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Teaching for Conceptual Understanding:

Ratios and Proportional Relationships

Professional Development

ParticiPant Workbook Sampler

2

For Professional Development resources and programs, visit www.pearsonpd.com.

Pearson School achievement Servicesteaching for conceptual Understanding: ratios and Proportional relationshipsParticipant Handbook

Published by Pearson School achievement Services, a division of Pearson, inc. 1900 E. Lake ave., Glenview, iL 60025

��3HDUVRQ��,QF��RU�LWV�DI¿OLDWHV�all rights reserved.Printed in the United States of america.

iSbn 115504iSbn 97814026-3885-5

Teaching for Conceptual Understanding: Ratios and Proportional Relationships© 2013 Pearson, Inc.

3

Table of Contents

teaching for conceptual Understanding: ratios and Proportional relationships overview . . . . . . . . . . . . . . . . . . . . . . . 4

Section 1: the Progression of ratios and Proportional relationships . . . . . . . 7

Section 2: building the Foundation for ratios and Proportional relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Section 3: Making connections within Mathematics . . . . . . . . . . . . . . . . . . . . 31

Section 4: Planning Lessons with ratios and Proportional relationships . . . 35

5HÀHFWLRQ�DQG�&ORVLQJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Jodi and Erik

Please delete everything from this point on except for pages 18, 19, and 20


4

Teaching for Conceptual Understanding: Ratios and Proportional Relationships Overview

Agenda

introduction

Section 1: the Progression of ratios and Proportional relationships

Section 2: building the Foundation for ratios and Proportional relationships

Section 3: Making connections within Mathematics

Section 4: Planning Lessons with ratios and Proportional relationships

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Outcomes at the conclusion of this workshop, you will be able to

� �� DUWLFXODWH�WKH�OHDUQLQJ�SURJUHVVLRQV�QHFHVVDU\�IRU�VWXGHQWV�WR�FRQFHSWXDOO\�XQGHUVWDQG�ratios and proportional relationships;

� �� LGHQWLI\�VWUDWHJLHV�IRU�KHOSLQJ�VWXGHQWV�EXLOG�WKHLU�PDWKHPDWLFDO�XQGHUVWDQGLQJ�RI�UDWLRV�and proportional relationships;

� �� XVH�D�SODQQLQJ�WHPSODWH�WR�EXLOG�OHVVRQV�WKDW�VWUDWHJLFDOO\�VXSSRUW�WKH�FRQFHSWXDO�development of ratios and proportional relationships;

� �� LGHQWLI\�VWUDWHJLHV�WKDW�VXSSRUW�VLPXOWDQHRXV�GHYHORSPHQW�RI�FRQFHSWXDO�understanding and problem-solving skills with the intentional use of purposeful student VWUXJJOH��ÀH[LEOH�JURXSLQJ��DQG�RQJRLQJ�DVVHVVPHQWV��DQG

� �� DUWLFXODWH�FRPPRQ�PLVFRQFHSWLRQV�DV�RSSRUWXQLWLHV�IRU�VWXGHQWV¶�FRQFHSWXDO�understanding of ratios and proportional relationships.


5

Introduction and Overview

Content Warm-upProblem 1: tammy bought 3 widgets for $2.40. at the same price, how much would 10 widgets cost?

Problem 2: tammy bought 4 widgets for $3.75. How much would a dozen widgets cost?


6

Section 1: The Progression of Ratios and Proportional Relationships

Exploring the Standards

RED:

YELLOW:

GREEN:

Impact Observations


7


The Importance of Ratios and Proportional Reasoning 1. _______________________________________________________________

_______________________________________________________________

2. _______________________________________________________________

_______________________________________________________________

3. _______________________________________________________________

_______________________________________________________________

4. _______________________________________________________________

_______________________________________________________________

5. _______________________________________________________________

_______________________________________________________________

Revisit Section 1 Big Questions� �� +RZ�DUH�\RX�UHVSRQVLEOH�IRU�WHDFKLQJ�QHZ�FRQWHQW�DW�\RXU�JUDGH�OHYHO"

� �� +RZ�GRHV�ZKDW�\RX�WHDFK�LPSDFW�VWXGHQW�XQGHUVWDQGLQJ�DW�WKH�QH[W�JUDGH�OHYHO"


8


5HÀHFWLRQ��7KH�-RXUQH\�%HJLQVWrite a short paragraph or story that describes the journey that your students will take during their formal study of ratios and proportional relationships.


9

Section 2: Building the Foundation for Ratios and Proportional Relationships

'H¿QLQJ�5DWLRWhat is a ratio?

create at least three scenarios that the ratio 3:1 can represent.


10


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Sorting Representations

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12





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9RFDEXODU\�DQG�6\PEROV�LQ��53��

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MisconceptionsList misconceptions that students might have about ratios that would prevent them from succeeding with this card sort.


14


Additive versus Multiplicative Reasoningcould 7 be the missing number in this proportion? 3 : 4 :: 6 :___ (Pearson achievement 6ROXWLRQV��

is it additive or multiplicative?

1. Janet and Jeanette were walking to school, each walking at the same rate. Jeanette VWDUWHG�¿UVW��:KHQ�-HDQHWWH�KDV�ZDONHG��EORFNV��-DQHW�KDV�ZDONHG��EORFNV��+RZ�IDU�will Janet be when Jeanette is at 12 blocks?

� �� /LVD�DQG�/LQGD�DUH�SODQWLQJ�FRUQ�RQ�WKH�VDPH�IDUP��/LQGD�SODQWV��URZV��,I�/LQGD¶V�FRUQ��LV�UHDG\�WR�SLFN�LQ��ZHHNV��KRZ�PDQ\�ZHHNV�ZLOO�LW�WDNH�IRU�/LVD¶V�FRUQ�WR�EH�UHDG\"

3. kendra and kevin are baking cookies using the same recipe. kendra makes 6 dozen and kevin makes 3 dozen. if kevin is using 6 ounces of chocolate chips, how many ounces will kendra need?

�9DQ�GH�:DOOH��.DUS��DQG�%D\�:LOOLDPV��


15


Models and Strategies for Problem Solving in 6.RP.3there are several different strategies listed in the standard that students can use to solve SUREOHPV�WKDW�LQYROYH�UDWLRV��:KHQ�VWXGHQWV�¿UVW�VWDUW�XVLQJ�WKHVH�VWUDWHJLHV��WKHLU�DSSURDFKHV�may be very in-depth and lengthy. as they become more familiar with the relationships and comparisons among equivalent ratios, their strategies will become more abbreviated and HI¿FLHQW��

�7KH�&RPPRQ�&RUH�6WDQGDUGV�:ULWLQJ�7HDP��

Tables of Equivalent RatiosUse a ratio table to solve the following problem:

7\OHU�DQG�,VDDF�KDYH�ERWK�YROXQWHHUHG�WR�EULQJ�VQDFNV�IRU�WKH�FODVV�¿HOG�WULS��7\OHU¶V�VQDFN�UHFLSH�FDOOV�IRU��FXS�RI�FKRFRODWH�FKLSV�IRU�HYHU\��FXSV�RI�QXWV��DQG�,VDDF¶V�VQDFN�UHFLSH�calls for 3 cups of chocolate chips for every 5 cups of nuts.

� �� :KHQ�ERWK�VWXGHQWV¶�EDWFKHV�RI�VQDFNV�KDYH�WKH�VDPH�DPRXQW�RI�FKRFRODWH�FKLSV��whose batch is “nuttier”?

� �� :KHQ�ERWK�VWXGHQWV¶�EDWFKHV�RI�VQDFNV�KDYH�WKH�VDPH�DPRXQW�RI�QXWV��ZKRVH�LV�PRUH�“chocolaty”?

3. if both tyler and isaac bring in the same number of cups of their snack, whose is “nuttier” and whose is more “chocolaty”?


16


Consider the following ratio:³)RU�HYHU\��FXSV�JUDSH�MXLFH��PL[�LQ��FXSV�SHDFK�MXLFH�´


now, read through the following ccSSM standards, and consider the connections between these standards as seen in the equivalent ratio tables and graphs below:

6.rP.3 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.

�1*$�&HQWHU�DQG�&&662��

6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship WR�RQH�DQRWKHU��ZULWH�DQ�HTXDWLRQ�WR�H[SUHVV�RQH�quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. analyze the relationship between the dependent and independent variables using graphs and tables, and relate WKHVH�WR�WKH�HTXDWLRQ��)RU�H[DPSOH��LQ�D�SUREOHP�involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

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17


Double Number Line Diagrams

Tape Diagrams



18


Solving Problems Using 6.RP.3 Models and Strategies

Problem A.

ricky racer can complete 3/4 of a race in 24 minutes. if he continues at that same rate, how long would it take him to complete the whole race? if he wanted to cut his time by 25%, how ORQJ�ZRXOG�LW�WDNH�KLP�WR�FRPSOHWH��RI�WKH�UDFH"��WDSH�GLDJUDP�RU�GRXEOH�QXPEHU�OLQH�

Problem B.

if two bags of candy cost $5, how much will 12 bags of candy cost? How much will 3 bags of FDQG\�FRVW"�:KDW�DERXW��EDJV�RI�FDQG\"��WDEOH�DQG�JUDSK�RU�HTXDWLRQ�


19


Problem C.

$W�WKH�IDUPHUV¶�PDUNHW��-RQHV�VHOOV�WRPDWRHV�DW��SRXQGV�IRU��ZKHUHDV�6PLWK�VHOOV�KLV�WRPDWRHV�DW��SRXQGV�IRU��:KLFK�IDUPHU�VHOOV�KLV�WRPDWRHV�DW�D�FKHDSHU�SULFH"��UDWLR�WDEOH�

Problem D.

to create her special paint color for the school project, Ms. Smith combines cups of red, blue, DQG�ZKLWH�SDLQW�LQ�D�UDWLR�RI��+RZ�PXFK�RI�HDFK�SDLQW�ZLOO�QHHG�WR�EH�PL[HG�WR�FUHDWH��JDOORQV�RI�SDLQW"��WDSH�GLDJUDP�


20


Percent as a Rate per Hundred in 6.RP.3c6ROYH�HDFK�SUREOHP�XVLQJ�D�GRXEOH�QXPEHU�OLQH��:KHQ�\RX�¿QLVK�DOO�IRXU�SUREOHPV��JR�EDFN�and solve each one using a different method. try to use a different approach for your second method for each problem.

$QQ�DQG�&DUPHQ�HDFK�ZHQW�RQ�ELF\FOH�WULSV��$QQ¶V�ZKROH�WULS�ZDV��PLOHV�ORQJ��&DUPHQ¶V�whole trip was 300 miles long. Each girl rides 30 miles each day.

1. What percent of her total trip does each girl complete in a day?

2. after how many days will each girl complete 40% of her trip?

3. at the start, each girl had 12 pounds of food. Unfortunately, each girl used up 75% of WKDW�IRRG�LQ�WKH�¿UVW��RI�WKH�WULS��:KHQ�WKLV�KDSSHQHG�

a. How many miles had each girl ridden?

b. How many pounds of food did each girl have left?

4. 80% of the way through the trip, each girl will be joined by a friend. For how many miles will each girl ride with a friend?

�$PHULFD¶V�&KRLFH��


21


*UDGH��3URJUHVVLRQ�5HÀHFWLRQ

For Grade 6 teachers, how will you prepare to teach the ccSSM Grade 6 ratios and Proportional relationships concepts?

)RU�*UDGH��WHDFKHUV��ZKDW�ZLOO�\RX�H[SHFW�\RXU�LQFRPLQJ�*UDGH��VWXGHQWV�WR�NQRZ�DERXW�ratios and proportions?


22


Ratios with Rational Numbers in 7.RP.17R�PDNH�6XSHU�6RXU�&DQGLHV�\RX�PL[��FXS�OHPRQ�MXLFH�IRU�HYHU\��FXS�VXJDU��,I�\RX�ZDQW�WR�PDNH��WRWDO�FXSV�RI�FDQG\�PL[��KRZ�PDQ\�FXSV�RI�OHPRQ�MXLFH�DQG�VXJDU�ZLOO�\RX�QHHG"

'H¿QLQJ�3URSRUWLRQDO�5HODWLRQVKLSVWhat is a proportional relationship?


23


Recognizing Proportional Relationships in 7.RP.2aExample #1

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Example #2

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Nonexample #1

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Nonexample #2

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24

Graphing Proportional Relationships 1. two of these four tables of values are ratio tables and two are not.

a. Which two tables are ratio tables? (this means that they represent a SURSRUWLRQDO�UHODWLRQVKLS��

� E�� :ULWH�D�IRUPXOD�LQ�WKH�IRUP�RI�\� �N[�IRU�WKH�WZR�SURSRUWLRQDO�UHODWLRQVKLSV�

� F�� 8VH�\RXU�IRUPXODV�WR�¿QG�WKH�PLVVLQJ�YDOXHV�LQ�WKH�WDEOHV�WKDW�UHSUHVHQW�proportional relationships.



25


2. Sketch and label graphs that represent the four relationships in problem 1.


26

3. Say how you can determine whether two quantities are proportional to each other. Use your results from problems 1 and 2 to support your answer.

4. a relationship can have two variables and still not be proportional. Say why.

5. Look again at the graphs you sketched in problem 2. What do you observe when you compare the graphs that represent proportional relationships to the graphs that represent non-proportional relationships?

6. in proportional relationships, when one variable equals zero, the other variable must also equal zero. Say why, using what you know about the tables, formulas, and graphs of proportional relationships to support your answer.

7. Sometimes, a proportional relationship has meaning for only some values. Say why, DQG�JLYH�DQ�H[DPSOH�WR�VXSSRUW�\RXU�UHDVRQLQJ�

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27


Conceptual Understanding of Multiple Representations

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Chart Graph

Equation Words


28


Multistep Problems

Different Context, Same Models and Strategies

the jacket keri wants to buy is 20% off this week. if she buys the jacket this week she will have to pay $140. How much would she have had to pay if she bought the jacket last week? 1RZ��LI�VKH�ZDQWV�WR�ZDLW�XQWLO�QH[W�ZHHN�WR�EX\�WKH�MDFNHW��WKH��VDOH�SULFH�ZLOO�LQFUHDVH�E\��+RZ�PXFK�ZLOO�VKH�KDYH�WR�SD\�QH[W�ZHHN"


29


Revisit Section 2 Big Questions� �� :KDW�W\SHV�RI�SUREOHPV�ZLOO�\RXU�VWXGHQWV�QHHG�WR�VROYH�LQ�RUGHU�WR�VROYH�SUREOHPV�

with ratios and proportions at a conceptual level?

� �� +RZ�GR�WKH�LQVWUXFWLRQDO�VWUDWHJLHV�WKDW�\RX�FKRRVH�WR�XVH�KHOS�\RXU�VWXGHQWV�GHYHORS�a conceptual understanding of ratios and proportions?


30

5HÀHFWLRQ��:LOO�'Rto help your students develop a conceptual understanding of ratio and proportions you will do the following:

1.

2.

3.

4.

5.

6.



31

Section 3: Making Connections within Mathematics

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Similar Triangle Applications

try to solve each problem in two ways—use ratios within the triangles, and use ratios between the triangles.

� �� $W�WKH�DLUSRUW��WKHUH�LV�D�ÀDJ�SROH��$W�D�FHUWDLQ�WLPH�HDFK�GD\��WKH�VKDGRZ�RI�WKH�ÀDJSROH�PHDVXUHV��P��DQG�D�PDQ�ZKR�LV��FP�WDOO�KDV�D�VKDGRZ�WKDW�PHDVXUHV�191 cm.

+RZ�KLJK�LV�WKH�ÀDJSROH"

remember to convert measurements to the same units.

2. Directly outside your hotel window, there is a very tall palm tree. You have heard that a hurricane may be coming, and you are concerned that the tree will fall down and smash your window. You make some direct measurements to help you determine whether you are in danger.

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32

a stick measuring 1 yard has a shadow of 2.5 ft at the same time the tree has shadow of 29 ft.

the distance between the base of the tree and the hotels is 12 yards.

are you safe?

� �� +RZ�GR�WKHVH�SUREOHPV�H[WHQG�VWXGHQWV¶�XQGHUVWDQGLQJ�RI�UDWLRV�DQG�SURSRUWLRQDOLW\"

2. How do these tasks support simultaneous development of problem solving skills and conceptual understanding?

3. What misconceptions or confusions might students bring to this problem set?

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Section 3: Making Conncetions within Mathematics



33

CCSSM Connections SearchUse the space below to make as many connections as possible to the ratios and Proportional relationships standards.

Ratios and ProportionalRelationship Standards

Connected Standards


34

Revisit Section 3 Big Questions� �� +RZ�FDQ�\RXU�VWXGHQWV¶�XQGHUVWDQGLQJ�RI�UDWLRV�DQG�SURSRUWLRQV�LPSDFW�WKH�WHDFKLQJ�

and learning of standards in other content areas?

� �� +RZ�ZLOO�\RX�KHOS�\RXU�VWXGHQWV�PDNH�FRQQHFWLRQV�EHWZHHQ�UDWLRV�DQG�SURSRUWLRQV�and topics that they will study when you address other standards?



35

Section 4: Planning Lessons with Ratios and Proportional Relationships

Classroom Resources BrainstormWhat resources can you provide students to help them become successful with solving problems involving ratios and proportional relationships?


36

Template for Success

Four Steps to Planning Effective Mathematical Instruction

When you plan a mathematics lesson, you start by selecting a set of problems or tasks for VWXGHQWV�WKDW�DGGUHVV�D�SDUWLFXODU�VWDQGDUG�RU�FOXVWHU��GHSHQGLQJ�RQ�WKH�OHVVRQ��<RX�KDYH�WKH�mathematics goal of the lesson in mind from the start of the planning process and, hopefully, you have also thought about the questions, “How will i know that students have met this goal? What evidence will i see in their work to support that claim?” Unfortunately, mathematics OHVVRQV�GRQ¶W�DOZD\V�JR�DV�SODQQHG��2QH�ZD\�WKDW�\RX�FDQ�KHOS�WR�HQVXUH�WKDW�\RXU�OHVVRQV�will go as planned is by planning for an effective mathematical discussion at the end of the class..

in addition to the Standards for Mathematical content, you must also be mindful of the Standards for Mathematical Practice. as you begin the planning process, you must consider which of the mathematical practices will be highlighted in the lesson and keep those practices LQ�PLQG�DV�\RX�WKLQN�DERXW�WKH�PDWKHPDWLFDO�GLVFXVVLRQV�DQG�H[SHULHQFHV�WKDW�\RX�ZDQW�WR�happen as a result of the lesson.

Good mathematical discussions do not just happen. they are the result of careful thought and planning. the goal is to keep the discussion focused on the mathematics of the lesson. the four-step process outlined below is designed with that goal in mind.

� �� :RUN�WKH�SUREOHP�VHW�RU�WDVNV�\RXUVHOI��7KLQN�DERXW�DOO�WKH�GLIIHUHQW��FRUUHFW��VROXWLRQV�WKDW�VWXGHQWV�PLJKW�SURGXFH��)RU�H[DPSOH��LI�WKH�SUREOHPV�DVN�VWXGHQWV�WR�FRPSDUH�two ratios, you�PLJKW�QDWXUDOO\�WKLQN�RI�¿QGLQJ�FRPPRQ�GHQRPLQDWRUV��EXW�VWXGHQWV�FRXOG�¿QG�GHFLPDO�HTXLYDOHQWV��XVH�EHQFKPDUN�QXPEHUV��RU�FURVV�PXOWLSO\�LQVWHDG�

2. think about the likely misconceptions and missteps. Group them with the correct solution methods that would result if the student had not taken the “misstep.”

3. once you have an idea about the different approaches and missteps that students might take, revisit the question, “What is the important mathematics that i want students to understand after working these problems or tasks?” ask yourself whether WKHVH�SUREOHPV�RU�WDVNV��DV�WKH\�DUH�SUHVHQWO\�ZRUGHG��ZLOO�UHVXOW�LQ�VWXGHQW�ZRUN�WKDW�brings this mathematics to the forefront.

a. Look back at the possible solution approaches. is there any way that students might solve these problems correctly that will not support the mathematics of the lesson? thinking back to the problems that ask students to compare two ratios, if the goal of the lesson is to have students understand that the SURFHGXUH�IRU�¿QGLQJ�FRPPRQ�GHQRPLQDWRUV�ZRUNV�EHFDXVH�RI�WKH�LGHQWLW\�SURSHUW\�RI�PXOWLSOLFDWLRQ�EXW�DOO�RI�WKH�VWXGHQWV�¿QG�GHFLPDO�HTXLYDOHQWV��LW�ZLOO�EH�GLI¿FXOW�WR�EULQJ�WKH�PDWKHPDWLFDO�³SXQFK�OLQH´�EDFN�WR�WKH�GHVLUHG�HQG�

Section 4: Planning Lessons wih Ratios and Proportional Relationships


37

b. Will the problems work as they are written, or do you need to think about UHZRUGLQJ��FKDQJLQJ�WKH�GLUHFWLRQV��RU�HYHQ�¿QGLQJ�D�GLIIHUHQW�WDVN�RU�SUREOHP�set? Yes, unfortunately, after all this work, sometimes you need to rethink the assignment at this point in the process.

4. once you are happy with the problem set or tasks, answer the question, “How will i UHDFW�LI�VWXGHQWV�GR�[��\��RU�]"´�IRU�HDFK�RI�WKH�DSSURDFKHV�RU�PLVVWHSV�\RX�LGHQWL¿HG�LQ�Steps 1 and 2.

a. What questions will you ask?

� E�� +RZ�ZLOO�\RX�OHDG�WKH�GLVFXVVLRQ�VR�WKDW�WKH�PDWKHPDWLFV�LV�FODUL¿HG"

c. What if you do not see the mathematics that you want to emphasize in the student work? Will you redirect student efforts, or can you ask a student to take a particular approach?

7KLV�VRXQGV�RYHUZKHOPLQJ�ZKHQ�\RX�¿UVW�VWDUW��EXW�ZLWK�SUDFWLFH��LW�EHFRPHV�DXWRPDWLF�WR�think this way. and, once you have done these things, you basically have your lesson planned!



38

Problem or task

%HIRUH�\RX�EHJLQ� :KDW�LV�WKH�LPSRUWDQW�PDWKHPDWLFV�WKDW�\RX�ZDQW�VWXGHQWV�WR�XQGHUVWDQG�DIWHU�FRPSOHWLQJ�WKH�SUREOHP�V��RU�WDVN�V�"

What Standards for Mathematical Practice do you want to highlight?

:LOO�WKH�SUREOHP�V��RU�WDVN�V��UHVXOW�LQ�VWXGHQW�ZRUN�WKDW�ZLOO�EULQJ�WKH�PDWKHPDWLFV�WR�WKH�IRUHIURQW"�,I�QRW��ZKDW�adjustments need to be made?

:RUN�WKH�SUREOHP�V��\RXUVHOI��,GHQWLI\�SRVVLEOH��FRUUHFW��VROXWLRQ�methods.

identify likely misconceptions related to correct solution methods.

How will you react if students do this?



39

Revisit Section 4 Big Questions� �� :KDW�NH\�HOHPHQWV�PXVW�\RX�EH�PLQGIXO�RI�ZKHQ�\RX�SODQ�OHVVRQV�WKDW�LQYROYH�UDWLRV�

and proportional relationships?

� �� +RZ�ZLOO�\RXU�OHVVRQ�SODQQLQJ�SURFHVV�FKDQJH�WR�LQFRUSRUDWH�WKHVH�NH\�HOHPHQWV"



40

5HÀHFWLRQ�DQG�&ORVLQJ

)LQDO�5HÀHFWLRQ��7DNLQJ�$FWLRQcreate three statements that describe actions that you will take in your classroom based on WRGD\¶V�ZRUNVKRS�

1. _________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

2. _________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

3. _________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________


41

References

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———. 2009a. Ramp-Up to Algebra: Ratio and Proportionality��:DVKLQJWRQ��'&��$PHULFD¶V�choice.

———. 2009b. Ramp-Up to Pre-Algebra: Ratios and Graphs��:DVKLQJWRQ��'&��$PHULFD¶V�&KRLFH�the common core Standards Writing team. 2011. “Progressions for the common core State

6WDQGDUGV�LQ�0DWKHPDWLFV��GUDIW��´�Tools for the Common Core��EORJ��$FFHVVHG�0DUFK��2013. KWWS��FRPPRQFRUHWRROV�¿OHV�ZRUGSUHVV�FRP��FFVVBSURJUHVVLRQBUSB��B��B��B��pdf.

Grouws, Douglas a., and kristin J. cebulla. 2000. Improving student achievement in mathematics. accessed March 21, 2013. KWWS��ZZZ�LDRHG�RUJ�¿OHV�SUDF��H�SGI�

national center on Education and the Economy, the. 2002. Numbers from Zero to One. :DVKLQJWRQ��'&��$PHULFD¶V�&KRLFH��

1DWLRQDO�*RYHUQRUV�$VVRFLDWLRQ�&HQWHU�IRU�%HVW�3UDFWLFHV��1*$�&HQWHU��&RXQFLO�RI�&KLHI�6WDWH�6FKRRO�2I¿FHUV��&&662��³&RPPRQ�&RUH�6WDWH�6WDQGDUGV�IRU�0DWKHPDWLFV�´�Washington, D.c.: national Governors association center for best Practices, council of chief 6WDWH�6FKRRO�2I¿FHUV��$FFHVVHG�0DUFK�� http://www.corestandards.org/assets/ccSSi_Math%20Standards.pdf.

Pearson achievement Solutions. 2007. Teaching and Understanding Rational Numbers: Key Idea 3: A Ratio Shows a Comparison by Division: Facilitator Guide. Glenview, iL. Pearson achievement Solutions.

Van de Walle, John a., karen S. karp, and Jennifer bay-Williams. 2013. Elementary and Middle School Mathematics: Teaching Developmentally. boston, Ma: Pearson Education, inc.

Using your smart phone, tablet, or laptop Internetbrowser, open the Pearson Customer SatisfactionSurvey or scan the QR code to open the link for avery brief survey.

Please select these answers for the following survey questions:

Q: Please select the product or service for which you are completing the survey. [Choose: Professional Development]

Q: How did you hear of this survey? [Choose: Workshop]

Q: Your Workshop ID is: __________________________

http://www.pearson.com/makethegrade

Using your smart phone, iPad, or laptop Internet browser, open the Pearson Customer Satisfaction Survey or scan the QR code to open the link for a very brief survey. http://www.pearsonschool.com/makethegrade Home page/choose: Professional Development How did you hear/choose: Workshop Workshop ID: Insert CRF

(to be provided by your facilitator)

Documents

Teaching for Conceptual Understanding: Ratios and ...assets.pearsonglobalschools.com/asset_mgr/versions/2013-07/EBE4… · Consider the following ratio: ³)RU HYHU\ FXSV JUDSH MXLFH