Elementary Mathematics: Attributes, Problem Structures, Strategies, and Algorithms

Attributes and Classification

• Attribute: A feature or characteristic used to describe an object.

  • Non-mathematical attribute: E.g., the color of a shape.

  • Mathematical attribute: E.g., the size of angles in a shape.

• Attributes are used in elementary grades for classifying shapes.

Classification Games

1. "I'm Thinking of a Block"

• Objective: One person thinks of an attribute block, and the other asks "yes" or "no" questions to identify it.

• Example Questions: "Is the block a quadrilateral?" "Is it big or small?" "What color is it?"

• Informative Questions: Questions about "sides, equal lengths, big or small, color" tend to provide the most information for identification.

2. Attribute Train

• Objective: Create a train of attribute blocks where each subsequent block differs from the preceding one by a predetermined number of attributes (e.g., color, size, shape).

• One-Difference Train: Each block must differ by exactly one attribute from the previous block.

  • Example: If the first block is a large blue square:

    • Second block could be a small blue square (size difference).

    • Third block could be a small red square (color difference).

    • Fourth block could be a small red triangle (shape difference).

• Variations: Can play with a one-difference, two-difference, or three-difference train.

• Challenge: Use all blocks in the attribute set.

Understanding Differences and Similarities

• Activities often involve distinguishing items that are "different" versus "the same" based on attributes.

• The goal is to help students match easier by clarifying how items vary or are consistent.

• Example classifications: Grouping "all circles no matter the size or color" or "everything that's blue."

• Key attributes for classification: Sides, objects, symmetry, length.

Problem Structures: Addition and Subtraction

Mathematical story problems are classified by their structure.

Join Problems

• Involve an action of increasing a quantity.

• Types of Unknowns:

  • Result Unknown: A quantity is increased, and the final total is unknown.

    • Example: $3 + 2 = ext{ ?}$ (If John has 3 apples and gets 2 more, how many does he have?)

  • Change Unknown: A starting quantity is known, the final total is known, but the amount added is unknown.

    • Example: $3 + ext{ ?} = 5$ (John has 3 apples, then Mark gave him some more. Now John has 5 apples. How many did Mark give him?)

    • Example from transcript: $27 + ext{ ?} = 40$

  • Start Unknown: The initial quantity is unknown, but the amount added and the final total are known.

    • Example: $ext{ ?} + 2 = 5$ (Mary had some cookies, then ate 2. Now she has 5. How many did she start with? - No, this is Separate. Wait. The example given in the transcript is w + 9 = 23 which fits Join, Start Unknown)

    • Example from transcript: $ext{ ?} + 9 = 23$

Separate Problems

• Involve an action of decreasing a quantity (subtraction).

• Types of Unknowns:

  • Result Unknown: A quantity is decreased, and the remaining amount is unknown.

    • Example: $13 - 5 = ext{ ?}$ (I had 13 stickers and gave away 5. How many do I have left?)

  • Change Unknown: A starting quantity is known, the final amount is known, but the amount taken away is unknown.

    • Example: $13 - ext{ ?} = 5$ (I had 13 stickers and gave some away. Now I have 5. How many did I give away?)

    • Example from transcript: $40 - ext{ ?} = 27$

  • Start Unknown: The initial quantity is unknown, but the amount taken away and the final amount are known.

    • Example: $ext{ ?} - 2 = 9$ (I had some cookies, and ate 2. Now I have 9. How many did I start with?)

    • Example from transcript: $ext{ ?} - 6 = 16$

Part-Part-Whole Problems

• Focus on the relationship between parts that make up a whole, without an implied or actual action of joining or separating.

  • Whole Unknown: Two parts are known, and the total whole is unknown.

    • Example: Mary has 3 red apples and 4 green apples. How many apples does she have in all?

  • Part Unknown: The whole and one part are known, and the other part is unknown.

    • Example: Mary has 7 apples. 3 are red, and the rest are green. How many green apples does she have?

Compare Problems

• Involve comparing two quantities to find a difference.

  • Difference Unknown: Two quantities are known, and the difference between them is unknown.

    • Example: Kevin has 5 stickers. James has 13 stickers. How many more stickers does James have than Kevin? ($13 - 5 = ext{ ?}$ or $5 + ext{ ?} = 13$)

Addition Strategies

Strategies are approaches to compute quickly and easily, often mentally.

1. Counting

• Finger Counting: A basic method for young children.

• Counting On/Counting Down: Used for both addition and subtraction.

2. Doubles Strategy

• Relating an addition problem to known doubles facts.

• Example: For $7 + 8$

  • $(7 + 7) + 1 = 14 + 1 = 15$

  • $(8 + 8) - 1 = 16 - 1 = 15$

3. Making 10 Strategy

• Decomposing one addend to make a 10 with the other addend.

• Example: For $6 + 9$

  • $9 + 1 = 10$ (take 1 from 6)

  • $6 - 1 = 5$

  • Then, $10 + 5 = 15$

• Mathematical Concept: This strategy utilizes the Associative Property of Addition ($(A+B)+C = A+(B+C)$) to regroup numbers for easier computation (e.g., $6+9 = (5+1)+9 = 5+(1+9) = 5+10 = 15$).

4. Decomposition/Recomposition

• "Taking apart" and/or rearranging numbers to make a problem easier to solve.

• Splitting Notation: ^ indicates decomposing, V indicates joining.

• By Place Value: Breaking addends into their place value components.

  • Example: $47 + 19$

    • $(40 + 7) + (10 + 9)$

    • $(40 + 10) + (7 + 9)$

    • $50 + 16 = 66$

• To Make Friendly Numbers: Breaking apart one addend to combine with the other to create a number with a zero in the ones place (a "friendly number").

  • Example: $47 + 19$

    • $47 + (3 + 16)$ (decomposing 19 into 3 and 16 to make 47 a 50)

    • $(47 + 3) + 16$

    • $50 + 16 = 66$

  • Alternatively: $(46 + 1) + 19$ (decomposing 47 instead)

    • $46 + (1 + 19)$

    • $46 + 20 = 66$

5. Compensation Strategy

• Changing a problem to make it easier by adding a number to one addend and later subtracting it (or balancing it out between addends) so the sum is not affected.

• Example: $37 + 159$

  • Make 37 a "friendly number" by adding 3: $(37 + 3) = 40$

  • Add 1 to 159 to make it 160: $(159 + 1) = 160$

  • Since 3 was added to 37 and 1 was added to 159, a total of $3+1=4$ was added. To compensate, subtract 4 from the sum.

  • $(37 + 3) + (159 + 1) - 4 = (40 + 160) - 4 = 200 - 4 = 196$

• Mathematical Concept: This strategy relies on the property that $(A+x) + (B-x) = A+B$ or $(A+x) + (B+y) - (x+y) = A+B$. It's related to the idea of maintaining the balance of the expression.

Addition Algorithms

An algorithm is a step-by-step procedure for solving computations using pencil and paper, applicable to all problems of a certain type. They were developed in the 17th century as an alternative to tools like the abacus, to promote efficient and accurate computation.

1. U.S. Standard Addition Algorithm

• A common paper-and-pencil method.

• Steps: Add digits column by column, starting from the ones place, and regrouping as necessary.

• Key Idea: Regrouping (instead of "Carrying")

  • The term "regrouping" emphasizes that equivalent quantities are being exchanged (e.g., ten ones are regrouped into one ten).

  • The term "carrying" can be confusing as it might suggest moving digits based on face value rather than the quantity they represent.

  • Example: For $28 + 15$

    1. Add the ones: $8 ext{ ones} + 5 ext{ ones} = 13 ext{ ones}$. (Regroup 13 ones into 1 ten and 3 ones.)

    2. Write down 3 in the ones column.

    3. Regroup the 1 ten to the tens column (above the 2 and 1).

    4. Add the tens: $1 ext{ ten} + 2 ext{ tens} + 1 ext{ ten} = 4 ext{ tens}$. (The regrouped 1 represents 1 ten, not 1 one.)

    5. Write down 4 in the tens column.

    6. Result: 43.

• Modeling Regrouping: Can be modeled with base ten blocks (dots for ones, lines for tens, squares for hundreds) to visually demonstrate that 10 ones are indeed equivalent to 1 ten, etc.

2. Partial Sums Algorithm

• Involves adding the values in each place value column separately and then summing these partial sums.

• Steps:

  1. Add the ones digits.

  2. Add the tens digits (remembering their place value, e.g., 7 + 9 becomes $70 + 90$).

  3. Add the hundreds digits (e.g., $400 + 300$).

  4. Sum the partial sums obtained from each place value.

• Example: For $476 + 398$

  • $6 + 8 = 14$ (ones)

  • $70 + 90 = 160$ (tens)

  • $400 + 300 = 700$ (hundreds)

  • $14 + 160 + 700 = 874$

• Order of Operations: It does not matter if you start from the ones place (right) or the hundreds place (left) when using partial sums, due to the Commutative and Associative Properties of Addition.

3. Lattice Algorithm

• An alternative algorithm often learned in elementary school.

• Steps:

  1. Draw a grid or lattice with diagonal lines.

  2. Write the addends, one above and one to the right of the grid.

  3. For each pair of digits being added, write their sum in the corresponding square, with the tens digit above the diagonal and the ones digit below.

  4. Sum the numbers along the diagonals, starting from the bottom right, carrying over to the next diagonal as needed.

• Mathematical Concept: The diagonals inherently account for place value, as each diagonal represents a specific place value column (ones, tens, hundreds, etc.).

4. Low-Stress Algorithm

• Useful for adding more than two addends.

• Steps:

  1. Start adding digits in the ones column.

  2. Any time a sum reaches 10 or more, draw a small bar (representing 10) and write down only the ones digit of the partial sum below. Continue adding the next digit to this new ones digit.

  3. Once the column is finished, write the final ones digit. Count the number of bars drawn in that column and add this count to the next place value column (the tens column).

  4. Repeat the process for the tens, hundreds, and subsequent columns.

• Benefit: This streamlines mental calculation by breaking down larger sums into smaller, manageable chunks, avoiding larger multi-digit intermediate sums.

Properties of Addition

• Associative Property of Addition: The sum of several addends does not change when the addends are grouped differently.

  • Example: $(1 + 2) + 3 = 1 + (2 + 3)$

• Commutative Property of Addition: The sum of two or more addends does not change when they are in a different order.

  • Example: $2 + 3 = 3 + 2$

Subtraction Strategies

Children use various strategies to solve subtraction problems.

1. Counting Strategies

• Counting Down (Take Away): Starting from the minuend and counting back the subtrahend quantity.

  • Example: For $13 - 5$, count down from 13: $12, 11, 10, 9, 8$. Result is 8.

  • Number Line Model: Start at 13, make 5 jumps to the left, landing on 8.

• Counting Up (Adding Up/Missing Addend): Starting from the subtrahend and counting forward to the minuend, keeping track of the amount added.

  • Example: For $13 - 5$, count up from 5 to 13: $5 o 6 (+1), 6 o 7 (+1), ext{…}, 12 o 13 (+1)$. Total added is 8.

  • Number Line Model: Start at 5, make jumps to the right until 13 is reached. The sum of the lengths of the jumps is the difference (e.g., $5 o 10 (+5), 10 o 13 (+3)$. Total $5+3=8$).

• Why both work: Subtraction can be viewed as finding a missing addend (e.g., $A - B = ext{ ?}$ is equivalent to $B + ext{ ?} = A$).

2. Decomposition Strategy

• Breaking apart numbers to make subtraction easier, often by place value or friendly numbers.

• Example: For $56 - 32$

  • $(56 - 30) - 2 = 26 - 2 = 24$

3. Compensation Strategy

• Changing the numbers in a subtraction problem to make it easier, while ensuring the difference remains constant or is adjusted correctly.

• Example of Constant Difference: For $53 - 18$

  • Add 2 to both the minuend and subtrahend to make the subtrahend a friendly number (multiple of 10).

  • $(53 + 2) - (18 + 2) = 55 - 20 = 35$

• Mathematical Concept: This works because adding or subtracting the same amount from both the minuend and subtrahend does not change the difference: $(A+x) - (B+x) = A - B$ and $(A-x) - (B-x) = A - B$. This is sometimes called the Constant Difference Property.

• Example from transcript: For $63 - 49$, Catherine's method: $(63 - 50) + 1 = 13 + 1 = 14$. This is not constant difference, it's adjusting the subtrahend to a friendly number (50) and then adding back the amount that was 'over-subtracted'.

4. Partial Differences Algorithm

• Subtracting in chunks, often from left to right.

• Example: For $1004 - 726$

  • $1004 - 700 = 304$

  • $304 - 20 = 284$

  • $284 - 6 = 278$

• Number Line Model: Represented by sequential jumps backward on a number line.

Subtraction Algorithms

1. U.S. Standard Subtraction Algorithm

• A common paper-and-pencil method.

• Key Idea: Regrouping (instead of "Borrowing")

  • Use "trade" or "regroup" instead of "borrow" to accurately reflect the mathematical action of exchanging equivalent quantities (e.g., 1 ten for 10 ones). The term "borrowing" implies that the quantity taken must be returned, which is not the case in place value regrouping.

  • Example: For $236 - 89$

    1. Start at the ones place: Can't subtract 9 from 6. Regroup 1 ten from the tens place. The 3 tens become 2 tens, and the 6 ones become 16 ones.

    2. Subtract ones: $16 - 9 = 7$. Write 7 in the ones column.

    3. Move to the tens place: Can't subtract 8 tens from the remaining 2 tens. Regroup 1 hundred from the hundreds place. The 2 hundreds become 1 hundred, and the 2 tens become 12 tens.

    4. Subtract tens: $12 - 8 = 4$. Write 4 in the tens column.

    5. Move to the hundreds place: Subtract 0 hundreds from the remaining 1 hundred. $1 - 0 = 1$. Write 1 in the hundreds column.

    6. Result: 147.

• Modeling Regrouping: Base ten blocks or drawings (short-hand notation: dots for ones, lines for tens, squares for hundreds) help visualize the exchange of equivalent quantities.

• Regrouping Across Zeros: When a digit in a place value from which you need to regroup is zero (e.g., hundreds place in 407), you must regroup from the next higher non-zero place value (e.g., the thousands place).

2. Equal Additions Algorithm

• This algorithm applies the Constant Difference Property by adding the same amount to both the minuend and the subtrahend to simplify the subtraction (e.g., to avoid regrouping in standard algorithm, or to make the numbers easier to work with).

• Example: For $703 - 187$

  • To make the subtrahend a friendly number (200), add 13 to both the subtrahend and minuend.

  • $(703 + 13) - (187 + 13) = 716 - 200 = 516$

• In standard visual representation for subtraction: This often looks like adjustments made to the digits in specific columns in both the minuend and subtrahend.

  • Example: For $345 - 178$

    1. To subtract 8 from 5: Add 10 to the 5 (making it 15). To maintain constant difference, add 1 to the 7 (tens place of subtrahend).

    2. Now we have 15 - 8 = 7.

    3. In the tens place: We need to subtract 8 (original 7 plus the added 1) from 4. Add 10 to the 4 (making it 14). To maintain constant difference, add 1 to the 1 (hundreds place of subtrahend).

    4. Now we have 14 - 8 = 6.

    5. In the hundreds place: We need to subtract 2 (original 1 plus the added 1) from 3.

    6. Now we have 3 - 2 = 1.

    7. Result: 167.

Vocabulary for Subtraction

• Minuend: The number from which another number is to be subtracted.

  • Example: In $5 - 1 = 4$, $5$ is the minuend.

• Subtrahend: The number which is subtracted from another.

  • Example: In $5 - 1 = 4$, $1$ is the subtrahend.

• Difference: The result of subtracting one number from another.

  • Example: In $5 - 1 = 4$, $4$ is the difference.

Large Numbers and Place Value

Our number system makes numbers more manageable by breaking them into groups called periods.

Periods of Numbers

• Trillions Period

• Billions Period

• Millions Period

• Thousands Period

• Ones Period

Powers of Ten

• Million: $1,000,000 = 10^6$

• Billion: $1,000,000,000 = 10^9$

• Trillion: $10^{12}$

• Quadrillion: $10^{15}$

Contextualizing Large Numbers

• Understanding very large numbers (e.g., one million, one billion) requires relating them to real-world contexts or visual representations (e.g., the weight of a million M&Ms, the height of a stack of 100 sticks).

• Example Calculations for 1,000,000 M&Ms:

  • If you eat 6 M&Ms per minute, 8,929 per day, 62,500 per week, and 16,372 per hour (this rate for an hour is inconsistent with minutes/day/week), it would take approximately $203 ext{ days}$ (or $2737 ext{ days}$ if eating 365 per day) to eat a million M&Ms.

  • 1,000,000 ext{ M&Ms} / 32 ext{ grams/bag} imes 0.89 ext{ extdollar{}/bag} = 17,115 ext{ extdollar{}} (cost estimation, details are vague).

  • If you eat 365 M&Ms per day: $1,000,000 / 365 ext{ days/year} imes 1 ext{ year} = 2739.7 ext{ days}$. Approximately $7.5 ext{ years}$.

• Number Line Representation: Number lines can represent very large numbers, with appropriate scaling (e.g., increments of 250,000,000). The perception of