# Uncategorized

## Teaching Angles

We are surprised every year (re-surprised?) at the most common mistakes fifth graders make around angles. What’s NOT surprising: The difficulty of choosing WHICH of the 2 numbers you encounter on the protractor… Is is 60º or 120º? What IS surprising

## The Case for Withholding Algorithms (For a While!)

In 1998, Tom Carpenter and his colleagues documented grades 1–3 students’ use of invented strategies and standard algorithms. The vast majority of students in the study used some invented strategies. The researchers found that students who used invented strategies before learning standard algorithms showed better understanding of place value and properties of operations than those …

## Decimal Multiplication – a Visual Approach

Our 8th grade teachers tell us that of the most confusing ideas for students is the difference between linear units (the sides of a rectangle) and quadratic units (the area of a rectangle). Students routinely confuse x,  x-squared, and x-cubed, without realizing what each represents. This concept SHOULD reach back to a conceptual development in …

## Teaching Decimals with Decimal Squares

The most visual, most fun way to learn decimals is with decimal squares. We use decimal squares we bought from https://decimalsquares.com  several years ago.  They show decimals in tenths, hundredths and thousandths, and the equivalencies are obvious. (see below) The website has fun interactive games, too, but you need to download the Shockwave app to …

## And Quizzes? How are we doing?

Word doc:   Quiz 3 1. Fractions. Yay – we’re getting this!  Multiplication plus subtraction with borrowing. Example 3 below is one of the few with a mistake, and its last step is correct, it’s the first step that’s not. THIS  problem, however, lead us into the pit of confusion.  Only the last example below is …

## Homework: Goals and Structure

Recent Homework assignments:  (mostly available as Word documents) HW#1       HW#2         HW#3     HW#4     HW#5      HW#6   HW#7   HW#8 Solution Keys:  HW_keys#1to#8 What are our main goals in assigning homework? Provide thoughtful practice, not drill. Hopefully in an independent setting, without help, so we can see how the learning is going. To allow for long periods of …

## Experimenting with 3-act Activities

We’ve heard so much about 3-Act activities. Here’s this week’s experiment, with our verdict at the end. Day 1… Does \$1 million fit in a briefcase? Show this GIF: Ask… “Would \$1 million fit into a normal briefcase, and if so, could an average adult carry it? Assume the bills are all 100s. When students …

## Part 2 of “What Does Remediation Look Like?” Michael’s silence. Plus Exploding Dots.

In our last blog, we asked, “How do we humans react to confusion and difficulty?” Amelia claims grumpily “I don’t get the multiply and divide by 10 thing.” She tends to get grumpy when confused. Michael withdraws into silence. “Shall we get the blocks for this, Michael?” Silence. “Which part is confusing?” Silence. Sigh. Tomas …

## What Does Remediation Look Like? Aka “Bring Your Knitting”

How do we humans react to confusion and difficulty? Amelia claims grumpily “I don’t get the multiply and divide by 10 thing.” She tends to get grumpy when confused. Tomas minimizes his struggles. “I’m fine now. I was just confused on yesterday’s quiz. I’m good now.” Nicole writes notes to us on her quiz. “I …

## Looking at Quizzes- Part 2 … How are we doing?

Quiz 2, Trimester 2, as a Word Document: Question #1: Fractions.  Everyone was able to solve this problem correctly!  This is huge. Fractions matter.