# Rational Number Project

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5 Teaching Actions Comments squares. Then take 3 of the columns, tenths columns, minus that. So that would be point 25. There s half of the grid and 5 pieces and I take away 3 from the half and just keep the Still going to be 2 so don t bother with that. Point 3 is 3 tenths columns minus point zero five is 5 little squares. So 3 tenths columns minus 5 little squares (pauses to think) is point 25. Translations: Symbolic to verbal Symbolic to picture to verbal to symbolic Symbolic to picture Lesson RNP 204 5

6 Additional Teacher Notes Lesson The examples below show students use of the Addition/Subtraction boards. These examples came from student interviews conducted after completing the decimal lessons. Decimal Addition To solve using the board the student first showed.28 on the left grid as 2 orange strips and 8 yellow squares. He showed.06 as 6 yellow squares on the right grid. He then said, I take 2 of those to make whole, 3 tenths. Then if you minus 2 from these (6 yellow squares) you have 4 left. He shows the final answer on the bottom grid as 3 orange strips and 4 yellows. 6 RNP 204 Lesson

7 Notice that the student mentally operated on the display of the two numbers in the problem to determine how to represent the answer on the bottom grid. In the next example, the student solving the same addition problem physically acted out the addition on the grid and then showed the result on the bottom grid. To solve using the board the student first showed.28 on the left grid as 2 orange strips and 8 yellow squares. He showed.06 as 6 yellow squares on the right grid. The student then crosses off 2 of the yellow squares from 6-hundredths. She then shades in 3 orange strips and 4 yellows on the bottom grid. This is 28 so I shade in 2 oranges and 8 yellows. Then there s only 6-hundredths so shade in 6 then take away 2. And add them to that. I am going to make tens so I can make 3. And there is still 4 left over. Lesson RNP 204 7

8 Subtraction: In this example the student acts out the subtraction as take-away marking off the needed number of tenths on the grid on the left side To solve using the board the student first showed.67 on the left grid as 6 orange strips and 7 yellow squares. He showed.5 as 5 orange strips on the right grid. He takes away 5 tenths by crossing out 5 orange strips on the left grid. He then shades the answer on the bottom grid as orange strip and 7 yellow squares. And I take the 5 oranges away from this and I cross them out over here and then I know when I come down over here I have tenth and 7 hundredths. 8 RNP 204 Lesson

9 Another student showed the two amounts on the board but mentally subtracted 5-tenths and then showed the final answer on the bottom grid. OK so you would subtract 6 bars from 5 bars which would give you one left and there s 7 yellows, there s 7 squares and no squares so you would just put these squares over there and there you go. (When asked what his final answer was he stated: 7 hundredths. Notice that this student rewrote the problem vertically to show his work with the grid. He lined up the 5 tenths under the 6 tenths without having to rewrite.5 as.50. Lesson RNP 204 9

10 To solve using the board the student first showed.67 on the left grid as 6 orange strips and 7 yellow squares. He showed.09 as 9 yellow squares on the right grid. For this one you have to subtract 7 to 9 which I am going to just use a bar and you would add a little circle there and get rid of one of the bars so it would be 5 bars and then go, 2,,,,8 and there you go. He then shaded in 5 oranges and 8 yellow on the bottom grid. That would be 58 hundredths. The student reflected on the two representations for the numbers involved in the problem. He imagined taking away tenth and leaving yellow square. (He pointed to the grid on the left while explaining this). Students used the grids in different ways to solve these problems. The most important part of the grid is that it allows students to clearly represent the two numbers in the problem. Then the action on the two numbers can be done mentally or physically by writing on the boards. The bottom grid is used then to show the change after the action. We found that students were able to accurately record the problems using symbols in vertical format after only a few examples. When doing the class work for lesson, many students first showed the two numbers on the top part of the addition board and then accurately recorded the problem in vertical form. Students did not make whole number errors of misaligning the numbers even though we did not introduce the rule line up the decimal points. Once the problem was written in symbols, many students solved the problem symbolically and then recorded the answer using orange and yellow colors on the bottom grid. The translations: symbolic to picture to symbolic aren t always that straight forward. Often we found that students use symbolic understanding first and then connected back to pictorial or concrete 0 RNP 204 Lesson

11 representation. The reality is that multiple representations for the mathematical concepts/skills being developed enhance students understanding and the connections among different representations are varied. Below find examples of student s work from lesson. There are no indications on the addition/subtraction board that the student used them to act out the problem. But the problems are recorded correctly using symbols and correctly solved. Lesson RNP 204

12 Decimal Addition and Subtraction Board Student Page A Lesson /Teacher Page

13 Lesson /Teacher Page 2 37-hundredths plus 4-tenths 56-hundredths take away 9- thousandths and 6- hundredths plus 7-tenths than 2 Less than 2 than Less than than 2 Less than 2 than Less than than 2 Less than 2 and 58- hundredths take away 6- tenths than Less than than 2 Less than 2 73-hundredths take away 6- hundredths than Less than than 2 Less than 2

14 Lesson /Warm Up Imagine adding using fraction circles. Describe how you would solve this.! Imagine as a decimal, how much of 2 the 0 x 0 grid is shaded? Describe this amount if you used only orange. Imagine as a decimal, how much of 4 the 0 x 0 grid is shaded? How many orange bars and yellow squares would be shaded to show 4?

15 Addition/Subtraction Boards Lesson /Student Page A

16 Name Lesson / Student Page B Addition and Subtraction with 0 x 0 Grids Solve each problem using the grid. Record the problem with symbols showing the final answer. Problem Estimate Picture Symbols = than one or less than one? 37-hundredths and 3-tenths more than 2 or less than? 2 than one or less than one?

17 Name Lesson / Student Page C 5-hundredths minus 7- hundredths than or 2 less than? 2 than or 4 less than? = than or 2 less than? 2

18 Name Lesson / Student Page D 99-hundredths 88-hundredths than or 2 less than? 2 than or 4 less than? 4 57-hundredths minus 3-tenths than or 2 less than? 2 than or 4 less than? 4

19 Post Lesson Reflection Lesson ) Number of class periods allocated to this lesson: 2) Student Pages used: 3) Adaptations made to lesson: (For example: added extra examples, eliminated certain problems, changed fractions used) 4) Adaptations made on Student Pages: 5) To improve the lesson I suggest:

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