Multiplication by 2-Digit Factor – Scaffolded and Color-Coded

This was used for a student with ADHD who understood how to do the multiplication but would rush and repeatedly made simple mistakes. This grid and color-coding strategy was used as a means of slowing him down. He had to alternate between highlighting and writing each product for an individual multiplication of two digits. This turned out to be an effective way to teach multiplication by 2-digit factors, in general. Here is how this works.

First highlight the ones-digit in and the row for the product that results from the ones-digit. This helps unpack the place value and why the algorithm works. Note: use a lighter color of highlighter (you will see why).

Highlight the ones digit in the top factor. Multiply the ones digits. Write the product. This is where the student alternates, which can allow for thinking through the steps.

Continue highlighting and writing products using the ones digit from the bottom factor.

Now use a darker highlighter to highlight the tens digit in the bottom factor, as well as the tens row at the bottom. Because the 3 is in the tens place, we write a zero. This unpacks the place value.

As was done with the yellow highlighter, alternate between highlighting digits to multiply and write the product in the row below. The darker highlighter is used second to make it visible when drawing over the previously used lighter color.

For carrying (regrouping), the top row can be split and the color can be used for the digits that are carried.

Here is a link to the handout used for these photos. It contains the two problems shown in this post along with blank templates. Here is a link to another post that shows a scaffold I use to unpack the carrying of a digit in multiplication.

Modified Multiplication Table – Area Model Included

The idea is that the student will have to count squares and eventually skip count by how many rows. By doing so the student is more engaged (or less passive) in determining the product byy engaging the visual representation. I am interested in feedback and will revise if this could be useful.

Here is a link to the document.

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.

Matching Activity for Multiplication Word Problems (and Creating)

Here is a matching activity on a Google Slides file for various multiplication word problems and matching groups of items. The students use gallery view of the slides and sort them to match. Then they can change the background color with a different color for each word problem and groups. This allows them a visual to represent the problem and an opportunity to analyze the components of the word problems. Slide 2 shows a template of an editable group of objects to allow you to create additional slides.

Here is a Google Slide file for student to create their own groups.

Factors and Multiples

Below are images of artifacts I created for work on factors and Multiples. The first is a Jamboard (you make a copy and then edit). The second is a handout to introduce factors and multiples. Here is a Superteachersworksheets has these Venn Diagrams problems on handouts.

Multiplication Single Digits – Special Cases

There is easy skip counting for multiplying by 2, 5, 10. Skip counting for 3 and 4 are often more challenging. There is the finger rule and related rules for multiplying by 9. I posted about rules for 6s and 8s (and scaffolding for the rule for 9s). 7s work for students who watch football but otherwise those are tricky.

A team member and I discussed this situation and she pointed out that the student can simply skip count by the other factor instead of 7…except for 7×7 and 7×8, along with 8×8 being problematic. Here is how I am addressing these 3 special cases. I focus on leveraging what they know about the other problems and then adding or subtracting, e.g., below the student focuses on 6×7 then adds another 7. Similar approach with the other two cases.

Below is an excerpt from a WORD document with the scaffolding for this. It includes the same scaffolding for all three cases.

Below is how I implement. The student completes the scaffolded problem and then applies to the steps to the unscaffolded problem. I present the two together periodically and then fade the scaffolding and mix in the special cases with the other multiplication problems.

Multiplying by 6, 8, 9

If you find this helpful, maybe consider making a small donation to help build an accessible playscape for children with disabilities.

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.

Grumpy Cat Multiplication

If you find this helpful, maybe consider making a small donation to help build an accessible playscape for children with disabilities.

Here is a CRA approach to multiplication that can be individualized. I created this one for a student who loves cats.

Here is a link to the Jamboard. You make a copy by clicking on the 3 dots at the top right, then you can manipulate the items.

Here is how I use this Jamboard.

  • You have a slide with a problem (5×2) and the appropriate grouping (2 in this case).
  • Go to slide 1 and choose the appropriate number of items in the group (5 in this case).
  • Copy (use CTRL C) and paste the appropriate # of grouped items into the groups (boxes).
  • Click on the =? to enter the answer.
  • The slides have groups up to 6.
  • You can have students personalize by choosing their own image. They can paste the image repeatedly to create the grouped items then snip the grouped items as a single image.

CCSS MATH Coherence Map – 3rd Grade Groups of Objects to HS Slope

A hidden treasure is the Common Core of State Standards Math Coherence Map. It is an interactive flow chart that shows connections between the various standards.

  • If you are teaching math, you can see the connections between what you are teaching, what was taught previously, and how you are preparing students for their future math education.
  • If you are a special education teacher, you can see the progression of prerequisite skills. If you write IEP objectives for grade level standards, you can address the prerequisite standards and progress made through these prerequisites can show progress towards mastery of the IEP objectives.

In this post, I show the progression from 1st grade standard on the = sign and 2nd grade standard of repeated addition all the way to interpreting slope in high school math.

After clicking “Get Started” you will narrow your focus to the grade, the cluster, and then the math domain.

The flow chart shows connections between a selected standard and others, including prerequisites. In this case 8.F.B.4 – 8th grade content that is a prerequisite for the high school math work. Click on the 8.F.B.4 standard and it pops up (below right).

In turn, the 8th grade standard is connected to a 7th grade prerequisite regarding ratios and proportions.

Notice that the 7th grade standard, 7.RP.A.2, appears to be a dead end (bottom left). I picked up the path by clicking on Grade seen at the top left of the screenshot and made my back to that standard and the connections to prerequisites appeared. (Same happened in 3rd grade shown further down in this post.)

The path continues from ratio and proportions in 7th grade to unit rate in 6th grade, multiplication word problems and multiplication in elementary school.

I want to emphasize that students are working on unit rate and slope problems in ELEMENTARY SCHOOL! 3.OA.A.1 below addresses groups of objects model for multiplication and 4.OA.A.2 addresses word problems involving multiplication.

I was recently working with a student entering middle school on multiplication word problems. To unpack the word problems and the concept of multiplication in context, I had her draw (photo below) groups and groups of objects to help her identify the unit rate (although we don’t use that term yet). This work will impact her math education through the high school math and even into college (slope has been a common gap for the college students in the math courses I taught at various colleges and universities).

This approach I used with the student could be used for high school students, especially those with special needs.

Multiplication Problems as Intro to Unit Rates and Slope (follow up)

I previously related elementary school word problems with math topics in secondary schools. The photo below shows a method to help elementary school students unpack the multiplication problem, to help middle school students identify the unit rate, and to help algebra students identify slope (you can focus on simple problems like this as an entry point to the linear function type problems).

In advance of this method, a review is conducted on the representation of multiplication using the groups of items model (below). By drawing a picture for the two parts of the problem that have a number, the students are guided to break the problem into parts and then to unpack the parts. The “5 boxes of candies” is represented by squares (or circles if you prefer) with no items inside. The “each box holds 6 candies” is represented by a single square with 6 items (dots) inside.

In turn, the drawing of the group of items leads to the multiplication statement, “6 candies x number of boxes.” I prompt students to include the items with the number as sometimes they will write this statement as “6 x number of candies”. I point out that 6 and candies go together. As seen in the previous blog post, the next step in this problem would be to replace “number of boxes” with the quantity given and then compute.