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Lesson 5-2: Applying Different Types of Comparison – Discussion
Create a new context or revisit a context you’ve shared previously involving the comparison of two quantities. How might a student who is using multiplicative thinking describe the comparison? How does this language differ from a student using language of counting or additive thinking?
Share your reflection below along with any wonders you still have.
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23 Replies
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Years ago, when I taught elementary school, I used to do quick images with my 3rd graders. I am going to dig through my files and bring them out again not only because they’re fun and my middle schoolers will love them, but what a great way for students to express their multiplicative and additive thinking! What great discussions students will have– “I saw 2 groups of….” What an opportunity to explicitly point out the multiplicative and additive thinking. And then, to top it off, strategically choosing the images and then comparing the quantities to give students to play with the language when describing the images in relation to each other.
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Christina,
I use dot cards a lot with the interventions I have with 1st-3rd graders so the idea of using them in a different way (to promote additive or multiplicative thinking) is awesome. I often find myself trying to push the boundaries depending on the developmental readiness of the students and engage them even further with deeper prompts.
With that in mind, I can use a dot card with a group of 3 dots and compare it with a dot card that has three groups of the same group of 3 dots (so 9). With the right prompts, maybe responses could be, “B has 3 times the amount of dots than A” or “A is 1/3 the amount of B.”
Or is this a stretch?
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My fourth graders are working on powers of ten. I think it would be good to have them compare the numbers (10, 100, 1000) using a picture such as this. The place value chart is not helping some of them.
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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?
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While not a response to the question prompt, I just wanted to drop a note and say how helpful it is to see kids actually engaged in these tasks! As much as I love everything I’ve learned from MMM, this is often a missing piece I feel would help me exponentially in successful implementation with my own students. And kudos to Kyle for being willing to show us your own adorable children! 😄
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Glad you enjoyed it! The tough part is permissions for using students in the lessons – but it is something we are hopefully able to do more of moving forward! 🙂
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As I stated in other posts, I realized that I need to change a question and use actual items to help kids solve/create reasoning for their answers. In algebra we often have student graphing with slope and using factions. In order to help the students understand how to graph and why the line is moving up (or down) the way it does, I would bring out the paper like you used with your children to help the students understand why factions (less than one) take so long for the line to rise since they need more parts. If break it up into section such as squares and triangles, the students would be able to see why one slope is bigger than the other and why it rises at a faster rate.
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I wish there were a way to “like” other people’s comments because I am learning so much from the course but also from the comments as I see how other’s are applying the content.
The main reason I took this course was the dismay I felt last year when I was thrown into teaching Math 8 due to COVID and started to teach similar figures. I thought I could do a day review of proportions and then explain how it applied to similar figures, since proportions are Math 7 content in my district. My students had such a limited understanding of proportions that I had to revamp my plans for the week, but even then they continued to struggle, which impacted them big time when we hit slope.
I knew I had to do something different this year. Last year I started with equivalent fractions; this year I need to start with visuals just like the video shows. They need to see the relationship first before they just work with fractions.
I also love Brent’s idea of having them see the difference in slopes by comparing the triangles and squares made by different slopes. Let’s look at the triangle made by a slope 1/2 and compare it to a slope of 1/4. Let’s look at the triangle made by a line with the slope of 2/1 and compare it to the triangle with a slope of 4/1 and talk about rise and run. Definitely incorporating this into my classroom! Thanks!
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This reply was modified 1 year, 3 months ago by Lavon Heath.
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That is awesome to hear how impactful the learning has been – including the learning in the comments. Bravo!
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This reply was modified 1 year, 3 months ago by
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I’ve got 4 red jelly beans (which I enjoy) and 6 green jelly beans (which I don’t eat).
Unfortunately, I have 2 more of the green jelly beans than the red ones. I have 2 less red ones than green ones.
I have one and a half as many green jelly beans as red jelly beans.
My group of red jelly beans is two-thirds the amount of green jelly beans.
This is excellent practice – and I think I will keep going and thinking about area and length model contexts as well as the set model context I started with!
Thinking: HOW MANY? or HOW MANY more/less” are each a cue to count. That is additive thinking.
How many times more? or How much more? are cues to think of the relationship and to think of groups. That is multiplicative thinking. (Too bad we can’t call it “groupthink”! That would be more inclusive of the division concepts that are just as important as the multiplication concepts within.)
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Tia’s plant is 12 inches tall. Mike’s plant is 4 inches tall. How would you describe Mike’s plant in relation to Tia’s plant?
Mike’s plant could be iterated 3 times over Tia’s plant. Therefore, the height of Mike’s plant is one-third the height of Tia’s plant.
When a student uses multiplicative thinking, they are making use of the multiplication/division relationships between quantities to describe how one quantity is relative to another.
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Thanks for sharing this reflection, @nicole-jackson !
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This type of thinking continues to show up in scientific notation when you ask students how many times greater is the distance to Mars from Earth than the distance to the moon for Earth. My students struggle with this question because they do no have that multiplicative thinking come to them as naturally as it should.
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John earns $32,000 a year while Sarah earns $96,000 a year. what would you say about their salaries? ( You could get a discussion on why she earns more.)
Additive: He earns $64,000 less or she earns $54000 more
Multiplicative: He earns 1/3 of what she does or She earns three times as much.
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I also wish for language that didn’t emphasize the “more” situations, because naming additive vs. multiplicative thinking with my students further de-emphasizes subtraction and division, the poor cousins. I could see talking about part-part-whole thinking compared to proportional thinking instead.
Something I’m wondering about: I use “times” in the inverse direction, for example I would say that while the white square is 4 times bigger than the green square, the green square is 4 times smaller than the white square. Or I might say that wand B is 3 times shorter than wand A. I wonder if that language makes sense to kids or if it confuses them. We have talked a lot about tenths as 10 times smaller. We play a game where a student names an everyday object and then we all try to think of an object that is 10 times bigger and an object that is 10 times smaller or a tenth the size. Scaling up and down by 10 is so important to support their understanding of place value, so I spend time on that. Thoughts?
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Society consistently uses the “times smaller” language and it isn’t actually mathematically correct. I still catch myself doing it. In its place, I try to use “one forth the size of” language which again helps to promote that multiplicative thinking while also bringing in fractional thinking.
Great wonder here.
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I wonder about using decimals for the comparison to pull the powers of ten out of students…
0.8 compared to 0.08
Students could be given tens unit blocks, but since I have sixth grade they only get a small amount.
This could help to have them see the additive–lining up decimals compared to the multiplicative multiplying by 10 which is a movement of decimals perhaps.
I am not sure but I feel like we take this exploration to the back. Could it be because it is so hard to draw because of the powers of ten?
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I liked the activity covering the squares and am looking at where I might insert it into the near future. I know the students will enjoy it, but even more, I imagine it will highlight which students are thinking in what way, which is useful information when teaching this concept. Once again, the fractional language struck me and I realize that both I and the students could use this language more often to express relationships between numbers. I’ve been encouraging them to build their math vocabulary lately and to use specific terms to explain their understanding (always ensuring they actually understand what they’re talking about and not just spouting vocabulary. This is also a challenge sometimes!). Perhaps we need to focus more on the fractional terms soon!
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My current application of this topic is listed above, but I also love how Christine is applying this to place value. When I taught an intervention class, students didn’t really understand powers of 10, but I feel that helping them see proportions would give them a deeper understanding rather than just having them move the decimal.
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I spend a lot of time getting my students to be specific in their responses about math (verbal and written). It’s good to see that part of the understanding of math is closely related to the correct way to speak/write about it. I really like how this course is vocabulary specific in how it explains concepts. I believe that the students really benefit from that. Using sentence frames is a great way to help them practice.
There are so many connections throughout our curriculum where the concepts are intertwined that using many of these activities is super helpful to the students to learn a new concept, review previous concepts, and start to see where they connect.
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This reply was modified 1 year, 3 months ago by Kristin Snell.
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Glad you agree with the importance of embedding and using correct terminology with students. It can be hard at times for building that fluency with terms, however, it does certainly help build understanding over the long term. So worth it.
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This reply was modified 1 year, 3 months ago by
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The jellybean task I referenced earlier. How many green? The green compared to the whole? Is it 1/10? The whole of the jellybeans in 10 times bigger than the number of green jellybeans.
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My 4th graders are pretty familiar with whole number multiplicative comparisons, and generally think (are largely given opportunities to practice them). But I’m wondering about mixed number comparisons. Two and a half times taller, one and a third more cookies, etc.
I need to look into and think more about that idea.
