According to the Paying Attention To Proportional Reasoning document, “The essence of proportional reasoning is the consideration of number in relative terms, rather than absolute terms.”
So, what does it mean to use absolute thinking vs. relative thinking? We’ll start to unpack these ideas in this lesson!
What You’ll Learn
- What it means to use absolute thinking versus relative thinking.
Action Item
Be sure to view the discussion prompt below and engage in a reflection based on the prompts. Sharing your reflection by replying to the discussion prompt is a great way to solidify your new learning and ensure that it sticks instead of washing away like footprints in the sand.
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Lesson 2 – Absolute Thinking vs. Relative Thinking – Discussion
What does proportional reasoning mean to you and why is it important?
Reflect on how proportional reasoning connects to your context by referencing the content, curriculum, and grade level of the students you teach.
Share your reflections and any wonders you still have.
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74 Replies
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I’m really digging the openness of the example Kyle showed in this video. All kids can stake a claim to either scenario and there’s no wrong answer. As the teacher, I like how I don’t have to “hunt” around he class for a correct response, rather I can focus on students’ arguments and use of context for their claim instead.
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Fantastic, @Patrick-Kosal!
That’s definitely the idea. I’m happy to see that you’re working through this course. It’s a doozie and I’m excited to learn alongside you throughout it!
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Being 8 weeks into distance learning with fifth graders, I’m burnt out. BUT then I saw the, ‘What shape is more purple?” slide and got excited. I can’t wait to show my students and know how to guide them based on their responses. This will be a great way to really make sure the last three weeks of distance learning are productive and fun.
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Fantastic to hear! Did you get a chance to try it ?
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I am an instructional support teacher for grades 1 – 6 plus will be teaching gr. 7/8 math in the fall. The delving into proportional reasoning is going to be a great thing for me to do! I hope to become better at thinking “on my feet” because I will have a better understanding of this concept. I have always been the person who follows the steps and comes up with the “correct” answer. The idea of being creative and having discussions and debates about how to find the “correct” answer is still fairly new to me.
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I’m so excited that YOU are excited! Yay!
I completely agree – you’ll be upping your content knowledge big time. Can’t wait to continue reading your reflections!
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As it so happens financial literacy is a new curriculum. The example on the investment is the type of thinking that the students are expected to do so your examples was such a fitting one to my grade level e.g., grade 7 math.
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Yes! In Ontario, Financial Literacy has merges from the old curriculum into the new math curriculum! There is so much goodness we can pull from financial literacy. Glad to know that you find this example helpful!
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I teach all levels of middle school math. I am wondering how many of my kiddos would respond to the purple problem. If they would think in absolute or relative terms initially and if they would be able to switch gears to see the other perspective.
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It’d be great if you tried it as a warm up just to get a sense of what type of thinking your students use by “default”. If you do, be sure to share here!
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Both scenarios offered nonthreatening ways to tap into the students’ prior knowledge and get a glimpse of their frame of thinking. If students have answers that reflect absolute thinking we guide them towards relative thinking.
I wonder how difficult of a task will this be.
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When I saw the illustration of the two purple shapes, it reminded me of Piaget’s stages of cognitive development in mathematics, particularly the pre-operational 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 dimension-height, 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.
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Oh this is a great comparative example and a way to make this type of investigation very tangible and concrete for students. This is a great comparison!
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I’m sure we’re going to get to it, but at this point I’m wondering how do we move students along the continuum of proportional reasoning …. if we have older students who are still using additive thinking, how do we manage to get them starting thinking multiplicatively?
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I am in the same boat, teaching math intervention to 6th, 7th and 8th graders. One thing that jumps to mind is using models to build these situations– like the picture illustrated. The $400 amount = 4 $100 amounts. Build the model, then explicitly point out it’s 4x as much because there are 4 of them. Lots of building and drawing, lots of discussions.
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Proportional reasoning is all about the context and developmental readiness of the student (or perspective). These tasks are great low floor high ceiling tasks because most children can take part in the discussion and defend their answer depending on whether they are thinking relatively or in absolutes. Often times it is easy to forget about the Mathematical Practices we are trying to develop in our students because we are so focused on teaching them the standards (unfortunately regardless of whether they are developmentally ready for it) but by focusing on the MPs, like #3, “Construct viable arguments & critique the reasoning of others” students learn to be precise with their mathematical vocabulary and gain greater understanding of math concepts like proportional reasoning by hearing the different reasoning of their peers. Doing tasks like these two, anticipating the types of answers and misconceptions students will come up with, can provide us with important information about where the students stand on the learning continuum.
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Couldn’t agree more, @aaron-davis!
Glad that you see the value in teaching through problem solving and realize that math practices are much more important than the standards themselves.
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A big “Ah-ha!” moment I’ve had is this idea of additive vs. multiplicative thinking in my students. They come to me with very shaky multiplication facts, and where as the teacher, the multiplicative relationships jump out at me, I spend many hours lamenting, “If only they knew their multiplication facts, these relationships would jump out at them too!”
What I’ve come to realize is not necessarily their multiplication facts (or lack thereof) holding them back, but rather they are thinking in absolute or additive terms. They are using the reasoning they know and are sure about, and trying to cram this new way of thinking into it– it doesn’t fit, they’re frustrated, I’m frustrated, it’s a hot mess.
What I am really excited about is explicitly teaching the idea of additive vs. multiplicative thinking. Of using tasks like the “Which shape is more purple” to have these discussions with students so they become more aware of where their thinking is and more aware of shifting their thinking to the multiplicative.
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I’m in the same mindset of the “ah-ha” moment as I was watching this. Often, I feel that the lack multiplication is what is hindering my students ability but I must agree with you that it’s definitely more due to the fact of additive vs. multiplicative thinking. I’m hoping to build on these ideas for my grade 8 and 9 students for this year.
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So excited for your new realization. Please keep us posted on your progress with this!
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My neurons were definitely making some connections that I have wanted to make for a while. I absolutely love the ideas from “Which shape is more purple?” Many would say that they are equivalent which now I know exactly how they are thinking because many wouldn’t be able to articulate their thinking. As we move into the relative thinking, there would be a select few but many wouldn’t be able to tell me why. This group would be on the edge to fully understand the concept.
I can’t wait to bring these ideas forward when I am teaching fractions to my grade 8 students. So many don’t understand the reasoning behind having a common denominator and I have been trying to figure out a concrete themed to help them understand.
I’m currently pondering how I want to start my grade 9 math course in about 7 weeks from now. I know I want to build relative thinking and figure out what tasks would be best for it. 🧐
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Excited for the new learning you’ve built around additive and multiplicative thinking! The best part is that we are using one or the other quite often without even realizing it.
Some fun places to play can be percentages… the language we use can be additive or multiplicative…
Increased by 10% or a 110% increase is a land where this thinking is happening back and fourth often times without us realizing it.
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That’s interesting. I never thought about that when looking at percentages. Sometimes I think I am stuck in additive thinking. How empowering to understanding the “name” of the thinking and how that framework can help us see the difference between the concepts.
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Yes! Empowering is a great word to describe it. Being able to better notice and name different types of thinking can give you opportunities to ask students purposeful questions that van nudge them towards multiplicative thinking.
Thanks for sharing!
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I noticed that in this example, it is suggested that when a $100 investment grows to $400, this would be considered a 400% return. From an investment perspective, the amount gained from the original investment is $300, which would be a 300% return. I love doing stock market challenges to help my students better understand this important concept.
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Good eye! The key is always in the language.
If we are thinking multiplicatively, we’d say 400%. If it is additively, we’d say “300% gain”.
More on these two types of thinking later in this course!
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I was thinking the same thing, Greg. Then I saw Kyle’s reply and I’m looking forward to the deeper dive. I do find myself telling the kids to pay attention to the clues in the problem because math is a very precise language and situationally dependent.
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I can’t wait to try the which is more purple and investment question with my high school students. This is such a great quick assessment to know for sure where each is in their thinking ability and helps me to know where to go from there.
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Let us know how it goes!
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I can’t wait to share it with my class as well (8th).
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I love the question about investment. It really makes students think, discuss and justify their thought process. I would like to use this with my 8th grade students.
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Be sure to keep us posted on how it goes. Photos of student thinking are always welcome 🙂
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I feel like every teachers needs a reminder to focus on relative terms. I say this because if think in this manner, I believe that we as teachers would be better to help students and understand the concept. I believe when I think in a relative manner, I teach better and I am able to demonstrate the concept or issue in a variety ways.
I enjoy seeing the examples shown in this video and hope to use this in my class and others like that for future lessons. As one said, I believe I can look at students answers and can help understand the problem as I am trying to promote relative thinking not an precise answer.
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So true. Also, to help us keep that relative thinking front of mind, we must be able to notice and name the difference between relative and absolute thinking. That way, we can catch ourselves before we share our thinking and also plan our prompts strategically.
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I’m glad that I’m learning a name for what I try to teach my kids about math. Math is very situational, or as I’m learning, relative to the situation. I feel like too many of my kids look at each problem in absolute terms and are only seeking an answer. I feel like by looking at in it in relative terms, they would be more invested in the process and therefore able to transfer their thinking to new situations.
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I’m a 6th grade math teacher, trying to motivate virtual learners and draw them into discussions. They much prefer being silent observers, left wondering why they aren’t doing well in class. I like the thoughtfulness of these activities. They are less threatening since they seem to ask for an opinion instead of an answer that can be right or wrong. Middle schoolers do not like risking being wrong in front of their peers. Great way to draw them into thinking more deeply about relationships between numbers.
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Takes a lot of time and community building to create that trust! Keep at it!
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My friend had used the second example in our slides for our students and I hadn’t watched this lesson yet. I was able to put together the multiplicative relationship but now having more background knowledge I feel like I would be able to better guide my students to that relative thinking.
The interesting part is that I did have a student when I showed the example initially who asked if the money kept growing or if it was a one time scenario. It led to an awesome discussion on why it would have mattered and he ended driving the discussion toward the relative thinking.
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Fantastic!
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I liked both the examples in this lesson which allows students to think about open ended scenarios leading to higher level thinking without looking for correct answers. Students can think freely and justify their responses. It can also help us as teachers to see where their thinking is and we can lead our students to dig deeper into discussions based on their responses.
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Great take away! Keeping things a bit more open certainly helps us better understand where students are at in their developmental journey!
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I enjoyed the two examples – more purple and would you rather (investment) – because they were visually strong and they reminded me of the non-numeric problems I have used to teach about proportional relationships/relative thinking. These make such logical sense!
(Non-numeric problems often deal with coffee or drink mixes with the same amount of water – one has more coffee or lemonade mix in it — and so one has a stronger coffee or lemonade flavor. I saw these in an NCTM book – Great Tasks.)
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I love these examples! I taught 7th grade for 3 years and recently taught 8th & Algebra. I used bar diagrams with the 7th graders and they made connections, it was a positive experience for them. I will be trying the investment example with my students… they love money and the would you rather site is awesome to get their brains moving.
I hadn’t thought of absolute vs relative thinking before but I am so happy I can see the difference now. This will help me to look for misconceptions and think as a student before giving my lessons.
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Glad to hear it! Being able to notice and name the difference is so important in order to identify where students are in their mathematical thinking and to help nudge them along.
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I teach 8th graders and the example of which one is more purple would even be great for my students. Some of my students cling to absolute thinking. They are very weak with fractions. I think this would be a jumping off point for discussions and maybe begin the journey toward relative thinking.
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We use Singapore Math which strongly emphasizes bar models from third grade up. I’ll be curious to see how my sixth graders respond to the purple problem with that background as compared to the student who does not have that background being new to our school.
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I see the difference in how students at differing developmental levels will attack this problem. Absolute thinking will be the immediate response for those students who are in need of more support to transition to relative thinking strategies.
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I loved both of the problems presented. As someone else mentioned, there isn’t necessarily one correct answer, so the kids won’t get it “wrong”. I teach math intervention to middle schoolers and I find that the less threatened students feel with the work the more likely they are to dive in and attempt to reason through it. I will be interested to hear what they will answer to the purple problem. I am adding it to my slides now for Monday. 😀
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Hi there! I just watched lesson 2 and I love the “would you rather”. I’m definitely borrowing that from! However, I have a question: you state in the video that scenario 1 where you invest $100 and it grows to $400 is a 400% rate of return. Since the $100 is the principal, isn’t this only a 300% rate of return? I don’t use that language when I teach this math, but I would certainly have said that the $400 represents a 300% increase from the original $100 (which is equivalent to saying that the $400 is 400% of the original $100). Do you disagree with that?
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This is a great wondering and a great example of additive vs relative thinking.
When you say “it increased by”, that is additive thinking language.
When you say “it quadrupled” or “4x’ed” or “it is 400% of its original value”, that is multiplicative thinking language.
When we talk about investments, often times people interchange the type of thinking without really knowing it.
So my take away from this is – as long as you know the language, you can describe the change in quantity in an accurate way.
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I’m glad to have a new way to think about the student’s who initially answer $500 to the investment question and refuse to move off that answer. Realizing now that it’s just a way of thinking and providing a way to approach and help them to a better way of thinking will be a help when I try this scenario in class.
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Love it. So glad to hear you found some value in the learning.
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I really like the concept of using this simple would you rather question as a first look at proportions in 7th grade. Before we even begin to consider proportionality and growth, this would be a great way to help students begin to think multiplicatively (or relatively) instead of absolutely. I have never heard of this differentiation between absolute and relative, and it’s going to help me recognize a block in students’ thinking that I’ve noticed before, but had trouble naming.
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Noticing and naming is so important. Glad that you’re feeling more confident to notice and name this thinking that can become a huge barrier for students.
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A few thoughts as I continue to ponder absolute vs. relative thinking: firstly, I’ve seen kids who know some multiplication facts but still don’t really understand multiplication. There is so much pressure to get those math facts, but how they think should really be the focus! And thinking about relationships in math is key – not just about relationships between quantities, but also the relationships among concepts (like connecting fractions, decimals, percents and ratios). Making those connections will literally build more neural connections and strengthen students’ fundamental understanding. Traditional math instruction has not emphasized relationships, but now we’re more aware of the importance of seeing and making connections. So seeing relationships between quantities could be part of a larger shift for our students toward thinking more relationally in mathematics!
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Great points here and I couldn’t agree more. Connections and relationships are the glue to helping students make sense of the mathematics and to allow them to better retain what they’ve learned.
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I really like how you can see the same concept in the different lenses of relative vs. abstract. I know from my own children in middle school, it seemed that math was so rigid and right vs. wrong, but this time of thinking adds in math flexibility to move around and see math in different ways. Elementary students need this so badly so that they are more flexible in problem solving later on with more difficult skills.
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I took a PD course last year that mentioned additive vs. multiplicative thinking, but I didn’t really conceptualize the difference. Seeing your examples helped me to understand that. In fact, I was able to reflect back on students I have taught who still divided in 6th grade by counting by 1’s rather than groups and see that they hadn’t made that leap past additive thinking. What would you do to help those students?
Also, I currently teach 8th grade math but have so many students who can’t reduce fractions to their simplest form. I had never identified that as a lack of ability to reason with proportions. I hope that I am able to get some more resources to stimulate this kind of thinking.
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This purple pic is a good example of why visuals are so important.
The dollar increase as well – it’s a great open ended question to give.
I’m a K-12 math coach and have used problems like this in many grades. It’s quite interesting to see the upper grade kids not know how to defend their answer, and to make them go back to a visual to defend their answers.
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So true. Also, by using “area” like in the purple prompt, it nudges towards multiplicative thinking over additive since there is no explicit quantity to add/subtract.
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Both of these problems with work great in my 8th grade class. It will definitely help me see where the gaps in understand fall. I know I have a lot of work to do, but I am afraid this will let me know just how much. I am excited to see what else you have to share.
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Yes, this will definitely help you assess where students truly are. While this can feel overwhelming, you should note that you cannot possibly “catch all students up” in a short period of time. This should make your life easier (not harder) as you now will know where students are and what next step you can take that will be meaningful for students (not for pacing guides or standard lists).
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I truly enjoyed the “Would you rather” question to help students understand fractions while making sense of them in a real world activity. I can’t wait to try it in class. Thanking you as well for the link to the would you rather math site.
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I like the attention to precise language. Communicating thinking clearly is a huge part of building the relative thinking. Saying that one is more purple “relative to the whole” is great.
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It is so true. I was in a class yesterday and I really noticed how imprecise this group of students were. When I pressed for units or to reference the quantity they were describing, it was blank stares. Over time we can help all students to have a firm understanding of the math they are doing through precise language!
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It means seeing things in tandem and relative to eachother vs. absolute and static
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I love how the examples validate the contribution of students who are thinking absolutely and those that are thinking relatively.
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So glad to hear it! The reality is that all thinking is helpful and it is efficient for different students at different stages. So we must value it and help them to progress from their current place to the next place in that journey.
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What a great way to describe proportional thinking, in terms of relative thinking. I’ve never thought about it that way before. That’s my first aha! Many of my students are stuck in absolute reasoning, usually additive thinking is there first go to.
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I teach 4th grade and we use ribbon diagrams to teach multiplicative comparisons. Ribbon diagrams in some programs. But we do not directly make the connection to comparisons like the more purple or the which investment is a better deal. I do love the openness of the tasks. In fact I copied the purple graphic to use as an opener in class tomorrow. Just starting our study of fractions, it will make a great icebreaker.
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Love these 2 open ended tasks. I co-teach college and career math and want to incorporate BTC. I can see the investment question as a BTC task. In this class we can take the time we need to explore concepts. Keeping a list of good questions that will engage all types of students. Will check out wouldyourrathermath as well.
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I loved the examples provided in this video. It was great to see the exact questions we could pose to our students and then a breakdown of visuals/explanations to use as well.
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I often use a warm-up in which I might show a rectangle constructed from different colored square tiles or a figure from pattern blocks and ask students to reason through what fractional value a certain color is (related to the defined whole). However, the “more purple” example given in this lesson has me eager to try it from this direction instead and see what observations and conversations it elicits.
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After this lesson, I have a clearer understanding of multiplicative thinking/relative thinking vs absolute thinking and how relative thinking is at the root of fractional thinking. In our 6th grade curriculum, Ratios, Rates and Percents is our first Module. I am wondering what knowledge of equivalent fractions students should come with to 6th grade. Does relative thinking reveal itself in previous grades, and it so, how?
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- Would You Rather? [WEBSITE]
