Math Facts Journal
The blog of MathFactLab
The 3 Times Table: How to Teach x3 Multiplication Facts
Looking for best practices in teaching the 3 times table? You've come to the right place. In this blog series, I am working through a recommended sequence of instruction for teaching multiplication/division facts along with the appropriate strategies at each level. This post is the fifth in the series, and today we are looking at the x3 facts, the second level of 'derived' facts in MathFactLab's multiplication and division fact program. If you've missed previous posts, you might wish to start with the first in the series here.
MathFactLab is dedicated to helping students develop math fact fluency through strategies, multiple-models and structured, individualized practice.
How to Teach the 3 Times Table:
x3 Multiplication and Division Facts
You might be wondering why the x3 facts come after the x4 facts in this instructional sequence. The answer is quite simple: In my experience, students have more difficulty with the x3 facts than with the x4 multiplication facts. For example, the addition necessary to construct 3 x 7 (14 + 7) is typically more difficult for students than the addition required for 4 x 7 (14 + 14). Also, by having students learn the x4 facts first, students are given landmarks on either side of the x3 facts (the x2 and x4 facts) as they work to learn them.
Let's get started with our first - and most important - strategy.
Double Plus One: 3 groups = 2 groups + 1 group
Students can construct the x3 facts by taking combining two foundational facts: the x2 and x1 multiplication facts. At MathFactLab, we model this using both dice and ten frames. Both of these strategies provide hints to help our students get there. Be mindful of the fact that 3x7, 3x8 and 3x9 can be more challenging for students.
Note: We introduce 3x9 at this level but do not assess it until students reach the x9 level. That is because 3 groups = 2 groups + 1 group is not a particularly efficient strategy for learning 3x9.
Other Strategies for x3 Multiplication Facts
Skip Counting/Number Lines
While not as simple as skip counting by two, ten or five, elementary students should still be able to fluently skip count by threes. This skip counting helps kids recognize quickly which whole numbers belong in the set of multiples of three and which do not.
Area Models
Students should have multiple mental models of any basic math fact. One of these foundational models, is the area model. Having students work with area models when learning the x3 multiplication facts will not only help them to solve, but also to visualize the fact and make connections with the other facts.
At MathFactLab, we show our area models horizontally and vertically, so a student can see 3 rows of six is equal to 6 rows of three.
Open Arrays for Division
Open arrays are a great way to help students develop a deeper understanding of the inverse relationship between multiplication and division. MathFactLab's open arrays show the length of one dimension of a rectangle, its area, and a question mark for the other dimension. This helps students to realize that finding a missing factor is division.
As the problems are presented both as division (24 ÷ 3 = __ ) and missing factor multiplication ( 3 x __ = 24), students realize that when they are practicing their division facts, they are also practicing multiplication facts.
Bar Diagram Division
I wish when I was in school my teachers had taught using bar diagrams. Bar diagrams illustrations can make the strategy for solving a complex word problem suddenly seem simple and straightforward, and in our case, can help a student who might be a little shaky about what exactly division is.
It's important when practicing division facts to practice using both of the factors of the fact triangle as divisors. So in the case of the 3-7-21 fact triangle, it's important to practice both 21 ÷ 3 and 21 ÷ 7.
Conclusion
While the idea of x3 may seem simple, in my experience, mastering them takes probably more work than any of the preceding levels. Even if we leave 3x9 for a later level, 3x7 and especially 3x8 can be stumbling blocks. One way to help kids over or around those stumbling blocks is by providing multiple strategies - or pathways - to solving these multiplication facts. We're less likely to get lost when we know multiple routes to our destination.
At MathFactLab, we take a fact family triangle approach to mastering the basic multiplication facts. When students learning the multiplication facts, think in terms of multiplication fact triangles, they division facts take hardly any effort: instead of identifying the apex of the fact triangle, division just requires the student to identify the missing factor at one of the bottom corners.