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Norms

- Listen as an Ally
- Value Differences
- Maintain Professionalism
- Participate Actively

4/3/2014 • page 2

Our Goals for this afternoon

- Recognize what makes a good task.
- Recognize how Standards for Practice mandate better ways of managing instruction.
- Importance of the relationship between multiplication and division.

Think of a Number

Many people have a number that they think is interesting. Choose a whole number between 1 and 25 that you think is special.

- Record your number.
- Explain why you chose that number.
- List three or four mathematical facts about your number.
- List three or four connections you can make between your number and your world.

There is no other decision that teachers make that has a greater impact on students’ opportunity to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages the students in studying mathematics.

Lappan and Briars (1995, pg. 138)

“What task can I give that will build student understanding?”

rather than

“How can I explain clearly so they will understand?”

Grayson Wheatley, NCCTM, 2002

Multiplication and Rectangles

Make as many different rectangles as you can using 12 square-inch color tiles.

What did you notice?

- Were the rectangles the same?
- Were the rectangles different?
- How would you describe your rectangle?
- Does that description fit someone else's rectangle?

Multiplication and Rectangles

Find all the rectangles you can make with 18 tiles.

Record your rectangles on grid paper.

What do you notice?

Which numbers have rectangles with 3 rows? List them from smallest to largest.

Which numbers have rectangles with 2 rows?

List them from smallest to largest.

Which numbers on the chart are multiples of 4? (have rectangles with 4 rows)

What do you notice?

Which numbers on the chart are multiples of 5?

How many different rectangles can you make with 5 tiles?

How many with 7 tiles?

List the prime numbers between 1 and 25

Are all odd numbers prime? Explain

Let’s look at the number nine!

What do you notice?

What other numbers have rectangles that are squares?

What is the next largest square after 25?

If my rectangle has a total of 18 squares tiles and 3 rows of tiles…

How many tiles are in each row?

Write a number sentence for this rectangle.

What standards in third and fourth would this task address?

How do these standards build on what fifth grade does?

Criteria Area in Third Grade

Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division.

Operations and Algebraic Thinking 3.OA

Represent and solve problems involving multiplication and division.

3.OA.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.

3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1

3.OA.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = ÷ 3, 6 × 6 = ?.

Operations and Algebraic Thinking 3.OA

Understand properties of multiplication and the relationship between multiplication and division.

3.OA.5 Apply properties of operations as strategies to multiply and divide.2Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by

3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

3.OA.6 Understand division as an unknown-factor problem.

For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

2Students need not use formal terms for these properties.

Operations and Algebraic Thinking 3.OA

Multiply and divide within 100.

3.OA.7 Fluently multiply and divide within 100, using strategies such as the

relationship between multiplication and division (e.g., knowing that 8 ×

5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end

of Grade 3, know from memory all products of two one-digit numbers.

Things to Think About…

What does Computational Fluency mean?

Developing fluency requires a balance and connection between conceptual understanding and computational proficiency. On the one hand, computational methods that are over practiced without understanding are often forgotten or remembered incorrectly…On the other hand; understanding without fluency can inhibit the problem-solving process. (PSSM, Page 35)

Things to Think About…

How do students demonstrate Computational Fluency?

Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. The computational methods that a student uses should be based on mathematical ideas that the student understands well, including the structure of the base-ten number system properties of multiplication and division and the number relationships. (PSSM, page 152)

Things to Think About…

Is there such a thing as effective drill?

There is little doubt that strategy development and general number sense are the best contributors to fact mastery. Drill in the absence of these factors has repeatedly been demonstrated as ineffective. However, the positive value of drill should not ne completely ignored. Drill of nearly any mental activity strengths memory and retrieval capabilities.

(Van de Walle)

Things to Think about…

What about timed test?

Teachers who use timed test believe that the test help children learn basic facts. This makes no instructional sense. Children who perform well under time pressure display their skills. Children who have difficulty with skills, or who work more slowly, run the risk of reinforcing wrong learning under pressure. In addition, children can become fearful and negative toward their math learning

(Burns 2000, p.157)

Operations and Algebraic Thinking 4.OA

Gain familiarity with factors and multiples.

4.OA.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.

“What task can I give that will build student understanding?”

rather than

“How can I explain clearly so they will understand?”

Grayson Wheatley, NCCTM, 2002

Math Notebook

- What skill did you review and practice?
- What strategies did you use while playing the game?
- If you were to play the game a second time, what different strategies would you use to be more successful?
- How could you tweak or modify the game to make it more challenging?

Standards for Mathematical Practices

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

Mathematical practices describe the habits of mind of mathematically proficient students…

- Who is doing the talking?
- Who’s doing the thinking?
- Who is doing the math?

Let’s revisit your poster

Now that you have worked through the various tasks, what additional numbers, pictures and words could you add to your poster to further illustrate the Mathematical Practice?

Think of a Number

Many people have a number that they think is interesting. Choose a whole number between 1 and 25 that you think is special.

- Record your number.
- Explain why you chose that number.
- List three or four mathematical facts about your number.
- List three or four connections you can make between your number and your world.

4/3/2014 • page 54

DPI Mathematics Site

http://math.ncwiseowl.org

Mathematics Wikki

http://maccss.ncdpi.wikispaces.net/Summer+Institute

Plus/Delta

- Please include on the back of the plus/Delta handout topics that you would like to see addressed or discussed during the webinars.
- November 17th
- January 10th
- February 9th
- March 8th

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