Teacher: Would anyone like to share their solution with the class? (TR7)
(Brandon raises his hand)
Brandon: 6/9 is bigger than 10/12 because 3/9 is bigger than 2/12. (SR4)
Teacher: What do you mean that 3/9 is bigger than 2/12? (TR3C)
Brandon: 1/9 is a bigger piece than 1/12. So 3/9 is going to be bigger than 2/12. (SR6)
Teacher: Comment? Suzanne. (TR7)
Suzanne: I disagree. (SR4)
Teacher: Why do you disagree with Brandon? (TR3J)
Suzanne: Because you have to compare 10/12 to 6/9, not only 2/12 to 3/9. (SR4)
Teacher: Could you come up to the board and show us a picture of what you are saying.
(Suzanne comes up the board and draws the picture below)
Suzanne:
Suzanne: So you can see that 10/12 is bigger than 6/9.
Brandon: But
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Emily is saying that 6/9 needs 3 more big eggs to fill the carton. Whereas, 10/12 needs 2 little eggs to fill the carton. So, like Brandon was saying 3/9 is bigger pieces than 2/12. But since, 2/12 has smaller pieces then it requires less to fill the whole. So, 10/12 would be the bigger fraction because it’s closer to its whole since there are less, smaller pieces. (SR6)
Teacher: Can anyone figure out how you would use a number line to help support your thinking? (TR1)
Jimmy: Can I draw it on two number lines, or does it have to be on one number line? (SR3)
Teacher: What does everyone think? (TR7)
Rebecca: I think it is better to draw two number lines for each fraction and compare to 1 on each number line. (SR4)
Teacher: Why do you think it is better to have two number lines, Rebecca? (TR3J)
Rebecca: Because then you can put each fraction on its own number line and still compare to one. So it doesn’t matter how you split it up one the number line. It just might make it easier to look at each of the fractions side by side. (SR6)
(Rebecca goes up to the board)
Rebecca:
Rebecca: So now you can see that 10/12 is bigger than 6/9 when comparing both fractions to 1. Also, Brandon can see that 3/9 is bigger than 2/12, but that only makes that fraction farther away from 1. So 10/12 would still be bigger.
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So we must keep in mind what we are comparing the fractions to. Brandon, you did a great job realizing that 3/9 is bigger than 2/12, but we need to remember to relate those fractions back to the original problem.
Part 4: “Book ends” for your lesson
4a. Launch:
Materials: Small whiteboards and markers.
Give each student a whiteboard and a marker. Write 2 fractions on a larger classroom whiteboard. Ask the students to write which faction they think is larger on their smaller whiteboard. The students only have 3-5 seconds to produce an answer. The teacher will call on students to explain their thinking without telling the students which answer is correct. This way, the teacher is able to gauge where the students thinking is at before the lesson begins. Remind the students that there are many strategies when comparing two fractions in order to motivate their thinking.
4b. Closure:
The lesson will be ended with a journal entry. Each student has their own math journal where they can write their thoughts down at the end of the lesson.
Question; Does everyone understand that the size of the pieces is important to consider when the fractions that are being compared are under the
For most people who have ridden the roller coaster of primary education, subtracting twenty-three from seventy is a piece of cake. In fact, we probably work it out so quickly in our heads that we don’t consciously recognize the procedures that we are using to solve the problem. For us, subtraction seems like something that has been ingrained in our thinking since the first day of elementary school. Not surprisingly, numbers and subtraction and “carry over” were new to us at some point, just like everything else that we know today. For Gretchen, a first-grader trying to solve 70-23, subtraction doesn’t seem like a piece of cake as she verbalizes her confusion, getting different answers using different methods. After watching Gretchen pry for a final solution and coming up uncertain, we can gain a much deeper understanding for how the concept of subtraction first develops and the discrepancies that can arise as a child searches for what is correct way and what is not.
Joe’s diagnostic assessment started out strong completing the two digit numbers with confidence. Moving on to the three digit numbers cracks in Joe’s understanding started to appear. Joe showed a good understanding of reading, writing and ordering two and three digit numbers. The knowledge of some more, some less and the ability of partitioning two digit numbers were also clear. Joe acknowledged that 36 less 10 is 26 because, “if you take away one bundle you are left with two bundles and six single ones.” The partitioning of the number 36 was achieved by splitting it into 26 and 10. Up until the point of some more, some less in the three digit numbers section Joe was displaying a strong understanding of place
Fractions have been a around long enough for me to understand that I do not like them, but they play a significant part in simplifying, for some, division of goods or time. There is no one person who can be credited with the invention of fractions, but their use has been traced back as early as 1000 BC, in Egypt--using the formula to trade tangibles, currency, and build pyramids.
Developing Fractions Sense Workbook B is a collection of 45 lessons and is suitable for teaching fractions in the fourth grade. The workbook begins with the recognition and description of the unit fraction, including representations of the unit fraction using pattern blocks, an area model, and a number line. For each of the next 44 lessons, there is an instructional section followed by exercises. Included are review questions at the end of the exercises to help students to make connections between old and new concepts. At the end of each lesson is a summary statement of the idea taught in the lesson. The lessons are carefully sequenced such that only one concept is addressed per lesson and each concept builds on the previous one. Also, the
Announcements signal the end of time to work on the bell ringer, and after announcements Ms. Schreyer leads the class in checking their work. After morning work is completed, the students begin their science block, then the students had their technology special. During this time, Ms. Schreyer had a planning period. When the students return from technology, a few students leave for a pull out emotional support class, a student from second grade joins the class, and the rest of the students have math class. After math class ends, the students went to lunch as I completed my time in the classroom. On Thursdays, I arrive a few minutes into math class, typically as they finished checking homework from the night before. I observed the remainder of the math lesson until it is time for lunch and recess. During recess most weeks, Ms. Schreyer's classroom was the workroom for students who did not complete their homework or lost recess time for whatever reason. After recess the students have a bathroom break, then switch classes. Ms. Schreyer's homeroom students move on to writing, and a new group of students came to math class. This class has 18 students, and included the students who receive
Fraction Differences First Sequence To begin with I looked at the first sequence of fractions to discover the formula that explained it. As all the numerators were 1 I looked at the denominators. As these all increased by 1 every time, I figured that the formula was simply [IMAGE] as the denominators corresponded to the implied first line as shown in this table below: nth number 1 2 3 4 5 6 7 8 Denominators 1 2 3 4 5 6 7 8 I shall call this Formula 1 (F1) for easy reference. Second Sequence [IMAGE] [IMAGE] [IMAGE] [IMAGE] [IMAGE] Again I decided to discount the numerator as it was 1, and I decided to concentrate on the differences between the denominators rather than the ‘fractions’.
I posed that the class has guinea pigs as pets. We have two cages for them. Each student will put them into the cages. I observed what they did. Then, I asked how many they put in each cage. Which cage has more? Which cage has less?
the other, nor is it possible to determine if a > b by the same amount that c >
Ratios traditionally measure the most important factors such as liquidity, solvency and profitability, as well as other measures of solvency. Different studies have found various ratios to be the most efficient indicators of solvency. Studies of ratio analysis began in the 1930’s, with several studies of the concluding that firms with the potential to file bankruptcy all exhibited different ratios than those companies that were financially sound.
3. Give an example of how/when you use fractions (including addition, subtraction, multiplication, division, and or ordering of) in your day to day activities outside of math class.
Countless time teachers encounter students that struggle with mathematical concepts trough elementary grades. Often, the struggle stems from the inability to comprehend the mathematical concept of place value. “Understanding our place value system is an essential foundation for all computations with whole numbers” (Burns, 2010, p. 20). Students that recognize the composition of the numbers have more flexibility in mathematical computation. “Not only does the base-ten system allow us to express arbitrarily large numbers and arbitrarily small numbers, but it also enables us to quickly compare numbers and assess the ballpark size of a number” (Beckmann, 2014a, p. 1). Addressing student misconceptions should be part of every lesson. If a student perpetuates place value misconceptions they will not be able to fully recognize and explain other mathematical ideas. In this paper, I will analyze some misconceptions relating place value and suggest some strategies to help students understand the concept of place value.
Usually in the game, only integers are used to make a result of 24. However, in some difficult 24 game questions, fraction calculation is required. For example: 2, 5, 5, 10 can be calculated as (5 − 2 ÷ 10) × 5 which is 24 over 5 multiplied by 5.
The assessors are presented with three products, two of which are identical and the other one different. The assessors are asked to state which product they believe is the odd one out.
The prominence of numeracy is extremely evident in daily life and as teachers it is important to provide quality assistance to students with regards to the development of a child's numeracy skills. High-level numeracy ability does not exclusively signify an extensive view of complex mathematics, its meaning refers to using constructive mathematical ideas to “...make sense of the world.” (NSW Government, 2011). A high-level of numeracy is evident in our abilities to effectively draw upon mathematical ideas and critically evaluate it's use in real-life situations, such as finances, time management, building construction and food preparation, just to name a few (NSW Government, 2011). Effective teachings of numeracy in the 21st century has become a major topic of debate in recent years. The debate usually streams from parents desires for their child to succeed in school and not fall behind. Regardless of socio-economic background, parents want success for their children to prepare them for life in society and work (Groundwater-Smith, 2009). A student who only presents an extremely basic understanding of numeracy, such as small number counting and limited spatial and time awareness, is at risk of falling behind in the increasingly competitive and technologically focused job market of the 21st Century (Huetinck & Munshin, 2008). In the last decade, the Australian curriculum has witness an influx of new digital tools to assist mathematical teaching and learning. The common calculator, which is becoming increasing cheap and readily available, and its usage within the primary school curriculum is often put at the forefront of this debate (Groves, 1994). The argument against the usage of the calculator suggests that it makes students lazy ...
The ratio for length to width of rectangles is 1.61803398874989484820. The numeric value is called “phi”.