Here is a Jamboard to introduce volume and units of volume. (See photo at very bottom for making a copy to edit.)

The students start with building an animal pen and shading in the space inside. The hands on approach and connection to prior knowledge of a fenced in area for animals sets the stage for actual measurement units in subsequent slides.

The photos below show how students will count out meters and square meters, adding a formal layer to the fence they built previously.

The following slide provides an entry point to understanding volume and units for volume. The students count out cubes, building on the counting of meters and square meters. The cubes were created using WORD Paint 3D. Here is an article I used to create these. For the grid that is tilted, I used functions on WORD – see this document. (I could have used Paint again.) I then show them the prism that is created but I am not discussing shapes yet to keep the focus on the concept of volume.

I then have students recreate the volume using NCTM’s Illuminations activity called Cubes.

Finally, I show examples of volume and move from cubic units to liters (litres – as I was initially teaching this lesson to a 5th grade class in India).

To introduce ratio tables, I draw upon a relevant prior knowledge for a child. Food, especially pizza is a go to context for me.

Use a CRA approach by using manipulatives (concrete), pictures (representational), and numbers (abstract). The ratio can be changed and other contexts can be used (e.g., $3 per slice and use dollar bills and the slices).

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. This is useful for developing number sense.

I also present items to purchase that are of interest to the student – the image below was used with a student who loves Minecraft.

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!

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.

For years over many settings from middle school to college I have witnessed students struggle to make sense out of inequalities like x < 4. Not only is the concept of an inequality of an inequality challenging, merely reading the symbol is problematic. This post shows an concept based approach to making sense 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.