Here is a link to a Jamboard for presenting the transformation (you click on the negative sign and rotate and slide it).
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.
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.
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
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.
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.
Finally, references to thumbs is replaced with the integers values. The thumbs tokens are maintained to allow for continued concrete representation.
To access the Jamboard you must make a copy.
I have frequently encountered the presentation of absolute value as a positive value or opposite. This is part of the repertoire of memory devices we (certainly I have in the past) use as a short cut to learning how to do the steps for a problem. The meaning of the absolute value of a number is it’s distance from 0 (below).
Below is an image of a Do Now or Initiation handout I use to introduce absolute value. From the start I focus like a laser on the meaning of distance for absolute value. I start with a situation that may be prior knowledge for them. Then take a step towards the mathy part with the numbers and slowly make my way to the symbol.
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!
All images were generated on this jamboard.
I have algebra students, in high school and college, who struggle with evaluating expressions like 2 – 5. This is a ubiquitous problem.
I have tried several strategies and the one that is easily the most effective is shown below. When a student is stuck on 2 – 5 the following routine plays out like this consistently.
- Me: “What is 5 – 2?”
- Student pauses for a moment, “3”
- Me: “So what is 2 – 5?”
- Student pauses, “-3?”
- Me: Yes!
I implemented this approach because it ties into their prior knowledge of 5 – 2. It also prompts them to analyze the situation – do some thinking.
I have posted on how to effectively provide support for current math topics. Here is an example (below) of how support can focus on both the current topic and prerequisite skills.
For example, on the 22nd in this calendar the current topic is solving equations. The steps for solving will include simplifying expressions and may involve integers. The support class can address the concept of equations, simplifying and integers which are all prerequisite skills from prior work in math.
This approach allows for alignment between support and the current curriculum and avoids a situation in which the support class presents as an entirely different math class. For example, I recently encountered a situation in which the support class covered fractions but the work in the general ed classroom involved equations. Yes, equations can have fractions but often they do not and the concepts and skills associated with the steps for solving do not inherently involve fractions.
The topic of imaginary numbers is one of the most abstract and therefore difficult math topics to teach in algebra. Here is how I introduce it to students (emphasis that this is only an introduction).
I write 1, 2, 3… on the board (see photo above) and explain to the student “at some point in life you learned to count on your fingers, 1, 2, 3…” These are called the Natural numbers.
Then I explain, “later you were told that no cookies means ZERO cookies. Zero is a new type of number. We call 0, 1, 2, 3… the Whole numbers. You learned a new type of number.”
This continues, “A little later on you were told you could have half a cookie and so you learned about a new type of numbers called fractions.”
This continues with negatives. Then I explain that all these number types can be found on the number line. We call the set of all of these numbers Real Numbers.
I conclude with “Now we have a new type of numbers that are not found on the number line. These are called imaginary numbers. Just like before you had number types you had before and now you have a new one to learn.”
The point of this approach is to help the students understand that a new number set simply builds on previous number sets. Also, the students have encountered this situation before.
This example involves adding integers which is a major challenge for many students. There are two strategies present in the photo.
Color coding is an effective way to break down a concept into parts. Here red is used for negative numbers and yellow for positive. The numbers are written in red and yellow with colored pencils.
The chips are a concrete representation. Typically integers are only presented in number form and often with a rule similar to the one below. The strategy is to count out the appropriate number of red chips for the negative number and yellow chips for positive number. Each yellow chip cancels a red chip and what remains is the final answer. If there are two negative numbers then there is no canceling and the total number of red chips is computed (same with positive and yellow).
- + + = +
- – – = –
- + – use the bigger number…
Rules are easy to forget or mix up especially when students learn the rules for multiplying integers. Concrete allows students to internalize the concept as opposed to memorize some abstract rule in isolation.