Simplify Rational Monomial Expressions

This post provides details about a handout for simplifying rational monomial expressions. It incorporates a couple strategies to make the simplification of rational monomial expressions more accessible. The strategies include address prerequisites skills ahead of time, chunking, and scaffolding. This incrementally walks the students through the steps.

The Pages of the Handout

The handout has 3 pages.

  • Page 1 is an initiation with two parts. There is a review of prerequisite skills aligned with the new topic. There is also a preview of the new topic with scaffolding to separate the factors into individual fractions.
  • Page 2 provides a Before and Now to draw upon student prior knowledge of simplifying using exponents rules. This is followed by scaffolded steps to separate the expression into individual fractions for each type of term (e.g., Xs). This provides a load reduction for what the student has to focus on.
  • Page 3 involves negative and 0 exponents with an additional step to address each.

Access to Handout

Here is a link to the handout.

Complete the Square for Vertex Form

This post provides a handout that guides students through the various steps for completing the square to transform an equation into vertex form. Students are guided through each step in isolation.

Overview

Students are presented each step in a separate chunk of the lesson. Then the steps are chained together, with scaffolding that is faded. This is a different approach than presented in a previous post. The chunks, examples, and scaffolding help make students more independent in completing the work. This frees up the teacher to provide more 1 on 1 support.

Chunks of the Lesson

The initiation addresses prerequisite skills: factoring, perfect squares, fractions, and doubles. In lieu of having students divide by 2, I focus on identifying fractions that add to the linear coefficient as you will see in the second page.

Desmos Activity to See Completing the Square

To introduce completing the square, I recommend a visual activity like this one from Desmos.

The students identify the constant that results in a perfect square. They do so by identifying doubles that result in the linear coefficient (e.g., 6 = 3 + 3). The examples help guide them through this process. This section could be presented after a hands on activity on

Students are then tasked with factoring perfect squares in isolation, including those with fractions. The doubles are modeled for whole numbers first, generalized to fractions.

At this point, the students have identified the constant to complete the square and then factored expressions. The next sections have students complete the square and then factor in equations. Note that the equations are structured as a step after the students would have subtracted the original constant, leaving the quadratic and linear terms on the right.

The last section chains all the steps together, first with scaffolding then without. Additional practice would be generated with other handouts that have problems in isolation.

Access to the Handout

Here is a link to the handout.

Fraction Multiplication with Cookies

Fractions are challenging. Multiplying fractions is really challenging! This post presents a Google Jamboard to introduce students to the concept of multiplication of fractions.

Overview

The artifact is chunked to incrementally move from multiplication of whole numbers to whole number and fraction to multiplication of fractions. The representation of multiplication as number of objects in a group times number of groups is the structure used throughout. Cookies on a plate is the context used to draw upon prior knowledge and make the idea more concrete.

This serves as an introduction. Each chunk can be followed by practice before moving on to the subsequent chunk.

Prior Knowledge

The Jamboard starts with a representation of multiplication as groups of objects, first with the number of objects in a group and the number of groups. This is presented first as cookies per person to connect to prior knowledge. Then presented per plate as the plate is subsequently used to model the fractions.

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Fractions

First, whole number times a fraction is presented. This allows for a connection to prior knowledge and introduces fractions in this representation. There are still 6 cookies per group, but now there is only 1/2 a group.

The students can move the cookies onto the plate to see the group of objects. Then they can cut the group in half.

To help make sense of the fractions used in the multiplication of two fractions, the fractional parts of the cookies are presented first.

For multiplication of fractions, the process is the same. There is 1/4 of a cookie in each group, then there is 1/2 a group. As was done previously, 1/2 the group is removed. Conceptually, you can explain to the students that they have 1/4 of a cookie and they split it with a friend.

Access to Jamboard

Here is a link to the Jamboard. You need to make a copy to access it.

Skip Counting Scaffolded

I work with students from elementary school to college. At all levels there are many students who struggle with multiplication. When I work with a student on multiplication I focus on skip counting as opposed to multiplication facts. Skip counting connects to multiplication as repeated addition, which is the foundation for scale and proportion. To do this, I scaffolded the skip counting and connect it to prior knowledge. This post provides details for this approach, which is presented in a handout.

Overview

The first page of the handout provides an overview.

Sections of the Handout

Note: the image of the 10s shows mistakes that are not in the actual handout.

Access to the Handout

Here is a link to the document. As stated in the overview on page 1, the elements of this handout may be cropped out and used individually. At least, the scaffolded rows can be covered to prevent the students from copy from previous rows completed.

I am interested in feedback on how to make this more useful or effective.

Telling Time in 2nd Half Hour

When the minute hand passes 30 minutes and approaches a full hour, the hour hand gets closer to the next hour. This can be problematic for students. Many see the hour hand close to the next hour and think that is the hour. This post presents an approach to address this with students.

At vs Almost 10:00

Here is an entry point for telling time in the 2nd half hour, when the hour hand appears to closer to the next hour. The clocks below contrast what 10 and a time getting closer to 10. This allows a focus on the minute hand. We can contrast the minute hand showing 10:00 vs getting close to 10:00. This is the first chunk in a sequence of mini-lessons.

The Judy clock is dynamic and allows us to move the minute hand closer and closer to the next hour, 10:00. We can have the student “see” time getting closer to the next hour but not being the next hour yet.

Practice with Time in 2nd Half Hour

Math-aids.com website provides choices for the times situations you want to create for a handout. In the image below, the clocks all have time in the 2nd half hour (with 5 minute increments). This can be used as follow up practice to the instruction shown above. Have the student do the following:

  • Identify what hour it is almost, e.g., “almost 10:00”
  • Identify the actual hour, e.g., “it is still 9:00.”
  • Count out the minutes in isolation.
  • Chain minutes and hours for time.

Introduction to Adding Integers

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

Prior Knowledge

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.

Transition

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.

thumbs from Educlips on TPT

Adding Integers

Finally, references to thumbs is replaced with the integers values. The thumbs tokens are maintained to allow for continued concrete representation.

Accessing Jamboard

To access the Jamboard you must make a copy.

Equal Parts of Fractions – Intro

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.

CCSS Coherence Map

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.

Handout

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.

Intro to Slope of a Line

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.

MathBootCamps

Handout

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.

Elapsed Time with Number Line and Clocks

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.

Make a copy of the Jamboard in order to use it.

Time on an Analog Clock – a Chunked Approach

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.