Chapter 18: Ratios, Proportions, and Proportional Reasoning by https://padlet.com/kmakura/ci9z98gt6o8j Darby McShane, Desirea Hoover, Jordan Edwards & Krysta Makura en-us 2017-03-21 16:21:37 UTC 2021-09-22 21:44:22 UTC hello@padlet.com https://padlet-assets.s3.amazonaws.com/icons/Growing.png RATIOS kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163615878 2017-03-30 07:26:49 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163615878 Ratios kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163616252 A number that relates to two quantities or measures within a given situation with a multiplicative relationship
  •  One of the four critical areas in 6th grade
  • Students’ understanding of multiplicative reasoning and multiplicative comparisons is critical
  • Closely related to fractions, but have distinct differences (i.e., ratios are not part-to-part). However, should be thought of as overlapping concepts (Ratio of cats to dogs in pet store vs. ratio of cats to pets in pet store.
  • Should be taught as relations that involve multiplicative reasoning rather than simply recording symbols (6:5).
    • Forming a ratio is a cognitive task, rather than a writing task

]]> 2017-03-30 07:29:08 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163616252 Four Types of Ratios kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163616502 2017-03-30 07:30:33 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163616502 Part-to-Part kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163616540 A ratio that relates one part of a whole to another part of the same whole
  • Students in an after-school program: 6 girls, 5 boys
Not fractions, but can be written using a fraction bar
  • 6/5: A ratio of 6 to 5, not six-fifths
Occurs across curriculum (slope, odds)
]]> 2017-03-30 07:30:49 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163616540 Part-to-Whole kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163617203 A ratio that compares a part of a whole, to the whole
  • Number of girls in an after-school program: 6/11
Can be thought of as a fraction: six-elevenths 
Includes percentages and probabilities
]]> 2017-03-30 07:34:38 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163617203 Ratios as Quotients kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163617435 Example: Limes are on sale and you can buy 5 limes for $1.00; The ratio of money for limes is $1.00 for 5 limes, which means the cost per lime is $0.20.

]]> 2017-03-30 07:36:13 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163617435 Ratios as Rates kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163617838 Rates are ratios between two measurements with different units (miles per gallon, square yards of wall and gallon of paint, flowers per bouquet) or relationships between two units of measure (centimeters per inch, inches per foot, feet per mile).
They also represent an infinite set of equivalent ratios
  • Example: A swimmer’s rate of laps swam is 30 laps in 1 day- equivalent ratios would be 60 laps in 2 days, 90 laps in 3 days, etc.

]]> 2017-03-30 07:38:26 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163617838 Two Ways to Think of Ratios kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163618415 2017-03-30 07:41:59 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163618415 Multiplicative Comparison kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163618623
  • Introduced in 4th grade (CCSS)
  • Foundational to understanding ratios
  • Comparing two different units (ratios as rates) is a 6th grade expectation (i.e., 8 roses per bouquet)
  • 6th graders are also expected to solve real-world problems related to ratios  

    • ]]>     2017-03-30 07:43:23 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163618623     Composed Unit kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163618847
  • Thinking of the ratio as one unit; 5 limes for $1.00 is a unit
  • Using the base unit, one can think of infinite multiples
  • Example: 10 limes for $2.00, 15 limes for $3.00, etc.
  • Composed units can be partitioned, enabling the ability to solve for any amount
  •  Example: 3 limes for $0.30, 1 lime for $0.20 

    • ]]>     2017-03-30 07:44:36 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163618847     PROPORTIONAL REASONING kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163619791 2017-03-30 07:49:17 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163619791 kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163619907 Uses: investigating interest, taxes, and tips or connecting figures, graphs, and slopes.
     
    Students start using proportional reasoning when they are involved in, one-to-one correspondence, place value, fractional concepts, and multiplicative reasoning.
    ]]>     2017-03-30 07:50:00 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163619907     Students must know: kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163619984 2017-03-30 07:50:26 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163619984 kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163620071 1. The difference between proportional and nonproportional situations
    • Is the comparison due to an additive, multiplicative, or constant?
      • Additive (walks the same amount): Janet and Jeanette walk at the same pace. When Janet is at 6 blocks, Jeanette has started and is at 2 blocks. How far will Jeanette be when Janet is at 12 blocks?
      • Constant(: Lisa and Linda both plant corn. Lisa plants 4 rows and Linda plants 6 rows. Linda’s corn is ready in 8 weeks, how long is Lisa’s till it’s ready?
      • Multiplicative: Kendra is making 6 dozen cookies and Kevin is making 3 dozen. Kevin uses 6 ounces of chocolate chips. How many ounces will Kendra use?
      • For students to distinguish between the different types of questions, they must make distinctions about each.
     
    A ratio is a number that expresses a multiplicative relationship that can be applied to a second situation in which the quantities are measured similarly.
    ]]>     2017-03-30 07:50:54 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163620071     kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163620756 2. Additive and Multiplicative Comparisons in Story Problems
    Using either additive or multiplicative reasoning in the wrong situation can lead to wrong answers. An assessment sheet might be useful in the category.
    Often, students will notice an answer could be out of place when the wrong comparison is used.
    ]]>     2017-03-30 07:54:21 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163620756     kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163621044 3. Covariation
    When two different quantities (ratio) vary together.
    Ex. 5 mangos cost $2.00. If there are 10 mangos, the cost will be ____. Once you know the new price or new number, you can figure out the last variable.
    ]]>     2017-03-30 07:55:47 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163621044     kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163621251 A. Within and between ratios
    Within ratio: two measures in the same setting (ex. mangos to price)
    Between ratio: two corresponding measures in different situations (ex. price of 5 mangos compared to price of 10 mangos)
    These ratios are popular in 8th grade and students explore similarities with geometric shapes.
    ]]>     2017-03-30 07:57:07 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163621251     kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163621330 B. Measurement and geometry
    Scale drawings can be a useful tool in seeing connections between multiplicative and proportional reasoning. Slope is the ratio between 2 values. If two figures are proportional then any corresponding linear dimensions will have the same scale factor.
    ]]>     2017-03-30 07:57:35 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163621330     kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163621497 C. Algebra
    Proportional situations are linear situations. Graphing equivalent rations is a powerful way to illustrate this concept. The ratio or rate is the slope of the graph.
    Having two sized shapes start in the same corner is useful for finding the ratio between the two shapes.
    Graphing ratios can be challenging for students. Some might have trouble using which points to graph. A good way to help students make sense of unit rates and proportional reasoning is to use real-life contexts and gathering data.
    ]]>     2017-03-30 07:58:44 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163621497     STRATEGIES FOR SOLVING PROPORTIONAL SITUATIONS kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163621982 2017-03-30 08:01:30 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163621982 Tape or Strip Diagrams kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163633640 2017-03-30 08:56:39 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163633640 kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163633705 Tape or strip diagrams can be great visuals that help students visualize multiplicative relationships.
    • Defined as “drawings that look like a segment of tape, used to illustrate number relationships. Also known as strip diagrams, bar models or graphs, fraction strips, or length models.”
    • Example: The ratio of boys to girls in the class is 3 to 4.
      • Once the ratio is provided students can set up their sketches in different ways. 
    ]]>     2017-03-30 08:56:56 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163633705     Double Number Line Diagrams kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634043 2017-03-30 08:58:20 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634043 kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634098 Double number line diagrams are similar to tape or strip diagrams but they might not show proportions. They must however have the two lines labeled, just as tape diagrams should be.
    • Example: Comparing the height between a zoo animal and the students’ height.
    ]]>     2017-03-30 08:58:31 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634098     Percents kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634246 2017-03-30 08:59:06 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634246 kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634261 Percent problems can be solved by using all sorts of strategies that have been talked about in this chapter.  
    •  Double number line is a great way to solve percent situations
      •  It can help students figure out the unknown
      •  After double number line is set up, the diagram can be used as a proportion
      •  One line would represent the measures of the problem
      •  The second line would represent the value in terms of percents
    •  Using the double number line not only shows modelling for part-whole, but also increase-decrease situations, as well as comparisons between two distinct quantities
    •  Linear models also do not restrict students from thinking about percents that are larger than 100 since the line can represent more than 100 rather than a circle
    ]]>     2017-03-30 08:59:11 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634261     Equations kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634476 2017-03-30 09:00:12 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634476 kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634503 Traditionally, textbooks taught students to set up equations of two ratios, and proportions b solving for the unknown (x) by cross-multiplying and solving for x.
    •  Students should be encouraged to reason in order to find the missing value rather than just using the cross-multiply algorithm
    1. Create a Visual Model: ask students to illustrate a problem that shows what is covarying instead of just setting up proportions. Provide visual cues to set up the proportions in order to support a wide range of learners.
    2. Solve the Proportion: unit rate can be found by dividing the amount by number and then multiplying by amount total. Scale factors can be used as well to solve problems. If using the cross-multiply, students need to understand in order to complete problems with more challenging numbers and unit rates/scale factors that are not easy to calculate.
    ]]>     2017-03-30 09:00:17 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634503     TEACHING PROPORTIONAL REASONING kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634858 2017-03-30 09:01:57 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634858 kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634900 1. Composed unit and multiplicative comparison should be used to build understanding of rations. A better understanding of multiplicative comparisons leads to a better understanding of rate, a strategy that can then be applied to proportions. ]]> 2017-03-30 09:02:11 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163634900 kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163635116 2. Students should be able to distinguish between proportional and non-proportional comparisons by giving examples of each and discussing differences.]]> 2017-03-30 09:02:59 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163635116 kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163635224 3. Ratio and proportion tasks should be provided in a wide range of contexts. This includes situations involving measurements, prices, geometrics and other visual contexts as well as all sorts of rates. ]]> 2017-03-30 09:03:27 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163635224 kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163635313 4. Students should be engaged in a variety of strategies for solving proportions =. Specifically use ration tables, visuals, question and graphs in order to solve problems. Make sure students apply reasoning strategies. ]]> 2017-03-30 09:03:51 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163635313 kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163635507 5. Recognize that symbolic or mechanical methods (cross-product algorithm) do not develop proportion reasoning and shouldn’t be introduced until students have many experiences with intuitive and conceptual methods. ]]> 2017-03-30 09:04:36 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163635507 kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163724556
  •  These strategies all students to use simple reasoning, as well as sophisticated strategies.
  •  Most commonly know strategy is cross products
  •  Should only be done after proportional problems with reasoning done using tape or diagram stripes.
  •  Multiple solutions strategies help students solve proportional reasoning problems.
  • These strategies are:
    •   Rate
    •  Scaling up or down
    •  Scale factors
    •  Ratio Tables
    •  Graphs
    •  Equation
    •  Percent 
  • First three are the most intuitive and the ones that students usually begin with when solving problems with reason.

    • ]]>     2017-03-30 14:40:35 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163724556     Rates and Scaling Strategies kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163730313 2017-03-30 14:56:20 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163730313 kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163730387 Rate is a quantity that is measured with another measured quantity.
    • Example: a rate of speed of 60 miles an hour.
    Unit rate method: for example- dividing the price of three apples by 3, then multiplying the unit of $0.80 per unit by 10. Will give the answer.
    • Ex: Tommy bought 3 apples for $2.40. At the same price, how much would 10 apples cost?
      • Unit rate would be three apples by 3 and $0.80 per unit by 10.
    Scale factor is the ratio of the model measurements to the actual measurement in the simplest form.
    Buildup strategy is usually done within the scale factor. 
    • Normally done on whole numbers and when the numbers are compatible.
    • Pizza Party example: Two camps of scouts are having a pizza party. The bear camp ordered enough so that every 3 campers would get 2 pizzas.  The raccoon camp ordered enough so that there would be 3 pizzas for every 5 campers. Which troop will have the most pizza?
    ]]>     2017-03-30 14:56:34 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163730387     Ratio Tables kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163731526 2017-03-30 14:59:35 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163731526 kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163731576
  • Great way to organize problems.
  • Organizes the two variable quantities that are related.
  • Numbers are compared or used to find a missing number
  • Connected to seeking the concept of an equivalent ratio rather than only trying to find the missing number
  • Should be taught prior to the cross product method because this allows the students to better understand what ratios are.
  • Ratio tables allow uses multiplicative relationship 
    • It is a relationship where two quantities can be expressed as multiples of each other. It can be generalized as y = ax, where y and x are multiples of each other.
    • ]]>     2017-03-30 14:59:41 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163731576     kmakura https://padlet.com/kmakura/ci9z98gt6o8j/wish/163732411 A study was conducted to determine how students reason in various proportional tasks. It was conducted to see if it was developmental or instructional factors that were related to proportional reasoning.]]> 2017-03-30 15:02:06 UTC https://padlet.com/kmakura/ci9z98gt6o8j/wish/163732411