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Lesson 5-1: Introduction to Multiplicative Thinking – Discussion
How does the language of multiplicative thinking differ from the language of additive thinking?
Share your reflection below along with any wonders you still have.
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26 Replies
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It’s the idea of one thing relating to another. The visual for this was fantastic. In 6th and 7th grade we might want to keep the visual here but also move it to the abstract through a ratio table like below. I can definitely see how this language is important and each move we make during a lesson can have a big impact on their future learning.
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The languague begins to differ when we start to unitize smaller groups to create larger quantities. I don’t know how often I’ve said “unitize” in the class, but it certainly earmarks a transition from additive to multiplicative.
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Even though quantities are still being compared, explicitly pointing out the relationship between the two is the key idea. ” ____times as long”, differs from “____ more than”; I think students immediately go for the additive thinking; I’m really excited about teaching them the purposeful shift into multiplicative.
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Additive thinking uses words such as “more than” or “less than”.
Multiplicative thinking uses the word “times” (if something is larger) and “of” (if something is smaller)
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Fantastic. Good short and sweet ways to keep the thinking organized in your mind.
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This is a great way to think of it. Well said!
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I think Vanessa’s definition is spot on.
I’m wondering now…my fourth-grade students can add and subtract but do not understand multiplicative thinking. When I say something is 10 times as big as or 1/10 the size of, they do not understand what I am saying. I’m thinking I need to use this visual and we need to back up a little. Thanks!
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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).
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I like these sentence frames! It’s perfect to help student build understanding of the connection between them. So many middle/high school students struggle with these concepts.
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Yes Jeanette, those sentence frames are wonderful. It clearly shows Vanessa’s explanation:
Additive uses “more than/less than”
Multiplicative thinking uses “times” and “of” because we are relating 2 objects to the size of one another.
When I set up enrichment tasks for the students at my school, I’ll be sure to include sentence frames like these. It is so important to develop their specific math vocabulary as early as possible.
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One of the biggest things was the visual of how you can convert back and forth with the reciprocal. In my opinion, seeing the relationship between these two vertically because of the visual/number line combination really made me see the relationship better. In other words, its the parts aspect of multiply that the bigger difference (in my opinion).
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The language of multiplicative thinking is in RELATIVE terms and taps into the operations of multiplication and division. (“three times as long as”, “one-half the length of.”)
In multiplicative thinking there is a reference/ relationship piece to the language. You are using one thing to describe another.
The language of additive thinking is in ABSOLUTE terms and taps into addition and subtraction.
In additive thinking you are using an absolute quantity to describe the attribute.
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The language of multiplicative thinking denotes number relationships and patterns between quantities.
Whereas, the language of additive thinking indicates greater or less than relationships between quantities, however, in absolute terms.
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The language changes to reflect the operation.
How much more than, how much shorter than etc. Comparing two numbers and asking how much you took away or gave to the other. Multiplicative language is how many times greater or what fractional part of something would give a sense of multiplication or division. Find the part versus how many parts.
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Love it! Nice work and good on you for getting it out in the open to help Solidify your understanding.
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I think the responses above have covered it. Additive thinking is in the absolute difference while multiplicative looks at relationships between objects. I love the use of the inverse for describing multiplicative inverses.
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Now we’ve hit upon the major struggle that my students seem to have.
One big difference is that multiplicative thinking is “relative.” I think students think of counting and additive thinking as very concrete (whether or not they actually are) but struggle when they have to think in a more relative term.
“The wand is 6 units long” seems pretty concrete. But, “The wand is three times as long as the other” is now a relationship that causes them to rethink their “counting” strategies. If they don’t successfully make this switch to a comparative relationship, I believe they get stuck here.
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One of the key differences is the shift from absolute reasoning to relative reasoning. Where you are comparing the magnitude of things with additive thinking, when you shift into a multiplicative sense, you rely on the relationship between one thing to another. This is the idea of counting something as ” X times of Y”, where Y is the new unit or other relative piece. It’s a pretty subtle and interesting shift!
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Having read the responses before mine, I thought what can I add?
My connection to what I have experienced in class. Many times I have asked students what relationship the numbers have when using multiplication or division concepts and not realized I truly was using a math concept or term because it is so subtle. Many of my struggling learners didn’t see a relationship….because it was relative and not absolute. Now, I understand why they didn’t understand. Oof!
I think I can work to visually get them to see the relationship because I have tools that can act as a step ladder to get my students into the concept. It is hard to see the missing steps when you can just “see” the relationship.
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This reply was modified 10 months, 1 week ago by Amy Johnson.
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I love that you’re “seeing” how you can work with students visually to help them make those multiplicative relationships draw out. Fantastic !
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This reply was modified 10 months, 1 week ago by
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Rather than comparisons by counting using words like more and less by themselves, we’re looking for how many times more or less. It was a good reminder to use fractions to when comparing smaller to larger models. I wonder if we need to also be using language such as equal groups when talking about multiplicative thinking?
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I agree that the above discussion is helpful. As I have worked with struggling students, I could see that they were using additive thinking but wasn’t sure how to direct them into multiplicative thinking. Now I can direct them by helping them use relational words such as “of” and “times,” and of course I will take them back to visual models.
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After reading the thread before I started my reply, I agree with all of it. It sometimes makes me wish I was still in the elementary setting so I could really push the visual/concrete representations of math. It is so necessary when trying to connect what they already know to the abstract of what I’m teaching in 7th grade.
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Earlier math is definitely a ton of fun when we are pushing for understanding using strategies and models. Never fear; there is a ton of visualizing we can do with grade 7 content as well! 🙂
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I love how well these concepts build on each other. And how well some of the ‘earlier’ concepts (additive thinking) introduce and expose multiplicative thinking. When I taught grade 2/3 I would show them where they were actually doing multiplication and show them the equation for it and it would blow their minds and they would get so excited.
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This goes back to the unit thinking in past modules. Three times as big is unitizing to the smaller wander. 1 to 3 so we can then say 3 times bigger. Thinking of groups instead of adding on units to get to the longer wand.
