Slope is one of the most important topics in algebra but perhaps one of the most challenging topics. One issue is that students often think of the formula as opposed to the idea of slope. The images shown below are for a handout that unpacks the idea or concept of slope.
The images show sections of a handout. Slope is a measurement of a graphed line. It measures steepness and also indicates direction. The handout starts by drawing on prior knowledge of measurements for newborns.
Prior knowledge of steepness and up and downhill are invoked.
How to measure steepness is introduced through a focus on stairwells.
The measurement of the length and height of steps leads into the rise and run of a hill. I use “right” and “up” or “down” which focus on direction. The terms “rise” and “run” can be addressed subsequent to the introduction. This part of the handout is scaffolded to reduce task demand and focus on the concept.
Finally, the students are provided a line and apply the ratio procedure. Note: the slopes are not the same as in the prior section.
Jamboard
An accompanying Jamboard will eventually be shared in this space.
Telling time is challenging for many students. This is likely a function of the abstract nature of time is. You cannot see or touch it. You experience observe it through a clock. Elapsed time is more abstract and challenging. An entry point to elapsed time may be student experience with walking from one point to another. This post details the a Google Jamboard that leverages this prior knowledge to present elapsed time.
The images below are from a handout to introduce elapsed time. This a revised version of another handout I created. The sequence in chunked to incrementally present additional elements. A number line is used to model, first on Jamboard then on a handout, then clocks are introduced. The first problem has an exact hour on the second clock to make it more simple but to still include minutes.
The clocks were created on math-aids.com, which has a page to allow you to choose times to be represented on clocks. They create clocks with color coded hands, which I follow with highlighters on the handouts and Jamboard.
First, the identify the the upcoming whole hour and marks the hands with highlighters or colored pens or pencils.
Determine the number of minutes to the hour.
Identify the whole hour preceding the second time and marks accordingly.
Determine the number of minutes from the whole hour to the second time.
Use the green marks used to identify the whole hours and determine how many hours passed.
I did not create a spot to write the answer to cut back on visuals.
The first page provides an introduction to the use of the number line without having to process the clocks.
Mark the whole hours.
Determine the number of minutes preceding and following the whole hours.
Determine the number of hours that passed.
A Jamboard is used to model the first 4 problems to engage the students kinesthetically and to unpack the concept. The students can do a Jamboard slide then work on the matching problem on the handout. (See photo at bottom for access.)
On the handout, I addressed the minutes of both clocks before determining hours. The Jamboard person can be used to flesh out the concept of time passing as the person walks. As a result, I suggest determining the hours before the minutes on the second clock as the person walks the entire way. When you return to the handout, you can reference the person walking the last 10 minutes and even show the students the Jamboard again when you do those minutes before determining hours.
The post shows a lesson to introduce factoring out a GCF. Factoring (and multiplication proficiency) are challenging for many students.
Overview
The handout is chunked to incrementally add task demands. A Jamboard is used as a complement to flesh out the concept of a factoring by allowing engaging the student kinesthetically, with scaffolding as guidance. (See image at bottom for how to make a copy to access it.)
Initiation
The sequence starts with a review of factoring and distributive property. The boxes are a preview of the scaffolding for the full factoring to be presented in subsequent chunks.
Jamboard
The Jamboard slides aligned with the first chunk has students pull apart the variables with exponents to help them understand the meaning of the exponents during factoring. For example, there are two Xs overlaid on the x squared. Similarly, there are two 5s, one overlaying the other to allow them to be distributed.
Handout
The next chunk in the lesson is factoring out with the terms written with the multiplication symbol written in between. The problems are aligned with the distributive problems to help with the concept of “reverse distribution.” The same problems are presented again in box form as the boxes will be used to support the full factoring. Using the same problems helps the students understand how the boxes work. The Jamboard reinforces the distribution by allowing the movement of the 5s out of the binomial, including with the box method.
The last chunk combines the factoring of individual terms and the factoring out. This is conducted with the box methods for support, and then the boxes are faded. The scaffolding starts by writing the expression with parentheses because that step is easy to overlook when explaining it to students and can be confusing for students.
Here is what the final version looks like.
Note: the handout has blank templates for copying and pasting if you want to create your own handout.
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).
The Jamboard incorporates scaffolded handouts. The compare problems has two separate scaffolded sections. The first is to unpack the concepts of difference and compare, followed by writing a math sentence.
Here is an introductory Jamboard to help students visualize and conceptualize change situations. Here is a video you can show to help students see movement and to get an idea of how to implement.
Note: this Jamboard is static to allow me to use the image from Clever Cat Creations.
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).
Plotting points is challenging for some students, even in high school. This approach uses the analogy of setting up a ladder for the x-value and climbing the ladder for the y-value.
Jamboard with Ladder
This activity is conducted on a Google Jamboard with moveable objects (see photo at the bottom on how to access it).The activity draws on prior knowledge, which allows for the steps for positioning the point point to be meaningful. The x-axis is established as the ground, with an image of green grass superimposed over the axis. The ladder is climbed, moving toward the sun. This aligns the ordered pair with the axes. It also allows for color coding on coordinate planes on subsequently used handouts.
Move the ladder to the appropriate location on the ground.
Position the dot at the bottom of the ladder on the x-value.
Move the dot up the ladder to the appropriate y-value.
Pull the ladder away.
Jamboard without Ladder
Next, the ladder is faded but the color is maintained. 3 is green so move along the grass to the 3. Then yellow 5 so move up 5, towards the sun.
Finally, the colored numbers are retained but the grass and sun are faded. References to the green grass and sun can be used as necessary, even with highlighters on the Jamboard.
Handout
The Jamboard can transition to a handout with matching coordinate planes. The grass and sun are faded. If a student is stuck, you can return to the Jamboard to model the problem.
Students can use highlighters as necessary to replicate the grass and sun numbers. The highlighters can be faded to result in a regular plotting a point problem.
