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.


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.

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.

Multiplying by 6, 8, 9

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I have found success with students using a double skip method for 6s and 8s. The 6s use every other 3 and the 8s, every other 4. Below is how I scaffold the process by drawing upon prior knowledge.

Clearly, the 9s rule is widely known and used. Here is how I scaffold it to support the students who will struggle with the mental process for implementing the rule. This guides them. Also, I introduce scaffolding at the start as Universal Design for Learning (UDL) to help more students learn it at first which reduces the amount of 1 on 1 support needed.

Here is a link to the handout with these scaffolds.

Hack for Multiplication (and division) Facts

A common method to learn multiplication facts is through skip counting. In turn, this is a means of learning division facts (see next paragraph). The challenge for many students is they struggle to learn the skip count routines or cannot engage brute force memorization effectively (e.g., have a working memory deficit).

The challenge with multiplication by skip counting is keeping track of two sets of numbers while memorizing the order of the skip counting. That is another example of the rubbing belly and patting head phenomena in math where one extra task demand undermines the process.

A hack I use to scaffold this process to reduce the task demand during the learning process is to provide rows from a multiplication chart (below) for the facts of focus (3s and 4s in this example). The same approach can be used for division facts, e.g., in the image below right I have the student choose the row of the divisor (3) and then skip count to until reaching the dividend (12). The idea is the student has less task demands while learning the process and seeing the number pattern. This allows for more repetitions or rehearsal.

For students more severely impacted by a disability or who simply struggle with the patting head and rubbing belly of skip counting, the appropriate times table row can be provided for each problem to allow the student to circle (below). This allows for a hands on approach with even less task demand. You could also laminate the rows to make them reusable in lieu of several consumable ones requiring more paper. I like the consumable as I use that for data collection.

Adding Money First Steps

The following shows steps to introducing the concept of the value of money and of adding coins.

The concept of a dime is presented as 10 pennies (see below). The dime is compared to a penny, nickel and quarter using these representations. Repeated use of these representations leads into an intuitive understanding of the coins.dime and ten penny bag

Next is determining the value of multiple coins. The place to start is with pennies, which is relatively easy as the number of pennies represents the value. The next step is to count dimes because counting by 10s is relatively easier than counting by 5s or 25s.

Dr. Russell Gersten is a guru in special ed. At a presentation at the 2013 national Council for Exceptional Children he explained that number sense is best developed using the number line. With this in mind I created a CRA approach using the number line.

First, the student lines up the dimes on the number line (see photo below) then skip counts to determine the cardinal value, which is the value of the coins. money number line dimes

Upon demonstrating mastery of counting dimes, the student moves from using coins (concrete) to a representation – see photo below.


This approach is used for nickels and then a combination of nickels and dimes (corresponding blog post forthcoming).