How to teach the multiplication facts

As a math coach one of the things I hear the most from upper elementary and middle school teachers is “What do I do with my students who don’t know their multiplication facts?” I hear this from third grade teachers fourth grade teachers fifth grade teachers all the way up to whatever grade level I work with. It just seems like we recognize in third grade how challenging it is to get students to learn their multiplication facts from memory, but we’re never able to solve it in fourth, fifth, or sixth grade so we are definitely not going to blame the third grade teachers at all.

What’s going on with the system that is preventing kids from finally learning their multiplication facts? There’s lots of advice out there…lots of books you can buy, but in this blog post I’m going to give you an idea based on an NCTM article I read. If you want a copy of the article, please consider joining NCTM…it is worth it.

What is fluency?

Let’s begin by quoting the CA Math Standards: “By the end of Grade 3, know from memory all products of two one-digit numbers.” (3.OA.C.7)

Notice the word is “memory” and not “memorized”. Knowing one’s multiplication facts from memory is the result of repeated experience working with the number facts in a variety of ways such that eventually the facts is known from memory. Learning facts through memorization denies students the opportunity to develop a deep understanding of the meaning of multiplication in the first place.

How is fluency developed?

Students need to progress through the different levels of mathematical reasoning:

Counting strategies which lead to additive thinking, which leads to multiplicative thinking

Research tells us that students generally do not retain their multiplication facts because traditional teaching approaches too often attempts to move from Phase 1 directly to Phase 3, with only a brief superficial attention to Phase 2.

While we are happy any time a student can correctly say that 8 x 7 is 56. What the teacher REALLY is interested in is HOW the student arrived at 56. Was it through counting strategies? Additive thinking? Or Multiplicative thinking?

We want multiplicative thinking.

Here are examples of solving 8 x 7 using each of the three levels of developmental reasoning:

The research is clear that we need to spend more time teaching students explicit development of reasoning strategies and how students can derive multiplication facts from known facts. In one study, students learned their multiplication facts better when they were taught strategies like using arrays and building upon known facts versus just repetition and memorization. On average students gained 7 times more multiplication facts per minute of strategy intervention compared to repetition intervention. This study used 350 students in grades 3, 4 and 5. The students spent 10-15 minutes per day over 5 weeks in one of the interventions.

What might this mean in YOUR classroom? Please take down the skip counting posters. Those skip counting posters prevent students from moving toward the more sophisticated multiplicative thinking…Instead students remain stuck using only additive thinking.

Strategic sequencing of facts and strategies

It is common for teachers to work through the facts one factor at a time. First they begin with the 1s, and then the 2s, and then the 3s and so on. However, it is more effective to introduce the multiplication facts starting with the least difficult facts and then grouping additional facts based on the strategies students will use to derive the facts.

Start with the easiest facts of 1, 2, 5, and 10. Provide numerous opportunities for students to explore these facts until they know the products from memory…not memorization. Then introduce students to the square facts 2×2, 3×3, 4×4, etc. How might a student learn 6×6 using the foundational facts?

Start with understanding 6×6 can be represented with a 6-by-6 array. Then notice the array can be cut into two smaller arrays: 5×6 and 1×6.

The 1s, 2s, 5s, 10, and squares are the foundational facts.

Students will learn the remaining facts using strategies derived from knowing these initial foundational facts. Students will represent the facts using a variety of visual representations: array, tape diagram, etc.

Adding or Subtracting a group

For example, students can use the “adding or subtracting a group” strategy to build upon the 2s, 5s, and 10s in order to find the 3s, 4s, 6s, and 9s.

For example, 6×7 can be expressed as 5×7 plus one more 7.

Similarly, 6×8 can be expressed as 5×8 plus one more 8.

Halving and then Doubling

Any time the student is working with a fact with at least one of the factors being even, this is is a good time to consider using the “halving then doubling” strategy.

For 8×6, we think of of rectangle that is 8 units tall and 6 units wide. Then this rectangle can be split into halves, create two identical smaller rectangles each with an area of 24. Adding 24 + 24 gives us the answer for 8×6.

Similarly, 6×7 can be thought of as “I know half of six is 3 and 3×7 is 21. So I double 21 to get 42 which is 6×7.”

Using a Nearby Square

This strategy works with multiplication facts such as 7×8 or 7×6 or 8×9. Students builds upon the known square product to derive the product.

For example, 7×8…

Decomposing a factor

The final strategy is to decompose one of the numbers to create known facts and then adding them together to get the final answer.

For example 8×7. The eight can be decomposed to 5 and 3 to find the two partial products of 5×7 and 3×7. Then add the 35 and 21 to get 56.

Deliberate and fun practice

Now that you have explicitly taught all the Phase 2 strategies, it is time for students to practice their multiplication facts in engaging ways. I love games as one way for students to practice their multiplication facts. I’ll share one old-school game and one new-school game.

Old-school game to practice

Product Game is similar to Connect-4 but students use paperclips to choose factors. The resulting factor is claimed by the player whose turn it is. Here are the directions…

1. Player 1 puts a paperclip on a number in the factor list. No square on the product grid is marked with Player 1’s color because only one factor has been marked; it takes two factors to make a product.
2. Player 2 puts the other paper clip on any number in the factor list (including the same number marked by Player 1) and then shades or covers the product of the two factors on the product grid.
3. Player 1 moves either one of the paper clips to another number and then shades or covers the new product.
4. Each player, in turn, moves a paper clip and marks a product. If a product is already marked, the player does not get to mark for that turn, The winner is the first player to mark four squares in a row — up and down, across, or diagonally.

Download a game board here.

New-school game to practice

Number Hive is an online version of Product Game previously shared.

https://play.numberhive.org 

Do you want to learn more about the strategic sequence of facts and strategies? Watch my 25-minute video detailing the steps.

https://youtu.be/WfWavldQCdE?si=4OS-41QPq0MrIQJt

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