MAT104 Blackboard Assignment: 6
Sample Student Response:
I feel that these lessons, directed at teaching fifth graders to add fractions with unlike denominators, are appropriate. Children’s introduction to fractions often occurs outside of school. They hear adults use fractions in various situations. For example:
· I’ll be back in half an hour.
· There is a quarter moon tonight.
· It’s a quarter past one.
· The recipe says to add two-thirds of a cup of water.
Therefore, these two lessons build on children’s previous experiences and help children clarify the ideas they have encountered in their own lives. Students can learn easier if they are using something that is familiar to them. A clock is an object that is used in every day life routines. Using the clock encourages students to incorporate their own real life experiences to their knowledge of math. This method makes it a more meaningful and fun learning experience. The students can also use mental computation skills of fractions.
The lessons meet NCTM standards and the methods help children become autonomous learners. Both lessons offer children an opportunity to work in large and/or small group settings. By doing this, children have the opportunity to communicate their ideas and thinking. When children are structured in small groups, they are able to receive immediate feedback about their ideas. Children learn to become active learners with the teacher guiding and facilitating the instructions.
My lessons do not offer the children rules to perform the algorithm of adding fractions with unlike denominators to help them understand the concepts. I provide concrete materials related to real-life situations. Many opportunities are offered for children to make sense of fractions, use fractional language, and learn to represent fractions with standard symbols. In my lessons children deal with fractions concretely and in the context of real life before they focus on symbolic representations. My lessons also meet parents concerns. The familiarity of the clock helps the parents to extend help to their child at home during homework time.
Day 1 Lesson Plan
Grade: 5
Topic: Understanding and adding clock fractions
Aim: Students will represent and add fractions with unlike denominators formed through the use of an analog clock.
Prior Knowledge:
1. Students have explored the relationship among fractions (½, ⅓, ¼, ⅙, ¹⁄₁₂, (¹⁄₆₀).)
2. They have used different combinations to make a whole.
3. They have had opportunities to create fractional parts on geoboards.
4. They have been able to order unit fractions with the same whole number.
5. They understand and are able to read analog clocks.
6. They are able to add fractions with like denominators.
Materials Needed:
1. Large clock face
2. Equivalents’ chart
3. Student Sheet I
4. Individual analog clocks
Motivation:
Children will look at a large non-operating clock. The minute hand is pointing to 12. The hour hand moves around to show the time.
Teaching Method: (In large group)
1. The teacher will move the ‘hour’ hand from 12 to 12. How many hours have been shown (12 hours)? The teacher will move the hour hand from 12 to 1 (shows 1:00). How many hours have moved?
2. Ask children what fraction of the way around the clock has gone (¹⁄₁₂)
Why is it ¹⁄₁₂? Allow students to explain.
3. Give another example. “When the hour hand moves from 12:00 to 3:00 what fraction of the way around the clock has it gone? (¼) or (³⁄₁₂)… Add ³⁄₁₂ to the Equivalents chart beside ¼.
4. Now using only the hour hand to indicate a fraction of the full clock rotation, what will the new fraction be, if we add ¹⁄₁₂ and ¼? Can this be done? What will be an easy way for us to see this? Is there an equivalent fraction for ¹⁄₁₂ or ¼? There is an equivalent fraction for ¼. It is ³⁄₁₂. How many hours are indicated by ¹⁄₁₂ (1 hour)? How many hours are indicated by ³⁄₁₂ (3hours)? Now when we add the two fractions, what is the sum? ¹⁄₁₂ + ³⁄₁₂ = ⁴⁄₁₂. How many hours is this (4 hours)? Can everyone see that? The answer is ⁴⁄₁₂ which when you look at the clock, you can say that it is also ⅓ because 4 hours are equal to ⅓ of the rotation around the clock face.
5. Have children now imagine that the hour hand is broken and the minute hand moves. The teacher will move the minute hand from 12 – 12. How many ‘minutes’ have been shown (60 minutes)? When the minute hand is moved from the 12 to the 3 (shows 12:15), how many minutes have gone by (15 minutes)? What fraction is that of an hour or 60 minutes? Allow students to explain.
Children may suggest ¼ or ¹⁵⁄₆₀. If no one suggests ¹⁵⁄₆₀, try to elicit this answer by telling them to think of this problem in terms of minutes. This is another equivalent fraction. How many minutes has the hand moved out of the number of minutes in an hour? How do we write this fraction? Add ¹⁵⁄₆₀ to the
Equivalents chart beside ¼.
6. Give the children this problem to work out. This time think about what happens if we have the minute hand start at the 12 and move one third of the way around the clock face. Where would the hand end up (on the 4, or 20 minutes)? Think about how many minutes out of 12 the hand has moved? What fraction represents ⅓ on the clock? (²⁰⁄₆₀). Now if we move the minute hand of the clock from the 4 to the 6 – how many more minutes have passed (10 minutes)? What fraction is 10 minutes on the clock’s face? It is ¹⁰⁄₆₀ or ⅙. Now if we want to add ⅙ and ⅓, how can we do it? Look at the Equivalents chart and then ask the students to use a similar procedure that was used with adding fractions represented by the hour hand that was just done. Hopefully, from the Equivalents chart the students can say that ⅙ equals 10 minutes or ¹⁰⁄₆₀, and ⅓ equals 20 minutes or ²⁰⁄₆₀. When added together the answer is ³⁰⁄₆₀. When you look at the clock, what fraction is ³⁰⁄₆₀? It is ½. Ask if anyone can draw a conclusion as to how it was possible to add fractions with unlike denominators? (We needed to find a common denominator first, and then we were able to add.)
Drills/Problems:
Hand out individual clocks to each child. Distribute a copy of Student Sheet I – Clock Fractions to each student. Explain that the arrows inside the clocks show the rotation of the hour hand. Have the students write fractions that tell how far the hand has moved. Under each clock they will write all the fraction names they know for that interval (equivalent fractions).
Circulate and observe students while they work. Notice what strategies they are using to find fractions of the full circle. As students find fractions, write them on the board. Students express these fractions in term of halves, thirds, fourths, sixths, and twelfths less than 1. Point out that all fractions that name the same amount are equivalent. They will create and write one addition problem using fractions with unlike denominators.
What can go wrong?
· Students do not have a complete understanding of the clock.
· Students lack the necessary social skills to work in groups
· Students do not have a complete understanding of fractions.
· They are not able to correctly identify equivalent fractions.
· They do not equate minutes or hours as being a fractional part of a total.
Homework:
Students will review Clock Fraction Student Sheet I. In class, fractions were formed using the ‘hour’ hand (twelfths). They will create two addition problems using the ‘minute’ hand on the clock (sixtieths)
What can go wrong?
· Parents do not immediately understand how the clocks are used to learn fractions.
· Students do not understand what equivalent fractions are.
· Students do not understand that unlike fractions must have a common denominator before they can be added together.
Day 2 Lesson Plan
Grade: 5
Topic: Adding Fractions on the Clock
Aim: Students will use the clock to add fractions with different denominators.
Prior Knowledge:
- Students have already practiced working with fractions on a clock during yesterday’s lesson.
- Students have an understanding of equivalent fractions.
Materials Needed:
- Overhead projector
- Large clock face
- Equivalents Charts
- Student Sheets I & II
- Individual clocks
- Chart Paper
Motivation:
Draw attention to the clock with the stationary minute hand. Tell the students that the clock face will be used to add fractions. Move the hand one third of the way around the clock (3:00). Then move it one sixth more and ask students where it will end up? What fraction has it moved altogether? Write the problem on a transparency on the overhead: ⅓ + ⅙ =
Teaching Method: (In large group)
1. Referring to this problem. Encourage students to talk together and find more than one way to think about the problem. Students may refer back to their completed worksheet 1. Students will share their approaches with the whole class. Allow students to demonstrate their thinking using the Large Clock transparency.
2. If students explain it in terms of hours, ask if anyone thought about it in minutes.
For example: In hours: ⅙ is 2 hours and ⅓ is 4 hours, and 2 and 4 make 6. Six is halfway around.
In minutes: ⅙ is 10 minutes and ⅓ is 20 minutes, so together they make 30 minutes which makes half an hour.
3. On chart paper, start a list of fraction addition problems. Write the equations student just solved.
⅓ + ⅙ = ½
Drill/Practice: (Work with a partner)
1. Write a few more examples on the board for students to do.
For example: ¼ + ½ = ¼ + ⅓
2. Circulate and ask students to explain how they came up with the answers.
Elicit different strategies and approaches.
3. Then ask children how could they add 1 ¾ + 2 ⁵/₁₂ =
Encourage students to work together and share with other students their strategies for finding the answer.
2. Tell children that when using the clock model to add fractions, there is no need to convert fractions to a common denominator. This happens automatically. For example: ¼ + ⅓ =
Reason: moving ¼ of the way around is 3 o’clock. Moving ⅓ is 4 hours more, or 7 o’clock. That’s equivalent to moving ⁷⁄₁₂ of the way around the clock
What can go wrong?
· Students do not have a complete understanding of the clock.
· Students lack the necessary social skills to work in groups
· Students do not have a complete understanding of fractions.
· They are not able to correctly identify equivalent fractions.
· They do not equate minutes or hours as being a fractional part of a total.
Homework:
Give out Clock Fractions Addition Problems Student Sheet 2. Children will make up and solve three addition problems. They will need to show their work on the clock face and record their strategies in solving each problem.
What can go wrong?
· Parents do not immediately understand how the clocks are used to learn fractions.
· Students do not understand what equivalent fractions are.
· Students do not understand that unlike fractions must have a common denominator before they can be added together.
Analysis of lessons 1 & 2
Aim: When using an object such as an analog clock, that is familiar to students and is constantly used by them in their daily routines, they incorporate their own real-life experiences to the learning of math. This makes it more meaningful and hopefully easier to understand. This satisfies NCTM recommendations.
Motivation: NCTM recommendations – Allows students to understand measurable attributes of objects and the units, systems and processes of fractions. The fact that the teacher is explaining and modeling in front of the class addresses the parents’ concern about children receiving teacher-directed instruction.
Teaching Method: NCTM: Addresses the use of visual models to relate equivalent fractions to one another. It is an appropriate tool that allows for students to do mental computation of fractions by moving the hands of the clock. Children are required to talk to one another and to share their approaches with the class.
Drills and Problems: Addresses NCTM recommendations. Students learn best when they discover, understand and integrate knowledge through independent exploration and team work.
Homework: By reviewing the Student Review Sheet, parents can become aware of the method used to learn fractions. They may then see that their child is more enthusiastically learning through a mix of information and styles providing them information. This meets the NCTM standards and the addresses the parents’ concerns about children getting drill and practice.
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