1Teaching Module 1: Preparing To Teach Proportional Reasoning
Proportional reasoning has been called both the capstone of elementary mathematics and the cornerstone of higher mathematics (Lesh, Post, Behr, 1988).
Teaching Module 1 addresses the following:
Proportional reasoning in the classroom and its place in the Common Core Mathematics Standards
The related teacher resources available at the Scale City website
What students should know and be able to do to understand proportional reasoning
2Reflecting on teaching proportional reasoning
Take a few minutes to answer the following questions:
What does proportional reasoning mean to you?
What is the place of proportional reasoning in the Common Core Mathematics Standards for grades six through eight?
How might you engage students in learning about proportional reasoning so that it is enjoyable as well as understandable?
NOTE: There will be multiple boxes like the one on this page throughout the teaching modules for you to record answers and reflections. Once you click on the save button, the text will be stored and available to print when you complete the entire module. If you are discussing the questions with a partner or professional learning community, you might just type a very brief summary of your discussion in the box. If you are going through the module individually, you should type a longer reflection.
Compare your initial thoughts to the critical areas for instructional time for grades six through eight in the Common Core State Standards. Click on the links below to read about these.
Scale City has a number of resources to help middle grades students develop their proportional reasoning abilities. Before using the modules with a class, it is helpful to assess what students already know about ratio, proportion, and scale.
Meaning of scale
Video: 1m 44s
What do you expect middle grades students to think of when you mention scale?
Click "view" on the left to watch a video of two seventh grade students explaining what scale means to them.
For personal reflection or discussion with a partner or a professional learning community, answer the following questions:
What, if anything, was surprising about these students’ replies?
What do you think Lily meant when she mentioned age?
Do you think that Laura’s mention of a scale drawing means that she understands scale?
Record your reflection in the box below and click the save button when you are finished.
Use the link below to download the Vocabulary List from Scale City for more information about the meaning of scale, ratio, proportion, and other related terms: The Road to Proportional Reasoning: VOCABULARY LIST (PDF)
4Prerequisite understandings and skills
Proportional reasoning involves the ability to make multiplicative comparisons among relationships. In elementary school, comparisons often involve addition and subtraction.
Additive comparisons continue to be important, but as students move into intermediate grades and middle school, they should begin to use multiplicative reasoning (involving both multiplication and division). Along with other prerequisite skills and knowledge, multiplicative reasoning is essential for the development of a deep understanding of proportional reasoning.
For personal reflection or discussion within a professional learning community, do the following activity:
List learning targets that you would expect students to have mastered before understanding proportional reasoning.
Write your list in the space below and click the save button when you are finished.
The remainder of this module consists of brief videos of students answering questions about some of the knowledge and skills required to understand proportional reasoning. The students who participated in the videos include three rising 7th graders and one rising 8th grader. All the students have been taking accelerated mathematics classes. Three are enrolled in Algebra 1 in the coming school year.
Before viewing each video, think about what your students might understand about the topic. As you view the video, consider whether the students seem to have mastered the topic or whether misunderstandings still exist. Think about questions, teaching strategies, and concrete models you might use to help students develop these understandings and skills.
If you are working through this module on your own or with a learning partner or community, consider these questions before and after viewing the video:
What might you do differently?
What might you do next?
What additional suggestions do you have?
For more information about some of these prerequisite skills and understandings, click on the link to the Getting Ready Review of Fractions, Decimals and Percents. Handouts that may be used as pre-assessments are also included here.
Read the fourth, fifth, and sixth grade standards for multiplication and division of whole numbers from the Number and Operations in Base Ten and Number System domains in the Common Core Mathematics Standards.
Alternative algorithms for division and multiplications
Video: 4m 49s
Think about why these standards for multiplication and division are critical for developing proportional reasoning.
Click "view" on the left to see Harper’s understanding of multiplication and division. Consider the following questions:
How well has Harper mastered the multiplication and division standards above?
What gaps do you notice in her understanding? What would you do to help her fill these gaps?
Write your thoughts in the space below and click the save button when you are finished.
6Division of whole numbers: Part 2
Traditional long division algorithms
Video: 1m 05s
Now click on the video to view Laura as she discusses division and completes the computation for 200 ÷ 13. Listen carefully to her use of vocabulary for division and subtraction. Consider the following questions:
What is the correct quotient?
How does Laura’s understanding of division compare to Harper’s?
What gaps does Laura have? What do you notice about her understanding of remainders? What might you suggest next?
Write your thoughts in the space below and click the save button when you are finished.
7Simplifying fractions
Read the fourth grade standard for extending understanding of fraction equivalence and ordering from the Number and Operations – Fractions domain in the Common Core Mathematics Standards.
Think about how the standards in the previous section on multiplication and division relate to simplifying fractions and determining whether two fractions are equivalent. Also, consider how this relates to the standard above on extending understanding of fraction equivalence.
Simplifying fractions
Video: 5m 25s
Click on the video on the left to watch Harper simplify the fraction 500/1000. Before viewing, think about how you would like to have Harper simplify this fraction. Consider the following questions:
Do you think that Harper has a conceptual understanding of what it means to simplify a fraction? Why or why not?
What does Harper understand about simplifying fractions? What would you do to strengthen this understanding on a concrete level as suggested by the standard above?
The fractions that Harper is working with are all equivalent to ½. What questions might you ask to extend her understanding to other equivalent fractions?
Write your thoughts in the space below and click the save button when you are finished.
8Developing an understanding of proportional relationships: Part 1
Ratios are often expressed in fraction form although ratios and fractions are not equivalent. Ratios frequently compare one part to another part and fractions often compare one part to a whole, but this is not always the case. For example, the ratio 2:5 might be used to compare the number of boys in a group to the total group of 5 children. In this case, the ratio of 2:5 compares part of the group (2 boys) to the whole group of 5. Fractions might also represent a measure (such as ¾ of an inch) or an operation (such as taking a third of a recipe) and these ideas are not typically thought of as ratios.
There are many similarities between ratios and fractions, however. In this section, consider how students apply what they know about finding equivalent fractions to finding equivalent ratios and solving a problem involving a proportion.
Students sometimes think that whenever they see three numbers in a problem and must find a missing fourth number, they should set up a proportion and solve it by cross-multiplying. It is important that students be able to recognize whether the situation represents a proportional relationship and that they do not just plug numbers into a poorly understood template.
9Developing an understanding of proportional relationships: Part 2
Equivalent ratios
Video: 1m 34s
Click on the video on the left to view Harper as she finds equivalent ratios for 6:16. Consider the following questions:
Why does Harper initially say that there are no more equivalent ratios after finding that 3:8 is equivalent to 6:16?
Later Harper states that there are an infinite number of equivalent ratios. What has caused this change?
Write your thoughts in the space below and click the save button when you are finished.
10Developing an understanding of proportional relationships: Part 3
Solving a proportion
Video: 1m 17s
Using everyday proportions
Video: 3m 46s
A proportion is a relationship of equality between two ratios. Think about how finding equivalent ratios and solving a proportion are similar to finding equivalent fractions. What do you want students to know about how they are alike and how they are different?
Click on the video at the left to view Lily as she solves the proportion 2/7 = 6/n. Consider the following questions:
How does Lily initially go about solving the proportion? Why do you think that cross-multiplying is her first choice?
What method does Lily decide would be easier than cross-multiplying? How might you get students to consider a variety of methods for solving these problems?
Write your thoughts in the space below and click the save button when you are finished.
Click on the video on the left to view Lily as she thinks about an everyday application of proportional reasoning. Consider the following questions:
Why does Lily say that you cannot solve the proportion 2/3 = n/4?
Do you think that Lily has an intuitive understanding of the proportional relationship in this situation? Why or why not?
At the end of this clip, Lily seems to understand that one pound of apples would cost $1.50. She has still not completed the original problem. What would you ask next to help her determine how many pounds of apples she could get for $4 if 2 pounds of apples cost $3? What models might you suggest?
Write your thoughts in the space below and click the save button when you are finished.
For more information about helping students develop proportional reasoning, see the books for teachers listed on page 3 of the Resources in the Teacher’s Diner. You can download the complete PDF with the link below:
Some modules in Scale City require an understanding of multiplication of fractions, decimals, and mixed numbers since scale factors or the amounts that are being scaled up or down are not always whole numbers.
Read the fifth grade standard below from the Number and Operations - Fractions domain.
The distributive property of multiplication over addition is quite useful for understanding multiplication of a mixed number by a whole number. Think about models you would use and questions you would ask to help students understand this.
Multiplying mixed numbers
Video: 2m 38s
Click "view" on the left to see Chris as he tries to make sense of multiplying a mixed number by a whole number. Consider the following questions:
What error does Chris make when multiplying 2½ by 2?
Why is Chris asked to convert the measurement from 2½ feet to 30 inches?
Chris uses multiplication to divide 60 by 12. What does this say about his understanding of the relationship between multiplication and division?
What questions might you ask next to determine if Chris understands the application of the distributive property of multiplication over addition to the multiplication of a mixed number by a whole number?
What visual or concrete models might you use to help Chris understand this property?
Write your thoughts in the space below and click the save button when you are finished.
In each of the modules in Scale City, students will be multiplying and dividing fractions, mixed numbers, and decimals. Watch for any misunderstandings as they solve these problems.
12Dividing fractions
Adults, as well as students, often have difficulty understanding division when the divisor is not a whole number. Note the following sixth grade standard from the Number System.
Check Yourself
Click the link below for a simple quiz. Close the window when you are finished to return here. Quiz: Dividing Fractions
Dividing fractions
Video: 1m 42s
Click on the video clip on the left to view Lily as she tries to make sense of dividing by ½. If you are viewing the video with a learning partner or community, discuss the following.
If the scale is 1:24, what height on the actual person is represented by ½ inch on the scale model? How does Lily use this as a model for 3½ ÷ ½? Is this a correct application of division by ½?
What questions might you ask next to help Lily extend her understanding of division by a fraction or decimal?
Write your thoughts in the space below and click the save button when you are finished.
Understanding the links between division and multiplication and applying division of fractions and decimals to real-life situations are essential to the videos and interactives in Scale City. Again, ask questions to determine understandings and misunderstandings.
13Understanding percent: Part 1
Read the following sixth and seventh grade standards related to understanding percent.
Understanding links between fractions, percents, and proportions is a critical life skill and many adults apply this understanding in everyday shopping situations. Some adults, however, do not have a good grasp of percent concepts. For example, they cannot understand why a product is not free if it has been marked down 50% and then the store has an “additional 50% off” sale.
Check Yourself
Click the link below for a simple quiz. Close the window when you are finished to return here. Quiz: Understanding percent
14Understanding percent: Part 2
Connecting fractions, and percents
Video: 3m 35s
Click on the video clip on the left to view Lily as she makes sense of finding the percent that is equal to 1/3. If you are viewing the video with a learning partner or community, discuss the following.
What does Lily understand about converting a fraction to a percent?
How is converting a fraction to a percent similar to finding equivalent fractions?
How is cross-multiplying related to understanding the relationship between fractions and percents?
What models might you use to help Lily develop a deeper, more concrete, understanding of percent?
Write your thoughts in the space below and click the save button when you are finished.
For activities designed to strengthen early concepts of fractions, decimals, and percents, explore the video, interactive simulations, lesson plans, assessments, and resources in the Dinosaur World module. Note the percent tape that students can use as they compare the height of the boy to the height of the dinosaurs.
Read the standards below, noting that students should begin to understand ordering fractions as early as third grade and that this understanding should be extended to ordering all fractions and to ordering decimals to the hundredths place by fourth grade.
Think about difficulties students might have in ordering decimals with a different number of decimal places.
Check Yourself
Click the link below for a simple quiz. Close the window when you are finished to return here. Quiz: Ordering decimals
16Ordering decimals: Part 2
Ordering decimals, part 1
Video: 1m 54s
Ordering decimals, part 2
Video: 3m 22s
Ordering decimals, part 3
Video: 1m 35s
Click on the video clip on the left to view Harper as she attempts to order decimals written in tenths and hundredths. If you are viewing the video with a learning partner or community, discuss the following.
At the end of this clip, do you think Harper understands ordering decimals?
Watch the second video as Harper continues to discuss her understanding of ordering decimals.
At the end of this clip, do you think Harper has a better understanding of ordering decimals?
Watch the final video as Harper continues to discuss her understanding of ordering decimals.
What do you think Harper was thinking when she stated that ordering decimals is another example of an inverse relationship?
What suggestions do you have for helping Harper deepen her sense of ordering decimals? What models might you use and what questions might you ask?
Write your thoughts in the space below and click the save button when you are finished.
For applications of ordering decimals, see the activities, lesson plans, videos, interactives, and assessments in the Kentucky Horse Park module.
Adults often struggle to understand the metric system of measurement since it is still not commonly used in everyday life in the United States, in spite of its ease of use and its importance to scientific investigations and international commerce. Read the fourth and fifth grade standards below, noting that students are expected to convert units within the metric system in fourth and fifth grade.
Mnemonic devices are often used to help students learn the order of metric prefixes. A popular phrase is, “King Henry died by drinking chocolate milk.” The first letter of each word stands for “kilo, hecto, deka, base, deci, centi, and milli”.
Using metric measures
Video: 6m 32s
Click "view" on the left to see Harper as she explains her understanding of the use of this mnemonic. If you are viewing the video with a learning partner or community, discuss the following.
Does Harper’s use of this mnemonic deepen her understanding of metric conversions?
How well do you think Harper understands conversions within the metric system?
How is this related to her understanding of decimals?
What might you do to strengthen this understanding?
Write your thoughts in the space below and click the save button when you are finished.
For applications of metric measurements and conversions, see the activities, lesson plans, interactives, videos, and assessments in Mural World and Miniature Land.
On the individual module pages, you’ll find a Navigation Guide and Printable Forms for each interactive. Your students can use the forms to write down answers to the questions posed in the interactives.
Scale City Sign (Image) Scale City graphics by Lounge Lizard Worldwide
Screen 2:
An Introduction to Scale City (Document)
Written by Dr. Linda Sheffield, Regents Professor Emerita of Mathematics Education at Northern Kentucky University
Screen 3:
Meaning of scale (Video)
Paul Petrey, director
Esther Tattershall, camera
Roger Tremaine, audio
Marianne Mosley, editor
The Road to Proportional Reasoning: VOCABULARY LIST (Document)
Written by Dr. Linda Sheffield, Regents Professor Emerita of Mathematics Education at Northern Kentucky University
Screen 4:
Getting Started (Document)
“Getting Started,” “Handouts 1 and 2” from the lesson plan for Dinosaur World written by Mary Henson
Screen 5:
Alternative algorithms for division and multiplications (Video) Paul Petrey, director
Esther Tattershall, camera
Roger Tremaine, audio
Marianne Mosley, editor
Screen 6:
Traditional long division algorithms (Video)
Video: Traditional long division algorithms
Paul Petrey, director
Esther Tattershall, camera
Roger Tremaine, audio
Marianne Mosley, editor
Screen 7:
Simplifying fractions (Video) Video: Simplifying fractions
Paul Petrey, director
Esther Tattershall, camera
Roger Tremaine, audio
Marianne Mosley, editor
Screen 8:
Quiz: Developing an understanding of proportional relationships (Interactive) Adobe Flash development by Drew Hunt
Screen 9:
Equivalent ratios (Video)
Paul Petrey, director
Esther Tattershall, camera
Roger Tremaine, audio
Marianne Mosley, editor
Screen 10:
Solving a proportion (Video) Paul Petrey, director
Esther Tattershall, camera
Roger Tremaine, audio
Marianne Mosley, editor
The Road to Proportional Reasoning: RESOURCES (Document) “Road to Proportional Reasoning Resources” written by Dr. Mary Duncan
Screen 11:
Multiplying mixed numbers (Video) Paul Petrey, director
Esther Tattershall, camera
Roger Tremaine, audio
Marianne Mosley, editor
Screen 12:
Quiz: Dividing Fractions (Interactive)
Adobe Flash development by Drew Hunt
Dividing fractions (Video)
Paul Petrey, director
Esther Tattershall, camera
Roger Tremaine, audio
Marianne Mosley, editor
Screen 13:
Quiz: Understanding percent (Interactive)
Adobe Flash programming by Drew Hunt
Screen 14:
Connecting fractions, and percents (Video) Paul Petrey, director
Esther Tattershall, camera
Roger Tremaine, audio
Marianne Mosley, editor
Screen 15:
Quiz: Ordering decimals (Interactive) Adobe Flash development by Drew Hunt
Screen 16:
Ordering decimals, part 1 (Video) Paul Petrey, director
Esther Tattershall, camera
Roger Tremaine, audio
Marianne Mosley, editor
Ordering decimals, part 2 (Video) Paul Petrey, director
Esther Tattershall, camera
Roger Tremaine, audio
Marianne Mosley, editor
Ordering decimals, part 3 (Video) Paul Petrey, director
Esther Tattershall, camera
Roger Tremaine, audio
Marianne Mosley, editor
Screen 17:
Using metric measures (Video) Paul Petrey, director
Esther Tattershall, camera
Roger Tremaine, audio
Marianne Mosley, editor
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Clicking on "View" at the left under "Teacher Activity" takes you to Teaching Module 1: Preparing To Teach Proportional Reasoning. This interactive teaching module addresses the necessary knowledge and skills students should master in order to understand the mathematics of proportional reasoning and scaling.
Overview
What do your students already know about ratio, proportion, and scale? Do they have the knowledge and skills necessary to understand the mathematics of proportional reasoning? Teaching Module 1 focuses on eight skills students should master before exploring Scale City:
Division of whole numbers
Simplifying fractions
Developing an understanding of proportional relationships
Multiplying mixed numbers
Dividing fractions
Understanding percent
Ordering decimals
Metric measurement
The module features brief videos of students answering questions that enable you to gauge their knowledge and skills in the above areas. As you view the videos, you will be asked to consider whether the students seem to have mastered the topic or whether misunderstandings still exist. You'll see one teacher's approaches to help the students develop these understandings and skills, and you'll reflect on the questions and teaching strategies you might use in similar situations.
The content of this teaching module is presented in multiple ways:
Video
Video links include a title and length. Clicking on the link will open up a new window. If you encounter problems such as stopping or skipping, the video may need more time to buffer. Just pause the video until buffering is complete.
Discussion/Reflection Questions
This teaching module includes multiple opportunities for you to write answers to reflection questions.
If you are using this resource with a partner or professional learning community, you might just type a very brief summary of your discussion in the box. If you are going through the module individually, you should type a longer reflection.
You will have an opportunity to save your responses by clicking on the "save notes" button. The text will be stored and available to print when you complete the entire module.
Quiz Questions
Multiple choice questions provide an opportunity for you to check your own understanding of the mathematics. Click on the link under "Check yourself" and the question will pop-up in a new window. The questions are written with teachers in mind—they aren't very easy! Each question includes an explanation for each of the answer choices given.
Common Core State Standards
The new mathematics standards are provided for each of the mathematics skills explored in Teaching Module 1. Click on the link and a window pops-up with the standards that correspond to the skill and grade level.
Scale City Resources
Links to the Scale City resources referenced in this teaching module are provided as needed.
Academic standards correlations on Teachers' Domain use the Achievement Standards Network (ASN) database of state and national standards, provided to NSDL projects courtesy of JES & Co.
We assign reference terms to each statement within a standards document and to each media resource, and correlations are based upon matches of these terms for a given grade band. If a particular standards document of interest to you is not displayed yet, it most likely has not yet been processed by ASN or by Teachers' Domain. We will be adding social studies and arts correlations over the coming year, and also will be increasing the specificity of alignment.
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