## Plotting Points Introduction

Plotting points is surprisingly challenging for some students. Here is an approach originated by one of my former math teacher candidates in a methods class I taught. This approach uses the analogy of setting up a ladder.

First, determine where to position the ladder, then climb the ladder. (brilliant and not my idea). Plot the point on the ladder, then pull the ladder away. The context includes green grass for the x and yellow for y because the y axis extends to the sun. This is shown on a Google Jamboard with moveable objects (you can make a copy to edit and use on your own).

Next, fade the ladder but keep the color – note the color of the numbers in the ordered pair. 3 is green so move along the grass to the 3. Then yellow 5 so move up 5, towards the sun.

Now, keep the the colored numbers and still refer to the green grass (faded) and sun (faded).

Finally, on a handout students can use highlighters as necessary to replicate the grass and sun numbers. The highlighters can be faded to result in a regular plotting a point problem.

## Counting Money – Jamboard

If you have a student who is learning to count money, here is a virtual set up to do so. I suggest having the student do a test run by moving coins into a box and bills into a box. It is easy to duplicate each item by clicking on the item to duplicate it.

If it works, you can insert images of items to purchase. Note, I start with just pennies or just \$1 bills and incrementally add additional currency. I also present items to purchase that are of interest to the student – the image below was used with a student who loves Minecraft.

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## Multiplication with Integers: a Meaning Making Approach

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.)

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.

## Scientific Notation – an Introduction

In a previous post I asked readers to identify a math topic that they wanted help unpacking. Scientific Notation was cited. Here is my approach to unpacking this topic.

My first step in presenting a new topic is meaning making. For scientific notation, the underlying idea is NOTATION – “the act, process, method, or an instance of representing by a system or set of marks, signs, figures, or characters.” We can represent numbers in different ways, one of which is scientific notation. This is useful to represent very large or very small numbers (as happens in science). It is useful because in lieu of writing out a bunch digits, the power of 10 can be used as a shortcut. In the image above you see that 4.5 x 104 has two parts, the decimal and the 10s.

Before I get into these big or small numbers, I address the concept of notation because that word is in the topic. To introduce a concept, I typically start with a related topic that is relevant for students. In this case it is money. To mirror the two parts of scientific notation, I list the bills and how many of each. In the left image below, I show both parts and pair combinations that are the same value (a single \$10 bill and ten \$1 bills). I then show how I can convert a single \$10 bill by dividing by 10 and then multiplying the number of bills by 10 (middle image). This previews the steps used in scientific notation. Then (right image) I show the same approach for dollars and cents (which previews decimals). Note: to help flesh out the dollars and cents I would first use the linked Jamboard.

The image below left keeps the concept of money, but the images are faded. The students are still working with money and how many but now with numbers only. The middle image introduces decimals, but the same steps are used (divide by 10 and multiply by 10).

Finally, the matched pairs shown in the previous handout pages (images above) are presented with an explanation of the parts of scientific notation (below left). I explain the idea of scientific notation as a special way to write numbers, list the two parts, and then I show examples by circling the ones in each pair (bottom left) that fit the criteria. Then they identify numbers that are written in scientific notation (below right).

Following this introduction lesson, I would explain the applications (linked above) and go into more detail on how to rewrite the number in scientific notation.

## Introduction to 2 Variable Inequalities

Previously, I shared how I use a Google Jamboard to introduce 1 variable linear equalities with a focus on conceptual understanding. I use the same approach for the 2 variable version (example problem below).

A conceptual gap that typically arises is the students do not understand what the shading represents. This is what I am addressing from the start using a Jamboard. First, the focus is on understanding the inequality and identifying a single point that works (below).

The next step is for students to determine more points that are solutions for the inequality, with no equal to part. (below).

The equal to part is addressed separately (below).

The equal to and the greater parts previously addressed are combined together.

The inequality is will be expanded to include an operation (+ 2) with a focus on the equal to part first.

The greater than with no equal to is addressed.

Then the equal to and greater than are addressed sequential. The equal to results in dots in a straight line and in lieu of plotting all the points, a line is drawn (building on the intro to 1 variable inequalities). This is followed by the greater than part and shading in lieu of plotting all of the dots above. THIS is where they gain an understanding of what the aforementioned shading is.

Finally, the dashed line is addressed by showing, as was done with the 1 variable inequalities, that there is a cutoff point that is not part of the solution set so in lieu of plotting a bunch of open circles, a dashed line is drawn.

## Life Skills Math – Not So Easy

As I wrote previously, shopping is dense with math tasks as are grocery stores. Here are some division situations that are sneaky challenging and require a student to know when and why to divide before even reaching for the calculator. I will use these to help illustrate the fact that life skills math is not simply counting money or using a calculator to add up prices. There is a great deal of problem solving and thinking skills that need to be developed.

For example, if a student has \$60 to spend on gifts for her 3 teachers the student needs to understand that she can spend up to \$20 per teacher (before even talking about taxes).

An entry point for division can involve a dividing situation the students intuitively understand, e.g., sharing food. Start with 2 friends sharing 8 Buffalo wings evenly (below).

This can lead into the 3 teachers sharing the \$60 evenly (below). In turn, this can be followed by the online shopping shown above.

This approach can be used to develop an understanding of unit cost (cited in the shopping is dense post). Start with a pack of items to allow the students to see the cost for a single item before getting into unit cost by ounces, for example.

I have had success with teaching these division related concepts using sheer repetition as much of our learning is experiential learning. Using a Google Jamboard as shown in the photos allows for the repetition.

## Introduction to Inequalities

For students with special needs, the teacher speaking “math” to students sounds like the teacher from the Peanuts cartoons.

This is apparently the case when students are learning about inequalities such as x < 4 because I have seen many high school and college students struggle with this topic. The challenge is that teachers are often focused on the math symbols and steps as opposed to the math concepts. In contrast, below are Google Jamboard slides I use (you can make a copy and edit) to introduce the concept of inequalities.

First, I start with a topic of interest and possibly prior knowledge for the students (age to get a drivers license – below). I present the idea of an inequality in context before I show any symbols. In this case, students identify ages that “work”.

Then I introduce the symbol (below). In this case, I include equal to for the inequality (x > 16 vs x > 16). The students plot the same points then we discuss that there are many other ages that work. These ages are called solutions. We put a closed circle on all of the solutions. Then discuss that ages are not exactly whole numbers so we can plot points on all the decimals. Then we discuss that the solutions keep going to the right so we keep drawing dots to the right. There are so many dots we draw a “line” instead of all the dots.

Then we do the same steps for a situation in which the number listed (52 in the case below) is NOT a solution. The students put dots closer and closer to the number but cannot put a closed circle on 52 as a solution (top photo below). Then we present the symbols and talk about the number as a cutoff point that we get really close to but cannot touch. Therefore we use an open circle to show the number is NOT a solution.

This introduction can be followed by problems on a handout, ideally with context then without.