# 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.

|
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.

|
Math.5.NF.B.4b

Common core State Standards

• Math:  Math
• NF:  Numbers & Operations--Fractions
• B:  Apply and extend previous understandings of multiplication and division
• 4b:
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

A Passion for Fractions
Lesson Objective: Multiply a fraction by a fraction
Grades 3-5 / Math / Reasoning
Math.Practice.MP2 | Math.Practice.MP3 | Math.5.NF.B.4b

#### Thought starters

1. Why does Ms. Pittard present students with a variety of solutions?
2. How does critiquing solutions help students develop an understanding of multiplying fractions?
3. What can you learn from Ms. Pittard about engaging all students?

Recommended (0)
Fantastic video! The teacher allowed the students processing time, think time, talking time and solving time. She guided without interrupting their work. I love the knowledge part that you want them to take away. Wow!
Recommended (0)
Thank you SO much for this video. You have an enormous class and you had them all funneled to focus and collaborate without giving them the answer! Amazing facilitation. You have inspired me to learn how I can better prepare my students.
Recommended (0)
Thank you so much for this math video. I'm a 5th grade teacher, and LOVE the ELA curriculum, but math has always been something that has been a little more difficult for me to explain to my kiddos. I will continue watching these videos to get ideas on how to better instruct my students during math!
Recommended (0)
The approach you used in the lesson kept me engaged through the video because I wanted to see what all the excitement was about. I truly love the time you provide for the students to think before sharing their thoughts, then allowing for transitions based on the conversations and body language from your class. This video will be in my go to model for math on Monday. Passion will always keep students learning and engaged. Thank you!
Recommended (1)

#### Transcripts

• A Passion for Fractions Transcript-Final Edited Program
Luna Productions
Program: Passion For Fractions
Teacher: Becky Pittard

Voice identification: Teacher in Classroom/Teacher

A Passion for Fractions Transcript-Final Edited Program
Luna Productions
Program: Passion For Fractions
Teacher: Becky Pittard

Voice identification: Teacher in Classroom/Teacher in Interview:
Student (for any/all student who speaks)

Teacher in Classroom: I am Becky Pittard, I teach 4th and 5th grade at Pathways Elementary school in Ormon Beach, Florida. Today’s lesson: we’re going to focus on a passion for fractions.

Teacher in Interview: We build the lesson around a story that the children were going to run our track here at school which is ¾ of a mile long.

Teacher in Classroom: After you run 2/3rds of that ¾ of a mile of that track we’re going to post a riddle and now we’re wondering how much of a mile is that 2/3rds of the way around the track?

Teacher in Interview: The students’ mission was to figure out what part of a mile is that 2/3rds mark.

Teacher in Classroom: Your job this morning is to look at these representations and find out which ones are accurate, they’re true, they’re good, they’re useful, and which ones are not accurate.

Teacher in Interview: Four different representations were given to the students because different people see drawings differently. So one representation might make sense to one child but the other representation might make sense to another child. So that was another reason for putting all four on one sheet of paper, giving them different entry points to understanding this idea.

Teacher in Classroom: What are you trying to find out? Give it to me simply? V___? What are you trying to find?

Student: We’re trying to find out how much the 2/3rds of the 3/4ths is. So we’re trying to find out where the riddle will be in the whole mile.

Teacher in Classroom: Ah, OK. So that is what we are trying to find out. What information in our story is going to help you find that out? Miss Courtney?

Student: Definitely the fractions, that way you know what numbers you’re dealing with.

Teacher in Classroom: OK, which fractions are …is that?

Student: The 2/3rds and the 3/4ths

Teacher in Classroom: OK, very good. All right then. This is going to be your think time alone.

Teacher in Interview: After we’ve analyzed the story and I feel that the children understand their mission with this story then I encourage them to use individual think time. I worry that children immediately start talking to their peers and one child does all the thinking and the other child does very little, so this individual think time gives each child time to think and reason before someone speaks to them.

Teacher in Classroom: What does the 3/4ths represent in our story? So when you go back to the story itself what is 3/4ths?

Student: That’s how much the track is of the mile.

Teacher in Classroom: OK, and what is 2/3rds? Why these …why was this colored in?

Student: Because that’s one you’re going to ….that’s when the runner completes the distance.

Teacher in Classroom: OK, good job

Student: I don’t think they times-ed it correctly.

Teacher in Interview: Several of the students went directly to the numbers and did not focus on the drawing. That often happens with intermediate students especially those who’ve had instruction in working with algorithms and working with numbers and not trying to conceptualize what is actually happening in the mathematics.

Teacher in Classroom: I just want to specifically know which part is wrong in that?

Student: I don’t really know how _____ 6/9ths

Teacher in Classroom: OK, then you want to go back to the drawing. Don’t let the numbers distract you. We’re really looking at the drawings themselves. Using your pencil show me what is one fourth of this rectangle.

Teacher in Interview: After the children have individual think time and I usually judge when to stop that because I will start seeing a lot of children who are stuck.

Student: I don’t understand it

Teacher in Interview: So I give the children the choice.

Teacher in Classroom: Let me give you the opportunity. If you would like to continue working by yourself you may do so. If you want to turn to people at your table, you may do so. Turn and talk about it with somebody at your table and see where you are.

Student: Because over here it has ____ four but it’s not split into three pieces like these three are.

Student: I still have one question …..

Teacher in Interview: I think many of us when we grew up we were taught to memorize steps and we were taught to memorize algorithms. This lesson is completely contrary to that because the expectation is for children to reason, to think, to understand.

Student: In this drawing the 1/3rd of the 3/4ths is equal to the 1/4th of the whole mile

Teacher in Interview: Part of the challenge in the reasoning in the math that we did today was that the children had two separate wholes to reason about and they were also reasoning within the number system we call fractions. That itself bumps up the complexity of their reasoning.

Student: So 2/3rds lines up right here and that’s ….if you take the whole mile is 2/4ths of a whole mile and 2/4ths is the same as ½ so that’s why I think it’s correct

Student: Do you understand that?

Student: Not at all.

Student: Courtney, do you understand the last one, how they did that ….

Student: Yes, I understand

Student: They went too far ____. They have 4 and then it says 3, but it keeps going

Student: What do you mean it keeps going?

Teacher in Interview: When you give them the freedom to engage their thinking, when you value what they do so they are empowered to think, I think the children love it and they develop that passion for doing mathematics because they see it as riddles.

Teacher in Classroom: Kiley, would you come up and talk to us about this first drawing. How does that represent the story?

Teacher in Interview: The first solution strategy, the drawing itself, was very simple. It split the mile into quarters, and then it simply counted three of those to be the 3rds of our 2/3rds, and so the answer became two of those 3rds.

Student: And so that’s where they should see the riddle.

Teacher in Classroom: Where is the whole? Where is the mile?

Student: Like right here

Teacher in Classroom: OK, then we said our track was 3/4ths of that mile. Would you point where is 3/4ths of that mile in the representation. OK. So we said that the riddle was going to be shown 2/3rds of the way around that 3/4ths. So where is 2/3rds, where is that riddle going to be?

Student: At 2/3rds

Teacher in Classroom: OK. Do you have other questions for Kiley? She’s explaining it very well.

Student: Emily?

Student: I got confused on what the stripes and the orange pieces meant.

Student: And these two they colored like that because at the end of them is where you would see the riddle

Teacher in Interview: When a student asked about the colors in the drawing and she was confused about the colors in the representation, that was a moment of ah-ha for me, that I didn’t understand what the children were thinking, that I had assumed that they would know that.

Teacher in Classroom: Then can anybody tell us what do the boxes that don’t …that are orange but don’t have blue stripes in them represent? B___?

Student: The only orange blocks, like blocks, yeah …They represent the ……They represent a part of the third, at the final thirds because the riddle is only on 2/3rds of them and there’s a last third to it. So you still need to include that one third.

Teacher in Classroom: Edward, do you think you can explain this one?

Student: The reason I think it’s wrong is because they counted up the three parts of each of the 3/4ths. And 6/9ths is not of a mile, it’s of the track. There’s actually three more parts here and if you do 3 times 4 it’s 12. It would actually be 6/12ths of a mile and they only have 6/9ths, and 6/12ths is also ½. They have 6/9ths of a mile, which is wrong

Teacher in Classroom: How many of you thought that representation was wrong? Whoa! That drawing is wrong. Edward did a great job of explaining, but I’m going to ask him to go that one more time and can you rephrase for us one more time, why is this drawing wrong?

Student: The drawing itself or the math?

Teacher in Classroom: The drawing itself. What is the drawing itself missing?

Student: It doesn’t show that this mile is divided into 3 parts, it basically shows like it’s not there at all, but when you say “of a mile” you have to take into account how big the total mile is.

Teacher in Classroom: So this person just forgot to think about the whole. What is the whole? One mile. They though the track was the whole, and that’s not our whole, our whole is one mile. So excellent reasoning ladies and gentlemen that said that this one was wrong. I want you to stop and look at all of those representations for just a moment. What is the same about all of the correct representations? Ashley?

Student: They all have like 3/4ths that are shaded in for the track

Teacher in Classroom: Absolutely. So they all acknowledge the length of the track. Aaron?

Student: They all show the one whole mile

Teacher in Classroom: _____?

Student: They all split into thirds and like with the end because they left the part that wasn’t the part with the track

Teacher in Classroom: OK, please turn to your template and let’s identify what piece of knowledge do I want you to take away from this lesson. I would like for you to take away the piece of knowledge that Jonah was alluding to. When you multiply fractions less than 1 you are taking a part of a part.

Teacher in Interview: When I summarized the learning for this lesson I included the phrase fractions less than 1 because what was happening today in the mathematics was peculiar to fractions less than 1 and we know that someday the children will multiply fractions greater than 1.

Teacher in Classroom: Are we going to do math all day, guys?

Students: YEA!

Teacher in Interview: It is part of my teaching to help children develop a passion about math and then a sense of responsibility about their own learning. Those two feelings go together. They’re important for children because it makes life more interesting to care about what you do and to solve puzzles, but also it will contribute to their life. I believe they can find more successful careers if they understand and have a passion for math.

(end)

#### School Details

Pathways Elementary School
Ormond Beach FL 32174
Population: 714

Data Provided By:

Becky Pittard

TCH Special

### Student Centered Civic Discussion & Deliberation

Grades 6-12, All Subjects, Civic Engagement

TCH Special

Grades 6-12, All Subjects, Civic Engagement

TCH Special

### The Importance of High Quality Discussions

Grades 6-12, All Subjects, Civic Engagement

Teaching Practice

### Three Ways to Encourage Student Collaboration

All Grades / All Subjects / Collaboration

TCHERS' VOICE

Class Culture

TCHERS' VOICE

Lesson Planning

TCHERS' VOICE