Multiplication Fact Progressions

Multiplication Fact Learning Plan for 3rd/4th:

- INTRODUCE

- Use number talks/problem strings to introduce various mental strategies for deriving multiplications
- Use visual models (e.g. arrays or area models) to support conceptual understanding and the making of connections (is there a better way to phrase this?)
- Each new strategy goes with a batch of multiplications

- REINFORCE

- Games or other activities that reinforce strategies

- PRACTICE

- Ask kids to create cards for each batch of multiplications
- Kids practice by deriving or (once deriving is easy) trying to remember multiplications from their decks. Or games or anything else.

- EXTEND

- Use additional number talks to extend these strategies beyond one digit by one digit facts

- Homework???

One way this is different from what I have done before is that many of the fact “families” are smaller than what I have used. I have given kids 10 per family plus the “turnaround fact” for each (3 x 5 and 5 x 3). I am interested to see how it goes with fewer facts in a family. It might be nice, especially if some of their practice includes writing out how they derived it or drawing an image to show how they derived it.

So here’s a way I might do this: Wait til a strategy arises / provide opportunities for it to arise. Name it and put it on our anchor charts. As kids start to use it, ask them to come up with some other facts they could derive in the same way. Then they practice those facts. Then they add more facts or start a new strategy.

Hmmm…. Record-keeping. Seems like it will be hard. Maybe each kid has a Multiplication Table to keep track of which facts they are practicing or have practiced. And a list of the strategies they are using and the associated facts? (Heidi)

Strategies:

Strategy: Close to Doubling/Skip Counting (3 x 3 and 3 x 7)

Strategy: Halving a Factor (5s) - Round 1

Strategy: Halving a Factor (5s) - Round 2

Additional Strategy for Fives: Doubling a Factor

Strategy: Double Double - Round 1

Strategy: Double Double - Round 2

Strategy: Double Double Double - Round 1

Strategy: Double Double Double - Round 2

Strategy: Doublings associated with 3 x 3

Strategy: Building Down 9s from 10s

Source: https://core.nthurston.k12.wa.us/common-pdfs/origo-mult-diiv-pdfs/last-facts-mult.pdf

See also: http://www.skitsap.wednet.edu/cms/lib/WA01000495/Centricity/Domain/509/Doubling.pdf

I haven’t formally practiced these because they’re boring and repetitive to show up in a deck of practice cards. But maybe it’s good to have crazy easy ones in for confidence, depending on the group??

In the beginning of 3rd Grade, many of my students don’t have strong doubling strategies. They won’t necessarily break apart numbers by place, and they often lack fluency in single-digit addition strategies. For that reason, doubles seem like a good place to start for multiplication.

First card set - Less than or equal to 10

2 x 2 | |

3 x 2 | 2 x 3 |

4 x 2 | 2 x 4 |

5 x 2 | 2 x 5 |

10 x 2 | 2 x 10 |

Second card set - Greater than 10

6 x 2 | 2 x 6 |

7 x 2 | 2 x 7 |

8 x 2 | 2 x 8 |

9 x 2 | 2 x 9 |

3 x 10 | 10 x 3 |

4 x 10 | 10 x 4 |

5 x 10 | 10 x 5 |

6 x 10 | 10 x 6 |

7 x 10 | 10 x 7 |

8 x 10 | 10 x 8 |

9 x 10 | 10 x 9 |

10 x 10 |

3 x 3 is just so useful, and Origo considers it a “last fact.” I can’t imagine leaving it towards the end of the learning progression, though.

Likewise, 3 x 7 is a “last fact,” but I don’t see why it should be last considering how useful it is for deriving other “last facts” and how often it shows up.

3 x 3 | |

3 x 7 | 7 x 3 |

3 x 10 is 30, so 3 x 5 is 15. This means that we need to practice halving before using this strategy.

Even ones are easier for this strategy.

4 x 5 | 5 x 4 |

6 x 5 | 5 x 6 |

8 x 5 | 5 x 8 |

Odd multiples of 10 are tougher for halving.

3 x 5 | 5 x 3 |

5 x 5 | |

7 x 5 | 5 x 7 |

9 x 5 | 5 x 9 |

3 x 5 is 15, so 6 x 5 is going to be 15 + 15.

3 x 5 | 5 x 3 |

6 x 5 | 5 x 6 |

4 x 5 | 5 x 4 |

8 x 5 | 5 x 8 |

3 x 2 is 6, so 3 x 4 is 6 + 6.

2 x 8 is 16, so 4 x 8 is 32.

3 x 4 | 4 x 3 |

4 x 4 | |

7 x 4 | 4 x 6 |

The facts in this batch build on the facts from Round 1. Round 1 included 3 x 4, 4 x 4 and 7 x 4. While students can use the same double doubling strategy for these facts, separating them in this way also gives them the opportunity to double a factor, and derive these facts in that way.

EX: 3 x 4 is 12, so 6 x 4 is 24. 4 x 4 is 16, so 8 x 4 is 32.

6 x 4 | 4 x 6 |

8 x 4 | 4 x 8 |

9 x 4 | 4 x 9 |

3 x 2 is 6, 12, 24.

8 x 2 is 16, so 32, 64.

3 x 8 | 8 x 3 |

8 x 8 |

As with Double Double, separating this into two rounds increases the possibilities for students to use other strategies for these facts. So, if kids know 3 x 8 = 24, then they can also deduce that 6 x 8 is 24 + 24. They might relate 7 x 8 to 6 x 8, though that’s tougher.

7 x 8 | 8 x 7 |

6 x 8 | 8 x 6 |

9 x 8 | 8 x 9 |

3 x 3 is 9, so 3 x 6 is 9 + 9 = 18.

6 x 6 is double 18.

3 x 6 | 6 x 3 |

6 x 6 |

10 x 3 is 30, so 9 x 3 is 27. 10 x 9 is 90, so 9 x 9 is 81. Note that 3 x 9 and 6 x 9 are related.

9 x 3 | 3 x 9 |

9 x 6 | 6 x 9 |

9 x 7 | 7 x 9 |

9 x 9 |

There are lots of ways to derive these, none of which are particularly natural. 7 x 6 should build on 7 x 3, 7 x 7 is memorable for whatever reason. 7 x 6 is the hardest fact, I’m pretty sure. Off the cuff, here is how I’ve seen kids succeed with these:

- 7 x 3 is 14 + 7
- 7 x 6 is 21 + 21
- 7 x 7 is just 49.

7 x 6 | 6 x 7 |

7 x 7 |