GEOMETRY,
MEASUREMENT,
ESTIMATION,
PROBLEM-SOLVING
Session 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
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.
Students should learn to see 8x16 as 8x10 + 8x6. Area problems help with this.
Distributive property using the area
model to do
mental computation
16
10
+
6
8
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: Visualization
Students 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: Analysis
Students 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 Deduction
Students 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 shapes
3.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 shapes
4.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
Angles
4.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
Right triangles, isosceles triangles, acute triangles,
scalene triangles, obtuse triangles, equilateral triangles
Classification of triangles
Chris’ Bates class
Types of quadrilaterals
Classification of quadrilaterals
4.G.2 Classify twodimensional figures
based on the presence or
absence of parallel or
perpendicular lines…
Relationships among shapes
5.G.4 Classify two-dimensional figures in a hierarchy based on
properties.
Relationships among shapes
5.G.4 Classify two-dimensional figures in a hierarchy based on
properties.
Quadrilaterals
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
ounces
Practice 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.
45 sq. ft.
? ft.
9 x ___ = 45
45 ÷ 9 =
___
9 ft.
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.
Finding Volume
Sugar cubes and boxes
at your table
Annenberg Learner - Surface Area and 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.
Finding Volume
Illuminations - Interactive Simulation of Volume
Toy Chest
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
12 x 15 x 30
3x4 x 3x5 x 3x10
3x2x2 x 3x5 x 3x2x5
Teaching through Problem Solving
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.
A 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.
Estimating
T-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 62
Use the first strategy that comes into your head.
Then figure out a different estimation strategy, perhaps
using different numbers.
Estimate 708 ÷ 27
Use 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