Unit 4

Multiplication and Area

 

Grade 3

Math

Unit Length and Description:

 

25 days

By Unit 4, students are ready to explore properties of operations and investigate area. Students measure the area of a shape by finding the total number of same-size units of area, e.g. tiles, required to cover the shape without gaps or overlaps. When that shape is a rectangle with whole number side lengths, it is easy to partition the rectangle into squares with equal areas and multiply.

Standards:

 

Major Cluster Standards

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

3.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.

3.MD.6

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

3.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.

d. Recognize area as additive. Find the areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

Mathematical Practices

MP.2

Reason abstractly and quantitatively. Students build toward abstraction starting with tiling a rectangle, then gradually moving to finishing incomplete grids and drawing grids of their own, then eventually working purely in the abstract, imaging the grid as needed.

MP.3

 

Construct viable arguments and critique the reasoning of others.

As students compare solution strategies, they construct arguments and critique the reasoning of their peers. This practice is particularly exemplified in daily Application Problems and problem-solving specific lessons in which students share and explain their work with one another.

MP.6

Attend to precision. Students precisely label models and interpret them, recognizing that the unit impacts the amount of space a particular model represents, even though pictures may appear to show equal sized models. They understand why when side lengths are multiplied the result is given in square units.

MP.7

 

Look for and make use of structure. In this module, patterns emerge as tools for problem solving. Students make use of structure as they utilize the distributive property to establish the 9 = 10 – 1 pattern, for example, or when they check the solution to a fact using units of 9 by making sure the sum of the digits in the product adds up to 9. They make use of the relationship between multiplication and division as they determine unknown factors and interpret the meanings thereof.

MP.8

Look for and express regularity in repeated reasoning. Students use increasingly sophisticated strategies to determine area over the course of the module. As they analyze and compare strategies, they eventually realize that area can be found by multiplying the number in each row by the number of rows.

Instructional Outcomes

3.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.

·         I can define “unit square”.

·         I can define 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.

·         I can find the area of a plane figure using unit squares.

·         I can cover the area of a plane figure with unit squares without gaps or overlaps.

 

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

·         I can measure areas by counting unit squares.

·         I can use unit squares of cm, m, in, ft, and other sizes of unit squares to measure area.

 

3.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.

·         I can find the area of a rectangle by tiling it in unit squares.

·         I can find the side lengths of a rectangle in units.

·         I can compare the area found by tiling a rectangle to the area 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.

·         I can multiply side lengths to find areas of rectangles.

·         I can solve real world problems using area.

·         I can use arrays to represent multiplication problems.

 

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.

·         I can use an array to multiply.

·         I can find the area of a rectangle by modeling the distributive property using multiplication and addition.

·         I can use tiling to find the area of rectangles using the distributive property.

 

d. Recognize area as additive. Find the areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

·         I can find areas of rectangles.

·         I can add area of rectangles.

·         I can recognize that areas of each rectangle in a rectilinear (straight line) figure can be added together to find the area of the figure.

·         I can separate a polygon into rectangles to find the area of each rectangle to solve real world problems.

·         I can separate polygons into non-overlapping rectangles.

 

Enduring Understandings:

 

·         The space inside a rectangle or square can be measured in square units.

·         There are several strategies that we can use for finding area:

o   Multiply side lengths

o   Break apart and distribute

o   Find the area of smaller rectangles and add them together

o   Find the area of a large rectangle and subtract the area of a smaller rectangle.

·         It is important to know the best strategy to use for the problem.

 

 

Essential Questions:

 

·         What is the area?

·         Why is it important to know area in real life?

·         What strategies can I use to determine the area of an object?

·         How is area used in the world?