Denise Lenihan
9/26/12
Mathematics
W-L Ch. 11 #1-6
1. Three ways one can count a set of objects are; by ones, counting each individual unit in order, by groups and singles, counting how many groups and then how many singles, and by tens and ones, counting ten, twenty, thirty, and then the ones, twenty one. These methods of counting can be used to coordinate concepts and oral and written names for numbers by asking students how to say the numbers and how to symbolically write the numbers. Students will then see both the amount a certain number will look like and how to write it, ten for example would look like ten one units or written, 10. Conceptually, students will be able to make those connections and form better understandings of the numbers.
2. Three types of physical models for base-ten concepts include; groupable models, those that clearly reflect the relationships of ones, tens, and hundreds, pregrouped/trading models, those that cannot be taken apart or put together, and nonproportional models, those that are not physically larger or smaller but representational. There is much significance in the difference between these models because they should be used at different times and at different progressive levels with different activities. As students become more familiar with the beginning groupable models, they can advance to pregrouped models that help provide a different understanding. Pregrouped models help combine multiplicative understanding with a positional system. Furthermore, as students understand that 10 units make “a ten”, they can use nonproportional models that help grasp a different understanding to reach a relational place-value understanding. In addition, students who are struggling in computation should be directed towards nonproportional models, thus emphasizing the difference between the other models.
3. Students can learn to write two- and three- digit numbers in a way that is connected to the base-ten meanings of ones, tens, and hundreds by first using base-ten language. Teachers can help students write the correct numbers if they model and say the correct form orally as it seems appropriate. Students can learn to write the numbers as they connect to the base-ten meanings by using place-value mats, whereby they can see the ones, tens, or hundreds place divided into sections.
4. The hundreds chart can be used to identify and use place-value concepts by putting up the chart in front of the class and having students recognize patterns or missing numbers. It can be used to identify specific patterns that use place-value concepts such as, the number in a column all end with the same number, in a row, the numbers count form left to right but the first digit (tens digit) stays the same, in a column the first number (tens digit) goes up by ones, or count by tens going down the right hand column. The students can practice with a blank chart as well and have them find the correct places for numbers.
5. Landmark numbers are multiples of 10, 100 and occasionally other numbers such as multiples of 25. Students should have a confident relationship with their landmark numbers that they use to develop number sense and place value. The teacher can use the hundreds chart of a number line to explore further the different relationships students may have with landmark numbers. One activity using a number line would be to label 0 to 100 at opposite ends and mark one point on the line with a question mark. Students must try to guess the number the question mark represents by using estimation from other numbers along the line.
6. Place-value concepts and computation skills and be developed at the same time by connecting procedural concepts such as addition and subtraction. These concepts relate and lay the foundation for being able to access computation skills with more efficiency and success. Landmark numbers can easily be broken apart, and groups can be visually noticed and understood. One activity is to list on a sheet different numbers that when added together equal 100, by groups of 5. For example, the numbers would be in a mixed order, but 85 would have to match 15, 75 would have to match 25, and 55 would have to match 45. Another activity is to use egg cartons that have two rows of five eggs, color the eggs red or blue and see how many different combinations add up to 10 eggs; 6 red plus 4 blue eggs, or 7 red plus 3 blue eggs.