There is a delineated sequence for teaching multiplication over the years, including repeated addition, set modeling, arrays, single digit etc (below). It exists to build conceptual understanding of the multiplication facts that are at some point memorized by many students. When I work with students who are a more than a year behind in the sequence for multiplication, I find that programming for these students to help them catch up sometimes involves shortcuts such as a reliance on rehearsal or resorting to use of the multiplication table in isolation. I am not against use of the table or narrowing the focus, but am promoting a more comprehensive approach.

Here is a sequence, on a Jamboard, I used for a recent student who was struggling for a long time with multiplication (explanation of each step shown below images). The student was interested in Minecraft so I used Minecraft items such as stone bricks and a wagon. I would spend as much time on each step, as necessary.

Count out the total number of stone bricks. This allows an assessment of how the student counts: by 3s or individually. If individually, I would prompt the student to count by 3s.

Add 3 + 3

Show a short video on the wagon (this adds interest and gives the students a bit of a break)

Present the bricks in 2 groups of 3, in context of 2 wagons with 3 bricks each.

Present the same problem as a multiplication problem but with the image for one of the factors in lieu of two numbers.

Use the multiplication table to skip count.

Present additional multiplication problems for independent attempts. The student completed both problems independently, without the table. For him this was a major success.

The follow up to this would be to assess his ability to do higher groups of 3s and groups of other numbers. For some students, I work on mastery of individual numbers before moving on. This builds confidence and allows for fluency in the process of skip counting out to the appropriate number. NOTE: I don’t worry about rote memorization of the facts but of fluency in the process of skip counting out the answers.

For students who are older, I sometimes recommend that the student be presented problems with visuals but then use a calculator to compute. This can develop conceptual understanding and also address the working memory and other related issues that undermine learning math facts.

Carrying the TENS digit in a multiplication problem is a sticking point for many students. To address this, I use a task analysis approach to zero in on the step of identifying the product for the ONES as a prelude to carrying.

In the example below, 5 and 4 are in the ONES place and the product is 20. The task analysis steps involved:

compute the product

identify the digits in the product

identify the digit in the ONES

identify the digit in the TENS

Understand that the TENS digit must be carried to the TENS column

By creating a place holder for the product and scaffolding it to differentiate between the TENS and the ONES, the student can visualize the product. This reduces the demand placed on working memory. Once mastery with the place holder is demonstrated, it can be faded (and used as necessary as part of corrective feedback).

NOTE: I started this mini-lesson for a student with ADHD by having him warm up with problems without carrying. Also, extra line below the 60 and 20 are used for multiplying by 2 digit numbers (next in the sequence).

I was recruited to help a middle school student who is having a very rough time at this time in his life. It was shared with me that he likes Marvel superheroes and he is struggling with counting money and multiplication. Below are some ideas I presented for a test run and photos of the items I ordered for these suggested activities.

For multiplication

Put the heroes (or villains) in groups of 2 and have him count out 4 groups and compute. Use different groups and number per group. (IGNORE the numbers on the cards)

Get a group of 10 villain cards. Pretend heroes have to travel in groups of 2 and ask how many groups to get 10 heroes to fight the 10 villains. (IGNORE the numbers on the cards). Variations of this.

After gets the idea of groupings, focus on the number on the cards and show him two 5s and have him compute. Variations of this.

Play a game where he draws two cards and has to multiply the cards (start with very low numbers or maybe show him a 2 card and he has to pick another card to multiply by 2.

For Money

Tell him he earn money to buy these figures, one at a time – a monetary version of a token economy. Have him rank them by his favorite to least favorite and come up with a price for each with his favorite figures costing more. Start with the least favorite and make the price such that with a little practice he could count out the coins to pay for it. Maybe 17 cents with dimes and pennies. He has to count out the money correctly and independently to actually buy the item.

Other options

If he needs work with addition you can play WAR in which 3 cards are played and each person adds to find the total. For subtraction do the same with 2 cards.

You can play subtraction in which one person has superheroes and the other has villains. In order for a villain to win a villain card has to be higher than a hero card by 3 or more.

You can write an 11, 12 and 13 on the J, Q, K cards respectively. All the games can be presented though Direct Instruction – I do, we do, you do. The You do can be used as daily progress monitoring. If he needs prompting this can be recorded. This can be used for your progress reports. Attached is a data sheet I use for activities.

The idea is that the student will have to count squares. By doing so the student is more engaged (or less passive) in determining the product and has to engage the visual representation.