Series: AFT CCSS Math

Math.Practice.MP2

Common core State Standards

  • Math:  Math
  • Practice:  Mathematical Practice Standards
  • MP2:  Reason abstractly and quantitatively.


    Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize--to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Download Common Core State Standards (PDF 1.2 MB)

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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.4.NF.B.4b

Common core State Standards

  • Math:  Math
  • 4:  Grade 4
  • NF:  Number & Operations--Fractions
  • B:  Build fractions from unit fractions
  • 4b: 
    Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.


    a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).


    b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)


    c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
    <br />
    Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, 100.

Download Common Core State Standards (PDF 1.2 MB)

Multiplying Whole Numbers & Fractions
Lesson Objective: Extend understanding of multiplication as repeated addition
Grades 3-5 / Math / Reasoning
Math.Practice.MP2 | Math.Practice.MP3 | Math.4.NF.B.4b

Thought starters

  1. How did Ms. Spies create opportunities for students to share and learn from others?
  2. What did students learn by critiquing the teacher's solutions?
  3. How did Ms. Spies help her students further their understanding of multiplication as repeated addition?
35 Comments
My 4th Grade team will love this!
Recommended (0)
To tell you the truth~I had to stop and think about how I would solve this math problem. I came up with several solutions AFTER I saw the children engaging in conversation! Yes~summeritis! But I did figure it out! This is a wonderful activity! I always get inspired to be a better teacher by watching videos of teachers being awesome themselves! Thank you for this thought-provoking fabulous post!
Recommended (0)
Yeah, this happens to me all the time. I set up a lesson to encourage students to multiply and they inevitably solve the problem with repeated addition. I find that in subsequent lessons, if I bump the number values, students more often will use multiplication in their reasoning. So, in this case I'd pose the question, "And how much ribbon would I need for 10 garden boxes?" Hmm. 40 x 2/3 or 2-2/3 x 10? Do you think that that would nudge them to see it as a multiplication problem? (I realize that the second is multiplication of a whole number and a mixed number, but with their diagrams, they could do it. Don't you think?)
Recommended (1)
Brilliant! I also agree with Ms. McCrackin about being able to extend the lesson.
Recommended (0)
Theresa, I think that would be an excellent next step. Students spent a great deal of time working with partial products, arrays, and the area model for multiplication so the transition to a larger number may push their thinking. The 4th grade standard really stresses multiplying a whole number and fraction and seeing the relationship to decomposing the fraction into unit fractions (if not already). One thing I learned from this was to use a context that maintains the fraction. Using 2/3 of a yard was easily converted to 2 feet, not requiring the use of fractions in the calculation.
Recommended (0)

Transcripts

  • Multiplying Whole Numbers & Fractions Transcript
    Title: Amy class_V9 final offline edit-
    Runtime: 00:09:27

    Teacher in Interview: I'm Amy Spees, I

    Multiplying Whole Numbers & Fractions Transcript
    Title: Amy class_V9 final offline edit-
    Runtime: 00:09:27

    Teacher in Interview: I'm Amy Spees, I teach 4th grade gifted at Cypress Creek Elementary School and our lesson today had students multiplying a whole number and fraction.

    Teacher in Classroom: Alight now you know we started our square foot garden so I thought well, what can we try and do so we can identify those crops and have some little pizazz to it. Ribbons. Ribbons came to mind because then they can flow in the breeze…

    Teacher in Interview: This lesson was designed to follow students that have been working extensively with unit fractions, being able to compare fractions and equivalent fractions on a number line, they’ve had some practice working with addition with fractions as well. We wanted to try now and take that next step and see what they would do if a fraction quantity was repeated, and if they would build upon their prior knowledge and experience with multiplication and extend that to fractions.

    Teacher in classroom: I figured out that this scrap really was the perfect size for each corner, and it just so happens to be two thirds of a yard long. Sebastian, how many corners does our garden have?
    Student: Four.
    Teacher: Four corners, so we’ve got to figure out how much ribbon we're going to need for our garden so we can have some decorated corners to label our crops. So if you would, go ahead, and if you haven’t already lets add today’s problem to your math notebooks, don’t forget to record todays date, and I’d like you to spend some time just first on your own …
    Teacher in Interview: The initial flow of the lesson was to really provide students with a contextual situation where they had to try and struggle through and come up with a strategy. I gave them some independent think time first just to see what their initial thoughts on how to handle that problem were. The next crucial price was to have them discuss their solutions together giving equal voice to all.
    Student: I got actually two and one third, right, because there is one third left right here.
    Student: Oh okay can you explain how you did it?
    Student: Well I did my thing by drawing a number line from zero to 12 and I split it up into thirds which would be 12 divided by four equals three.
    Teacher in classroom: I also want to try and create an environment where students felt safe and comfortable even though their reasoning may not have been correct.
    Student: Then I landed on eight and I covered two of the thirds
    Teacher in classroom: And I would circulate and make sure that that kind of discussion was still going to be in a coaching situation rather than a punitive one.
    Student: And then the third ribbon would be this, and then the fourth ribbon would be here. Oops yeah, it is two thirds
    Student: It’s okay you just made a mistake.
    Student: Yeah, I did three feet and then two thirds of a yard would have to be two feet, and we should …since there’s one on each corner we have to multiply two feet by four, which is eight yards.
    Student: Two yards … two and two thirds of a yard.
    Student: You’d have to purchase nine
    Student: No you could purchase…
    Student: Three yards
    Student: Yeah about three yards
    Student: Yeah, okay three yards
    Teacher in Interview: The question asks them to decide how much ribbon was needed and many of them I think lacked the experience of buying partial yards or partial pounds and felt it needed to be an actual whole number. So several of the groups went ahead and tried to round to the nearest yard rather staying with the exact amount.
    Student: It’s three because um
    Teacher: Okay, three what?
    Student: Three yards, or nine feet, because a yard is three feet
    Student: Well we just wrote down that ribbon is only sold by the yard and one yard is three feet but you need like two thirds of a yard, which is 32 feet so if you multiply at two feet, or two thirds of a yard, by the four corners of the square foot garden, you get eight feet.
    Teacher: Tell me again why did you multiply two times four?
    Student: Because there’s four corners on the square foot garden.
    Teacher in classroom: The lesson ended up taking a slightly different turn because of student responses. My goal was to try and see whether or not they would make that link between repeated addition with fractions and our previous work with multiplication and whole numbers.
    Teacher in classroom: I wanted to ensure that we at least reached that point where students saw the repeated addition could be then be translated into multiplication.
    Student: We added two thirds four times for each of the corners
    Teacher: You added two thirds, four times, for each of those corners and you came up with?
    Teacher in classroom: I did make an adjustment towards the end when I recognized that no student’s solution actually showed that so instead of just providing another situation where they would use similar strategies with the addition I provided them with some cards showing possible solutions, and I told them that these were teacher strategies they had used to solve the same problem.
    Teacher: So let’s take a look at one at a time and tell me your thinking, so which one, how about Erin, do you want to start with yours.
    Student: It says four right here and the actual answer is three yards.
    Teacher: Why do you think they put a four there, what strategy were they trying to use?
    Student: An array
    Student: I think it was an array.
    Teachers: So if we setup an array what would this quantity be?
    Student: Four, the four units
    Teacher: Oh could that be the four corners?
    Student: Yes
    Teacher in classroom : This particular card showed an array model and at first students were kind of confused, they thought that the four was the solution and when I asked them, well does that look like any other strategy we had used in the past, they were then able to identify oh, that’s the array.
    Teacher: Did somebody else say eight thirds somewhere before if we looked at their charts? Look at team C, what do they do? When they added two thirds, plus two thirds, plus two thirds, plus two thirds?
    Teacher in classroom: The idea was to show various ways that multiplication can be represented and then connecting that to the fractions.
    Teacher: So this group is using?
    Student: Multiplication
    Teacher: Multiplication. And they did get the same solution as you all? Eight thirds which is the same as
    Student: Two and two thirds
    Teacher: Hm, so would this be a yes or a no?
    Student: Yes
    Teacher: That’s a yes, alright Ms. Erin?
    Student: This one is correct because they do add two thirds and two thirds, and two thirds, and two thirds and they have eight thirds right there.
    Teacher: How did they figure out the eight? Go ahead Erin
    Student: They did four times two equals eight
    Teacher: Why do you think they did that? Go ahead
    Student: Because if you cover up the three in the line, its four twos
    Teacher in classroom: One student recognized the repeated addition situation with two thirds, two thirds, two thirds, and two thirds
    Teacher: Why didn’t you include the three?
    Teacher in classroom: And she then said though, if we cover up the threes it’s really just four sets of two
    Student: When you do that, that’s multiplication
    Teacher in classroom: And that’s indicating that we can see the repeated addition being used at multiplication
    Student: Because when you add fractions it’s the bottom number stays the same if they’re all the same number
    Teacher in classroom: And that’s when their discussion pulled out that that numerator is determining the pieces we're dealing with whereas the denominator is the size that has been partitioned from the whole.
    Teacher: I love her thinking. She is saying we're only going to need to multiply the numerator, but why is that?
    Student: Because you’re not going to have to change the bottom because that’s the whole, you only have that many. You can change the top because you can have as many as them, as you need, but you have to keep the bottom the same because that’s as many as in one whole.
    Teacher: So that’s telling us how many pieces the whole has been divided into, as you said…
    Teacher in classroom: I think an important element during this transition time of common core is to recognize that not all of our students will come to us as prepared as common core would like. My students did not have as extensive work with unit fractions in grade three as they will in the future, so I did have to spend more time working with that to try and help get ready for this particular lesson.
    Teacher: Interesting, does that match then, the work we see underneath?
    Student: Yes
    Teacher in classroom: Students were able to then recognize, hey, wait that top number just tell us how many pieces we’re dealing with. The denominator is showing us how many parts that whole has been divided into. So we don’t multiply the denominator by that whole number because that’s just determining the size, and that really was a crucial moment in the lesson to have that explanation come out.
    Teacher: Why didn’t we multiply the four times three there?
    Teacher in classroom: Our next steps will be to really solidify what was learned from today and comparing those strategies so the students have a chance to articulate whether or not those indeed were correct. At this stage we're simply multiplying a whole number and a fraction. When they move onto 5th grade, then they’re taking a look at part of a part to where they’re multiplying a fraction by a fraction.

    (end)

School Details

Cypress Creek Elementary School
6100 South Williamson Boulevard
Port Orange FL 32128
Population: 787

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Amy Spies

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