MeasurementData

Geometry

Wiki with resources and PowerPoint:http://

elementary-math-resources.wiki.inghamisd.org/home

1. What are the key questions that focus student's thinking and organize their learning?

2. What are the key activities that promote thinking?

3. What are the key strategies that have proven effective for learning about fractions, area and perimeter, and geometric shapes?

Overview

• Review of addition/subtraction and multiplication/division – Concepts AND Procedures

• Measuring length, volume, mass, time• Geometric shapes and their attributes• Fraction concepts

• First, time for 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.

Diagnostic Assessments

http://math.wiki.inghamisd.org/home

Math Work Stations, Debbie DillerDeveloping Number Concepts, Kathy RichardsonTeaching Student-Centered Mathematics, Van de Walle and LovinGeorgia Common Core lessonsNew York Common Core lessons

Measuring length in 2nd and 3rd grade

According to the CCSS, what length-measuring activities should 2nd grade students be doing?Do you have the tools they need?

What about measuring lengths in 3rd grade? Look at the graphing standards (3.MD.4) What tools are needed?

(see the wiki for half-inch and quarter-inch rulers)

0 1 2 3 4 5

Shoe 1 inches

Shoe 2 inches

Shoe 3 4 inches

Shoe 4 inches

inches

x xx x x

Line plot of measurements

How is 2.MD.9 different from 3.MD.4? Rulers (black line masters) are on the wiki.

Measuring area in 2nd and 3rd grade

What area-measuring activities should 3rd grade students be doing?What tools are needed?

Measuring mass and volume

3.MD.2 Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). (Excludes compound units such as cm3 and finding the geometric volume of a container.) What tools are needed? What might be measured?

3.MD.2 continued: Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. (Excludes multiplicative comparison problems involving notions of “times as much.”)Develop 4 such problems.

Telling time2.MD.7 Tell and write time from analog and digital clocks to the nearest five minutes, using am and pmHow do children come to learn that “5” on the clock means “25 minutes after the hour”?

3.MD.1 Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.What tools do you need for measuring time intervals in minutes? How can number lines be used for solving elapsed time problems?

See Georgia Measurement Units – 2nd grade page 57

Field trip to the zoo Georgia 3rd grade Measurement pp. 84-91

Using money

2.MD.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ (dollars) and ¢ (cents) symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?

What does it take to solve these problems?

Develop three more problems.

Web pages with money activity ideas are on the wiki.

Area standards3.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).

Area of a rectangle

• Fill this shape with tiles so that none are overlapping. Count how many tiles you have.

• Can you find a quick way to count the tiles?

7

4

Area standards3.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.

3

4

A shape puzzle

• Fill this shape with two different color tiles or counters, making two rectangles.

• Represent the total number of tiles or counters with a number sentence that uses multiplication.

7

4

A shape puzzle

• Fill this shape with two different color tiles or counters, making two rectangles.

• Represent the total number of tiles or counters with a number sentence that uses multiplication.

4 x 7 = 4 x 5 + 4 x 2

Area standards3.MD.7 Relate area to the operations of multiplication and addition. (cont’d)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.

4 x 7 = 4 x 5 + 4 x 2

7 = 5 + 2

4

Area standards3.MD.7 Relate area to the operations of multiplication and addition. (cont’d)

d. Recognize area as additive. Find 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.

Scaled Picture Graph

• How many books did Nancy read?• How many more books did Juan read than Nancy?

Bar Graph with Analysis

How is 2.MD.10 different from 3.MD.3?

Geometric Shapes

• Find all the different combinations of pattern blocks that can be used to make the hexagon.

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≠

Recognizing and analyzing shapesK.G.1 Describe objects in the environment using names of shapes…

K.G.2 Correctly name shapes regardless of their orientations or overall size.

K.G.4 Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other attributes (e.g., having sides of equal length).

Found on Peace, Love and First Grade, a blog by “Laura”Original on Ashley Hughes Teachers Pay Teachers store

Color the Shapes• Color all the

triangles green.• Color all the

rectangles red.• Color all the circles

blue.• Are there some

shapes you haven’t colored?

Shapes All Around Us

The Shape of Things, by Dayle Ann Dodds

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

=

Attributes1.G.1 Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); for a wide variety of shapes; build and draw shapes to possess defining attributes.

2.G.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. (Sizes are compared directly or visually, not compared by measuring.)

AttributesGive each student a set of shapes. Have them brainstorm ways to describe the shapes. Record their responses on chart paper. Guide students to look for ways other than color and size when describing the shapes such as by number of sides, number of corners, or no corners. Make a class chart.

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

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

Attributes3.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

Measuring and classifying angles starts in 4th grade

Why can we say that a rectangle is a category of shapes?

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

Shape sorts1. Take any shape. Tell one or

two things you find interesting about the shape.

2. Choose two shapes. Find something alike and something different about the two shapes.

3. The group selects one shape and places it in the center of the workspace. Find all other shapes that are like this shape according to the same rule.

4. Do a second sort with the same target shape but using a different property.

Shape sorts5. Groups share their sorting

rules with the class and show examples. Everyone draw a new shape that will also fit in the group according to the same rule.

6. Do a “secret sort” by selecting about 5 shapes that fit a secret rule, leaving some similar shapes in the pile. Others find similar shapes and try to guess the rule.

2.G.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Match the Rule

AttributesIn your small group, look at the pictures of the shapes on your page. List all the attributes you can find that all the shapes share.

• Sides• Corners (Measurement and classification of angles starts in 4th grade)

(Diagonals and symmetry are introduced in 4th grade)

The rhombus has 4 sides4 cornersopposite sides are the same length

What’s my shape?One student will come to the front of the room and reach into a bag of shapes. The student will pick out one shape, but keep her/his hand in the bag so no one can see the shape. The student then describes the shape, talking about attributes. After describing the shape as completely as possible, the student says “What’s my shape?” All of the other students think silently about what the shape might be and put their thumb up on their chest when they have an idea. The teacher calls on several students, drawing each shape they suggest on the board. When enough shapes have been drawn, the first student pulls the shape out and names it.Teacher-led discussion can continue about how the target shape matches or doesn’t match the shapes on the board.

Build a Shape• Read The Greedy Triangle (on wiki)• Ask students if they can make shapes with their bodies and a

piece of yarn.• Use straws, pipe cleaners, or other manipulatives to create a

triangle, rectangle, square and trapezoid. Model how you connect the straws and pipe cleaners to create a shape (sample below). Read The Greedy Triangle again. Have students create the shapes as you come to each shape in the book.

Georgia Common Core first grade workbook p. 29

Georgia lesson plans on wiki

Build a cube

“It is often difficult for students to visualize as it requires a coordination of both two and three-dimensional shapes. Activities which require students to think about, manipulate, or transform or a shape mentally will contribute to students overall visualization skills.”

Making Shapes

What can students learn from using Geoboards?

National Library of Virtual Manipulatives

Pattern Blocks - Composing

Pattern Blocks - Composing

National Library of Virtual Manipulatives

Math Playground

Tangrams

Fractions of a shape2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Fraction Fill In Game Students will work with a partner to play Fraction Fill In to develop proficiency with fractions. To use spinner, put a paperclip in the middle. Hold it in place with the tip of the pencil. Have the student thump the paper clip to spin and see where it lands.

pp. 55-56 Georgia 1st grade lessons

Sharing a Cake• pp. 58-61 Georgia grade 2 unit 5

Key Strategies for Fractions

Base early understanding on fair shares• Cookies, candy bars,

geometric shapes

Use representations of different kinds• Circles, rectangles, bars, number lines

pp. 63-70 gr. 2 unit 5

Key Strategies for FractionsDevelop the meaning of a fraction as a number (a place on the number line); connect a point on the number line to a fraction of a whole through the meaning of denominator and numerator

Use real-world examples including measurement• 1st grade: measure by placing unit bars on an object without

gaps.• 2nd grade: measure using rulers; know feet and inches• 3rd grade: solve problems involving perimeter of polygons;

measure using rulers to 1/2 inch and 1/4 inch.

Key Strategies for Fractions

Develop estimation skills for comparing fractions by basing comparisons on benchmark fractions such as 1/2.

Which is larger? Explain your reasoning. Use manipulatives or drawings.

Fractions – 3rd Grade Goal

How do each of the 3rd grade CCSS for fractions contribute to being able to do this?3.NF.1 – use manipulatives to model 3/8 and 5/8 (A)3.NF.2 – use a number line to show 3/4 and 3/5 (B)3.NF.3 a, b – show how 3/4 and 6/8 are equivalent using manipulatives and drawings (A)3.NF.3 d – solve the problems above and explain your reasoning (B)

Number Talks questions:• Draw a picture to show 3/4.• Which is larger, 3/4 or 1/2? How would you prove this?• How much more is needed to make 1? How do you know?• Draw a picture to show 3/5. How did you decide where to put 3/5?• Which is larger, 3/5 or 1/2? How would you prove this?• How much more is needed to make 1? How do you know?• How would you show or explain which is larger?

Let’s try some more: 1/2 and 1/3. 1/4 and 2/4. 4/6 and 4/8.

Dolphin Racing Game

Fractions – 3rd Grade Goal

Which Common Core standards can this help students learn?

The 10 Instructional Shifts

1. Incorporate ongoing cumulative review into every day’s lesson.

2. Adapt what we know works in our reading programs and apply it to mathematics instruction.

3. Use multiple representations of mathematical entities.

4. Create language-rich classroom routines.5. Take every available opportunity to support the

development of number sense.

6. Build from graphs, charts, and tables.7. Tie the math to such questions as: “How big?” “How

much?” “How far?” to increase the natural use of measurement throughout the curriculum.

8. Minimize what is no longer important.9. Embed the mathematics in realistic problems and

real-world contexts.10. Make “Why?” “How do you know?” “Can you

explain?” classroom mantras.

1. What are the key questions that focus student's thinking and organize their learning?

2. What are the key activities that promote thinking?

3. What are the key strategies that have proven effective for learning about fractions, area and perimeter, and geometric shapes?

At least one big idea for each topic

• Measuring and estimating length, volume and mass

• Tell and write time and solve problems• Solve problems involving money• Draw graphs, use measurement data• Shapes and their attributes