Category Archives: Concrete-Representation-Abstract

Unit Cost and Actual Shopping

I previously shared that grocery shopping has a lot of tasks that are overlooked. One is working with unit costs. There are two math tasks related to unit cost, interpreting what a unit cost is and computing the total cost for buying multiple items.

When I take a student into a grocery store to work on unit costs, where is what I do. I start with a pack of items (photo on the left) and ask the student to compute the cost for 1 item, in this case, “what is the price for the pack of chew sticks, how many in a pack, how much for 1 chewstick?” Then I prompt the student to compute the total if he buys 3 packs. This allows the student to differentiate between the two tasks. The cost per items is easier to grasp and then is followed with the same prompts for a jar of sauce (below right).

I then have the student compare unit costs for the large vs the small jars and ask, “do you want to pay $4.99 per ounce or $5.99 per ounce.” This language is more accessible than “which is a better deal?” You can work towards that language eventually.

These tasks can be previewed at school with a mock grocery store. The price labels can be created on the computer.

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Division of Fractions with Cookies

Division of fractions may be one of the most abstract concepts in middle school math. Here is an approach to address the concept using a Google Jamboard (you can make a copy which allows you to edit it), which would be a foundation for the ensuing steps. I will preface this approach by stating the obvious. Because this is very abstract and challenging for students, the approach is more complex – no royal road to dividing fractions.

To unpack this concept I start with the concept of division itself. One interpretation is distributing a collection of items into equal groups to determine how many items in each group. That lends itself well to dividing by a fraction. In the example below, I show 6 cookies divided into two groups to get 3 cookies per group. That is the goal, identify the per group amount.

Then we introduce a fraction. 6 divided by 1/2 can be stated in the group context as 6 cookies for half a plate or for half a group.

But we want a whole plate, a whole group. How do we get that? We need another half group which ends up revealing that we multiply by 2. (Keep in mind that the goal here is to unpack the concept and not so much the actual steps yet.)

Now we can turn our attention to the full dividing fractions situation. The approach is the same as the whole number divided by a fraction; we start with the fractional item in the fractional group. Then we build the whole plate (group) which results in building the whole cookies. At the end I take a stab at showing the mathy steps but I am unsure how I would unpack the steps at this point – again, focusing on the concept in this activity. I think I would not show the steps and have the students simply do hands on building a whole group, by manipulatives and subsequently by drawing.

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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.

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Intro to Reading Scatterplots – Used Mustangs

One step in reading and analyzing scatterplots is simply identifying what the dots on the graph represent. If students do not understand the dots (including the position) how can they analyze. An approach I have used is start by having students create their own scatterplot for mileage and price of used cars they shop for on Carmax.com. This allows them experience the scatterplot from a data and context point of view.

Then I present the scatterplot of used Ford Mustangs on a Jamboard (image above) with ads for two used Mustangs along with a cutout of each car. The goal is to help the students understand the reasoning behind the position of each dot.

First, I take the cutout of the first car and “drive it” along the x-axis (top 3 photos in gallery below). This helps them understand the horizontal axis placement. Then I move the car up to the appropriate price (bottom row left). Finally, I replace the car cutout with the bigger blue dot that was placed by the ad with the car. We then discuss that a dot can be used to represent that car and the location on the scatterplot is based on the two values in the ordered pair (which can be typed into the ( , ) in the Jamboard next to each car.

The same steps are used for the other Mustang (see it “driving” along the x-axis below).

The next step would be to identify additional points on the scatterplot. I then revisit driving the cars and show that driving the car more miles results in a lower price and driving the car less miles results in a higher price. Finally, we discuss that this is a general trend but that it is not always true for each car. I highlight a couple points where one of the cars has more miles and a higher price (below). This leads into a discussion about additional factors influencing price.

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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.)

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.

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Simplifying not so Simple Equation Solving

Several special ed teachers identified solving multi-step equations as the most challenging math topic to teach in middle school math. Here is my approach to teaching multi-step equations like 3m + 4m + 1 = 15. .

First, I use a task analysis approach to break down the math topic like we cut up a hotdog for a baby in a high chair. MOST of the steps involved are prior knowledge or prerequisites skills. I present these in a Do Now (warm up, bell ringer, initiation) – see image below. This allows me to fill in the gaps and to lay the foundation for the lesson. The prerequisite skills include simplifying expressions and solving 2 step equations. I also present meaning for the equation with a relevant real life problem that is modeled by this equation. By attempting the walkathon problem without the “mathy” approach, the students will more likely understand the equation and why they add 3m and 4m.

After reviewing the Do Now I use Graspable Math, which is a free online application that allows users to enter their own expressions and equations. These can be manually simplified and solved by moving parts around. Here is a tutorial on how to do this. This allows them to manually work with the simplifying and the equation before working on the handout, in a concrete-representational-abstract approach.

This is followed by a scaffolded handout with the use of color coding. I have student work on the first step in isolation as that is the new step (the other steps are prior knowledge and were addressed in the Do Now). This avoids all the work on the other steps that can result in sensory overload and allows me to address mistakes in the new content immediately.

This handout can have the equations removed and be used as a blank template to follow. In turn this would be followed with regular solving worksheets.

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Introduction to Unit Cost (Unit Rate)

Unit rate (e.g., hamburger meat on sale for $2.39 per pound or you make $13 per hour) is an incredibly important topic in middle and high school. First, unit rates and unit costs are common in life. Second, in the Common Core State Standards math categories you can see that Ratios and Proportions (which includes unit rate) are a 6th and 7th grade topic and are then replaced by Functions in 8th grade. Below is a photo showing a graph of a function you can see that the slope in an application is a unit rate.

The unit rate is also conceptually challenging whether it is in a function or is a unit cost at the store. This is a major sticking point for many students in special ed who have fallen behind. To address this, I used the approach below.

First, I present a pack of items the student likes (4 pack of Muscle Milk for this student). Use a Jamboard to show a 4 pack and the price of the 4 pack (photo on left). Then I “pull out” the 4 individual bottles and divide the $8 among the bottles to show $2 for each bottle. Finally, I have the student shop for packs of items at a grocery store or Amazon and compute the price for 1 item using a mildly scaffolded handout.

I Follow the same steps for ounces or pounds but show how 4 oz is divided into single ounces (in lieu of a pack divided into single items). Then the student shops for items that can easily be divided to get a unit cost.

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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.

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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.

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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.

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