In this unit, students will:

● Understand concepts of area and relate area to multiplication and addition.

● Find the area of a rectangle with whole- number side lengths by tiling it.

● Multiply side lengths to find areas of rectangles with whole-number side lengths in context of solving real world and mathematical problems.

● Construct and analyze area models with the same product.

● Describe and extend numeric patterns.

● Determine addition and multiplication patterns.

● Understand the commutative property’s relationship to area.

● Create arrays and area models to find different ways to decompose a product.

● Use arrays and area models to develop understanding of the distributive property.

● Solve problems involving one and two steps and represent these problems using equations with letters such as “n” or “x” representing the unknown quantity.

● Create and interpret pictographs and bar graphs.

The understanding of and ability to use multiplication and division is the basis for all further mathematics work and its importance cannot be overemphasized. As students move through upper elementary grades and middle school, the foundation laid here will empower them to work with fractions, decimals, and percents.

Area is a measure of the space inside a region or how much it takes to cover a region. As with other attributes, students must first understand the attribute of area before measuring.

The concept of multiplication can be related to the area of rectangles using arrays. Students need to discover that the length of one dimension of a rectangle tells how many squares are in each row of an array and the length of the other dimension of the rectangle tells how many squares are in each column.

Using this model, students should be able to create arrays to solve real-life problems involving multiplication and apply this concept with addition, subtraction, and division to solve equations involving two steps or more to find the solution.

STANDARDS FOR MATHEMATICAL PRACTICE (SMP)

This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.

Students are expected to:

1. Make sense of problems and persevere in solving them. Students make sense of problems involving area.

2. Reason abstractly and quantitatively. Students demonstrate abstract reasoning by connecting area with multiplication and arrays.

3. Construct viable arguments and critique the reasoning of others. Students construct and critique arguments regarding area by creating or drawing arrays or area models to prove answers.

4. Model with mathematics. Students use arrays or area models to find area.

5. Use appropriate tools strategically. Students use tiles and drawings to solve area problems.

6. Attend to precision. Students use vocabulary such as area, array, area model, and dimensions with increasing precision to discuss their reasoning when solving area problems.

7. Look for and make use of structure. Students compare rectangles with the same area but different dimensions and look for patterns in the shapes of the rectangles.

8. Look for and express regularity in repeated reasoning. Students will notice that arrays and multiplication can be used to solve area problems.

****Mathematical Practices 1 and 6 should be evident in EVERY lesson. ***

CONTENT STANDARDS

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

MGSE3.OA.8. Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

MGSE3.OA.9. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.2 For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

‡ See Glossary, Table 3

Represent and interpret data.

MGSE3.MD.3. Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

MGSE3.MD.4. Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.

Geometric Measurement: understand concepts of area and relate area to multiplication and to addition.

MGSE3.MD.5. Recognize area as an attribute of plane figures and understand concepts of area measurement.

a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.

b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.

MGSE3.MD.6. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

MGSE3.MD.7. Relate area to the operations of multiplication and addition.

a. 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.

b. Multiply side lengths to find areas of rectangles with whole number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

c. Use tiling to show, in a concrete case, that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

For more detailed information about unpacking the content standards, unpacking a task, math routines and rituals, maintenance activities and more, please refer to the Grade Level Overview.

BIG IDEAS

● Area models are related to addition and multiplication.

● Area covers a certain amount of space using square units.

● When finding the area of a rectangle, the dimensions represent the factors in a multiplication problem.

● Multiplication can be used to find the area of rectangles with whole numbers.

● Area models of rectangles and squares are directly related to the commutative property of multiplication.

● Rearranging an area such as 24 sq. units based on its dimensions or factors does NOT change the amount of area being covered (Van de Walle, pg 234). Ex. A 3 x 8 is the same area as a 4 x 6, 2 x12, and a 1 x 24.

● A product can have more than two factors.

● Area in measurement is equivalent to the product in multiplication.

● Area models can be used as a strategy for solving multiplication problems.

● Some word problems may require two or more operations to find the solution.

ESSENTIAL QUESTIONS

● How can area be determined without counting each square?

● How can the knowledge of area be used to solve real world problems?

● How can the same area measure produce rectangles with different dimensions? (Ex. 24 square units can produce a rectangle that is a 3 x 8, 4 x 6, 1 x 24, 2 x 12)

● How does understanding the distributive property help us multiply large numbers?

CONCEPTS/SKILLS TO MAINTAIN

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas. ● Addition, Subtraction, Multiplication, Division

● Skip counting

● Relationship between addition and multiplication

● Two-dimensional plane figures

● Understanding of arrays

● Solving one-step word problems

● Factors of products

● Commutative Property of Multiplication

● Distributive Property of Multiplication

Fluency: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Fluent problem solving does not necessarily mean solving problems within a certain time limit, though there are reasonable limits on how long computation should take. Fluency is based on a deep understanding of quantity and number.

Deep Understanding: Teachers teach more than simply “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives. Therefore, students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of foundational mathematics concepts by applying them to new situations, as well as writing and speaking about their understanding.

Memorization: The rapid recall of arithmetic facts or mathematical procedures. Memorization is often confused with fluency and automaticity. Fluency implies a much richer kind of mathematical knowledge and experience.

Number Sense: Students consider the context of a problem, look at the numbers in a problem, make a decision about which strategy would be most efficient in each particular problem. Number sense is not a deep understanding of a single strategy, but rather the ability to think flexibly between a variety of strategies in context.

Fluent students:

• flexibly use a combination of deep understanding, number sense, and memorization.

• are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and making meaning from them.

• are able to articulate their reasoning.

• find solutions through a number of different paths.

For more about fluency, see: http://www.youcubed.org/wpcontent/uploads/2015/03/FluencyWithoutFear-2015.pdf and: https://bhi61nm2cr3mkdgk1dtaov18-wpengine.netdna-ssl.com/wp-content/uploads/nctm-timedtests.pdf

STRATEGIES FOR TEACHING AND LEARNING

Adapted from North Carolina Dept. of Public Instruction Teaching Resources

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Students gain a full understanding of which operation to use in any given situation through contextual problems. Number skills and concepts are developed as students solve problems.

Problems should be presented on a regular basis as students work with numbers and computations.

Researchers and mathematics educators advise against providing “key words” for students to look for in problem situations because they can be misleading. Students should use various strategies to solve problems. Students should analyze the structure of the problem to make sense of it. They should think through the problem and the meaning of the answer before attempting to solve it.

Encourage students to represent the problem situation in a drawing or using manipulatives such as counters, tiles, and blocks. Students should determine the reasonableness of the solution to all problems using mental computations and estimation strategies.

Students can use base–ten blocks on centimeter grid paper to construct rectangular arrays to represent problems involving area.

Students are to identify arithmetic patterns and explain these patterns using properties of operations. They can explore patterns by determining likenesses, differences and changes. Use patterns in addition and multiplication tables.

Represent and interpret data.

Representation of a data set is extended from picture graphs and bar graphs with single-unit scales to scaled picture graphs and scaled bar graphs. Intervals for the graphs should relate to multiplication and division within 100 (product is 100 or less and numbers used in division are 100 or less). In picture graphs, use multiplication fact values, with which students are having difficulty, as the icons. For example, one picture represents 7 people. If there are three pictures, students should use known facts to determine that the three pictures represent 21 people. The intervals on the vertical scale in bar graphs should not exceed 100.

Students are to draw picture graphs in which a symbol or picture represents more than one object. Bar graphs are drawn with intervals greater than one. Ask questions that require students to compare quantities and use mathematical concepts and skills. Use symbols on picture graphs that students can easily represent half of, or know how many half of the symbol represents.

Students are to measure lengths using rulers marked with halves and fourths of an inch and record the data on a line plot. The horizontal scale of the line plot is marked off in whole numbers, halves or fourths. Students can create rulers with appropriate markings and use the ruler to create the line plots

Geometric measurement– understand concepts of area and relate area to multiplication and to addition.

Students can cover rectangular shapes with tiles and count the number of units (tiles) to begin developing the idea that area is a measure of covering. Area describes the size of an object that is two-dimensional. The formulas should not be introduced before students discover the meaning of area.

The area of a rectangle can be determined by having students lay out unit squares and count how many square units it takes to completely cover the rectangle completely without overlaps or gaps. Students need to develop the meaning for computing the area of a rectangle. A connection needs to be made between the number of squares it takes to cover the rectangle and the dimensions of the rectangle. Ask questions such as:

● What does the length of a rectangle describe about the squares covering it?

● What does the width of a rectangle describe about the squares covering it?

The concept of multiplication can be related to the area of rectangles using arrays. Students need to discover that the length of one dimension of a rectangle tells how many squares are in each row of an array and the length of the other dimension of the rectangle tells how many squares are in each column. Ask questions about the dimensions if students do not make these discoveries. For example:

● How do the squares covering a rectangle compare to an array?

● How is multiplication used to count the number of objects in an array?