By Karen S. Karp, Sarah B. Bush, and Barbara J. Dougherty · 2016. 8. 18. · states that Kate has 2 cookies to divide among herself and two friends, then the portion for each person

RULES 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. Dougherty

That Expire

1318 August 2014 • teaching children mathematics | Vol. 21, No. 1 www.nctm.org

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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I magine the following scenario: A primary teacher presents to her students the fol-lowing set of number sentences:

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 inter-pret 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.

T h e C o m m o n C o r e State Standards (CCSS) for Mathematical Practice advocate for students to become problem solvers who can reason, apply, just i fy, and ef fect ively

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 over-

generalize it.4. Provide counterexamples, noting when the

rule is not true. ALI

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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 con-tent standards in which the rule no longer “always” works.

Thirteen rules that expire1. 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 refl ect 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 key-words 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, computa-tional 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

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).

3. You cannot take a bigger number from a smaller number. Students might hear this phrase as they fi rst 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 applica-tion or word problems involving contexts that include integers, students learn that this “rule” is not true for all problems. For example, a gro-cery 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 nega-tive number, (4º – 22º = –18º). Expiration date: Grade 7 (7.NS.1).

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 counter-examples. 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

14

13

112

MathType 2

8 4 2 or 4 812

MathType 3

14

25

58

MathType 4

12

MathType 5

12

2 = 4

17

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÷ = ÷ =

÷ =

÷

This is also the case when one of the factors is a negative number and the other factor is positive, such as –3 × 8= –24. Expiration date: Grade 5 (5.NF.4 and 5.NBT.7) and again at Grade 7 (7.NS.1 and 7.NS.2).

5. Subtraction and division make numbers smaller. This rule is commonly heard in grade 3: both subtraction and division will result in an answer that is smaller than at least one of the

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numbers in the computation. When numbers are positive whole numbers, decimals, or frac-tions, subtracting will result in a number that is smaller than at least one of the numbers involved in the computation. However, if the subtraction involves two negative numbers, students may notice a contradiction (e.g., –5 – (– 8) = 3). In division, the rule is true if the num-bers are positive whole numbers, for example:

MathType 1

14

13

112

MathType 2

8 4 2 or 4 812

MathType 3

14

25

58

MathType 4

12

MathType 5

12

2 =

17

× =

÷ = ÷ =

÷ =

÷ 14

However, if the numbers you are dividing are fractions, the quotient may be larger:

MathType 1

14

13

112

MathType 2

8 4 2 or 4 812

MathType 3

14

25

58

MathType 4

12

MathType 5

12

2 =

17

× =

÷ = ÷ =

÷ =

÷ 14

This is also the case when dividing two nega-tive factors: (e.g., –9 ÷ –3 = 3). Expiration dates: Grade 6 (6.NS.1) and again at Grade 7 (7.NS.1 and 7.NS.2c).

6. You always divide the larger number by the smaller number. This rule may be true when students begin to learn their basic facts for whole-number divi-sion and the computations are not contextu-ally based. But, for example, if the problem states that Kate has 2 cookies to divide among herself and two friends, then the portion for each person is 2 ÷ 3. Similarly, it is possible to have a problem in which one number might be a fraction:

Jayne has

MathType 1

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13

112

MathType 2

8 4 2 or 4 812

MathType 3

14

25

58

MathType 4

12

MathType 5

12

2 =

17

× =

÷ = ÷ =

÷ =

÷ 14

of a pizza and wants to share it with her brother. What portion of the whole pizza will each get?

In this case, the computation is as follows:

MathType 1

14

13

112

MathType 2

8 4 2 or 4 812

MathType 3

14

25

58

MathType 4

12

MathType 5

12

2 =

17

× =

÷ = ÷ =

÷ =

÷ 14

Expiration date: Grade 5 (5.NF.3 and 5.NF.7).

7. Two negatives make a positive. 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 inte-gers, such as in –5 + (–3) = –8. Expiration date: Grade 7 (7.NS.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 num-bers (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 under-standing of terms and their partial understand-ing 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 con-text 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

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relationships are fixed. For example, the rela-tionship between 3 and 8 is always the same. To determine the relationship between two num-bers, 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 mag-nitude. 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).

12. Please Excuse My Dear Aunt Sally. This phrase is typically taught when students begin solving numerical expressions involv-ing multiple operations, with this mnemonic serving as a way of remembering the order of operations. Three issues arise with the appli-cation of this rule. First, students incorrectly believe that they should always do multipli-cation 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

Some commonly used language “expires ” and should be replaced with more appropriate alternatives.

Expired mathematical language and suggested alternativesWhat is stated What should be stated

Using the words borrowing or carrying when subtracting or adding, respectively

Use trading or regrouping to indicate the actual action of trading or exchanging one place value unit for another unit.

Using the phrase ___ out of __ to describe a fraction, for example, one out of seven to describe

MathType 1

14

13

112

MathType 2

8 4 2 or 4 812

MathType 3

14

25

58

MathType 4

12

MathType 5

12

2 = 4

17

× =

÷ = ÷ =

÷ =

÷

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 num-ber 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.

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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. An equal sign is a relational symbol. It indi-cates that the two quantities on either side of it represent the same amount. It is not a signal prompting the answer through an announce-ment 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 languageIn addition to helping students avoid the thir-teen 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 mathemat-ics (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 dateOne characteristic of the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) is to have fewer, but deeper, more rigor-ous standards at each grade—and to have less

overlap and greater coherence as students progress from K–grade 12. We feel that by using consistent, accurate rules and precise vocabu-lary 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 pre-pare students and allow them to have smoother transitions moving from grade to grade. Addi-tionally, 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 stan-dards, 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.”

REFERENCESBoaler, 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 Offi cers. 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.

Other rules that expireWe 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.

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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.” Educa-tional 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, [email protected], 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.

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Engage Students in Learning: Fawn Nguyen

Mathematical Practices [email protected]

An NCTM Interactive Institute for Grades K-8 Twitter:

@fawnpnguyen

Atlanta, July 11-13, 2016 Blog: fawnnguyen.com

CCSS Grade 6: Understand ratio concepts and use ratio reasoning to solve problems.

CCSS.MATH.CONTENT.6.RP.A.1

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 real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

CCSS.MATH.CONTENT.6.RP.A.3.B

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?


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

CCSS.MATH.CONTENT.6.RP.A.3.A

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.

CCSS.MATH.CONTENT.6.RP.A.3.C

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.

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

http://www.corestandards.org/Math/Content/6/RP/A/1/


http://www.corestandards.org/Math/Content/6/RP/A/3/b/


http://www.corestandards.org/Math/Content/6/RP/A/3/a/

http://www.corestandards.org/Math/Content/6/RP/A/3/c/

http://www.corestandards.org/Math/Content/6/RP/A/3/d/




@fawnpnguyen


quantities.

CCSS Grade 7: Analyze proportional relationships and use them to solve real-world and mathematical problems.


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.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.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.


Recognize and represent proportional relationships between quantities.

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.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.


Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions,


http://www.corestandards.org/Math/Content/7/RP/A/2/a/

http://www.corestandards.org/Math/Content/7/RP/A/2/c/


http://www.corestandards.org/Math/Content/7/RP/A/2/b/

http://www.corestandards.org/Math/Content/7/RP/A/2/d/





@fawnpnguyen


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 Question Topic

1. Reasoning with ratios involves attending to and coordinating two quantities.

How does ratio reasoning differ from other types of reasoning?

Ratios

2. A ratio is a multiplicative comparison of two quantities, or it is a joining of two quantities in a composed unit.

What is a ratio?

3. Forming a ratio as a measure of a realworld attribute involves isolating that attribute from other attributes and understanding the effect of changing each quantity on the attribute of interest.

What is a ratio as a measure of an attribute in a realworld situation?

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 the corresponding values of the quantities change.

What is a proportion? Proportions

http://www.nctm.org/store/Products/Developing-Essential-Understanding-of-Ratios,-Proportions,-and-Proportional-Reasoning-for-Teaching-Mathematics--Grades-6-8/


Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning Grades 68 (source: NCTM)

Essential Understanding Question Topic

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 factor, then the other quantity 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.

What are the key aspects of proportional reasoning?

Proportional Reasoning

8. A rate is a set of infinitely many equivalent ratios.

What is a rate and how is it related to proportional reasoning?

9. Several ways of reasoning, all grounded in sense making, can be generalized into algorithms for solving proportion problems.

What is the relationship between the crossmultiplication algorithm and proportional reasoning?

10. Superficial cues present in the context of a problem do not provide sufficient evidence of proportional relationships between quantities.

When is it appropriate to reason proportionately?

http://www.nctm.org/store/Products/Developing-Essential-Understanding-of-Ratios,-Proportions,-and-Proportional-Reasoning-for-Teaching-Mathematics--Grades-6-8/


Task or Question EssentialUnderstandings

Common Core StateStandards

What is a fraction? 4Write a sentence with a fraction in it (not the word “fraction,”but an actual number) to describe something in the room. 4What is a ratio? 2Write a sentence with a ratio in it (not the word “ratio,” but anactual number) to describe something in the room. 2Three-fifths of the sand went through a sand timer in 18minutes. 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 thesand timer? Show two different ways to solve this problem.

2, 8

PoW: Ration Ratios 2, 3To make Luscious Lilikoi Punch, Austin mixes 1/2 cup of lilikoipassion fruit concentrate with 2/3 cup water. If he wants to mixconcentrate and water in the same ratio to make 28 cups ofLuscious Lilikoi Punch, how many cups of lilikoi passion fruitconcentrate and how many cups of water will Austin need?

8

Dan Meyer's video Expected Value (blue/purple circle) 2, 3Taufique has two bags, each containing some red and somewhite gumballs. In bag A, there are 10 red and 15 whitegumballs. In bag B, there are 6 red and 8 white gumballs.

If Taufique reaches in without looking, from which bag is hemore likely to pull out a red gumball?

Draw a diagram to represent how you solved the problem, andexplain it.

2, 3, 4

How does a probability of pulling a white gumball from a bagwith 3 white and 2 red gumballs compare to the probability ofpulling a white gumball from a bag with 12 white and 8 redgumballs?

3Which road is steeper, one with a 4 percent grade for 0.5 milesor one with a 4 percent grade of 0.25 miles? 3Are the fractions 6/9 and 10/15 equivalent? 7Jonnine had a board. She cut and used 2/5 of the board forbracing. She measured the piece used for bracing and found itto be 3/4 foot long. How long was the original board?

7, 9

Two friends play a game of coin toss. A wins if 5 heads comeup first, and B wins if 5 tails come up first. Each person puts in$10 bet; winner takes all of $20.

Game must end, however, when A has 2 heads and B has 4tails. How should they split the money?

2

As much as possible, use only mental arithmetic to determinewhich is larger, 14/29 or 15/31? 9MARS Lesson: Using Proportional Reasoning

Handout Ratios & Proportions (1) 1, 2, 7Handout Ratios & Proportions (2) 2, 7Handout Ratios & Proportions (3) 3, 8, 4

https://docs.google.com/document/d/1rmE562s8nsv7MXRIWLRRRiNgWk7O0qNg3LLYuYthUPU/edit?usp=sharing

https://docs.google.com/document/d/1fM0aVuNYPnlN4W3PmCAjz7Lv6jbumIhVopWVQLWB_MI/edit?usp=sharing

https://docs.google.com/document/d/1nkRzBJ30kJIDm27UrmcC7UC8FClJpGIY2_m5ZuufOrU/edit?usp=sharing


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?

http://mathforum.org/


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.

http://donsteward.blogspot.com/

http://www2.stetson.edu/~efriedma/weight/



Ratios and Proportions (2) (source: http://donsteward.blogspot.com/)

http://donsteward.blogspot.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?

https://www.amazon.com/Challenge-Elementary-Middle-School-Student/dp/0967991552/ref=sr_1_5?ie=UTF8&qid=1467244664&sr=8-5&keywords=Edward+Zaccaro

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

Teacher guide Using Proportional Reasoning T-1

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. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. 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.


BEFORE THE LESSON

Assessment task: Mixing Drinks (15 minutes) Have students complete this task, in class or for homework, a few days before the formative assessment lesson. This will give you an opportunity to assess the work and to find out the kinds of difficulties students have with it. You should then be able to target your help more effectively in the lesson that follows.

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.

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.

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.

Student Materials Proportional Reasoning S-1 © 2013 MARS, Shell Center, University of Nottingham

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 or different to Friday’s fizzy orange?

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.


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

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).

• Will this fizzy orange mixture taste slightly less orangey than Friday’s and Saturday’s fizzy orange?

Provides little mathematical explanation • Can you use math to explain your answer?

Completes the task correctly The student needs an extension task.

• Can you find a fizzy orange mixture that is more orangey than Friday’s fizzy orange but less orangey than Saturday’s fizzy orange?


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:

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.)

Proportional Reasoning Projector Resources

Which is strongest?

P-1

Card 1:

Card 2:

Card 3:


Display Slide P-2 of the projector resource:

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:

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:

While students are working, you have two tasks: to notice their approaches to the task and to support student problem solving.

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.


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.

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.

P-3


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:

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.


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.

P-4


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! Now display Slide P-6 of the projector resource showing Sifi’s reasoning and ask:

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; is correct for the right-hand mixture, for the number of soda juice boxes per orange juice box, but the left-hand mixture is .

Now use Slide P-7 of the projector resource to display Alex’s reasoning and ask:

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

orange not orange (it is in the ratio of 3:4

(orange:soda)) and the right-hand mixture is

orange not (ratio 4:5). Since is less than , the right-hand mixture will be slightly more orangey (but it may be hard to tell this small difference in practice!)

411

311

73

43

94

45

37

49


Sifi’s Reasoning

P-4

In both cases, for every orange there is 1 14

soda, so they will both taste the same.

The Pythagorean Theorem: Square Areas Projector Resources

Emmanuel’s Reasoning

P-2

Both of these have one more soda than orange, so they will taste the same.


Alex’s Reasoning

P-5

In the first case, 34

of the whole mixture is

orange, whereas in the second case 45

is

orange so they will taste different.


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.


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 orange and Saturday’s mixture is orange. Comparing these fractions to see which will taste the most orangey ( compared with ) 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 orange, which is slightly less orangey than Friday’s and Saturday’s mixture. This means that for every liter of orange, 1 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. 25

of the drink is apple juice.

b. 25


c. 13


Mixture c is the least appley drink. Qaylah should mix, for every liter of apple, 2 liters of soda.

37

49

37 =

2763

49 =

2863

25

12


Collaborative task: The correct matching/ordering from least orangey to most orangey (with ratio of orange to soda also given) is as follows:

1:3

1:2

3:5

2:3

3:4

4:5

1:1

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.

For example, they may choose to shade it in the ratio of 5 orange: 6 soda and place it between cards C and M.


Card Set: Orange and Soda Mixtures

A

B

C

D

E

F

Half of the mixture is orange

G

For every orange there are 2 sodas

H For every orange there is 1

41 soda

I

One fourth of the mixture is orange

J

32 of the mixture is

soda

K

For every orange there is 1

31 soda

L

For every soda there is

32 orange



A

B

C

D

E

F


G


H

Orange : Soda = 4 : 5

I


J

32 of the mixture is

soda

K

For every orange there is 1

31 soda

L

For every soda there is

32 orange

Student materials Using Proportional Reasoning S-1 © 2015 MARS, Shell Center, University of Nottingham

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.



A

B

C

D

E

F


G


H

Orange : Soda = 4 : 5

I


J

3

2

of the mixture is

soda

K

For every orange

there is 13

1 soda

L

For every soda

there is 3

2 orange


Card Set: Orange and Soda Mixtures (continued)

M

N

O

P

Card Set: Blank Cards

Shade in: Shade in:

Shade in: Shade in:


Mixing Drinks (revisited)

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:

a. For each liter of soda mix 2

3

liters of apple juice.

b. Mix apple and soda in the ratio 2 : 3.

c. 2

3

of the mixture is soda, the rest is apple juice.

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.

To make 6 liters of Fruit Fizz, mix 1 liter of raspberry juice, 2 liters of

apple juice and 3 liters of soda

Using Proportional Reasoning Projector Resources

Which is strongest?

P-1

Card 1:

Card 2:

Card 3:


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.

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.

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.

P-4


Emmanuel’s Reasoning

P-5

Both of these have one more soda than orange, so they will taste the same.


Sifi’s Reasoning

P-6

In both cases, for every orange there is 1 14

soda, so they will both taste the same.


Alex’s Reasoning

P-7

In the first case, 34

of the whole mixture is

orange, whereas in the second case 45

is

orange so they will taste different.

© 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.

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

Documents

By Karen S. Karp, Sarah B. Bush, and Barbara J. Dougherty · 2016. 8. 18. · states that Kate has 2 cookies to divide among herself and two friends, then the portion for each person