Sunnyside School District

Math Training6-5-14

Conceptual Lessons

Focus 1 & 2; Mathematical Shifts & Practices; June 2014

Good Morning

1. What are the most important things to look for in a performance task?

2. What comes to your mind when you think about teaching conceptual understanding? What would it look like in a classroom?

Analyze a Task

1. Each table is assigned Task A or Task B.

2. Work with your table to choose a solution path.

3. Meet up with someone from a table with the other task and discuss:

1. How are the two tasks different?

2. What was the discussion at your table focused on when solving?

3. Which of these is the better task? Why?

CCSS Math Shifts

1. Focus

2. Coherence

3. RigorProcedural Skill and FluencyConceptual UnderstandingApplication of Math Standards

Math Practice Standards

1. Make sense of problems and persevere is 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.

Math Practices 7 and 8

Read Math Practices and discuss:

1. How would these look in a classroom?

2. How is MP 8 different from MP 7?

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Initial Instruction

Balance BetweenConcepts Procedures

8

Concepts and Procedures

Doing something correctly, in itself, does not indicate understanding.

Conceptual Understanding

• Students can explain HOW the math works.

• Students can explain why THEY are using a specific strategy.

• Students can derive formulas and/or shortcuts on their own.

• Students can explain the meaning of math symbols and numbers.

Teaching Conceptually

Two Methods:

1. Create conceptual activities where students discover an understanding.

2. Increase the depth of teacher questioning in your classroom during lessons AND REQUIRE all students to answer them.

Rigor (DOK) of Teacher Questions

Level 1 Questions:

1. Simplify using the order of operations.

2. Solve: 5x + 3 = 168

Level 2 Questions:

3. Represent this word problem with an equation or drawing.

4. Write (in words) your steps to solve the problem.

Level 3 Questions:

5. How do you know that your answer is correct?

6. Why can’t you add fractions with unlike denominators?

From Teacher Questions

Consider this:

• Did one student answer the question?

• Did the teacher answer the question?

• Did ALL of the students answer the question?

Reflection

How can you help teachers plan for asking higher level questions in their math class?

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Understanding Examples

Examples:

1. The real-world situation determines how a remainder needs to be interpreted when solving a problem.

2. Ratios give the relative sizes of the quantities being compared, not necessarily the actual sizes.

3. Equivalent fractions are found by multiplying the fraction by a form of one because one doesn’t change the value.

Let’s Try One4.NBT.3

Use place value understanding to round multi-digit whole numbers to any place.

1. What are some understanding that students need when learning this standard?

Conceptual Activity

Once the understanding is identified, teachers can create an activity to help students discover it.

Must include an essential question at the end to ensure that all students mastered it.

Conceptual ActivityMultiplying Fractions

Complete the problems below:

2 x 4 =

½ x 4 =

½ x ¼ =

Draw a picture of each problem that represents how you get from the factors to the product.

Write the question (in words) that each equation is asking. The same question has to work for all the multiplication problems.

On Your Own

When multiplying fractions less than one, why are the products smaller than the factors?

Definitions and formulas

Perimeter is the size of something given by the distance around it.

The formula for perimeter of a rectangle is:

P = 2 (L + W)

1. Why does that formula represent perimeter of a rectangle?

2. How can you write the formula in another way?

Area Definition and Formula

Area is the measure of the space inside a region or how much it takes to cover the region.

Formula for area of a rectangle:A = L x W

3. Draw a 3 inch by 2 inch rectangle on a piece of paper and show that the area is 6 inches squared.

Student Engagement

You are creating a garden and you have 30 ft of fence.

1. Create 4 rectangular drawings of your garden where each drawing has different lengths and widths (draw to scale).

2. Calculate the area for each of your gardens.

3. What pattern do you notice about the areas?

Answer the following on a note card:

• What is the relationship between the shape of a rectangle and it’s area?

• What is the relationship between area and perimeter of a rectangle?

De-brief

• What did you notice about the lesson?

• How would it have been different if you didn’t have to answer those questions on the note card?

• My concept was: The maximum area for a given perimeter of a rectangle is when the shape is closest to a square. • Did your answer get close to this?

Area/Perimeter Activity

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All of the gardens to the right have the same perimeter of 30 ft.

What do you notice about the areas?

Gardens:

14 x 1 Area = 14 sqft

8 x 7 Area = 56 sqft

9 x 6 Area = 54

sqft

12 x 3

Area = 36 sqft

Closure

1. How did it feel to be a student during a conceptual lesson?

2. What was challenging when creating a conceptual lesson?

Breaking apart a Standard

Read the standard and determine:

1. The required procedures/skills

2. The understandings

3. Application Expectations

1.OA.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

Example

4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

Skills: Compare decimals; use symbols correctly; Able to draw visual model of a decimal.

Concepts: The size of the whole determines the comparison; Explain comparison based on size of each piece; Explain every comparison.

Application: None

Details: Decimals only to hundredths, only 2 decimals at once.

Standards Study

Read the standards assigned to your table to determine:

• Procedures and skills required of students

• Understandings (explanations) required of students

• Application (word problems) required of students

• Details

Closure