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.
The concept of fractions as some number of equal parts begins in 1st grade per the Common Core (image below). There are students who struggle with the idea of equal parts and this could undermine student work in subsequent topics. The activity cited in this post is designed to develop the concept of equal parts.
Jamboard with Sharing Slides
The following images are from a Jamboard used as an introduction to equal parts activity (see photo at the end for access). The activity is chunked to incrementally present more of the ideas underlying equal parts. The use of the Jamboard can be viewed in a FB Reel and on YouTube.
First, the idea of equal is addressed by presenting a situation in which two students are sharing candy. Partitioning out pieces alludes to the set notation of fractions.
The idea of sharing equal amounts transition to sharing a single candy that can be broken into parts. The candy bar image is actually two images of parts. The a non equal sharing is used to unpack equal parts. This is continued for a circular shape and a triangular-ish shape.
Jamboard with Mathy Slides
There are additional slides to do more “mathy” work with equal parts. First, the students are asked to choose the shape that was cut into equal part (rectangle, circle, triangle). Then the students partition the shapes but with a dotted line as scaffolding.
Each shape can be connected to the food images from above. For example, the student may intuitively understand that a pizza is cut from the crust to the tip. I use pizza fractions to unpack the need for common denominators, which reinforces the significance of the concept of equal equal parts cited previously.
Here is an image of an accompanying worksheet. It draws upon the images from the Jamboard and follows the same sequence.
Accessing the Jamboard
The image below shows how to make a copy of the Jamboard in order to use it.
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.
An accompanying Jamboard will eventually be shared in this space.
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.
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.
Below are photos from multiple lessons to introduce multiplication. They are combined into a single document. I use a task analysis approach to first develop conceptual understanding of multiplication as repeated addition. This is followed by skip counting and then using skip counting to multiply. The lessons are not necessary completed in a single day.
Lesson 1 focus is to unpack repeated addition vs simple addition to build on prior knowledge.
Lesson 2 focus is to unpack arrays by identifying rows and columns which are the factors in a multiplication problem. It builds on the previous lesson with repeated addition of groups that are then converted into arrays of items and then into arrays of circles and squares.
Lesson 3 transitions from repeated addition to skip counting (with a future focus of multiplication by skip counting vs fact memory).
Lesson 4 combines skip counting and the rows and columns of arrays into a multiplication sentence.
The nature of the task analysis approach is a sequence of topics building towards the objective of multiplying single digit numbers. Mastery of each of the steps or lessons can be recorded as progress towards mastery of the overall objective. Below is an excerpt from a Google Sheet that is used to record such progress. This can be shared with the team, including parents.
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.