Grade 3: 3.MD.C.5-7, Geometric measurement: understand concepts of area and relate area to multiplication and addition

Lesson Plan: 3.MD.C.7-7a, Using Tiling to Find Area (This lesson should be adapted, including instructional time, to meet the needs of your students.)

Background Information
Content/Grade Level / Mathematics/Grade 3
Unit/Cluster: / Geometric Measurement: 3.MD.C: Understand concepts of area and relate area to multiplication and addition
Essential Questions/Enduring Understandings Addressed in the Lesson /
  • Why do area formulas work?
  • How do we find the area of shapes for which we do not know or cannot recall a formula?
  • How can we count the squares more efficiently?
  • How could we get a more precise approximation of the area?
  • Area is a measure of the space inside a region or how much it takes to cover a region.

Standards Addressed in This Lesson / 3.MD.C.7: Relate area to the operations of multiplication and addition.
3.MD.C.7a: Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
It is critical that the Standards for Mathematical Practice are incorporated in ALL lesson activities throughout the unit as appropriate. It is not the expectation that all eight Mathematical Practices will be evident in every lesson. The Standards for Mathematical Practice make an excellent framework on which to plan your instruction. Look for the infusion of the Mathematical Practices throughout this unit.
Lesson Topic / Finding the area of a rectangle by tiling it and providing an opportunity for students to develop their own formula (rule) for finding the area of a rectangle.
Relevance/Connections /
  • Perimeter, multiplication, fluency maintenance of addition and subtraction facts
  • Thinking about a real problem in the classroom
  • 3.OA.A: Represent and solve problems involving multiplication and division.
  • 3.OA.C: Multiply and divide within 100.

Student Outcomes /
  • Students will use tiling to determine the area of a specific rectangle.
  • Students will generate a rule (formula) for finding the area of a rectangle.
  • Students will discuss the importance of a standard unit of measure.

Prior Knowledge Needed to Support This Learning / Students already have a solid background that area is the interior measurement of a plane figure and by counting the inside “unit squares” they will be able to know the area of a given figure (3.MD.C.5a and 3.MD.C.5b).
Method for determining student readiness for the lesson / Use Warm Up as a pre-assessment (see below).
Learning Experience
Component / Details / Which Standards for Mathematical Practice(s) does this address? How is the Practice used to help students develop proficiency?
Warm Up / Materials Needed:
  • 12 square tiles for each student
Using 12 square tiles, students will create an array using their 12 tiles. Discuss the different configurations that they could make. / SMP 4: Model with mathematics. – Students use the 12 to make various arrays.
SMP 2: Reason abstractly and quantitatively – Students discuss the different configurations that they made using the 12 tiles.
Motivation / Pose a real-world mathematical problem for students to solve.
I would like to create a reading corner in our classroom with 1’ x 1’ carpet squares. I don’t want to spend more money than I have to and I don’t want to come up short or have too many extra. How can I accurately find out how many carpet tiles I need?
Discuss how we can begin to solve this problem. What information do we still need?
Activity 1
UDL Components
  • Multiple Means of Representation
  • Multiple Means for Action and Expression
  • Multiple Means for Engagement
Key Questions
Formative Assessment
Summary / Materials Needed:
  • 5 x 7 inch index card (one per student)
  • Post it notes of different sizes
  • Base ten unit cubes
  • Hundreds chart
  • Cheese-it crackers
Note: avoid using one-inch color tiles at this time.
  • Resource Sheet 1: Exit Slip: Similarities & Differences
Rather than having us all work in the same corner, I will give you a piece of paper that will represent our reading corner.
Distribute a 5 x 7 index card to each student. This will represent the reading corner on a smaller scale. Each student will be given different “unit squares” [sticky notes, cheddar cheese crackers, base-10 unit cubes (ones), 100s chart (students can place the index card on top of the chart and trace around its perimeter), or any other square manipulative]. Avoid using tiles at this time. Your goal is that students will see the need for a standard unit of measurement. Student should cover the index card with their manipulatives.
Discuss findings. Encourage discussion and debate.
  • Why did some students end up with a different answer?
  • What do we notice about the size of the unit square and the area found? (The larger the unit, the smaller the area and the smaller the unit, the larger the area.)
  • Are there situations where you would want to use a smaller square unit? (Finding the area of a small object – the area of a footprint) When would it be appropriate to use a large square unit? (Finding the area of a large object – the area of a football field.)
Students should realize the importance of using a common unit of measure. Since the teacher will be purchasing carpet tiles that are 1’ x 1’, we should agree on one unit of measure for our index cards.
Formative Assessment: Distribute Resource Sheet 1: Exit Slip: Similarities & Differences and allow time for students to record their thinking.
What are some of the similarities and differences that we saw when we covered our index cards using different manipulatives? Why do you think these differences occurred? / SMP 1: make sense of problems and persevere in solving them – Students use tiling to explore the need for a standard unit of measure.
SMP 4: Model with mathematics – Students use square units to determine the area of the index card.
SMP 3: Construct viable arguments and critique the reasoning of others – Students compare the area measurements of each other and discuss why they are different.
Activity 2
UDL Components
  • Multiple Means of Representation
  • Multiple Means for Action and Expression
  • Multiple Means for Engagement
Key Questions
Formative Assessment
Summary / Materials Needed:
  • 12 one-inch tiles per student
  • 5 x 7 inch index card (one per student)
  • Resource Sheet 2: Exit Ticket: Rules for Finding Area
Based on the discussion from Activity 1, students will know the importance of using a standard unit of measure.
Yesterday we discussed how we were going to tile one section of our classroom with carpet tiles. I was able to purchase 12 carpet tiles since they come in a pack of 12. I didn’t buy any more packages, because I don’t want too many extra or not enough.
Distribute 12 one-inch square tiles to each student and the index card from the previous lesson (this represents the 12 carpet squares already purchased). Have students explore ways of finding the area using the tiles. Students will quickly realize they don’t have enough tiles to cover the whole area. Pose questions to elicit discussion.
  • What are some solutions to not having enough tiles? It wouldn’t make sense to buy more tiles right now, so what can we do?
The discussion should lead the students to figuring out that the area of a rectangle can be found in a variety of ways. Have the students come together to discuss their solutions and to find commonalities and differences between different strategies.
Here are a few potential strategies you might see in your classroom. Remember, do not explicitly tell the students a formula for finding area of rectangles!
  • Students can use the additive property to figure out the remainder of space not covered.

“I knew one row has 7 tiles and there can be 5 rows so
7 + 7 + 7 +7 + 7= 35.”
  • Students can use the distributive property to figure out the remainder of the space not covered.

“I know 10 makes 2 columns so 6 columns gave me 30. There was one more column left over, so that would be 5 more for a total of 35.”
  • Students can find the lengths of the sides to determine the area.

“I knew one side measures 5 and the other measures 7, so 5 x 7 = 35.” / SMP 1: Make sense of problems and persevere in solving them – Students explore ways to find the area when they don’t have enough tiles to cover the index card.
SMP 3: Construct viable arguments and critique the reasoning of others – Students explain strategies they used to determine the area of the index card.
SMP 8: Look for and express regularity in repeated reasoning – Students used their knowledge of repeated addition or the distributive property to make sense of the problem and find the solution.
Closure / Go back to the classroom problem regarding the classroom reading corner.
If our reading corner is 5’ x 7’ and each tile is 1’ x 1’, how many carpet tiles do I need to purchase? If the tiles come in packs of 12, how many packages do I need to get?
Formative Assessment: Distribute Resource Sheet 2:Exit Slip: Rules for Finding Area
What would be a good rule for finding the area of a rectangle if you do not have enough square units to cover the entire area? Explain and provide an example. /
  • Construct viable arguments and critique the reasoning of others.
  • Attend to precision.

Supporting Information
Interventions/Enrichments
  • Special Education/Struggling Learners
  • ELL
  • Gifted and Talented
/ -Use previous exit slip to sort students into groups. Possible activities for the following groups:
  • Struggling students- Give more tiles, then slowly remove them so they can answer the question.
  • Enrichments- Using Resource Sheets 3A&B “Uh-Oh” Cards, give students a problem with the wrong solution. Ask them to explain what went wrong and how to fix it. For example, instead of the response of 35, they say 12. What did they do wrong? Example 2: Instead of the response of 35, they said 20. What could they have done wrong? Example 3: Student said they got an area of 24. Provide advice to a student that said 24. What did they do wrong?
If you did not have square tiles and only had rectangular tiles that were 1 x 2, how could you use them instead?
Materials / -Square tiles
-5 x 7 index cards per student (one per student)
-Post it notes of different sizes
-Base ten unit cubes
-Hundreds chart
-Cheese-it crackers
-12 one-inch tiles per student
-5 x 7 inch index card (one per student)
-Resource Sheet 1: Exit Slip: Similarities & Differences
-Resource Sheet 2: Exit Ticket: Rules for Finding Area
-Resource Sheets 3A&B: “Uh-Oh” Cards - Extension questions
Technology / Elmo or other projector, if available for student use when sharing solutions and modeling arrays
Resources
(must be available to all stakeholders) / See Resource List in Unit Plan

Resource Sheet 1 Exit Slip: Similarities & Differences

Name ______Date ______

What are some of the similarities and differences that we saw wen we covered our index cards using different manipulatives?

Why do you think these differences occurred?

Resource Sheet 2 Exit Ticket: Rules for Finding Area

Name______Date______

What would be a good rule for finding the area of a rectangle if you do not have enough square units to cover the entire area? Explain and provide an example.

Resource Sheet 3A “Uh-Oh” Cards

Resource Sheet 3B “Uh-Oh” Cards

July 17, 2013 Page 1 of 11