GEOMETRY, MEASUREMENT, ESTIMATION, PROBLEM-SOLVINGSession 3

Goals for the day• Apply multiplication, division and fraction skills to other

topics: Measurement, estimation, graphing, mental math• Consider the developmental levels of geometric thinking• Plan lessons to promote problem-solving• Re-consider the key strategies for all math lessons

Sharing

Concepts and Procedures• As a review, what do students need to know conceptually

in order to add two-digit numbers? 24 + 51 = ___• What do students need to know conceptually to multiply

single-digit numbers? 5 x 6 = ___• What do students need to do in order to get good at these

two things (after they learn the concepts?)

develop procedures, practice

Students should learn to see 8x16 as 8x10 + 8x6. Area problems help with this.

3.OA.6 Understand division as an unknown-factor problem. For example, divide 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Distributive property using the area model to do mental computation

16

8

10 + 6

80 48 128

Kakooma by Greg Tang

Geometric Shapes• Sort your shapes by any criteria that make sense to you.

See if you can come up with two different sortings. Explain your rule. Use the attributes in 4.G.2.

van Hiele Levels of Geometric Thought

Level 0: VisualizationStudents recognize and name figures based on the global, visual characteristics of the shape. Students at this level are able to make measurements and even talk about the properties of shapes, but these properties are not abstracted from the shape at hand. It is the appearance of a shape that defines it for a student. A square is a square “because it looks like a square.”

Other visual characteristics may include “pointy,” “fat,” “sort of dented in.” Classification of shapes at this level is based on whether they look alike or different.

from Van de Walle and Lovin, 2006

≠

Shapes by grade

• K: squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres

• 1st: rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles

• 2nd: triangles, quadrilaterals, pentagons, hexagons, and cubes

• 3rd: rhombuses

• 4th: parallelogram is implied by classifying figures based on parallel lines

van Hiele Levels of Geometric Thought

Level 1: AnalysisStudents are able to consider all shapes within a class rather than a single shape. By focusing on a class of shapes, students are able to think about what makes a rectangle a rectangle (four sides, opposite sides parallel, opposite sides equal, four right angles, etc.) Irrelevant features (e.g. orientation or size) fall into the background.

Students begin to appreciate that a collection of shapes goes together because of its properties.

from Van de Walle and Lovin, 2006

=

van Hiele Levels of Geometric Thought

Level 2: Informal DeductionStudents are able to develop relationships between and among properties of shapes. They recognize sub-classes of properties: “If all 4 angles are right angles, it is a rectangle. Squares have 4 right angles, so squares must be rectangles.”

from Van de Walle and Lovin, 2006

Relationships among shapes3.G.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

Quadrilaterals

Why can we say that a rectangle is a category of shapes? What’s a 3rd grade definition of rectangle?

Relationships among shapes4.G.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. (Do all right triangles look alike?)

2-dimensional figures - polygons

triangles quadrilaterals

http://www.engageny.org/resource/grade-4-mathematics-module-4

Angles4.MD.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement…

4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

Angle-measure applet

Which wedge is right? 4th grade

https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx

Measuring angles

• Georgia Common Core Formative Assessment lessons

https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx

Is this possible?1. A triangle with one right angle and one obtuse angle?

Draw it:

2. A triangle with three acute angles?

3. A triangle with exactly two acute angles?

4. A triangle with two obtuse angles?

5. What kind of figures can you draw with two right angles?

Types of triangles

Classification of triangles

Right triangles, isosceles triangles, acute triangles, scalene triangles, obtuse triangles, equilateral triangles

Chris’ Bates class

Types of quadrilaterals

Classification of quadrilaterals

4.G.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines…

Relationships among shapes5.G.4 Classify two-dimensional figures in a hierarchy based on properties.

Relationships among shapes

Quadrilaterals

5.G.4 Classify two-dimensional figures in a hierarchy based on properties.

Measurement• Measure length in length, mass, liquid volume, time.• Solve word problems involving measurements and money

involving simple fractions or decimals.• Measure or calculate the perimeter and area of

rectangles.• Display measurement data in line plots.

Line plot of measurements

Metric Measurement… Really???4.MD.1 Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example: Know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), etc.

4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

Look at it as an Opportunity!Practice multiplication… How many inches are in 18 feet? Organize information in tables… A conversion chart of pounds to ouncesPractice measuring things… What units should we use to measure this table, feet, inches, meters, centimeters? How long is it, exactly?Practice addition and subtraction of different “bases”… 4 hrs 10 min + 5 hrs 24 min, 5.24 kg + 2.83 kg

Engage NY Mathematics Lesson Plans 4th Gr. Module 2, p. 2.A.4

Next level of complexity…4.MD.1 …Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit…5.MD.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step real world problems.

This is an opportunity to work with decimals and fractions (15 inches = 1 ¼ feet). HOWEVER, this is difficult for many students, and the metric system is very rarely used by elementary kids.

What problems would you have your students do?

More Opportunity!4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

What’s the corresponding problem involving perimeter?A page of these kinds of problems would require students to write the corresponding equation pair and fill in both blanks.

45 sq. ft.

9 ft.

? ft. 9 x ___ = 45 45 ÷ 9 = ___

Sugar cubes and boxes at your table

Annenberg Learner - Surface Area and Volume

Finding Volume

5.MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

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

b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

5.MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

5.MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent three-fold whole-number products as volumes, e.g., to represent the associative property of multiplication.

b. Apply the formulas V =(l)(w)(h) and V = (b)(h) for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

Illuminations - Interactive Simulation of Volume

Finding Volume

Generate possible dimensions for other toy chests that have a volume of 5400 cubic inches. Here is one:12” high by 15” deep by 30” wide 9 x 15 x 30 = 4050

Toy Chest

12 x 15 x 30

3x4 x 3x5 x 3x10

3x2x2 x 3x5 x 3x2x5

A restaurant is open 24 hours a day. The manager wants to divide the day into work shifts of equal length. Show the different ways this can be done. The shifts should not overlap, and all shifts should be a whole number of hours long.

Teaching through Problem SolvingA classroom floor will be covered with 200 square feet of carpeting. The length of the room is 25 feet. What is the width of the room?

A field trip lasts for 2 ½ hours.

Smarter Balanced Assessment

1) Scan the section from van de Walle and summarize three key points.

2) Share with your partner.3) Discuss pros and cons.4) Develop one problem that would take more than 3

minutes to solve.

EstimatingT-shirts with the school logo cost $6 wholesale. The Pep Club has saved $257. How many t-shirts can they buy for their fund-raiser?

Describe the steps you would take to get an exact answer.Do as many steps as you can in your head. When you stop, ask yourself, is this a good estimate?

Estimate 438 x 62Use the first strategy that comes into your head.Then figure out a different estimation strategy, perhaps using different numbers.

Estimate 708 ÷ 27Use a multiplication strategy to approximate how many 27s are in 708.Any other estimation strategies come to mind?

Computational Estimation• As you peruse the reading, look for several things and

make lists:• Big Ideas, the most important concepts about teaching

estimation• Great activities that you’d like to try

• Read up through p. 249. Scan the rest of the section.

• Be ready to share one of each.

Bridges supplement 5th grade, p. 231