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Lesson 4-1: From Spatial to Counting Comparisons – Discussion
Create a context with two quantities. How might a student who is counting describe the comparison?
Share your reflection below along with any wonders you still have.
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34 Replies
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I never thought of how kids moved from the concrete with dinosaurs and the additive thinking of concreteness fading to move to tiles for the linear path which leads to a number line. Our elementary teachers don’t introduce a number line until 3rd or 4th grade and I think that is a mistake.
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When giving a child 2 differing quantities, adults often seem to ask, “Which is more?” So it’s not surprising that kids use the word “more” more often than “less”. It was interesting to me when I heard the statement that this reflects in student comfort working with addition/subtraction, and multiplication/division later on.
The word “less” doesn’t seem as common in everyday language as “more” is.
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There are 4 geese swimming in a slough. 6 more come to swim. How many geese are there altogether?
For a student who is counting, they would probably take 4 “counters” and put another 6 with the first 4. Then they would count 1,2, 3..etc up to 10
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Middle school students come to my math intervention class with so many gaps. All of these lessons and videos are just really stressing the importance of giving my students these experiences and opportunities;
I absolutely have students who just need the experience of counting objects. Given 100s of items, what strategies do they implement to count? how do they keep their count?
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Yes, we can all relate!
Are you asking which strategies they should use or where to begin? Just want to clarify.
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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.
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This is what the counting looks like.
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This reply was modified 2 years, 3 months ago by Jeanette Cox.
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I’ve never seen this dot counting lining up with individual object counting. Love this thanks for the image to make it visual.
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This reply was modified 2 years, 3 months ago by
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I had not thought of comparing as being “this one has this many less.” My students are taught to talk through everything and I’ve heard them say this but didn’t really realize it’s importance. I can see how making sure students realize this is an important step in the process.
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Glad to hear we’re all learning something new every day!!
It is so easy to hear something in class, but not immediately see the relevance. I know I missed these things for the majority of my career :(. Wish I could get those years back, but I know that they were necessary to get me to where my thinking is now…
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Jan has 7 candies. Her sister gives her 4 more. How many candies does she now have?
7 is (a quantity of) 3 and 4. I know that 4 plus 4 is 8, and if i count on 3….9, 10, 11.
I find that 11 is the total number of candies.
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Love how you’ve broken down how a student might work through this problem through hierarchical inclusion and composing/decomposing number! Nice job!
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Since we have been using currency in my Grade 5 class. This video inspired me to compare unlike coins to ask which is more or less. I would line up pictures of each coin. This is would be great activity before working with place value.
Which is More or Less?
a) dimes: 10 – 10 – 10 – 10 – 10 – 10
b) quarters: 25 – 25
Change up the types of coins, from more or less move to a number line comparison, then ask how much more or less.
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Solid! What a great way to leverage making connections across concepts. Sort of spiralling ideas together. Nice job.
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In my interventions I have been working a lot with dot cards with my students to build up their mental bank of numbers. I would flash a card like the one attached:
They would say the total and then 2 and 7 make 9. An additional prompt sometimes would be, “How many more to make 10?”
Other times I would show them some dot pattern cards and ask, “How are they alike? How are they different?” or “Which one Doesn’t Belong?”
Students may say A does not belong because it does not have a group of 5. If they are having trouble seeing that A and B have the same quantity, nudging the kids to make an a measured comparison, by lining them up with counters, may help them. Then I can ask them, how many less does A and B have than C. With these images, they tend to not need to line them up though, especially when comparing B and C because they are “measuring” using the dot shapes.
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This reply was modified 2 years, 2 months ago by Aaron Davis.
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Great thinking here. It can be really hard to try and preplan the purposeful questions and pivot in the moment based on what they say (or don’t say). You’ve set yourself up nicely here.
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This reply was modified 2 years, 2 months ago by
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Problem: Your friend has 5 candies. You have 8 candies. Who has less candies? I could see students lining up the candies to compare in a straight line. Older students might make groups and take candies from each other to see who has candies remaining.
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Great thoughts.
I love how we can make a connection to zero pairs and integers when students basically “cancel” candies to see how much of one group is left over.
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In thinking about how I could relate this to my 8th graders, I would create a story about two kids collecting candy during a night of trick or treating. The first kid got 5 pieces to start from their parents, then another 4 pieces at the next house and 4 at the next and 4 at the one after that one. The second kid got 3 pieces from his parents but got 6 pieces from the next 3 houses he visited. Demonstrate with visuals who has the most.
This can be done in so many ways visually to demonstrate the counting which could lead to discussion of patterns and how determine who has the most. Counting would be used and comparison. After going through this part, we could have a discussion about the relationships and how it could lead to future predictions which is 8th grade focus.
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Love it. Contextually rich and asking students to model thinking is really important!
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If a student had $10 in bank and he or she withdrew $12, what is the balance? Something like this is taught in the upper grades for students to make connection with borrowing money, debt, etc.
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In a bowl of jelly beans, we might ask a student to count her favorite color jelly bean and then count her least favorite. Student will say which one has more.
This lesson resonated with me in a big way around the discussion of more and less comparisons and how we tend to focus on the “positive” more. You know what, though?
Less is so much more in life!!!! Less pollution, less over-crowding, less disease, less noise, etc, etc. It was interesting to me that this idea of how we preference “more” over “less” then leads to a preference for addition over subtraction and maybe even multiplication over division. Makes me realize that we need MORE educating and LESS indoctrinating!!!!
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Over the years working with students in intervention groups, many upper elementary, and even some of my sixth graders still use tally marks to help themselves with computation. It really shows their need for more activities to count with concrete objects and notice and create patterns. An example activity might be to use dominos to pose questions to students comparing each half of the domino. Which has more? Which has less? How many more? How many less? How many altogether? How many more to make ten? How many more to make twenty?
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Awesome examples. Thanks for sharing!
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I think that this is an issue with my Middle School students. Early education is so important. It sets the very foundation. I think I need to lower my floor for my students so I can bring back some of this sense making and not have my student lost because they never developed this foundation.
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A man went to the lumber yard and bought a board. He asked them to cut it for him. One piece was 2.7 meters and the other was 2.07 centimeters. Which board was shorter? how do you know? For my sixth graders.
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This reply was modified 1 year, 4 months ago by Anne Sheeter.
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This reply was modified 1 year, 4 months ago by
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I’m thinking of a ‘game’ style comparison using pairs of dice. I can see having students roll the dice and discuss the idea of more or less. This can be ramped up by using different sided dice. (D10, D20, etc.). It will give them hands on and make it fun.
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If my students subtract 13 from 8, they start by using red and black counters and they line up the zero pairs. Then they count to see that 5 red counters remain. We then use number lines but eventually want to make the jump to the abstract, but I still have students who count up on their fingers from 8 (or who count backwards from 13, depending upon the student.). How do I help them make the jump from counting to additive thinking?
Also, I was interested in the idea of using number tiles before transitioning to number lines. I teach middle school and hadn’t seem that but I do have a few students who don’t seem entirely comfortable with number lines.
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Moving from counting to skip counting is a helpful step. A number line as a tool can be helpful to skip count by 2s (or more) and to begin building the ability to “see” more additive relationships. This can only be done by suggesting / nudging students towards it though. The tool is the key to enabling them to get there…
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As many of the people responding to this post, I was struck by the more vs. less language as well. I have often wondered why thinking through subtraction and division was alway harder for students than adding or multiplying. This is much clearer for me now.
Students might compare quantities of stickers or erasers by lining them up with 1-1 correspondences. Actually this reminds me of how my mom and I count the score when we play “Ticket to Ride”. We compare similar trains so we only have to count the ones that are left over. I don’t know if this is an exact example using mathematical strategies, but it reminds me of that! Perhaps we need to move beyond comparison and use another strategy… 😉
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28 points and 31 points
4 touchdowns with the extra point compared to 4 touchdowns with the extra point and a field goal
Football–the American version
Can that still be counting if it is a student who really loves the game but not multiplication? I feel like I have some boys who don’t recognize 4 times 7 but would immediately see that 28 points is 4 touchdowns with the extra point. I see them counting 1 touchdown and extra point = 7 points.
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Yeah I see teachers using ten-frames, for example, to have kids count the dots then make the addition sentence for making 10. (3+ __ = 10).
I think the two go hand in hand and whenever you’re making simple addition sentences, the corresponding subtraction equation should be included.
Kids can see there were 10 but then 7 got taken away, etc. and I have 3 but I need 7 more to make 10.
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I like the “Which One Doesn’t Belong” tasks with a group of coins in each quadrant. Students have to count coins as one way to compare the groups and then they start putting unit values to coins as they compare.
