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.
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.
If you have taught algebra, you have likely experienced this error. We know many students will make this type of error and we can help many students avoid it by being proactive.
Below are excerpts from a Google Jamboard that can be used to unpack the underlying concept of the division or simplifying shown above. First, start with prior knowledge students can relate to, presented as manipulatives.
Then move, in a CRA fashion, a step towards more symbolic representation of the concept.
Finally, represent the situation in symbolic form. The focus here is to show the problem as two separate division problems to emphasize that both terms are divided. Then write out the simplified expression below.
The approach shown above is an entry point to simplifying rational expressions, with the same type of common errors we see there as well.
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.
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.
The photo below is courtesy of Robert Yu, Head of Lego Education China, as shared by Jonathan Rochelle, Director for Project Management at Google.
The use of Legos shown here is a classic (and wicked clever) example of manipulatives.
Before writing the actual fractions students can use drawings as shown below. The sequence of manipulatives, drawings then the actual “mathy” stuff constitutes a Concrete-Representational-Abstract (CRA) model. Concrete = manipulative, Representational = picture, abstract = symbolic or the “mathy” stuff.
Here is an easy way to create and implement strategy to unpack place value for students (created by one of my former graduate students). I suggest using this after manipulatives and visual representations (drawing on paper) in a CRA sequence. It is hands on but it includes the symbolic representation (numbers). Hence is another step before jumping into the mathy stuff.
Simplifying expressions (see photo below) is one of the most challenging algebra tasks for many students receiving special education services. A major problem is that it is typically presented as symbol manipulation…addressed in very symbolic form.
My approach is to make math relevant and more concrete. Below is a scaffolded handout I use to help unpack the concept. In the handout I start with items the student intuitively understands, tacos and burritos or tacos and dollar bills. In the top left of this handout the student is asked how many tacos he or she has. 3 tacos eventually is written as 3T. See next photo to see how the handout is completed as NOTES for the students.
As I work with the problems below I remind the student that the “T” stands for taco so “3T” stands for 3 tacos. This takes the student back to a more concrete understanding of what the symbols mean.
To address negatives I use photos of eating a taco or burrito. “-2T” is eating 2 tacos.
So “3T – 2T” means I have 3 tacos and ate 2. I have 1 taco left… 1T. For students who may need an even more concrete representation, use actual tacos or other edible items.
Here’s a common word problem used for linear functions and equations (y=mx+b):
There are 6 inches of snow on the ground. Snow is falling at a rate of 2 inches per hour. Write a linear equation showing total snow as a function of time (in hours). The equation would be y=2x + 6.
Often the word problems like this are presented on a sheet of paper in isolation as an attempt to make the math relevant and to develop conceptual understanding. For students who have trouble with conceptual understanding, words on paper are likely too abstract or symbolic to allow applications like the one above to be meaningful.
The real life application is useful if presented more effectively. Here’s an approach to use the same scenario but in a more relevant and meaningful presentation. The photo above shows the current amount of snow – call it 6 inches. Students can be shown the photo to allow for a discussion about accumulation and for their estimates of the amount of snow shown. The photo below shows an excerpt from a storm warning. Showing this warning and a snow fall video can allow for a discussion about rate of snow fall and the purpose for storm warnings. Combined, this approach can lead into the above word problem.
Once the application is presented students can be asked to compute snow levels after 1 hour, 2 hours etc. Then they can be asked to determine how long it would take for the accumulation to reach 18 inches (the prediction for the day this post was composed). After computing the answers WITHOUT the equation the students can be shown how to use the equation – the “mathy way.”
Slope is one of the the most important topics in algebra and is often understood by students at a superficial level. I suggest introducing slope first by drawing upon prior knowledge and making the concept relevant (see photo above). This includes presenting the topic using multiple representations: the original real life situation, rates (see photo above) and tables, visuals, and hands on cutouts (see photos below).
A key aspect of slope is that it represents a relationship between 2 variables. Color coding (red for hours, green for pay) can be used to highlight the 2 variables and how they interact – see photo above and below.
The photo below can be used either in initial instruction, especially for co-taught classes, or as an intervention for students who needs a more concrete representation of a rate (CRA). The clocks (representing hours) and bills can be left in the table for or cut out.