This problem involves doubling. Knowing their doubles (and halves) to at least 20 enables students to solve some patterning, number and simple fraction problems. Students need to see clearly that the sum of a double is equivalent to product of that number multiplied by 2.
Grandpa has five pockets in his jacket. In one pocket he has one chocolate. In another pocket he has two chocolates. In another he has four chocolates. In yet another he has double that many and in the fifth pocket he has double that many again.
How many chocolates does Grandpa have in his jacket?
- Play a game of "Guess what’s in the jacket pocket?" to introduce the problem. Answer yes or no to guesses until the students determine there are chocolates in the pocket.
- Pose the problem for the students to solve. Check that the students understand doubles.
What does it mean to double a number?
What is double 12 (24)? If you didn't just know that, how did you work it out?
- Tell the students that you want them to try to work out the answers in their heads.
- As the students work ask questions that focus on their use of mental strategies for addition.
How did you work out how many chocolates were in the jacket?
How did you keep track of the chocolates?
Are you convinced that your answer is correct? Why are you?
- Share solutions.
This problem could be posed in many ways. The doubling might be the amount of money you earn on successive days. It might be the number of students or pets living in neighbouring houses. It might also be the number of points that you score in successive games.
Extension to the problem
- What if grandpa had three times as many chocolates in each pocket as he did in the pocket before? Assume that the first pocket had just one chocolate.
- What if grandpa had 93 chocolates. What is the smallest number of chocolates he had in any pocket?
Grandpa’s pockets hold 1, 2, 4, 8 and 16 chocolates. So he has 31 altogether.
Solution to the extension:
- The number of chocolates is 1 + 3 + 9 + 27 + 81 = 121.
- Probably the easiest way to do this is by guess and check. But a smarter way is to notice that if grandpa’s smallest pocket had 5 chocolates, he would have 5 + 10 + 20 + 40 + 80 = 155 = 5(1 + 2 + 4 + 8 + 16) so this is 5 times the 31 he had in the original problem. So how many times does 31 go into 93? The answer is 3. So the smallest number of chocolates that grandpa has in one of his pockets is 3.