Elementary Mathematics Benchmarks, Grade 1
Number (N)
N.1.1 Understand and use number notation and place value up to 100.
a. Count to 100 by ones and tens.
- Group objects by tens and ones and relate written numerals to counts of the groups by ones, and to counts of the groups by tens.
b. Read and write numbers up to 100 in numerals.
- Understand and use numbers up to 100 expressed orally.
- Write numbers up to 10 in words.
c. Recognize the place value of numbers (tens, ones).
- Recognize the use of digit to refer to the numerals 0 through 9.
- Arrange objects into groups of tens and ones and match the number of groups to corresponding digits in the number that represents the total count of objects.
N.1.2 Compare numbers up to 100 and arrange them in numerical order.
a. Arrange numbers in increasing and decreasing order.
b. Locate numbers up to 100 on the discrete number line.
- Understand that on the number line, bigger numbers appear to the right of smaller numbers.
Note: The discrete number line is not the continuous number line that will be used extensively in later grades, but a visual device for holding numbers in their proper regularly spaced positions The focus in grades K-2 is on the uniformly spaced natural numbers, not on the line that connects them. However, for simplicity, in these grades the discrete number line is often called the number line.
- Use the number line to create visual representations of sequences.
Examples: Even numbers, tens, multiples of five.
- Understand and use relational words such as equal, bigger, greater, greatest, smaller, and smallest, and phrases equal to, greater than, more than, less than, and fewer than.
c. Compare two or more sets of objects in terms of differences in the number of elements.
- Use matching to establish a one-to-one correspondence and count the remainder to determine the size of the difference.
- Connect the meanings of relational terms (bigger, etc.) to the order of numbers, to the measurement of quantities (length, volume, weight, time), and to the operations of adding and subtracting.
Example: If you add something bigger, the result is bigger, but if you take away something bigger, the result will be smaller.
N.1.3 Add, subtract, compose, and decompose numbers up to 100.
a. Be able to solve problems that require addition and subtraction of numbers up to 100 in a variety of ways.
- Know addition and subtraction facts for numbers up to 12.
- Add and subtract efficiently, both mentally and with pencil and paper.
Note: Avoid sums or differences that require numbers greater than 100 or less than 0.
- Be able to explain why the method used produces the correct answer.
Note: Any correct method will suffice; there is no reason to insist on a particular algorithm since there are many correct methods. Common methods include "adding on" (often using fingers) and regrouping to make a 10.
Examples: 6 + 8 = 6 + 4 + 4 = 10 + 4 = 14;
or 6 + 8 = 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14; - Add three single-digit numbers.
Examples: 3 + 4 + 1 = ?; 7 + 5 + 3 = ?
- Understand and solve oral problems with a variety of phrasing, including how many more or how many fewer.
- Know how to use a calculator to check answers.
b. Understand how to compose and decompose numbers.
- Identify and discuss patterns arising from decompositions.
Example: 8 = 7 + 1 = 1 + 7 = 6 + 2 = 2 + 6 = 5 + 3 = 3 + 5 = 4 + 4;
9 = 8 + 1 = 1 + 8 = 7 + 2 = 2 + 7 = 3 + 6 = 6 + 3 = 5 + 4 = 4 + 5. - Represent decomposition situations using terms such as put together, add to, take from, break apart, or compare.
c. Use groups of tens and ones to add numbers greater than 10.
- Using objects or drawings, add the tens, add the ones, and regroup if needed.
Note: Grouping relies on the commutative and associative properties of addition. Examples in early grades foreshadow more formal treatments later. The vocabulary should await later grades.
Examples:
(a) 17 + 24 = 17 + 23 + 1 = 17 + 3 + 20 + 1 = 20 + 20 + 1 = 40 + 1 = 41.
(b) 58 + 40 = 50 + 8 + 40 = 50 + 40 + 8 = 98.
(c) 58 + 6 = 50 + 8 + 6 = 50 + 14 = 50 + 10 + 4 = 60 + 4 = 64.
(d) 58 + 26 = (58 + 2) + (26 - 2) = 60 + 24 = 84.
d. Create and solve addition and subtraction problems with numbers smaller than 20.
- Create and discuss problems using drawings, stories, picture graphs, diagrams, symbols, and open equations (e.g., 4 + ? = 17).
- Use the (discrete) number line to illustrate the meaning of addition and subtraction.
- Express answers in a form (verbal or numerical) that is appropriate to the original problem.
- Always check that answers are intuitively reasonable.
Measurement (M)
M.1.1 Measure length, weight, capacity, time, and money.
a. Use rulers, scales, and containers to measure and compare the dimensions, weight, and capacity (volume) of classroom objects.
Note: The concepts of addition and order are intrinsic in quantities such as length, weight, and volume (capacity). So measuring such quantities provides an independent empirical basis for understanding the properties of numbers that is different from simple counting.
- Round off measurements to whole numbers.
- Recognize the essential role of units in measurement and understand the difference between standard and non-standard units.
Example: An inch or foot marked on a ruler is a standard unit, whereas a paperclip or a classmate's foot used to measure is a non-standard unit.
- Represent addition by laying rods of different lengths end to end, combining items on a balance, and pouring liquids or sand into different containers.
- Estimate lengths with simple approximations.
- Understand and use comparative words such as long, longer, longest; short, shorter, shortest; tall, taller, tallest; high, higher, highest.
b. Tell time from analog (round) clocks in half-hour intervals.
- Use the expressions "o’clock" and "half past."
c. Count, speak, write, add, and subtract amounts of money in cents up to $1 and in dollars up to $10.
- Know the values of U.S. coins (penny, nickel, dime, quarter, dollar bill).
- Use the symbols $ and ¢ separately (e.g., $4, 35¢ instead of $PS.4.35).
- Use coins to decompose monetary amounts given in cents.
Example: 17¢ = one dime, one nickel, and two pennies = 10 + 5 + 1 + 1; or 17¢ = three nickels and two pennies = 5 + 5 + 5 + 1 + 1.
- Understand and solve money problems expressed in a different ways, including how much more or how much less.
Note: Avoid problems that require conversion from cents to dollars or vice versa.
Probability and Statistics (PS)
PS.1.1 Use picture graphs to pose and solve problems.
a. Interpret picture graphs in words (orally) and with numbers.
- Answer questions about the meaning of picture graphs.
b. Create picture graphs of counts and measurements from collected or provided data.
- Represent data both in horizontal and vertical forms.
- Label axes or explain what they represent.
- Pose and answer comparison questions based on picture graphs.
Geometry (G)
G.1.1 Recognize, describe, and draw geometric figures.
a. Identify and draw two-dimensional figures.
- Include trapezoids, equilateral triangles, isosceles triangle, parallelograms, quadrilaterals.
Note: Be sure to include a robust variety of triangles as examples, especially ones that are very clearly not equilateral or isosceles.
- Describe attributes of two-dimensional shapes (e.g., number of sides and corners).
b. Identify and name three-dimensional figures.
- Include spheres, cones, prisms, pyramids, cubes, rectangular solids.
- Identify two-dimensional shapes as faces of three-dimensional figures.
c. Sort geometric objects by shape and size.
- Recognize the attributes that determined a particular sorting of objects and use them to extend the sorting.
Example: Various L-shaped figures constructed from cubes are sorted by the total number of cubes in each. Recognize this pattern, then sort additional figures to extend the pattern.
- Explore simultaneous independent attributes.
Example: Sort triangular tiles according to the four combinations of two attributes, such as right angle and equal sides.
G.1.2 Rotate, invert, and combine geometric tiles and solids.
a. Describe and draw shapes resulting from rotations and flips of simple two-dimensional figures.
- Identify the same (congruent) two-dimensional shapes in various orientations and move one on top of the other to show that they are indeed identical.
- Extend sequences that show rotations of simple shapes.
b. Identify symmetrical shapes created by rotation and reflection.
c. Use geometric tiles and cubes to assemble and disassemble compound figures.
- Count characteristic attributes (lines, faces, edges) before and after assembly.
Examples: Add two right triangles to a trapezoid to make a rectangle; create a hexagon from six equilateral triangles; combine two pyramids to make a cube.
Algebra (A)
A.1.1 Recognize and extend simple patterns.
a. Skip count by 2s and 5s and count backward from 10.
b. Identify and explain simple repeating patterns.
- Find repeating patterns in the discrete number line, in the 12 x 12 addition table, and in the hundreds table (a 10 x 10 square with numbers arranged from 1 to 100).
Note: Use examples based on linear growth (e.g., height, age).
- Create and observe numerical patterns on a calculator by repeatedly adding or subtracting the same number from some starting number.
c. Determine a plausible next term in a given sequence and give a reason.
Note: Without explicit rules, many answers to "next term" problems may be reasonable. So whenever possible, rules for determining the next term should be accurately described. Patterns drawn from number and geometry generally have clear rules; patterns observed in collected data generally do not.
A.1.2 Find unknowns in problems involving addition and subtraction.
a. Understand that addition can be done in any order but that subtraction cannot.
- Demonstrate using objects that the order in which things are added does not change the total, but that the order in which things are subtracted does matter.
- Use the fact that a + b = b + a to simplify addition problems.
Examples: 2 + 13 = 13 + 2 = 15 (by adding on);
7 + 8 + 3 = 7 + 3 + 8 = 10 + 8 = 18.Note: The relation a + b = b + a is known as the commutative property of addition. It reduces significantly the number of addition facts that need to be learned. However, the vocabulary is not needed until later grades.
- Demonstrate understanding of the basic formula a + b = c by using objects to illustrate all eight number sentences associated with any particular sum:
Example: 8 + 6 = 14, 6 + 8 = 14; 14 = 8 + 6, 14 = 6 + 8;
14 – 8 = 6, 6 = 14 – 8; 14 – 6 = 8, 8 = 14 – 6.
A.1.3 Understand how adding and subtracting are inverse operations.
a. Demonstrate using objects that subtraction undoes addition and vice versa.
- Subtracting a number undoes the effect of adding that number, thus restoring the original. Similarly, adding a number undoes the action of subtracting that number.
Example: 2 + 3 = 5 implies 5 – 2 = 3, and 5 – 2 = 3 implies 2 + 3 = 5.
- Use the inverse relation between addition and subtraction to check arithmetic calculations.
Note: Addition and subtraction are said to be inverse operations because subtraction undoes addition and addition undoes subtraction. However, this vocabulary is not needed until later grades.
Caution: Subtraction is sometimes said to be equivalent to "adding the opposite," meaning that 5 – 3 is the same as 5 + –3. Here the "opposite" of a number is intended to mean the negative of a number. However, since negative numbers are not introduced until later grades, this formulation of the relation between addition and subtraction should be postponed.