to fractions? 5. Ratios can be meaningfully reinterpreted as quotients. How are ratios related to division? 6. A proportion is a relationship of equality between two ratios. In a proportion, the ratio of two quantities remains constant as What is a proportion? the corresponding values of the quantities change. Topic Ratios Proportions Engage Students in Learning: Fawn Nguyen Mathematical Practices [email protected] An NCTM Interactive Institute for Grades K­8 Twitter: @fawnpnguyen Atlanta, July 11­13, 2016 Blog: fawnnguyen.com Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning Grades 6­8 (source: ​NCTM​) Essential Understanding Question 7. Proportional reasoning is complex and involves understanding that ­­ a. Equivalent ratios can be created by iterating and/or partitioning a composed unit. b. If one quantity in a ratio is multiplied or divided by a particular What are the key aspects factor, then the other quantity of proportional reasoning? must be multiplied or divided by the same factor to maintain the proportional relationship; and c. The two types of ratios ­­ composed units and multiplicative comparisons ­­ are related. 8. A rate is a set of infinitely many equivalent What is a rate and how is ratios. it related to proportional reasoning? 9. Several ways of reasoning, all grounded in sense making, can be generalized into algorithms for solving proportion problems. 10. Superficial cues present in the context of a problem do not provide sufficient evidence of proportional relationships between quantities. What is the relationship between the cross­multiplication algorithm and proportional reasoning? When is it appropriate to reason proportionately? Topic Proportional Reasoning Engage Students in Learning: Fawn Nguyen Mathematical Practices [email protected] An NCTM Interactive Institute for Grades K­8 Twitter: @fawnpnguyen Atlanta, July 11­13, 2016 Blog: fawnnguyen.com Task or Question
What is a fraction?
Write a sentence with a fraction in it (not the word “fraction,”
but an actual number) to describe something in the room.
What is a ratio?
Write a sentence with a ratio in it (not the word “ratio,” but an
actual number) to describe something in the room.
Essential
Understandings
4
4
2
2
Three-fifths of the sand went through a sand timer in 18
minutes. If the rest of the sand goes through at the same rate,
how long does it take for the rest of the sand to go through the
sand timer? Show two different ways to solve this problem.
2, 8
PoW: Ration Ratios
2, 3
To make Luscious Lilikoi Punch, Austin mixes 1/2 cup of lilikoi
passion fruit concentrate with 2/3 cup water. If he wants to mix
concentrate and water in the same ratio to make 28 cups of
Luscious Lilikoi Punch, how many cups of lilikoi passion fruit
concentrate and how many cups of water will Austin need?
Dan Meyer's video Expected Value (blue/purple circle)
8
2, 3
Taufique has two bags, each containing some red and some
white gumballs. In bag A, there are 10 red and 15 white
gumballs. In bag B, there are 6 red and 8 white gumballs.
If Taufique reaches in without looking, from which bag is he
more likely to pull out a red gumball?
2, 3, 4
Draw a diagram to represent how you solved the problem, and
explain it.
How does a probability of pulling a white gumball from a bag
with 3 white and 2 red gumballs compare to the probability of
pulling a white gumball from a bag with 12 white and 8 red
gumballs?
Which road is steeper, one with a 4 percent grade for 0.5 miles
or one with a 4 percent grade of 0.25 miles?
Are the fractions 6/9 and 10/15 equivalent?
Jonnine had a board. She cut and used 2/5 of the board for
bracing. She measured the piece used for bracing and found it
to be 3/4 foot long. How long was the original board?
3
3
7
7, 9
Common Core State
Standards
Two friends play a game of coin toss. A wins if 5 heads come
up first, and B wins if 5 tails come up first. Each person puts in
$10 bet; winner takes all of $20.
2
Game must end, however, when A has 2 heads and B has 4
tails. How should they split the money?
As much as possible, use only mental arithmetic to determine
which is larger, 14/29 or 15/31?
9
MARS Lesson: Using Proportional Reasoning
Handout Ratios & Proportions (1)
Handout Ratios & Proportions (2)
Handout Ratios & Proportions (3)
1, 2, 7
2, 7
3, 8, 4
Engage Students in Learning: Fawn Nguyen Mathematical Practices [email protected] An NCTM Interactive Institute for Grades K­8 Twitter: @fawnpnguyen Atlanta, July 11­13, 2016 Blog: fawnnguyen.com PoW: Ration Ratios (source: ​mathforum.org​) Problem​: At a recent math conference, lunch was provided for the participants. To be sure that there was enough food for everyone, the kitchen staff made more lunches than there were people attending. In fact, the ratio of prepared lunches to people was 7:5. Because they anticipated a large number of vegetarians at the conference, the staff made 2 vegetarian lunches for every 3 non­vegetarian lunches. It turned out that the ratio of non­vegetarians to vegetarians at the conference was 3:4. What was the ratio of vegetarian lunches to vegetarians? Engage Students in Learning: Fawn Nguyen Mathematical Practices [email protected] An NCTM Interactive Institute for Grades K­8 Twitter: @fawnpnguyen Atlanta, July 11­13, 2016 Blog: fawnnguyen.com Ratios and Proportions (1) (source: ​http://donsteward.blogspot.com/​) 1. Ratios as unequal sharing: 2. Shade the rectangles in the given ratios: 3. Erich Friedman's ​weight puzzles​: Place weights 1 to 5 so that this mobile balances. Engage Students in Learning: Fawn Nguyen Mathematical Practices [email protected] An NCTM Interactive Institute for Grades K­8 Twitter: @fawnpnguyen Atlanta, July 11­13, 2016 Blog: fawnnguyen.com Engage Students in Learning: Fawn Nguyen Mathematical Practices [email protected] An NCTM Interactive Institute for Grades K­8 Twitter: @fawnpnguyen Atlanta, July 11­13, 2016 Blog: fawnnguyen.com Ratios and Proportions (2) (source: ​http://donsteward.blogspot.com/​) Engage Students in Learning: Fawn Nguyen Mathematical Practices [email protected] An NCTM Interactive Institute for Grades K­8 Twitter: @fawnpnguyen Atlanta, July 11­13, 2016 Blog: fawnnguyen.com Engage Students in Learning: Fawn Nguyen Mathematical Practices [email protected] An NCTM Interactive Institute for Grades K­8 Twitter: @fawnpnguyen Atlanta, July 11­13, 2016 Blog: fawnnguyen.com Engage Students in Learning: Fawn Nguyen Mathematical Practices [email protected] An NCTM Interactive Institute for Grades K­8 Twitter: @fawnpnguyen Atlanta, July 11­13, 2016 Blog: fawnnguyen.com Ratios and Proportions (3) (source: Edward Zaccaro’s ​Challenge Math​) 1. It took 6 people 8 days to build a brick wall. The construction crew needs to build an identical wall but needs it done faster. If they add 2 people to the crew, how long will the brick wall take to build now? 2. A crew of 8 people finished ¼ of a tunnel through a mountain in 30 days. If they added 2 more workers, how long will it take them to finish the tunnel? 3. Luke paints a car in 6 hours while Daniel paints the same car in 3 hours. If they work together, how long will take them to paint the car? CONCEPT DEVELOPMENT
Mathematics Assessment Project
CLASSROOM CHALLENGES
A Formative Assessment Lesson
Using Proportional
Reasoning
Mathematics Assessment Resource Service
University of Nottingham & UC Berkeley
For more details, visit: http://map.mathshell.org
© 2015 MARS, Shell Center, University of Nottingham
May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license
detailed at http://creativecommons.org/licenses/by-nc-nd/3.0/ - all other rights reserved
Using Proportional Reasoning
MATHEMATICAL GOALS
This lesson unit is intended to help you assess how well students are able to reason proportionally
when comparing the relationship between two quantities expressed as unit rates and/or part-to-part
ratios. In particular, it will help you assess how well students are able to:
•
•
•
Describe a ratio relationship between two quantities.
Compare ratios expressed in different ways.
Use proportional reasoning to solve a real-world problem.
COMMON CORE STATE STANDARDS
This lesson gives students the opportunity to apply their knowledge of the following Standards for
Mathematical Content in the Common Core State Standards for Mathematics:
6.RP:
Understand ratio concepts and use ratio reasoning to solve problems.
This lesson also relates to all the Standards for Mathematical Practice in the Common Core State
Standards for Mathematics, with a particular emphasis on Practices 1, 2, 3, 4, and 6:
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
INTRODUCTION
The lesson unit is structured in the following way:
• Before the lesson, students work individually on an assessment task designed to reveal their
current understanding and difficulties. You then review their solutions and create questions for
students to consider, in order to improve their work.
• After a whole-class introduction, students work in groups, putting diagrams and descriptions of
orange and soda mixtures into strength order. Students then compare their work with their peers.
• Next, in a whole-class discussion, students critique some sample work stating reasons why two
mixtures would or wouldn’t taste the same. Students then revise and correct any misplaced cards.
• After a final whole-class discussion, students work individually either on a new assessment task,
or return to the original task and try to improve their responses.
MATERIALS REQUIRED
•
•
•
Each student will need a mini-whiteboard, pen, and eraser, and a copy of Mixing Drinks and
Mixing Drinks (revisited).
Each small group of students will need the cut-up Card Set: Orange and Soda Mixtures and Card
Set: Blank Cards, a sheet of poster paper and a glue stick.
You may wish to have some orange juice and soda for mixing/tasting but this is not essential.
TIME NEEDED
15 minutes before the lesson, a 100-minute lesson (or two 55-minute lessons), and 15 minutes in a
follow-up lesson. Timings given are approximate and will depend on the needs of your class.
Teacher guide
Using Proportional Reasoning
T-1
BEFORE THE LESSON
Assessment task: Mixing Drinks (15 minutes)
Have students complete this task, in class or for
Mixing Drinks
homework, a few days before the formative
When Sam and his friends get together, Sam makes a fizzy orange drink
assessment lesson. This will give you an
by mixing orange juice with soda.
On Friday, Sam makes 7 liters of fizzy orange by mixing 3 liters of orange
opportunity to assess the work and to find out
juice with 4 liters of soda.
On Saturday, Sam makes 9 liters of fizzy orange by mixing 4 liters of
the kinds of difficulties students have with it.
orange juice with 5 liters of soda.
You should then be able to target your help
1. Does the fizzy orange on Saturday taste the same as or different to Friday’s fizzy orange?
If you think it tastes the same, explain how you can tell.
more effectively in the lesson that follows.
If you think it tastes different, does it taste more or less orangey? Explain how you know.
Give each student a copy of Mixing Drinks.
Introduce the task briefly, helping the class to
understand the task:
This task is about making a fizzy orange
drink by mixing different quantities of
orange and soda.
You are going to compare how orangey the
drinks will taste, as well as working out the
amount of orange and soda needed to make
fizzy orange with a similar orangey taste.
2. On Sunday, Sam wants to make 5 liters of fizzy orange that tastes slightly less orangey than
Friday’s and Saturday’s fizzy orange. For every liter of orange, how many liters of soda should be
added to the mixture? Explain your reasoning.
It is important that, as far as possible, students
answer the questions on the sheet without assistance. If students are struggling to get started, ask
questions that help them understand what they are being asked to do, but do not do the task for them.
Students should not worry too much if they cannot understand or do everything, because there will be
a lesson related to this, which should help them. Explain to students that by the end of the next lesson
they should expect to answer questions such as these confidently; this is their goal.
Student Materials
Proportional Reasoning
© 2013 MARS, Shell Center, University of Nottingham
S-1
Assessing students’ responses
Collect students’ responses to the task. Make some notes on what their work reveals about their
current levels of understanding and their different problem-solving approaches.
We suggest that you do not score students’ work. Research suggests that this will be
counterproductive, as it will encourage students to compare their scores and distract their attention
from what they can do to improve their mathematics. Instead, help students to make further progress
by summarizing their difficulties as a series of questions. Some suggestions for these are given in the
Common issues table on the next page. These have been drawn from common difficulties observed in
trials of this unit.
We recommend that you:
•
write one or two questions on each student’s work, or
•
give each student a printed version of your list of questions, and highlight the questions for each
individual student.
If you do not have time to do this, you could select a few questions that will be of help to the majority
of students and write these questions on the board when you return the work to the students in the
follow-up lesson.
Teacher guide
Using Proportional Reasoning
T-2
Common issues
Suggested questions and prompts
Reasons additively rather than multiplicatively
For example: The student states that the fizzy
orange tastes the same on Saturday as it did on
Friday because one more liter of orange and one
more liter of soda has been added and these just
‘cancel each other out’ (Q1).
Or: The student states that the fizzy orange tastes
the same on Saturday as it did on Friday because
both mixtures contain one more liter of soda than
orange (Q1).
• How could you use math to check that the
addition of a liter of orange and a liter of soda
has no effect on the taste?
• What would happen to the taste if a liter of
orange and a liter of soda were added to 1 liter
of soda?
• If 3 liters of fizzy orange was made in the
same way, by mixing 1 liter of orange with 2
liters of soda, would this taste the same also?
Sole focus on orange as the ‘active’ ingredient
For example: The student thinks that Saturday’s
fizzy orange will taste more orangey than
Friday’s, because it has more orange in it than
Friday’s has (Q1).
• How much soda is in Saturday’s fizzy orange?
How much soda is in Friday’s fizzy orange?
What do you notice?
• Is how orangey the fizzy orange tastes
determined by the number of liters of orange it
contains?
Sole focus on soda as the diluting ingredient
For example: The student thinks that Saturday’s
fizzy orange will taste less orangey than Friday’s,
because it has more soda in it than Friday’s has
(Q1).
• How much orange is in Saturday’s fizzy
orange? How much orange is in Friday’s fizzy
orange? What do you notice?
• If 5 liters of fizzy orange were made by mixing
4 liters of soda with 1 liter of orange, would it
also taste more orangey than Saturday’s fizzy
orange?
Provides an explanation based on one mixture
only
For example: The student states that Saturday’s
fizzy orange will taste less orangey than Friday’s,
because the mixture contains less orange in it than
soda (Q1).
• Does Friday’s fizzy orange contain more
orange than soda or more soda than orange?
• How can you compare the taste of Saturday’s
fizzy orange to the taste of Friday’s fizzy
orange?
Makes incorrect assumptions
• Will this fizzy orange mixture taste slightly
less orangey than Friday’s and Saturday’s
fizzy orange?
For example: The student thinks that on Sunday,
Sam should mix 1 liter of orange with 4 liters of
soda because 2 liters of orange with 3 liters of
soda will taste the same as Friday’s and
Saturday’s fizzy orange (Q2).
Or: The student assumes that for every liter of
orange two liters of soda are required (Q2).
Provides little mathematical explanation
• Can you use math to explain your answer?
Completes the task correctly
• Can you find a fizzy orange mixture that is
more orangey than Friday’s fizzy orange but
less orangey than Saturday’s fizzy orange?
The student needs an extension task.
Teacher guide
Using Proportional Reasoning
T-3
SUGGESTED LESSON OUTLINE
Whole-class introduction (10 minutes)
Give each student a mini-whiteboard, pen, and eraser. Remind the class of the assessment task they
have already attempted.
Recall what we were working on previously. What was the task about?
In today’s lesson we are going to consider different mixtures of orange and soda used to make
fizzy orange and think about which ones taste more/less orangey.
Display Slide P-1 of the projector resource:
Which is strongest?
Card 1:
Card 2:
Card 3:
Projector Resources
Proportional Reasoning
P-1
Each of these three cards describes a fizzy orange mixture.
The diagrams on cards 1 and 3 show the amount of orange and soda in the mix (where the
shaded boxes represent the orange and the dotted boxes represent the soda) and card 2 gives a
description of the fraction of the fizzy orange mixture that is orange.
Working on your own, on your mini-whiteboard, write the card numbers in order from least
orangey to most orangey. [Card 3, Card 1, Card 2.]
Give students a few minutes to work on this before asking to see their whiteboards. If there are a
range of responses within the class, collate them on the board and hold a whole-class discussion.
Spend a few minutes discussing the strategies used to compare the three cards.
Explain to students that they are going to be working in groups on a similar activity putting cards in
order of strength from least orangey to most orangey.
Individual think time, then collaborative work: Orange and Soda Mixtures (30 minutes)
Before students work collaboratively, it can be helpful to give students individual ‘thinking time’.
This allows everyone to have time to construct ideas to share and avoids the conversation being
dominated by one student.
Organize students into groups of two or three. Give each group the cut-up Card Set: Orange and
Soda Mixtures, a sheet of poster paper, and a glue stick.
On these cards there are descriptions of fizzy orange mixtures.
Some cards show the number of orange and soda juice boxes in the mixture, some contain a
written description of the mixture and some show empty juice boxes which you will need to shade
in (color orange juice boxes and draw dots for soda.)
Teacher guide
Using Proportional Reasoning
T-4
Display Slide P-2 of the projector resource:
Individual think time
Your task is to work with your partner to put the
cards in order of strength, from least orangey (on
the left) to most orangey (on the right).
1.  Look at the cards and think about ways you
could carry out this task.
2.  Write your ideas on your mini-whiteboards.
There is no need for students to order the cards during this individual activity.
When students have had sufficient time to think about the task:
Projector Resources
Proportional Reasoning
P-2
First, take turns to explain to each other your ideas for how to carry out the task.
Ask questions if you do not understand your partner’s explanation.
Take a few minutes to come up with a joint plan of action.
Display Slide P-3 of the projector resource and explain how students are to work together on the task:
Working together
1. 
Work together to put the cards in order of strength, taking turns
with the work.
a.  Explain decisions to your partner.
2. 
If you think more than one card describes the same fizzy orange
mixture, group them together.
a.  If a group of cards does not contain a juice box card, then
shade in one of the Cards M - P.
3. 
When you both agree where each card should go and why, glue
them onto your poster. On your poster, explain your decisions.
While students are working, you have two tasks: to notice their approaches to the task and to support
student problem solving.
Projector Resources
Proportional Reasoning
P-3
Make a note of student approaches to the task
Listen and watch students carefully. In particular, notice how students make a start on the task, where
they get stuck, and how they overcome any difficulties.
Do they begin with what they think is the strongest or weakest mixture or do they just pick a random
card? Do students compare orange to soda (e.g. for every orange there are 2 soda) or orange to
mixture (e.g. ½ the mixture is orange). When they discover cards that are of equal strength, how do
they justify this to one another? Do they use fractions, decimals, percentages, ratios or proportions?
Do they switch between different descriptions? How do they go about shading cards M to P?
You can use this information to focus a whole-class discussion towards the end of the lesson.
Support student problem solving
As students work on the task support them in working together. Encourage them to take turns and if
you notice that one partner is doing all the ordering or that they are not working collaboratively on the
task, ask a student in the group to explain a card placed by someone else in the group.
Teacher guide
Using Proportional Reasoning
T-5
Try not to make suggestions that push students towards a particular approach to the task. Instead, ask
questions to help students clarify their thinking. The following questions and prompts may be helpful:
Which mixture do you think is the most orangey? Why?
How do you know that this mixture is more orangey than that one?
Why does this card come here?
Encourage students to write on the cards.
If several students in the class are struggling with the same issue, you could write one or two relevant
questions on the board and hold a brief whole-class discussion. For example, if students are using
‘additive’ rather than ‘multiplicative’ reasoning; e.g. thinking that 3:5 (Card B) is the same as 4:6
(Card E) you could ask:
Why do you think that these will taste the same?
Can you think of another fizzy orange mixture that will also taste the same? How do you know?
Students who finish early with the cards in the right order could be given cut-up Card Set: Blank
Cards and asked:
Can you invent a card that would go in between these two?
Can you invent a card that would go in the same place as this one?
What would you add to this mixture to make it taste like this mixture?
Sharing work (15 minutes)
Give students the opportunity to compare their work by visiting another group. It is likely that some
groups will not have ordered all the cards but a comparison can still be made as to whether students
consider a particular card to be more orangey or less orangey than another. It may be helpful for
students to jot down on their mini-whiteboards their agreed order of the cards before they visit
another group.
Show Slide P-4 and explain how students are to share their work:
Sharing work
1.  One person from each group get up and visit a different
group.
2.  If you are staying with your poster, explain your card
order to the visitor, justifying the placement of each
card.
3.  If you are the visitor, look carefully at the work and
challenge any cards that you think are in the wrong
place.
4.  If you agree on the placement of the cards, compare
your methods used when ordering.
Projector Resources
Proportional Reasoning
P-4
Extending the lesson over two days
If you are taking two days to complete the unit then you may want to end the first lesson here. At the
start of the second day, allow time for students to remind themselves of their work before moving on
to discuss their ordering of the cards as a whole-class.
Whole-class discussion (25 minutes)
Now hold a brief whole-class discussion in which students discuss their ordering. Draw attention to
significant differences between the ordering that particular groups have arrived at.
Teacher guide
Using Proportional Reasoning
T-6
Were there any disagreements when you compared your work? Someone give me an example.
What reasoning did you each give?
Was different math used to figure out the ordering? [E.g. orange to soda or orange to mixture]
Once you have had a chance to compare reasons given, spend some time exploring conflicting
reasoning/conclusions when comparing the following two fizzy orange mixtures:
Display Slide P-5 of the projector resource to show Emmanuel’s reasoning and ask:
What do you think about Emmanuel’s
reasoning? Is he right or wrong? Why?
Students should be suspicious of this kind of
‘linear’ reasoning by now and if they are not you
could explore what would happen if you
continued the pattern to the left (2 orange and 3
sodas, 1 orange and 2 sodas etc.). Taking one
more step to the left we would have no orange
and 1 soda. There is still ‘one more soda than
orange’ but everyone will agree that this will not
taste orangey at all!
Emmanuel’s Reasoning
Both of these have one more soda than
orange, so they will taste the same.
Projector Resources
Now display Slide P-6 of the projector resource
showing Sifi’s reasoning and ask:
Sifi’s Reasoning
What do you think about Sifi’s reasoning? Is
she right or wrong? Why?
Sifi’s method is better than Emmanuel’s because
she is thinking proportionately, but she has made
an error; 1 14 is correct for the right-hand mixture,
In both cases, for every orange there is 1
for the number of soda juice boxes per orange
juice box, but the left-hand mixture is 1 13 .
Projector Resources
orange not
4
5
(ratio 4:5). Since
3
7
is less than
3
of the whole mixture is
4
4
orange, whereas in the second case is
5
orange so they will taste different.
4
9
4
9
P-4
In the first case,
orange (it is in the ratio of 3:4
(orange:soda)) and the right-hand mixture is
Proportional Reasoning
Alex’s Reasoning
What do you think about Alex’s reasoning?
Is he right or wrong? Why?
Alex has come to the correct conclusion about
the mixtures not tasting the same but his method
contains an error. The left-hand mixture is 73
3
4
1
4
soda, so they will both taste the same.
Now use Slide P-7 of the projector resource to
display Alex’s reasoning and ask:
orange not
P-2
The Pythagorean Theorem: Square Areas
, the right-hand mixture will be slightly more orangey
Projector Resources
Proportional Reasoning
P-5
(but it may be hard to tell this small difference in practice!)
Teacher guide
Using Proportional Reasoning
T-7
Finally, you might want to ask:
Did you use any of these methods? Which ones?
Did you use any other methods? What were they?
What do you think now about all of these methods?
Poster review (10 minutes)
Students now have an opportunity to reconsider the ordering of their cards:
Now that you have had a chance to compare and discuss your work and we have looked at what
Emmanuel, Sifi and Alex have said, you might like to have another look at your poster and decide
in your groups whether you are still happy with where you have placed the cards.
If you think a card is in the wrong place, draw an arrow on your poster to where you think it
should go.
While this is happening, encourage students to voice their reasoning for the movement of a card.
Whole-class discussion (10 minutes)
You may want to finish with a brief whole-class discussion in which students discuss their ordering
and talk more generally about what they have gained from the lesson.
Did you change your ordering after we talked together about it? Why / Why not?
How confident are you with your ordering now?
What have you learnt today about how you get mixtures that taste the same or different?
Use your knowledge of the students’ group work to call on a wide range of students for contributions.
Follow-up lesson: reviewing the assessment task (15 minutes)
Give students their responses to the original assessment task Mixing Drinks and a copy of the task
Mixing Drinks (revisited). If you have not added questions to individual pieces of work then write
your list of questions on the board. Students then select from this list only those questions they think
are appropriate to their own work.
Look at your original responses and the questions [on the board/written on your paper]. Answer
these questions and revise your response.
On your mini-whiteboard make some notes on what you have learned during the lesson. Now
have a go at the second sheet: Mixing Drinks (revisited). Can you use what you have learned to
answer these questions?
If students struggled with the original assessment task, you may feel it more appropriate for them to
revisit Mixing Drinks rather than attempting Mixing Drinks (revisited). If this is the case give them
another copy of the original assessment task instead.
If you are short of time you could give this task for homework.
Teacher guide
Using Proportional Reasoning
T-8
SOLUTIONS
Assessment task: Mixing Drinks
1. The ratio of orange to soda on Friday is 3:4, which is not equal to the ratio of orange to soda on
Saturday (4:5), so the fizzy orange mixtures will not taste the same. Friday’s mixture is 37 orange
and Saturday’s mixture is
orangey ( 37 =
27
63
4
9
orange. Comparing these fractions to see which will taste the most
compared with
4
9
=
28
63
) reveals that Saturday’s fizzy orange mixture will taste
more orangey. However, students may comment that even though Saturday’s fizzy orange is
stronger than Friday’s, it is likely that you would not be able to taste any difference because the
difference is only very slight.
2. If Sam mixes 2 liters of orange with 3 liters of soda, the mixture will be
2
5
orange, which is
slightly less orangey than Friday’s and Saturday’s mixture. This means that for every liter of
orange, 1 12 liters of soda should be added to the mixture.
Assessment task: Mixing Drinks (revisited)
1. The completed table is as follows: (missing values are identified in bold)
Amount of Raspberry
Juice (liters)
Amount of Apple
Juice (liters)
Amount of Soda
(liters)
Total Amount of
Fabulous Fruit Fizz
(liters)
1
2
3
6
0.5
1
1.5
3
2
4
6
12
2. a.
2
of the drink is apple juice.
5
b.
2
of the drink is apple juice.
5
c.
1
of the drink is apple juice.
3
Mixture c is the least appley drink. Qaylah should mix, for every liter of apple, 2 liters of soda.
Teacher guide
Using Proportional Reasoning
T-9
Collaborative task:
Card Set:
and SodatoMixtures
The correct matching/ordering
fromOrange
least orangey
most orangey (with ratio of orange to soda also
A
B
given) is as follows:
1:3
C
D
E
F
Half of the mixture
is orange
1:2
G
3:5
For every orange
there are 2 sodas
I
A
2:3
For every orange
there is 1 1 soda
4
J
One fourth of the
mixture is orange
K
2
3
of the mixture is
soda
Card Set: Orange and
Soda Mixtures
L
For every orange
there is 1 1 soda
B
3
C
For every soda
there is 2 orange
3
D
Student Materials
3:4
H
Proportional Reasoning
© 2013 MARS, Shell Center, University of Nottingham
E
S-2
F
Half of the mixture
is orange
G
4:5
For every orange
there are 2 sodas
I
H
J
One fourth of the
mixture is orange
1:1
K
For every orange
there is 1 1 soda
3
L
Orange : Soda
=4:5
2
3
of the mixture is
soda
For every soda
there is 2 orange
3
Card N has been designed so that it cannot be shaded to be equivalent to any of
the other cards. Students should shade the card with a number of orange/soda
juice boxes of their choice (between 1 and 10) and then place it in the
appropriate place based on how orangey the mixture is.
Student Materials
Proportional Reasoning
© 2013 MARS, Shell Center, University of Nottingham
S-2
For example, they may choose to shade it in the ratio of 5 orange: 6 soda and
place it between cards C and M.
Teacher guide
Using Proportional Reasoning
T-10
Mixing Drinks
When Sam and his friends get together, Sam makes a fizzy orange drink
by mixing orange juice with soda.
On Friday, Sam makes 7 liters of fizzy orange by mixing 3 liters of orange
juice with 4 liters of soda.
On Saturday, Sam makes 9 liters of fizzy orange by mixing 4 liters of
orange juice with 5 liters of soda.
1. Does the fizzy orange on Saturday taste the same as Friday’s fizzy orange, or different?
If you think it tastes the same, explain how you can tell.
If you think it tastes different, does it taste more or less orangey? Explain how you know.
2. On Sunday, Sam wants to make 5 liters of fizzy orange that tastes slightly less orangey than
Friday’s and Saturday’s fizzy orange. For every liter of orange, how many liters of soda should be
added to the mixture? Explain your reasoning.
Student materials
Using Proportional Reasoning
© 2015 MARS, Shell Center, University of Nottingham
S-1
Card Set: Orange and Soda Mixtures
A
B
C
D
E
F
Half of the mixture
is orange
G
H
For every orange
there are 2 sodas
I
Orange : Soda
=4:5
J
One fourth of the
mixture is orange
K
For every orange
there is 1 1 soda
3
Student materials
2
3
of the mixture is
soda
L
For every soda
there is 2 orange
Using Proportional Reasoning
© 2015 MARS, Shell Center, University of Nottingham
3
S-2
Card Set: Orange and Soda Mixtures (continued)
M
N
Shade in:
O Shade in:
Shade in:
P
Shade in:
Card Set: Blank Cards
Student materials
Using Proportional Reasoning
© 2015 MARS, Shell Center, University of Nottingham
S-3
Mixing Drinks (revisited)
To make 6 liters of Fruit Fizz,
mix 1 liter of raspberry juice, 2 liters of
apple juice and 3 liters of soda
1. Complete the table below with the amounts of raspberry juice, apple juice and soda needed to
make the different quantities of Fruit Fizz. The mixture must taste exactly the same each time.
Amount of Raspberry
Juice (liters)
Amount of Apple
Juice (liters)
Amount of Soda
(liters)
Total Amount of
Fruit Fizz (liters)
1
2
3
6
1
12
2. Here are three ways to make apple fizz:
2
liters of apple juice.
3
b. Mix apple and soda in the ratio 2 : 3.
a. For each liter of soda mix
c.
2
of the mixture is soda, the rest is apple juice.
3
Qaylah wants to mix the least appley drink. Which mixture should she choose?
For every liter of apple, how many liters of soda should she add to the mixture?
Explain your reasoning.
Student materials
Using Proportional Reasoning
© 2015 MARS, Shell Center, University of Nottingham
S-4
Which is strongest?
Card 1:
Card 2:
Card 3:
Projector Resources
Using Proportional Reasoning
P-1
Individual think time
Your task is to work with your partner to put the
cards in order of strength, from least orangey (on
the left) to most orangey (on the right).
1.  Look at the cards and think about ways you
could carry out this task.
2.  Write your ideas on your mini-whiteboards.
Projector Resources
Using Proportional Reasoning
P-2
Working together
1. 
Work together to put the cards in order of strength, taking turns
with the work.
a.  Explain decisions to your partner.
2. 
If you think more than one card describes the same fizzy orange
mixture, group them together.
a.  If a group of cards does not contain a juice box card, then
shade in one of the Cards M - P.
3. 
When you both agree where each card should go and why, glue
them onto your poster. On your poster, explain your decisions.
Projector Resources
Using Proportional Reasoning
P-3
Sharing work
1.  One person from each group get up and visit a different
group.
2.  If you are staying with your poster, explain your card
order to the visitor, justifying the placement of each
card.
3.  If you are the visitor, look carefully at the work and
challenge any cards that you think are in the wrong
place.
4.  If you agree on the placement of the cards, compare
your methods used when ordering.
Projector Resources
Using Proportional Reasoning
P-4
Emmanuel’s Reasoning
Both of these have one more soda
than orange, so they will taste the
same.
Projector Resources
Using Proportional Reasoning
P-5
Sifi’s Reasoning
In both cases, for every orange there is 1
1
4
soda, so they will both taste the same.
Projector Resources
Using Proportional Reasoning
P-6
Alex’s Reasoning
3
of the whole mixture is
4
4
orange, whereas in the second case is
5
In the first case,
orange so they will taste different.
Projector Resources
Using Proportional Reasoning
P-7
Mathematics Assessment Project
Classroom Challenges
These materials were designed and developed by the
Shell Center Team at the Center for Research in Mathematical Education
University of Nottingham, England:
Malcolm Swan,
Nichola Clarke, Clare Dawson, Sheila Evans, Colin Foster, and Marie Joubert
with
Hugh Burkhardt, Rita Crust, Andy Noyes, and Daniel Pead
We are grateful to the many teachers and students, in the UK and the US,
who took part in the classroom trials that played a critical role in developing these materials
The classroom observation teams in the US were led by
David Foster, Mary Bouck, and Diane Schaefer
This project was conceived and directed for
The Mathematics Assessment Resource Service (MARS) by
Alan Schoenfeld at the University of California, Berkeley, and
Hugh Burkhardt, Daniel Pead, and Malcolm Swan at the University of Nottingham
Thanks also to Mat Crosier, Anne Floyde, Michael Galan, Judith Mills, Nick Orchard, and Alvaro
Villanueva who contributed to the design and production of these materials
This development would not have been possible without the support of
Bill & Melinda Gates Foundation
We are particularly grateful to
Carina Wong, Melissa Chabran, and Jamie McKee
The full collection of Mathematics Assessment Project materials is available from
http://map.mathshell.org
© 2015 MARS, Shell Center, University of Nottingham
This material may be reproduced and distributed, without modification, for non-commercial purposes,
under the Creative Commons License detailed at http://creativecommons.org/licenses/by-nc-nd/3.0/
All other rights reserved.
Please contact [email protected] if this license does not meet your needs.