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Unit 7 Proportional ReasoningAlgebraic and Geometric Thinking

8.8.131Unit 7 Proportional ReasoningIntroduction:This section of the unit is very important in connecting mathematical concepts covered thus far as they relate to proportional reasoning, specifically equivalent fractions, division of fractions, relating unlike units of measure and expressions that relate two quantities in a multiplicative way. Proportional Reasoning is a key unit that is foundational to deeper understanding and required as a link to Algebraic and Geometric Thinking.

NumerationQuantity/MagnitudeBase TenEqualityForm of a NumberProportional ReasoningAlgebraic and Geometric Thinking 2007 Cain/Doggett/Faulkner/Hale/NCDPILanguageThe Components of Number Sense

2Background:This is one of the Number Sense Components that we will be covering in Unit 7. Proportional reasoning is more than just setting up ratios and cross-multiplying!!In this unit we hope you will see that proportional reasoning is pervasive throughout the curriculum from 3rd grade on as students begin to think multiplicatively.DiscussWhere does proportional reasoning fit in the curriculum? What topics, what grades?What has been participants experience with proportional reasoning? Have them think back to their own experiences and those of their students.Allow 2-3 minutes to talk in their table groups and then report out to the large group.Proportional Reasoning Defining the Concept

Definingthe ConceptDiagnosisWhere the Research Meets the Road Classroom Application3Proportional ReasoningProportional reasoning involves a multiplicative relationship between two quantities.

Proportional reasoning is one of the skills a child acquires when progressing from the stage of concrete operations to the stage of formal operations.4Key Concepts:Multiplicative is KEY here. We need to understand that in 3rd grade we begin this journey in earnest as we introduce multiplication and build deeper understanding of fractions.Students have misconceptions that are developmental. When they are young student, they can only attend in a univariate, fashion. Terminology: Univariate is only being able to attend to one quantity and not two. Example if you pour orange juice from a big container into two different size containers and ask which one is orangier students will say that the smaller container is orangier. They are attending only to the fact that the container is smaller, so the orange must be stuffed in there. They arent sophisticated enough to think about the proportions in the orange juice itself. This develops with instruction and age as Piaget would suggest, but the type of INSTRUCTION is KEY as Vygotsky would suggest!Reference: Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning; Grades 6-8, NCTM 2010

Why does this work?If Sam can mow one lawn in 2 hours, how many lawns can he mow in 8 hours?Write a proportional relationship that represents the situation.

Write a non-example.

5Engage in Related ActivityHave participants in table groups decide which, if any or all of these proportions are set up correctly. Once they have done so, have them arrive at a non-example. Allow 3-5 minutes for this activity. Debrief regarding LearningUse the Handout in Trainer Notes with the proportion shown. Hold up the copy of the proportion.Turn the card 90 degrees and read the new proportion.Turn the card again 90 degrees---read the proportionTurn the card one last time 90 degrees.Why does this work?(You are maintaining the multiplicative relationship that exists within and between the numbers in the relationship.)Once the proportion is set up correctly, the paper can be turned to show the between and within proportions.Non-example: x/8 = 2/1 The unknown number of lawns can be related to the number of lawns that could be mowed in 8 hours, but not in the same way that 2 hours relates to 1 lawn. The multiplicative relationship between and within is not maintained.Language

FUNCTIONAL The relationship of two values in the SAME measure space. We utilize this internal multiplicative relationshipto maintain proportionality and/or similarity.

SCALAR The relationship of values from two DIFFERENT measure spaces. We utilize thiscorresponding multiplicative relationship to move between different SCALES or to extrapolate from one situation to another.

Terminology:Be clear about these words. We recommend you emphasize the words Functional and Scalar and what those factors do for you. Do they maintain the shape/relationship (functional) or do they allow you to take that shape/relationship and move to a bigger or smaller size (scalar).

Key Concepts

In other words if I am taking a picture and want to get a frame with a border, the Functional relationship ensures that I get the correct shape. By maintaining this relationship, I am able to ensure that the picture is not distorted.

On the other hand, the Scalar relationship is what allows me to enlarge from the picture size to the size of frame I want.This also applies to proportions that arent a shape. For example, this could relate to the number of hours it takes to mow a certain number of lawns.

BUT because we know the functional and scalar relationships go hand in hand, at the symbolic level you can choose to use either one based on which has the friendlier numbers. For example, if you wanted green paint that required two quarts of yellow paint for every three quarts of the blue paint, if we were going to use six quarts of blue paint, how much yellow would we need? In this situation, using the scalar relationship would be friendlier. The value of one in the scalar function 2/2, whereas the functional relationship would require multiplication by a mixed number (1 ) or fraction (2/3).

6Measure SpaceFunctional within one object or story or measure space Books Cost

3 9

12 ?Scalar between two objects or stories or measure spaces BooksCost

3 9

12 ?

The relationship betweentwo corresponding valuesin the DIFFERENT measure spaces (situations).The relationship of the values within the SAME measure space (situation).Use of Schematic Representations in Improving Childrens Understanding of Proportional Reasoning Problems Pyday and Nunes, University of OxfordA measure space involves relating two quantities. In the functional, the relationship can involve part-to-part (for example books to cost or fiction to non-fiction books) or part-to-whole (fiction books to total books). In a scalar relationship, you are comparing like units, whether it is a part or a whole. For example, you are comparing books to books, cost to cost, nonfiction books to nonfiction books, or total books to total books).

This slide serves to reinforce the concepts of Functional (Within) and Scalar (Between) proportional reasoning. The quantities within the oval represent the within one object or story or measure space.

In the slide above, the measure space is circles. The horizontal arrows represent the functional relationship and the vertical arrows represent the scalar relationship.7Measure SpaceFunctional within one object or story or measure space Situation Books Cost 3 9

12 ? Scalar between two objects or stories or measure spaces Situation Books Cost 3 x3 9

12 ?The relationship between two corresponding values in the DIFFERENT measure spaces (situations).The relationship of the values within the SAME measure space (situation).Adapted from "Use of Schematic Representations in Improving Childrens Understanding of Proportional Reasoning Problems Pyday and Nunes, University of Oxfordx4Reformat this slide to include the labels (books / cost).

There exists an internal multiplicative relationship within a measure space and a corresponding multiplicative relationship between two measure spaces.

In the example above on the left hand side, the functional relationship represents a unit rate. For every one book I pay three dollars. Conversely, for every three dollars I can buy one book. Therefore, if I buy three book, the cost is nine dollars.

In the example above on the right hand side, the scalar relationship allows me to increase or decrease quantities while maintaining proportionality. I am able to do so, because I can multiply by different forms of one which will change the form of the proportional relationship. For example, if I increase the number of books from three to twelve I know that my form of one should be 4/4. 8FunctionalUse what is known within a situation to figure out missing values for an analogous situationA hamster eats 12 scoops of food in 4 days. How much food will the hamster need for 7 days?Use of Schematic Representations in Improving Childrens Understanding of Proportional Reasoning Problems Pyday and Nunes, University of Oxford1 3X 3 2 6 Number of DaysNumber of Scoops 3 9 7 ? 4 12In this situation, we are comparing the relationship of two unlike quantities (days and scoops). This relationship is within the given situation( or the same measure space). (i.e. 4 days/12 scoops or 12 scoops/4 days)

As we begin to look at the multiplicative relationship within the values we can see that the number of scoops is always equal to three times the number of days. The number of days I have is always equal to one third the number of scoops.

For the example of days to scoops, the unit rate is three (for every one day, I have three scoops). This unit rate represent the multiplicative relationship that will remain constant in the functional relationship. You can move to a different measure space and to maintain the proportional relationship, the number of scoops will be the number of days multiplied by a factor of three. In this case, this constant multiplicative relationship is what results in equality.

We can now take our story and create a function table, a graph, and a linear equation. Look at the connection above, when the double number line is turned vertically, it looks remarkably like a function table.

The linear function is ----(Scoops = 3 times the number of days).. Y = 3x (y=scoops and x =number of days)

9ScalarUse the relationship of known corresponding parts of an analogous situation to extrapolate and find the new space missing partEmily bought 4 balloons and paid $ 10 for them. She went back and bought 12 more balloons. How much did she have to pay for the 12 balloons?

Use of Schematic Representations in Improving Childrens Understanding of Proportional Reasoning Problems Pyday and Nunes, University of Oxford412Balloons 10 ?Money ($)X 3X 3In this particular scenario, we are relating the balloons and money relationship to another situation. Specifically, we want to relate the quantities in the first situation to corresponding quantities in the second situation. There is a scalar operator of 3 that will transform these quantities of the same type. (i.e. balloons to balloons and money to money). When you look above, you can see the multiplicative relationships exist from 3 balloons to 12 balloons by a factor of three, as well as from 10 dollars to 30 dollars by a factor of three. When both relationships have the same multiplicative relationship, we have created a form of one (3/3).

Likewise, if we move in the opposite direction, from 12 dollars to 4 dollars, we are doing so by a factor of 1/3. When moving from 30 dollars to 10 dollars, we are also doing so by a factor of three. Again, we have created another form of one (1/3 / 1/3).

10Functional Use the relationship existing within theoriginal measure space (situation)

3 feet long = ? feet long 2 feet wide ? feet wide

Here, the length is 3/2 times the size of the widthThe width is 2/3 times the size of the length

To create a correlate in a second space we consider the relationships from the same space:

To build a second space length from a second space width we multiply by 3/2To build a second space width from a second space length we multiply by 2/3Give examples for participants to go from a second measure space width to that corresponding length and vice versa.

i.e. Given a length of 9, find the width. Solution: 2/3 (9) = a width of 6.Given a width of 6, find the length. Solution: 3/2 (6) = a length of 9.

Show the next slide, with the solutions

11Rate of Change

The change in length for a unit of change in width. WidthLengthThe functional relationship also 12Rate of Change

The change in width for a unit of change in length.

LengthWidthChange to fractional notation13Scalar Use the relationship that exists between two different measure spacesA rectangle has a length of 3 feet and a width of 2 feet. How wide is a similar rectangle that is 3 feet longer in length than the original rectangle? 3 feet long 2 = 6 feet long 2 feet wide 2 ? feet wide

Our scalar operator is 2.Again, we are looking at comparing like quantities from one situation the relationship of feet in length to feet in width to corresponding quantities in a second situation.Geometry Connection We are able to find corresponding values of similar triangles using the scalar relationship.14Bubble Gum (pieces)

Cost (cents)

Grant26Eli8?Grant and Eli each bought the same kind of bubble gum at the same store. Grant bought two pieces of gum for six cents. If Eli bought eight pieces of gum, how much did he pay?You Try. Use Functional and then Scalar to figure out Elis cost Have a table discussion around the guiding question regarding scalar, functional, between and within.

15Functional Method: SLOPEThe relationship of the values within the SAME measure space (situation).Constant Rate of Change

2G : 6P = 8G : 24P

2G : 8G = 6P : 24P

Proportional Reasoning: Student Misconceptions and Strategies for Teaching, PCK Tools, CAMS and COREWithin here refers to the relationship stated in the situation or problem that relates two values (i.e. gumballs and pennies).

The unit rate can be connected to the unitizing structure of division, e.g., if we pay 6 cents for 2 pieces of bubble gum, the price per piece is 3 cents

However, we want to discuss the multiplicative relationship. We can talk about this relationship and build off division of fractions as related to the bean party where dividing is equivalent to multiplying by the reciprocal.

e.g., the multiplicative relationship between bubblegum and price is 3x = y or price and bubblegum is 1/3 y = x

Pieces of bubblegum times 3 = price or price times 1/3 = pieces of bubble gum

Bubble gum Price per piece (Between Cases) Total Cost 3 3 3 6 3 9 3 12etc.

The functional method is more difficult as students need to be able to see the fixed constant multiplicative relationship between the two values.16Scalar Method: MULTIPLICATIVE IDENTITYThe relationship between two corresponding values in the DIFFERENT measure spaces.Scalar operator Form of ONE

2G : 6P = 8G : 24P

2G : 8G = 6P : 24P2G x 3 = 6P8G x 3 24PProportional Reasoning: Student Misconceptions and Strategies for Teaching, PCK Tools, CAMS and COREBetween refers to two different, but proportionally related situations or problems. We are comparing the same type of quantitysame type of unit in two different measure spaces. In this case, we are comparing gumballs to gumballs and pennies to pennies. ---Between two ratios is an external relationship of quantities. There is a fixed multiplicative relationship.

Here, we can connect the scalar operator back to the identity property of multiplication. We are showing different forms of a number through the scalar method.

Essentially we are taking the unit rate and showing this in different forms. If 1 piece of bubblegum costs 3 pennies, then 2 will cost 6 pennies, etc. If we want to know how many pennies it would cost for 8 gumballs, we could multiply by the value of 1, 4/4 (changing the unit rate to a different form) to arrive at our solution.

Children have a clear preference for the scalar method as it is intuitive to their build up thinking of number. This method is additive in nature and is considered to be preproportional by Piaget since the additive solution lacks multiplication although the process/ solution can be connected to multiplication as levels of proportional thinking mature.

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DiagnosisDefiningthe ConceptDiagnosisWhere the Research Meets the Road Classroom Application18Proportional Reasoning---Is foundational to higher mathematicsDoes not always go hand-in-hand withsymbolic representationHas four levels of solution strategies0 (use of additive strategies, solution arrived at by luck)1 (use of pictures, models, manipulatives)2 (use of level 1 strategies and multiplication and division strategies as well)3 (use of cross-multiplication or equal ratios)19Key Concepts:Reference:Langrall and Swafford (2000) have classified students proportional reasoning strategies into four levels. Levels of Proportional Reasoning Strategies-ConnectionsLevel 0: Student is able to make a lucky guess or find that by adding, the proportion works for the first time they try to establish the relationship. (Example: 6 Students/1 Principal) If a student can see that adding five to the number of principals will produce a rule for the first situation in a T-Chart, then they might infer that this rule will work for all cases.Level 1: Students may use pattern blocks to show that 3 hexagons: 2 squares, then they can demonstrate that 6 hexagons has the same proportional relationship to 4 squares. They may be able to represent this relationship with pictures as well.Level 2: Students begin to see the multiplicative relationship between and within their proportions. Remind participants this works for division as well since dividing by 5 is the equivalent to multiplying by 1/5.Level 3: The use of equal ratios = 2/4 and the cross multiplication procedure is where teachers traditionally begin exploration of this concept. Cross-multiplication is not proportional reasoning, but becomes the most efficient way for students to solve most proportions.

Establishing a conceptual base for proportional reasoning is critical to Algebraic and Geometric Thinking! Say no to Just Do It!.

Student IssuesStudents do not analyze the relationships among symbols.

Students use the cross-product method even when the within or between relationships are obvious.20Read and DiscussAsk participants to reflect on these two student issues based on the the information covered in the previous slides. Key Question: Why does it matter that we resolve these student issues and reach the students when the opportunities first arise for deeper understanding in proportional reasoning. Possible Answer: Students need the deeper conceptual understanding in the early stages in order to anchor them in the proportional reasoning connected to Algebraic Thinking. If we just memorize procedures, then we have nothing to fall back on when our memory fails us. We need to always stay connected to the concrete aspect of a concept for deeper understanding to remain intact. Where the research meets the roadDefiningthe ConceptDiagnosisWhere the Research Meets the Road Classroom Application21We view proportional reasoning as a pivotal concept. On the one hand, it is the capstone of childrens elementary school arithmetic; on the other hand, it is the cornerstone of all that is to follow.(Lesh, R., Post, T., & Behr, M. (1988)22Quotations:This is a very powerful quote when we think about the ramifications of what happens to our students who do not receive a strong foundation in proportional reasoning. (Lesh, R., Post, T., & Behr, M. (1988)

Four Major IssuesEmphasis on procedures vs. developing sense of the meaning of rational numbersWe teach as if Level 3 were the access point, rather than Level 1Use of representations in situations where rational numbers and whole numbers are easily confused.The inherently confusing nature of fractional notation to represent proportions (2 for every 5, 2 out of 5, and 2/5) Joan Moss and Robbie Case 1999

23BackgroundMoss and Case developed a innovative rational number curriculum that began with percentages. Reference:Developing Childrens Understanding of the Rational Numbers: A New Model and Experimental CurriculumThey felt that students everyday knowledge of percentages and intuitive ideas could foster powerful understanding in terms of the rational number system.Key Concepts:Emphasis on procedures came at the expense of understanding.Emphasis was on the adult formal understanding rather than about the students informal knowledge about fractions. DiscussWhat opportunities to we miss in helping students to understanding proportional relationships rather than having them just learn procedures and focus on adult formal understanding? Allow for participant responses. If the opportunities listed below are not mentioned they should be reviewed later in the presentation as indicated. (Possible answers include: repeated patterns (blue, blue, red repeated four times to create a larger unit); intuitive proportional reasoning (i.e. For each dollar I can buy three pencils.)The proportional nature of equivalent fractionsTranslating equations such as for every Principal there are 6 studentsMeasurement relationships that are multiplicative such as for every yard there are three feet.

Visuals of examples of last two bullets are provided on the next two slidesTeachers use of shaded parts of a figure and the total number of parts as separate entities proved confusing to students.Teachers need to understand that fractional notation is not obvious to students.

The inherently confusing nature of fractional notation to represent proportionsJoan Moss and Robbie Case 1999

1/6 or fraction of stable unit like money (9/20) (Story One)VERSUS1 out of 6 or proportion ofa changing unit (8/30)(Story Two)Key Concepts:Moss and Case found a major issue which was the use of representations in situations where rational numbers and whole numbers are easily confused. Tie this back in to how we present fractions. Do we emphasize the unit as 6ths? What is the whole? The use of other vehicles than pie charts is critical. Various representations should be used to present fractions, always emphasized that the wholes represent the same quantity. Students arent given the tools to evaluate whether a fraction notation is being used as a fraction a place on the number line or as a proportion a relationship that is not on the number line.The inherently confusing nature. We have experienced that. This is exactly what we grappled with with story one and story two in Unit 4. There was a different algorithm because in one story we used fractional notation to describe a stable unit situation a true fraction. In the other situation we used fractional notation to describe a changing unit situation a true proportion. In both cases it is fine to use fractional notation, BUT we must understand the context to understand the algorithms.

24.RatiosFractionsPart to wholePart to Part3:2:1Rational Number 5/8 Operator5/8 of 40As a measure 5/85 + 1/2This Venn diagram helps you to see where ratios and fractions are alike and not alike. Reference: Developing Essential Understanding of Ratios, Proportions and Proportional Reasoning; Grades 6-8, NCTM)Key Concepts:Ratios: We can talk about ratios as a part to a part and a part to a whole. For example, we can say that there are four girls for every five boys (part to part) or we can say there are four girls out of nine students. (part to whole) With ratios, we can compare more than two parts. For example, we can say that our salad dressing recipe calls for 3 parts oil to 2 parts vinegar to 1 part water. (3:2:1) The Golden Ratio is 5 + is an example of a ratio that is an irrational number which is not a fraction. Fractions: We can talk about fractions as a part to a whole, but fractions can also be used as operators such as 5/8 of 40 and are rational numbers. In addition they can be located on a number line as a measure. According to Moss and Case, rational number notation is not transparent for students and we cannot just simply define the notation at the beginning of a lesson. Further exploration as seen in this diagram is critical to avoiding confusion of the concepts regarding ratios and fractions.

25FRACTIONS MATTER:Teaching Fractions must include connections to Proportional Reasoning!

What is my Unit?I cant make sense of a fraction unless I know what my comparative unit is!How does my portion size affect my servings?26Key Concepts: CONNECTIONS TO prior learning in Foundations of Mathematics How do the details of our instruction of fractions affect students proportional reasoning!Notice how not discussing the UNIT denies students the ability to process the difference between proportions that have a fluid unit size and fractions with a stable unit. Their ability to comprehend these related, but different, mathematical structures affects their ability to READ and COMPREHEND mathematical stories.Notice how we bury proportional reasoning by saying things like flip and multiply! We could be talking through Proportions! Portion size and servings are proportional reasoning AND it connects to how fractions behave.

SolutionsGreater emphasis on meaningGreater emphasis on the proportional nature of fractionsGreater emphasis on Level 1 andLevel 2 solution strategiesGreater development of diagram literacyJoan Moss and Robbie Case 1999

27Read and Discuss:Ask participants about the difficulties involved in the implementation of these strategies. (with table partners for 2-3 minutes)Possible answers: time factor, background of students previous learning, teacher understanding of the concept

Proportional ReasoningDevelops hand in hand with Quantity and Magnitude.Is a skill the child obtains when moving from the concrete to the formal operational stage of development.Can relate equal lengths of unlike units.Can be conceptually understood through diagram literacy.

28Key Concepts:Piagets theory of intellectual development states that proportional reasoning is one of the skills a child obtains when moving from the concrete operations stage to the formal operations stage. (Piaget & Beth, 1966) Piagets Theory of Intellectual DevelopmentStudents need time to develop and practice diagram literacy with proportional relationships as they move from concrete to representational stages . (Deizman and English 2001)Reference:Diezmann, C.M. & English, L.D. (2001) Promoting the use of diagrams as a tool for thinking. In A.A. Cuoco & F.R. Curcio (Eds.). The roles of representation in school mathematics (pp.78-89). National Council of Teachers of Mathematics: Reston, VADietzman and English (1999) found that a diagram can serve as a means to "unpack" the structure of a problem and lay the foundation for its solution but children need to be taught diagram literacy. This means knowing about a diagram's use and being able to use that knowledge appropriately.We will explore diagram literacy in an upcoming slide.

Classroom ApplicationDefiningthe ConceptDiagnosisWhere the Research Meets the Road Classroom Application294th Grade NAEP ItemA fourth grade class needs five leaves each day to feed its two caterpillars.How many leaves would it take to feed twelve caterpillars?Solve this problem without cross multiplication.

30Engage in Related ActivityAllow participants to work on this item alone for about five minutes. Participants may draw pictures, diagrams, etc. to solve this problem. They should do so intuitively as fourth grade students have not had any experience with formal proportional reasoning. This is what Deizman and English refer to as diagram literacy. Debrief regarding LearningAllow participants to report out as time permits. Possible correct answer: For every 5 leaves I can feed two caterpillars, so if I have 12 caterpillars that is six groups of 2 caterpillars. So six groups of five leaves equals 30. Participants could diagram this proportional relationship by showing 2 caterpillars with 5 leaves coming off the caterpillars. Six groups of 2 caterpillars would give students 12 caterpillars---so they could then count the leaves or add groups of five or multiply 6 x 5 until all six groups have been accounted for.Number lines are great tools as well to show the relationship. ALSO connect back to the within and between/scale factor relationships. So with the caterpillar example what is the within relationship? (How many leaves the caterpillars in their diet per daythe unit rate)What would you use the Scale Factor for? (Projecting how much you will need for a given number of days).

Paper Clip Chains - AIMS Activity

Engage in Related ActivityYou can use the AIMS Activity (Paper Clip Chains) if you have time. See Activity in the trainer notes. This activity has been reproduced to use in this training only. You must purchase the rights to duplicate Paper Clip Chains for your training groups. AIMS Activities provide understanding to mathematical concepts and good questioning strategies as well.Directions, Component of Number Sense type questioning and debriefing of the activity is provided in the activity itself Debrief regarding Learning Make note that this is the type of activity that encourages same-action and is also the reason we insist on this being a K-12 training. Same-action and activities such as this allows you to do one activity but to challenge some students to represent the problem algebraically. At the same time, students who are still trying to access proportional reasoning can get conceptual repetitions and practice with the concept of comparing two numbers and making sense of the idea that for the bigger unit Ill need less, for the smaller unit size Ill need more K-12 connections As an instructor you want to know where this concept has been to help you support students who may be struggling AND where this concept will go in order to support students who can make further connections. NOT a DIFFERENT LESSON but real connections up through the curriculum.31Helping the ProcessStrong instruction will help students to find the strategies that will involve the fewest computations.Students who feel comfortable with a variety of solution strategies will increase their conceptual understanding, as well as their procedural fluency with proportions.32Read this aloud.Explore Proportions through SimilarityEnlargementsReductionsScale factorsIndirect Measurement33DemonstrateSee Trainer Notes for Unit 7: Similarity Examples (for ways to illustrate these situations)Connects the Components of Number Sense: Magnitude to GeometryEngage in Related ActivityHave participants play with these ideas in table groups by creating problems if time permits for about five minutes.Students Need Opportunities toReason about proportional situationsFind unit ratesConstruct equivalent ratiosLink ratios, percent and fractions

34Key Concepts:Bullet 1 relates to the caterpillar and leaves NAEP problem, as well the Paper Clip Chains.Bullet 2 is having students ask, How much per item?Bullet 3 = 2/4 (Communicating the scale factor involved in this process is paramount in having students understanding the multiplicative nature of proportions. Fractions can actually be graphed in the first quadrant of a coordinate grid. See the AIMS book Proportional Reasoning for more details in explaining this type of connection. Bullet 4 What is the connection among these forms of the number? Students are intuitive about the relationships, so let them explore.

Bridging to Level 3Requires a strong understanding of the components of proportional reasoning:Change between equivalent ratios is multiplicative, not additiveThe multiplicative change is constantThe relationship between ratios is the scale factor (If you multiply one ratio by the scale factor, the result is the second ratio.)35Read and DiscussIn table groups, allow participants to give an example of each of these three statements regarding the components of proportional reasoning for about five minutes. Share out a few examples as time permits. Possible Examples:Bullet 1: Equivalent fractions have a multiplicative change as we move from one fraction to an equivalent. i.e. 1/3 is equal in value to 3/9. The multiplicative value of one used to arrive at 3/9 is a value of one that is 3/3.Bullet 2: I can count on the multiplicative relationship to be unchanging. If for every yard I have, I have 3 feet, then if I have 2 yards, I will have 3 x 2, 6 feet. Bullet 3: If I am going to create an equivalent ratio, then I will need to maintain the same between constant of proportionality. i.e. If I want my doll house to be proportional to my actual house in width and height, then I need to look at the multiplicative relationship of the original house to the doll house and find the scale factor. If 1 cm = 1 foot, then if my house is 30 feet tall, then my doll house would be 30 cm tall.

What Component of Number Sense can help us explain why cross multiplication works?Show how you maintain EQUALITY through the proof of ad = bc through the use of properties. See hidden slide for instructions, but this proof is much more effective if you walk through it with participants via a smart board or an Elmo. 36Is cross multiplication just a piece of magic?Teach students that when you multiply equal quantities by the same value, their products will be equal.

111

137Click slides to walk the participants through the process. Use this process to explain to participants the way that we use mathematical properties to arrive at ad=bc.The first part of the problem shows procedural cross multiplication.Clicking on the statement Teach students that when you multiply equal quantities by the same about, their products will be equal. begins the sense making way through properties.Track the flow of the problem so that you can explain each step. When you are multiplying by b, you are maintaining a balanced system since both sides of the equation are being multiplied by the same quantity. That same quantity b/b can be simplified to 1 which simplifies the left hand side of the equation to a. In a like manner, you can multiply both sides of the equation by d and then d/d simplified to one, which leaves ad=bc, which shows the magic of cross multiplication. Problems that Encourage Proportional SenseAdapted from Esther Billings Mathematical Teaching in the Middle School NCTM38The next slides illustrate key ideas from this article.

John drove 60 miles in 2 hours. If he continues to drive at this speed, how long will it take him to drive 40 additional miles?

39Read and DiscussHow will students solve this problem involving proportional reasoning? (Perhaps by cross multiplying depending on their experience.)Engage in Related ActivityHave participant arrive at a solution individually. Encourage them not to cross multiply! Have them present their various solutions as time permits. Various methods will be demonstrated. (bar models each bar represents 20 minutes and 10 miles) (a double number line where the units are each divided into thirds) (a clock ---circular number line could be used with various compared values) (charts that resemble double number lines)Debrief regarding LearningIs it just an exercise in cross-multiplication? (No, students should seize the opportunity to think about the fact that they will be traveling less than 2 hours extra to travel the extra 40 miles. This type of problem can reinforce time measurement. What if students see that for every 60 miles it will take 120 minutes, which means that if I double the miles, I double the minutes. So 40 additional miles will take 80 minutes more for a total of 200 minutes which is 3 hours and 20 minutes.)This is making sense of the concept rather than 60:2 = 40:x and cross multiplying to get 1 1/3 hours and then adding in the 2 hours and then oh my goodness what is a third of an hour??? You will still arrive at 3 hours and 20 minutes, but without the understanding involved in the intuitive strategy that students need to feel comfortable with at this stage of the concept.In the following situations it is indicated which carafe contains the stronger coffee. Determine which carafe will contain the stronger coffee after the alterations have been made. Explain how you came to your answer.40These carafe situations come from Ester Billings article.Engage in Related ActivityHave participants reason about each scenario, one at a time and then seek participant feedback for each scenario that follows.

Carafe B contains weaker coffee than Carafe A. Add one spoon of instant coffee to carafe A and one cup of water to carafe B.ABWhich carafe will have the stronger coffee?41Debrief regarding LearningAdding water weakens B and adding coffee strengthens A which was already stronger anyway. This is obvious to students.Carafe A and Carafe B contain coffee that tastes the same. Add one spoon of instant coffee to both carafe A and B.ABWhich carafe will have the stronger coffee?42Debrief regarding LearningStudent may reason that doing the same thing to both carafes means that the coffee will remain the same strength when one spoon of instant coffee is added to both.Do we add the same thing to both sides? Does that keep the solutions equivalent strength? The faulty thinking is disrespecting the volume relationship. We are not thinking about the two types of quantity here---strength of the coffee and the volume of the coffee. Only being able to attend to one quantity at a time is univariant proportional reasoning.

Trashketball and Carafes?Trashketball is a proportion too

How would you use student knowledge of Trashketball to support their understanding of the carafe situation?

DiscussHow would you use student knowledge of Trashketball to support their understanding of the carafe situation? (in table groups for 2-3 minutes)

Debrief regarding LearningOne kid has made 40 out of 80 shots and another has made 30 out of 60both hitting at a rate of 50%. If more shots are made out of the original shots taken, who is going to do better? There is an analogy here with our carafes. What if each kid made five more shots out of their original shots attempted? The kid hitting five more shots out of 80, 45 out of the 80 would not be as strong a shooter as a kid shooting and making 35 out of 60 shots. In the carafe situation, the equal amount added is to unequal volumes is analogous to the five additional shots made per player out of the different number of shots taken by each kid. The 80 and the 60 represent the volumes of the carafes. Equal amounts added to unlike volumes does not produce equal taste or an equivalent ratio of shots!43Carafe A contains weaker coffee than Carafe B. Add one spoon of instant coffee to carafe A and carafe B.ABWhich carafe will have the stronger coffee?44Debrief regarding LearningCarafe B will remain stronger. A is a weaker shooter and makes a shot and B is a stronger shooter. Which shooter is stronger?Carafe B contains the stronger coffee. Add one spoon of instant coffee to Carafe A and 1 cup of water to Carafe B.ABWhich carafe will have the stronger coffee?45Debrief regarding LearningThere is no way of knowing. Participants will be puzzled by the fact that this relationship cannot be defined.There is simply not enough information.There is not enough information to know which shooter has the better rate of shooting either. Exposing students to nonnumeric quantities will strengthen their proportional reasoning.46Read this statement.ALSO MAKE CONNECTIONS TO OTHER DISCUSSIONS SUCH AS TRASHKETBALL!Why dont kids think proportionally?Because we dont build on their intuitive thinking.Because its hard for them.Because we dont teach it very well.Because we dont make connections between different forms of numbers.Because its hard for us47Read and DiscussIt is very difficult for adults to make changes and adjustments to what we learned.Perimeter around the earthImagine there is a string that goes around the earth and represents the earths perimeter at its widest arc. Now increase that string by 1 meter. How far away would the string now be from the earths surface?48Engage in Related ActivityHave participants work in groups of 4 or 5. Ask them to discuss and come up with an answer to share with the whole group. Call on groups for answers. Anticipate that most groups will argue that the string will be right next to the earth, sort of a negligible amount above the ground.MAKE SURE PARTICIPANTS DO NOT SEE ANY SLIDES PAST THIS ONE. IF THEY SEE PI IT TENDS TO CHANGE THEIR DISCUSSION.Lets use visuals to help us think this through49Engage in Related ActivityAllow 5 to 10 minutes for this activity Each group should have different size circles. These represent a slice at an equator of a sphere.In addition to the circles, you will need meter sticks, string and scissors.Give each group a different size/color circle and a piece of string. Have them cut (or have it pre-cut) a piece of string that is the perimeter of their circle Plus 1 meter.Direct them to take the string and encircle their circle with the string. They will then measure the distance from their sphere slice (circle) to the string. This represents what is happening in the string around the earth problem. Each group is to do this with their different sized circles. Remember that they begin with the perimeter of their own circle and add the same amount (1 meter).Directions follow on the next slide. MAKE SURE PARTICIPANTS DO NOT SEE ANY SLIDES PAST THE NEXT ONE. IF THEY SEE PI IT TENDS TO CHANGE THEIR DISCUSSIONAt Your Tables---Use the string to represent the circumference ofyour circleAdd 1 meter (100 cm) of string to the circumferenceCut your string. The length of your string represents the circumference of your circle plus 100 cm.Use your string to make a larger circle aroundyour original circle. (Both circles should have the same center.)Measure and record the distance from the edge ofthe new circle to the edge of the original circle. (in cm) About how much of a meter does this amount represent?

Debrief regarding LearningHave each group tell you what the distance was from their sphere to their string around the earth. Each group will give an answer that will be approximately 1/6th of a meter. (about 16 cm)Discuss with group why? Why did we all get essentially the same answer? Did we make a mistake? Is their a constant at work here? What is going on?Elicit answers from the group. Have them think about Pi. Ask them if they can figure this out using algebra?Have the participants look at each others circles to note the measure of the distance from the edge of the original circle to the edge of the new circle. It is a good idea to record the circle color and the measures given by each group.

50Time for Lunch!

Have participants notice the PI before they leave for lunch. 51Prototype for lesson construction

Touchable visualDiscussion:Makes sense Of concept12Learn to Record these ideasV. Faulkner and DPI Task Force adapted from GriffinSymbolsSimply record keeping!Mathematical StructureDiscussion of the concreteQuantityConcrete display of concept52Debrief regarding LearningOkay, so we have had an AHA with this topic at the quantity or concrete, touchable level. Whats next? We need to make sense of what is happening here and finally connect it to the symbolic. Hold onto to your forksTheres

53Lets look at this situation again considering the idea of Pi.

Engage in Related ActivityHow many diameters does it take for you to get around your circle (circumference)?Have participants try this with their circle.Each group will need about three diameter lengths to get around their circumference. (This relationship can be done the other way around as well--get a circumference and see how many diameters you can get out of the circumference.Debrief regarding Learning It does not matter how big the circle is, does it? This ratio is a constant ratio! NO matter how big or small your circle this relationship will hold. We call this ratio, this relationship, Pi. It is very close to 3. The important thing is the relationship, not the precise number (3.14). Our instruction tends to emphasize the NUMBER and not the RELATIONSHIP. So we end up not really being able to USE or UNDERSTAND circles in the same way.

54Circle RelationshipsIf given dd = C

If given r(2r) = C

If given CC = dC 2 = r 55Key Concepts:Think of as About 3. Think through these relationships and what they mean. By thinking about as about 3, we do a BETTER job of communicating the real meaning and importance of --that it helps us to understand the RELATIONSHIPS/PROPORTIONS within a CIRCLE.Emphasize the RELATIONSHIPS!Questioning Techniques: Developing Number SenseIf a circle has a circumference of 12, about what is its diameter?If a circle has a diameter of 10, about what is its circumference?If a circle has a circumference of 32, about what is its diameter? Its radius?If a circle has a radius of 4, about what is its circumference?56DiscussConsider your questioning and implementation emphasis. Generally if we have a problem with regard to ratio we pull out a calculator and start plugging in numbers to get an answer. This is a product of our in-class emphasis.What if instead we focused on the relationships involved? The class instruction would look something like this: Read first line-- Okay lets picture this. If the circumference is 12 then how long is its diameter? Remember the circumference is always about 3 times bigger than the diameter--so how long is the diameter?Read next line: Okay, a diameter of 10. Picture that now. So three of them will make the circumference. So how long is the circumference? Continue---Also note that as students get fluent with this you begin to get more precise and say Is it a little more or a little less than 30? Etc. This is the essence of proportional thinking. This thinking gets developed with this questioning technique. It does not get developed with plug and chug! It also develops the students Number Sense and allows them to begin to recognize if their answers make sense.

Proportional ThinkingLets reinforce what we know about proportional relationships to make sure we are ready to reason with our string around the earth problem57$225$200 $175$150$125$100$75$50$25$00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Lawns MowedYou have a lawn mower and your parentswill pay for maintenance and gas if you makeuse of your summer by starting a lawnmowing business.

You will make $25 for each lawn you mow.

This means that, if I tell you how many lawnsyou mow you can figure out how much you earned OR if I tell you how much your earned, you can tell me how many lawns you mowed. Each point has TWO pieces of information!

Plot the following points withyour small group:You mowed 0 lawns (0, ___)You mowed 1 lawn (1, ___)You mowed 2 lawns (2, ___)You mowed 3 lawns (3, ___)You earned $100 dollars (___, 100)You earned $200 dollars (___, 200)Look at the point that I plotted.Did I plot it correctly? How do you know?58Engage in Related Activity This slide gets printed for the participants. Participants work with hard copy in their small table groups.Read the scenario as a whole group. Say: Take a look at your worksheet that you will work on in your group.When you are done, one of the questions I will ask you isDoes this situation represent a relationship that VARIES DIRECTLY? Be prepared to answer that question.Terminology:When two variables are related in such a way that the ratio of their values always remains the same, the two variables are said to be in direct variation. In simpler terms if one quantity is twice another then they vary directly. Have participants work on worksheet to fill out the graph.When done (should not take more than about 3-5 minutes) review the answers with whole group. The points will be graphed in the next slide.

The next slide is hidden in the presentation and should not be printed on their PowerPoint. Unhide the slides in order to demonstrate the relationships.$225$200 $175$150$125$100$75$50$25$00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Lawns MowedI have a lawn mower and will mowLawns this summer for$25.00 a lawnMoney earned = $25 Lawns mowedY = mX59Demonstrate

Okay lets look at this relationship. I have a lawn mower and will mow lawns this summer for $25.00 a lawn. My parents are paying for my gas and for the lawn mower. So I have no EXTRA costs.Do you see that the these two parts of the story (lawns and cost) vary directly and therefore, when you plot them, they form a line?Each time you click a point will come up. Click again on the bozo point that does not belong, and it will go away.Next Say: Later on we will be doing Dollar Deals. You will see how we have the same sort of relationship here that we will have when we shop at a store with standardized pricing. Here, instead of shopping, we are earning money, but we are earning it at a steady rate. Click to get money earned text box up.Do you see that this relationship can be written algebraically (click for last box: y = mx). We can write it like this because we know that how many lawns we mow will vary directly because of our set price. We will do more with this later today. For now, I just need to make sure that you see that the amount of money I make = $25 times the lawns I mow. Make sense? Do you see how we can predict exactly how much we will earn based on how many lawns we mow? Even if we mow 1,000 lawns, or 876 lawns for that matter, we can figure it out.Note: Make sure that everyone is comfortable with this! It is critical that they focus, not on The Answers, but on this idea of a linear relationship that can be plotted. Now, back to the earth---

60Can we use our new understanding of the mathematical structure, the RELATIONSHIPS THAT ARE BUILT INTO A CIRCLE, to make sense of our string problem?CIRCUMFERENCE30

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00 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100RADIUSPlot these points:Radius of 5 (5, ~___)Radius of 10 (10, ~ ___)Circumference of 72 (~___, 72)

On the Graph we have plotted(~16, 100) and what other point?(____, _____).

Look at the two points thatare plotted. Using these two points answer the following:

For every 100 change in Circumference,about what will be the change in Radius?

What is the slope of this linear relationship? X = Y = 100(~16, 100)CIRCUMFERENCE250

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00 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100RADIUSPlot these points:Radius of 5 (5, ~___)Radius of 10 (10, ~___)Circumference of 72 (~___, 72)

On the Graph we have plotted (~16, 100) and what other point? (____, _____).

Look at the two points thatare plotted. Using these two points answer the following:

For every 100 gain in Circumference,about what will be the gain in Radius?

What is the slope of this linear relationship? X(~16, 100)(32, 200)= ~16= 100100 ~16= ~6Y63This slide is hidden and should not be printed in participant handouts.

Discuss(2-3 minutes in table groups)Okay. So what about the relationship between the RADIUS and the CIRCUMFERENCE?Talk in your groups and see if you can come up with an EQUATION in the form of y = mx for the relationship between the Radius and the Circumference. Click for y = mx on PowerPointWhat did you get?DemonstrateYes, because the radius is half of the diameter, we basically have to double it and THEN multiply by about three. So we are multiplying by 2 and then by 3. This is equivalent to a factor of ~6. So if you know the radius you multiply by 6 to get the circumference. It takes about 6 radii to go around the circle. Do you see this?If any groups appear to not understand this, take the time to explain it further until you see a big AHA.DemonstrateSo if we plot these points you will see that the Y or outcome, or circumference in this case varies directly with the radius and is 6 times bigger than the size of your radius. On our graph it would look like this: Click points and name them as you click through.Notice how a radius of 5, for instance, gets us a circumference of 30. That makes sense, right? Radius of 5 means a diameter of 10 and then times Pi (about 3) equals 30.5 x 2 x 3 = 30. or, 5 x 6.BUT REMEMBER that we can also go backwards with this, right? So if I know that I have a circumference of 30, what is my Radius? What if I have a circumference of 24, what is my Radius? How did you get that?Okay, so you can figure out your radius by dividing your circumference by 6, right? Will that always work?YES. IT WILL ALWAYS WORKMake sense?Do you see that there are linear relationships BUILT INTO the circle. It is still a circle, but we know that the circumference is directly related to the size of the diameter or radius. Got it?NOTE: Again, make sure that everyone is with you and that they see this relationship.And finally, what is our slope here? 6 right, or 6/1. For every gain of one unit in our Radius we get a 6 unit gain in Circumference. The Rise over Run, or slope, then, is 6/1.CIRCUMFERENCE30

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00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20RADIUSWhat is the relationship betweenThe RADIUS and the Circumference?Circumference = ~6(Radius)y = mxSlope ~6/130565Sogiven the equation for our relationship between radius and circumference, lets find the missing value.If we have a circumference of 30, what is our radius? ~ 5 right? The diagram demonstrates this relationship. 30/~ 6 = ~ 550?6650/about 6 = ~ 48/6 so we know it is ~ 8, right?100?67100/about 6 is about 16 right?We figured that 50 was about 8, so if we double the circumference to 100 the radius will be just double that 8, right?50 + 100?68Whats this?50/about 6 + 100/about 6 =about 8 + 1about 6 = about 24, right?

100 + 100?69Now, we know that this one has a circumference of 200. But lets write that in the form of our 100 that we just had, plus 100. What will our radius be?100about /6 + 100/about 6, right?So thats about 32.Bring the groups attention to the fact that we are using 6 as an approximation for 2 because they are pretty close to equal.Remember we are really finding the relationship that exists when Circumference divided by 2Pi but are rounding that number to 6. 2Pi is about 6 right? 2 * 3.14 is about 6? So we are using circumference/ 6 to keep it simple and to emphasize the relationships. So I am saying that 2 = ~ 6. I am using them interchangeably. Got it?Lets try another one (next slide)1,000,000 + 100?70DiscussWow. Big number here, but we know that it is just the total Circumference divided by 6 right?In this form, it is natural for us to do these one at a time. So if I have 1,000,000/6 that is about 160,000, right? (calculator answer = 166,666,67). (orwith making sense of the concept- (100 x 10 x10 x10 x10)/6 + 100/6 = about 16 x 10 x 10 x 10 x 10 + 16 = about 160,016)Okay, so we can figure out our radius easily by dividing by 6, right? No matter what the size of our circle, right?Hmmmm What else do you notice?No matter how big or small our circle is, if we add 100 extra units to it, those extra units will affect the radius in the same way, right?Wow! Do you see how that relates to our string problem? We are lengthening our circumference and seeing how that affects our radius. Because these two things vary directly, it does not matter how big the circle is100 unit increase in circumference will always increase the radius by 16 units!Heres a regular old circle With our added meter of circumference we have 16 cm of added radius.Circumference is 50 cm71Heres what will happen if it is really big!With our added meter of circumference, we still have just 16 cm of added radius.72What if it gets really small?Still, when we add a meter of circumferencewe still have just 16 cm of added radius.And what if we engage in a thought experiment? What if our circle goes downto a circumference of 0?73What if we have a circumference of 0?With, a circumference of 0 wed have no circle at all. We can still add our meter of string to nothing. It will go around a point and wed STILL be 16 cm away from our center, and have a radius of ~16cm74Key Concepts:Do you see that we can certainly add a meter of string to nothing at all? But then, what is the radius of the circle made by that string? It is, of course, ~16cm!

Pi and the earth

Pi is a ratioPi is a constant12V. Faulkner and DPI Task Force adapted from GriffinSymbolsRecord keeping: GeneralizationMathematical StructureSense-making of the concreteQuantityConcrete display of conceptc/(2*Pi) + 100/ (2* Pi)

75DiscussDidnt you need all three of these aspects of number to make sense of the situation? Do you see that you would need continued repetitions and practice NOT JUST at the symbolic level to continue to develop your understanding? Do you see how deep the connections are between geometry and algebra?

NumerationQuantity/MagnitudeBase TenEqualityForm of a NumberProportional ReasoningAlgebraic and Geometric ThinkingLanguageThe Components of Number Sense

2007 Cain/Doggett/Faulkner/Hale/NCDPI76Key Concepts:Good instruction on this topic will utilize virtually all of the components of number sense to communicate these important ideas!If we had not gone back to remind you about common denominators and connected that to you in almost a one-to-one correspondence way, would you have gotten lost? Had we not thoroughly explored proportion, would you have been lost?If we had not emphasized when we were merely changing a form--would you have gotten lost?But you did not get lost! We kept it all tied together through tapping into all of these elements of understanding number!

Houston, we have a problem77If time allows, have participants make some connections. How has proportional reasoning and the other components of number sense begun to help us think about how we can change our instruction to help us deal with our cultural problem. ALSO use this idea to discuss your role as the person who has to get the rocket ship back to earth. You are Houston! Foundations of Math helps you be the technician who figures out how to get students to understand the math so you can get them where they need to go! Its not okay to say well the student just doesnt have what it takes. Imagine if they would have said that in Houston! We can get the kids through the atmosphere.Connect to students through concrete connections and pictures. You can get them through the atmosphere this way! Their ability to utilize symbolic shortcuts will develop over time as they internalize how to think proportionally.Start where they are and with what they have. DEVELOP their thinking based on what they know. Do not bypass that development in order to have them do harder problems with less understanding. They will not make it through the atmosphere that way!Visual structures and Connections, Connections, Connections! Algebraic and Geometric Thinking78Unit 7 Algebraic ThinkingIntroduction:This section of the unit is very important in connecting the mathematical concepts covered in proportional reasoning to algebraic thinking. Proportional Reasoning is a key Component of Number Sense that is foundational to deeper understanding and required as a link to Algebraic Thinking. In the previous section, we have covered the proportional nature of comparing two unlike quantities in a multiplicative way as it is tied to direct variation. Participants had the opportunity to reason informally through the various activities to include the caterpillar and leaves, the paper clip chains and the carafe situations, They were also given the opportunity to transition to more formal proportional reasoning with the lawn mower problem and the Circumference and Radius relationship activity.

NumerationQuantity/MagnitudeBase TenEqualityForm of a NumberProportional ReasoningAlgebraic and Geometric ThinkingLanguageThe Componentsof Number Sense

2007 Cain/Doggett/Faulkner/Hale/NCDPI79Background:This is one of the Number Sense Components that we will be covering in Unit 7. Algebra is fundamentally very concrete and can be tied into Quantity and Magnitude as well as other Components of Number Sense. DiscussWhat has been participants experience with algebra? Have them think back to their own experiences and those of their students.Allow 2-3 minutes to talk in their table groups and then report out to the large group.Algebraic Thinking Defining the Concept

Definingthe ConceptDiagnosisWhere the Research Meets the Road Classroom Application80Defining the ConceptThe study of the letter x?

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82Tracy is still in concreteville. When Tracy took her Algebra Class---she tries this problem. What do you think that her answer might be?

83Tracy is still operating in a very concrete world! She found X, did she not?Think about algebra as---a systematic way of expressing generality and abstractionthe syntactically guided transformation of symbols.

(Adding It Up, National Research Council) 84Key Concepts:This is the definition from the National Research Council in the publication Adding It UpThe key here is the idea that we can SYSTEMATICALLY represent real quantities in the universe and we can extend this to more abstract postulations.Part of algebra is manipulating symbols. But what you know from the research that we have presented and the research you have read, that this really is the last stage in the development of a mathematical concept. What is the quantity and mathematical structures underneath these symbols? We will explore that in this unit.

In simpler terms, algebra is Using what we know to find out what we dont know.Analyzing patterns and relationships.Making predictions, modeling situations, and solving problems.

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86Key Concepts:As you are developing students sense of algebra through shopping, which we will do later today, you want to emphasize for them that all of these very different looking things are just DIFFERENT FORMS OF THE SAME EQUATION. Different forms of the SAME STORY!!!

DiagnosisDefiningthe ConceptDiagnosisWhere the Research Meets the Road Classroom Application87From Elementary School Arithmetic, a student brings---A desire to execute operations rather than use them to represent relationships.The use of an equal sign to announce a result rather than to show equality.A use of inverse or undoing operations to solve a problems.A perspective of letters as representing unknowns but not variables.88Read these issues regarding the transition from elementary arithmetic to formal algebra.Reference:Refer to the article in the trainer notes in Unit 5, Childrens Understanding of Equality: A Foundation for Algebra to enrich your understanding of Bullet 1 and Bullet 2.Reference:Adding it Up: Helping Children Learn Mathematics by the National Research CouncilChallenges Students Experience with AlgebraTranslate word problems into mathematical symbols Describe, paraphrase, or explain Link the concrete to a representational to the abstract Remember vocabulary and processes Show fluency with basic number operations Maintain focus Show written work89Read and DiscussRead through these challenges and have participants connect their own experience or those of their students.

At the Elementary Level, Students with Disabilities Have Difficulty with:Solving problems (Montague, 1997; Xin Yan & Jitendra, 1999)Visually representing problems (Montague, 2005)Processing problem information (Montague, 2005)Memory (Kroesbergen & Van Luit, 2003)Self-monitoring (Montague, 2005)

90Read and DiscussRead through these difficulties and those on the next slide and have participants connect their own experience or those of their students.Terminology/ExamplesVisually representing problems (Caterpillars and Leaves)Processing problem information (Principals and Students)Memory D'Esposito defines Working Memory this way: the temporary retention of information that was just experienced but no longer exists in the external environment, or was just retrieved from long-term memory. Working Memory is what enables us to keep several pieces of information active while we try to do something with them. Working Memory is quite purposeful -- we hold all these pieces of information together in order to solve a problem or carry out a task.Self-monitoring --The conscious process of watching one's own reading for problems and difficulties in order to successfully employ fix-up strategies.At the Middle School Level, Students with Disabilities Have Difficulty:Meeting content standards and passing state assessments (Thurlow, Albus, Spicuzza, & Thompson, 1998; Thurlow, Moen, & Wiley, 2005)Mastering basic skills (Algozzine, OShea, Crews, & Stoddard, 1987; Cawley, Baker-Kroczynski, & Urban, 1992)Reasoning algebraically (Maccini, McNaughton, & Ruhl, 1999)Solving problems (Hutchinson, 1993; Montague, Bos, & Doucette, 1991)91Read and DiscussRead through these difficulties and those on the previous slide and have participants connect their own experience or those of their students.

Where the Research Meets the RoadDefiningthe ConceptDiagnosisWhere the Research Meets the Road Classroom Application92NAEP TaskGive the value of y when x = 3.

x1347ny8111493Reference:Adding it Up: Helping Children Learn Mathematics by the National Research CouncilThis example from the NAEP reveals the shortcomings of traditional school algebra. Most students could recognize the pattern of adding 7 and use it when x =3 (69% success rate for students with one year of algebra and 81% success rate for students with two years of algebra)They were less successful however when asked in a subsequent problem to derive y, given x= n (41% and 58% respectively)3X +7 = YWrite a story problem for this equationHow would your students do?

7th grade?8th grade?Algebra 1?Algebra 2?Calculus?94None of the students, 0/41, at the beginning of the semester for ALGEBRA 2 students in a district and school in North Carolina were able to write an appropriate word problem for this equation. Engage in Related Activity

If time permits, allow participants to write of story problem for this equation. After Dollar Deals, this task will be a great deal easier and you will want to revisit the task.

Algebra is --A gateway to higher levels of mathematics education.Abstract thinking required for our full participation in our democratic society and technological driven world.

Dudley and Riley 199895Our students should not only know how to do algebra, but should also see and interpret algebraic representations in the newspaper and envision how to use algebra to organize their own world. They should know not only what was being analyzed through an algebraic model but also what is NOT being modeled.

For students to meaningfully utilize algebra, it is essential that instruction focus on sense making, not symbol manipulation. Throughout their mathematical careers, students should have opportunities to reflect on and talk about general procedures performed on numbers and quantities.

Battista and Brown (1998) 96Teachers should engage children in learning about the general principles of mathematics in conjunction with learning about arithmetic. The trick is, if you have not been taught this way--how do you teach it to the kids? That is why the training that you are doing here and continued exploration is critical to improving your work with students. Your understanding will revolutionize how you teach, and perhaps particularly how you will teach algebra.

Concrete to Representational to AbstractAlgebra is fundamentally very concrete!Like all math, Algebra is just a way to model concrete realities, actions and situations.Why do we bypass the concrete reality of algebra and go directly to symbolic manipulation? 97Remember Sharon Griffins quote that Mathematics is not about number, mathematics is about quantity. This does not all of a sudden go away when we hit Algebra! Mathematics, even Algebra, is still about underlying quantities!! Classroom ApplicationDefiningthe ConceptDiagnosisWhere the Research Meets the Road Classroom Application98Relating QuantitiesTeaching Experience, Number of Years you plan to continue to teach(Red Dots)Number of letters in your first name, Number of First Cousins(Green Dots)Distance in miles you drove,Time traveled in Minutes(Blue Dots)99Engage in Related ActivityCreate an ordered pair for each of the three situations in the slide with each of the two pieces of information for each situation.i.e. (5,25) I have taught for 5 years and plan to teach for 25 more yearsRemind participants to move along the horizontal axis from the origin first and then on the vertical axis. Point out that they need to read the information on the both the x and the y axis. Have large size graphs prepared so that the participants can place their sticky dots on the appropriate location on the coordinate grid. Make certain that each participant has one red, one blue and one green sticky dot. Lets look at the ordered pairs that you graphed on the coordinate grids. Discuss with participants the following:

Teaching experience and number of years you plan to continue to teach has negative correlation. The number of letters in your first name and the number of first cousins has no correlation.The distance that you drove in miles and the time you traveled has a positive correlation.

Guiding Questions:What type of correlation does the data show? (Teaching Experience, Years to Teachnegative correlation, Number of Letters in First Name, Cousins, no correlation, Distance you Drove, Timepositive correlation)Can we predict another point on the graph? What story could our outliers tell us? (someone is tired of teaching or loves teaching, someone is speeding or shopping along the way, someone comes from a very large family)Relating QuantitiesChildren make natural connections between numbers.Children need many opportunities to investigate relationshipsvisuals provided as numerical data is comparednon-examples to help set the limits of a concept and to fully structure their understanding.100What does the relationship mean?How are data related or not related?What new data might come in at a later time? How could you use the relationship to predict another data point?What could have happened if?What happens between data points?What is the essence of this pattern?

101These are questions that students need to be able to ask about how quantities are related. These thought processes involve higher levels of mathematical understanding.Other ExamplesNumber of Pets , Number of family membersGrade level, Average ageNumber of miles driven, gas in tankAge/Grade on Report Card in MathLength of foot, heightAmount of trouble you get into, parent happiness factor

102These are other examples of numbers that may be related in some way.Remember to throw in non-examples, situations where there is no pattern, no relationship, so that students can see the power of knowing that two things do not have a relationship. Non-examples help solidify the idea of the pattern. Otherwise students might begin to assume that all sets of two things do not have a correlation or pattern, just like our cousins and number of letters in first name graph. You need to see what something is NOT in order to better understand what it is.Ordered Pairs can have Predictive Value--But they do not always represent a Cause and Effect

Example: Vocabulary, Height

103Key Concepts:To be literate in algebraic thinking, students need to also understand that predication and causality are two different things. Ordered pairs can have predictive value, but there is not always a cause and effect relationship between the two things.Does increasing vocabulary mean that we grow in height?Both are related to age, but one does not cause the other.

Prototype for Lesson Construction

Touchable VisualDiscussion:Makes Sense of Concept12Symbols:Simply record keeping!Mathematical StructureDiscussion of the concreteQuantity:Concrete display of conceptLearn to Record these IdeasV. Faulkner and DPI Task Force adapted from Griffin104Again, the teacher can use this prototype for lesson construction to organize their planning.Even Algebra can fit this mold!(av. cost)(items) + gas = grocery $mx + b = y

12SymbolsSimply record keeping!Mathematical StructureDiscussion of the concreteQuantity:Concrete display of conceptFaulkner adapting Leinwald, Griffin

How do we organize data to make predictions and decisions? Why is the slope steeper for Fancy Food then Puggly Wuggly? What about gas cost? Is there a point at which the same items would cost the same at both stores?, etc.. 105Information listed in the ovals above include. Briefly remind participants about each of the ovals, but dont go into a great deal of detail as this information will be revisited in the Dollar Deals activities. QuantityShoppingMathematical StructureWe need to be sure that these are the types of questions we are asking--questions that bring meaning to the algebra:How do we organize data to make predictions and decisions? Why is the slope steeper for Fancy Food then Puggly Wuggly?What about gas cost? Is there a point at which the same items would cost the same at both stores?, etc..

Symbols(average cost) (items) + gas = grocery $mx + b = y

Common Core Standards: AlgebraSeeing Structures in Expressions

Creating Equations

Reasoning with Equations106Key Concepts:If you use this approach, we promise you that your students will think and act as the CCS (Common Core Standards) asks them too! It will be HOW they approach the algebra!REACH ALL STUDENTS! But as soon as we start talking about reaching all students, and about deepening understanding and getting students to be critical thinkers, Many of you begin to think, perhaps begrudgingly, about what I call the most stressful word in Education today: Differentiation. We need to think about Sameationone activity that reaches all students at levels of thinking. The various shopping activities that follow will allow for reaching all students. Algebra through ShoppingThe following lesson units/materials are all copyrighted with Wake County Public Schools/Valerie Faulkner.These have also been incorporated into Walch Educational publications, Intro to High School Math and Foundations of MathematicsFurther training in these units is available through DPI.The reality that shopping can be modeled with algebra is, of course, public domain.Steve Leinwand also presents ideas using average cost from shopping register tickets and speeding ticket formulas to develop algebraic concepts.107Read through this slide.Note Foundations of Math co-developer and trainer Valerie Faulkner developed these, but note that she does NOT make any money for the sale of these texts. They are the property of Wake County and Walch Educational (She developed them on Wake County time). Unit 1Concept Development Dollar Deals

108Key Concepts:

In Dollar Deals students explore the identity function. And as they explore this most basic of functions (a 1:1 correspondence) they learn the ideas of mathematical models, the power behind a Cartesian graph and a t-chart, and they develop critical vocabulary. They do this all by shopping at Dollar Deals!Of course the teachers guiding questioning techniques and activities is critical to this success AND the teachers willingness to keep the algebra simple while the class learns and explores the structures of algebra.

Dollar Deals Learning Objectives

Dollar Deals12SymbolsSimply record keeping!Math StructureDiscussion of the concreteQuantity:Concrete display of concept1 2 3321x f(x) y

x=y 1x = y 1x = y+0Ratio;Forms of same info;1:1;Identity function109Key Concepts:

Vocabulary development and algebraic structures are being taught with the Dollar Deals basic situation.Also when you plot points make sure to say to students (again and again) There are two pieces of information for every ordered pair.Also, empowering students to see a connection between these symbolic representations and real meaning. Discussions can be had about complicated concepts BECAUSE the situation is simple. The focus is on the structural and vocabulary development. If you move too quickly into more complicated examples, you lose your opportunity to discuss these concepts with the students at a more concrete level at which they are able to understand the concept.No Tax, No Tricks:One Dollar per One Item.Dollar Deals

Where EVERYTHING is just one dollar.110Engage in Related Activity See Dollar Deal Trainer Notes in Unit 7. Key Concepts:We dont develop student understanding of algebraic equations as this basic of level with understanding. X and Y vary directly and we need to set that foundation from the beginning. We tend not to begin here because we learned procedurally and we begin with what we learned. Terminology:The Identity propertythe identity of the number does not change. 1 x = y illustrates the Identity Property of Multiplication (by 1) x + 0 = y illustrates the Identity Property of Addition (of 0).

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00 1 2 3 4 5 6 7 8 9 10(5 items, $5)If you buy 5 items it will cost 5 dollars

111111Move student through the story of Dollar Deals, to the detail of one shopper and then back to the main idea of the generalized equation for Dollar Deals.Develop FROM the STORY, not the other way around. Start with where the brain lives with the comparison, with the story, with Filling up your Shopping Bag! Connect this to the idea from the research about the way the brain processes number. We need to connect to this part of the brain if students are going to be able to process and remember algebra deeply.

X = Y112112Key Concepts:Only once the student accesses the idea of a linear relationship through seeing the structure can they begin to process the generalized equation. Guiding Question:What might be confusing for our secondary students with this equation? (The idea that x and y equal the same value might be difficult depending on the students conceptual understanding of equality.)

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00 1 2 3 4 5 6 7 8 9 10(5 items, $5)(7 items, $7)(2 items, $2)(0 items, $0)(10 items, $10)The shoppers are the details.

The EquationX=Y is the Main Idea of the story!113113Key Concepts:So now, as we complete the details of different shoppers, we understand the STRUCTURE of the Relationship. This allows us to REASON with the ideas and to CREATE our own algebra stories.Give Credit to Dr. Chris Cain for emphasizing Story Structure:Kids struggle going back and forth from a point to the equation. Why?Because we dont make it clear to them that the generalized equation is the Main Idea of the story and the points are the DETAILS!The price at the store is the general equation or contract, the points are individual shoppers! Understanding this structurally makes a huge difference in how students process the DIGITS!There are two pieces of information in the story for each point. (number of items and the cost). Have students write this out so they can go from the details to the main idea of the story---the equation. It is more difficult to figure out the main idea from the details than finding the details from the main idea. Unit 2Dollar Deals vs. Puggly Wuggly

114Key Concepts:Slowly add more complicated mathematical issues. At the next store we are charged $2.00 per item. How does this affect slope? What will this look like to your wallet? On a Cartesian plane? In your math model? On a T-chart?

Puggly Wuggly vs. Dollar Deals Learning Objectives

12SymbolsSimply record keeping!Math Structure:Discussion of the concreteQuantity:Concrete display of concept1 2 3321x f(x) y

?Coefficient and slopeHow does cost affect slope?Why does PW have a steeper slope?How do you model PW?2x=y115These are the learning objectives for Puggly Wuggly vs. Dollar Deals.Math StructureCoefficient and slopeHow does cost affect slope?Why does PW have a steeper slope?How do you model PW?

Bag day sale!

All items you can fit in 1 bag: 2 dollars per item!

Puggly Wuggly Super Sale!

116Engage in Related ActivityHave a few shoppers shop at Puggly Wuggly for items. Discuss the differences in the two models. Allow participants at their tables to discuss how the story differs from the previous story and write a math model as an equation for the Puggly Wuggly Super Sale. Guiding Question(s) Is everyones model the same in form? In representing the correct shopping scenario?2x = y; y = 2x; y = 2x + 0The next slide gives a graphic representation of the two stories and the two math models for the Dollar Deals and Puggly Wuggly Super Sale. 10

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00 1 2 3 4 5 6 7 8 9 10Discussion of slope takes on moremeaning: The pricesare STEEPER

Cost goes up more quickly!(2 items, $2)(2 items, $4)117117DemonstrateThis slide illustrates the Puggly Wiggly vs. Dollar Deals Scenario in a different form of the story. Emphasize how the context helps students to see the main idea of the story and how the slope is critical to that main idea. Also how the context allows the vocabulary to be developed

Unit 3Puggly Wuggly vs. Fancy Food

118Engage in Related ActivityKey Concepts:Continue to solidify students understanding and then build on this understanding with further considerations.For instance, it is important to discuss what we are NOT measuring? What does and does not get quantified when we compare information?We need to develop the students ability to understand and read graphs and other information presented through numbers.Guiding Question(s)We are only measuring what?What about quality?How well the workers are treated?How pleasant the shopping experience is?Tie-in with Numeracy and reading charts and graphs in textbooks and newspapers. What is and IS NOT being measured?

Puggly Wuggly vs. Fancy Foods Learning Objectives

12SymbolsSimply record keeping!Math Structure:Discussion of the concreteQuantity:Concrete display of concept1 2 3321x f(x) y

?Coefficient and slopeHow does cost affect slope?What does it mean that the slope lines meet at 0?How do you model FF and PW?2.5x=y4x = y119These are the learning objectives for Puggly Wuggly vs. Fancy Foods.Math StructureCoefficient and slopeHow does cost affect slope?What does it mean that the slope lines meet at 0?How do you model FF and PW?

No salejust our usual price: $2.50 per item

Puggly Wuggly

120Engage in Related Activity Participants will be comparing the two shopping stories on the next slide in relationship to Dollar Deals and Puggly Wuggly Super Bag Day Sale scenarios. When the student is at the point where they really understand the basic structure, ask them to create their own story and model the scenario. How does their slope compare to Dollar Deals and the other shopping scenarios? You can pay now, or you can pay laterThe best food at the worst prices!All organic food$4 per itemFancy Foods

121Key Concepts:Fancy Foods may actually be cheaper in the long run when other factors are entered. Is it really cheaper per item in the long run? What are we analyzing with our linear equation? ONLY cost per item. We are not analyzing other factors. With more sophisticated math, mathematicians CAN analyze more factors. Keep this in mind. The linear equation is the foundation for most all of the mathematics that comes after it.Other Forms of the Number Relationship!X(Items)Y=2.5xPWY=4xFF12.5425837.51241016512.520

122Key Concepts:

Again reinforce the concept that information can be represented in many different forms.If you were the manager at these stores what form of the information might you want to track your cashiers?Unit 4Feeding the DumDee

123Key Concepts:

This unit explores the idea of the constant and what that does to your cost even if you dont buy any items once you get to the store.

Feeding the DumDee Learning Objectives

12SymbolsSimply record keeping!Verbal:Mathematical StructuresQuantity:Concrete display of concept1 2 3321x f(x) y

Why do thelines intersect?When is it cheaper to shop at fancy foods?What is a Y interceptand what is aconstant?2.5x=y4x = y

124In this case, we use travel costs (cost of gas for your car, bus fare) to create a more complex, yet still very understandable scenario.

Math Structure:

Why do the lines intersect?When is it cheaper to shopAt Fancy Foods?What is a Y intercept and what is a constant?

Feeding the DumDeeWhat Does it Look Like?

125Key Concepts:This unit introduces the idea of a constant and why that constant is called the y-intercept: Make sure to use the word constant and to connect it to the idea of the y-intercept consistently.Engage in Related ActivityBegin the unit with a little play, wherein both teachers have a role. See DumDee script in Trainers Notes.One lives next door to the Fancy Foods and is shopping there. The other teacher lives 10 miles from the Puggly Wuggly and is shopping there. But the DumDee, which only gets 7 miles to the gallon, will eat $9.00 of gas on the trip. That has to be added to the cost of the groceries. Thats your constant.Go to the next slide for an animation of the skit. Click on the points as they are mentioned by the participants in the story.

Thanks to James ONeal and Travis Blackwell for the idea for this skit as a way to communicate this topic.30

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00 1 2 3 4 5 6 7 8 9 10126126This slide animates the Dum Dee scenarioSelect two volunteers from the audience.ASK some of these questions with the graph still displayed.At what point will both shoppers incur the same cost for the same number of items? ( , ) (6,24)When is it maybe a good idea to just walk next door to Fancy Foods to shop? Why? (When you need just a few items. The total cost would be less.)When is it maybe a good idea to drive to Puggly Wuggly? Why? (When you need more items. The total cost would be less.)Why might someone decide to shop at Fancy Foods anyway?(They live right next door, prefer organic food or love the shopping experience.)When a linear equation is in the form y=mx +b, the b is called the Y-intercept. Why? (It is where the point crosses the y-axis and represents, in this case, in additional cost added to the cost per item times the number of items.)What is th