Number and Operations: Fractions (CCSSM)

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Common Core Content Domain - Number and Operations: Fractions

Contributors: Myuriel von Aspen, Erika Espinoza, Lina Gil, Angela Khai, Ann Kim, Brittany Kirby, Bernadette Saldana, Cecilia Shang, Jennifer To, Melanie Wong, Jessica Yen, Marissa Young

Contents

Meaning of Fractions

Common Core Progression for the Meaning of Fractions

  • 2nd Grade:
    • Students use fraction language to describe wholes (e.g. two halves, three thirds, four fourths) and equal partitions of circles/rectangles (e.g. halves, thirds, half of, etc.).
  • 3rd Grade:
    • Students understand two important aspects of a fraction: the whole and its equal parts.
    • Students understand that a unit fraction is one equal part of a partitioned whole with equal parts (e.g. the unit fraction 1/4 is one equal part of a partitioned whole with 4 equal parts).
    • Students understand that a fraction is composed of a combination of the same unit fraction (e.g. the fraction 2/3 is composed of 2 parts of the unit fraction 1/3).


Curriculum Examples

2nd Grade: Using fraction language to describe wholes and equal partitions of circles/rectangles

Describing wholes and equal partitions of circles/rectangles


Area models (both circles and rectangles) can be used to assist students in using fraction language to describe them in terms of wholes or equal partitions. When presented with area models like the blue ones here then, they are able to describe them as two halves (in relation to its whole) or state that one partition is half of its whole (in relation to its equal partitions).

  
Circle & Rectangle.jpg
Images were created using: eManipulatives (www.eduplace.com)


3rd Grade: Defining and specifying a fraction’s whole and equal parts

Defining the whole


When teaching fractions, it is important to explain that the whole of a fraction can also be defined as a collection/set of objects, not just a figure in a representation. Posing questions and utilizing diagrams like the ones here can help illustrate this concept.

  
Defining the Whole.jpg
Image from: Math Expressions, Grade 3


Specifying the whole


During instruction, it is also crucial to specify the whole in a representation. As illustrated in the diagram here, there are two possible answers to the question of what fraction is represented by a shaded area in an area model when the whole is specified in two different ways.

  
Specifying the Whole.jpg
Image from: Progressions for the Common Core State Standards in Mathematics (draft)


Clarifying the meaning of equal parts


Although the idea of equal parts may be intuitive (students can see when a shape is divided into parts and figure that its parts are equal because of their same size), the possible student misunderstanding of equal parts illustrated here must be addressed before moving forward in instruction.

  
Unequal Parts.jpg
Image from: North Carolina Dept. of Public Instruction: Instructional Support Tools for Achieving New Standards, 3rd Grade Mathematics


It is therefore important to clarify the meaning of equal parts as parts with equal measurements. Exposing students to different area representations of a fraction, such as the ones seen here, can help.

  
Different Area Representations - Equal Parts.jpg
Image from: Progressions for the Common Core State Standards in Mathematics (draft)


3rd Grade: Understanding a unit fraction as one equal part of a partitioned whole with equal parts

Using visual representations to build conceptual understanding of unit fractions


Unit fractions (fractions with numerator 1) enable students to build their conceptual understanding of fractions. They come to understand that a unit fraction is formed when a whole is divided into equal parts and one part is taken (e.g. if a whole is partitioned into 5 equal parts, then each part is 1/5 of the whole). The use of representations such as area models and diagrams (e.g. circles, rectangles, number line diagrams, etc.) like the ones referenced here provides students with visual aids that help them identify the unit fraction of each whole.

  
Unit Fraction.jpg
Image created using eManipulatives: www.eduplace.com
Partitioning Fractions: Rich Task

Problem: You and your friend want to share a sandwich. You decide to cut your sandwich creatively, where you get the shaded pieces and your friend gets the non-shaded pieces. However, your friend believes you got more! How can you convince your friend that you each got half of the sandwich?

sandwiches

Launch

What can I do to make students want to solve this task?

  • Input student’s names

What information should I give at the outset? How should I provide that information?

  • Give students the cut-out pieces of the different sandwiches, tell them to represent the picture using the cut-out pieces
  • Manipulate the cut-outs to show half

During

What should I look for in student work?

  • How students rearranged the shapes to make half
  • With explanations and recordings of their responses

What misconceptions are likely to occur?

  • May not see all these representations as ‘half’
  • Mis-”rearrange” the shapes

What kinds of solutions might I want to share?

  • Multiple ways that half could be presented
  • Different manipulations of half

Post-Task

How will I structure the discussion?

  • Pair-share with the first, second, then third representations
  • Draw out students’ explanations of their reasoning
  • Have a student share their explanation + representation, then manipulate the shape and ask if it still shows half
  • Confirm whether half is still represented

What will you include in the discussion?

  • Extending the discussion: “How do you know it’s half (even though it doesn’t look)?

How will I ensure that the whole class is on board?

  • Extension: instead of using squares, students will show their knowledge using pie cut-outs

3rd Grade: Understanding a fraction as a combination of the same unit fraction

Using unit fractions to build fractions


Students expand their knowledge of fractions to include the meaning of fractions as combinations of the same unit fraction. Students realize that unit fractions are the basic building blocks of fractions, for combining a number of the same unit fraction makes a fraction (e.g. putting 3 of the 1/4s together makes the fraction 3/4). In this curriculum, students reinforce their understanding of the meaning of fractions by completing problems that ask them to identify the unit fraction and use it to write its related fraction, as seen in the problems displayed here.

  
Meaning of Fraction.jpg
Image from: Math Expressions Student Activity Book, Grade 3


Curriculum Resources

1. New Zealand Numeracy Development Project Book 7: Teaching Fractions, Decimals, and Percentages - http://www.nzmaths.co.nz/sites/default/files/Numeracy/2008numPDFs/NumBk7.pdf

  • This resource provides teachers with different strategies and activities that can be used to help students develop their conceptual understanding of fractions.

2. ConceptuaMath: http://www.conceptuamath.com/fractions.html

  • This website provides free fraction tools and sample lessons for teachers to help students explore the meaning of fractions and other fraction-related concepts.

Technology Resources

1. Cross the River: http://www.hbschool.com/activity/cross_the_river/

  • This website provides students with quick fraction identification practice. Students are shown different area models with shaded equal parts, and they must choose the fraction that corresponds to the area model.

2. Vectorkids: fractions: http://www.vectorkids.com/vkfractions.htm

  • This is another noteworthy resource for students to practice identifying fractions based on area models.

3. Bowling for Fractions: http://www.hbschool.com/activity/bowling_for_fractions/

  • This website allows students to reinforce their understanding of fractions by having them color in parts of a given whole and then input its corresponding fraction.

4. Find Grampy: http://www.visualfractions.com/FindGrampy/findgrampy.html

  • In this math applet, students are asked to find Grampy who hides poorly behind a rectangular hedge that is divided into equal parts. With the help of Grammy, students must input the fraction that corresponds to Grampy’s position behind the hedge in order to find Grampy.

5. Jelly Golf: http://www.fuelthebrain.com/Game/play.php?ID=215

  • In this interactive golf game, students are asked to move a golf club in a circular motion according to a given fraction so as to hit the golf ball. Incorrect moves lead to more golf ball hits and thus more practice with fractions.

6. Glog: http://www.glogster.com/ Johanning, J., & , (2013). Developing algorithms for adding and subtracting fractions. Mathematics Teaching in the Middle School, 18(9), 527-532.

  • A Glog is an interactive poster that can be accessed online for free for use in the classroom. Students are able to use this electronic resource to share and discuss the information of their topic as they process it. By using this Glog, students are thoroughly engaged as they are continually clicking on videos, links, and websites to learn more.

7. Smartboard Applet

Enhancing Mathematical Learning in a Technology-Rich Environment Jennifer Suh, Christopher Johnston, and Joshua Douds November 2008, Volume 15, Issue 4, Page 235

  • Teachers describe working collaboratively to plan mathematics lessons in a technology-rich environment. Addressing the needs of their diverse students, in particular, English Language Learners and students with special needs, the authors discuss how a technology-rich learning environment influences critical features of the classroom. They use classroom examples to show how technology tools amplify opportunities for extending mathematical thinking

8. Digital Camera

Orr, J., Suh, J. (2013). A picture is worth 100 math ideas. Teaching Children Mathematics, 19 (7), 458- 460.

  • Teachers Jennifer Orr and Jennifer Suh use cameras in their classrooms to be used in mathematical tasks. In the lower grades, students take pictures of objects that match the number their group is given to represent. Once pictures are taken to represent numbers 1-100 a movie is created with the pictures where the students get to narrate the pictures. In the upper grades, students can take pictures or bring in pictures to be used for mathematical tasks. For example, they can find the area of an object within the picture.


9. Applets Son, J. (2012). Fraction multiplication from a Korean perspective. Mathematics Teaching in the Middle School, 17(7), 388-393.

  • Using an activity called Modeling Mathematics with Tech-knowledgy, these professors explored mathematics concepts and various related models and representations in both a preservice elementary methods course and an in-service teacher content course focused on numbers. This activity involved five processes: 1. exploring multiple mathematics applets focused on one math concept, 2. evaluating multiple models in terms of their meaning, math fidelity, affordances and constraints, 3. designing and planning a technology- enhanced math lesson, 4. teaching and assessing using a technology- enhanced math lesson, 5. reflecting on the lesson implemented . Based on their observations and recording of the reflections, pedagogy, and assessment notes of teachers, they concluded that the challenge for teachers is learning to select the best instructional tools and create a learning environment to effectively integrate technology for learning.

Assessment Items

Assessment Item Example #1


This mathematics assessment item is one of many that the Smarter Balanced Assessment Consortium plans to incorporate into the upcoming Common Core State Standards assessments. It reveals what students are expected to be able to do at a more complex level in relation to simple fractions, examining whether each fraction can take different forms.

  
Assessment1.jpg
Image from: Smarter Balanced Assessment Consortium: Mathematics Item Specifications Grades 3-5


Assessment Item Example #2


This is another assessment item that aligns with the Common Core State Standards for Mathematics. As seen in the image, students are expected to identify the fraction that corresponds to each point on the string.

  
Assessment-2.jpg
Image from: NYSED.gov

Equivalent Fractions

Lead Contributor: Cecilia Shang

Common Core Progression for Equivalent Fractions

  • 3rd Grade: Understanding two fractions are equivalent as the same size or same point on a number line. Also recognize and generate simple equivalent fractions (including whole numbers) and explain why they are equivalent.
  • 4th Grade: How one fraction is equivalent to another fraction by using visual fraction models in attention to the number and the size of parts when they are actually equal. Also compare fractions that have different numerators and different denominators.

Curriculum Examples

3rd Grade: Finding Equivalent Fractions that are equal

Equivalent Fractions that are equal

When teaching equivalent fractions that are equal, it is important for students to see that the same point or model are equal even when the size of pieces (not overall size of part) have changed.

  
3rd grade pic.png
Image from: Common Core Progression: Fractions, Grade 3


Students start by experimenting equivalent fractions on a number line diagrams, they will notice that the same point can be labeled with different fractions. They will see that they are equal to each other like 1/2 = 2/4 = 3/6 = 4/8. These are equivalent fractions.

1/2 = 2/4, 4/6 = 2/3, 1/4 = 2/8

Another important equivalent that students need to understand is seeing and recognizing whole numbers as fractions, exp. 2 = 2/1= 4/2 = 6/3 = 8/4 ….

1= 1/1, 2/2, 3/3, 4/4, …..

Equivalent Fractions: Rich Task
  • Main Contributors: Angela Khai and Ann Kim

Word Problem:

Day One: Angela and Ann are sisters who go to different schools. The school day is just as long at Angela’s school as at Ann’s school. At Angela’s school, there are 6 class periods of the same length each day. Ann’s day is broken into 3 class periods of equal length. How long is each class period? Use a stack bar graph and/or a number line to represent your findings.

Day Two: One day, both of their schools started late (teachers had a meeting). Angela only had four classes and Ann only had two. Ann claims her school day was shorter than Angela’s was because she had only two classes on that day. Is she correct?

Launch

What can I do to make students want to solve this task?

  • Replace names with student names in the classroom
  • Goal is now relatable -- class periods, student names

What information should I give at the outset? How should i provide that information?

  • The periods are divided by subject area -- 10:00 - 11:00, you go to word study, that is the same thing as a “period”
  • Tools available: white paper, pencil, unifix cubes, a clock
  • UNPACK:
    • First Day: Figure out how long Angela’s class periods are and Ann’s class periods are and make sure that both equal the same total hours. Pose: If Angela’s classes were each 30 minutes, then how long is each period of Ann’s classes? Now try to figure out if Ann’s class periods were each 2 hours long, then how long is each period of Angela’s classes?
    • Second Day: Think back to yesterday’s lesson. What were our findings? Pick a time length period, and use those findings to solve this problem. How long was Angela in school? How long was Ann in school? Was Ann correct?

During

What should I look for in student work?

  • The time length of both schools are the same
  • Strategies used
  • Their justification of their work

What misconceptions are likely to occur? How will I field them?

  • Students simply look at the numbers instead of delving deeper into the problem (misunderstanding the task)
  • Students think that each period is the same length for both schools → Give another example using the stacked bar graph and/or number line

What kinds of solutions might I want to share?

  • Students who used different visual models that show the equal time lengths
  • Students who used fractions to solve the problem
  • Students who used numbers and models to show their thinking

Post-Task

How will I structure the discussion?

  • Students will pair-share their thinking first, and then a whole discussion will occur
  • We will start with students who used visuals to represent their findings and then move on to the more abstract strategies, and then ending with students who used fractions to solve the problem.

What will you include in the discussion (big ideas, mathematical connections)

  • Equivalent Fractions
  • Ask: What was the most efficient strategy used? Why was it efficient? What were your steps on how you solved the problem (What was your thinking?)?

How will I ensure that the whole class is on board? (Engagement, closure, assessment)

  • Agree/Disagree (thumbs up, thumbs down)
  • Disagree → Agreeing students will have to convince the students who disagreed, or vice versa

- All students must defend their thinking by providing concrete proof

  • Assessment: Informal discussion in class and their work on the assignment
Equivalent Fractions: Number String
  • Main Contributors: Lina Gil, Angela Khai, Ann Kim, and Melanie Wong

Number Strings Lesson

Task example: 3.NF.3: Number Strings

Problem Set 1. ½ = 2/4 2. ½ = 2/6 3. 3/6 = 4/8 4. ¼ = 2/8 5. 3/12 = ¼ 6. Challenge: What is equivalent to 5/4?

Actions

Write up each problem on the board one at a time, and have students discuss their strategies prior to showing the subsequent problem. Spend time probing each student’s strategy so that his/her thinking is made explicit (see below for sample classroom discourse around this problem set). This will allow students the opportunity to consider the strategies presented and will prompt them to think about relationships between the problems as they go along.

As students verbally describe their strategies, write down their strategies on the chart paper. Using a double number line or a clock (see solution section below for example) will help students visualize the relationship between numbers. Pure verbal explanations can sometimes be difficult for children (and even adults) to understand. Having a visual representation allows more students to understand, take part in the discussion, and make use of the representation for further problems.

Behind the Numbers

The first problem is the scaffold to the second. Assuming that children know how to partition a shape into 2 and 4 equal parts, this set encourages them to build from this understanding. The second true/false question, asks the students to show their thinking of ½ and 2/6, the students will partition a line into 2 equal parts and 6 equal parts, they will have to determine if the statement is true (it’s false) and which fraction part is larger. The second problem builds off the first problem because the first fraction has been partitioned already. The third true/false statement has the students working with denominators they may not have been exposed to but since the second problem already had the students partitioning into 6 equal parts, the third problem builds off of it. The fourth and fifth true/false statement are two equivalent fractions of ¼ in different forms. This allows the students to be exposed that there is not just one equivalent fraction for a given fraction but equivalent fractions are all the same. The challenge problem has the students thinking of equivalent fraction greater than 1.

A Visit to the Classroom

Teacher: Here’s our first warm-up problem: True or false- ½ and 2/4 are equal? Thumbs-up when you have an answer. Sara?

Sara: It’s true. ½ and 2/4 are equal.

Teacher: (Draws the following representation of Sara’s strategy on a number line against the whiteboard.)

Ak 1.jpg

Teacher: Does everyone agree with Sara? Okay. Let’s try the next one: ½ = 2/6?. Eliannah?

Eliannah: It’s false because if you split the 2/6 into 6 equal parts, the 2 parts, is less than where the ½ is.

Teacher: Let’s see what that looks like on the number line. (Records the jumps in a different color on Sara’s number line.)

Ak 2.jpg

Teacher: What do you think? Does Elliannah’s strategy work? Okay, let’s try the next one, true or false: 3/6 = 4/8. Show me with thumbs-up when you’re ready. Kade.

Kade: It’s true, I think. 3 is half of 6 and 4 is half of 8. So they’re both half.

Teacher: Did someone have a strategy to prove it is true? Julie?

Julie: If you split first one into 6 equal parts and the second fraction into 8 equal parts, they’ll both be at the same point.

Ak 3.jpg

Teacher: Did anyone notice something similar about the 3/6 and 4/8 compared to some of the other problems?

Dylan: They’re all half! Like what Kade said.

Teacher: Good! Now, let’s look at this problem (write ¼ = 2/8 on the board). True or false - ¼ = 2/8? Turn to a partner and tell them what you think. (after some time) Sara?

Sara: My partner and I think it’s true because four goes into eight twice and one goes into two twice. If you split a number line into eight parts and jump two times, it looks the same as if you split a line into four pieces and jump once.

Teacher: draws Sara’s explanation. Give me a thumbs up if you notice something about this problem and the previous question. Kyle?

Fractions 4.jpg

Kyle: I noticed that 2/8 is half of 4/8. If I look at the drawing, 2/4 would equal 4/8 and 2/4 would be the same as 3/6!

Teacher: If you agree with Kyle, then give me a thumbs up. Thank you Kyle!

Teacher: Lets try another one, true or false: 3/12 = ¼ ?

Robert: It’s true because 3/12 is still ¼ when you draw it on the number line. 3/12 is the same as ¼

Teacher: Show me a thumbs up if you agree with Robert or a thumbs down if you disagree with Robert (Teacher draws the number line)

Fractions 5.jpg

Teacher: What do you notice between this problem and the previous problem? What conclusions can you draw? (Give students time to think)

Teacher: Pair-Share with your partner (Give students time to share)

Teacher: Grace, what did you and your partner talk about?

Grace: We noticed that 3/12 is equal to 2/8 because it’s also ¼

Teacher: Oh interesting, do we all agree with Grace? Thumbs up or thumbs down?

Teacher: Great, I have one more problem I want to show you. (Writes 5/4 on the board)

Teacher: It is not a true or false question. What is the equivalence to this fraction, 5/4? Try to remember what we’ve done so far. I’ll give you all a minute to think about it. If you think you have an idea, show your thumbs up.

Teacher: Why don’t you all share with your partner, tell them what you think. Lucy, what did you and your partner think about?

Lucy: Maddie and I said that 5/4 is ¼ added 5 times.

Teacher: So if I were to draw a number line, what should it look like? David, can you draw what you think?

(David draws on the white board)

Fractions 6.jpg

Teacher: David, can you explain your thinking?

David: Like what Maddie and Lucy said, 5/4 is ¼ added 5 times. So each line represents another ¼

Fractions 7.jpg

Teacher: Does anyone else have anything to add onto David’s idea? Jenny?

Jenny: I thought that 5/4 can also be 1 whole and ¼, because 4 parts make 1 whole. Then there is ¼ left over.

Teacher: (adds onto David’s drawing)

Fractions 8.jpg

Teacher: Thumbs up if you agree with Jenny and David. So what is the equivalence of 5/4?

Class: 1 whole and ¼ !

Teacher: (writes on board “1 ¼”) So now, we see that 5/4 is equivalent to 1¼.

4th Grade: Finding Two Equivalent Fractions that are equal

Students will need to know how to find if two fractions are equivalent.

Is 2/3=4/6?

There is procedural and conceptual knowledge, as teachers we want to teach conceptual in that if you broke up 4/6 with counters of 4 red and 2 more yellow will see there are 3 groups and 2 groups are red counters showing 2/3 and 4/6 are equivalent.

Finding two fractions that are equivalent

Students should be able to understand that 2/3 can be broken into smaller pieces into 6 with 4 of them colored and are of equal area

  
4th grade pic.png
Image from: Common Core Progression: Fractions, Grade 3


Students also understand that multiplying or dividing the numerator and denominator by the same non-zero number can give them the equivalent of the original fraction. (procedural)

2/3 (2x2/3x2)= 4/6


4th grade students use their understanding of equivalent fractions to compare fractions with different numerator and different denominator. When the size of parts are different to each other (or the numerator and the denominator are different).


Compare 5/8 and 7/12, which is larger?

60/96= (12x5/12x8) 56/96=(7x8/12x8)


So... 7/12 is smaller than 5/8

Curriculum Resources

Fraction Concepts http://enlvm.usu.edu/ma/nav/toc.jsp?sid=_shared&cid=emready@fraction_concepts&bb=published

  • Offers full lesson plans, worksheets, and activities on part-whole fractions.

Technology Resources

Cyberchase (PBS) www.pbs.org/teachers/search/results.html?

  • Videos on equivalent fractions and activities.

Fraction Bars (Math Playground) http://mathplayground.com/Fraction_bars.html

  • This program allows the user to explore fractional parts, concepts of numerator, denominator, and equivalency

Fraction Track http://standards.nctm.org/document/eexamples/chap5/5.1/index.htm

  • A game involving equivalent-fraction concepts.

Assessment Items

  • Smarter Balanced Assessment Consortium: [1]
    • A website that has Common Core resources and sample assessment questions.

Comparing Fractions

Lead Contributor: Myuriel von Aspen

Common Core Progression for Comparing Fractions

  • 2nd Grade:
    • Students compare lengths using a standard measurement unit.
  • 3rd Grade (3.NF.3d):
    • Students recognize that comparing two fractions is only valid when referring to the same whole.
    • Students compare two fractions with the same numerator or the same denominator by reasoning about their size.
    • Students use the symbols >, =, < to record the results of comparisons and justify their conclusion (e.g. by using a visual fraction model).

Curriculum Examples

3rd Grade: Referring to the same whole

Using circle models to illustrate the importance of referring to the same whole


It is critical that students understand the importance of referring to the same whole when they are comparing fractions. Using concrete models helps the student visualize this concept. These circle models illustrate the problem that arises when students compare two fractions that do not refer to the same whole. As they are learning to compare fractions, it is useful to show them an example such as this one, to illustrate the error in reasoning that 1/4 is greater than 1/2.






  
Same whole number.jpg
Image From: Progressions for the Common Core State Standards in Mathematics

3rd Grade: Comparing two fractions with the same numerator

Using fraction tiles to compare fractions with the same numerator


Using fractions tiles based on the same whole is a concrete way for students to compare fractions with the same numerator based on their size. For instance, students can observe that 3/4 is greater than 3/5 because each 1/4 unit fraction is larger than each 1/5 unit fraction. After much exploration with concrete models, students can start verbalizing reasoning on their own about fractions with the same numerator. For example, they can explain that when you break up a whole, the more pieces you break it into (bigger denominator), the smaller each piece becomes. Therefore, for two fractions with the same numerator, the one with the smaller denominator is the larger fraction because each piece is bigger.

  
Fraction Tiles.jpg
Image From: The Roll Out Fractions Game: Comparing Fractions


Using number lines (length model) to compare fractions with the same numerator


Using two parallel number lines is a representational way that can be used to compare fractions that have the same numerator. For example, students can observe that 2/6 is greater than 2/8 because the segment from 0 to 2/6 is longer than the segment from 0 to 2/8. Students will eventually note that for fractions with the same numerator, the fraction with the smaller denominator is greater because the unit fraction of the one with the smaller denominator is greater than the unit fraction of the one with the larger denominator (e.g. 2/6 is greater than 2/8 because 1/6 is greater (or longer) than 1/8). Remember that at this stage, the standard is asking students to compare by reasoning about the size (area or length).


  
Number Lines.png
Image From: Math Expressions 3rd Grade

3rd Grade: Comparing two fractions with the same denominator

Using a circle model to compare fractions with the same denominator


Fraction circles (either concrete or representational) can help students compare fractions with the same denominator. For instance, they can take 5 pieces out of the fraction circle that is divided into 8 pieces and observe that 5/8 is greater than the remaining 3/8 because it is made up of more unit fractions.

  
Fraction Circle.png
Image From: helpingwithmath.com


Using a number line (length model) to compare fractions with the same denominator


A number line is a good model to solidify the concept that fractions are numbers. It also helps students visualize the partitioning of a segment of length 1 into equal parts (with the same denominator). With a number line, a student can compare two fractions with the same denominator. For example, a student can observe that 1/6 is smaller than 5/6 because the segment from 0 to 1/6 is shorter than the segment from 0 to 5/6 since it is made up of only 1 unit of 1/6 as opposed to 5 units of 1/6.


Students can also start observing that a fraction that is to the left of another fraction on a number line is smaller because the segment from 0 to the fraction is shorter than from 0 to the fraction that is more to the right. This, of course, applies only to positive numbers, but at this point, these are the only numbers that that are being studied.

  
Number line to compare.png
Image From: Wu, H. (2011). "Phoenix Rising"


Using a number line (length model) to compare fractions smaller and greater than 1 and with the same denominator.

A number line is also helpful for students to visualize how some fractions can be greater or smaller than 1. Students can notice that 5/3 is greater than 1 because it is made up of 5 units of 1/3, whereas 1 is made up of only 3 units of 1/3.

  
Number line greater than 1.png
Image From: "Wu, H. (2011) Teaching Fractions according to the CCS


Using an area model to compare fractions with the same denominator


An area model provides visual support for students to conceptualize that when the denominator is the same, the larger the numerator, the larger the fraction because it is made up of more pieces. In the example at the right, a student can observe that each of the pieces represents 1/6. They can also note that 5/6 is larger than 3/6 and 1/6 because 5/6 is made up of 5 units of 1/6, whereas 3/6 is only made up of 3 units, and 1/6 is only made up of 1 unit.

  
Area Model.png
Image From: IllustrativeMathematics.org

3rd Grade: Using >, =, < and justify the conclusion by using visual fraction models

Using >, =, < to compare fractions


Since 1st grade, students have been using the symbols >, =, and < to compare whole numbers. In third grade, they continue to use these symbols but within the context of fractions. Once students understand the concept of equivalency and are able to compare fractions, they can begin to compare fractions in a more abstract way by using the >, =, and < symbols. Students should be able to demonstrate their understanding of using >, =, and < when comparing fractions by answering questions such as the one on the right.


  
Using symbols.png
Image From: Math Expressions 3rd Grade


Justifying conclusions by using visual fraction models


Students are not only expected to use >, =, and < when comparing fractions, but also expected to explain their reasoning using visual representations. When asked to compare three fractions such as 2/3, 2/5, and 2/7, students should be able to use an area model such as the one on the right to explain their reasoning. A student's reasoning might be as follows (from IllustrativeMathematics.org):

"When the numerators are the same, that means we have the same number of pieces. A larger denominator means a smaller piece. If we have the same number of pieces but the pieces are smaller, we will have a smaller total amount. We can shade in each bar as shown [on the right] to illustrate." A student can then state that 2/3 < 2/5, or 2/7 > 2/3, or any other combination based on their reasoning above.


  
Area model same numerator.png
Image From: IllustrativeMathematics.org

Curriculum Resources

New Zealand Numeracy Development Project Book 7: Teaching Fractions, Decimals, and Percentages - http://www.nzmaths.co.nz/sites/default/files/Numeracy/2008numPDFs/NumBk7.pdf

New Zealand Numeracy Project - Material Masters: http://www.nzmaths.co.nz/material-masters?parent_node=

  • Material Masters 4-19 and 4-20 provide masters for fraction pieces and fractions on geoboards.

The Roll-Out Fractions Game:Comparing Fractions Media:Fractions_Concrete_to_Abstract.png

  • This table presents ways in which a teacher and students can work with fractions from concrete to representational (pictorial) to abstract or any combination.

Teaching Fractions According to the Common Core Standard: http://math.berkeley.edu/~wu/CCSS-Fractions.pdf

  • Article by H. Wu from UC Berkeley

Math-Aids.com: http://www.math-aids.com/Number_Lines/Fractions_Number_Lines.html

  • This website allows teachers to create and print out number lines with any given denominator.

Technology Resources

Fraction Models: http://illuminations.nctm.org/ActivityDetail.aspx?ID=11

  • This website allows the student to explore different representations for fractions including improper fractions, mixed numbers, decimals, and percentages. Additionally, there are length, area, region, and set models. Adjust numerators and denominators to see how they alter the representations and models. Use the table to keep track of interesting fractions.

Assessment Items

Smarter Balanced Assessment Item
Grade 3 Claim 1: Concepts and Procedures
DOK Level: 2: Basic Skills & Concepts


  
Compare fractions assessment1.png
Image From: SmarterBalanced.org


Smarter Balanced Assessment Item
Grade 3 Claim 3: Communicating Reason
DOK Level: 3: Strategic Thinking & Reasoning


  
Compare fractions assessment2.png
Image From: SmarterBalanced.org

Adding and Subtracting Fractions

Lead Contributor: Brittany Kirby

Common Core Progression for Adding and Subtracting Fractions

  • 4th Grade:
    • Students add and subtract fractions with like denominators.
    • Students add and subtract whole numbers and fractions, including addition and subtraction of mixed numbers with like denominators and conversion of mixed numbers to improper fractions.
  • 5th Grade:
    • Students add fractions in which both denominators need to be changed.
    • Students solve word problems involving estimation, by using visual fraction models, equations, benchmark fractions and number sense to assess reasonableness.


Curriculum Examples

4th Grade: Adding and subtracting fractions with the same denominator

The common meaning of addition

Students should understand that the meaning of addition is the same for both fractions and whole numbers. Just as the sum of 4 and 8 can be seen as the length of the segment obtained by joining together two segments of lengths 4 and 7, so the sum of ⅔ and 8/5 can be seen as the length of the segment obtained by joining together two segments of lengths ⅔ and 8/5.


  
Fractions as length 2.png
Image From: Progressions for the Common Core


Composing and decomposing fractions

Students use the above insight that fraction addition encompasses the same concept as whole number addition to understand that fractions are built up from unit fractions (e.g. 5/3=1/3+1/3+1/3+1/3+1/3). Students can then begin decomposing and composing fractions in this way, to bridge the gap between the meaning of fractions and the process of adding and subtracting fractions.

  
Number line.png
Image From: Math Expressions 4rd Grade, Volume 2
By this reasoning, students see that if we add fractions with the same denominator, we get a fraction with the same denominator whose numerator is the sum of the numerators.
  
Addition algorithm.png
Image From: H. Wu

4th Grade: Working with whole numbers and fractions

Computing sums of whole numbers and fractions

Students should use their knowledge of equivalent fractions to convert whole numbers to equivalent fractions with the same denominator as the fraction addend (if adding 7 to to ⅕, 7=35/5). They should understand that mixed numbers are the sum of the whole number and the fractional part.




  
Whole numbers and fractions.png
Image From: Progressions for the Common Core
Adding and subtracting fractions with like denominators

The first example here shows how models can be used to help students see that fractional parts including more than a whole can be added together similarly to the way unit fractions can be added. Students may use various strategies of adding the whole numbers and the fractional parts as shown in the example, using their knowledge of composing and decomposing numbers. The second example shows how similar visual models can be used to for mixed number subtraction.




  
Adding mixed numbers with like denominators NC.png
Image From: North Carolina Department of Public Instruction
  
Mixed numbers with like denominators from NC.png
Image From: North Carolina Department of Public Instruction


Converting mixed numbers to fractions

Converting a mixed number to a fraction should not be viewed as a separate technique to be learned by rote, but simply as a case of fraction addition. This example shows how the solver determined the improper fraction and then converted to a mixed number. Models such as these can be used to show both conversions (from mixed numbers to improper fractions and vice versa).




  
Improper to mixed NC.png
Image From: North Carolina Department of Public Instruction

5th Grade: Adding fractions in which both denominators need to be changed

It is NOT necessary that students find the least common denominator. This skill could be taught as enrichment later on as a skill for simplification of computation, but is not required to add fractions with unlike denominators. Instead, students may find a common denominator by multiplying both denominators or seeing a common multiple of both denominators, first using visual fraction models (area models, number lines, etc.) to build understanding before moving to the algorithm. This ability involves the concept of equivalent fractions described in early sections. Students may choose to use the clock model to explain their approaches to solving a problem such as ⅓ + ⅙.
  
Clock model.png
Image From: North Carolina Department of Public Instruction

5th Grade: Solving fraction word problems

Using models to represent word problems

Students may solve the following problem using an area model as shown: Jerry was making two different types of cookies. One recipe needed 3/4 cup of sugar and the other needed 2/3 cup of sugar. How much sugar did he need to make both recipes?


  
Area model unlike denominators.png
Image From: North Carolina Department of Public Instruction
Using estimation to assess reasonableness

Students should also learn to use estimation to assess the reasonableness of their answers. A possible estimation strategy for solving the same problem is shown here. This student is using 1/2 as a benchmark, a common and beneficial estimation strategy.


  
Jerry estimation.png
Image From: North Carolina Department of Public Instruction

Curriculum Resources

  • Instructional Support Tools, 4th grade Math (North Carolina Dept of Public Instruction): [2]
    • This resource "unpacks" each of the Common Core standards, giving examples and explanations to support teacher understanding of the new curriculum.
  • Instructional Support Tools, 5th grade Math (NC Dept of Public Instruction): [3]
    • This resource gives similar information to that above, for 5th grade.
  • Fourth Grade Fractions (North Carolina Dept of Public Instruction: [4]
    • This document has several lesson plans for fractions with instructional materials attached.

Technology Resources

  • Thinking Blocks Website: [5]
    • This website allows students to practice addition and subtraction of fractions of different types through online bar diagrams.

Assessment Items

  • Smarter Balanced Assessment Consortium: [6]
    • This website has Common Core assessment resources, as well as some sample test questions.

Fraction Multiplication

Multiply a fraction or a whole number by a fraction

Main contributors: Erika Espinoza, Marissa Young, Bernadette Saldana, Jessica Yen
California Common Core State Standards
  • 5.NF.4-Multiply a fraction or a whole number by a fraction


Number Strings Lesson-Fractions

Problem Set
12 ÷ 2
24 ÷ 4
½ × 12
¼ × 24
Jessica made a cheesecake for her class. She needed to divide the cake into 24 equal slices for all the students to eat; however, not everyone wanted a slice. If only ¾ of the cheesecake was eaten. How many slices of cheesecake were left?
Actions
Write up each problem on the board one at a time, and have students discuss their strategies prior to showing the subsequent problem. Spend time probing each student’s strategy so that his/her thinking is made explicit (see below for sample classroom discourse around this problem set). This will allow students the opportunity to consider the strategies presented and will prompt them to think about relationships between the problems as they go along.
As students verbally describe their strategies, write down their strategies on the chart paper. Using an open number line or a number line created by a train of cubes (see solution section below for example) will help students visualize the relationship between numbers. Pure verbal explanations can sometimes be difficult for children (and even adults) to understand. Having a visual representation allows more students to understand, take part in the discussion, and make use of the representation for further problems.
Behind the Numbers
The first problem is the scaffold to the second. Assuming that children know the pattern of dividing by 2, this set encourages them to build from this understanding. The second expression has the solution value equal to the solution value of the first one, and students may see the connection between the first and second problem by seeing that 12 ÷ 2 is 6 and 24 ÷ 4 is also 6. The third expression introduces a fraction of ½ , which is multiplied by a whole number of 12. Students can offer to look at this problem or rephrase this problem in a different way. Instead of saying ½ times 12, students could say ½ of 12 and ask themselves: what is half of 12? The following problem also has a solution of 6. Students can repeat the problem aloud just as they did for the previous one and say: ¼ of 12 is what? They have the opportunity to think of this problem by applying what they did to the previous one. The last problem is a word problem with the same concept. Since students already had the opportunity to solve for ¼ of 12, students can think about the remainders, which is ¾ and ask them what whole number is left?
A Visit to the Classroom
Teacher: Here’s our first warm-up problem: 12 ÷ 2. Thumbs-up when you have an answer. Marissa?
Marissa: It’s 6. I know that 12 ÷ 2 because I thought about half of 12. I thought about a pizza.
Teacher: (Draws the following representation of Marissa’s strategy on the whiteboard using a fraction pie).


Fractions Erika1.png


Teacher: Does everyone agree with Marissa? Okay. Let’s try the next one: 24 ÷ 4. Bernadette?
Bernadette: I did half more of the other problem. I know 24 is double 12, so I thought about two pizzas and divided each one by 4 and added them together, so 12 divided by 4 is 3 plus 3 is 6.
Teacher: Let’s see what that looks like on a fraction pie. (Uses the same fraction pie of 12 parts and divides both by 4 by shading in the parts).


Fractions Erika2.png


What do you think? Does Bernadette’s strategy work? It does save a lot of time if we think of doubles and dividing both by 4. Okay, let’s try the next one, ½ × 12. Show me with thumbs-up when you’re ready. Erika.


Erika: It’s 6, I think. I thought of it like the first problem. Instead of saying “times,” I said ½ of 12 in my head and it’s the same thing! So, ½ × 12 is 6.
Teacher: Nice, you broke apart the 12 into 2 like the first problem and you knew that was half. Okay, let me record your strategy.



Fractions Erika3.png


Teacher: Did anybody do it a different way? Did anyone not use a fraction pie as a strategy?
Jessica: I used a number line. Same idea, but I visualized a line: 0 to 12. I know the midpoint is 6.



Fractions Erika4.png



Teacher: Nice, it’s helpful to see the fractions in a number line too. Now let’s try ¼ × 12. Again, show me with thumbs-up when you’re ready. Collin?
Collin: I thought about Erika’s strategy and also used “of” instead of “times,” so that I don’t get confused. So I thought of it as: ¼ of 24. It’s kind of like the second problem too. Bernadette divided it into four equal parts, but she used two fraction pies of 12, so I did the same thing. It was helpful to think of the clock too. If I took ¼ parts from each clock and added them together, it gives me 6.
Teacher: Okay, I see you’re also making a connection with the previous problems. Let me draw it out how I think you envisioned it with the clock.


Fractions Erika5.png

Teacher: Yes, it is helpful to think of the clock. I see that ¼ of one clock and ¼ of another makes 6, which is half of 12 and the clock has 12 equal parts. Okay now let’s take that same thought and think about it with food. Let’s say a cheese cake. Look at the following word problem and think about what the question is asking you to do? (Show the word problem on the document camera).
Jessica made a cheesecake for her class. She needed to divide the cake into 24 equal slices for all the students to eat; however, not everyone wanted a slice. If only ¾ of the cheesecake was eaten. How many slices of cheesecake were left?


Marissa: Oh, I see. It’s asking us to state how many slices of pizza are left. Since ¾ of the pizza were eaten, then that means there are ¼ pieces left, so that means there are 6 slices of cheesecake left.
Teacher: Okay, good. I see that you continued to make the connection with the previous problem. Can you see a pattern?
Marissa: Yes, the solution to all the problems was 6. I also realized that dividing the whole numbers we used and multiplying fractions is the same thing.
Teacher: Okay, I hear what you’re saying. So, this is what I hear you saying: multiplying fractions by a whole number is like finding parts of the whole number. Okay, good. Let’s keep this in mind when we move on to the next lesson.


Illustrative Math on Multiplication of Fractions

Problem Set 1: Connor and Makayla Discuss Multiplication
http://www.illustrativemathematics.org/illustrations/321
Main contributors: Erika Espinoza, Marissa Young, Bernadette Saldana, Jessica Yen
Bernadette said, "I can represent 3×2/3 with 3 bars of chocolate each of length 2/3."

Fractions Erika6.png

Jessica said, “I know that 2/3×3 can be thought of as 2/3 of 3. Is 3 copies of 2/3 the same as 2/3 of 3?”
1. Draw a diagram to represent 2/3 of 3.
2. Explain why your picture and Bernadette’s picture together show that 3×2/3=2/3×3.
3.What property of multiplication do these pictures illustrate?
Launch
What can I do to make students want to solve this task?
In order to make this problem more accessible to students, and make them want to solve this task, the names of the students in the word problem will be changed to names of students in the class. Instead of having rectangles in the problem, it will be changed to something that is more appealing to students such as chocolate bars. The numbers used in the problem can also be changed to accommodate the level at which students are performing in multiplying fractions.
What info should I give at the outset? How should I provide that information?
Before students are given time to work on the problem individually, the problem will be written on the board and presented to the whole class. The teacher will ask the student to reflect on the problem, reread the problem multiple times to figure out what the problem is asking them, and to analyze the problem. As a class, students will share out their thoughts.
During
What should I look for in student work?
For student work, the teacher will be looking for diagrams, drawings, and explanations as to the student’s reasoning for solving the task as they did. Students must be able to justify why they drew their diagram the way they did, and why their diagram and the one provided exhibit the commutative property.
What misconceptions are likely to occur? How will I field them?
Misconceptions that are likely to occur include: students transferring their knowledge of multiplication of whole numbers thus believing that their product will be larger than the factors; the wording in the problem may have students confused between the multiplication sign (x) and the word “of” (i.e. “I know that 2/3 x 3 can be thought of as 2/3 of 3.”); students may not know how to interpret the diagram provided in the problem; some students might already know the traditional algorithm for multiplying fractions, and they may not be able to explain how they got their answer or how to represent it through a visual representation. In order to combat misconceptions, it would be beneficial to have the student talk through their strategies because sometimes when they are spoken aloud students will catch their own mistakes. Another strategy would be to pair up students who have similar strategies so they can discuss with each other their reasoning; the students may have a better way of explaining it to each other rather than the teacher.
What kinds of solutions might I want to share?
It would be beneficial to share a range of solutions including solutions that progress from simple to more complex, ones that show common misconceptions, strategies where the student’s thinking is visible, and solutions that illustrate conceptual and procedural understanding. By presenting student strategies that progress from simpler to more complex, students at different levels of computation may be able to push towards a higher form for solving mathematical tasks. Students may see that someone has a similar strategy, but there is a “twist” to it; they may be inclined to try that strategy, which will advance their multiple ways of solving and thinking. Solutions that include common misconceptions are a great way to help students that are almost at the point of breakthrough. They just need to see where their solution may not work quite well and be able to edit their strategy. Another solution worth sharing are ones where students are using drawings and words to explain their thinking, and if someone were to look at their work it would almost be like a step-by-step guide to their solving strategy. When students have built the conceptual understanding of multiplying fractions, they should be able to see how the traditional algorithm works.
Post-Task
How will I structure the discussion? What will you include in the discussion (big ideas, mathematical connections)
The discussion will start with a think-pair-share so that students have time to discuss and revise strategies as needed after talking to a partner. Once students have had time to discuss with a partner, the group will engage in a whole class discussion. Students will be chosen to present strategies and will be given feedback from classmates and the teacher. The order of students presenting their strategies will be determined upon some sort of progression from simple to more complex strategies. The discussion will also include any misconceptions that students may continue to have. Misconceptions can be worked out as a class in a way that will hopefully provide the student with the misconception with a solution to their misunderstanding. As part of the discussion, the teacher may choose to take a stance on a solution strategy for solving multiplication with fractions. The goal is to have students argue that the strategy will not work, or will not work in all instances, and they will need to provide support for their reasoning. This is will allow students to strengthen their understanding of their strategies, and it will help the teacher see if the students are able to explain their strategies. Ideas to be discussed will include Parts B and C of the task. Students will need to be able to explain their drawing and show that concept of the commutative property.
Big idea
How will I ensure that the whole class is on board? (Engagement, Closure, Assessment)
During the pair or group shares, the teacher will monitor the classroom, then do a whole class discussion with students sharing their strategies with the class. The teacher will also collect student samples of the task that include their strategies and reasoning for the way they solved the problem. Throughout the lesson, during individual work, think-pair-shares, and the whole class discussion, the teacher will be providing guiding questions for the students to build off of if she feels the students need the assistance. Strategies from the whole class discussion will also be recorded on poster paper that will be hung up for future reference. The following day, the teacher can pose an “Entrance Slip” where students must use what they learned about multiplying fractions in the previous day’s lesson. The teacher can then look through the slips and determine which students may need extra guidance.


Problem Set 2: To Multiply or Not To Multiply?”
http://www.illustrativemathematics.org/illustrations/609
Main contributors: Erika Espinoza, Marissa Young, Bernadette Saldana, Jessica Yen
Launch
Some of the problems below can be solved by multiplying 1/8×2/5, while others need a different operation. Select the ones that can be solved by multiplying these two numbers. For the remaining, tell what operation is appropriate. In all cases, solve the problem (if possible) and include appropriate units in the answer.
1. Two-fifths of the students in Erika’s fifth grade class are girls. One-eighth of the girls like chocolate ice cream. What fraction of Anya’s class consists of girls who like chocolate ice cream?
2. A huge classroom is in the shape of a rectangle 1/8 of a mile long and 2/5 of a mile wide. What is the area of the classroom?
3. There are 2/5 of a cake left. If Bernadette eats another 1/8 of the original cake, what fraction of the original cake is left over?
4. In Jessica’s fifth grade class, 1/8 of the students are boys. Of those boys, 2/5 have red hair. What fraction of the class is red-haired boys?
5. Only 1/20 of the guests at the party wore both red and green. If 1/8 of the guests wore red, what fraction of the guests who wore red also wore green?
6. Marissa was planting a garden. She planted 2/5 of the garden with potatoes and 1/8 of the garden with lettuce. What fraction of the garden is planted with potatoes or lettuce?
7. At the start of the trip, the gas tank on the car was 2/5 full. If the trip used 1/8 of the remaining gas, what fraction of a tank of gas is left at the end of the trip?
8. On Monday, 1/8 of the students in Ms. Yen’s class were absent from school. The nurse told Ms. Yen that 2/5 of those students who were absent had the flu. What fraction of the absent students had the flu?
9. Of the children at Ann’s daycare, 1/8 are boys and 2/5 of the boys are under 1 year old. How many boys at the daycare are under one year old?
10. The track at school is 2/5 of a mile long. If Lina ran 1/8 of the way around the track, what fraction of a mile did she run?
What can I do to make students want to solve this task?
In order to make this rich task engaging for students, we decided to use their names within the word problems. Since this problem is multi-layered, there are vast opportunities to embed at least one-third of the students’ names. We also decided to change the context of the word problems to become more relatable to the class. For instance, instead of posing the problem, “A farm is in the shape of a rectangle 1/8 of a mile long and 2/5 of a mile wide. What is the area of the farm?”, we adjusted it so that that the task is to find the area of the classroom, which the students can visually see in their immediate surroundings. Thus, the word problems appeal to the students’ experiential and background knowledge.
What info should I give at the outset? How should I provide that information?
To begin the problem, the teacher will first pose the main question to the class on the whiteboard. “Some of the problems below can be solved by multiplying 1/8×2/5, while others need a different operation. Select the ones that can be solved by multiplying these two numbers. For the remaining, tell what operation is appropriate. In all cases, solve the problem (if possible) and include appropriate units in the answer.” The teacher will first read aloud the problem to the students, then have the students individually analyze the problem for a couple of minutes. Students will be asked for their observations about the prompt, what they notice, and their planned strategy in solving this problem. After giving students time to reflect on the question, the teacher will ask students to pair-share their observations with an elbow partner. The teacher will circulate the classroom and listen in on how students are decomposing the problem and what strategies they will be using. Lastly, the teacher will ask for student volunteers to share their discussions and interpretations of the problem. The teacher will facilitate the discussion and clarify any misconceptions about the problem before students begin the rich task. When students are ready to begin the task, the teacher will read the remaining corresponding problems and ask students to work in pairs.
During
What should I look for in student work?
Students will be working in pairs to complete this rich task, which will be completed on a large butcher paper. Within their work, we are expecting students to be solving each of the subproblems both procedurally and conceptually. Hence, students will understand that there is an equal emphasis on the importance of algorithms and creating visual representations. Also, we want students to be able to justify their answers, specifically providing evidence as to why the problem does/does not involve the operation of multiplication.
As students are working with partner, we will be listening for the usage of academic language within their discussions. For instance, we are expecting students to reason with one another that when they are multiplying fractions by a whole number, the end product is smaller than the original whole number.
What misconceptions are likely to occur? How will I field them?
When students are multiplying or choosing the operations for the problems, misconceptions are likely to occur. This may include just multiplying the numbers together without analyzing what the problem is asking. To field them we will ask guiding questions (i.e. How do you know?, How do we know when to multiply them?, How are these different from the previous problems?) and direct them towards keywords, concepts, drawings, or manipulatives.
What kinds of solutions might I want to share?
The types of solutions the teacher should ask a couple of students to share out should include common misconceptions, simple solutions, and more complex solutions. Before the students present, the teacher should ask students for permission to share out their work and to think about how they would present their strategy and thinking to the class. The teacher will then ask students to present their work in a progression from simple and more conceptual work which the entire class can easily understand, to more abstract and complex solutions. Misconceptions should be utilized so that students can analyze where these ideas originated from and why it might be easy to make this mistake. True understanding of the multiplication of fractions can be understood in depth if students are able to articulate why some of the subproblems does not include the usage of the multiplication operation, and how multiplication and division differ. Additionally, there needs to be a discussion on the students’ conceptual understanding so that students will understand the meaning behind the symbols and algorithms of multiplying fractions.
Post- Task
How will I structure the discussion?
The whole class discussion will begin with the common misconceptions about fractions and multiplication. This includes what they struggled with and any questions they might have regarding misconceptions. The teacher will then highlight the simple solutions, while gradually showing the different steps and strategies in order to display the more complex solutions. Throughout this process, students will present their solutions and different strategies.
What will you include in the discussion (big ideas, mathematical connections)?
The important connection to show the students is the process of translating the fraction word problems into the correct operation to use (especially multiplication). Questions like “What were similarities between the problems that could be solved using multiplication? and What were the differences between the multiplication problems and the other operations problems?” may help students know when to use multiplication in fractions, understanding wholes and fractions, and mathematical connections to properties like the commutative property and using other operations.
How will I ensure that the whole class is on board? (Engagement, Closure, Assessment)
To ensure that the whole class is on board with this problem, the teacher will circulate the classroom to look for any off-task behavior. The teacher will also take anecdotal notes on the different types of strategies students are using so that there is a wide array of strategies that can be presented to the class. As an informal assessment, the teacher will ask probing questions to ensure that the students are critically thinking about the problem and to help students who may be struggling with the task. The teacher should also look at the visual representations and check if students are able to portray each sub-problem accurately.
In regards to the closure, the teacher will ask students do an extension activity where they will be creating their own multiplication problem regarding fractions and whole numbers. They will then trade their word problem with a different elbow partner and ask the partner to solve the problem using both procedural and conceptual work. This activity can serve as an exit slip because the teacher will be able to clearly see whether students know how to set up a multiplication problem, what exactly their problem will be looking for, and if they can solve their partner’s multiplication fraction problem that might have a different sentence structure than their own.

Decimals

Fraction Division

Multiplication as Scaling

Annotated References

Empson, S. (2011). Equal sharing and the roots of fraction equivalence. Teaching Children Mathematics, 7(7), 421-425.

  • This article explores how upper elementary students invent strategies to solve equivalence problems.

Gould, H. T. (2011). Building understanding of fractions with lego bricks. Teaching Children Mathematics, Retrieved from http://faculty.tamucc.edu/sives/1350/tcm2011-04-498a.pdf

  • This article discusses using art to teach fractions, decimals, and percent equivalents. The task was for students to use a hundredth grid and cover the grid with unit squares, and then the students were required to make fractions according to how many colored unit squares were on the hundredth grid. The findings were that students were able to conceptually understand fractions, and the task engaged students in the learning through art.

Hodges , T. E., Cady, J. S., & Collins, R. L. (2008). Fractions representation: Not-so-common denominator among textbooks. Mathematics Teaching in Middle School, 14(2), 78-84.

  • This article uncovers three different textbooks to identify the various fractions representations used to support student learning. Researchers focus their study on three areas: representation mode, models, and problem content. They found that all textbook problems lack the support of real-world context, visual models, and representations, and majority of the problems use symbols and numeral fractions.

Johanning, J. (2013). Developing algorithms for adding and subtracting fractions. Mathematics Teaching in the Middle School, 18(9), 527-532.

  • This article is that though teachers are attempting to balance procedural and conceptual understanding for students, research cautions that simply having students use any type of visual representation does not carry meaning. The author states that fraction problems that ask students to create visual representations should engage students in mathematical reasoning. Thus, the author gives an example of a Lands Problem that asks sixth grade students to discover for themselves how fraction equivalences looks like conceptually.

Ortiz, E. (2006). The roll out fractions game: Comparing fractions. Teaching Children Mathematics, 13(1), 56-62.

  • This article details how the "roll out fractions game" can be played in class to help students learn to compare fractions.

Petit, M. M., Laird, R., & Marsden, E. (2010). They "get" fractions as pies; Now what? Mathematics Teaching in the Middle School, 16(1), 5-10.

  • This article reveals the importance of using multiple models to help students develop their conceptual understanding of fraction concepts. The authors contend that the use of the such models will help students transition to the use of reasoning and flexible strategies to solve problems involving fractions.

Phelps, K. (2012). The power of problem choice. Teaching Children Mathematics, 19(3), 152-157.

  • Over the span of three days, Katherine Phelps uses individual, paired, and whole class time for students to develop strategies for adding and subtracting fractions with unlike denominators through the use of differentiated number choices when working with word problems. On the first day students work individually, on the second they work in a partnership to discuss strategies that they used to solve the problems, and on the third day students participate in a whole class discussion.


Scaptura, C., Suh, J., & Mahaffey, G. (2007). Using art to teach fraction, decimals, and percent equivalents. Mathematics Teaching in the Middle school, 13(1), 24-28.

  • This article discusses using art to teach fractions, decimals, and percent equivalents. The task was for students to use a hundredth grid and cover the grid with unit squares, and then the students were required to make fractions according to how many colored unit squares were on the hundredth grid. The findings were that students were able to conceptually understand fractions, and the task engaged students in the learning through art.

Son, J. (2012). Fraction multiplication from a Korean perspective. Mathematics Teaching in the Middle School, 17(7), 388-393. Focusing on the Korean perspective on how fractions are introduced in textbooks, there are three steps that diagrammed problems in Korean textbooks: 1. Developing the meaning of the operation using real-life contexts. 2. Developing strategies for computing. 3. Practicing strategies and strategy selection.There are four things that gave students conceptual and procedural understanding in the Korean textbook: 1. developing the meaning of the operation using a real-life context; 2. developing strategies for computing; and using and applying strategies. 3. Develop multiple computational strategies. 4. Develop conceptual understanding and procedural fluency simultaneously.


The Common Core Standards Writing Team. (2011). Progressions for the common core state standards in mathematics (draft). 1-13.

  • This document provides an overview of the progressions for the Common Core State Standards in Mathematics, specifically with respect to the following domain: grades 3-5 Number and Operations - Fractions.

Vinogradova, N., & Blaine, L. (2013). Sweet work fractions. Mathematics Teaching in the Middle School, 18(8), 484-491.

  • Sweet Work Fractions discusses a game to play in a classroom to reinforce comparing fractions. There are 3 tables at the Maximum Chocolate Party (MCP). One, two, and three chocolate bars, identical in size and flavor, are on the first, second, and third tables, respectively.

Wu, H. (2011). Phoenix Rising: Bringing the Common Core State Standards for Mathematics to life. American Educator, 35(3), 3-13.

  • This article among other topics provides information about unit fractions and addition of fractions.

Wu, H. (2011). Teaching fractions according to the Common Core Standards. Retrieved from: http://math.berkeley.edu/~wu/CCSS-Fractions.pdf

  • This article deals specifically on how to teach fractions according to the new CCSSM requirements. It has many examples, definitions, and illustrations.