Welcome to Unit 5 seminar!

Welcome to Unit 5 seminar! Here is the agenda for this evening… 1. Make announcements and ask/answer any questions anyone has thus far in the course 2. Review the project requirements for Unit 6 3. Discuss the Unit 5 readings. Announcements: Your unit 5 projects are due Tuesday I will return your grades to you by Sunday or earlier! A class email goes out when I have graded every assignment for each unit. Are there any other questions or comments before we begin?

Project Unit 5: • Imagine you are a paraprofessional assigned to a fourth-grade classroom. Today you are working with Bonnie and Emanuel, helping them to master the concept of fractions. The cooperating teacher, Mrs. Villacengo, has asked you to share some examples of fractions in each of the three models (see your readings and pay close attention to tonight’s seminar): • Region or Area Models • Length or Measurement Models • Set Models • Explain one example for EACH of these three models that you could use to illustrate the fractions. Be sure to specifically identify which fraction you are illustrating (3/4, 5/10, 1/8, etc.). Consider how you might use visual aides (pie pieces, geoboards, paper-folding, rulers/numberlines, etc.) for visual learners or manipulatives for a “hands-on” approach for tactile learners. What are some other ways you modify your approach based on the needs/developmental level of your students?

Unit 5 Project continued • Now, Mrs. Villacengo is asking you to work with two new students, Mary Kate and Ashley, to help them practice computation with fractions. She gives you the following problem to go over with them: • 3 3/4 divided by 2 • You see that she has taught the rest of the class the common algorithm of “invert-and-multiply”, which looks like this: (Our field trip tonight will help you learn how to invert and multiply).3 3/4 divided by 2 = 15/4 x 1/2However, having had much experience with the problem-solving approach and knowing what you know about developmentally appropriate practices, you think the girls will have a better understanding if you can make this problem more meaningful. You could put it into a context, such as a word problem, or come up with a way for them to use it in their own experiences.

More for Unit 5 Project • Discuss how you would go about working on this problem with the girls. Consider what you read about the importance of using the following: • Informal Exploration • Estimation • Hands-on tasks to put the work in a context • Opportunities for students to discuss their progress

A possible outline might look like this… • Start with an opening paragraph in which you introduce the topics you will address in your paper. (1 paragraph) • Share your examples using three models and discuss how you can use the three models to teach fractional concepts (3-4 paragraphs) • Share how you will help 2 new students using a word problem to solve the problem using the algorithm invert and multiply • Explain how you would teach the 2 new students use one of these approaches (Informal Exploration, Estimation, Hands-on tasks to put the work in a context, or Opportunities for students to discuss their progress) (1 paragraphs) • End with a closing paragraph, in which you summarize the main points of your paper (1 paragraph). • Your project should be approximately three (3) typed pages (not including the title page and the reference page).

How can you use Word effectively? 1. Click on Insert 2. Click on Chart Pick the chart that works for you!

Fractions • Research has found that American students tend to be very weak in fractions. • “The rules and ideas around fraction computation are meaningless without a better understanding of fractions than that which students have been developing up to this time” (Van De Walle, 2007). • Textbooks tend to take a traditional approach to teaching fractions. • Many teachers may also have a poor understanding of fractions since they experienced the same type of education in fractions. • Why, in your opinion, are fractions so confusing to students?

Main Ideas about Fractions • There are different models—fractions of a whole, fractions of a set, etc., A whole means different things depending on the context of the problem—it could be a pie, an apple, a basketball court, a set of cars, a set of shapes, etc. , • They are parts of a whole number—if students don’t fully understand whole numbers, they are probably going to struggle to understand fractions. • Multiplying makes them smaller, dividing makes them bigger. A smaller denominator means a larger piece and a larger denominator means a smaller piece, etc. • One whole is not a standard whole—one whole can be a mile, a group of jelly beans, a medium pizza, a large pizza, etc.

Region/Area Models: fractions that are based on an area or region. • Examples: circular pie pieces, rectangular regions, geoboards, drawings on graph paper or dot paper, pattern blocks, paper folding. • What are some real world examples of region/area models? A = L x W for rectangles A = (1/2) B x H for triangles. Calculate building materials

Length/Measurement Models: lengths are compared instead of area. • Examples: fractions strips or cuisenaire rods, line segment drawings, folded paper strips • What are some real world examples of length/measurement models?

Set Models: a set of objects in which subsets make up the fractional parts • Examples: blocks, counters, different colored beans, etc • What are some real world examples of set models?

Diagrams and Fractions • How do diagrams of fractions support students’ understanding of fractions? • Diagrams also help students focus again, not on how to solve problems, but being able to solve in a meaningful, effective way for the individual.

Denominator and Numerator • The denominator names the kind or denomination of a fractional part. It is also a divisor of the whole.” In other words, the denominator tells how many parts make up the whole. • “The numerator is a counter or enumerator of the fractional parts named by the denominator.” In other words, it tells the number of pieces that are selected. Let’s take a look at some pictures…

Shapes, denominator, and numerator • How many pieces is this shape divided into? • That is the denominator. • How many pieces are shaded? • That is the numerator.

Examples • Let’s do some more examples with written fractions… • What is the numerator in this fraction? • What is the denominator in this fraction?

Symbolism and Formulas • Before introducing this type of symbolism (written fractions), make sure that children have multiple experiences with hands-on or visual representations of fractions. • In teaching fractions, why should you show models before numbers and symbols? • Why is it "dangerous" to rush to rules or formulas for computation with fractions?

Overdependence • The text specifically covers formulas you may be accustomed to, but warns not to rely on them because an overdependence on formulas may interfere with conceptual understanding. So, a student may know how to add and subtract fractions with a formula, but does not know how to explain the answer they got or how to check it. • The book covers strategies for teaching the formulas, but the focus, as with everything else, is an understanding of WHY these formulas work

Multiplication of fractions Let’s take this example: • ½ x ¼ = 1/8. Think of this number sentence as what is ½ of 1/4 . That is what multiplication of fractions is. Here we go... • Look at the shape. It is divided into fourths. So, I have the ¼ part. • Now I need to take ½ of one of the fourths. • If I cut this fourth in half, it would look like this... • I will cut the rest of the shape so we can see what fraction we are now working with. • If you count the parts, you will see that we have 8 pieces in the shape. • That means half of one of the fourth pieces equals 1/8. Therefore, ½ x ¼ = 1/8

Division of Fractions • Let’s take a new example... • Let’s take the problem 1/4 divided by 2. • The algorithm would have us invert the 2 to make ½ and multiply it by 1/4. But the algorithm does not explain why this works. • ¼ divided by 2 = • ¼ x ½ = 1/8 • Here is an example. • We will start with ¼ of a pizza. The problem says to divide one fourth in half. So, I will take this piece and cut it in half. • I have half of a fourth, which is 1/8. (You can see that if I cut the rest of the fourths, it would make eighths.) So, ½ of ¼ is 1/8. • 3 3/4 divided by 2 = 15/4 x 1/2

Fractional Anchors • In teaching whole numbers, we use anchors of 5 and 10. We also use anchors in teaching fractions as well. • For fractions, the important anchors are 0, 1/2 and 1 whole. To know to which of these benchmarks a fraction is closest or if it is more or less than 1 or 1/2 are very valuable ideas. Number Line 0…1/2…1…1 ½…2…

Number Lines

Understanding Models and Written Fractions • We want children to develop an understanding of these anchors, both as models and as written fractions. • For example, if we show children a shaded part of a fraction, we want them to be able to identify if it is closer to 0, ½, or 1 whole. • The same is true for written fractions. For example, it is important for children to be able to determine that 9/10 is closest to one whole.

Equivalent Fractions • Equivalent fractions are ones that are equal, but may have different denominators.

Activity… • To develop a conceptual understanding of equivalent fractions; the same quantity can • have different fraction names. • Encourage children to look for patterns in equivalent fractions. Let’s take an example…

6/9 and 2/3 “Two students looked at this picture. Each saw a different fraction. Kyle saw 6/9, but Terri said she saw 2/3. How can they see the same drawing and yet each see different fractions? Which one is right? Why?”

Improving Fraction Understanding • To improve fraction understanding, the author suggested the following strategies… • present tasks in context • connect to whole number • use estimation and other informal methods • use models when exploring operations • How do story problems aid in students’ understanding of fractions?

Connections • Why is connecting to the whole number and estimation so helpful in understanding fractions? • What types of activities develop estimation skills?

How do we invert and multiply? • Invert and Multiply http://www.mathplayground.com/howto_divide_fractions.html What is the answer? 2/3 divided by 3/5= If I was your student how would you explain what to do?

Virtual Field Trip • Fraction Pieces • http://nlvm.usu.edu/en/nav/frames_asid_274_g_2_t_1.html?open=activities&from=category_g_2_t_1.html

Something to think about… • Let’s move onto the Unit 5 reading… • Why is it helpful for students to focus on relating fractions back to whole numbers? • How can you go from concrete to abstract in teaching fractions? How should we be preparing students for this early on? • It is important to stress the understanding that as you have "more parts," those parts get smaller. How could we promote this understanding? • Do a lot of work with fractions before get into computation (include estimating) • An understanding of basic concepts of fractions also leads to an understanding of ratios, percents and proportions.

Way to go! • Excellent job in seminar! You contributed actively in class and added insights on teaching fractions. Thank you! More Resources: http://khe2.acrobat.com/p93203473/

Welcome to Unit 5 seminar!