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Geometry and measurement help us represent in an orderly fashion what we see in our world. Whether we are cooking or cartooning, shopping or shipping, painting a canvas or a wall, designing an addition for a house or a play yard for preschool, we continually bump up against these mathematical organizers. Lifelong learners should know and understand these interconnected and symbiotic mathematical domains.
Adult learners who attend basic mathematics classes at any level share a wealth of pragmatic experience surrounding geometric and spatial concepts. They have probably built a bookcase, laid out a garden, applied wallpaper or tiled a floor, all the while discovering informally the rules which formally govern the study of geometry itself
Geometry and measurement often spark a renewed interest in mathematics for those students who have been turned off for some reason or heretofore have felt unsuccessful with mathematics learning. Investigating problems that involve geometry and measurement broadens all students' mathematical understanding and engages them as they explore mathematical ideas.
Hands-on, interactive investigations using nonstandard and standard units help adult basic education students develop an understanding of the many measurable attributes of physical objects. Measurement sense including length, time, temperature, capacity, weight, mass, area, volume, and angle will benefit from this approach. This realistic approach helps build an accessible measurement vocabulary and a meaningful comprehension of what it means to measure.
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Learners engage in problem solving within adult contextual situations by communicating, reasoning, and connecting to:
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Standard G-1. Use and apply geometric properties and relationships to describe the physical world and identify and analyze the characteristics of geometric figures,
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Standard G-2. Use transformations and symmetry to analyze mathematical situations,
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Standard G-3. Specify locations and describe spatial relationships using coordinate geometry and other representational systems,
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Standard G-4. Understand measurable attributes of objects and the units, systems, and processes of measurement and apply appropriate techniques, tools and formulas to determine measurements.
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Outline of Learning Levels
Level 1. Beginning Adult Numeracy
See “How to Use This Document (Teacher’s Guide) and (Connecting Curriculum, Instruction and Assessment),” pages 8-10.
At this time, the Massachusetts ABE Test for Math does not assess students’ knowledge at Level 1.
Strand: Number Sense
Learners engage in problem solving within adult contextual situations by communicating, reasoning, and connecting to the following standards:
Standard 1N-1. Represent and use numbers in a variety of equivalent forms in contextual situations | Benchmark: At this level an adult will be expected to: | | Examples of Where Adults Use It |
1N-1.1 Count reliably forward and backward up to 20 items.
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1N-1.1.1 Demonstrate an understanding that if items are rearranged, the numbers stay the same
1N-1.1.2 Count forward and backward from ten or less
1N-1.1.3 Count forward and back from 11-20 |
Counting children in a group to make sure no one is missing
Counting dollar bills to pay for a purchase
Counting items at the grocery express line
Using the remote channel tuner for a TV
Watching a digital timer on a microwave count down the time
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1N-1.2 Recognize odd and even numbers up to 100.
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1N-1.2.1 Demonstrate an understanding that even numbers represent amounts that can be paired
1N-1.2.2 Demonstrate an understanding that odd numbers represent amounts that when paired have one remaining
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Identifying the number of possible couples at a dance or a dinner party
Recognizing when house numbers go up in odd or even numbers
Finding a room in a hospital or hotel
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1N-1.3 Read, write, and compare numbers from 0 up to 100.
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1N-1.3.1 Explain how the position of a digit signifies its value
1N-1.3.2 Demonstrate an understanding of directionality in reading numbers and comparisons from left to right.
1N-1.3.3 Explain what each digit in a two-digit number represents, including the use of zero as a place holder
1N-1.3.4 Distinguish between greater than and less than, and recognize between-ness when comparing numbers
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Telling which address falls in a given block, knowing the first number on the block
Writing a money order for a whole dollar amount (no change) |
1N-1.4 Using a 100 chart, skip count by 2’s, 5’s, and 10’s.
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1N-1.4.1 Know the multiples of 2, 5, and 10 to 100
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Counting nickels and dimes
Finding the amount of money in a small stack of $2, $5, or $10 bills
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| | Benchmark: At this level an adult will be expected to: | Enabling Knowledge and Skills | Examples of Where Adults Use It |
1N-2.1 Demonstrate an understanding of different meanings of addition (e.g. counting on, combining) of numbers up to 20.
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1N-2.1.1 Add by counting on (e.g. four objects plus three objects can be totaled by counting on three more than four (or five, six, seven), or counting on four more than three (or four, five, six, seven)
Demonstrate an understanding that combining two amounts into one larger total is adding.
1N-2.1.2 Use objects, pictures, or tallies to show addition
1N-2.1.3 Demonstrate the ability to visualize grouping of objects
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Paying a twelve dollar amount by using a ten dollar bill and two ones
Figuring hours of work or sleep by using fingers to count
Figuring hours of sleep by joining the hours slept before and after midnight
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1N-2.2 Demonstrate an understanding of subtraction as taking away or separating from numbers up to 20.
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1N-2.2.1 Subtract by counting back (e.g. taking away four of seven objects by counting back--six, five, four, three)
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Figuring how much of $20 is left while paying out $14
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1N-2.3 Demonstrate an understanding of how addition and subtraction relate to each other.
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1N-2.3.1 Add back to check subtraction (e.g. 10 – 6 = 4, 6 + 4 = 10)
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Making change (e.g. for a twenty dollar bill, by counting on from the price to $20)
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| | Benchmark: At this level an adult will be expected to: | Enabling Knowledge and Skills | Examples of Where Adults Use It |
1N-3.1 Know all pairs of numbers with a total of 10.
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1N-3.1.1 Combine amounts that add to 10 without having to count
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Adding using mental math
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1N-3.2 Add numbers with totals to 20.
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1N-3.2.1 Use the operation of addition and related vocabulary (e.g., add, sum of, total, plus, etc.)
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Calculating totals, e.g., five reams of paper in a full box plus three packs on the shelf
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1N-3.3 Subtract single-digit numbers from numbers up to 20.
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1N-3.3.1 Use the operation of subtraction and related vocabulary (e.g. difference, take away, less than)
1N-3.3.2 Know subtraction facts for pairs of numbers with totals to 10 (e.g. 10 – 6 = 4)
1N-3.3.3 Know how to add back to check subtraction (e.g. 10 – 6 = 4, and 6 + 4 = 10)
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Working out the shortfall in numbers, e.g. eggs for a recipe, plants to fill a display tray, cups to serve visitors
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1N-3.4 Double whole numbers to 10.
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1N-3.4.1 Know doubles of numbers to 10
| Finding the cost of tickets for an amusement ride for two children.
Planning fare for round trip subway travel at $1 a token |
1N-3.5 Finding half of whole numbers up to 20.
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1N-3.5.1 Know doubles of numbers to 10
1N-3.5.2 Demonstrate the ability to separate amounts in two piles
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Sharing the cost of pizza between two people.
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1N-3.6 Use a calculator to check calculations using whole numbers.
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1N-3.6.1 Identify the signs for addition, subtraction, equals
1N-3.6.2 Recognize the numerals 0 – 9
1N-3.6.3 Demonstrate an understanding of the order to key in numbers and operators
1N-3.6.4 Demonstrate the ability to clear the display, and recognize that this should be done before starting a new calculation
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Finding the total score for a card game
Finding the total price of 3 items ordered from a menu
Finding the change for a purchase
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Strand: Patterns, Functions, and Algebra
Learners engage in problem solving within adult contextual situations by communicating, reasoning, and connecting to the following standards:
Standard 1P-1. Explore, identify, analyze, and extend patterns in mathematical and adult contextual situations | Benchmark: At this level an adult will be expected to: | Enabling Knowledge and Skills | Examples of Where Adults Use It |
1P-1.1 Sort up to 20 objects or lists by color, shape, number, letter, or size.
| 1P-1.1.1 Identify attributes of objects and classify such as shape, size, number and/or size |
Sorting laundry
Sorting bottles for recycling facility
Sorting telephone numbers by area code and figuring which are long distance calls
Shelving stock
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1P-1.2 Recognize and create simple repeating patterns (e.g. color, rhythmic, shape, number, and letter) and identify the unit being repeated.
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1P-1.2.1 Count forward and back by 1's from 1 to 20
1P-1.2.2 Read and write whole numbers from 1 to 100
1P-1.2.3 Skip count by 2’s, 5’s, and 10’s from 1 to 100
1P-1.2.4 Identify odd and even
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Knowing on which side of the hall or street a room or a house is
Counting pennies or 1 dollar bills
Counting nickels or five dollar bills
Counting things 2 at a time
Counting dimes or 10 dollar bills
Counting beats in music
Designing a necklace and describing the assembly rule
Laying tile on a floor
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Standard 1P-2. Articulate and represent number and data relationships using words, tables, graphs, rules, and equations | Benchmark: At this level an adult will be expected to: | Enabling Knowledge and Skills | Examples of Where Adults Use It |
1P-2.1 Explore basic number relationships (e.g., find all the ways numbers to 10 can be written as sums).
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1P-2.1.1 Know all pairs of numbers with totals to 10
1P-2.1.2 Decompose numbers into sums of smaller numbers 17 = 10 + 7
1P-2.1.3 Demonstrate an understanding that 2 + 3 and 3 + 2 yield the same sum; therefore, they are counted once in a list
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Playing card games
Preparing for further study
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Standard 1P-3. Recognize and use algebraic symbols to model mathematical and contextual situations | Benchmark: At this level an adult will be expected to: | Enabling Knowledge and Skills | Examples of Where Adults Use It |
1P-3.1 Use and interpret +, -, and = to represent combining, taking away, and equivalence.
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1P-3.1.1 Demonstrate recognition that + represents operations of combining
1P-3.1.2 Demonstrate recognition that - represents operations of separation
1P-3.1.3 Demonstrate recognition that = represents vocabulary such as: is equal to, is the same as, and gives you.
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Using a four-function calculator to find the total whole dollar amount of a grocery bill
Using a calculator to find how much change you get from a $20.00 bill
Helping children with homework.
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1P-3.2 Understand simple number sentences such as: 9 + 1 = 10 and ___ + 5 = 10 and 8 - 3 = ___ where the ___ represents a missing amount.
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1P-3.2.1 Demonstrate an understanding that an underlined blank space represents a missing value in addition and subtraction equations
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Helping children with homework.
Test taking when seeking employment |
1P-3.3 Make statements of inequality e.g.:
2 is less than 10
10 is greater than 8
99 is less than 100
6 + 5 10
| 1P-3.3.1 Explain that directionality of reading numbers and expressions moves from left to right |
Helping children with homework
Test-taking when seeking employment |
Standard 1P-4. Analyze change in various contexts | Benchmark: At this level an adult will be expected to: | Enabling Knowledge and Skills | Examples of Where Adults Use It |
1P-4.1 Describe qualitative change, such as lengthening or decreasing hours of daylight, or rising or falling of temperature over time.
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1P-4.1.1 Observe physical change over time
1P-4.1.2 Compare changes which go up or increase with those which go down or decrease
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Discussing weather patterns
Describing seasons, daylight savings time, or tides |
Strand: Statistics and Probability
Learners engage in problem solving within adult contextual situations by communicating, reasoning, and connecting to the following standards:
Standard 1S-1. Collect, organize and represent data | Benchmark: At this level an adult will be expected to: | Enabling Knowledge and Skills | Examples of Where Adults Use It |
1S-1.1 Gather data to answer posed questions.
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1S-1.1.1 Demonstrate that observing and asking relevant questions and counting gathered responses can produce answers
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Planning a neighborhood party
Planning what kind of pizza or sandwiches to order for an employee luncheon
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1S-1.2 Group objects or responses by a single criterion.
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1S-1.2.1 Demonstrate an understanding of the concept of categories by grouping items by shape, size, color, or yes or no responses
1S-1.2.2 Know how to count each category for subtotals up to 20
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Keeping track of who will or will not attend party
Sorting stock by size
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Standard 1S-2. Read and interpret data representations | Benchmark: At this level an adult will be expected to: | Enabling Knowledge and Skills | Examples of Where Adults Use It |
1S-2.1 Identify graphs in available resources.
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1S-2.1.1 Explain how graph is a visual representation
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Reading a graph in an ad or poster
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1S-2.2 Extract simple information from a list or two-column table.
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1S-2.2.1 Identify how lists can be ordered in different ways (e.g. alphabetically, numerically, or randomly)
1S-2.2.2 Make a 1-1 correspondence within a row in charts with two columns
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Checking items against a stock list
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1S-2.3 Read values on a bar graph up to 100.
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1S-2.3.1 Skip-count by 2, 5, or 10
1S-2.3.2 Demonstrate an understanding and that the height of the bar is equal to the amount on the axis across from it
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Reading a nutrition graph in a health poster
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1S-2.4 Make comparative statements about relative values on a bar graph.
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1S-2.4.1 Explain how comparative statements such as greater than or less than can be made based on the height of the bars
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Conversing about information contained in newspapers and magazines
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1S-2.5 Connect simple graphs and tables to arguments or statements.
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1S-2.5.1 Demonstrate how to locate titles
1S-2.5.2 Explain that titles indicate subject matter
| Reading a chart or graph in a health pamphlet. |
Standard 1S-3. Describe data using numerical descriptions, statistics, and trend terminology | | | | |
Not applicable at this level.
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| Standard 1S-4. Make and evaluate arguments and statements by applying knowledge of data analysis, bias factors, graph distortions, and context | | | | |
Not applicable at this level.
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Standard 1S-5. Know and apply basic probability concepts | Benchmark: At this level an adult will be expected to: | Enabling Knowledge and Skills | Examples of Where Adults Use It |
1S-5.1 Discuss events as likely or unlikely.
| 1S-5.1.1 Develop an understanding that while some events are impossible, some are certain to happen, and in other events some are more likely to occur than others |
Deciding whether or not to carry an umbrella
Making the call when flipping a coin
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Strand: Geometry and Measurement
Learners engage in problem solving within adult contextual situations by communicating, reasoning, and connecting to the following standards:
Standard 1G-1. Use and apply geometric properties and relationships to describe the physical world and identify and analyze the characteristics of geometric figures | Benchmark: At this level an adult will be expected to: | Enabling Knowledge and Skills | Examples of Where Adults Use It |
1G-1.1 Recognize, name, describe and compare common basic 2-D shapes (square, circle, rectangle, triangle) using everyday language (straight, curved, etc.).
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1G-1.1.1 Identify the names of shapes
1G-1.1.2 Demonstrate an understanding that shape is independent of size and orientation
1G-1.1.3 Show two triangles or two rectangles in different positions and sizes
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Identifying things (e.g. a curved road, a straight highway, a rotary)
Recognizing the shape and meaning of a triangular yield sign and other shapes in buildings and everyday structures
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1G-1.2 Understand the conventions for naming a rectangle by its length and width.
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1G-1.2.1 Demonstrate an understanding that the longer side is called the length.
1G-1.2.2 Demonstrate an understanding that the shorter side is called the width.
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Purchasing window shades or coverings
Describing a rectangular photo or frame; or a room size by its length and width
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Standard 1G-2. Use transformations and symmetry to analyze mathematical situations | Benchmark: At this level an adult will be expected to: | Enabling Knowledge and Skills | Examples of Where Adults Use It |
1G-2.1 Estimating where a line of symmetry falls in a basic shape.
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1G-2.1.1 Demonstrate an understanding concepts of sameness or half-ness
1G-2.1.2 Divide a figure in half
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Cutting a cake in half
Folding objects |
Standard 1G-3. Specify locations and describe spatial relationships using coordinate geometry and other representational systems | Benchmark: At this level an adult will be expected to: | Enabling Knowledge and Skills | Examples of Where Adults Use It |
1G-3.1 Use the cardinal directions to describe where one location is relative to another.
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1G-3.1.1 Know the convention that is North is the opposite direction from South and that East and West are opposite
1G-3.1.2 Explain the difference between vertical and horizontal
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Reading a road sign or route sign which uses north or south, east or west
Making a simple map with cardinal directions
Locating offices, apartments that are labeled with cardinal directions
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1G-3.2 Understand and use location prepositions and everyday language of position appropriately.
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1G-3.2.1 Know the meaning of terms such as left, right, bottom, top, down, up, behind, over, through, etc.
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Assembling a piece of furniture from a diagram
Giving oral directions for getting from one place to another
| Standard 1G-4. Understand measurable attributes of objects and the units, systems, and processes of measurement and apply appropriate techniques, tools, and formulas to determine measurements | Benchmark: At this level an adult will be expected to: | Enabling Knowledge and Skills | Examples of Where Adults Use It |
1G-4.1 Show equivalent amounts of money using different bills and coins.
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1G-4.1.1 Know coin & bill names and values
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Getting out money to pay at the register
Verifying change given at a store
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1G-4.2 Read, record, and use date concepts in common formats.
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1G-4.2.1 Know the months and corresponding numbers, days of week
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Completing forms (birth date, etc.)
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1G-4.3 Read, record, and understand time of the day.
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1G-4.3.1 Count to 60 by 5’s and 10’s
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Reading a bus schedule that uses AM and PM
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1G-4.4 Read analog and digital clocks.
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1G-4. 4.1 Demonstrate an understanding that each hour of digital time is read to 59 minutes
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Looking at clock outside a bank and know if one is on time
| | 1G-4.5 Compares familiar quantities, length, mass, capacity, time, temperature, using informal comparative language and methods (e.g. taller, heavier, smallest). |
1G-4.5.1 Explain how the suffixes –er, -est, and how, more, less, and too will change the quantity
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Sorting by size to organize a kitchen cabinet
Understanding a child’s growth chart |
1G-4.6 Read a ruler to the nearest whole inch.
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1G-4.6.1 Line up the edge of a ruler to measure an object
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Measuring the length and width of photo
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1G-4.7 Begins to develop personal reference points of measure (one’s height, weight).
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1G-4.7.1 Demonstrate a general recognition of common heights and weights for women, men and children
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Give one’s height or weight on a medical form
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1G-4.8 Find the perimeter of rectangles up to 20 units.
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1G-4.8.1 Know that the two lengths are of equal measure and the two widths are of equal measure
1G-4.8.2 Know that the perimeter of a rectangle is equal to the total of the four sides
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Buying weather stripping
Buying wood for a picture frame or baseboard
Finding the length of fencing around a garden
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