This post presents a Google Jamboard manipulative activity to help scaffold the act of subtraction which helps unpack the concept of subtraction.
The Jamboard can be individualized with Google Images. The can allow for context. In this example, maybe the context is there are 7 players and 4 have an injury or on COVID protocol and have to sit out.
The artifact also incorporates scaffolding, color coding, and manipulatives. Subtraction is an operation, which invokes a verb. The kinesthetic aspect of the manipulatives helps to unpack the concept of subtraction.
Write the problem, using color.
Circle the starting amount in the row as the same color as the initial number.
Populate the row with the images of interest to the student.
I have found simplying expressions to be one of the most challenging Algebra 1 topics. This post shows a scaffolded handout approach to simplifying.
Scaffolding Like Terms
I have attempted to provide a deeper understanding of “like terms” in this post. This handout may be a useful follow up or it may be the entry point for simplifying.
The scaffolded handout focuses attention on the problem being an expression and on unpacking what simplifying and like terms mean. This is followed by a sequence of steps to address each mental and written step.
An effective strategy is to color code, showing which terms are like terms.
Word problems are challenging for many students. Writing a system of equations to model a word problem has unique challenges. This post provides details about a scaffolded handout with color coding can unpack the process for generating the appropriate system of equations.
Unpacking the Word Problem
A mistake I have witnessed over the years is students mistakenly using given values for both equations. In the problem below, students are far more likely to generate the equation for the yellow part: 2x + 3y = 24. The challenge is that the blue part has only 1 number so students will often write 2x + 3y = 10, using the dollar quantities a second time.
By highlighting the two parts of the word problem with given values, the students can match parts of the word problem with respective equations. The scaffolding separates the parts, and the color allows for matching.
The rest of the problem is prior knowledge with the students using one of the methods for solving. The scaffolding continues to lower the task demand by reducing the need to remember all the steps. This allows them to focus more bandwidth on the new steps.
This post details a scaffolded approach for multiplying multi-digit numbers by 2-digit numbers. It was originally created for a student with ADHD who understood how to do the multiplication but would rush and repeatedly made simple mistakes. It is useful for all students.
This grid and color-coding strategy was used as a means of slowing him down. He had to alternate between highlighting and writing each product for an individual multiplication of two digits. This turned out to be an effective way to teach multiplication by 2-digit factors, in general. Here is how this works.
First highlight the ones-digit in and the row for the product that results from the ones-digit. This helps unpack the place value and why the algorithm works. Note: use a lighter color of highlighter (you will see why).
Highlight the ones digit in the top factor. Multiply the ones digits. Write the product. This is where the student alternates, which can allow for thinking through the steps.
Continue highlighting and writing products using the ones digit from the bottom factor.
Now use a darker highlighter to highlight the tens digit in the bottom factor, as well as the tens row at the bottom. Because the 3 is in the tens place, we write a zero. This unpacks the place value.
As was done with the yellow highlighter, alternate between highlighting digits to multiply and write the product in the row below. The darker highlighter is used second to make it visible when drawing over the previously used lighter color.
For carrying (regrouping), the top row can be split and the color can be used for the digits that are carried.
Here is a link to the handout used for these photos. It contains the two problems shown in this post along with blank templates. Here is a link to another post that shows a scaffold I use to unpack the carrying of a digit in multiplication.
Telling time on an analog clock is challenging for many students, especially some with special needs. I worked with a middle school student with a disability one summer and after a few lessons he scored 100% over two days on telling time. Below shows the progression I used with him. I used a task analysis approach of breaking the task into smaller steps and chunking the steps to introduce an additional task demand incrementally.
I use math-aids.com worksheets for time and for many topics because it provides dynamic worksheets in which users can choose features. This helps to enable implementation of a task analysis and chunking approach.
The first chunk is whole hour time, which is an option on math-aids. The clocks produced by have color coded hands, with green for hours and red for minutes. I use use additional color coding through highlighters because the handouts are likely printed in B&W and because it engages students kinesthetically.
The second chunk is time with minutes under 5 minutes. Math-aids allows a user to choose specific times and will create clocks with those times. To provide a visual aid, students can write out the numbers on the handout or they can be printed or handwritten on the master copy. You can snip out the clocks shown, paste into a WORD document and then add the numbers. Note: I do not jump to half hours and quarter hours until last. I want students to focus on hours and minutes. This is analogous to counting money. I don’t introduce cents after talking about half a dollar.
The next chunk is time with minutes between 6 and 10 minutes. This is how I introduce the 5 minute mark and have them count on from 5 (see the 5 in the middle bottom). This leads to all the tic marks for 5s.
The 5s are the entry point to navigating the entire clock. I introduce the tic marks for 5s without the numbers for the hours. Students can be prompted to draw in red minute hands for a given numbers of minutes in 5s. If you want to make a handout for this, save the image below and crop.
The students are then given the minute hands and are prompted to identify the minutes as a multiple of 5s (you don’t say “multiples” but can say “in 5s”). An option is to have them highlight the minute hands at first then fade the highlighting. They do this first with no numbers for hours then with numbers for hours – an option on math-aids.
Then students are asked to tell time by identifying the fives and then counting on (as they did with time with minutes under 10). Here are some options.
You can have them highlight.
They can focus on just the 5s and write the multiple of 5 preceding the minute hand.
They can count on from the 5s multiple they wrote, e.g., “15, 16, 17”
Focus first on minutes below 15 as the hour hand is close to the hour (top row in image below). Then address time with minutes between 16 and 30 because the hour hand has moved further away from the hour and it starts to get tricky for students to determine the hour. To address this, you can shade in the tic marks between the hours. Notice that I do not shade the subsequent hour. This also sets the stage for time when the hour hand is close to the subsequent hour (next chunk).
Time with the minute hand on the left side is tricky because the hour hand is close to the subsequent hour. The aforementioned shading can help. I also find it useful to have the student notice that the hour hand is not at the 12 yet, but almost. You can have students draw the marking and word as I did below.
Finally, there is a need to generalize. You can print images of clocks from a Google Images search into a handout and use the same strategies from above. This would be followed by actual clocks.
Here are images from a handout that serves as an introduction to piecewise functions. The focus is to develop conceptual understanding of piecewise before attempting to graph independently. The work is divided into chunks to reduce the level of task demand at a given time in the process.
The first section has an application and introduces the idea of pieces addressed together. The graph is discrete which allows the students to see the points in lieu of looking at lines.
Pages 2 and 3 provide scaffolding for graphing. Page 2 presents the sections separately. Page 3 pulls them together, but with the intervals physically separated into columns over the graph.
Finally, the students are introduced to the function notation, with additional scaffolding. They are also asked to identify y-values for given x-values in function notation to help connect x-values with different pieces.
The idea is that the student will have to count squares and eventually skip count by how many rows. By doing so the student is more engaged (or less passive) in determining the product byy engaging the visual representation. I am interested in feedback and will revise if this could be useful.
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.
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.