Linear functions are major work in middle school mathematics, but before students can graph slope or write equations, they must understand what a function truly represents. In this episode, we trace the vertical development of functional thinking from 6th grade ratio reasoning to 7th grade proportional relationships and into 8th grade functions. You’ll hear how unit rates and the constant of proportionality evolve into slope, and why understanding how two quantities vary in tandem is the foun...
Mar 1
11 min
How proportional reasoning prepares students for modeling Over the past three months, we’ve built something intentional. Geometry helped students see structure. Number systems helped them understand magnitude. Ratios helped them recognize relationships. In this episode, we bring it all together. Proportional reasoning is not the end goal. It’s the bridge. Before students ever graph a line or write an equation, they must see patterns in how quantities vary together. We explore how ratio thinki...
Feb 22
11 min
Rates, Tables, Tape Diagrams, and the Coordinate Plane Ratios are relationships. But proportional reasoning is what happens when students learn to use those relationships to solve problems. In this episode, we move from identifying ratios to reasoning with them. You’ll explore how tape diagrams, ratio tables, double number lines, and the coordinate plane make proportional relationships visible before formulas ever appear. Geometry shows up through alignment, iteration, and scaling. Number sen...
Feb 15
19 min
Before students can work flexibly with ratios, they must be able to answer a more fundamental question: What exactly are we comparing, and why? In this episode, we zoom in on the core comparison structures that sit beneath ratios, proportions, geometry, and algebra: part-to-part relationships, part-to-whole relationships, and unit reasoning. You’ll hear how these ways of thinking develop over time, how they connect back to geometry through reference points, structure, and scale, and why the n...
Feb 8
16 min
Why ratios are about how quantities move together This episode launches our Connecting Math series by reframing ratios as relationships rather than calculations. Instead of treating ratios as fractions or procedures to memorize, we explore how ratios describe how two quantities vary together and why that way of thinking must be developed over time. Building on December’s focus on Seeing Math and January’s work around Understanding Math, this episode connects geometric reasoning and number sen...
Feb 1
16 min
How strong number reasoning prepares students for ratios and algebra Throughout January, we’ve explored rational numbers, negative numbers, distance, value, fractions, decimals, and division. On the surface, it may seem like this month was about numbers. But this work didn’t begin in January. In December, we focused on seeing math—using geometry to help students notice structure, reason about space, and make sense of relationships before symbols ever appeared. Those same ideas carried forward...
Jan 25
18 min
Building understanding instead of teaching tricks Many students reach middle school able to perform fraction and decimal procedures without truly understanding what those numbers represent. In this episode, we slow down and reconnect fractions, decimals, and division to meaning. We explore why fractions should be understood as division first, how the number line supports flexible movement between representations, and why spatial reasoning matters when students place, compare, and reason about...
Jan 18
21 min
Strategies that clarify the number line for every learner Negative numbers are often taught through rules that don’t stick. In this episode, we return to meaning. We explore how students have been reasoning about space, direction, and position on the number line since the earliest grades, and why negative numbers are an extension of that work—not a new concept. By grounding integer operations in movement and distance, this episode shows how students can reason about direction, magnitude, and ...
Jan 11
18 min
When students struggle with fractions, decimals, and integers, it’s often assumed they’re missing skills. In reality, they’re missing understanding. This episode opens our Understanding Math series by focusing on how students make sense of rational numbers as quantities that have value, direction, and position—not just symbols to manipulate. We explore how early experiences with counting, comparing, and representing numbers develop over time, why the number line is such a powerful organizer o...
Jan 4
13 min
This episode closes our geometry focus by showing how similarity and scale serve as the bridge into ratios, proportions, and later functions. We explore what makes figures similar, how scale factor represents a preserved relationship rather than a formula, and why students need visual experiences with enlargement and reduction before working with numbers. Along the way, we address real classroom challenges, including unfinished learning, limited instructional time, and the pressure to move st...
Dec 28, 2025
16 min
Load more