How important is fractional equivalence? Important enough for another model! These models are fantastic for helping students develop a natural understanding of equivalent fractions.  Listen in as Kim and Pam discuss how to use a clock model for denominators uncommon to the money model.
Talking Points:

• Use adding and subtracting fractions to help think and reason about fraction equivalence
• Using the word "over" for fractions is NOT helpful
• Equivalent fraction examples on a clock model
• A real student's math journey

Jordan Using the Clock Model 

In this episode Pam and Kim finish up their money Problem String from the last episode. But first, Pam wanted to reflect a little on how she was less than helpful when Kim was struggling to think about nickels. We all learn and grow, and self-reflection is super important!
Talking Points:

• Do you need time, or do you need help? (see episode 40)
• Once kids have an answer, do we force them to find another way?
• For which denominators is thinking about money helpful?
• Ways to think about 2 2/5 minus 9/20
• The reciprocal relationships between fractions and coins can be confusing
• Fractional equivalence is the goal by asking students to add and subtract fractions

It's time to talk about everyone's two favorite things: fractions and money! Fraction equivalence is a super important concept in mathematics, and it is best developed naturally in a familiar context. In this episode Pam and Kim model using coin denominations to  reason about fractional equivalence. 

Talking Points:

• If you understand fractional equivalence, you don't have to memorize any rules for operators of fractions
• Examples of using the money model to think about fraction equivalence, and bonus, fraction operations as well:
• Using decimals and fraction notation interchangeably
• Kim's uncomfortable relationship with nickels

Let's keep talking fractions! In this episode Pam and Kim discuss an interesting rich task from Cathy Fosnot to help students reason about fractions within the context of sharing fairly. Listen in to learn about the outcomes teachers should look for when teaching fractional sense in a rich contextual problem.
Talking Points:

• Splitting the wholes into unit fractions 
• Splitting the leftovers into smaller unit fractions and re-unitizing
• Patterns when comparing same-denominator fractions vs same-numerator fractions

See Ep 103 for more discussion on Rich Tasks

Additional Sources:  Field Trips and Fund-Raisers

How young can young learners begin to experience fractions? Younger than you think! In this episode Pam and Kim describe simple ways we can help young students begin to develop fractional understanding. Pay attention to how natural it can be for young learners to learn math!
Talking Points

• Struggles that young learners may experience with "half"
• Simple classroom activities to have students fair shares in early grades as they practice counting
• Experiences and conversations in halving even and odd amounts
• Standards and fractions of an object vs fraction of a set
• Important unit fractions and resources
• Doubling and halving simultaneously
• Simple experiences parents can use to help their kids develop fractional understanding 

Additional sources:

• Beads, Shoes and Making Twos
• Games for Early Number Sense

Time to get into fractions! There is so much to learn about teaching fractions and in this episode we wanted to lay some important ground work.  All too often we limit students by giving them experience with only one or two of the important ways fractions can be interpreted. When we help students fully understand fractions by using all of the interpretations, their ability to think proportionally skyrockets!
Talking Points

• What is a rational number?
• The 5 Interpretations of Rational Numbers (Credit to Susan Lamon)
• Solving fractions with reasoning!
• Shifting between the 5 interpretations as we solve
• Applying all 5 interpretations to one problem

Though some people think Kim and Pam's voices are hard to tell apart, teaching Real Math has nothing to do with a particular style or even personality type. We believe in seeing the humanity in students and empowering them with mathematical confidence!
Talking Points:

• Pam and Kim's results on the Enneagram Test.
• 2 important things - 
  1. Teaching Real Math is more about beliefs about students and mathematics than personality
  2. Leverage your beliefs to get support in teaching Real Math.

If there's one thing that often comes up in teachers' conversations, it's standardized tests. And Pam and Kim are not afraid to tackle the tough conversations! In this episode Pam asks Kim about her experience with standardized tests, and how teaching Real Math helped Kim's students to succeed. Also don't forget to send in questions for our special 50th episode! We're so excited!
Talking Points:

• Can students succeed on tests if all they know is Real Math?
• Everyone, in every demographic, can learn more Real Math than fake math.
• How much time do students need to prepare for tests? 
• Suggestions for test prep.
• How teachers view the test and their actions speak loudly
• It's all about "know your content, know your kids".

Learn about numberless word problems here: https://bstockus.wordpress.com/numberless-word-problems/

In this final episode in our series on homework, Pam and Kim discuss what types of homework teachers should send out, as well as some important pitfalls to avoid. 
Talking Points:
Pen and paper homework

• Games as homework and things to consider
• Computer activities as homework and things to consider
• Communication is critical to honor family time when assigning homework
• Using homework as a teaching tool, not for the sake of work
• Be sure you are sending the right message about mathematics when looking at returned homework.
• How the goal of Real Math helps teachers make decisions about homework

It's part two of our series on homework! Last time we heard Kim's perspective on when and what kinds of homework to give. Now it's Pam's turn. As a teacher for both high school and university classes, Pam gives an insightful look into when homework is necessary, and how we can always allow the humanity of our students to have precedence.
Talking points:

• What does the research say about homework? How does it vary by grade level?
• What is the purpose of homework?
• How much homework should students have?
• What topics should homework cover?
• The role of students' humanity in grades and homework.