The Product and Quotient Rules are challenging largely because there are few ways to present them in an accessible format. This post outlines an approach using manipulatives on a Google Jamboard.
Jamboard
The Jamboard activity involves using moveable variables as manipulatives to provide a hands on approach to learning the rules. The first slide presents the idea of exponents as repeated multiplication. The Product and Quotient Rules use the manipulatives approach to unpack the underlying concepts of the rules. Note: there are set problems followed by blank templates. Here is a FB Reel and a YouTube video showing how this works.
Operations on integers and integers in general is challenging for many students. Negative numbers are abstract. Whole numbers and fractions can be represented with images. The activity presented draws upon student prior knowledge of thumbs up and down in a vote to make negative more accessible.
The following images are from a Jamboard. Here are accompanying videos on FB Reels and Youtube showing how this works. There are 3 sets of images (or chunks) loosely following a CRA appoach and all referring to the same two situations.
Prior Knowlege drawing upon a classroom setting (concrete)
Transition using thumbs (representational)
Introduction of adding integers, using thumbs (more abstract but still supported by
Prior Knowledge
The activity starts with a couple of classroom votes using thumbs up and down.
This is followed by scaffolding to focus on how the voting works through a comparison of the quantity of thumbs up vs down.
Transition
This section introduces thumbs as counters for integers, which is a common instructional strategy (yellow for positive and red for negative). The scaffolding is the same.
Here is a Jamboard (see photo at very bottom for how to make a copy to edit) that presents like terms as visual manipulatives and then eases into the symbolic form – the “mathy” stuff. The following shows each slide as is and how it looks after completion.
Start with two groups of common items, pull them together because they are alike, and compute the total. This slide is designed to introduce the students to how the manipulatives work for this Jamboard and to introduce them to the concept of like terms.
This slide introduces actual like terms. It is the “mathy” representation of the previous slide and is the most basic form of simplifying algebraic expressions. (The phrase, simplifying algebraic expressions, can be introduced later to allow the focus to remain on “like terms.”)
This slide introduces different types of terms at a conceptual level and how they are rearranged into like groups. It also introduces the use of a binary operation between the two groups of terms (vs adding everything together as the students are wont to do). This slide also establishes color coding by like terms, which is useful for when the work shifts from concrete form (manipulatives) to abstract form (written symbols).
This is the mathy version of grouping by like terms. This slide is crucial as it presents the Associate Property with the binary operation symbols acting for a moment as unary operations. In other words, the addition symbol follows the 2b, the 4t, and the 1b before latching on to another term. It is easier to address this while all operations are addition. Present the original problem as an algebraic expression and not individual pieces “3t plus 2b plus 4t plus 1b”. NOTE: I keep the coefficient of 1 to reduce task demand – one less thing to think about. I address the implied coefficient of 1 after students have had ample exposure to like terms.
The use of 1 dollar bills is intended to introduce constants. I fluctuate between whether to write the terms for the bills with or without units (4 dollars vs 4).
At this point, I suggest giving students independent practice with expressions that have addition only before moving on to the following slide.
This slides shows how I introduce subtraction and negative terms. The image shows a woman eating a taco, hence it cancels one of the tacos.
I present the eating 1 taco image as a negative, with the terms separated as opposed to being an expression. This allows the students to see the “-” symbol as a unary operations (negative) and then as a binary operations (subtraction). In other words, they see the symbol “attached” to the term. It is a prelude to the use of the Associative Property with subtraction.
The new image is of a person eating a burrito. The slide introduces the concept of the negative term in the Associate Property.
**The students are now presented with the what is likely the most challenging aspect combining like terms, which is the “-” fluctuating between being a unary and binary operation. The original problem is again presented as an algebraic expression, “2b minus 2t plus 4t minus 1b”. The minus 2t is converted into negative 2t while the minus 1b is converted temporarily converted into negative 1b as it is moved, but then is converted back into minus 1b. This is the opportunity to unpack this situation. You redo the problem with the 4t minus 2t and negative 1b + 2b (and add the plus symbol) as part of the discussion. You can also duplicate this slide and revise into different problem.**
The next two slides have the color and the “?” sticky note faded. They can duplicate an existing sticky note to record the final answer (or you can add in the “?”).
Finally, students are presented static algebraic expressions. I return to color coding as this is a support they can take with them to handouts. Eventually, the color is faded on the handouts, but they can still use shapes to indicate the different types of terms.
In my experience, many students struggle with translating word problems modeled by a proportional relationship or a linear function into a algebraic expression or equation. (Image below.)
Here is a link to a Jamboard that can be used to introduce the algebraic expression to model the unit rate. (See image below on how to copy and edit.)
Here is how you can use this to introduce modeling the word problem.
Start with the unit rate concept. In this case there is $45 “in” every hour. This is modeled in slide 1 (top 2 photos).
The next 2 photos show slide 2 in which the student duplicates the $45 image and fills 2 hours, with $45 “in” each. They complete the multiplication expression by multiplying by 2.
This is followed by the same steps for 3 hours (photo bottom left) and sequentially to 6 hours.
In the last slide there are no hours shown because the # of hours is unknown. This leads to using “X” to represent the unknown NUMBER of hours (I don’t let students get away with x=hours) and finally the algebraic expression (bottom photo).
Here is a CRA approach to multiplication that can be individualized. I created this one for a student who loves cats.
Here is a link to the Jamboard. You make a copy by clicking on the 3 dots at the top right, then you can manipulate the items.
Here is how I use this Jamboard.
You have a slide with a problem (5×2) and the appropriate grouping (2 in this case).
Go to slide 1 and choose the appropriate number of items in the group (5 in this case).
Copy (use CTRL C) and paste the appropriate # of grouped items into the groups (boxes).
Click on the =? to enter the answer.
The slides have groups up to 6.
You can have students personalize by choosing their own image. They can paste the image repeatedly to create the grouped items then snip the grouped items as a single image.
The game is played on a Jamboard. There are moveable game pieces on the left (Lego figures chosen to mirror the players – no hair is me), along with movable bills. There is a white rectangle partially covering the cashier’s money. It is a moveable rectangle I use to reveal the money when the cashier pays out money to a player. The money is subsequently covered again. When money is paid, the appropriate number of bills are moved to the cashier’s counter. Change can be computed and given. (technical note: you can click on an object on Jamboard to change its order, e.g., click on the bills to move them back, behind the rectangle.)
The game is a version of the Allowance Game, which is appears to be a version of Monopoly. The goal is to simulate budgeting and real life spending situations in an interactive and gamified way. The spaces can be revised to cater to the interests and reality of the players. The activities are all ones that I have used in isolation with students I help. The game can be played online and with multiple players who need to learn consumer math topics. (When you share the Jamboard with others, you can make them editors which allows them to move pieces.)
Players start with money in a bank account (center of the board) and then roll a virtual die and move accordingly. If a player does not have enough money for a spending activity, the activity cannot be completed. For some experiences, they are limited in what they can spend, e.g., buying a birthday present. (For rent, I will use an IOU until the player makes enough money – obviously a lot to possibly unpack in this scenario.) Spaces have an activity that falls into one of three categories:
earn money at a job by rolling dice for the number of hours worked
have a static money experience, e.g., get $20 from birthday or spend $12 on tooth brush
have a dynamic money experience, e.g., spend money on Amazon or attend a baseball game and the player goes to a related website (for example, a player buys a Red Sox ticket and a YouTube video of highlights of a game is shown – maybe 2 minutes)
A couple notes: I left the START space empty and am thinking I will move the find a job activity to that space. As of this posting I did not have students find a job yet and simply opened a link to a Target store site for employment and showed them a job ad. I think I will start the game with each player finding a job and rolling the die to earn money at the start.
Below are the steps I use to create and revise the game. If you have any suggestions, please post in the comment section.
Here is a link to a master copy of the game on a Google Doc. It can be copied and then revised. I store my game pieces on here as well.
Here is a link to a master copy of a WORD document I use to position the board with a cashier to create the image shown on the Jamboard.
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 below) 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). I 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. 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.
