0 and Negative Exponents – Scaffolded

This post presents a scaffolded and meaning making approach to exponents that are 0 or negatives.


The slide show below presents all 4 pages.

  • The handout starts with an initiation to preview the prerequisites for what is presented in the lesson. It also introduces a chart that will be used for discovery.
  • Page 2 presents a discovery activity of following a pattern of dividing by 2 down to the 0 exponent. The concept of exponents is presented as the number of occurrences of the base. This leads to the idea of a 0 exponent indicating the base is no longer present, but there is still 1.
  • Similarly, on the 3rd page the pattern of dividing continues into negative exponents to show the resulting fractions. The negative exponents are then presented as reciprocals.
  • For terms with multiple factors (e.g., 5x vs just x) the students are presented steps to write the factors separately. This unpacks the reason why the negative exponent acts only on one of the factors (unless both are grouped with parentheses).

Access to Handouts

Here is a link to the student handout, and a link to the teacher handout.

Customized Number Lines for Handouts

Per request, I created a short video showing how I create customized number lines on WORD. This post also includes a link to a WORD document with 3 customized number lines: time, money with negatives, and miles.

Elapsed time

The image below is from a post on elapsed time. I wanted to create different time scales to match clocks I could create on math-aids.com.

The Video

In the video I show how I created the time number line. In the top image below, you can see the table highlighted. I then show how I copy and paste the number line and then edit to create units with money, with negatives.

Here is a screenshot of the video. You can see the number line in an early stage of development. Below the image is a link the video.

Link to video


Below is an image of the three customized number lines. Here is a link to the handout, which is in WORD format to allow you to revise to suit your work with students.

If you find this helpful, please consider making a small donation to a fund to build an accessible playground at a camp site for individuals with disabilities.

REVISED: Scaffolded Handouts for Solving

I previously posted scaffolded handouts for solving 1 step linear equations. I added a couple steps to create a new handout. In it I flesh out ALL of the steps, including identifying the inverse, the number to eliminate, simplifying the expressions to get the identity (0 or 1) and then simplifying again to eliminate the identity (see images below).

Enrichment Resources for Math

When I work with advanced or talented and gifted students, I recommend that enrichment be used to “go wide, not deep.” The point is not to simply move the students vertically, but challenge them to go deeper with current content or use critical thinking activities.

Here is an online resource of problems from the Paul ErdÅ‘s International Math Challenge that will indeed challenge almost any student (check the problem below for 3rd and 4th graders).

Here is another online resource of problems from the Art of Problem Solving (AoPS). The link is to a search posts that address practice problems related to the American Mathematics Competition for 8, 10, 12. Below is an AMC 8 practice problem.

Intro to Multiplication – A Sequence of Lessons

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.

Lesson 5 uses skip counting to multiply, first with arrays and groups, then as multiplication problems. Here is the link to a post about the Grumpy Cat Jamboard cited in the document.

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.

Elapsed Time – Scaffolded Handout

Several elementary teachers shared that elapsed time was the hardest topic to teach. Here is a scaffolded handout to help compute elapsed time. The elapsed time setting is presented with two clocks, starting and stopping time. Below is an image for one of the more advanced pages of the handout. Here is how the strategy works.

  • The time is divided into minutes and hours.
  • The students identify how may minutes are needed in the first clock to get to the next hour.
  • Then they identify how many minutes are present in the second clock.
  • Finally, students determine the change in the hours.
  • I did not include a spot to place the total elapsed time as the focus is to identify how to break up the problem into parts.

The handout starts with a focus on identifying the minutes leading up to the the next whole hours and the minutes after the last whole hour. The task demand is increased incrementally with whole hours only, then only one or the other clock having minutes then both clocks having minutes.

Introduction to Absolute Value – Distance from Home

Here is a Jamboard to introduce the concept of absolute value (make a copy and you can edit – see photo at bottom of this post).

Start with prior knowledge of how many houses a child is from home. Emphasize that the number of houses is positive.

Change the setting from a row of houses to the number line and refer to the distance from home.

Change from distance to absolute value and emphasize that the symbol indicate distance from home and is called absolute value.

The house is removed and introduce the concept of the distance from 0.

Finally, convert from boys to the numbers and the distance from 0 for a number. Emphasize that the distance is positive, even for negative numbers.

To make a copy of the Jamboard.