No Series: Related Problems: Reasoning About Addition

Math.Practice.MP3

Common core State Standards

  • Math:  Math
  • Practice:  Mathematical Practice Standards
  • MP3:  Construct viable arguments and critique the reasoning of others.

    Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and--if there is a flaw in an argument--explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Download Common Core State Standards (PDF 1.2 MB)

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Math.Practice.MP8

Common core State Standards

  • Math:  Math
  • Practice:  Mathematical Practice Standards
  • MP8:  Look for and express regularity in repeated reasoning.

    Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Download Common Core State Standards (PDF 1.2 MB)

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Math.1.OA.B.3

Common core State Standards

  • Math:  Math
  • 1:  Grade 1
  • OA:  Operations & Algebraic Thinking
  • B:  Understand and apply properties of operations and the relationship between addition and subtraction
  • 3: 
    **Apply properties of operations as strategies to add and subtract. Students need not use formal terms for these properties. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6
    • 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 =
    • (Associative property of addition.)**

    Download Common Core State Standards (PDF 1.2 MB)

Related Problems: Reasoning About Addition

Lesson Objective: Compare and contrast a related set of problems
Grade 1 / Math / Argumentation
6 MIN
Math.Practice.MP3 | Math.Practice.MP8 | Math.1.OA.B.3

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Discussion and Supporting Materials

Thought starters

  1. What kinds of problems work best for this activity?
  2. How are students encouraged to learn from each other throughout the lesson?
  3. What do student learn from analyzing related problems?

6 Comments

  • Private message to hazel outley
Absolutely love this. Excellent lesson. The "private think time" is a good way to maintain a quiet atmosphere. Reviewed prior days lesson. Presented the day's lesson and allowed "turn and talk" followed with sharing with whole group. Students were able to step forward to explain their reasoning. Presented with a challenge and were able to successfully predict the outcome. Wrap up included a review.
Recommended (0)
  • Private message to Shazia Qazi
My mentee and I watched this video and we loved the different forms of interaction of the teacher and the students. We also noticed that the students could use the strategies with different yet similar concept problems!
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  • Private message to Kristin Hutchison
I love that this teacher listens to his students while they have turned and talked. This seems to drive his questioning and instruction! Great formative assessment!
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  • Private message to Bridgett Carter
Awesome!!! I will do this at home with my grandson.
Recommended (0)
  • Private message to Randee Scott
Great teacher! Awesome talk time with the first graders. I'm going to implement this lesson next week. Thank you! ?
Recommended (0)

Transcripts

  • Reasoning About Addition Through Related Problems Transcript

    Speaker 1: Ladies and gentlemen, who remembers what we did yesterday when we looked

    Reasoning About Addition Through Related Problems Transcript

    Speaker 1: Ladies and gentlemen, who remembers what we did yesterday when we looked at some patterns and numbers? And then we came up with a rule or an idea that we had about the pattern we saw. You folks remember that?

    Children: Yeah.

    Speaker 1: Yeah. The mission of today's lesson was to get kids to begin to reason and justify arguments.

    It's really, really helpful if you do private think time so everybody has a chance to think about it first.

    They're looking for similarities and differences between the problems sets. Also identifying patterns, reasoning through that. Really the goal is to get them to share, to talk out loud and to explain their thinking.

    Ready for the next one?

    Children: Yes.

    Speaker 1: Okay. Santiago.

    Santiago: Eleven. Eleven. [inaudible 00:01:02] eleven!

    Speaker 1: I have one more for you.

    Children: Eighty. [crosstalk 00:01:08] Five plus seven equals ...

    Speaker 1: All right, McKenna?

    McKenna: Twelve.

    Speaker 1: All right. Can I just get a thumbs up if you've noticed something about those numbers up there? Taking a look at them? Okay. You turn and talk to the person next to you and share what do you notice up here? What do you notice?

    Children: [crosstalk 00:01:27] Ten, eleven, twelve and five, six, seven. Cause it's the same in the rolls. One and five are the same.

    Speaker 1: You notice in the rows they all are the same? Cool.

    What do you noticing about the ones?

    Children: I noticed that the next one's going to be the one and the next one's going to be the five.

    Speaker 1: So you're already thinking about what the next one's going to be.

    So, in terms of noticing. What are we noticing up here? I heard some good things. Juliete what did you notice? I heard what you shared.

    Juliete: I noticed there that there is a lot of fives.

    Speaker 1: A lot of fives. Look at those. All right. Leah, what are you noticing?

    Leah: I notice a lot of plus and equals.

    Speaker 1: Plus and equal. Woo. Great noticing everybody.

    Christi: I notice that starting by ten, eleven, twelve.

    Speaker 1: Okay. Interesting. So Christi just noticed a ten, eleven, twelve.

    The students did a really nice job of beginning to notice different things. Noticing the fives, noticing that they were adding. The equal sign. Noticing rows. And so I felt really good about that and then as they began to see the patterns it was really cool to see them begin to make sense of the problem.

    Children: Five, six, seven.

    Speaker 1: So what number do you thinks going to go here Kim?

    Kim: Eight.

    Speaker 1: Let's try that. If we were to do that, is there anything else we could predict up here?

    Children: Nine.

    Speaker 1: Lawson?

    Children: Nine.

    Speaker 1: Okay, so I hear nines.

    Children: Five. Five.

    Speaker 1: If I was to add this, what would be the next number that goes there?

    Children: Thirteen. Ten, eleven, twelve, thirteen.

    Speaker 1: Oh. So you're using the pattern ten, eleven, twelve, five, six and seven.

    A string of related problems is where the students are looking to see a rule or a pattern that's occurring so that they're able to justify a reason through that and then generalize that as a rule.

    One of the things that you've noticed, that several of you have noticed, is that this is staying the same. These are all fives. And then what's happening here? We're adding one.

    Children: Adding one.

    Speaker 1: And then what's happening here?

    Children: Adding one. If it was nine plus five it would be fourteen.

    Speaker 1: Okay.

    Children: And then five plus ten would be fifteen.

    Speaker 1: So if we're thinking about this, if this stays the same and we add one and then our answers changing by one, or the totals changing by one ... If I told you five plus twelve equals seventeen, what is five plus thirteen?

    Children: Eighteen.

    Speaker 1: Woah. No way.

    Children: Yes. Yes way. Yes way.

    Speaker 1: Christi, how did you do that?

    Christi: If five plus twelve equals seventeen then five plus thirteen is equal eighteen because thirteen is after twelve.

    Speaker 1: Thirteen is after twelve. Thirteen is one more than twelve. Can we say that? Thirteen is

    Children: One more than twelve.

    Speaker 1: That takes us back to here. One more. If we add one more. What's happening?

    I think that this type of learning with talking to each other, listening to each other, sharing their thinking transfers to all areas of school. It's not just math. They really feel empowered by the tasks that were in front of them today. That's really what's at the heart is when you have those moments. When that student feels confident enough to get up and share with the world who they are and what they're thinking. And their thinking matters and that you can create a space where their thoughts are validated and important and heard. Really, really just good for the soul.

    Ladies and gentlemen, thank you so much for your thinking. You did a marvelous job of thinking about this problem and looking for patterns in everything that you noticed. I was really, really impressed how you noticed that they were the same and how things were moving by one. We'll have to figure out a way to write that down so that we can continue to come back and revisit it.

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