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.
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.
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!
There is a delineated sequence for teaching multiplication over the years, including repeated addition, set modeling, arrays, single digit etc (below). It exists to build conceptual understanding of the multiplication facts that are at some point memorized by many students. When I work with students who are a more than a year behind in the sequence for multiplication, I find that programming for these students to help them catch up sometimes involves shortcuts such as a reliance on rehearsal or resorting to use of the multiplication table in isolation. I am not against use of the table or narrowing the focus, but am promoting a more comprehensive approach.
Here is a sequence, on a Jamboard, I used for a recent student who was struggling for a long time with multiplication (explanation of each step shown below images). The student was interested in Minecraft so I used Minecraft items such as stone bricks and a wagon. I would spend as much time on each step, as necessary.
Count out the total number of stone bricks. This allows an assessment of how the student counts: by 3s or individually. If individually, I would prompt the student to count by 3s.
Add 3 + 3
Show a short video on the wagon (this adds interest and gives the students a bit of a break)
Present the bricks in 2 groups of 3, in context of 2 wagons with 3 bricks each.
Present the same problem as a multiplication problem but with the image for one of the factors in lieu of two numbers.
Use the multiplication table to skip count.
Present additional multiplication problems for independent attempts. The student completed both problems independently, without the table. For him this was a major success.
The follow up to this would be to assess his ability to do higher groups of 3s and groups of other numbers. For some students, I work on mastery of individual numbers before moving on. This builds confidence and allows for fluency in the process of skip counting out to the appropriate number. NOTE: I don’t worry about rote memorization of the facts but of fluency in the process of skip counting out the answers.
For students who are older, I sometimes recommend that the student be presented problems with visuals but then use a calculator to compute. This can develop conceptual understanding and also address the working memory and other related issues that undermine learning math facts.
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.
Money is intuitive for many students, even when the underlying math is not. For example, I often find that students who do not understand well the concept of Base 10 place value do understand $10 and $1 bills. With this in mind, I created a virtual scaffolded handout that builds on student intuitive understanding of the bills through the use of $10 and $1 bills to represent regrouping. Here is a video showing how I use it.
In the photo below, at the top, a $10 bill was borrowed into the ones column. The reason is that $7 needed to be paid (subtracted) but there were only five $1 bills. In the photo below, bottom, the $10 bill was converted into ten $1 bills. On the left side of the handout, the writing on the numbers shows the “mathy” way to write out the borrowing.
Once the student begins work with only the numbers, the $10s and $1s can be referenced when discussing the TENS and ONES places of the numbers. This will allow the student to make a connection between the numbers and their intuitive, concrete representation of the concept.
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 and, in my view, should be taught by activating prior knowledge. My 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.
I start by showing 2 different outlines and asking which is “taller” and stick use that term until the student gets the idea. This isolates the focus to comparing items.
Then I redo all the comparisons using the term “taller” and when the student makes a selection, 10 in this case, I reply “yes, 10 is MORE!” and have them repeat “more.”
Finally, I redo the comparisons asking which is more and for improper responses I ask which is taller then restate that the taller item is “more.”
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.
I was recruited to help a middle school student who is having a very rough time at this time in his life. It was shared with me that he likes Marvel superheroes and he is struggling with counting money and multiplication. Below are some ideas I presented for a test run and photos of the items I ordered for these suggested activities.
Put the heroes (or villains) in groups of 2 and have him count out 4 groups and compute. Use different groups and number per group. (IGNORE the numbers on the cards)
Get a group of 10 villain cards. Pretend heroes have to travel in groups of 2 and ask how many groups to get 10 heroes to fight the 10 villains. (IGNORE the numbers on the cards). Variations of this.
After gets the idea of groupings, focus on the number on the cards and show him two 5s and have him compute. Variations of this.
Play a game where he draws two cards and has to multiply the cards (start with very low numbers or maybe show him a 2 card and he has to pick another card to multiply by 2.
Tell him he earn money to buy these figures, one at a time – a monetary version of a token economy. Have him rank them by his favorite to least favorite and come up with a price for each with his favorite figures costing more. Start with the least favorite and make the price such that with a little practice he could count out the coins to pay for it. Maybe 17 cents with dimes and pennies. He has to count out the money correctly and independently to actually buy the item.
If he needs work with addition you can play WAR in which 3 cards are played and each person adds to find the total. For subtraction do the same with 2 cards.
You can play subtraction in which one person has superheroes and the other has villains. In order for a villain to win a villain card has to be higher than a hero card by 3 or more.
You can write an 11, 12 and 13 on the J, Q, K cards respectively. All the games can be presented though Direct Instruction – I do, we do, you do. The You do can be used as daily progress monitoring. If he needs prompting this can be recorded. This can be used for your progress reports. Attached is a data sheet I use for activities.