Here is a link to a Jamboard for presenting the transformation (you click on the negative sign and rotate and slide it).

Here is a link to a video showing how this transformation. This video addresses the underlying concept of the transformation by showing an IOU being cancelled.

The effort to provide intervention to fill in gaps is challenging for different reasons. One reason is the effort to balance support for current content while filling in gaps. This post shows an example of how to fill in gaps while working through the current topic.

Overview

Various rubrics used to assess teacher instruction includes an effort to build on or connect to prior knowledge. If the student has gaps with prior knowledge, the lesson becomes less accessible for students with the gaps. Previously, I addressed how to support both current content and fill in gaps. The idea is to systematically fill in gaps by addressing prerequisite skills as they arise in new lessons.

Example

The handout out below shows an example of how this can play out. The first page is used as a do now for the content presented on page 2. If you are teaching a student how to solve 1 step equations and are moving into integers, page 1 is a a means of supporting the new content while filling in possible gaps. The first image shows the student will need to evaluate -13 – 3 as part of the solving in the lesson. This can be addressed in the do now, as shown in the 2nd image, page on the right. (Notice all the problems on page 1 are steps to solve on page 2 problems.) This is useful for students with special needs and for differentiation.

Operations on integers and integers in general is challenging for many students. Negative numbers are abstract. Whole numbers and fractions can be represented with images. The activity presented draws upon student prior knowledge of thumbs up and down in a vote to make negative more accessible.

The following images are from a Jamboard. Here are accompanying videos on FB Reels and Youtube showing how this works. There are 3 sets of images (or chunks) loosely following a CRA appoach and all referring to the same two situations.

Prior Knowlege drawing upon a classroom setting (concrete)

Transition using thumbs (representational)

Introduction of adding integers, using thumbs (more abstract but still supported by

Prior Knowledge

The activity starts with a couple of classroom votes using thumbs up and down.

This is followed by scaffolding to focus on how the voting works through a comparison of the quantity of thumbs up vs down.

Transition

This section introduces thumbs as counters for integers, which is a common instructional strategy (yellow for positive and red for negative). The scaffolding is the same.

Adding Integers

Finally, references to thumbs is replaced with the integers values. The thumbs tokens are maintained to allow for continued concrete representation.

I previously tackled the difference between the ” – ” symbol used to represent a negative and a subtraction problem. This post gets into multliplication through the context of buying a Wendy’s frosty.

One comment to preface what is presented here. Negative numbers are abstract and challenging for many students. Multiplying by negatives is even more so. The approach presented here for multiplying gets a little complex, which is inherent to the topic. In other words, this is an involved process as “there is no royal road to geometry.” What I present here is a path I develop over time, as seen in the sequence below.

First, review of a couple building blocks. Multiplication can be represented at groups of items. 2 x 3 can be represented as 2 groups of 3 $1bills, e.g., you bought 2 Frosties for $3 each.

Negative in terms of money can mean you owe money. Hence, -3 means you owe $3, e.g., you order a frosty and owe $3.

If you change your mind and cancel the frosty, the $3 you owe is cancelled and you get your money back. +$3.

If you order 2 frosties, you owe $3 and another $3, which is -3 + -3. (You owe $6 or -6.)

Cancel those two frosties and you get your money back. -(-6) is now +6

2 x -3 means you means you have 2 negative 3s – repeated addition, or -3 + -3.

If you ordered the 2 frosties and owe $6 then cancel, you get your $6 back or +6.

In mathy terms, cancelling the 2 x -3 is written as -(2 x -3) or -2 x -3.

The – for the -2 can be held out front to focus on 2 orders of frosties or 2 x -3. That was covered previously. Then the extra negative cancels that order so you get your money back. And so you have multiplying two negatives!