Using Manipulatives in Geometry
Using manipulatives in the classroom is essential when introducing geometry to students. When I was in elementary school I always struggled with math. I am a hand- on learner. Seeing how things work out visually helps me grasp the concept. In chapter 10, they introduce the classification of quadrilaterals. There are many shapes that they students are expected to learn. I like the idea of having children find the shapes in the classroom or at home—help them see how the shapes mirror things in real life. Or perhaps the children could do an activity where they would make shapes out of licorice pieces. In this activity the children would be able to create all kinds of shapes on their own. The teacher could call out different shapes that they had learned earlier and the students would then construct them as a review. Another activity students could do is create pictures using geometric shapes. Students could cut out shapes using construction paper and create their own creatures. After they have completed their picture the children could write a few sentences about their animal. All of these activities make geometry “real.”
Sunday, August 22, 2010
Monday, August 16, 2010
MAR 3: Levi Johnson
Activity: Let’s Balance
Source: Utah Education Network
Url: http://www.uen.org/Lessonplan/preview.cgi?LPid=14418
1. What introductory information is necessary for children to have prior to starting this activity?
- Add one digit numbers
- Subtract one-digit numbers
- General number sense
2. What grade level/s is appropriate for this activity? Please use appropriate justification for your answers here.
This activity is identified in the lesson plan as second grade. However, it fits into the MN first grade standards. It is a beginning level activity for introducing the idea of algebra. Students learn how to balance an equation using simple manipulatives.
3. MN Standards Addressed
1.2.2.1 Represent real-world situations involving addition and subtraction basic facts, using objects and number sentences
1.2.2.3 Use number sense and models of addition and subtraction, such as objects and number lines, to identify the missing number in an equation such as:
4. How will you engage students with different learning styles?
I like this activity because it engages students with a variety of learning styles:
- For students who enjoy paper pencil work, it includes a worksheet
- For students who learn by doing and working kinesthetically—it involves manipulatives
- For students who learn by talking about what they are learning, it includes a small group activity.
5. How does this activity connect to the real world for students?
I like this activity because it uses real world situations to help students see what is really invisible—algebra. The students begin by learning the idea of “balancing” two sides by using their arms to balance items in both hands. This begins the idea of an equation and an equal sign. I like how students actually “see” what happens when the equation (or arms) or unbalanced. The activity helps them get the idea that solving an equation is filling in an unknown to make it balanced.
6. Why is this activity with its concepts important for student learning?
This activity lays the foundation for more complex problem solving. If students do not understand the idea of creating “balance” on both sides of the equal sign, they will have difficulty solving in the future.
The activity also introduces the idea of drawing a “T” to solve the problem. This too is a fundamental skill in helping students solve algebraic problems. I used the same process in high school!
7. What adaptations can be made to the activity?
This activity can be adapted by using more or less complex numbers. More advanced students might be able to think in two digit numbers. Less advanced students could work with smaller numbers. Using manipulatives helps students who are struggling see the answer appear as they solve the problems.
8. What are your comments on this activity? Would you use it in your classroom?
I think this is a fun activity to try in the classroom. I was surprised to see algebra standards in the first grade class. However, doing an activity like this seems entirely doable and would work to help lay the groundwork for more complex reasoning that comes later.
ACTIVITY
Summary:Students will explore both equal and non-equal number sentences.
Main Curriculum Tie: 2nd Grade - MathematicsStandard 2 Objective 2Model, represent, and interpret number relationships using mathematical symbols.
Materials:ExplorationFor each group:
· Balance
· Manipulatives (bears, cubes, Unifix® cubes, blocks, etc…)
Balance the Scale For each group:
· Deck of cards, with face cards removed, or number cards (0-10)
For each student:
· Balance the Scale worksheet
Number Balance
· Number balance
· Paper
· Pencil
To Equal or Not to Equal
· Spinner with numbers 1, 2, and 3
· To Equal or Not to Equal worksheet
· Pencil
Attachments
· balance_scale.pdf
· equal.pdf
Background For Teachers:Students need to understand that an equation is a relationship between numbers where both sides of the equation are equal. The mathematical situation is represented by the equal (=) sign. Students also need to understand what it means when a number sentence is not equal on both sides. When a number sentence is not equal on both sides, the not equal (≠) sign is used.
Intended Learning Outcomes:1. Demonstrate a positive learning attitude.5. Understand and use basic concepts and skills.6. Communicate clearly in oral, artistic, written, and nonverbal form.
Instructional Procedures:Invitation to Learn
Have a student stand with his/her arms out straight (look like a balance scale).
Add one book (novels or basals work best) at a time to each side and observe how the student’s arms change with each book that is added. Discuss what happened when we put a book on the balance/student’s arm? What would happen if we only put the books on one side?
Instructional Procedures
Exploration
1. Have students free explore with the balance and manipulatives.
2. Have a class discussion on what they observed using the balance. They need to build, discuss, and write equations while working with the balance and manipulatives.
3. Have students build various equations using the different manipulatives.
Example: 4 red bears + 3 blue bears = 4 blue bears +3 red bears
When using manipulatives, make sure they are all the same size and weight. (Don’t use the family bears.)
2 red dinosaurs + 2 yellow dinosaurs = 3 red dinosaurs + 1 yellow dinosaur
Balance the ScaleStudents play in groups of two to four.
1. Each player is dealt 6 cards. The rest of the cards are placed facedown in a pile.
2. Each player chooses any 4 cards from his/her hand to place on the Balance the Scale worksheet.
3. Students need to balance the scale by placing their cards in addition problems that create an equation (equal on both sides). (e.g., 2 + 5 = 4 + 3 or 1 + 3 = 2 + 2) If using face cards, an ace equals 1 and 0 is shown by leaving a square blank (e.g., 6 + 3 = 9 + __).
4. If the student can create a true equation, they earn 1 point. Each student takes a turn to complete round one. All cards from round one are placed in a discard pile. If the student can’t create a problem, they place their cards in the discard pile.
5. On every turn, each student is dealt 6 cards from the original pile. If you run out of cards, shuffle the discard pile and continue to play. The game continues until a student reaches the score of 10.
Number Balance
1. Place the balance where all students can see it.
2. Place a weight on one side of the scale. Give a student a weight to place on the other side that will balance the scale (e.g., 8 = 8). Write the equation on the board. Model other examples as needed.
3. Place one weight on the balance and ask a student to place a weight on the other side that will balance the scale without using the same number. Write the statement on the board (e.g., 6 ≠ 4). Review the not equal symbol and the number statement and ask whether the number statement is true (e.g., yes it is true, 6 does not equal 4). Ask students how we can balance the scale.
4. If students don’t come up with the idea to add another weight to make an equation, give a weight to another student and ask if s/he can now balance the scale (e.g., 6 = 4 + 2). Continue with multiple examples. It is possible to add multiple weights to both sides.
To Equal or Not to Equal
1. Have a class discussion on equations and number sentences using the not equal sign (e.g., 6 + 2 = 8 and 6 + 2 ≠ 10).
2. Write several examples until students understand the symbols and how to use them.
3. Students play with a partner. The first player spins the spinner and writes his/her number on the recording sheet on any of the four blank spaces. The student spins a total of four times, filling in a blank space each time.
4. The student needs to fill in the sign that makes his/her number sentence true (e.g., 3 + 1 ≠ 1 + 1 or 2 + 3 = 2 + 3).
5. The partner states, “I agree that 3 + 1 ≠ 1 + 1.” If the number sentence is an equation (e.g., 2 + 1 = 1 + 2), they earn one point. If the number sentence is not equal (e.g., 2 + 1 ≠ 3 + 1) they earn two points. If player one made an incorrect number sentence, no points are earned.
6. Play moves to the second player and continues until the To Equal or Not to Equal worksheet is complete or time runs out.
Extensions:
· Students can also play the games using subtraction.
· Students can use a balance to check their equations. This may be helpful for students who need a reinforcement strategy.
· Math journal
· Write and illustrate equations using the counters. Explain your work.
· Class book: Using stamps, stickers, or small die cut shapes, have students illustrate an equation.
· Using the number of the day, write equations where there are multiple addends on both sides (e.g., If the number of the day is 15, an equation could read 7 + 7 + 1 = 5 + 5 + 5).
Family Connections
· Using materials at home, create number sentences that are equal on both sides.
· Students teach the symbols = and ≠ to a family member.
Assessment Plan:
· Observe students while they are participating in any of the activities.
· Have students demonstrate that they can write, illustrate, and solve various problems using the symbols = and ≠.
· Discussion and journal entries: “What does it mean if something is not equal? What does the word ‘equation’ mean? Why do we need the = sign (or ≠ sign)?”
Bibliography:Research Basis
Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking Mathematically; Integrating Arithmetic & Algebra in Elementary School. Portsmouth, NH: Heinemann.
This book documents the widespread misunderstanding of the equal sign by students in grades 2-6. The text includes a series of true/false questions to help students begin to unpack their misunderstandings and help them develop the real meaning of this symbol. (Research was compiled at the Wisconsin Center for Educational Research.)
Source: Utah Education Network
Url: http://www.uen.org/Lessonplan/preview.cgi?LPid=14418
1. What introductory information is necessary for children to have prior to starting this activity?
- Add one digit numbers
- Subtract one-digit numbers
- General number sense
2. What grade level/s is appropriate for this activity? Please use appropriate justification for your answers here.
This activity is identified in the lesson plan as second grade. However, it fits into the MN first grade standards. It is a beginning level activity for introducing the idea of algebra. Students learn how to balance an equation using simple manipulatives.
3. MN Standards Addressed
1.2.2.1 Represent real-world situations involving addition and subtraction basic facts, using objects and number sentences
1.2.2.3 Use number sense and models of addition and subtraction, such as objects and number lines, to identify the missing number in an equation such as:
4. How will you engage students with different learning styles?
I like this activity because it engages students with a variety of learning styles:
- For students who enjoy paper pencil work, it includes a worksheet
- For students who learn by doing and working kinesthetically—it involves manipulatives
- For students who learn by talking about what they are learning, it includes a small group activity.
5. How does this activity connect to the real world for students?
I like this activity because it uses real world situations to help students see what is really invisible—algebra. The students begin by learning the idea of “balancing” two sides by using their arms to balance items in both hands. This begins the idea of an equation and an equal sign. I like how students actually “see” what happens when the equation (or arms) or unbalanced. The activity helps them get the idea that solving an equation is filling in an unknown to make it balanced.
6. Why is this activity with its concepts important for student learning?
This activity lays the foundation for more complex problem solving. If students do not understand the idea of creating “balance” on both sides of the equal sign, they will have difficulty solving in the future.
The activity also introduces the idea of drawing a “T” to solve the problem. This too is a fundamental skill in helping students solve algebraic problems. I used the same process in high school!
7. What adaptations can be made to the activity?
This activity can be adapted by using more or less complex numbers. More advanced students might be able to think in two digit numbers. Less advanced students could work with smaller numbers. Using manipulatives helps students who are struggling see the answer appear as they solve the problems.
8. What are your comments on this activity? Would you use it in your classroom?
I think this is a fun activity to try in the classroom. I was surprised to see algebra standards in the first grade class. However, doing an activity like this seems entirely doable and would work to help lay the groundwork for more complex reasoning that comes later.
ACTIVITY
Summary:Students will explore both equal and non-equal number sentences.
Main Curriculum Tie: 2nd Grade - MathematicsStandard 2 Objective 2Model, represent, and interpret number relationships using mathematical symbols.
Materials:ExplorationFor each group:
· Balance
· Manipulatives (bears, cubes, Unifix® cubes, blocks, etc…)
Balance the Scale For each group:
· Deck of cards, with face cards removed, or number cards (0-10)
For each student:
· Balance the Scale worksheet
Number Balance
· Number balance
· Paper
· Pencil
To Equal or Not to Equal
· Spinner with numbers 1, 2, and 3
· To Equal or Not to Equal worksheet
· Pencil
Attachments
· balance_scale.pdf
· equal.pdf
Background For Teachers:Students need to understand that an equation is a relationship between numbers where both sides of the equation are equal. The mathematical situation is represented by the equal (=) sign. Students also need to understand what it means when a number sentence is not equal on both sides. When a number sentence is not equal on both sides, the not equal (≠) sign is used.
Intended Learning Outcomes:1. Demonstrate a positive learning attitude.5. Understand and use basic concepts and skills.6. Communicate clearly in oral, artistic, written, and nonverbal form.
Instructional Procedures:Invitation to Learn
Have a student stand with his/her arms out straight (look like a balance scale).
Add one book (novels or basals work best) at a time to each side and observe how the student’s arms change with each book that is added. Discuss what happened when we put a book on the balance/student’s arm? What would happen if we only put the books on one side?
Instructional Procedures
Exploration
1. Have students free explore with the balance and manipulatives.
2. Have a class discussion on what they observed using the balance. They need to build, discuss, and write equations while working with the balance and manipulatives.
3. Have students build various equations using the different manipulatives.
Example: 4 red bears + 3 blue bears = 4 blue bears +3 red bears
When using manipulatives, make sure they are all the same size and weight. (Don’t use the family bears.)
2 red dinosaurs + 2 yellow dinosaurs = 3 red dinosaurs + 1 yellow dinosaur
Balance the ScaleStudents play in groups of two to four.
1. Each player is dealt 6 cards. The rest of the cards are placed facedown in a pile.
2. Each player chooses any 4 cards from his/her hand to place on the Balance the Scale worksheet.
3. Students need to balance the scale by placing their cards in addition problems that create an equation (equal on both sides). (e.g., 2 + 5 = 4 + 3 or 1 + 3 = 2 + 2) If using face cards, an ace equals 1 and 0 is shown by leaving a square blank (e.g., 6 + 3 = 9 + __).
4. If the student can create a true equation, they earn 1 point. Each student takes a turn to complete round one. All cards from round one are placed in a discard pile. If the student can’t create a problem, they place their cards in the discard pile.
5. On every turn, each student is dealt 6 cards from the original pile. If you run out of cards, shuffle the discard pile and continue to play. The game continues until a student reaches the score of 10.
Number Balance
1. Place the balance where all students can see it.
2. Place a weight on one side of the scale. Give a student a weight to place on the other side that will balance the scale (e.g., 8 = 8). Write the equation on the board. Model other examples as needed.
3. Place one weight on the balance and ask a student to place a weight on the other side that will balance the scale without using the same number. Write the statement on the board (e.g., 6 ≠ 4). Review the not equal symbol and the number statement and ask whether the number statement is true (e.g., yes it is true, 6 does not equal 4). Ask students how we can balance the scale.
4. If students don’t come up with the idea to add another weight to make an equation, give a weight to another student and ask if s/he can now balance the scale (e.g., 6 = 4 + 2). Continue with multiple examples. It is possible to add multiple weights to both sides.
To Equal or Not to Equal
1. Have a class discussion on equations and number sentences using the not equal sign (e.g., 6 + 2 = 8 and 6 + 2 ≠ 10).
2. Write several examples until students understand the symbols and how to use them.
3. Students play with a partner. The first player spins the spinner and writes his/her number on the recording sheet on any of the four blank spaces. The student spins a total of four times, filling in a blank space each time.
4. The student needs to fill in the sign that makes his/her number sentence true (e.g., 3 + 1 ≠ 1 + 1 or 2 + 3 = 2 + 3).
5. The partner states, “I agree that 3 + 1 ≠ 1 + 1.” If the number sentence is an equation (e.g., 2 + 1 = 1 + 2), they earn one point. If the number sentence is not equal (e.g., 2 + 1 ≠ 3 + 1) they earn two points. If player one made an incorrect number sentence, no points are earned.
6. Play moves to the second player and continues until the To Equal or Not to Equal worksheet is complete or time runs out.
Extensions:
· Students can also play the games using subtraction.
· Students can use a balance to check their equations. This may be helpful for students who need a reinforcement strategy.
· Math journal
· Write and illustrate equations using the counters. Explain your work.
· Class book: Using stamps, stickers, or small die cut shapes, have students illustrate an equation.
· Using the number of the day, write equations where there are multiple addends on both sides (e.g., If the number of the day is 15, an equation could read 7 + 7 + 1 = 5 + 5 + 5).
Family Connections
· Using materials at home, create number sentences that are equal on both sides.
· Students teach the symbols = and ≠ to a family member.
Assessment Plan:
· Observe students while they are participating in any of the activities.
· Have students demonstrate that they can write, illustrate, and solve various problems using the symbols = and ≠.
· Discussion and journal entries: “What does it mean if something is not equal? What does the word ‘equation’ mean? Why do we need the = sign (or ≠ sign)?”
Bibliography:Research Basis
Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking Mathematically; Integrating Arithmetic & Algebra in Elementary School. Portsmouth, NH: Heinemann.
This book documents the widespread misunderstanding of the equal sign by students in grades 2-6. The text includes a series of true/false questions to help students begin to unpack their misunderstandings and help them develop the real meaning of this symbol. (Research was compiled at the Wisconsin Center for Educational Research.)
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