First, students will learn how to multiply two, three, and four-digit numbers by one-digit numbers, by expanding numbers, or equivalently, using the area model. For example, 5831 = 5000 + 800 + 30 + 1. So \(7 \cdot 5831 = 7(5000 + 800 + 30 + 1).\)
Multiplying with area model: 6 x 7981 by Khan Academy
Multiply 3- and 4-digits by 1-digit with area models by Khan Academy
Next, students will learn how to multiply multi-digit numbers by one-digit numbers, using Genaille-Lucas rulers.
Genaille-Lucas Rulers by D!NG
Print either set of rulers found here, then practice multiplying some numbers with them.
I would love to program an applet for Genaille-Lucas rulers. Should have snap points to make it easy to place the rulers with precision. Also, there should be a tray from which you can drag onto the canvas whatever rulers you please. For example, if you're doing 3 * 2222, you could drag one 3-ruler and four 2-rulers onto the workspace. By selecting a ruler and pressing Del, you can remove a ruler from the workspace. Another nice feature would be a basic zoom in and out by using the scroll-wheel.
After that, students will learn how to multiply two-digit numbers, using concrete and visual methods.
Next, students will learn how to multiply two-digit numbers, using the partial products method.
Watch these Khan Academy videos:
Do these Khan Academy exercises:
Next, lead this demonstration. Instead of introducing it as learning the 17 times table, I would introduce it as skip counting by 17, from 0 to 170. I think this demonstration is valuable, as it shows students how to memorize a small number of just about anything. It may also increase their confidence in their ability to learn.
Next, give your students these challenges:
Follow the Numbers by NRICH: Order of digits doesn't matter, their initial sum will be the same. For example, 73 and 37 have the same initial sum. In fact, many numbers will have the same initial sum. For example, 4, 13, 22, 31, and 40, all have the same initial sum. From this observation, we can figure out where the numbers 1-100 go by following 18 possibilities, because 1 has the smallest initial sum, which is 1, and 99 has the largest initial sum, which is 18. Some of these 18 will have the same journey, for the reasons previously stated. These are: 1 and 10, 2 and 11, ..., 9 and 18. We also know where a number goes once we find a number we've seen before. Following each journey in turn, remembering that our 1-18 are the initial sums, we have
- 1, 2, 4, 8, 16, 14, 10, ...
- 3, 6, 12, ...
- 5, 10, ...
- 18, 36, 18, ...
Thus, every number gets caught in 1 of 3 patterns:
- 2, 4, 8, 16, 14, 10, 2, ...
- 6, 12, 6, ...
- 18, 18, ...
This strategy can be used to find the patterns for the other two proposed rules as well:
- Add the digits, then multiply by 3.
- Add the digits, then multiply by 5.
In each of the problems below, you're given 4 cards, each containing a single digit. Place the 4 cards to make two two-digit numbers, that give the largest possible product.
Here's the answer.
Conclude by leading this investigation:
Using Least 1s by MathPickle
4.NBT.B.5: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.