• Use connecting objects, e.g., cubes, to measure attributes of distance (length, height) by counting the number of objects needed to measure.
• Use a scale to compare the weight of two objects.

• Select representation of more and less, short and long, heavy and light; tall and short.

• Model-Lead-Test ("Watch me…do together….you try")*
• Least-to-Most prompts (e.g., "Start by putting the shortest item in place like this…")*

• Graphic organizer with more or less representations to prompt understanding
• Unifix cube wands to compare lengths (same size same color – additional cubes in another color)
• Balance scales
• Weight scales
• Rulers
• Yard sticks

• Use connecting objects, e.g., cubes, to measure attributes of distance, length, and height
• Use a scale to compare the weight of two objects.

• Select representation of more and less, short and long, heavy and light; tall and short.
• Apply understanding that if object 1 is longer/heavier than object 2 and object 2 is longer/heavier than object 3, then object 1 must be longer than object 3.

• Model-Lead-Test ("Watch me…do together….you try")*
• Least-to-Most prompts (e.g., "Start by putting the shortest item in place like this…")*

• Sequencing Template for short - tall with graphic representations to prompt understanding
• Number line
• Measuring tools

• Understand that smaller units will require more to measure an object than using larger units to measure the same object.
• Use connecting objects, e.g., cubes, to measure attributes of distance, length, and height.

• Select the number that represents the number of units used to measure the length of an item.
• Understand the concept of more and less.

• Model-Lead-Test ("Watch me…do together….you try")*
• Least-to-Most prompts (e.g., "Start by filling in the first row of the template with paperclips like this…")*
• Teacher uses the measuring template for an object being measured and student counts as each paperclip is added to the template (repeat for markers) until the item has been measured. Student may use an electronic counter. The teacher asks which item required more to measure the object and which required less to measure the object.

• Measuring template (i.e., length of item being measured: 1^st row segmented into sections for paperclips; 2^nd row segmented into sections for markers)
• Measuring stick made of Unifix cubes

• Counting up to --- objects
• Identify the beginning and end point that needs to be measured.
• Recognize that an object can be measured by lining up multiple objects of the same size without gaps or overlaps.

• Select the numeric symbol that represents the number of units used to measure the length of an item.

• Model-Lead-Test ("Watch me…do together….you try")*
• Least-to-Most prompts (e.g., "Start by placing a paperclip next to the item like this…")*
• Teacher places copies of the original paperclip along the side of an object being measured and student counts as each copy is placed until the item has been measured. Student may use an electronic counter.

• Measuring template (i.e., length of item being measured –segmented into sections the size of paperclips)
• Measuring stick made of Unifix cubes

• Understand that smaller units are part of larger units within the same system (12 inches = 1 foot).
• Identify the smaller and/or larger unit (e.g., inches are smaller than feet).
• Understand that they should use the unit of measure that will require fewer units to measure objects.

• Select the numeric symbol that represents the number of units that make up a larger unit of measure.
• Select representation of larger units of measure within a system of measurement.

* This should be to "nearest whole unit" – in grade 2 we do not measure fractions/parts of a unit

• Model-Lead-Test ("Watch me…do together….you try")
• Least-to-Most Prompts*
• Teacher identifies an object to be measured. "I can measure with inches or feet, which unit will require less to measure the table?" "Which unit would I use to measure the room?"
• Use collapsible ruler to show how smaller units make up a larger unit.

• Foldable ruler
• Measuring stick made of Unifix cubes
• Talking ruler
• Ruler
• Color coded units (inches=red, feet=blue, yards=green)

• Understand that smaller numbers are part of larger numbers (e.g., 5 can be made of 4 and 1 or 3 and 2, etc.).
• Within the same system of measurement identify the smaller or larger unit—inches are shorter than feet, feet are shorter than yards, etc.

• Select the numeric symbol that represents the number of units that make up a larger unit of measure.
• Understand that multiple units make up a larger unit of measure (12 inches in 1 foot).

• Model-Lead-Test ("Watch me…do together….you try")*
• Least-to-Most prompts (e.g., "Put an inch on the ruler like this…")*
• Teacher places inches on the ruler, student counts as inches are added to the ruler, may use an electronic counter.
• Use collapsible ruler to show how smaller units make up a larger unit.

• Foldable ruler
• Measuring stick made of Unifix cubes
• Interactive whiteboard or other technology

• Understand that smaller numbers are part of larger numbers (e.g., 5 can be made of 4 and 1 or 3 and 2, etc.).
• Identify the smaller unit (e.g., inches are smaller than feet).
• Count up to 20 objects.
• Recognize that when we compare lengths we want to answer "How much longer is object 1 than object 2?"

• Represent the numeric symbol by breaking the whole into the corresponding number of parts.
• Select the numeric symbol that represents the differences.

• Model-Lead-Test ("Watch me…do together….you try")*
• Least-to-Most prompts to use the foldable ruler*
• Teacher represents the problem using Unifix cubes (measuring stick) taken away from the whole. Student counts as cubes are removed from the stick. Student may use an electronic counter.

• Foldable ruler
• Measuring stick made of Unifix cubes (use two colors to reflect those taken away)
• Talking ruler

• Count up to 20 objects.
• Identify the area on a surface (e.g., piece of paper).
• Recognize that area can be determined by covering a rectangular area with square tiles that have no gaps or overlaps.
• Use square tiles to cover a rectangle.
• Count the number of tiles to determine the area.
• Decompose rectangles within a rectilinear figure.

• Select the numeric symbol that represents the number of squares used to find area of a figure.
• Count to find the area of a rectangle when given a picture or array.

• Model-Lead-Test ("Watch me…do together….you try")*
• Use Least Intrusive Prompts (e.g., "Put a tile on like this…")*
• Teacher does the tiling. Student counts as tiles are taken off. Student may use an electronic counter.

• 1 inch tiles
• 1 inch tiles that are numbered
• Raised grid with numbered squares
• Raised frame and raised grid
• Hand tally counter or software that counts
• Interactive whiteboard or other software that allows the student to move tiles on or off the figure
• Counting sheet that allows students to mark the tiles that have been counted

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.

• Decompose a rectilinear figure into rectangles
• Identify the perimeter of a rectilinear figure
• Identify the area of a rectilinear figure

• Understand the following concepts and vocabulary (pictures/symbols): area, perimeter, length, width, side, +, -, X, ÷

• Task analysis (solving problems using formulas); isolate each step of the solution process
• Model-Lead-Test ("Watch me…do together….you try")*
• Least-to-Most prompts*
• Relate a story problem to everyday life/relevant context

• Premade formula worksheets
• Calculator
• Foldable ruler
• Conversion charts (inches to feet, feet to yards)
• 1 inch tiles
• Raised grid with squares numbered
• Graph paper or Grid paper (virtual or with raised lines, on overhead transparencies, etc.)
• Graphic representation of square and rectangle
• Interactive whiteboard, PowerPoint, or other visual demonstrating how squares change to rectangles when 2 sides are elongated

• Match like units of measurement within a measurement system (e.g., hours to minutes, inches to feet).

• Use various strategies to add, subtract, multiply, and divide.
• Use a pictorial representation of a ratio to make conversions.

• Task analysis for problem solving
• Model-Lead-Test*
• Least-to-Most prompts*
• Provide a calendar. The teacher says there are seven days in one week and counts out each day (1-7) and points to the calendar. Say, "Show me one week." Say, "There are seven days in one week for a ratio of 7:1 (days: week). So, how many days are in three weeks?" "If you have to write two book reports per week, how many book reports will you write in four weeks?"
• Use plastic fraction bars to make equivalent measurements. For example, shade a picture a ruler into twelve portions and use the fraction bars to visually illustrate the equivalent of 6 inches (6/12) and one foot (12/12).
• Students can solve a one-step problem by using manipulatives and/or incorporating symbolic numeral cards to correspond to a concrete model. For example, the teacher can give a problem such as "The bookshelf is 2 feet long. There are 12 inches in one foot. How many inches is the bookshelf altogether?" Then have students solve this problem by using objects. The students count out 24 objects.

• Calendar
• Calculator
• Counters and graphic representation of ratios and fractions
• Worksheet with partially completed formula
• Interactive whiteboard or PowerPoint
• Balance or scale
• Clock
• Counting tiles
• Cups and buckets to measure volume

• Recognize that in the same system, I can measure the same object with 2 different units (e.g., I can measure the height of a desk in both inches and feet).

• Understand the following concepts and vocabulary: conversion, inch, foot, yard
• Understand standard units and abbreviations (e.g., feet=ft)

• Multiple exemplar training (e.g., "This is an inch, this is an inch…this is not an inch, show me an inch.")*
• Task analysis steps to convert from inches to feet using a table
• Teach student to use proportions (e.g., 12:1, 12 inches = 1 foot) to convert the same measurement from one unit to another.
• Measure length using one inch increments (how many) and one foot increments (how many).
• Have students place the U.S. unit cards/representations in order from smallest to largest.

• Conversion table, adapted or un-adapted measuring tools
• Calculator
• Counting blocks or manipulatives
• Counting mechanism (e.g., number line)
• Match measuring tool to unit (e.g., "Identify the tool to measure inches.")
• Software
• Rulers with limited measurement (e.g., only 1 inch and ½ inch tabs)

• Recognize simple shapes within a larger shape.
• Identify the dimensions (base, height, length, width, etc.) of smaller shapes.
• Multiply fractions and whole numbers.

• Given a picture identify the dimensions of 2-D and 3-D shapes.
• Understand the following concepts and vocabulary: polygon, trapezoid, pentagon, rectangles, squares, triangles, area.

• Task analysis to apply strategies for taking apart shapes
• Model-Lead-Test ("Watch me…do together….you try")*
• Least-to-Most prompts*
• Multiple exemplar training*
• Remind students how to find the area of an entire figure using their measurements. Demonstrate how they could break the figure into smaller rectangles and add the areas of the smaller rectangles.
• Ask student to use the squares from pattern blocks or color tiles (the side of the square should equal 1 inch). Ask the student, "What is the area of the square?" (1 sq. inch) Put 1 square to the right of the original square, and 1 square above it. The figure will look like this:
• Ask the student to find the area of the new figure. (3 sq. inches)

• 1 inch tiles
• Raised grid with numbered squares
• Grid paper
• 2-dimensional shapes (polygon, trapezoid, pentagon, rectangles, squares, and triangles)
• PowerPoint showing how simple shapes are combined to make complex shapes
• Interactive whiteboard
• Graphic representation of simple and complex shapes
• Use real-world examples (cutting a magazine clipping to fit onto a card that is a simple shape)

• Multiply using concrete objects.
• Divide using concrete objects.
• Use a ratio to solve a measurement conversion problem.

• Multiply whole numbers.
• Divide whole numbers.
• Use a pictorial representation of a ratio to solve problem.

• Task analysis for problem solving (formula)
• Model-Lead-Test*
• Least-to-Most prompts*
• Provide a calendar. The teacher says there are seven days in one week and counts out each day (1-7) and points to the calendar. Say, "Show me one week." Say, "There are seven days in one week for a ratio of 7:1 (days: week). So, how many days are in 3 weeks?"
• Rates such as miles per hour, ounces per gallon and students per bus should be reinforced. Using ratio tables develops the concept of proportion. Compare equivalent ratios; present real-life problems involving measurement units that need to be converted; represent measurement conversions with models such as ratio tables, T-charts or double number line diagrams.

• Premade function table
• Conversion chart
• Calendar
• Calculator
• Counters and graphic representation of ratios

• Identify the radius and diameter of a circle.
• Multiply decimals and whole numbers.

• Recognize the meaning of terms used in formulas as labeled representations related to circles.
• Understand the following concepts and vocabulary: circumference, area, pi, and radius.

• Task analysis with formula
• Model-Lead-Test ("Watch me…do together….you try")*
• Least-to-Most prompts*
• Say, "Here is a circle. Here is the circumference." Trace the circumference with your finger. Ask the student, "Show me circumference."
• Use picture cards and number sentences with formulas.

• Calculator
• Graphic of circle
• Tiles to place inside of circle to represent area
• Interactive whiteboard or other software
• Rolling counter, string, or yarn to measure circumference
• Assistive Technology
• Real-world materials

• Recognize attributes of a 3-dimensional shape.
• Multiply whole numbers, fractions, and decimals.

• Recognize that volume of 3-D shapes can be found by finding the area of the base and multiplying that by the height.
• Understand the following concepts and vocabulary: volume, cylinder, cone, height, radius, circumference, cube, sphere, side, pi

• Task analysis for applying formula
• Model-Lead-Test*
• Least-to-Most prompts*
• Fill cylinders and cones with water or rice to illustrate volume. Describe volume as what is "inside."
• Provide relevant, real-world examples and uses.

• Cones, cylinders, cubes, and spheres in differing sizes and textures
• Cardboard models that can be folded to make 3-dimensional shapes
• Partially completed formula
• Calculator

• Recognize how the space inside a figure increases when the sides are lengthened.
• Multiply whole numbers, fractions, and decimals.
• Apply formulas to find surface area, area, volume of figures.

• Given a picture, identify dimensions needed to calculate surface area, area, and volume.
• Compare greater than, less than, equal/same squares and rectangles in 2 and 3 dimensions.
• Understand the following concepts and vocabulary: similar, area, length, width, volume, square, rectangle, prism.

• Model-Lead-Test ("Watch me…do together….you try")*
• Least-to-Most prompts *
• Teacher does the tiling. Student counts as tiles are taken off. Student may use an electronic counter.
• Use scale drawings of geometric figures to connect understandings of proportionality and similarity and congruence

• Hollow square and rectangular boxes
• Blocks or tiles to place in the boxes
• 1 inch tiles
• Raised grid with numbered squares
• Interactive whiteboard that allows for student to move tiles on or off the figure
• Graph paper
• Dot paper
• Real-world materials

• Determine what units are used in problem (e.g., money, time, units of measurement, etc.).
• Match the action of combining with vocabulary (i.e., in all; altogether) or the action of decomposing with vocabulary (i.e., have left; take away, difference) in a word problem.

• Apply conversions of units while solving problems (e.g., Recognize that monetary units can be combined to equal other monetary units).
• Translate wording into numeric equation.

• Task analysis
• Model-Lead-Test *
• Least-to-Most prompts*
• Create relevant, story-based problems. For example, the story may be used to solve a problem about money and shopping at the grocery store. Use graphic organizers to provide students a means for organizing their work. Break down and isolate each step in solving the math task.

• $1, $5, and $10 bills
• Number line labeled with $1/unit, $5/unit, and $10/unit
• Calculator, software that counts, or other means of hand tallying
• Graph paper where each square equals a unit

• Understand concepts of area, volume, width, length, height, equals.
• Identify the unknown quantity when given an equation and labeled figure. (E.g., Provide a labeled prism and the equation V=L×H. Ask the student to draw/indicate the label on the prism to the letter in the equation.)
• Identify the unknown quantity when given an equation and labeled figure.
• Multiply and divide to find measurement.

• Given a picture, identify dimensions needed to calculate surface area, area, and volume.
• Understand formula representation (e.g., "h" in the equation means height).
• Use letters to represent numbers.
• Use letters to represent variables.
• Recognize symbols for equals, addition (+), and multiplication (×).

• Tiling/fill-in space and count
• Sequence: 1. Area 2. Volume 3. Missing attribute
• "If the area of a rectangle is 24cm² and it has a base of 6cm, what would the height be?"
• Task analysis with Least Intrusive Prompts
• Replace a letter (variable representing an unknown quantity) with a number or representation of a number (symbols, manipulatives).
• Provide a labeled prism and the equation V = L x W x H. Ask the student to draw/indicate the label on the prism to the letter in the equation.' 'Break down and isolate each step in solving the math task.
• Provide nets to be taken apart (unfolding) to illustrate three-dimensional objects. This process can also be used for the study of the surface area of prisms.

• Pre-made formula
• Use of calculator
• Manipulatives (2-D shapes, prism, cube (e.g., box))
• Counters (e.g., tally counter) and counting mechanism (e.g., number line)