Ratios and Proportions Content Module

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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.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

Plot the Course

http://www.worthwhilesmile.com/air-balloons-kaleidoscope/

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.

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.

Module Objectives

After viewing the content module, teachers will:

1. Set up ratios and proportions within real-life contexts
2. Solve ratios and proportions with and without algorithms
3. Find unit rate
4. Identify similar figures and their properties
5. Identify and create dilations of figures

Time for Take Off

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.

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).

Vocabulary

Proportion - an equation stating that two ratios are equal $two-thirds equals four-sixths$

Ratio - a comparison of two quantities, can be written in a variety of forms $(11\ to\ 20,\ 11:20\ or\ \frac{11}{20})$

Equivalent ratios - two ratios that are the same in their simplest form e.g. $\frac{1}{3}=\frac{2}{6}$

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

Ideas to support vocabulary learning

• Use visual representations

Floating on Air

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.

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

Finding Unit Rates powerpoint

• 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

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.

Sharing the Sky

Many hot air balloons floating together

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: Talking calculator Text-to-speech software or voice recordings to read aloud story problems Single message sequence voice–output devices to count aloud Captioning software that presents auditory information visually Provide tactile options: Object cues, using miniature objects or other tangible symbols to assist with problem comprehension and operations Create numbers and symbols out of tactile materials When demonstrating graphical linear representations of ratios, raise the grid by using glue over grid, puffy paint, or wiki sticks to represent proportional relationship (Students will need the grid raised to count x- and y-axis and another texture or height for the line representing the proportional relationship). Provide visual and manipulative options to scaffold representation of concepts: Color code problems and corresponding parts of calculator to support students correctly entering information Have students physically demonstrate ratios using manipulatives or concrete objects (e.g., pencils to students).	Reduce Physical Effort: Place materials on slant board or eye gaze array Display flip chart, interactive white board or other teaching materials at student eye level Student can scan an array of possible options and use a switch to select the answer Use computer representation of figures that can be manipulated with switch Demonstrating ratios verbally (e.g., 4: 3 can be demonstrated by saying beep beep beep beep: bam bam bam)	Illustrate through multiple media: Utilize interactive whiteboard Incorporate interactive websites that provide nonlinguistic tools for exploring math concepts: Illuminations http://illuminations.nctm.org/ActivitySearch.aspx Math Open Reference http://www.mathopenref.com/ There are many resources listed here: http://www.udlcenter.org/implementation/examples Use a talking calculator	Provide customized display of information: Consistent model by utilizing modes of communication used by students (point to symbols representing concepts, operations) Teacher model competent use of AAC during instruction

	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: 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 Student states answer by selecting picture or symbol Students can use talking calculator to assist with counting number of items in each ratio unit 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.	Provide options for responses/expression: Student selects numbers versus writing them Selection of correct answer is done after a model Ratios can be recognized without counting. Display ratios that are the same and one not the same and use the example/non example script to help students identify similar ratios. Rather than indicating ratios by number, with simple ratios, have students indicate proportional relations by stating same or not same. Optimize access to tools/ alternatives for responding: Provide symbols, objects, manipulatives, and pictures for matching/ student responses Use computer/interactive whiteboard to show ratios so student can interact using a switch or eye gaze.	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 Utilize a switch or adapted computer mouse Have student write answers with novel pencil or use a tablet computer 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)	Provide options for modes of communication: 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. Choose response by pointing to or selecting object or item Use a blink response select answer

	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: Digital/talking representations, videos, talking calculators Interactive websites Increase personal relevance: Use items that are familiar and reinforcing to students Incorporate high preference items into story problems, as well as student names Provide tactile options for engagement: Use concrete items of interest to demonstrate ratios and try to incorporate tactile surfaces for students with visual impairment	Recruit interest by increasing personal relevance: 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 Pair student with another student without a physical impairment and have them complete hands on activities together Use items of high interest when demonstrating ratios Have student do a scavenger hunt with favorite items to locate constant ratios (e.g., if student enjoys Legos, have her collect several Lego bricks that have the same number of studs). 1 brick : 8 studs	Recruit interest by providing choices: Digital/talking representations, videos, talking calculators Provide options for sustaining effort and persistence: Break tasks down to maximize student attention Use high interest items that demonstrate a constant ratio such as a favorite car (1 car 4 wheels), robot (1 robot two extending arms), or DVD (1 DVD to one hole in the center of the disk) Vary demands and materials to maintain interest Increase personal relevance: Use items that are familiar and reinforcing to students Incorporate high preference items into story problems, as well as student names 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).	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

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

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

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.

Ratios and Proportions Content Module Assessment

Ratios and Proportions Content Module Assessment Key