A B C

Solutions for Session 10, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6

 Problem B1 Students are using logical reasoning and their understanding of place value in base ten to help them interpret base five numbers and convert them to base ten. Students are also using mental mathematics and the order of operations to solve the problems. The students realize that they must choose the appropriate place value first, meaning using the exponents to find the greatest power of 5 that is less than the number they are converting.

 Problem B2 When students transfer what they know about base ten to another base, their understanding of place value is extended and deepened. Also, when students have an understanding of the base ten place-value system, their ability to do complex computations increases. They are more able to use mental mathematics to solve problems.

 Problem B3 Because students can picture what the manipulatives look like, they can understand that they first have to figure out what is the greatest power of 5 that is less than or equal to the number. This translates into dividing by the same power of 5 in the algorithm.

 Problem B4 Ms. Miles's lesson focuses on understanding numbers and ways of understanding numbers and number systems. For example, by converting from one base to another and vice versa, the students become more aware of the ways we represent numbers and the meanings underlying those representations, such as place value. They also make connections between the representations and are able to extend them when working with other bases.

Problem B5

 a. These two students imagine using the tiles to solve the problem. They correctly imagine choosing four 1 tiles for the digit on the right, three 5 tiles for the second digit, and two 25 tiles for the third digit. At this point, Ming Hui correctly adds what they have so far. Then Kenneth seems to stop thinking about the tiles and chooses 100, rather than 125, as the value of the next power of 5. Ming Hui appears to be happy with this answer. The students most likely made this error because they inadvertently switched to thinking in base ten, and therefore derived 100 as the next place value instead of 125. b. These students need a reminder of the values of each power of 5. You could suggest that they always make a place-value chart before doing conversions in either direction. For this problem, their chart would look like this: You could then suggest that they work from left to right when converting. This number would be (1 • 125) + (2 • 25) + (3 • 5) + (4 • 1) = 125 + 50 + 15 + 4 = 194.

Problem B6

 a. These students clearly understand place-value systems but are using an unconventional method to translate the numbers. Brad first assumes 342 is a base five number and correctly translates it to base ten. When corrected by Kent, he translates from each base ten value to its base five equivalent. So instead of working with the number 342 from the beginning, he starts with 300. He correctly states that 300 is 12 • 25, which in turn is (2 • 125) + (2 • 25). b. These students clearly understand the system. However, you might suggest that translating each place of a base ten number separately could become messy and lead to mistakes. These students might also benefit from a base five chart. They could use a chart like the one below to remind them of the powers of 5 that are the place values and then record and subtract the number of each power of 5 as they keep a running total, as shown: This tells us that the number is 2332 in base five.