Ratios and Proportions Content Module
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| − | + | {{BACK TO|[[Content Modules]]}} | |
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| + | =Ratios and Proportions: Skills covered in the module= | ||
* 6.ME.2a2 Solve one step real world measurement problems involving unit rates with ratios of whole numbers when given the unit rate | * 6.ME.2a2 Solve one step real world measurement problems involving unit rates with ratios of whole numbers when given the unit rate | ||
* 6.ME.1b4 Complete a conversion table for length, mass, time, volume | * 6.ME.1b4 Complete a conversion table for length, mass, time, volume | ||
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| + | =Plot the Course= | ||
[[File:Plotthecourse.jpg|thumb|x200px|link=http://www.worthwhilesmile.com/air-balloons-kaleidoscope/|http://www.worthwhilesmile.com/air-balloons-kaleidoscope/|alt=three hot air balloons floating in the sky]] | [[File:Plotthecourse.jpg|thumb|x200px|link=http://www.worthwhilesmile.com/air-balloons-kaleidoscope/|http://www.worthwhilesmile.com/air-balloons-kaleidoscope/|alt=three hot air balloons floating in the sky]] | ||
| − | + | ==The rationale== | |
Everyday people use ratios and proportions to problem solve in their life. Whether you are trying to determine how many gallons of paint to buy to cover a large space or estimate how many tanks of gas you might need for a long journey, the process for determining these variables uses the principles of ratio and proportion. In addition to everyday activities, there are many jobs that require a firm understanding of ratios and proportions such as construction, landscaping, and culinary skills. | Everyday people use ratios and proportions to problem solve in their life. Whether you are trying to determine how many gallons of paint to buy to cover a large space or estimate how many tanks of gas you might need for a long journey, the process for determining these variables uses the principles of ratio and proportion. In addition to everyday activities, there are many jobs that require a firm understanding of ratios and proportions such as construction, landscaping, and culinary skills. | ||
| − | + | ==Module Goal== | |
The goal of this module is to provide detailed instruction on the more difficult concepts using proportions and ratios to teachers of students with disabilities at the middle and high school level. This module promotes a mathematical understanding of these concepts so that a teacher can begin to plan how to teach the concepts to students. Additionally, this module will provide instructors with potential adaptations and modifications to consider when designing materials and instruction for students with severe disabilities. | The goal of this module is to provide detailed instruction on the more difficult concepts using proportions and ratios to teachers of students with disabilities at the middle and high school level. This module promotes a mathematical understanding of these concepts so that a teacher can begin to plan how to teach the concepts to students. Additionally, this module will provide instructors with potential adaptations and modifications to consider when designing materials and instruction for students with severe disabilities. | ||
| − | + | ==Module Objectives== | |
After viewing the content module, teachers will: | After viewing the content module, teachers will: | ||
# Set up ratios and proportions within real-life contexts | # Set up ratios and proportions within real-life contexts | ||
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# Identify and create dilations of figures | # Identify and create dilations of figures | ||
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| − | [[File:Timefortakeoff.jpg|thumb| | + | |
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| + | =Time for Take Off= | ||
| + | [[File:Timefortakeoff.jpg|thumb|A flame used to lift hot air balloon]] | ||
Understanding the vocabulary used within ratios and proportions is important for both teachers and students in planning and implementing math lessons. As a teacher, knowing and using the mathematical terms not only ensures your instruction stays true to the math content, but also will help with collaborating with other math teachers or content experts. When choosing which vocabulary to teach, it is most important that the teacher selects the most salient, important, or most frequently used vocabulary for each lesson. | Understanding the vocabulary used within ratios and proportions is important for both teachers and students in planning and implementing math lessons. As a teacher, knowing and using the mathematical terms not only ensures your instruction stays true to the math content, but also will help with collaborating with other math teachers or content experts. When choosing which vocabulary to teach, it is most important that the teacher selects the most salient, important, or most frequently used vocabulary for each lesson. | ||
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While providing vocabulary instruction, you may consider including pictures or objects to make the instruction more concrete for students with disabilities (See Ideas to support vocabulary learning below). | While providing vocabulary instruction, you may consider including pictures or objects to make the instruction more concrete for students with disabilities (See Ideas to support vocabulary learning below). | ||
| − | + | ==Vocabulary== | |
| − | + | {{Def|Proportion |an equation stating that two ratios are equal <math alt="two-thirds equals four-sixths">(\frac{2}{3}=\frac{4}{6})</math>}} | |
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| − | + | {{Def|Ratio |a comparison of two quantities, can be written in a variety of forms <math>(11\ to\ 20,\ 11:20\ or\ \frac{11}{20})</math>}} | |
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| + | {{Def|Equivalent ratios |two ratios that are the same in their simplest form e.g. <math>\frac{1}{3}=\frac{2}{6}</math>}} | ||
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| + | {{Def|Common denominator |a common multiple of the denominators}} | ||
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| + | {{Def|Least common denominator |the smallest common multiple of denominators}} | ||
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| + | {{Def|Cross products |product of numbers multiplied diagonally when comparing ratios}} [[File:Cross_Product.JPG]] | ||
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| + | {{Def|Similar figures |figures with the same shape but are not the same size}} | ||
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| + | {{Def|Corresponding sides |matching sides of polygons}} | ||
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| + | {{Def|Corresponding angles |angles in the same position in polygons}} | ||
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| + | {{Def|Dilation |enlargement or reduction of a figure}} | ||
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| + | {{Def|Scale factor |a ratio used to reduce or enlarge a figure}} | ||
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| + | ==Ideas to support vocabulary learning== | ||
* Use visual representations | * Use visual representations | ||
[[File:DilationExample.jpg]] | [[File:DilationExample.jpg]] | ||
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| − | [[File:Floatingonair.jpg|thumb| | + | |
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| + | =Floating on Air= | ||
| + | [[File:Floatingonair.jpg|thumb|A close up of one hot air balloon with a second floating in the distance]] | ||
Before you can begin teaching solving problems using ratios and proportions, you need a deep understanding of these mathematical concepts. Some of these concepts may be familiar to you. Below is a list of skills that should be covered at each grade level. For concepts that you need more information about, please view the accompanying PowerPoint presentations that will walk you through an example as well as make some suggestions for instruction. | Before you can begin teaching solving problems using ratios and proportions, you need a deep understanding of these mathematical concepts. Some of these concepts may be familiar to you. Below is a list of skills that should be covered at each grade level. For concepts that you need more information about, please view the accompanying PowerPoint presentations that will walk you through an example as well as make some suggestions for instruction. | ||
| − | + | ==Middle and High School== | |
In middle school skills include: | In middle school skills include: | ||
* 6.ME.2a2 Solve one step real world measurement problems involving unit rates with ratios of whole numbers when given the unit rate | * 6.ME.2a2 Solve one step real world measurement problems involving unit rates with ratios of whole numbers when given the unit rate | ||
| − | [[Media:Finding Unit Rates. | + | [[Media:Finding Unit Rates.pptx| Finding Unit Rates powerpoint]] |
* 6.ME.1b4 Complete a conversion table for length, mass, time, volume | * 6.ME.1b4 Complete a conversion table for length, mass, time, volume | ||
* 6.PRF.1c1 Describe the ratio relationship between two quantities for a given situation | * 6.PRF.1c1 Describe the ratio relationship between two quantities for a given situation | ||
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* 6.PRF.2b5 Use ratios and reasoning to solve real-world mathematical problems | * 6.PRF.2b5 Use ratios and reasoning to solve real-world mathematical problems | ||
* 7.ME.1d1 Solve problems that use proportional reasoning with ratios of length and area | * 7.ME.1d1 Solve problems that use proportional reasoning with ratios of length and area | ||
| − | [[Media:Solving Ratios Without Algorithms | + | [[Media:Solving ratios without algorithms.pptx| Solving Ratios Without Algorithms powerpoint]] |
* 7.PRF.1e1 Determine unit rates associated with ratios of lengths, areas, and other quantities measured in like units | * 7.PRF.1e1 Determine unit rates associated with ratios of lengths, areas, and other quantities measured in like units | ||
* 7.PRF.2a5 Use variables to represent two quantities in a real-world problem that change in relationship to one another | * 7.PRF.2a5 Use variables to represent two quantities in a real-world problem that change in relationship to one another | ||
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* 7.ME.2e2 Solve one step problems involving unit rates associated with ratios of fractions | * 7.ME.2e2 Solve one step problems involving unit rates associated with ratios of fractions | ||
* 7.PRF.1g1 Solve real-world multistep problems using whole numbers | * 7.PRF.1g1 Solve real-world multistep problems using whole numbers | ||
| − | [[Media:Solving | + | [[Media:Solving proportions.pptx| Solving Proportions powerpoint]] |
* 7.PRF.1g2 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities | * 7.PRF.1g2 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities | ||
* 8.PRF.1e2 Represent proportional relationships on a line graph | * 8.PRF.1e2 Represent proportional relationships on a line graph | ||
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In high school skills include: | In high school skills include: | ||
* H.ME.2b1 Determine the dimensions of a figure after dilation | * H.ME.2b1 Determine the dimensions of a figure after dilation | ||
| − | [[Media:Dilations. | + | [[Media:Dilations.pptx| Dilations powerpoint]] |
Great! Now that you have viewed the PowerPoint presentations most useful to you, the next section will provide some ideas to consider when planning for Universal Design for Learning. | Great! Now that you have viewed the PowerPoint presentations most useful to you, the next section will provide some ideas to consider when planning for Universal Design for Learning. | ||
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| − | [[File:Sharethesky.jpg|thumb|bottom]] | + | |
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| + | =Sharing the Sky= | ||
| + | [[File:Sharethesky.jpg|thumb|bottom|Many hot air balloons floating together]] | ||
| + | <font size=3>UNIVERSAL DESIGN FOR LEARNING</font> | ||
'''Some examples of options for teaching ratios and proportions to students who may present instructional challenges due to:''' | '''Some examples of options for teaching ratios and proportions to students who may present instructional challenges due to:''' | ||
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{| border=1 | {| border=1 | ||
| − | |width = "500 | + | |width = "500"| |
| − | |width = "500 | + | |width = "500"|'''Sensory Differences such as''' '''Blindness, Visual Impairment, Deafness, or Deaf/Blindness''' |
| − | |width = "500 | + | |width = "500"|'''Physical Disability or Motor Differences (such as weakness or motor planning difficulty)''' |
| − | |width = "500 | + | |width = "500"|'''Extremely limited evidence of experience/ skill or motivation/ attention''' |
| − | |width = "500 | + | |width = "500"|'''Lack of or extremely limited use of speech''' |
|- | |- | ||
| − | |width = "500 | + | |width = "500"|'''Options for Representation''' |
| − | |width = "500 | + | |width = "500"|'''Provide auditory options:''' |
* Talking calculator | * Talking calculator | ||
* Text-to-speech software or voice recordings to read aloud story problems | * Text-to-speech software or voice recordings to read aloud story problems | ||
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* Have students physically demonstrate ratios using manipulatives or concrete objects (e.g., pencils to students). | * Have students physically demonstrate ratios using manipulatives or concrete objects (e.g., pencils to students). | ||
| − | |width = "500 | + | |width = "500"|'''Reduce Physical Effort: ''' |
* Place materials on slant board or eye gaze array | * Place materials on slant board or eye gaze array | ||
* Display flip chart, interactive white board or other teaching materials at student eye level | * Display flip chart, interactive white board or other teaching materials at student eye level | ||
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* Demonstrating ratios verbally (e.g., 4: 3 can be demonstrated by saying beep beep beep beep: bam bam bam) | * Demonstrating ratios verbally (e.g., 4: 3 can be demonstrated by saying beep beep beep beep: bam bam bam) | ||
| − | |width = "500 | + | |width = "500"|'''Illustrate through multiple media:''' |
* Utilize interactive whiteboard | * Utilize interactive whiteboard | ||
* Incorporate interactive websites that provide nonlinguistic tools for exploring math concepts: | * Incorporate interactive websites that provide nonlinguistic tools for exploring math concepts: | ||
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* Use a talking calculator | * Use a talking calculator | ||
| − | |width = "500 | + | |width = "500"|'''Provide customized display of information:''' |
* Consistent model by utilizing modes of communication used by students (point to symbols representing concepts, operations) | * Consistent model by utilizing modes of communication used by students (point to symbols representing concepts, operations) | ||
* Teacher model competent use of AAC during instruction | * Teacher model competent use of AAC during instruction | ||
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|} | |} | ||
{| border=1 | {| border=1 | ||
| − | |width = "500 | + | |width = "500"| |
| − | |width = "500 | + | |width = "500"|'''Sensory Differences such as''' '''Blindness, Visual Impairment, Deafness, or''' |
'''Deaf/Blindness''' | '''Deaf/Blindness''' | ||
| − | |width = "500 | + | |width = "500"|'''Physical Disability or Motor Differences (such as weakness or motor planning difficulty)''' |
| − | |width = "500 | + | |width = "500"|'''Extremely limited evidence of experience/ skill or motivation/ attention. ''' |
| − | |width = "500 | + | |width = "500"|'''Lack of or extremely limited use of speech.''' |
|- | |- | ||
| − | |width = "500 | + | |width = "500"|'''Options for Expression''' |
| − | |width = "500 | + | |width = "500"|'''Vary the methods for response by:''' |
* Student states answer or scans raised numbers to select correct answer; use voice output devices for student to select the correct answer | * Student states answer or scans raised numbers to select correct answer; use voice output devices for student to select the correct answer | ||
* Provide manipulatives for student to respond or contribute to interaction | * Provide manipulatives for student to respond or contribute to interaction | ||
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* Give students a ratio with two different types of counters/ manipulatives for each ratio unit (i.e., 2:4 would be represented with 2 pennies and 4 bear counters). Have students create the same ratio with two different types of counters (i.e., 2 paper clips and 4 pencils). Provide student with response options or AAC device. | * Give students a ratio with two different types of counters/ manipulatives for each ratio unit (i.e., 2:4 would be represented with 2 pennies and 4 bear counters). Have students create the same ratio with two different types of counters (i.e., 2 paper clips and 4 pencils). Provide student with response options or AAC device. | ||
| − | |width = "500 | + | |width = "500"|'''Provide options for responses/expression:''' |
* Student selects numbers versus writing them | * Student selects numbers versus writing them | ||
* Selection of correct answer is done after a model | * Selection of correct answer is done after a model | ||
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* Use computer/interactive whiteboard to show ratios so student can interact using a switch or eye gaze. | * Use computer/interactive whiteboard to show ratios so student can interact using a switch or eye gaze. | ||
| − | |width = "500 | + | |width = "500"|'''Provide multimedia options for responses/expression:''' |
* Allow the student to make selections by pointing to, gazing at, or selecting answers on the interactive white board | * Allow the student to make selections by pointing to, gazing at, or selecting answers on the interactive white board | ||
* Utilize a switch or adapted computer mouse | * Utilize a switch or adapted computer mouse | ||
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* Students can demonstrate understanding of ratios by eye gazing to proportional ratios, using technology to create ratios (computer games or lessons that can be completed using a switch) | * Students can demonstrate understanding of ratios by eye gazing to proportional ratios, using technology to create ratios (computer games or lessons that can be completed using a switch) | ||
| − | |width = "500 | + | |width = "500"|'''Provide options for modes of communication:''' |
* Incorporate responses into student's AAC device or eye gaze array | * Incorporate responses into student's AAC device or eye gaze array | ||
* Phrase questions so that they require a "yes/no" response, these can easily be answered using an eye gaze, head turn, two switches, etc. | * Phrase questions so that they require a "yes/no" response, these can easily be answered using an eye gaze, head turn, two switches, etc. | ||
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* Use a blink response select answer | * Use a blink response select answer | ||
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{| border=1 | {| border=1 | ||
| − | |width = "500 | + | |width = "500"| |
| − | |width = "500 | + | |width = "500"|'''Sensory Differences such as Blindness, Visual Impairment, Deafness, or Deaf/Blindness''' |
| − | |width = "500 | + | |width = "500"|'''Physical Disability or Motor Differences (such as weakness or motor planning difficulty)''' |
| − | |width = "500 | + | |width = "500"|'''Extremely limited evidence of experience/ skill or motivation/ attention. ''' |
| − | |width = "500 | + | |width = "500"|'''Lack of or extremely limited use of speech.''' |
|- | |- | ||
| − | |width = "500 | + | |width = "500"|'''Options for Engagement''' |
| − | |width = "500 | + | |width = "500"|'''Recruit interest by providing choices:''' |
* Digital/talking representations, videos, talking calculators | * Digital/talking representations, videos, talking calculators | ||
* Interactive websites | * Interactive websites | ||
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* Use concrete items of interest to demonstrate ratios and try to incorporate tactile surfaces for students with visual impairment | * Use concrete items of interest to demonstrate ratios and try to incorporate tactile surfaces for students with visual impairment | ||
| − | |width = "500 | + | |width = "500"|'''Recruit interest by increasing personal relevance:''' |
* Ensure that engaging and high preference content is visible and accessible to student | * Ensure that engaging and high preference content is visible and accessible to student | ||
* Use figures that are large enough to accommodate the movements that the student is able to make | * Use figures that are large enough to accommodate the movements that the student is able to make | ||
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1 brick : 8 studs | 1 brick : 8 studs | ||
| − | |width = "500 | + | |width = "500"|'''Recruit interest by providing choices:''' |
* Digital/talking representations, videos, talking calculators | * Digital/talking representations, videos, talking calculators | ||
'''Provide options for sustaining effort and persistence:''' | '''Provide options for sustaining effort and persistence:''' | ||
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* Use personally relevant items. Consider a tangible token economy system which follows a ratio (e.g., for every 3 tokens, Johnny earns five minutes of listening to Bob Dylan; use CD's to represent minutes). | * Use personally relevant items. Consider a tangible token economy system which follows a ratio (e.g., for every 3 tokens, Johnny earns five minutes of listening to Bob Dylan; use CD's to represent minutes). | ||
| − | |width = "500 | + | |width = "500"|'''Recruit interest with modes of communication: ''' |
* Allow students to choose items or subjects that are relevant to them via AAC devices, symbols, or eye gaze array | * Allow students to choose items or subjects that are relevant to them via AAC devices, symbols, or eye gaze array | ||
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| + | =Prepare for Landing= | ||
[[File:Prepareforlanding.jpg|thumb|]] | [[File:Prepareforlanding.jpg|thumb|]] | ||
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In addition to collaborating with other educational professionals in your building, the following list of resources may also help provide special educators with ideas for activities or support a more thorough understanding of the mathematical concepts presented in this content module | In addition to collaborating with other educational professionals in your building, the following list of resources may also help provide special educators with ideas for activities or support a more thorough understanding of the mathematical concepts presented in this content module | ||
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| + | ==Additional Resources== | ||
* [http://www.homeschoolmath.net/teaching/proportions.php http://www.homeschoolmath.net/teaching/proportions.php] - this website provides explanations and real-world examples of how to apply ratios and proportions across different settings and situations | * [http://www.homeschoolmath.net/teaching/proportions.php http://www.homeschoolmath.net/teaching/proportions.php] - this website provides explanations and real-world examples of how to apply ratios and proportions across different settings and situations | ||
* [http://www.ncpublicschools.org/acre/standards/common-core-tools/ http://www.ncpublicschools.org/acre/standards/common-core-tools/] - this website provides an "unpacking document" for the Mathematics Common Core State Standards that helps teachers identify what is most important and the essential skills for each standard | * [http://www.ncpublicschools.org/acre/standards/common-core-tools/ http://www.ncpublicschools.org/acre/standards/common-core-tools/] - this website provides an "unpacking document" for the Mathematics Common Core State Standards that helps teachers identify what is most important and the essential skills for each standard | ||
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* [http://exchange.smarttech.com/search.html?subject=Mathematics http://exchange.smarttech.com/search.html?subject=Mathematics-] - this SMART board exchange has developed lessons by classroom teachers differentiated by grade level. You can also search by skill and/or state standards. | * [http://exchange.smarttech.com/search.html?subject=Mathematics http://exchange.smarttech.com/search.html?subject=Mathematics-] - this SMART board exchange has developed lessons by classroom teachers differentiated by grade level. You can also search by skill and/or state standards. | ||
* [http://www.google.com/url?q=http://www.ksde.org/LinkClick.aspx%3Ffileticket%3DVq9AjrFFWzE%253D%26tabid%3D3763%26mid%3D11170&sa=U&ei=8lB3Try4CJOltwfmq5DfDA&ved=0CBIQFjAA&usg=AFQjCNE_DzuxI_rhYkU0H1qpjuqmM9sjng Powerpoint Presentation] - this website provides a webinar about how to adapt materials for students who have visual impairments | * [http://www.google.com/url?q=http://www.ksde.org/LinkClick.aspx%3Ffileticket%3DVq9AjrFFWzE%253D%26tabid%3D3763%26mid%3D11170&sa=U&ei=8lB3Try4CJOltwfmq5DfDA&ved=0CBIQFjAA&usg=AFQjCNE_DzuxI_rhYkU0H1qpjuqmM9sjng Powerpoint Presentation] - this website provides a webinar about how to adapt materials for students who have visual impairments | ||
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| − | + | ==Module Assessments== | |
| − | '''Sample General Education lesson plans''' | + | Below are links to assessments for the purpose of self-checking your understanding of the content in this module. After answering the assessment questions, you may want to review the key to check your work. |
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| + | [[Media:Ratios and Proportions Content Module Assessment.pdf|Ratios and Proportions Content Module Assessment]] | ||
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| + | [[Media:Ratios and Proportions Content Module Assessment Key.pdf|Ratios and Proportions Content Module Assessment Key]] | ||
| + | <!-- | ||
| + | =='''Sample General Education lesson plans'''== | ||
Insert developed lesson plans here | Insert developed lesson plans here | ||
| − | '''Have an idea: Upload the lesson plans you've created here''' | + | |
| + | =='''Have an idea: Upload the lesson plans you've created here'''== | ||
Insert link for teachers to upload lesson plans | Insert link for teachers to upload lesson plans | ||
| − | '''Teacher's Corner: Blog with other teachers''' | + | |
| − | Insert forum or blog for teachers to share ideas | + | =='''Teacher's Corner: Blog with other teachers'''== |
| − | Up for a | + | Insert forum or blog for teachers to share ideas |
| + | :[[File:Upforachallenge.JPG|500px|center|Up for a Challenge]] --> | ||
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| + | |||
| − | [[Category: | + | [[Category: Content Modules]] |
| − | [[Category: | + | [[Category: Math]] |
Latest revision as of 16:07, 9 September 2015
BACK TO Content Modules
Contents |
[edit] Ratios and Proportions: Skills covered in the module
- 6.ME.2a2 Solve one step real world measurement problems involving unit rates with ratios of whole numbers when given the unit rate
- 6.ME.1b4 Complete a conversion table for length, mass, time, volume
- 6.PRF.1c1 Describe the ratio relationship between two quantities for a given situation
- 6.PRF.2a3 Use variables to represent two quantities in a real-world problem that change in relationship to one another
- 6.PRF.1c2 Represent proportional relationships on a line graph
- 6.PRF.2b3 Complete a statement that describes the ratio relationship between two quantities
- 6.PRF.2b4 Determine the unit rate in a variety of contextual situations
- 6.PRF.2b5 Use ratios and reasoning to solve real-world mathematical problems
- 7.ME.1d1 Solve problems that use proportional reasoning with ratios of length and area
- 7.PRF.1e1 Determine unit rates associated with ratios of lengths, areas, and other quantities measured in like units
- 7.PRF.2a5 Use variables to represent two quantities in a real-world problem that change in relationship to one another
- 7.PRF.1e2 Represent proportional relationships on a line graph
- 7.ME.2e1 Solve one step real world problems related to scaling
- 7.PRF.1f1 Use proportional relationships to solve multistep percent problems
- 7.ME.2e2 Solve one step problems involving unit rates associated with ratios of fractions
- 7.PRF.1g1 Solve real-world multistep problems using whole numbers
- 7.PRF.1g2 Use variables to represent quantities in a real-world or mathematical problems, and construct simple equations and inequalities to solve problems by reasoning about the quantities
- 8.PRF.1e2 Represent proportional relationships on a line graph
- H.ME.2b1 Determine the dimensions of a figure after dilation
[edit] Plot the Course
[edit] The rationale
Everyday people use ratios and proportions to problem solve in their life. Whether you are trying to determine how many gallons of paint to buy to cover a large space or estimate how many tanks of gas you might need for a long journey, the process for determining these variables uses the principles of ratio and proportion. In addition to everyday activities, there are many jobs that require a firm understanding of ratios and proportions such as construction, landscaping, and culinary skills.
[edit] Module Goal
The goal of this module is to provide detailed instruction on the more difficult concepts using proportions and ratios to teachers of students with disabilities at the middle and high school level. This module promotes a mathematical understanding of these concepts so that a teacher can begin to plan how to teach the concepts to students. Additionally, this module will provide instructors with potential adaptations and modifications to consider when designing materials and instruction for students with severe disabilities.
[edit] Module Objectives
After viewing the content module, teachers will:
- Set up ratios and proportions within real-life contexts
- Solve ratios and proportions with and without algorithms
- Find unit rate
- Identify similar figures and their properties
- Identify and create dilations of figures
[edit] Time for Take Off
Understanding the vocabulary used within ratios and proportions is important for both teachers and students in planning and implementing math lessons. As a teacher, knowing and using the mathematical terms not only ensures your instruction stays true to the math content, but also will help with collaborating with other math teachers or content experts. When choosing which vocabulary to teach, it is most important that the teacher selects the most salient, important, or most frequently used vocabulary for each lesson.
Below you will find a list of vocabulary included within this module. It may or may not be necessary to provide instruction for all terms as students may have learned them previously. Expressions are mostly covered in middle school so vocabulary for this content module has been combined. If you are a high school teacher and are not confident your students know some of these vocabulary terms, you may want to review and teach some unknown terms in the focus and review part of your lesson plan.
While providing vocabulary instruction, you may consider including pictures or objects to make the instruction more concrete for students with disabilities (See Ideas to support vocabulary learning below).
[edit] Vocabulary
- Proportion - an equation stating that two ratios are equal
- Ratio - a comparison of two quantities, can be written in a variety of forms
- Equivalent ratios - two ratios that are the same in their simplest form e.g.
- Common denominator - a common multiple of the denominators
- Least common denominator - the smallest common multiple of denominators
- Cross products - product of numbers multiplied diagonally when comparing ratios
- Similar figures - figures with the same shape but are not the same size
- Corresponding sides - matching sides of polygons
- Corresponding angles - angles in the same position in polygons
- Dilation - enlargement or reduction of a figure
- Scale factor - a ratio used to reduce or enlarge a figure
[edit] Ideas to support vocabulary learning
- Use visual representations
[edit] Floating on Air
Before you can begin teaching solving problems using ratios and proportions, you need a deep understanding of these mathematical concepts. Some of these concepts may be familiar to you. Below is a list of skills that should be covered at each grade level. For concepts that you need more information about, please view the accompanying PowerPoint presentations that will walk you through an example as well as make some suggestions for instruction.
[edit] Middle and High School
In middle school skills include:
- 6.ME.2a2 Solve one step real world measurement problems involving unit rates with ratios of whole numbers when given the unit rate
- 6.ME.1b4 Complete a conversion table for length, mass, time, volume
- 6.PRF.1c1 Describe the ratio relationship between two quantities for a given situation
- 6.PRF.2a3 Use variables to represent two quantities in a real-world problem that change in relationship to one another
- 6.PRF.1c2 Represent proportional relationships on a line graph
- 6.PRF.2b3 Complete a statement that describes the ratio relationship between two quantities
- 6.PRF.2b4 Determine the unit rate in a variety of contextual situations
- 6.PRF.2b5 Use ratios and reasoning to solve real-world mathematical problems
- 7.ME.1d1 Solve problems that use proportional reasoning with ratios of length and area
Solving Ratios Without Algorithms powerpoint
- 7.PRF.1e1 Determine unit rates associated with ratios of lengths, areas, and other quantities measured in like units
- 7.PRF.2a5 Use variables to represent two quantities in a real-world problem that change in relationship to one another
- 7.PRF.1e2 Represent proportional relationships on a line graph
- 7.ME.2e1 Solve one step real world problems related to scaling
- 7.PRF.1f1 Use proportional relationships to solve multistep percent problems
- 7.ME.2e2 Solve one step problems involving unit rates associated with ratios of fractions
- 7.PRF.1g1 Solve real-world multistep problems using whole numbers
Solving Proportions powerpoint
- 7.PRF.1g2 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities
- 8.PRF.1e2 Represent proportional relationships on a line graph
(See slide 4 in solving ratios without algorithms PowerPoint presentation)
In high school skills include:
- H.ME.2b1 Determine the dimensions of a figure after dilation
Great! Now that you have viewed the PowerPoint presentations most useful to you, the next section will provide some ideas to consider when planning for Universal Design for Learning.
[edit] Sharing the Sky
UNIVERSAL DESIGN FOR LEARNING
Some examples of options for teaching ratios and proportions to students who may present instructional challenges due to:
| Sensory Differences such as Blindness, Visual Impairment, Deafness, or Deaf/Blindness | Physical Disability or Motor Differences (such as weakness or motor planning difficulty) | Extremely limited evidence of experience/ skill or motivation/ attention | Lack of or extremely limited use of speech | |
| Options for Representation | Provide auditory options:
Provide tactile options:
Provide visual and manipulative options to scaffold representation of concepts:
|
Reduce Physical Effort:
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Illustrate through multiple media:
Illuminations http://illuminations.nctm.org/ActivitySearch.aspx Math Open Reference There are many resources listed here: http://www.udlcenter.org/implementation/examples
|
Provide customized display of information:
|
| Sensory Differences such as Blindness, Visual Impairment, Deafness, or
Deaf/Blindness |
Physical Disability or Motor Differences (such as weakness or motor planning difficulty) | Extremely limited evidence of experience/ skill or motivation/ attention. | Lack of or extremely limited use of speech. | |
| Options for Expression | Vary the methods for response by:
|
Provide options for responses/expression:
Optimize access to tools/ alternatives for responding:
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Provide multimedia options for responses/expression:
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Provide options for modes of communication:
|
| Sensory Differences such as Blindness, Visual Impairment, Deafness, or Deaf/Blindness | Physical Disability or Motor Differences (such as weakness or motor planning difficulty) | Extremely limited evidence of experience/ skill or motivation/ attention. | Lack of or extremely limited use of speech. | |
| Options for Engagement | Recruit interest by providing choices:
Increase personal relevance:
Provide tactile options for engagement:
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Recruit interest by increasing personal relevance:
|
Recruit interest by providing choices:
Provide options for sustaining effort and persistence:
Increase personal relevance:
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Recruit interest with modes of communication:
|
[edit] Prepare for Landing
Below you will find ideas for linking ratios and proportions to real-world applications, the college and career readiness skills addressed by teaching these concepts, module assessments for teachers, sample general education lesson plans incorporating the Universal Design for Learning framework, blog for teachers to share their ideas, and a place to upload and share lesson plans from teachers who completed this module.
One way to help assist in a special educator's development within this curricular area is through collaboration with other teachers in your building. Often these skills are practiced outside of a math classroom in other curricular areas as well as during everyday tasks like grocery shopping. Some activities with real world connection include:
- Make a scale model
- Find the best price during shopping (unit rate)
- Painting a house
- Cooking
In addition to the real-world applications of these measurement concepts, skills taught within this content module also promote the following college and career readiness skills.
Communicative competence: Students will increase their vocabulary to include concepts related to "ratios and proportions" In addition, they will be learning concepts such as: "enlarge", "reduce", and "scale".
Fluency in reading, writing, and math: Students will have an opportunity to increase their numeracy and sight word fluency while participating in problem solving related to "ratios and proportions" such as number recognition, counting, one-to-one correspondence, and reading concepts that include the use and understanding of descriptors related to size.
Age appropriate social skills: Students will engage in peer groups to solve problems related to "ratio and proportions" that will provide practice on increasing reciprocal communication and age appropriate social interactions. For example, students might work together with their peers to create a scale model of a building.
Independent work behaviors:
By solving real life problems related to "ratio and proportions" students will improve work behaviors that could lead to employment such as landscaping, culinary skills, construction, and other agricultural professions. When providing opportunities for real life problems leave some materials out and prompt/teach the students to determine who they should ask and what they should ask for to be able to solve the problem.
Skills in accessing support systems: At times, students will need to ask for assistance to complete activities related to "ratios and proportions" which will give them practice in accessing supports. Students will gain practice asking for tools such as talking calculators, a digital tape measure, or other manipulatives. They can ask a peer to complete the physical movements of the tasks they are not about to do themselves. Be sure to teach students to ask versus having items or supports automatically given to them.
In addition to collaborating with other educational professionals in your building, the following list of resources may also help provide special educators with ideas for activities or support a more thorough understanding of the mathematical concepts presented in this content module
[edit] Additional Resources
- http://www.homeschoolmath.net/teaching/proportions.php - this website provides explanations and real-world examples of how to apply ratios and proportions across different settings and situations
- http://www.ncpublicschools.org/acre/standards/common-core-tools/ - this website provides an "unpacking document" for the Mathematics Common Core State Standards that helps teachers identify what is most important and the essential skills for each standard
- http://www.xpmath.com/careers/topicsresult.php?subjectID=1&topicID=3- - not only does this website provide additional teacher support for teaching ratio and proportions, but this website also provides a link to why these skills are important to different professions and jobs
- http://www.cumbavac.org/Math.htm- - provides a variety of activities, work sheets, and web quests to use in your classroom
- http://www.mathforum.com/- - website specifically for teachers that provides a variety of ideas and activities to use in your classroom
- http://www.teachertube.com/ - Youtube for teachers! Simply search for your content area and this websites provides a variety of videos including videos of math experts working through math problems step by step (free registration required)
- http://www.k8accesscenter.org/training_resources/math.asp - this website not only provides some ideas and activities to use in your classroom, but also includes presentations and webinars from the North Carolina Department of Public Instruction about research-based strategies that have proven effective in teaching math for students with varying level of disability
- http://exchange.smarttech.com/search.html?subject=Mathematics- - this SMART board exchange has developed lessons by classroom teachers differentiated by grade level. You can also search by skill and/or state standards.
- Powerpoint Presentation - this website provides a webinar about how to adapt materials for students who have visual impairments
[edit] Module Assessments
Below are links to assessments for the purpose of self-checking your understanding of the content in this module. After answering the assessment questions, you may want to review the key to check your work.
