I had a fascinating exchange with my son last night. I was telling him a made up story, and at one point this character who is cursed and very old said that his age was “Four hundred times a score and one.” My son asked what a score was, and I told him twenty.
He then tried to figure out the person’s age. He started by asking what is four times twenty, and I told him he could figure that out. He counted up by fours to get eighty, and then said that four hundred times twenty is eight hundred. He thought for a moment and said that no, it must be eight thousand, which meant the final answer is eight thousand four hundred.
This makes me curious about how he understands the number twenty. He knows apparently that four hundred times something is one hundred times whatever four times the something is, although I am not clear he would explain it like that. However, he apparently did not use the fact that twenty times something is the same as ten times two times the something.
In this project, students attempt to create all of the possible array representations for each of the numbers smaller than a given number(which may depend on the amount of time they have to devote to this project).
For example 6 = 2 x 3 and 6 = 1 x 6, so two possible arrays for 6 are:
The objective after they have these arrays is to classify the numbers into groups based on whatever patterns they see in the structure of the arrays.
What observations do you think students might have when doing this project?
Screen-shot from one of the puzzles included in the block game.
I wrote this puzzle/game last year with the hope it could be used to help generate some thinking about area, multiplication, and addition.
Here are some questions you could use with the game.
Try solving the problems, if possible, in a variety of different ways. Which ways give you the most points? How much is each piece worth? What is a general strategy you can use to get as many points as possible? How do you know your general strategy is effective?
Note: This game does not currently work in Internet Explorer.
This idea comes from Malke Rosenfeld. There are other ways of arranging the tower, so while the diagram above should give you some ideas as to what a multiplication tower is, you and your students should adapt this project. Additionally, once it is built, you can use the tower to look for patterns in multiplication.
This problem allows for multiple possible solutions, and will require students to think about some assumptions they will have to make to solve this problem. You can easily vary the difficulty of the problem by changing the number of people invited to the party.
How many tables and how many chairs?
Ellen is planning a party for her friends. She has invited 100 of them, but she doesn’t know exactly how many of her friends will attend. She wants to put out tables for her friends, and she wants to put enough chairs at each table so that none of her friends has to sit alone. Assume that her friends will fill up each table as they arrive. How many tables should she put out, with how many chairs at each table?