Jeanette Cox
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How much a part of everyday life proportional relationships plays. In altering recipes, in determining area coverage per gallon of paint etc.

This course has reaffirmed for me the importance of K5 math instructional pedagogy founded on problembased learning rather than didactic teaching. And as an advocate of the 8 mathematical practices, “The Concepts Holding Your Students Back, has demonstrated each:
<b style=”fontfamily: inherit; fontsize: inherit; color: inherit;”>1. Make sense of problems and persevere in solving them <b style=”fontfamily: inherit; fontsize: inherit; color: inherit;”>2. Reason abstractly and <b style=”fontfamily: inherit; fontsize: inherit; color: inherit;”>quantitatively; <b style=”fontfamily: inherit; fontsize: inherit; color: inherit;”>3. Construct viable arguments and critique the reasoning of others; <b style=”fontfamily: inherit; fontsize: inherit; color: inherit;”>4. Model with mathematics; <b style=”fontfamily: inherit; fontsize: inherit; color: inherit;”>5. Use appropriate tools strategically; <b style=”fontfamily: inherit; fontsize: inherit; color: inherit;”>6. Attend to <b style=”fontfamily: inherit; fontsize: inherit; color: inherit;”>precision; <b style=”fontfamily: inherit; fontsize: inherit; color: inherit;”>7. Look for and make use of structure; <b style=”fontfamily: inherit; fontsize: inherit; color: inherit;”>8. Look for and express regularity <b style=”fontfamily: inherit; fontsize: inherit; color: inherit;”>in repeated reasoning
Another takeaway is the need for math curriculum specialists in districts all over the U.S. to know about MakeMathMoments and the wealth of resources, units and professional development available at their fingertips.
I’ve learned importance of engaging students in math at varying entry levels through noticing and wondering to spark their curiosity: I’ve relearned the importance of spatial reasoning and the concept of number as it relates to building the concept of units: I’ve confirmed the value of visual representations and models like the double number line: and have been reminded that the person doing the talking is doing the learning which means using math talks: Finally wrapping it all up is the interconnectedness of all the standards that lead to proportional reasoning.
Lastly, I’m pleased to say that in my work in math curriculum and consulting, I’ve had the good fortune to have learned from the best–Cathy Fosnot, Sherry Parrish, Pam Harris, Graham Fletcher, and Jon Orr and Kyle Pearce. Thank you.
 This reply was modified 1 year, 8 months ago by Jeanette Cox.

Like you, I love Ryan’s idea about beginning with a visual of a carton (or a glass much like your Hot Chocolate video) and a liter of milk and asking the students to Notice and Wonder. Write down their observations. And then provide another video or visual of actual pouring of the milk into a glass. Then estimating how many glasses of milk they think the liter can fill. (For multiple entry points students could determine the amount of milk in each glass (250ml) requiring measurement conversion. Visual drawing could be made to represent 4 glasses per liter and multiplicative thinking. Act two, would proceed as Ryan indicated with the pouring of the liter into the jug. Estimating how many liters will fill the jug. And wondering how many glasses could be filled from a jug. Here is where the double number line would be valuable as the next step of representation. (top line marking liters, and bottom line marking cups (1:4). Act three would show the six jugs being poured into one container alongside a carton (=250ml/glass). A composed unit ratio table would be appropriate:
 This reply was modified 1 year, 8 months ago by Jeanette Cox.
 This reply was modified 1 year, 8 months ago by Jeanette Cox.
 This reply was modified 1 year, 8 months ago by Jeanette Cox.

This course has been eyeopening in many ways. As I mentioned in my introduction, my work over the past two years has been devoted to writing math strategies for a program designed to serve the whole child creating teaching effectiveness, efficacy, and equity. Through this process I found the Mathematics Knowledge Network (KNAER) report and cited this in the Background Knowledge portion of Fractions Grades 35.
In 201112 the KNAER project highlighted five ways of thinking about fractions: as linear measures on a number line, partwhole relationships, partpart relationships, quotients, and operators (fraction as operator, as in 1/5 of 3 affects the ability of students to generalize and to work with unknowns, both of which are fundamental to algebra). Another area of importance is developing the concept of a â€˜unitâ€™ fraction to define a fraction. A fraction is considered a multiple of a unit fraction: â€˜One onethird and two more onethirds gives us three onethird units.â€™
Now I know why this was such an important inclusion. My only regret is that I would have had this course before writing this strategy. Also, I see where you are one of the contributors. No wonder this is outstanding work. https://www.knaerrecrae.ca/index.php/fr/knaernetworksmain/mathnetwork

Like Marianne, I am beginning to internalize the differences between the two different ratio tables–multiplicative comparison and composed units. Thank you for the realistic example of the Uncle Ben’s rice scenario and the importance of authentic uses of math in daily life. Love the engaging videos which lead to noticing and wondering which hook the learner. Then moving to the importance of estimation which becomes a talking point for students to share their thinking. All this is what good teaching looks like and sounds like.

I concur with Taneshaâ€™s and Marianneâ€™s comments regarding the richness of the coffee mug task as it addresses an array of objectives from multiplicative comparisons, fractional reasoning, fair sharing also known as partitive division as well as ratios, rates and graphing on the x and y axis. It is in the quality of the problem solving tasks teacherâ€™s select that determines the depth of the mathematics learned. As a learner, I found myself thinking about my thinking as I processed the varied models of problem solving the relationship between cups of water to mugs of coffee progressing from the concrete use of linking cubes, to the area models of cups of water to mugs, to the vertical number line representing the composed units of measure and then transferring to the ratio table which visually lends itself to the vertical scaling up and down as well as the horizontal relationships which lead to revealing the rates, resulting in the linear proportional relationship leading to the algebraic expression where y = kx or x = ky which lends itself to graphing the points on a two variable graph like a scatter plot.
I have benefitted greatly from these mindstretching scenarios.

Walmart is advertising a special for Irish Spring Original Deodorant Soap 3 bars (2 pack) Total of six for $10.00.
The constant of proportionality is 3:5
I can scale in tandem up and down. In order to find the rate I scaled up to get two packs with 12 bars for $20. To find the bar of soap per dollar you partitively divide 12/20ths reduced to 3/5ths of a bar of soap per dollar. If you want to know the dollar rate per bar of soap you partitively divide 20/12 or 5/3 and get $1. 666 or round to 1.67 per bar.T
 This reply was modified 1 year, 8 months ago by Jeanette Cox.
 This reply was modified 1 year, 8 months ago by Jeanette Cox.
 This reply was modified 1 year, 8 months ago by Jeanette Cox.

Rate would be more useful when looking at functional relationships; while ratios are comparing ‘this’ to ‘that.’
 This reply was modified 1 year, 8 months ago by Jeanette Cox.

As a former 7th grade math teacher, I would select the Circumference vs. Height task primarily because of what I know now vs. what I knew then. Because of the cognitive psychologists’ learning strategies, specifically interleaving, which suggests that we learn best when concepts are revisited in different contexts, the Circumference vs. Height task interleaves multiplication, multiplicative comparison, measurement, spatial reasoning, geometry and the meaning of pi as the relationship between circumference and diameter. Rich task, requiring perseverance, inquiry, cognitive dissonance.

Since my current status is that of a math per diem and a math strategy writer for Pathways4Learning, the ideas from these problem based lessons are so inline with the research and closely resemble the lesson approach recommended in the Ratio and Rates document written for NC 6th grade. The approach to a lesson is: Launch, Explore, Discuss. See the attached link and page 7 for the Alien task. https://www.nc2ml.org/wpcontent/uploads/2018/08/RatioTManual.pdf

I would use the following learning objective from NC 5th grade: Analyze patterns and relationships.
NC.5.OA.3 Generate two numerical patterns using two given rules.
â€¢ Identify apparent relationships between corresponding terms.
â€¢ Form ordered pairs consisting of corresponding terms. 
Since I’m responding to this discussion question after watching the entire series, it would definitely depend on my objective and grade level as well as the 3 teacher moves: Knowing where my students are coming from; knowing where they’re going and anticipating what my students will do. Now applying those ideas to 5th grade, I would definitely look at multiplicative comparison and quotative division using the double number line.

Multiplicative comparison as I now understand, is comparing two like units such as the height of a building in relation to the height of a flag pole. Whereas a composed unit comparison is a unit represented by two different things like number of ice cream cones to cost: if two ice cream cones cost 4.50 what is the cost of 10 ice cream cones and then the rate would be the cost of one.

Thinking of the piano keyboard the following ratio problem represents a multiplicative relationship of black keys to white keys per octave. Then reframed the ratio to a composed unit by comparing octaves to black keys and then determining the unit rate of black keys to octaves.
 This reply was modified 1 year, 9 months ago by Jeanette Cox.


This task is from Illustrative Math http://tasks.illustrativemathematics.org/contentstandards/4/OA/A/tasks/356 . It is a great site for all grade levels. This task is for grade 4:.OA.1 Here is the task for students: There are two snakes at the zoo, Jewel and Clyde. Jewel was six feet and Clyde was
eight feet. A year later Jewel was eight feet and Clyde was 10 feet. Which one grew
more? Then it gives the “Commentary” for teachers: The purpose of this task is to foster a classroom discussion that will highlight the difference between multiplicative and additive reasoning. Some students will argue that they grew the same amount (an example of “additive thinking”). Students who are studying multiplicative comparison problems might argue that Jewel grew more since it grew more with respect to its original length (an example of “multiplicative thinking”). This would set the stage for a comparison of the two perspectives. In the case were the students donâ€™t bring up both arguments, the teacher can introduce the missing perspective. 
Absolute thinking is looking only at the numbers. Like a cup of coffee cost $1.00 and then it was changed to $2.00. The absolute change is the difference between the two prices or $1.dollar. However, if we look at the relative cost of the increase it would require us to look at $1.00/$1.00 or 1 whole = 100% increase in price.

Additive comparison problems ask: How many more ?or How many less? while multiplicative compare problems ask: How many times as much? or How many times as many?

Here is where using sentence frames play an important role in developing the language of math beginning early in K when students begin comparing two objects:
(_________) is longer than/shorter than (__________).
Additive comparison: (Blank) is _____ more than (blank) or (________) is ____ less than (_____).
Multiplicative comparison: ( A ) is _____ times more than (B). or
(B) is (fraction) as big as (A).

These lessons engage students on multiple levels of cognitive learning strategies and in particular this lesson engaged in the “new” area of discovery for me–dualcoding which simply means engaging multiple senses particularly visual with verbal and video with audio. Here is a link to these cognitive strategies if anyone is interested. https://lovetoteach87.com/2019/05/02/examplesofdualcodingintheclassroom/ The videos from Kyle’s “Tap Into Teens” as well as Dan Meyer’s and Graham Fletcher are all wonderful lesson examples which as a former k5 math curriculum coordinator, I would include in various grade level math talks for problem solving. All this information is reaffirming and renewing.

Love that you are mixingup addition and subtraction which is called “interleaving” under the 6 cognitive learning strategies. This is particularly effective in mathematics and has shown effective results in longterm memory. So thanks. Now to a problem situation:
A teacher bought 20 pencils for her class. One week later she only had 14 pencils left. How many pencils did she give to her students?
Result known and start known change unknown.
for some reason image will not post.
 This reply was modified 1 year, 9 months ago by Jeanette Cox.
 This reply was modified 1 year, 9 months ago by Jeanette Cox.
 This reply was modified 1 year, 9 months ago by Jeanette Cox.
 This reply was modified 1 year, 9 months ago by Jeanette Cox. Reason: trying to upload image of illustration of problem
 This reply was modified 1 year, 9 months ago by Jon Orr.

In the example of the teddy bear counters and the color tiles the child could line the Teddy Bear counters one under each of the tiles and notice that there is one more Teddy Bear than color tiles so 5 is one more than four preparing the way for subtraction. Or the reverse, there is one less tile than there are Teddy Bears 4 is one less than 5. This could be written on an anchor chart for the comparison language. ____ is ___ more than _____;
_____ is ____ less than _____. Important sentence frame for comparison problems.


On the tray place the teddy bear counters and the tiles. Which is more and which is less? How do you know? First by direct visual comparing and then counting using one to one correspondence and cardinality that the last number said represents the quantity. The direct measure is placing each object under a dot and then saying the last number named as the indirect measure of the numeral 4. Heirarchial inclusion can be seen as two red and one yellow and one green are the same as four. The dots provide a number path leading to using a numberline.

I’m connecting my newest research challenge to your amazing math task such as the Green Screen task which engages students in two of the six strategies for effective learning–concrete strategy–in which you link an example to the idea you are studying; and interleaving–which links more than one cognitive challenge together as in converting cm to meters and calculating area. A task for using the different stages of the continuum of measurement involves younger students and determining which weighs more or less an apple or a pear.
Direct comparison: Use a bucket balance and place apple on one side pear on the other.
Indirect comparison: Place apple on one side of bucket balance and place cubes in other bucket until they balance and count the cubes. Then do the same with the pear.
Direct measurement: Use sticks of butter that weigh 4 oz. to measure each.
Indirect measure: Use a customary hexagonal mass measuring set .

Piggybacking off of Heidi’s example is this one for volume. This comes from Lessonresearch.net
Direct Comparison: Which bottle holds more?
Indirect Comparison: Pour liquid into one bottle then pour contents of that bottle into the opposite one to see whether it overflows and in that case the first bottle holds more; or if there is still room in the bottle than that bottle holds more.Indirect Measure: Use a smaller cup and fill it and count the number of cups that each container holds. Container a holds 4 cups while container b hold 3.5 cups.
Direct Measure: Volume of a liter is equal to a 10 x 10 x 10 cm cube.

The visual illustrations of both partitive and quotitive (measurement) division were very helpful. As a former curriculum coordinator one of the highlights of my experiences was working with parents who were often skeptical about the “new” ways math was being taught. To make a point for conceptual understanding vs. rote memorization of an algorithm, I would ask them if they recalled how to divide fractions. Often they would come up with “invert and multiply and don’t explain why.” Then we would have some fun with pattern blocks to demystify the why. Using a red trapezoid and comparing it to the yellow hexagon, they would identify it as a half of the yellow. Then we identified the blue rhombus as thirds. Now, let’s see how that algorithm works by taking the blue rhombii to determine how many can fit in the red trapezoid. Here it is demonstrated on BrainingCamphttps://www.youtube.com/watch?v=g5aET6HLMeI To illustrate a measurement division problem we would ask how many green triangles does it take to make onehalf using pattern blocks? https://youtu.be/oewp2S51FW4
In researching technology as a tool to teach division of fractions Conceptua Math is awesome. Click on link and you can see a demo using the line model http://www.conceptuamath.com/mathtools/

Currently, I am writing k5 math strategies for a software program designed to help teachers identify and match academic needs in their classroom to strategies that address learning styles as well as social/emotional needs. Through this process, I’ve had to dig into the research behind specific major content standards and geometry, fractions, and measurement are among the biggies. This module hits the mark when it comes to backing up the research. Here is some background info that went into the geometry strategies:
Spatial reasoning is involved in geometry, estimation and measurement, use of diagrams, graphs, and drawings, breaking fractions down into geometric regions, or conceptualizing mathematical functions. According to the recent NCTM Connections Standard (2000), it is the responsibility of the teacher to: 1) make the connection between mathematics and other subjects, and (2) help students see and experience how mathematical ideas interconnect and build on one another to produce a coherent whole. Research provides a rationale for a shift in focus in mathematics education towards the development of spatial reasoning skills. Battista, Casey and other researchers have found a relationship among older studentsâ€™ spatial skills and mathematics achievement.

The object being measured is “BMI” or body mass index. The specific attributes being measured are weight and height. The units used to measure each attribute will be Kilograms or pounds to measure weight divided by the height in meters squared or inches squared times 703. Since the U.S. uses customary units, I will be using pounds divided by inches squared times 703. This could be a crosscurricular connections between math and Physical Ed./health. Since there is a high incidence of obesity among children in the U.S. this measurement task would be beneficial. Here is a link for finding BMI https://www.youtube.com/watch?v=4LfANanF0Dg

During the past year and a half, I’ve been engaged in writing math strategies for an education firm. One of the major areas of this work has been researching and compiling effective strategies for building conceptual understanding of fractions from K5. Through researching the progressions of fractional reasoning my go to references were, Battista, Steffe, and Olive. They identified six levels of fractional reasoning or trajectories of mathematical thinking required to develop the ability to reason proportionally. Generally, third grade marks the introduction to multiplication and fractions. Later in the elementary grades, students need to begin developing strategies for understanding fractions, decimals and percents by using manipulatives, drawings and diagrams. Division is the most important concept related to fractions, decimals and percents and all these math concepts require proportional
reasoning.Through the elementary years, students begin experiencing a shift in mathematics concepts from additive to multiplicative situations. Although multiplicative concepts are initially difficult for students to
comprehend, aâ€”
mathematics curriculum must not wait â€¦
to advance multiplicative concepts, such as
ratio and proportion. These principles must
be introduced early when considering additive situations. (Post et al. 1993) This was taken from: Tobias, Jennifer M., Andreason, Janet B., Developing Multiplicative Thinking from Additive Reasoning. “Teaching Children Mathematics,” September, 2013, Vol.2. Issue 2. pp. 1029 
When I saw the illustration of the two purple shapes, it reminded me of Piaget’s stages of cognitive development in mathematics, particularly the preoperational stage in which he took 3 containers of similar size and poured the same amount of water into each, the child thought that because the container that was wider had a lower level of water it therefore, was less because they were only looking at one dimensionheight, and not width. Students looking at the two shapes are only looking at one dimension, the purple partition when they say they are the same (absolute value) like Piaget’s example, vs. viewing them relationally. Great task for assessing and asking probing questions to move students to the next level of thinking.

I appreciate your feedback. So help me clarify for my own understanding. On the venn diagram I placed composed unit so that is merged in the intersect indicating that it lends itself to finding rate. Would that be correct?

Love what Ryan and Valerie stated. Now I’ll try my best to synthesize the two together. Rate is comparing two of the same units through partitive division which is dividing a number into measured groups of the same quantity while quotative–is finding how many of one unit is in another. (yikes! this is mindboggling) not sure I’ve got this right.
 This reply was modified 1 year, 9 months ago by Jeanette Cox.

I related to your example of grocery shopping. Yesterday I was looking for Arborio rice and found one container which through “direct” measuring looked bigger than the other container and was more expensive so I naturally thought it contained more but when I picked up the packages their net weight was the same. How my spatial reasoning fooled me. And your example of kindergarteners loving to play in the sand and fill containers reminded me of some research I recently did when writing the background information for geometry in K2. Here is what quoted:
It is important to note the importance of spatial reasoning. It is recognized beyond the limits of geometry, and the existing literature provides a firm basis for a conclusion that spatial ability and mathematics share cognitive processes beginning early in development (Cheng and Mix, 2014 p. 3; Davis et al. 2015; Jones and Tzekaki 2016; K12, 2014, p. 3). Spatial reasoning seems to become crucial at the very beginning of math education.

I agree that creating your “own” units and calling them by a name is a genius idea.
One of my favorite math literature books for measuring is, “Measuring Penny” by Loreen Leedy. She uses some clever units of measure such as qtips, etc. Here is the link: https://www.youtube.com/watch?v=kH1Qh6bgq0 Certainly gets students thinking about why different units are necessary depending on the attributes of a dog: ears, paw width, tail length, weight, how high they can jump etc.