Multiplication Strategies

Young mathematicians’ first formal introduction to multiplication and division happens in 3rd Grade.

In our unit, Equal Groups, situations are presented in context. The situation usually requires students to identify the number of groups and the number of items within each group.

There are 5 tricycles. Each tricycle has three wheels. How many wheels are there in all?

Or, the situation may read,

Tricycles have three wheels. There are 5 tricycles. How many wheels are there in all?

Number of Groups: 5
Number of Items in Each Group: 3

Students are taught to represent the problem with both an addition equation and a multiplication equation to illustrate the connection, and use a variable for the missing piece of information.

Addition equation: 3+3+3+3+3=w
Multiplication equation: 5x3=w (read 5 groups of 3)

Students generally begin solving the multiplication situations with their prior knowledge of repeated addition.

Repeated Addition: 3+3+3=9 and 3+3=6, so 9+6=15.

Then, some students move into skip counting if they can easily skip count by that number. If the situation allows counting skip counting by 2’s, 5’s, and 10’s, students will almost always start with this strategy. If however, the situation has them skip counting by 8’s, then we’ll teach students to use 10 as their anchor. Skip count by 10 and go back 2.

Skip Counting: 3,6,9,12,15.

One tool that we teach students to use to organize their thinking when skip counting is a ratio table. Ratio tables help students keep track of the number of groups as they skip count.

During the unit, students are introduced to multiplication situations using arrays, too. An array is a rectangular arrangement with rows and columns.

To create a common language with our students, we have them give us the dimensions of the row first and then the column. The array above would be a 5x3.

There are many ways to solve the array. One of the things students do is skip count the array.
They can skip count the array be either the rows or columns.

Students naturally begin to see and explore the commutative property of multiplication. Though the situation is 5x3, they realize they can applying their number knowledge properties, and solve 3x5 instead (5,10,15 or 5+5+5=15).

Students’ next level of understanding develops when they recognize that they can decompose an array into smaller arrays to help them solve problems.

In this situation, if a student didn’t know the product of 5x3 quickly, they could decompose the array into (5x2) + (5x1) = 15.

This understanding is extremely important as students move into problems that are more difficult when they begin multiplication like 6x8. When in the early stages of developing automaticity, they may not know 6x8, but if they know (3x8), they can solve 2(3x8), or, if they know 6x4, they can solve 2(6x4).

Students’ ability to think flexibly with decomposing arrays in multiple ways, builds a strong foundation for fluency in multiplication. The skill allows students to attack any multiplication equation for which they don’t automatically have a product, and leads into being able to solve more difficult equations like 14 x 12.

After the unit, Equal Groups, students have a solid conceptual foundation and can think about multiplication flexibly. But, if we stop there, they may never become fully fluent. We continue to practice fact fluency with our combination club flash cards, by playing multiplication bingo, doing fluency clicker reviews, and doing a timed fluency snapshot several times a week. The fluency snapshots are presented by similar facts. (5’s and 10’s together) (2’s and 4’s and 8’s together) (3’s, 6’s, 9’s, 12’s together) (7’s) and (11’s). Presentation with similar facts promotes the conceptual understanding we build throughout this unit.

Our goal is for every student to leave third grade knowing each of their multiplication facts within three seconds. This foundational knowledge creates automaticity and will help them be successful in fourth grade as they embark on more complex multiplication problems like 49 x 58.

Free Fall Gizmo

Gizmos are interactive on-line simulations in Mathematics and Science. On Tuesday and Thursday, students worked on a Gizmo called Free Fall Tower.  The Gizmo allowed students to explore how quickly different objects fall in air and how they fall in a vacuum, with no air. They also observed the objects acceleration as they fell.  Students who reached Activity 3 also explored the terminal velocity of objects falling through air, with and without a parachute.   

This activity promoted enrichment on the concept of gravity, mass, and acceleration, but is not content that students will be held accountable to learn in Grade 3. Students had the option to add the recording sheet to their red homework folder if they wanted to complete the activity at home, or to their green science folder that stays at school. 

On-Line Science Textbook

Today we taught the students how to log in to their on-line Science textbook. The link is on the right hand side of our blog under Science Websites. The student's username and password is in the back cover of their planner. 

Gravity Lab Extension

During our study on gravity, we watched a video clip of Miss Russell skydiving. We thought it'd be interesting as an extension activity to show the historic Space Jump that occurred on Sunday and compare it to Miss Russell's experience. 

After we watched the video clip, we discussed the space suit used by Mr. Baumgartner, the capsule that he rode in, and the helium balloon he used to carry him into the stratosphere.

We then embarked on a little competition to offer basic exploration of weight and overcoming gravity, and the opportunity for students to engage in a trial and error method of problem solving.  Their challenge was to make a miniature hot air balloon that would hover in the air and move slowly toward the ceiling. If they added too much weight, gravity would be too strong. If they added too little weight, the balloon would rise too quickly. The group who had the miniature balloon that floated the slowest to the ceiling would claim victory.  There were four trials in all and watching the excited students as they planned, implemented, and launched their balloons was well worth the slightly organized chaos in our learning laboratory.

Examining Multiples Charts

In math, we have been examining multiples charts. We have highlighted multiples of all numbers 2-12.  Students have observed multiples charts that have relationships such as 3, 6, and 12 (see below). 

In class we have discussed the following types of questions:

How many groups of 3 does it take to make 30? ( _____ x 3 = 30)

How many groups of 6 does it take to make 30? ( _____ x 6 = 30)

What happens to the number of groups? Why? 

Is 100 a multiple of 3? Is 100 a multiple of 6? Is 100 a multiple of 12?

Below is an example of the work that students completed in class as part of their multiples study. 

Pictures of Things That Come in Groups

Yesterday, we had our second lesson on multiplication. The task required students to choose an item that comes in groups and draw several groups of the item. Then, they had to write a sentence about the three pieces of mathematical information in their picture. After that, they included an addition and multiplication equation to describe their picture. Our young mathematicians did a great job!

Things That Come In Groups

Yesterday, students began the study of multiplication and division. In order to understand these topics conceptually, it is important that they understand the meaning of multiplication and division which is the combining (and separating) of equal groups. Yesterday, students began brainstorming about things that come in groups (see chart below).  They also solved the story problems about equal groups. Check out our thinking below.