Base 10 blocks are a go to representation for place value. They are also easy to implement for addition or subtraction with place. With a group model, they are useful for multiplication and division. It is harder to model multiplication of multi-digit numbers with regrouping. This post presents a Google Jamboard with base 10 blocks on a scaffolded chart to provide such a model.
Fractions are challenging. Multiplying fractions is really challenging! This post presents a Google Jamboard to introduce students to the concept of multiplication of fractions.
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.
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.
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.
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.
This post details a scaffolded approach for multiplying multi-digit numbers by 2-digit numbers. It was originally created for a student with ADHD who understood how to do the multiplication but would rush and repeatedly made simple mistakes. It is useful for all students.
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.
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.
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.
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.
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.
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.