Ratios and Proportions Content Module

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'''Ratios and Proportions: Skills covered in the module'''
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* 6.ME.2a2 Solve one step real world measurement problems involving unit rates with ratios of whole numbers when given the unit rate
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* 6.ME.1b4 Complete a conversion table for length, mass, time, volume
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* 6.PRF.1c1 Describe the ratio relationship between two quantities for a given situation
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* 6.PRF.2a3 Use variables to represent two quantities in a real-world problem that change in relationship to one another
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* 6.PRF.1c2 Represent proportional relationships on a line graph
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* 6.PRF.2b3 Complete a statement that describes the ratio relationship between two quantities
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* 6.PRF.2b4 Determine the unit rate in a variety of contextual situations
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* 6.PRF.2b5 Use ratios and reasoning to solve real-world mathematical problems
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* 7.ME.1d1 Solve problems that use proportional reasoning with ratios of length and area
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* 7.PRF.1e1 Determine unit rates associated with ratios of lengths, areas, and other quantities measured in like units
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* 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.PRF.1e2 Represent proportional relationships on a line graph
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* 7.ME.2e1 Solve one step real world problems related to scaling
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* 7.PRF.1f1 Use proportional relationships to solve multistep percent problems
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* 7.ME.2e2 Solve one step problems involving unit rates associated with ratios of fractions
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* 7.PRF.1g1 Solve real-world multistep problems using whole numbers
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* 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
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* 8.PRF.1e2 Represent proportional relationships on a line graph
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* H.ME.2b1 Determine the dimensions of a figure after dilation
  
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'''Plot the Course '''
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[http://www.worthwhilesmile.com/air-balloons-kaleidoscope/-http://www.worthwhilesmile.com/air-balloons-kaleidoscope/]
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'''The rationale'''
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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.
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'''Module Goal'''
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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.
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'''Module Objectives'''
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After viewing the content module, teachers will:
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# Set up ratios and proportions within real-life contexts
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# Solve ratios and proportions with and without algorithms
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# Find unit rate
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# Identify similar figures and their properties
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# Identify and create dilations of figures
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'''Time for Take Off'''
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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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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.
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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).
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'''Vocabulary'''
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* Proportion- an equation stating that two ratios are equal (''2/3=4/6)''
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* Ratio- a comparison of two quantities, can be written in a variety of forms (11 to 20, 11:20 or ''11/20)''
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* Equivalent ratios- two ratios that are the same in their simplest form (e.g., ''1/3=2/6)''
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* Common denominator- a common multiple of the denominators
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* Least common denominator- the smallest common multiple of denominators
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* Cross products- product of numbers multiplied diagonally when comparing ratios
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(''2/3=4/6)''
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* Similar figures- figures with the same shape but are not the same size
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* Corresponding sides- matching sides of polygons
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* Corresponding angles- angles in the same position in polygons
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* Dilation- enlargement or reduction of a figure
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* Scale factor- a ratio used to reduce or enlarge a figure
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'''Ideas to support vocabulary learning'''
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* Use visual representations
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Dilation example:
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* Have students distinguish between a ratio and not a ratio
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11 to 20 11:20 ''11/20 7   8.5''
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* Have students identify corresponding angles and sides when comparing two polygons
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* Review similar and not similar figures
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'''Floating on Air'''
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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.
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'''Middle and High School'''
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In middle school skills include:
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* 6.ME.2a2 Solve one step real world measurement problems involving unit rates with ratios of whole numbers when given the unit rate
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(Insert finding unit rates PowerPoint presentation here)
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* 6.ME.1b4 Complete a conversion table for length, mass, time, volume
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* 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
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* 6.PRF.2b4 Determine the unit rate in a variety of contextual situations
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* 6.PRF.2b5 Use ratios and reasoning to solve real-world mathematical problems
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* 7.ME.1d1 Solve problems that use proportional reasoning with ratios of length and area
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(Insert solving proportions PowerPoint presentation here)
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* 7.PRF.1e1 Determine unit rates associated with ratios of lengths, areas, and other quantities measured in like units
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* 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.PRF.1e2 Represent proportional relationships on a line graph
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* 7.ME.2e1 Solve one step real world problems related to scaling
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* 7.PRF.1f1 Use proportional relationships to solve multistep percent problems
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* 7.ME.2e2 Solve one step problems involving unit rates associated with ratios of fractions
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* 7.PRF.1g1 Solve real-world multistep problems using whole numbers
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(Insert solving proportions PowerPoint presentation)
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* 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
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* 8.PRF.1e2 Represent proportional relationships on a line graph
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(See slide 4 in solving ratios without algorithms PowerPoint presentation)
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In high school skills include:
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* H.ME.2b1 Determine the dimensions of a figure after dilation
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(Insert dilations PowerPoint presentation here)
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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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'''Finding Unit Rates'''
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'''Solving Ratios without Algorithms'''
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'''Solving Proportions'''
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'''Dilations'''
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'''Sharing the Sky '''
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'''UNIVERSAL DESIGN FOR LEARNING'''
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'''Some examples of options for teaching ratios and proportions to students who may present instructional challenges due to:'''
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{| border=1
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|width = "500" style="background-color:#FFFFFF;"|
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|width = "500" style="background-color:#FFFFFF;"|'''Sensory Differences such as''' '''Blindness, Visual Impairment, Deafness, or Deaf/Blindness'''
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|width = "500" style="background-color:#FFFFFF;"|'''Physical Disability or Motor Differences (such as weakness or motor planning difficulty)'''
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|width = "500" style="background-color:#FFFFFF;"|'''Extremely limited evidence of experience/ skill or motivation/ attention'''
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|width = "500" style="background-color:#FFFFFF;"|'''Lack of or extremely limited use of speech'''
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|-
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|width = "500" style="background-color:#FFFFFF;"|'''Options for Representation'''
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|width = "500" style="background-color:#FFFFFF;"|'''Provide auditory options:'''
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* Talking calculator
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* Text-to-speech software or voice recordings to read aloud story problems
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* Single message sequence voice–output devices to count aloud
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* Captioning software that presents auditory information visually
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'''Provide tactile options:'''
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* Object cues, using miniature objects or other tangible symbols to assist with problem comprehension and operations
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* Create numbers and symbols out of tactile materials
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* 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).
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'''Provide visual and manipulative options to scaffold representation of concepts:'''
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* Color code problems and corresponding parts of calculator to support students correctly entering information
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* Have students physically demonstrate ratios using manipulatives or concrete objects (e.g., pencils to students).
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|width = "500" style="background-color:#FFFFFF;"|'''Reduce Physical Effort: '''
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* Place materials on slant board or eye gaze array
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* Display flip chart, interactive white board or other teaching materials at student eye level
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* Student can scan an array of possible options and use a switch to select the answer
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* Use computer representation of figures that can be manipulated with switch
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* Demonstrating ratios verbally (e.g., 4: 3 can be demonstrated by saying beep beep beep beep: bam bam bam)
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|width = "500" style="background-color:#FFFFFF;"|'''Illustrate through multiple media:'''
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* Utilize interactive whiteboard
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* Incorporate interactive websites that provide nonlinguistic tools for exploring math concepts:
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Illuminations
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[http://illuminations.nctm.org/ActivitySearch.aspx-http://illuminations.nctm.org/ActivitySearch.aspx]
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Math Open Reference
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[http://www.mathopenref.com/-http://www.mathopenref.com/]
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There are many resources listed here:
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[http://www.udlcenter.org/implementation/examples-http://www.udlcenter.org/implementation/examples]
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* Use a talking calculator
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|width = "500" style="background-color:#FFFFFF;"|'''Provide customized display of information:'''
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* Consistent model by utilizing modes of communication used by students (point to symbols representing concepts, operations)
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* Teacher model competent use of AAC during instruction
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|-
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|}
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{| border=1
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|width = "500" style="background-color:#FFFFFF;"|
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|width = "500" style="background-color:#FFFFFF;"|'''Sensory Differences such as''' '''Blindness, Visual Impairment, Deafness, or'''
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'''Deaf/Blindness'''
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|width = "500" style="background-color:#FFFFFF;"|'''Physical Disability or Motor Differences (such as weakness or motor planning difficulty)'''
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|width = "500" style="background-color:#FFFFFF;"|'''Extremely limited evidence of experience/ skill or motivation/ attention. '''
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|width = "500" style="background-color:#FFFFFF;"|'''Lack of or extremely limited use of speech.'''
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|-
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|width = "500" style="background-color:#FFFFFF;"|'''Options for Expression'''
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|width = "500" style="background-color:#FFFFFF;"|'''Vary the methods for response by:'''
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* Student states answer or scans raised numbers to select correct answer; use voice output devices for student to select the correct answer
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* Provide manipulatives for student to respond or contribute to interaction
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* Student states answer by selecting picture or symbol
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* Students can use talking calculator to assist with counting number of items in each ratio unit
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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. 
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|width = "500" style="background-color:#FFFFFF;"|'''Provide options for responses/expression:'''
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* Student selects numbers versus writing them
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* Selection of correct answer is done after a model
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* 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.
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* Rather than indicating ratios by number, with simple ratios, have students indicate proportional relations by stating same or not same.
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'''Optimize access to tools/ alternatives for responding:'''
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* Provide symbols, objects, manipulatives, and pictures for matching/ student responses
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* Use computer/interactive whiteboard to show ratios so student can interact using a switch or eye gaze.
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|width = "500" style="background-color:#FFFFFF;"|'''Provide multimedia options for responses/expression:'''
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* Allow the student to make selections by pointing to, gazing at, or selecting answers on the interactive white board
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* Utilize a switch or adapted computer mouse
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* Have student write answers with novel pencil or use a tablet computer
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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)
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|width = "500" style="background-color:#FFFFFF;"|'''Provide options for modes of communication:'''
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* Incorporate responses into student's AAC device or eye gaze array
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* 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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* Choose response by pointing to or selecting object or item
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* Use a blink response select answer
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|-
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|}
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{| border=1
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|width = "500" style="background-color:#FFFFFF;"|
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|width = "500" style="background-color:#FFFFFF;"|'''Sensory Differences such as Blindness, Visual Impairment, Deafness, or Deaf/Blindness'''
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|width = "500" style="background-color:#FFFFFF;"|'''Physical Disability or Motor Differences (such as weakness or motor planning difficulty)'''
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|width = "500" style="background-color:#FFFFFF;"|'''Extremely limited evidence of experience/ skill or motivation/ attention. '''
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|width = "500" style="background-color:#FFFFFF;"|'''Lack of or extremely limited use of speech.'''
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|-
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|width = "500" style="background-color:#FFFFFF;"|'''Options for Engagement'''
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|width = "500" style="background-color:#FFFFFF;"|'''Recruit interest by providing choices:'''
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* Digital/talking representations, videos, talking calculators
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* Interactive websites
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'''Increase personal relevance: '''
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* Use items that are familiar and reinforcing to students
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* Incorporate high preference items into story problems, as well as student names
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'''Provide tactile options for engagement:'''
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* Use concrete items of interest to demonstrate ratios and try to incorporate tactile surfaces for students with visual impairment
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|width = "500" style="background-color:#FFFFFF;"|'''Recruit interest by increasing personal relevance:'''
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* Ensure that engaging and high preference content is visible and accessible to student
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* Use figures that are large enough to accommodate the movements that the student is able to make
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* Pair student with another student without a physical impairment and have them complete hands on activities together
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* Use items of high interest when demonstrating ratios
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* 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).
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\[\[File:Insert Picture here.jpg\]\]\[\[File:Insert Picture here.jpg\]\]
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1 brick : 8 studs
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|width = "500" style="background-color:#FFFFFF;"|'''Recruit interest by providing choices:'''
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* Digital/talking representations, videos, talking calculators
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'''Provide options for sustaining effort and persistence:'''
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* Break tasks down to maximize student attention
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* 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)
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* Vary demands and materials to maintain interest
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'''Increase personal relevance: '''
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* Use items that are familiar and reinforcing to students
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* Incorporate high preference items into story problems, as well as student names
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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).
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|width = "500" style="background-color:#FFFFFF;"|'''Recruit interest with modes of communication: '''
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* Allow students to choose items or subjects that are relevant to them via AAC devices, symbols, or eye gaze array
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|-
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|}
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'''Prepare for Landing '''
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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.
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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:
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* Make a scale model
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* Find the best price during shopping (unit rate)
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* Painting a house
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* Cooking
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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.
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Communicative competence:
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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".
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Fluency in reading, writing, and math:
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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.
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Age appropriate social skills:
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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.
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Independent work behaviors:
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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.
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Skills in accessing support systems:
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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.
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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
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'''Additional Resources'''
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* [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
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* [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://www.xpmath.com/careers/topicsresult.php?subjectID=1&topicID=3-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
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* [http://www.cumbavac.org/Math.htm-http://www.cumbavac.org/Math.htm-] - provides a variety of activities, work sheets, and web quests to use in your classroom
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* [http://www.mathforum.com/-http://www.mathforum.com/-] - website specifically for teachers that provides a variety of ideas and activities to use in your classroom
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* [www.teachertube.com-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)'''
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* [http://www.k8accesscenter.org/training_resources/math.asp\#webinar-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
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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.
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* [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-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-] - this website provides a webinar about how to adapt materials for students who have visual impairments
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'''Module Assessments'''
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Insert assessment here
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'''Sample General Education lesson plans'''
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Insert developed lesson plans here
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'''Have an idea: Upload the lesson plans you've created here'''
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Insert link for teachers to upload lesson plans
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'''Teacher's Corner: Blog with other teachers'''
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Insert forum or blog for teachers to share ideas
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Up for a challengejpeg
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Adapt the following general education lesson plan; adapt, and upload. These lesson plans may be shared with higher education professionals developing strategies to provide meaningful academic instruction in mathematics to students with severe disabilities. 
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Insert blank lesson plan form with UDL chart here
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Insert link for teachers to upload lesson plans
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'''Ratios and Proportions Assessment '''
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# How do you solve problems using ratios and proportions?
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## Using an algorithm
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## Using proportional reasoning without an algorithm
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## Both a and b
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## None of the above
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# Which ratio is equivalent to ''1/7?''
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##  3/21
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##  14/24
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##  7/6
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##  22/100
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# A 16 oz box of cereal costs $5.49. How much are you paying per ounce?
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## ≈0.50/oz
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## ≈0.30/oz
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## ≈0.44/oz
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## ≈ 0.34/oz
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# Bethany's heart beats 225 times in 3 minutes. How many times does her heart beat per minute?
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## 70 times
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## 75 times
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## 78 times
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## 80 times
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''$180/(712 h)=$x/(20 h)''
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## $300
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## $325
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## $250
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## $295
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# Which is NOT a ratio
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## 2 to 5
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## 2:5
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## 2.5
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''2/5''
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#  What is the correct measurement for the missing angle?
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## 90°
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## 25°
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## 45°
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## 40°
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# Below is a table showing the price per pound for cashews at the grocery store.  What is the missing value?
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{| border=1
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|width = "1250" style="background-color:#FFFFFF;"|Weight (lb)
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|width = "1250" style="background-color:#FFFFFF;"|Cost
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|-
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|width = "1250" style="background-color:#FFFFFF;"|1
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|width = "1250" style="background-color:#FFFFFF;"|1.25
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|-
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|width = "1250" style="background-color:#FFFFFF;"|2
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|width = "1250" style="background-color:#FFFFFF;"|x
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|-
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|width = "1250" style="background-color:#FFFFFF;"|3
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 +
|width = "1250" style="background-color:#FFFFFF;"|5.00
 +
 +
|-
 +
|}
 +
 +
 +
# 2.25
 +
# 2.50
 +
# 1.75
 +
# 1.50
 +
 +
 +
# The scale factor for the dilation below is
 +
 +
 +
 +
 +
 +
## 2
 +
## 4
 +
## 6
 +
## 3
 +
 +
# Two figures are similar if….
 +
 +
## They have the same shape
 +
## They are the same size
 +
## They have the same angle measurements
 +
## A and C
 +
 +
 +
 +
'''Ratios and Proportions Assessment: Key'''
 +
# How do you solve problems using ratios and proportions?
 +
 +
## Using an algorithm
 +
## Using proportional reasoning without an algorithm
 +
## Both a and b
 +
## None of the above
 +
 +
Correct feedback: Yes, the answer is both a and b
 +
 +
Incorrect feedback: Sorry, the answer is a and b. Please review the solving ratios without an algorithm PowerPoint. 
 +
 +
# Which ratio is equivalent to ''1/7?''
 +
 +
#  3/21
 +
#  14/24
 +
#  7/6
 +
#  22/100
 +
 +
Correct feedback: Yes, the answer is  ''3/21 .''
 +
 +
Incorrect feedback: Sorry, the answer is  ''3/21. Please review the vocabulary for this module.  ''
 +
 +
# A 16 oz box of cereal costs $5.49. How much are you paying per ounce?
 +
 +
## ≈0.50/oz
 +
## ≈0.30/oz
 +
## ≈0.44/oz
 +
## ≈ 0.34/oz
 +
 +
Correct feedback: Yes, the answer is ≈0.34/oz
 +
 +
Incorrect feedback: Sorry, the answer is≈0.34/oz. Please review the finding unit rate PowerPoint.
 +
 +
 +
# Bethany's heart beats 225 times in 3 minutes. How many times does her heart beat per minute?
 +
 +
# 70 times
 +
# 75 times
 +
# 78 times
 +
# 80 times
 +
 +
 +
Correct feedback: Yes, the answer is 75 times
 +
 +
Incorrect feedback: Sorry, the answer is 75 times. Please review the solving proportions PowerPoint. 
 +
 +
''$180/(712 h)=$x/(20 h)''
 +
 +
## $300
 +
## $325
 +
## $250
 +
## $295
 +
 +
Correct feedback: Yes, the answer is $300.
 +
 +
Incorrect feedback: Sorry, the answer is $300. Please review the solving proportions PowerPoint.
 +
 +
# Which is NOT a ratio
 +
 +
## 2 to 5
 +
## 2:5
 +
## 2.5
 +
''2/5''
 +
 +
Correct feedback: That's right, 2.5 is not a ratio.
 +
 +
Incorrect feedback:  Sorry, 2.5 is not a ratio. Please review the vocabulary for this module.
 +
 +
 +
#  What is the correct measurement for the missing angle?
 +
 +
 +
 +
 +
 +
 +
## 90°
 +
## 25°
 +
## 45°
 +
## 40°
 +
 +
Correct feedback: The answer is 40°
 +
 +
Incorrect feedback.  The answer is 40°. Please go back and review the vocabulary for this module
 +
 +
# Below is a table showing the price per pound for cashews at the grocery store.  What is the missing value?
 +
 +
{| border=1
 +
|width = "1250" style="background-color:#FFFFFF;"|Weight (lb)
 +
 +
|width = "1250" style="background-color:#FFFFFF;"|Cost
 +
 +
|-
 +
|width = "1250" style="background-color:#FFFFFF;"|1
 +
 +
|width = "1250" style="background-color:#FFFFFF;"|1.25
 +
 +
|-
 +
|width = "1250" style="background-color:#FFFFFF;"|2
 +
 +
|width = "1250" style="background-color:#FFFFFF;"|x
 +
 +
|-
 +
|width = "1250" style="background-color:#FFFFFF;"|3
 +
 +
|width = "1250" style="background-color:#FFFFFF;"|5.00
 +
 +
|-
 +
|}
 +
 +
# 2.25
 +
# 2.50
 +
# 1.75
 +
# 1.50
 +
 +
Correct feedback, Great! The answer is $2.50
 +
 +
Incorrect feedback: Sorry, the answer is $2.50. Please review the solving ratios without algorithms PowerPoint.
 +
 +
 +
# The scale factor for the dilation below is
 +
*
 +
 +
 +
 +
 +
 +
 +
 +
 +
## 2
 +
## 4
 +
## 6
 +
## 3
 +
 +
Correct feedback: That's correct, the answer is 2
 +
 +
Incorrect feedback. Sorry, the answer is 2. Please review the dilations PowerPoint.
 +
 +
# Two figures are similar if….
 +
 +
## They have the same shape
 +
## They are the same size
 +
## They have the same angle measurements
 +
## A and C
 +
 +
Correct feedback: That's right, the answer is d. Two figures are similar if they have the same shape and angle measurements
 +
 +
Incorrect feedback: Sorry, the answer is D. Please go back and review the vocabulary for this module.
 +
'''General Education Math Lesson Plan'''
 +
'''Ratios and Proportions: Dimensional Analysis'''
 +
'''Source: '''Bennett, J.M., Burger, E. B., Chard, D. J., Hall, E., Kennedy, P. A…Waits, B. W. (2011). ''Mathematics.'' Austin, TX: Holt McDougal
 +
'''Standards: '''6.PRF.1c1 Describe the ratio relationship between two quantities for a given situation
 +
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
 +
'''Learning Outcome: '''Students will use ratios to convert from customary and metric units
 +
'''Materials: '''
 +
'''Activities:'''
 +
* Focus and Review: Review common conversions (e.g., ___ inches in 1 ft; ____ centimeters in 1 meter)
 +
* Lecture: Teacher works through a variety of problems using conversion factors explaining how to choose the correct factors when setting up ratios (e.g., 40 mph=
 +
′′40 𝑚𝑖𝑙𝑒𝑠1 ℎ𝑜𝑢𝑟 . Once students are successful setting up ratios, teacher demonstrates how to use the ratios to solve word problems (e.g., How many seconds in an hour).''
 +
* Guided Practice: Students work in pairs to complete five word problems from their math textbooks
 +
* Independent Practice: Students work five word problems using real-world application. Students are expected to pull essential facts from the story to create the ratio and solve.
 +
'''Activity: Create a universally designed version of the above lesson'''
 +
{| border=1
 +
|width = "1250" style="background-color:#FFFFFF;"|UDL Planning
 +
 +
|width = "1250" style="background-color:#FFFFFF;"|My ideas
 +
 +
|-
 +
|width = "1250" style="background-color:#FFFFFF;"|Representation- adaptations in materials (e.g., adapt for sensory impairments)
 +
 +
|width = "1250" style="background-color:#FFFFFF;"|Visual models of ratios; use a table and reasoning to solve problems instead of ratios; use a talking calculator to complete operations; color code ratio to indicate first, next, last; color code word problem to indicate where each fact should go when setting up the ratios
 +
 +
|-
 +
|width = "1250" style="background-color:#FFFFFF;"|Expression- how will student show learning (e.g., use of assistive technology; alternative project)
 +
 +
|width = "1250" style="background-color:#FFFFFF;"|Use a calculator or manipulative; student uses AT to scan possible answers; student indicates correct set up of ratio but is not responsible for completing the calculations
 +
 +
|-
 +
|width = "1250" style="background-color:#FFFFFF;"|Engagement- how will student participate in the activity
 +
 +
|width = "1250" style="background-color:#FFFFFF;"|Student can work in a pair during independent practice; alter word problems to make personally relevant (e.g., add student's name, change the context to be something familiar)
 +
 +
|-
 +
|}
 +
 +
'''Ratios and Scale Lesson Plan '''
 +
''Concept/principle to be demonstrated:''
 +
In nearly every construction occupation, ratio is used to determine scale, capacity, and usage. Ratio is critical to safety on the worksite, and in the finished product. A ratio is a comparison of two or more quantities, and can be expressed in several forms. Understanding is demonstrated by solving a variety of construction-related problems.
 +
''Lesson objectives/Evidence of learning:''
 +
* Identify and express ratios in several forms and in simplest terms.
 +
* Use different ratios to show the same scale/proportion of an object.
 +
* Compare and contract how different mathematical procedures could be used to complete a particular task.
 +
* Transfer mathematical vocabulary, concepts, and procedures to other disciplinary contexts and the real world.
 +
* Recognize and explain the meaning of information presented using mathematics.
 +
* Solve a variety of construction related problems.
 +
''How this math connects to construction jobs:''
 +
Ratios provide an easy way to compare two quantities. When a builder reviews blueprints prepared by architects, she or he checks the scale of the drawing (usually in a key, similar to a geography map) to determine the ratio to which the blueprint was drawn. This lesson will help students comprehend how ratios and proportions are used in construction.
 +
* '''Architects '''use ratios to draw blueprints to a scale that is easy for builders to interpret.
 +
* '''Engineers '''use ratios to test structural and mechanical systems for capacity and safety issues.
 +
* '''Painters '''use ratios to mix pigments to get a desired color.
 +
* '''Millwrights '''use ratios to solve pulley rotation and gear problems.
 +
* '''Operating Engineer'''s apply ratios to ensure the correct equipment is used to safely move heavy materials such as steel on worksites.
 +
''Teacher used training aids:''
 +
* Set of blueprints or other documents that show proportioned scale (i.e., road map)
 +
* Architecture scale (optional)
 +
''Additional online aid:''
 +
* Reference to [www.constructmyfuture.com-http://www.constructmyfuture.com/] website – ''Top 10 Construction Projects of the 20th Century ''pages (optional)
 +
''Materials needed per student:''
 +
* Calculator with √ key & memory +/- functions
 +
* Ratios and Scale Worksheet
 +
* Rulers (optional)
 +
* Graph paper for each student
 +
''Lesson Introduction:''
 +
Ratios are used in construction to design buildings to the desired scale; to communicate the scope of a project from an architect's desk to a worksite; and to accurately use and manage products. In today's lesson, we will first look at concrete mix as an example. It may not sound glamorous, but it's important a cement mason gets the ratio of concrete mix to water just right – too much water can reduce the strength of a foundation, which could lead to cracking and other serious structural safety issues. Other types of materials that construction workers regularly mix on the job site include paints, glues and adhesives, and gasoline.
 +
Ratios are used when an operating engineer calculates how much product can be hoisted in the air above a worksite. He or she must use the correct cabling and equipment to safely move materials, such as steel, in areas where other people are working.
 +
''Lesson Components:''
 +
# Look at structures listed in the Top 10 Construction Projects of the 20th Century webpage on [www.constructmyfuture.com-http://www.constructmyfuture.com/]; in reading the descriptions of building these famous structures, (World Trade Center, Hoover Dam, etc.) ask students in what steps of the building project do they think ratio would be important to know, and why.
 +
'''Note:'' '''''It is helpful to ask if any students have been to these famous structures, and what they observed. For example: Since 1937, 1.6 billion cars have crossed the Golden Gate Bridge in San Francisco – what decisions do you think designers of this bridge made to ensure the bridge would be safe? How does would ratio relate to these decisions?
 +
# A ratio is a comparison of two like quantities that are expressed in the same units of measure. A ratio takes on the form of a fraction; however, the final form of a ratio is not left as a fraction. It is written as a statement of the ratio relationship (this to that).
 +
''Examples to write on the board:''
 +
'''3 inches/5 inches '''(any ratio can be expressed as a fraction)
 +
3 inches/5 inches = '''3/5 '''(whenever possible, cancel identical units)
 +
'''3 : 5 '''(read aloud "the ratio of three to five")
 +
# A ratio written in either form can be reduced like a fraction.
 +
'''5 : 10 '''5 can be divided into both numbers (numerator & denominator)
 +
'''5 : 10 = 1 '''Complete the math
 +
'''5    5    2'''
 +
'''1:2 '''This is the simplified ratio in the referred format.
 +
# Order of the ratio is established by the problem statement. Placement of the numbers in the numerator and denominator is critical.
 +
''Examples to write on the board:''
 +
What is the ratio of 16 quarts to 5 gallons?
 +
'''16 qt : 5 gal '''Write the ratio.
 +
'''4 gal : 5 gal '''Change to the same units
 +
'''4 gal : 5 gal '''Cancel identical units
 +
'''4 : 5 '''Ratio is now in lowest terms. This could be used to measure 4 cups to 5 cups, or 4 quarts to 5 quarts.
 +
# Concrete mix is an example of how ratios can show the relationship of more than two quantities. Cement, sand and crushed stone are mixed in the ratio of 1 : 2 : 5 by weight. For every pound of cement used, two pounds of sand and five pounds of crushed rock are used. How much of each component are needed for 4000 pounds of concrete?
 +
'''1 + 2 + 5 = 8 '''There are 8 parts to the mix (denominator)
 +
'''1/8 '''There is one part of cement in the mix
 +
'''2/8 or 1/4 '''There are 2 parts of sand in the mix.
 +
'''5/8 '''This is the portion of the ratio that is crushed rock
 +
'''1/8 x 4000 = 500 lbs cement'''
 +
'''1/4 x 4000 = 1000 lbs sand'''
 +
'''5/8 x 4000 = 2500 lbs crushed rock'''
 +
'''500lbs cement + 1000 lbs sand + 2500lbs crushed rock = 4000 lbs total mixture'''
 +
# Show students the blueprint drawings and/or road map. Point out the key features of the blueprint or map, asking what these features are called (such as a map legend). Invite a student to review the blueprint or map, and tell the class the scale of the document. Explain this is a ratio used to make it possible to precisely draw and convey actual measurements in a usable document.
 +
# Architects and engineers use ratio in technical drawings and blueprints. By the way, blueprints are not always the color blue – before computer aided drafting, copies of building specifications were drawn using blue lines (hence the name which is still used, "blueprint"). Nowadays, most technical drawings and blueprints are reproduced in black and white.
 +
# Pass out 8 1/2" x 11" graph paper to students, asking them to orient the paper in landscape (11" sides being the top and bottom of the page). Have students draw a "legend" in the bottom right hand corner on each side of the paper, with these different scales:
 +
Side One: 1/4" = 1'0" (most graph paper boxes equal 1/4")
 +
Side Two: 1/4" = 5'0"
 +
Tell the students they will draw a basic, one story house "shell" plan (exterior walls only) to these different scales/ratios, with the final shell dimensions matching in both drawings (square footage, placement of windows and doors, etc.). When two ratios can be set equal to each other, a proportion is formed.
 +
Explain how this activity will help them understand the relationship between ratios and proportion as a way to communicate information and make decisions. Students can determine the overall square footage and shape they want for their house, but need to use these perimeters:
 +
Front door opening = 36"
 +
At least five windows = 28"
 +
A bay window = 24"x42"x24" (may need to draw this on the board)
 +
A sliding glass door = 6' in width
 +
A two-car attached garage with one or two doors
 +
For extra credit or homework, students can take home their drawings, and add interior rooms and features, such as fireplaces, sunken tubs, and other fun and creative additions.
 +
# Use '''''Ratios and Scale Worksheet '''''in class or as homework.
 +
 +
 +
'''Ratios and Scale Worksheet'''
 +
 +
Name ______________________________
 +
 +
Solve the following problems and reduce answers to simplest terms without units:
 +
'''Problem \#1'''
 +
3 feet : 6 inches
 +
'''Problem \#2'''
 +
25 / 80
 +
'''Problem \#3'''
 +
25 lb cement : 50 lb sand : 75 lb crushed rock
 +
'''Problem \#4'''
 +
3 rejections to 24 good welding joints
 +
'''Problem \#5'''
 +
The blueprint for a building is drawn to a scale of 1/4" = 1 ft. If the dimensions measure 6 1/2 inches by 11 inches on the print, what are the building dimensions?
 +
'''Problem \#6'''
 +
Two gears have 64 and 40 teeth. What is their ratio?
 +
'''Problem \#7'''
 +
What is the ratio of 3 yards to 12 inches?
 +
 +
 +
'''Ratios and Scale Worksheet: KEY'''
 +
Solve the following problems and reduce answers to simplest terms without units:
 +
'''Problem \#1'''
 +
3 feet : 6 inches '''36 : 6      6 : 1'''
 +
'''Problem \#2'''
 +
25 / 80 '''25 : 80      5 : 16'''
 +
'''            5    5'''
 +
'''Problem \#3'''
 +
25 lb cement : 50 lb sand : 75 lb crushed rock  '''25 : 50 : 75        1 : 2 : 3'''
 +
'''                                                                          25  25  25'''
 +
'''Problem \#4'''
 +
3 rejections to 24 good welding joints  '''3 : 24        1 : 8'''
 +
'''                                                            3    3'''
 +
'''Problem \#5'''
 +
The blueprint for a building is drawn to a scale of 1/4" = 1 ft. If the dimensions measure 6 1/2 inches by 11 inches on the print, what are the building dimensions?
 +
'''26' x 44' 1/4 : 6 1/2 (X4) 1: 26'''
 +
'''1/4 : 11 (X4) 1: 44'''
 +
'''Problem \#6'''
 +
Two gears have 64 and 40 teeth. What is their ratio? '''64 : 40 8 : 5'''
 +
''' 8    8'''
 +
'''Problem \#7'''
 +
What is the ratio of 3 yards to 12 inches? '''9 : 1'''
 +
 +
 +
''''''Activity: Create a universally designed version of the above lesson''''''
 +
{| border=1
 +
|width = "1250" style="background-color:#FFFFFF;"|UDL Planning
 +
 +
|width = "1250" style="background-color:#FFFFFF;"|My ideas
 +
 +
|-
 +
|width = "1250" style="background-color:#FFFFFF;"|Representation- adaptations in materials (e.g., adapt for sensory impairments)
 +
 +
|width = "1250" style="background-color:#FFFFFF;"|
 +
 +
|-
 +
|width = "1250" style="background-color:#FFFFFF;"|Expression- how will student show learning (e.g., use of assistive technology; alternative project)
 +
 +
|width = "1250" style="background-color:#FFFFFF;"|
 +
 +
|-
 +
|width = "1250" style="background-color:#FFFFFF;"|Engagement- how will student participate in the activity
 +
 +
|width = "1250" style="background-color:#FFFFFF;"|
 +
 +
|-
 +
|}
 +
 +
 +
 +
 +
[[Category: Math Content Module: Ratio and Proportion]]

Revision as of 10:38, 12 August 2013

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



[1] 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


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 (2/3=4/6)
  • Ratio- a comparison of two quantities, can be written in a variety of forms (11 to 20, 11:20 or 11/20)
  • Equivalent ratios- two ratios that are the same in their simplest form (e.g., 1/3=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

(2/3=4/6)

  • 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

Dilation example:



  • Have students distinguish between a ratio and not a ratio

11 to 20 11:20 11/20 7 8.5


  • Have students identify corresponding angles and sides when comparing two polygons




  • Review similar and not similar figures



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. 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

(Insert finding unit rates PowerPoint presentation here)

  • 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

(Insert solving proportions PowerPoint presentation here)

  • 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

(Insert solving proportions PowerPoint presentation)

  • 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

(Insert dilations PowerPoint presentation here)

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.

Finding Unit Rates

Solving Ratios without Algorithms

Solving Proportions

Dilations

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:
  • 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 [2]

Math Open Reference [3]

There are many resources listed here: [4]

  • 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).

\[\[File:Insert Picture here.jpg\]\]\[\[File:Insert Picture here.jpg\]\] 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

  • [5] - this website provides explanations and real-world examples of how to apply ratios and proportions across different settings and situations
  • [6] - 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
  • [7] - 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
  • [8] - provides a variety of activities, work sheets, and web quests to use in your classroom
  • [9] - website specifically for teachers that provides a variety of ideas and activities to use in your classroom
  • [www.teachertube.com-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)
  • [10] - 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
  • [11] - this SMART board exchange has developed lessons by classroom teachers differentiated by grade level. You can also search by skill and/or state standards.
  • [12] - this website provides a webinar about how to adapt materials for students who have visual impairments

Module Assessments Insert assessment here Sample General Education lesson plans Insert developed lesson plans here Have an idea: Upload the lesson plans you've created here Insert link for teachers to upload lesson plans Teacher's Corner: Blog with other teachers Insert forum or blog for teachers to share ideas Up for a challengejpeg


Adapt the following general education lesson plan; adapt, and upload. These lesson plans may be shared with higher education professionals developing strategies to provide meaningful academic instruction in mathematics to students with severe disabilities. Insert blank lesson plan form with UDL chart here Insert link for teachers to upload lesson plans

Ratios and Proportions Assessment

  1. How do you solve problems using ratios and proportions?
    1. Using an algorithm
    2. Using proportional reasoning without an algorithm
    3. Both a and b
    4. None of the above
  1. Which ratio is equivalent to 1/7?
    1. 3/21
    2. 14/24
    3. 7/6
    4. 22/100
  1. A 16 oz box of cereal costs $5.49. How much are you paying per ounce?
    1. ≈0.50/oz
    2. ≈0.30/oz
    3. ≈0.44/oz
    4. ≈ 0.34/oz
  1. Bethany's heart beats 225 times in 3 minutes. How many times does her heart beat per minute?
    1. 70 times
    2. 75 times
    3. 78 times
    4. 80 times

$180/(712 h)=$x/(20 h)

    1. $300
    2. $325
    3. $250
    4. $295


  1. Which is NOT a ratio
    1. 2 to 5
    2. 2:5
    3. 2.5

2/5

  1. What is the correct measurement for the missing angle?



    1. 90°
    2. 25°
    3. 45°
    4. 40°
  1. Below is a table showing the price per pound for cashews at the grocery store. What is the missing value?
Weight (lb) Cost
1 1.25
2 x
3 5.00


  1. 2.25
  2. 2.50
  3. 1.75
  4. 1.50


  1. The scale factor for the dilation below is



    1. 2
    2. 4
    3. 6
    4. 3
  1. Two figures are similar if….
    1. They have the same shape
    2. They are the same size
    3. They have the same angle measurements
    4. A and C


Ratios and Proportions Assessment: Key

  1. How do you solve problems using ratios and proportions?
    1. Using an algorithm
    2. Using proportional reasoning without an algorithm
    3. Both a and b
    4. None of the above

Correct feedback: Yes, the answer is both a and b

Incorrect feedback: Sorry, the answer is a and b. Please review the solving ratios without an algorithm PowerPoint.

  1. Which ratio is equivalent to 1/7?
  1. 3/21
  2. 14/24
  3. 7/6
  4. 22/100

Correct feedback: Yes, the answer is 3/21 .

Incorrect feedback: Sorry, the answer is 3/21. Please review the vocabulary for this module.

  1. A 16 oz box of cereal costs $5.49. How much are you paying per ounce?
    1. ≈0.50/oz
    2. ≈0.30/oz
    3. ≈0.44/oz
    4. ≈ 0.34/oz

Correct feedback: Yes, the answer is ≈0.34/oz

Incorrect feedback: Sorry, the answer is≈0.34/oz. Please review the finding unit rate PowerPoint.


  1. Bethany's heart beats 225 times in 3 minutes. How many times does her heart beat per minute?
  1. 70 times
  2. 75 times
  3. 78 times
  4. 80 times


Correct feedback: Yes, the answer is 75 times

Incorrect feedback: Sorry, the answer is 75 times. Please review the solving proportions PowerPoint.

$180/(712 h)=$x/(20 h)

    1. $300
    2. $325
    3. $250
    4. $295

Correct feedback: Yes, the answer is $300.

Incorrect feedback: Sorry, the answer is $300. Please review the solving proportions PowerPoint.

  1. Which is NOT a ratio
    1. 2 to 5
    2. 2:5
    3. 2.5

2/5

Correct feedback: That's right, 2.5 is not a ratio.

Incorrect feedback: Sorry, 2.5 is not a ratio. Please review the vocabulary for this module.


  1. What is the correct measurement for the missing angle?




    1. 90°
    2. 25°
    3. 45°
    4. 40°

Correct feedback: The answer is 40°

Incorrect feedback. The answer is 40°. Please go back and review the vocabulary for this module

  1. Below is a table showing the price per pound for cashews at the grocery store. What is the missing value?
Weight (lb) Cost
1 1.25
2 x
3 5.00
  1. 2.25
  2. 2.50
  3. 1.75
  4. 1.50

Correct feedback, Great! The answer is $2.50

Incorrect feedback: Sorry, the answer is $2.50. Please review the solving ratios without algorithms PowerPoint.


  1. The scale factor for the dilation below is





    1. 2
    2. 4
    3. 6
    4. 3

Correct feedback: That's correct, the answer is 2

Incorrect feedback. Sorry, the answer is 2. Please review the dilations PowerPoint.

  1. Two figures are similar if….
    1. They have the same shape
    2. They are the same size
    3. They have the same angle measurements
    4. A and C

Correct feedback: That's right, the answer is d. Two figures are similar if they have the same shape and angle measurements

Incorrect feedback: Sorry, the answer is D. Please go back and review the vocabulary for this module. General Education Math Lesson Plan Ratios and Proportions: Dimensional Analysis Source: Bennett, J.M., Burger, E. B., Chard, D. J., Hall, E., Kennedy, P. A…Waits, B. W. (2011). Mathematics. Austin, TX: Holt McDougal Standards: 6.PRF.1c1 Describe the ratio relationship between two quantities for a given situation 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 Learning Outcome: Students will use ratios to convert from customary and metric units Materials: Activities:

  • Focus and Review: Review common conversions (e.g., ___ inches in 1 ft; ____ centimeters in 1 meter)
  • Lecture: Teacher works through a variety of problems using conversion factors explaining how to choose the correct factors when setting up ratios (e.g., 40 mph=

′′40 𝑚𝑖𝑙𝑒𝑠1 ℎ𝑜𝑢𝑟 . Once students are successful setting up ratios, teacher demonstrates how to use the ratios to solve word problems (e.g., How many seconds in an hour).

  • Guided Practice: Students work in pairs to complete five word problems from their math textbooks
  • Independent Practice: Students work five word problems using real-world application. Students are expected to pull essential facts from the story to create the ratio and solve.

Activity: Create a universally designed version of the above lesson

UDL Planning My ideas
Representation- adaptations in materials (e.g., adapt for sensory impairments) Visual models of ratios; use a table and reasoning to solve problems instead of ratios; use a talking calculator to complete operations; color code ratio to indicate first, next, last; color code word problem to indicate where each fact should go when setting up the ratios
Expression- how will student show learning (e.g., use of assistive technology; alternative project) Use a calculator or manipulative; student uses AT to scan possible answers; student indicates correct set up of ratio but is not responsible for completing the calculations
Engagement- how will student participate in the activity Student can work in a pair during independent practice; alter word problems to make personally relevant (e.g., add student's name, change the context to be something familiar)

Ratios and Scale Lesson Plan Concept/principle to be demonstrated: In nearly every construction occupation, ratio is used to determine scale, capacity, and usage. Ratio is critical to safety on the worksite, and in the finished product. A ratio is a comparison of two or more quantities, and can be expressed in several forms. Understanding is demonstrated by solving a variety of construction-related problems. Lesson objectives/Evidence of learning:

  • Identify and express ratios in several forms and in simplest terms.
  • Use different ratios to show the same scale/proportion of an object.
  • Compare and contract how different mathematical procedures could be used to complete a particular task.
  • Transfer mathematical vocabulary, concepts, and procedures to other disciplinary contexts and the real world.
  • Recognize and explain the meaning of information presented using mathematics.
  • Solve a variety of construction related problems.

How this math connects to construction jobs: Ratios provide an easy way to compare two quantities. When a builder reviews blueprints prepared by architects, she or he checks the scale of the drawing (usually in a key, similar to a geography map) to determine the ratio to which the blueprint was drawn. This lesson will help students comprehend how ratios and proportions are used in construction.

  • Architects use ratios to draw blueprints to a scale that is easy for builders to interpret.
  • Engineers use ratios to test structural and mechanical systems for capacity and safety issues.
  • Painters use ratios to mix pigments to get a desired color.
  • Millwrights use ratios to solve pulley rotation and gear problems.
  • Operating Engineers apply ratios to ensure the correct equipment is used to safely move heavy materials such as steel on worksites.

Teacher used training aids:

  • Set of blueprints or other documents that show proportioned scale (i.e., road map)
  • Architecture scale (optional)

Additional online aid:

Materials needed per student:

  • Calculator with √ key & memory +/- functions
  • Ratios and Scale Worksheet
  • Rulers (optional)
  • Graph paper for each student

Lesson Introduction: Ratios are used in construction to design buildings to the desired scale; to communicate the scope of a project from an architect's desk to a worksite; and to accurately use and manage products. In today's lesson, we will first look at concrete mix as an example. It may not sound glamorous, but it's important a cement mason gets the ratio of concrete mix to water just right – too much water can reduce the strength of a foundation, which could lead to cracking and other serious structural safety issues. Other types of materials that construction workers regularly mix on the job site include paints, glues and adhesives, and gasoline. Ratios are used when an operating engineer calculates how much product can be hoisted in the air above a worksite. He or she must use the correct cabling and equipment to safely move materials, such as steel, in areas where other people are working. Lesson Components:

  1. Look at structures listed in the Top 10 Construction Projects of the 20th Century webpage on [www.constructmyfuture.com-http://www.constructmyfuture.com/]; in reading the descriptions of building these famous structures, (World Trade Center, Hoover Dam, etc.) ask students in what steps of the building project do they think ratio would be important to know, and why.

Note: It is helpful to ask if any students have been to these famous structures, and what they observed. For example: Since 1937, 1.6 billion cars have crossed the Golden Gate Bridge in San Francisco – what decisions do you think designers of this bridge made to ensure the bridge would be safe? How does would ratio relate to these decisions?

  1. A ratio is a comparison of two like quantities that are expressed in the same units of measure. A ratio takes on the form of a fraction; however, the final form of a ratio is not left as a fraction. It is written as a statement of the ratio relationship (this to that).

Examples to write on the board: 3 inches/5 inches (any ratio can be expressed as a fraction) 3 inches/5 inches = 3/5 (whenever possible, cancel identical units) 3 : 5 (read aloud "the ratio of three to five")

  1. A ratio written in either form can be reduced like a fraction.

5 : 10 5 can be divided into both numbers (numerator & denominator) 5 : 10 = 1 Complete the math 5 5 2 1:2 This is the simplified ratio in the referred format.

  1. Order of the ratio is established by the problem statement. Placement of the numbers in the numerator and denominator is critical.

Examples to write on the board: What is the ratio of 16 quarts to 5 gallons? 16 qt : 5 gal Write the ratio. 4 gal : 5 gal Change to the same units 4 gal : 5 gal Cancel identical units 4 : 5 Ratio is now in lowest terms. This could be used to measure 4 cups to 5 cups, or 4 quarts to 5 quarts.

  1. Concrete mix is an example of how ratios can show the relationship of more than two quantities. Cement, sand and crushed stone are mixed in the ratio of 1 : 2 : 5 by weight. For every pound of cement used, two pounds of sand and five pounds of crushed rock are used. How much of each component are needed for 4000 pounds of concrete?

1 + 2 + 5 = 8 There are 8 parts to the mix (denominator) 1/8 There is one part of cement in the mix 2/8 or 1/4 There are 2 parts of sand in the mix. 5/8 This is the portion of the ratio that is crushed rock 1/8 x 4000 = 500 lbs cement 1/4 x 4000 = 1000 lbs sand 5/8 x 4000 = 2500 lbs crushed rock 500lbs cement + 1000 lbs sand + 2500lbs crushed rock = 4000 lbs total mixture

  1. Show students the blueprint drawings and/or road map. Point out the key features of the blueprint or map, asking what these features are called (such as a map legend). Invite a student to review the blueprint or map, and tell the class the scale of the document. Explain this is a ratio used to make it possible to precisely draw and convey actual measurements in a usable document.
  2. Architects and engineers use ratio in technical drawings and blueprints. By the way, blueprints are not always the color blue – before computer aided drafting, copies of building specifications were drawn using blue lines (hence the name which is still used, "blueprint"). Nowadays, most technical drawings and blueprints are reproduced in black and white.
  3. Pass out 8 1/2" x 11" graph paper to students, asking them to orient the paper in landscape (11" sides being the top and bottom of the page). Have students draw a "legend" in the bottom right hand corner on each side of the paper, with these different scales:

Side One: 1/4" = 1'0" (most graph paper boxes equal 1/4") Side Two: 1/4" = 5'0" Tell the students they will draw a basic, one story house "shell" plan (exterior walls only) to these different scales/ratios, with the final shell dimensions matching in both drawings (square footage, placement of windows and doors, etc.). When two ratios can be set equal to each other, a proportion is formed. Explain how this activity will help them understand the relationship between ratios and proportion as a way to communicate information and make decisions. Students can determine the overall square footage and shape they want for their house, but need to use these perimeters: Front door opening = 36" At least five windows = 28" A bay window = 24"x42"x24" (may need to draw this on the board) A sliding glass door = 6' in width A two-car attached garage with one or two doors For extra credit or homework, students can take home their drawings, and add interior rooms and features, such as fireplaces, sunken tubs, and other fun and creative additions.

  1. Use Ratios and Scale Worksheet in class or as homework.


Ratios and Scale Worksheet

Name ______________________________

Solve the following problems and reduce answers to simplest terms without units: Problem \#1 3 feet : 6 inches Problem \#2 25 / 80 Problem \#3 25 lb cement : 50 lb sand : 75 lb crushed rock Problem \#4 3 rejections to 24 good welding joints Problem \#5 The blueprint for a building is drawn to a scale of 1/4" = 1 ft. If the dimensions measure 6 1/2 inches by 11 inches on the print, what are the building dimensions? Problem \#6 Two gears have 64 and 40 teeth. What is their ratio? Problem \#7 What is the ratio of 3 yards to 12 inches?


Ratios and Scale Worksheet: KEY Solve the following problems and reduce answers to simplest terms without units: Problem \#1 3 feet : 6 inches 36 : 6 6 : 1 Problem \#2 25 / 80 25 : 80 5 : 16 5 5 Problem \#3 25 lb cement : 50 lb sand : 75 lb crushed rock 25 : 50 : 75 1 : 2 : 3 25 25 25 Problem \#4 3 rejections to 24 good welding joints 3 : 24 1 : 8 3 3 Problem \#5 The blueprint for a building is drawn to a scale of 1/4" = 1 ft. If the dimensions measure 6 1/2 inches by 11 inches on the print, what are the building dimensions? 26' x 44' 1/4 : 6 1/2 (X4) 1: 26 1/4 : 11 (X4) 1: 44 Problem \#6 Two gears have 64 and 40 teeth. What is their ratio? 64 : 40 8 : 5 8 8 Problem \#7 What is the ratio of 3 yards to 12 inches? 9 : 1


'Activity: Create a universally designed version of the above lesson'

UDL Planning My ideas
Representation- adaptations in materials (e.g., adapt for sensory impairments)
Expression- how will student show learning (e.g., use of assistive technology; alternative project)
Engagement- how will student participate in the activity
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