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You will earn 2 hours of PD/EILA for completing this section!
You will earn 2 hours of PD/EILA for completing this section!
Getting Started Building Fraction Sense
Getting Started Building Fraction Sense
Recess at 20 Below
Recess at 20 Below
A great way to set up a unit, lesson, or task is to make a connection to literature. the book Recess at 20 Below by Cindy Lou Aillaud is a wonderful entry point for the Snow Day task below. Click on the link at the right for a reading of the book. Be sure to purchase your own copy if you are going to use it in the classroom! the original reference is below. There is also an updated version of the book (2019) available.
Aillaud, C. (2005). Recess at 20 below. Portland, OR: Alaska Northwest Books.Snow Day
Snow Day
Work through the Snow Day task from Illustrative Mathematics. Be sure to not look at the Commentary until after you work through the task!
Reflection Questions:
What fractions understandings seem to be important to know in order to work through the task?
How might you employ MTP 3 (Use and connect mathematical representations) in the facilitation of this task?
Why might Recess at 20 Below be an engaging hook for students to begin the Snow Day task lesson?
The Language of Mathematics
The Language of Mathematics
Click on the link to the left to watch this TEDx talk from Randy Palisoc titled "Math isn't hard, it's a language".
After watching, find KY.3.NF.1 in your KAS-M document.
How does the Kentucky standard compare to the one highlighted in the video?
How might thinking about the language of fractions inform your instruction?
Scaffolding the Definition of a Fraction
Scaffolding the Definition of a Fraction
The video on the right illustrates how to scaffold the definition of a fraction as written in KY.3.NF.1.
Watch the video.
Practice teaching the steps of the video using a model different from the rectangular area model shown in the video (e.g., circular or hexagonal area model, set model, linear or number line model).
Scaffolding the Definition of a Fraction.mp4CSA.mp3
CSA
CSA
Students should have opportunities to build procedural fluency from conceptual understanding (MTP 6). Be sure students have experiences with content that involves the concrete, semi-concrete, and abstract representations. You can integrate these together into your instruction and use explicit instruction (HLP 16) to model each of the representations and make connections between them (MTP 3). An outcome is that students can select and appropriately use tools in strategic ways (SMP 5).
These representations don't necessarily follow a liner path (i.e. concrete to semi-concrete to abstract), although this may be the starting point. Some students, while mainly working with abstract representations, may find it useful to draw pictures or use manipulatives periodically to support themselves in making sense of problems (SMP 1).
Students will move at different paces through these representations. Provide them with tasks (MTP 2) and pose purposeful questions (MTP 5) that will support them in moving from one representation to the next (SMP 1). Although a student can work with an abstract representation for one piece of content, this does not mean they can do that for all content areas or domains of mathematics.
Models of Fractions
Models of Fractions
Area
Number Line
Measurement/Linear
Set
Teaching Fractions Using Manipulatives
Teaching Fractions Using Manipulatives
Watch this quick video from National Center on Intensive Interventions.
It is important to use various models for developing fraction sense. Some models work well for certain situations, but not for others. Exposure to different models allows students opportunities to make meaningful connections between areas of mathematics.
Activity and Reflection Question:
Review your unit/lesson plans around fraction instruction. Make a tally of the number of times each model is presented. What do you notice in the data? How might this noticing inform your instruction?
Changing the unit or the whole isn’t just about pattern blocks, tangrams, fraction bricks, Cuisenaire Rods or any other math manipulative. It applies to everyday life. Let’s go to the grocery store.
I want an egg. Eggs aren’t usually sold as a single unit. They are sold by the dozen. If the whole is the dozen, what fraction is an egg? (1/12) What fraction are the 2 eggs I scrambled for breakfast? (2/12 or 1/6). What fraction are the 3 eggs I need for a cake? (3/12 or ¼) In some stores you can purchase eggs with 18 in a carton. Keeping the dozen as the whole, what fraction is that carton of 18 eggs? (1 ½)
To make this point better, you may want to bring in some concrete objects (dozen eggs, 18-egg carton, 6-egg carton, 6-pack of cola, case of cola, etc.)
Reflection Questions:
Write out some other purposeful questions (MTP 5) you could ask.
What other examples from the grocery store can you come up with?
What other real world examples might you use beyond food?
What is the Whole.mp3Click above to hear Mark talk through this section!
Click above to hear Mark talk through this section!
The most important question to ask when working with fractions is, "What is the whole?" Without knowing what the whole is, you can't interpret a fractional part. As Petite, Laird, & Marsden (2010) p. 41 write, "a fraction should always be interpreted in relation to the specified or understood whole." So, even prior to thinking about if parts are equal, you must identify the whole. Here's a way to get students thinking about this question. Present students with the question, "Which do you want, a whole candy bar or half of the candy bar?" Have them move to the side of the room based on their answer and justify (MTP 4, SMP 3) why they moved there. Start with the "whole" spokesperson, then move to the "half" group. When they are finished, give the a "whole" group a mini-sized candy bar and the "half" group half of a large candy bar. Ask them about their assumptions as they answered.
Diving Deeper into Fraction Sense-Making
Diving Deeper into Fraction Sense-Making
Pattern Block Fractions
Pattern Block Fractions
This activity will help to build fraction sense by using pattern blocks, specifically by examining how each shape is named as the whole changes. The activity is from Amplify and is called Equivalent Fractions with Pattern Blocks. Click on the link to view the activity. Be sure to also click the links in the site to get the full PDF documents that accompany the activity.
Reflection Questions (write answers in your Participant Handout):
Identify and articulate to yourself any areas of struggle for you as you worked through these activities. How did you work through that struggle? In what ways could that inform you as you help students with their productive struggle (MTP 7)?
What misconceptions might surface as students work through these activities?
What are some possible purposeful questions (MTP 5) you could ask students as you facilitate their learning with these activities?
Paper Folding
Paper Folding
Make a copy of the Paper Folding task from YouCubed.org. This task is adapted from Driscoll et al. in their book Fostering Geometric Thinking: A Guide for Teachers, Grades 5-10 (2007). Work through the task. Be sure to convince yourself that your reasoning is correct with each part!
Blank Squares for Paper Folding you could cut out and use with this activity.
Reflection Questions:
What would be the learning goal for this task (MTP 1, HLP 12)?
How might you implement this task in your classroom so as to develop SMP 3 (Construct a viable argument and critique the reasoning of others) with your students?
What are your hunches about some of the misconceptions that could surface as students work through the task?
Developing Equivalence
Developing Equivalence
1) Explore the simulation Fractions: Equality from PhET Simulations and the University of Colorado Boulder.
This is a free, open source site for math and science simulations. It does require you to register to get all of the available downloads for the simulations.Work through the simulation.
How might you use simulations such as these to help students look for express regularity in repeated reasoning (SMP 8)?
2) As you move into the abstract representation, here is an activity from NRICH Maths that students can do. This Fractions Jigsaw has a self-check feature built into the task.
Blank Fraction Tiles
Blank Fraction Tiles
In this video, Christina Tondevold uses blank fraction tiles to discuss teaching partitioning and iteration. She incorporates ways to move from concrete to semi-concrete to abstract while you teach.
Fraction Cards
Fraction Cards
You can make use of these Fraction Cards from Graham Fletcher for teaching fractions from the semi-concrete representation.
Partitioning
Partitioning
If you haven't already done so, create an account by REGISTERing with the Kentucky Numeracy Project's Intervention Guide. These are free resources provided to every teacher in Kentucky by the Kentucky Center for Mathematics. If you have created an account, log into it.
Find and download the following activities:
KNP #F7701.1 "Fraction Squares--I can Share"
KNP #F7701.3 "Fraction Squares--What Fraction Part am I?"
KNP #F7701.4 "Fraction Squares--Build it! Draw it!"
Work through the activities.
Record key insights from your learning in your Participant Handout.
Reasoning
Reasoning
Make a copy of the activities Reasoning About Fractions. These are adapted from achievethecore.org. Some blank templates that may be helpful are linked below.
Blank rectangles and circles.
Blank number lines
Reflection Questions:
How does a task like Activity 1 elicit student thinking (MTP 8) about fraction comparisons? What other problems might you pose to your students?
How are the problems posed in Activity 2 different from the ones in Activity 1? What strategies did you use to compare the fractions?
How did you think about the tasks in Activity 3? How might you need to modify/adapt the tasks for the students in your classroom (HLP 13)?
Addressing Misconceptions
Addressing Misconceptions
The slide deck on the right is an activity which provides you an opportunity to consider student misconceptions. The slides contain two student work samples for the Illustrative Mathematics task 3.NF Ordering Fractions.
Click on the slide deck to get started. Slide 1 contains the directions.
Is it Bigger?
Is it Bigger?
You present the following problem to your students:
What are some of the responses you might anticipate?
How might your address any misconceptions your students have?
Additional Resources
Additional Resources
3rd grade fraction cards.pdf3rd Grade Fraction Cards
3rd Grade Fraction Cards
This card set contains various rectangular area model representations for fractions with denominators 2, 3, 4, 6, 8.
4th grade fraction cards.pdf4th Grade Fraction Cards
4th Grade Fraction Cards
This card set contains various rectangular area model representations for fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12.
3_KDE_Representing_Fractions_on_a_Number_Line_Grade_3.pdf3rd Grade Formative Assessment Lesson
3rd Grade Formative Assessment Lesson
This formative Assessment Lesson focuses on representing fractions on a number line. It includes a pre- and post-assessment, lesson plan, and materials.
Click on the link to complete the Exit Slip for this section of the training. After you submit the form, you will be sent a PD certificate for 2 hours of learning. The certificate will come from Certify'Em. Please check your clutter and junk folders if it doesn't appear in your Inbox. If you don't receive a certificate within 24 hours, please email mark.helton@ckec.org.
Click on the link to complete the Exit Slip for this section of the training. After you submit the form, you will be sent a PD certificate for 2 hours of learning. The certificate will come from Certify'Em. Please check your clutter and junk folders if it doesn't appear in your Inbox. If you don't receive a certificate within 24 hours, please email mark.helton@ckec.org.
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