Sunnyside School District

Math TrainingModule 6

Conceptual Lessons

Good Afternoon

1. What challenges have you encountered when you ask students higher level questions?

2. How do you know what students understand about the math they are doing?

Identify the DOK LevelFor each question, identify the DOK Level as 1, 2, or 3:

1. Explain why area is measured in square units and volume is measured in cubic units?

2. Are these pieces bigger or smaller?

3. Tell me which answer is ridiculous and why.

4. You divide 402 by 3 and by 6. Without actually dividing, predict which quotient will be greatest. Explain your thinking.

5. Why does order matter when you subtract, but not when you add?

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

Between

Concepts Procedures

Teaching Conceptually

Two Methods:

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

2. Create conceptual activities/lessons in which students discover an understanding.

Conceptual Lesson

Step 1: Identify the understanding you want students to discover.

Step 2: Create a question that you will ask to assess whether students discover the understanding.

Step 3: Design a problem or a set of problems to lead to the understanding.

Step 4: Let students work together to solve and discuss.

Step 5: Ask the essential question to ensure all students get the understanding.

Step 1Identify the understandings:

• Sometimes they are written right in the standard.

• There are also understandings that you have to interpret from the standard.

• A few standards are skill based and therefore, have no understandings.

• Understandings should be:• Written as a statement• True----Always true• Conceptual• Important

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Understanding Examples

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

2. Counting a set in a different order, does not change the total.

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

4. 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.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.

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

Step 2

Once you identify an understanding that is important enough for a conceptual lesson/activity, then create a question to ask at the end of the activity.

Step 3 and 4

• Create a problem or set of problems.

• These can be teacher-led with time for student discussion or they can be partner activities.

• A good strategy is to connect simple known problems to more complex problems.

• Ask students to talk about “What do you notice?”

Step 5

Ask your high level question at the end of the lesson to ensure all students understand.

Require that students answer this independently and in writing.

Conceptual ActivityMultiplying Fractions

Complete the equations below:

2 x 4 =

½ x 4 =

½ x ¼ =

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

Consider how the size of the factors compare to the size of the product.

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)

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

Student Activity

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 the question at the end?

• 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

Create an Activity

Using the understandings that you created earlier, brainstorm an activity or lesson that you could do with students to help them discover the understanding.

4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.

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.

Closure

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

2. What was challenging when creating a conceptual lesson?

Standards Study

• K.G.1-6• 1.G.1-3• 2.G.1-3• 3.MD.5-8• 4.G.1-3• 5.G.1-4• 6.SP.1-5