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.
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
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.
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.
Finally, references to thumbs is replaced with the integers values. The thumbs tokens are maintained to allow for continued concrete representation.
Below are images from a Google Jamboard for a hands on introduction to ratios. (See image at the bottom for how to make a copy in order to use it.) The images are from Clever Cat Creations and provide a visual representation. The moveable items engage the students kinesthetically. It also helps unpack the concept of ratio as a comparison of two quantities as the students count out the quantities and represent them as numbers in a ratio. The scaffolding guides the process.
First, students move the terms to make a connection between the statement and the ratio.
Then the objects are counted and moved.
Then the ratio is written.
The quantities can be flipped to show an alternative ratio.
There is a blank to create your own and another with shapes.
You have to make a copy in order to move the pieces.
Below are images from a Jamboard and a handout that scaffold cross multiplying to solve a proportion. (See image at bottom to make a copy of the Jamboard.) This is an entry point, with a focus on how to write the ensuing equation. Solving would be a prerequisite skill so it is not addressed (but obviously would follow). This allows for less task demand placed on the students and for more time spent on the new steps.
The arrows and shading scaffold the cross multiplication step. Students move the terms from the proportion to the equation. This allows for kinesthetic engagement and helps students see how the equations are formed. The scaffolding for the equation guides students to writing the equation, which I have found a challenging step for some students. The equation is written first as factors to reinforce the idea of multiplication, then the students simplify for the second equation.
The handout draws upon the Jamboard and uses the same scaffolding. The template is blank to allow for use with other handouts. The students can copy problems from another handout and follow the scaffold to get to the equation. The steps and equation can be transferred over to the handout.
Here is a Jamboard to introduce volume and units of volume. (See photo at very bottom for making a copy to edit.)
The students start with building an animal pen and shading in the space inside. The hands on approach and connection to prior knowledge of a fenced in area for animals sets the stage for actual measurement units in subsequent slides.
The photos below show how students will count out meters and square meters, adding a formal layer to the fence they built previously.
The following slide provides an entry point to understanding volume and units for volume. The students count out cubes, building on the counting of meters and square meters. The cubes were created using WORD Paint 3D. Here is an article I used to create these. For the grid that is tilted, I used functions on WORD – see this document. (I could have used Paint again.) I then show them the prism that is created but I am not discussing shapes yet to keep the focus on the concept of volume.
I then have students recreate the volume using NCTM’s Illuminations activity called Cubes.
Finally, I show examples of volume and move from cubic units to liters (litres – as I was initially teaching this lesson to a 5th grade class in India).
In working with students who have fallen significantly behind in math, a common challenge is adding and subtracting. This can manifest in word problems or simple add and subtract problems. When this is the case, the first thing I check is whether the student understands conceptually what addition and subtraction are. Here is a Google Slides file that shows the approach I use. (You can make a copy and then edit.) I copy and paste slides for subsequent days so the Google Slides file services as a repository of the trials – data collection.
Each slide as the same format.
The operation in bold font.
The primary pile with the objects to work with.
The secondary pile with objects used for addition prompts.
The garbage can for objects discarded in subtraction prompts.
I like to use pennies as objects but will use Google Images of topics the student likes if I need something extra to keep their interest and attention. The student is tasked with performing an action with the objects and to distinguish between tasks to show discernment of what action to perform, relative to the prompt. There are 3 operations types: addition, subtraction, and sorting. The sorting is used simply as a distractor. Each image shows the original slide and a slide of the final product. The slides can be copied to for additional prompts. This first round focuses on common language that speaks to addition and subtraction.
In the next round, the sorting is removed and the language is focused on the terms add and subtract as a step in shaping understanding of the eventual symbols.
Finally, the actual symbols are used. If the student gets confused the previous language can be used as a prompt.
Here is an introduction to solving equations using a Jamboard (see photo at very bottom for how to make a copy). A seesaw is used to unpack the concept of an equation as two sides that are equivalent. The box is used to unpack the concept of a variable representing an unknown number (or oranges in this context). The form of a solution for an equation is established, with students revising to create other solutions. Here is a YouTube video and a FB Reel showing how it works.
Students are then provided a couple slides to match seesaw representations with actual equations. The matching provides a scaffold to support the connection between representations.
Then the seesaw and box representation is used to unpack the steps. Students are provided the steps as written directions on how to model the given solving steps with the seesaw.
The students are then provided the equation and seesaw representation, along with the solving steps provided as moveable pieces. Students slide pieces and move seesaw objects to make the connection between the two. Here is a link to a post with an updated version of this handout.
Finally, students are given the equation and tasked with completing all of the steps including the initial set up. This slide can be copied with new equations entered for additional practice (including having the variable on the right or writing the number before the variable (e.g., 5+m=8).
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.