Making Math Accessible for All Students
The Jamboard has images of basic base 10 blocks. The background provides side by side tables for numbers and for blocks. Additional blocks are set aside for regrouping. Here is a FB Reel and a YouTube video showing how to use this artifact.
Here is a link. You need to make a copy to access it
The first page of the handout provides an overview.
Note: the image of the 10s shows mistakes that are not in the actual handout.
Here is a link to the document. As stated in the overview on page 1, the elements of this handout may be cropped out and used individually. At least, the scaffolded rows can be covered to prevent the students from copy from previous rows completed.
I am interested in feedback on how to make this more useful or effective.
The student likes horseback riding. This can be useful to make the concept relative (10 horses live in a barn) and engaging. Here is a YouTube video and a FB Reel showing how this works. The key is 10 horses can “enter” the barn and disappear.
The Jamboard has a slide with a Legos version. This allows a nice transition from using actual Legos as the 1×1 blocks can connected to make a 10.
Here is a link to the Jamboard. You have to make a copy to use.
A visual of the hand releasing the balloon draws upon prior knowledge and provides a mnemonic for when to round up. The scaffolding provides support to focus on fewer parts of the problem. The supports are faded to incrementally increase task demand placed on the student.
The first page has maximum scaffolding with the student focusing only on which tens to round to. In turn, an a different element is the focus (e.g., writing the 0s then rounding) and then the scaffolding is increasingly faded.
Here is a link to a WORD document so you can change numbers AND add digits in the hundreds. Here it is in PDF for easier access.
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 related elementary school word problems with math topics in secondary schools. The photo below shows a method to help elementary school students unpack the multiplication problem, to help middle school students identify the unit rate, and to help algebra students identify slope (you can focus on simple problems like this as an entry point to the linear function type problems).
In advance of this method, a review is conducted on the representation of multiplication using the groups of items model (below). By drawing a picture for the two parts of the problem that have a number, the students are guided to break the problem into parts and then to unpack the parts. The “5 boxes of candies” is represented by squares (or circles if you prefer) with no items inside. The “each box holds 6 candies” is represented by a single square with 6 items (dots) inside.
In turn, the drawing of the group of items leads to the multiplication statement, “6 candies x number of boxes.” I prompt students to include the items with the number as sometimes they will write this statement as “6 x number of candies”. I point out that 6 and candies go together. As seen in the previous blog post, the next step in this problem would be to replace “number of boxes” with the quantity given and then compute.
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.
The Gutenberg printing press was revolutionary because it provided a faster way to share words. In turn, these words and how they were structured were representations of ideas used to make sense of the world around us.
Math is a language with words and other symbols that also makes sense of the world around us. We consume and know more math than we realize or allow ourselves credit for.
When buying the latest iteration of an iPhone, we may call forth algebra. How much will you pay if you buy an iPhone for $1000 and pay $80 a month for service? Well, that depends on how many months you will use this iteration before moving on to the next iPhone. The number of months is unknown so algebra gives us a symbol to represent this unknown number of months, x (or n or whichever letter you want).
Just as there is formal and informal English (or other language), we can engage algebra formally or informally. You don’t need to write an equation such as y = 1000 + 80x to figure out how much you will pay. You can do this informally, compute 80 times 10 months + 1000 on the calculator. Then try 80 times 12 months etc.
Math provides us a means of organizing and communicating ideas that involve quantities like the total cost for buying an iPhone.
The difficulty in learning math is that it is often taught out of context, like a secret code. In contrast, a major emphasis in reading is comprehension through meaning, such as activating prior knowledge (see below).
In fact, math absolutely can be taught by activating prior knowledge. An approach is to work from where the student is and move towards the “mathy” way of doing a problem.
Without meaning, students are mindlessly following steps, not closer to making sense of the aspects of the world that involve numbers.
The graphic organizer below is used to show the student the steps for addition. It also addresses the concept of addition (which I have addressed previously) as an act of pulling “together” two parts to form a whole.
The student is prompted to move the first part (set of coins in this case) to the number chart. This can be completed with 1 to 1 correspondence or with subitizing (identifying the number of items without counting). Then, the student is prompted to move the second part while counting on, e.g. 7, 8 etc. (as opposed to starting from the left and counting from the first coin: 1, 2 etc.). The chart scaffolds the counting on and allows the student to see the total as a magnitude.
It is important to first teach the students the “rules of the game”, i.e. how to use the graphic organizer. To do this have the student simply move the first part to the number chart then the second part. The student can also be prompted to state the addition problem (written at the top). When the student is fluent with these steps the counting on can be implemented.
The next step would be to replace the coins with the symbolic representation, numbers.
Comparing numbers is very challenging for some students and likely speaks to a major gap in their number sense. It is also very challenging to address effectively with these students. The photo below shows an entry point into I have used with success in helping such a student.
Even for these struggling students, taller and shorter are likely prior knowledge that students understand intuitively. The people outline on the vertical number chart leverages this intuitive understanding to compare numbers.
For a change of pace, I created a revised version of the card game WAR by using these outlines (no face cards). The cards are 5″x7″ which I purchased on Amazon. Here is a link to the WORD document with the outlines I cut out and taped to the cards. After a student shows success with this revised game, I play regular WAR but use the people outlines for feedback to make a correction or as a prompt as necessary, with the intent to fade their use.
