December 3, 2012 Presenter: Kathy Carpenter

kcarpenter@mathteachingtools.com

Goals for this Session

• Examine a progression of more of the content for Fractions.

• Discuss a variety of ways to teach standards.

• Discuss how to assess for student understanding.

• Housekeeping:

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Any questions you have can be submitted in the chat box during the webinar.

Shift From: To:

Focus

Laundry lists

Teaching a little bit of everything

every year

Concentrated areas that move in and

out of the curriculum and come alive

using the math’al practices

Coherence Disjointed topics (vertically and

horizontally)

Ideas that increase in sophistication

with each grade level, connections

within grade levels

Application Problem solving after content Content coming alive through

applications, purposefulness in student

learning

Conceptual Understanding

Jumping right into procedures Assessing for understanding at a

conceptual level

Procedural Skill

and Fluency

Drill drill drill Fluency that goes beyond rote

memorization and facilitates the use of

multiple strategies

Shifts Happen…

Today’s Third-grade Standard

• Number and Operations—Fractions: Develop an understanding of fractions as numbers.

3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

A. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

B. Recognize and generate simple equivalent fractions, e.g., 1/2=2/4, 4/6=2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.

• 3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

C. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3=3/1; recognize that 6/1=6; locate 4/4 and 1 at the same point of a number line diagram.

D. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Today’s Fourth-grade Standard

• Number and Operations—Fractions: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

3. Understand a fraction a/b with a>1 as a sum of fractions 1/b.

A. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

B. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model.

• 3. Understand a fraction a/b with a>1 as a sum of fractions 1/b.

C. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

D. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

Today’s Fifth-grade Standards

• Number and Operations—Fractions: Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

3. Interpret a fraction as division of the numerator by the denominator (a/b=a÷b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

•A. Interpret the product (a/b)×q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a×q÷b.

•B. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Third-grade Task: illustrativemathematics.org

Jon and Charlie plan to run together. They are arguing about how far to run.

Charlie says,

• I run 3/6 of a mile each day.

Jon says,

• I can only run 1/2 of a mile.

If Charlie runs 3/6 of a mile and Jon runs 1/2 of a mile, explain why it is silly for them to argue. Draw a picture or a number line to support your reasoning.

Third-grade task: illustrativemathematics.org

Bryce drew this picture:

Then he said, “This shows that ¼ is greater than ½.

• A. What was his mistake? Draw a picture that shows why ½ is greater than ¼.

• B. Which of these comparisons of ¼ with ½ are true? Explain.

¼ > ½

¼ < ½

¼ = ½

½ > ¼

½ < ¼

Fourth-grade task: illustrativemathematics.org

Use <, =, or > to compare the following sums:

• 1/2 + 1/4 ________ 1/3 + 1/5

• 1/3 + 1/2 ________ 1/3 + 1/4

Fifth-grade task: illustrativemathematics.org

A. Five brothers are going to take turns watching their family's new puppy. How much time will each brother spend watching the puppy in a single day if they all watch him for an equal length of time? Write your answer

Using only hours,

Using a whole number of hours and a whole number of minutes, and

Using only minutes.

B. Mrs. Hinojosa had 75 feet of ribbon. If each of the 18 students in her class gets an equal length of ribbon, how long will each piece be? Write your answer

Using only feet,

Using a whole number of feet and a whole number of inches, and

Using only inches.

Fifth-grade task: illustrativemathematics.org

• Label the points on a number line that correspond to 1/6, 2/6, 3/6, 4/6, and 5/6.

• Carefully cut out a strip of paper that has a length of 5/6.

Bring the ends of the strip together to fold the strip of paper in half. How long is half of the strip? Use your strip to mark this point on the number line.

What two numbers can you multiply to find the length of half the strip? Write an equation to show this.

• Unfold your paper strip so that you start with 5/6 again. Now fold the strip of paper in half and then in half again.

How long is half of half of the strip? Use your strip to mark this point on the number line.

What numbers can you multiply to find the length of half the strip? Write an equation to show this.

Based on today, what are 3 changes I can make to my class this school year?

Note: Work on “chunks” at a time to keep from getting overwhelmed and to measure the impact of the changes.

Next webinar: Early February (Mon), 3:30p