Functions are perhaps the most prevalent and important topic covered in secondary math, aside from maybe 1 variable linear equations. The concept of a mathematical function is challenging for many students. This post provides details about a meaning making approach to introducing functions.
The introduction is presented on a Google Jamboard, to allow for movement in the pairing of inputs and outputs. It starts with analogies pairing of items using a gumball machine and a Coke machine and proceeds incrementally towards the various representations. The functions are contrasted with examples of relationships that are not functions.
Slides of the Jamboard
Slides1 and 2 present the gumball and Coke machines. Students can move the items to see how a quarter can result in 2 different color gumballs while the Coke button results in only 1 output.
In slides 3 and 4, the use of an hourly wage introduces input and output with quantities. Slide 4 shows two different pay amounts for the same number of hours worked. This taps into prior knowledge.
The sequencing progresses through
Each includes an example and a non-example.
The last slide provides a sorting activity.
Access to Jamboard
Here is the link. To access the Jamboard, you need to make a copy.
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.
Tenths vs Tens…Hundredths vs Hundreds. Problematic for many students. I believe this is a conceptual problem. This post provides an approach to unpack the concepts through money in a scaffolded handout.
Money is likely prior knowledge for many if not most students, and is a relevant context. This handout attempts to leverage interest or knowledge of money to unpack decimal place values. In the first page, the concept of “tenth” is addressed with dimes as 1/10 of a dollar. Similarly, “hundredth” is addressed with pennies as 1/100 of a dollar. A key point to consider is that US monetary system base unit is a dollar. More on that in the other pages.
Hundreds to Hundredths
The handout aligns each place value with the appropriate currency. This is followed by writing each number in numeric form and then word form with the place value table as a guide. To enhance the word part, you can highlight the each place value in money, digit, and word in the same color (e.g., the “2” in yellow).
Also, note the shading. The dollar as the base unit is in the center and shaded the darkest. The tens and tenths are shaded the same as they are a factor of 10 from ones. (I don’t reference the term with students.) Same for hundreds and hundredths.
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.
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.
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.
I introduce key characteristics with parabolas and use the analogy of a rollercoaster. Riding once (and never a again) the Superman rollercoaster at Six Flags New England got me thinking about this. At one point the roller coaster hits ground level (a zero) and then goes underground (negative y values).
Here is a handout I use for the introduction. Here are images showing how I use the handouts. The table helps students visualize the x-values and y-values when looking at the graph. The rollercoaster provides extra context for the various characteristics, e.g., increasing means the rollercoaster is going up. Note the scaffolding by adding context clues for each characteristics.
I start with max height of the rollercoaster. I highlight the actual graph first, then the y-values in the table. Then point out we are looking for the x-values for what we highlighted.
The issue of highlighting the vertex for the increasing and decreasing values would be addressed when writing the interval. The idea of the rollercoaster at the tip top provides context to develop the concept.
Similarly, I start with the zeroes. Again, highlight the graph, then the y-values, then the x-values.
The issue of highlighting the zeroes for the positive and negative values would be addressed when writing the interval. The idea of the rollercoaster at the ground level provides context to develop the concept.
I has been effective to have students highlight the parts of the respective axis when discussing the domain and range (not discussed yet). A common challenge is understanding that the x-values continue to the right or left when it appears they simply go down. To address this, I use a very wide parabola to show more lateral movement.