I. Curricular Framework

Concept:

Balance

Essential Question:

How is the concept of Borrowing in subtraction a study in Balance?

Bridge:

Manipulate the Process

Content:

Subtraction: Borrowing, regrouping

Outcomes:

II. Standards Aligned

III. Instruction and Assessment

1. Connect: Connecting to the Concept Experientially

Objective: To introduce/reinforce concept of borrowing/regrouping during subtraction by allowing the children to physically be numbers.

Activity: Draw three large rectangles on the floor as shown in the diagram. Make the first and third rectangles large enough for nine children to stand in. Make the second rectangle large enough to comfortably hold twenty or more children. Draw corresponding rectangles on the chalkboard. Have nine children stand in the first rectangle. Have another child try to join the rectangle. Since the rectangle will be too crowded to hold the tenth child, have all ten children hold hands and move in a line to the second larger rectangle. Name the number formed by the string of ten children in the second rectangle and the empty first rectangle. Have additional children enter the first rectangle until nine children again fill the first rectangle. Name the number formed by the two rectangles each time a child enters the first rectangle. Once more, have another child try to join the first rectangle. Repeat the process of joining hands and moving to the second rectangle. We then have two groups of ten children holding hands in the second rectangle. Name the number. Have one more child enter the first rectangle. Name the number. Write the number in the corresponding rectangles on the chalkboard. Note: Piaget's Concept of Reversibility State that the time has come to take some away from our number, twenty-one. Have four children enter the third rectangle. Name the number. State that we are going to take four away from twenty-one. Write the number, four, in the corresponding rectangle on the chalkboard. Ask, "Do we have enough people in our first rectangle to take four away?" When the answer is given, ask, "What can we do?" Wait for a child to suggest taking some people out of the second rectangle. Have one group of ten from the second rectangle, while still holding hands, move to the first rectangle. After the group of ten move to the first rectangle, have them stop holding hands. State that the persons in the first rectangle are ones again. Ask, "How many children are in the first rectangle now?" When the answer is given, note that the first rectangle is over-crowded. Change the numbers in the rectangles on the chalkboard accordingly. Ask again, "Do we have enough people in our first rectangle to take four away?" When the answer is given, have each of the four children from the third rectangle take the hand of a child in the first rectangle and move away from the rectangles. Ask, "Was the number of children in the third rectangle the same as the number of children taken out of the first rectangle?" When the answer is given, have the four children from the first rectangle and the four from the third rectangle take their seats. Ask, "How many children are left in the first rectangle?" When the answer is given, ask, "How many children are left in the two rectangles?" After the answer is given, state the problem, "Twenty-one take away four leaves seventeen." Change the numbers in the rectangles on the chalkboard to correspond to the number of persons in the rectangles. Have the children take their seats.

Alternate Activity: For small classes and remedial groups. Draw three rectangles on the floor as shown in the diagram. Make the first and third rectangles large enough to hold nine large blocks. Make the second rectangle large enough to easily hold more than thirty blocks. Draw the corresponding rectangles on the chalkboard. Have the children take one block from a box of blocks and bring the block into the first rectangle one at a time (name the number of blocks in the first rectangle after each block is added) until nine blocks have been placed into the rectangle. Have another child try to add another block to the first rectangle. Since the rectangle will be too small to hold the tenth block in a single row the child will need to place the block on top of another block to remain in the rectangle. Note how crowded the rectangle is. Ask the children for a better solution. When a child answers that some or all of the blocks could be placed in the larger rectangle, have the child move all the blocks to the larger rectangle, placing the blocks in one column of ten. Name the number. Have the children continue to add blocks to the first rectangle one at a time (continue to name the number of blocks in the two rectangles after each block is added) until nine blocks have again filled the first rectangle. Repeat the procedure of adding another block and carrying the ten to the second rectangle. Name the number. Have a child add one more block to the first rectangle. Name the number. Write the number in the corresponding rectangles on the chalkboard. Note: Piaget's Concept of Reversibility State that the time has come to take some away from our number, twenty-one. Have four children each bring one block to place in the third rectangle. Name the number. State that we are going to take four away from twenty-one. Write the number, four, in the corresponding rectangle on the chalkboard. Ask, "Do we have enough blocks in our first rectangle to take four away?" When the answer is given, ask, "What can we do?" Wait for a child to suggest taking some blocks out of the second rectangle. Have a child move one group of ten blocks from the second rectangle to the first rectangle, stacking some blocks on top of each other. Ask, "How many blocks are in the first rectangle now?" When the answer is given, note that the first rectangle is overcrowded. Change the numbers in the rectangles on the chalkboard to correspond to the number of blocks in each rectangle. Ask again, "Do we have enough blocks in our first rectangle to take four away?" When the answer is given, have a child take one block from the first rectangle in one hand and one block from the third rectangle and, then, stand to the side. Continue this process until all the blocks in the third rectangle are gone. Ask, "How many children are standing at the side with blocks in their hands?" Then ask, "How many blocks are in the first rectangle now?" After the answer is given, state the problem, "Twenty-one take away four leaves seventeen." Change the numbers in the rectangles on the chalkboard to correspond to the number of blocks in the rectangles. Have the children who have been standing at the side place their blocks into the box and return to their seats. Note: Adapt these activities to the teaching of bases other than base ten for upper grade level students.

Assessment: Children's participation in the activity.

2. Attend: Attending to the Connection

Objective: To allow the children to discuss/examine what happened.

Activity: Talk about the experience of grouping and regrouping numbers. Let the children form other subtraction problems some of which require borrowing/regrouping.

Assessment: Participation and demonstrated level of understanding of the concept

Assessment, Phase One, Level of Engagement, Fascination:

3. Image: Creating a Mental Picture

Objective: To practice the concept with another set of hands-on materials.

Activity: Have the students manipulate base-ten blocks from the metric multibase activity blocks set. Give the students several problems to work similar to those acted out in section one.

Assessment: Working several problems correctly using the base-ten blocks.

Note: Multibase Activities are available in base four, five, and ten from Creative Publications, Palo Alto, California. Metric Multibase Blocks are also available from Creative Publications.

Assessment, Phase Two, Seeing the Big Picture:

4. Inform: Receiving Facts & Knowledge

Objective: To teach the algorithm of borrowing/regrouping in subtraction.

Activity: Teacher lectures using overhead projector and chalkboard. Teacher facilitates understanding of the student's textbook explanation of the concept and algorithm.

Assessment: Subtracting three of four problems correctly on paper using the algorithm for borrowing/regrouping.

Assessment, Phase Three, Success with Acquiring Knowledge:

5. Practice: Developing Skills

Objective: To give additional practice in the concept and algorithm.

Activity: Teacher's sequenced math worksheets, textbook problems, workbooks, and activity cards using the metric base-ten blocks. Homework is incorporated at this point.

Assessment: Correct use of base blocks to complete activity cards and for each level of sequenced worksheets selected a score of 90% accuracy on one worksheet.

Assessment, Phase Four, Success with Acquiring Skills:

6. Extend: Extending Learning to the Outside World

Objective: To allow the students to add a practical element to reinforce the need for learning the concept and algorithm.

Activity: Have the children make up some story problems using subtraction with borrowing/regrouping.

Assessment: Quality of story problems.

7. Refine: Refining the Extension

Objective: Quality of story problems.

Activity: Have the children make worksheets with borrowing/regrouping problems including some story problems to try on each other.

Assessment: Quality of worksheets.

8. Perform: Creative Manifestation of Material Learned

Objective: To share what they have learned. To reinforce the concept and algorithm learned.

Activity: Each student will give his/her worksheet to another student to complete. S/he will help the other student when needed, correct the other student's work, and provide the other student with feedback as to whether the concept and algorithm were mastered.

Assessment: Participation and sharing. Students who may need more work at this stage are identified, as well as, those who have a gift for tutoring their peers.

Assessment, Phase Five,Performance, Creative Use of Material Learned: