Elementary Math

Elementary Math

Organize your paragraphs using this mind map. It will help you map out all details related to a topic, so that you can organize them in coherent sentences.

Week 4

EXAM #2

Build Fractions/Manipulatives

• Fraction tiles (flat strips in pieces like 1/2, 1/3, 1/4, etc.)
• Fraction circles (often used because we naturally recognize missing portions of a circle)
• Fraction squares (growing in popularity for comparing fraction sizes)
• Pattern blocks (shapes that can be used for fraction activities)
• Colored counters or linking cubes (used for general modeling)
• Fraction rods and color rods (different sizes and styles, some with shaded boxes for fractions)

Understanding/Comparing Fractions

In a fraction:

• The numerator (top number) tells the number of pieces we have
• The denominator (bottom number) tells the size of the piece

1. Same Denominator

If the denominators are the same, just compare the numerators.

Example:

3/8 vs. 5/8 → 5/8 is greater (because 5 pieces is more than 3 when the size is the same)

2. Same Numerator

If the numerators are the same, the fraction with the smaller denominator is greater (because the pieces are bigger).

Example:

3/5 vs. 3/8 → 3/5 is greater (because fifths are bigger than eighths)

3. Missing Pieces

Look at how many pieces are missing to reach the whole amount.

Example:

5/6 is missing 1 piece (1/6)

7/8 is missing 1 piece (1/8)

Since 1/8 is smaller than 1/6, 7/8 is closer to 1 → 7/8 is greater

Week 3

Divisibility Rules/Prime Factorization

2 – Number ends in 0, 2, 4, 6, or 8

3 – Sum of the digits is divisible by 3

4 – Last 2 digits are divisible by 4

5 – Number ends in 0 or 5

6 – Number is divisible by both 2 and 3

9 – Sum of the digits is divisible by 9

10 – Number ends in 0

Factor Tree

• You start with the number and break it down into any two factors.
• Then you keep breaking down each branch until all the numbers are prime.
• Circle the prime numbers and multiply them to get the prime factorization.

Example:

36

→ 6 × 6

→ 2 × 3 and 2 × 3

Prime factors: 2 × 2 × 3 × 3

Division Alt Algorithms

Build/Show Division

Steps:

Step 1: Build the Total Using Units

Start with the number you're dividing (the dividend) and represent it using individual units.

Step 2: Make Equal Groups

Group the units based on the divisor. Keep grouping until you can't make a full group anymore.

Step 3: Count the Groups

The number of full groups is your whole-number answer.

Step 4: Leftover Units Become a Fraction

If some units are left over, write them as a fraction over the divisor.

Example 1: Small Numbers (Use Circles)

8 ÷ 3

• Use 8 units
• Make groups of 3
• → 3, 3 (2 full groups)
• Leftover: 2 units
• Answer: 2 2/3

Example 2: Larger Numbers (Base Ten)

63 ÷ 12

• Build 63 using longs (for tens) and units
• Group the units into sets of 12
• Leftover units that can’t make a full group become the fraction part

Why this helps:

• Makes division visual and hands-on
• Reinforces equal groups and the meaning of a remainder
• Shows how remainders can be written as fractions

Multiplication Alt Algorithms

1. Multiplication in Expanded Form

Break apart the number using place value, then multiply each part.

Example: 23 × 4

→ 23 = 20 + 3

→ (20 × 4) + (3 × 4)

→ 80 + 12 = 92

Why it helps:

• Makes it easier to multiply in steps
• Reinforces place value

2. Multiplication Left to Right

Multiply the biggest place value first, then move to the right.

Example: 23 × 4

→ 20 × 4 = 80

→ 3 × 4 = 12

→ 80 + 12 = 92

Why it helps:

• Supports mental math
• Matches how we read numbers

3. Area Model

In this strategy, each number is broken into its place value parts (tens and ones). You draw a box or grid and label the top and sides with those parts. Then, you multiply each pair of numbers and write the answers in each section of the box. Finally, you add all the partial products together to get the total.

Why it helps:

• Makes multiplication visual
• Reinforces place value
• Helps break big problems into smaller steps

4. Lattice Method

This strategy uses a grid to organize the multiplication of each digit. You draw a box divided into diagonals and label the digits of the numbers on the top and sides. Each box shows one multiplication, with tens written in the top triangle and ones in the bottom. After all parts are filled in, you add diagonally and combine the sums to get the final answer.

Why it helps:

• Keeps work organized
• Reduces errors when multiplying larger numbers
• Visual support for place value and regrouping

Build/Show Multiplication

1. Multiplication as Groups

This shows multiplication as a certain number of groups with the same number in each group.

Example: 3 × 4

• Think of it as 3 groups of 4
• You can draw 3 circles and put 4 dots inside each

Visual:

Circle 1 → ⚫⚫⚫⚫

Circle 2 → ⚫⚫⚫⚫

Circle 3 → ⚫⚫⚫⚫

Then count all the dots: 4 + 4 + 4 = 12

Why this helps:

• Great for early multiplication
• Reinforces repeated addition
• 
  

2. Multiplication as an Array

This uses rows and columns to represent multiplication.

Example: 3 × 4

• Draw 3 rows with 4 objects in each (or vice versa)

Visual:

⚫ ⚫ ⚫ ⚫

⚫ ⚫ ⚫ ⚫

⚫ ⚫ ⚫ ⚫

Count all: 3 rows × 4 columns = 12

Why this helps:

• Builds understanding of the area and structure
• Makes the commutative property (3×4 = 4×3) visual

3. Multiplication with Base Ten Blocks (Area Model)

This method uses place value to break apart numbers and multiply them in parts.

Example: 13 × 14

Step 1: Break each number into tens and ones

13 = 10 + 3

14 = 10 + 4

Step 2: Set up the box with one number on the top and one on the side:

  10  |  3

   ------------------

10| 100  | 30

4  |  40 | 12

Step 3: Multiply each part

• 10 × 10 = 100
• 10 × 3 = 30
• 4 × 10 = 40
• 4 × 3 = 12

Step 4: Add all the pieces together

100 + 30 + 40 + 12 = 182

Week 2

EXAM #1

Subtraction Alt Algorithms

1. Subtraction in Expanded Form

Break both numbers into tens and ones, then subtract each part.

Example:

74 – 52

→ 70 + 4 and 50 + 2

→ 70 – 50 = 20

→ 4 – 2 = 2

→ 20 + 2 = 22

Why it helps:

It shows exactly how much is being taken away from each place value.

2. Subtraction Left to Right

Start with the biggest place value and move left to right.

Example:

74 – 52

→ 70 – 50 = 20

→ 4 – 2 = 2

→ 20 + 2 = 22

Why it helps:

It matches how we read numbers and builds mental math confidence.

3. Equal Addends Strategy

Change both numbers by the same amount to make subtraction easier.

Example:

74 – 49

→ Add 1 to both numbers: 75 – 50

→ 75 – 50 = 25

Why it helps:

It keeps the difference the same while making the math easier to do in your head.

Build/Show Subtraction

How to Show Subtraction with Base Ten Blocks

We use base ten blocks (longs and units) to help students see what subtraction really means, taking away, regrouping, and comparing amounts.

Base Ten Blocks You’ll Use:

• Longs = 10
• Units = 1

Steps to Build and Subtract:

Step 1: Build the First Number

Use base ten blocks to build the number you're starting with.

Example: 42 – 18

• Build 42 → 4 longs (10s) and 2 units (1s)

Step 2: Subtract What You Can

Try to subtract the units (ones) and longs (tens) from what you built.

In this example:

• You need to take away 8 units, but you only have 2.
• So you regroup one long (10) into 10 units.

Now you have:

• 3 longs (30) and 12 units

Step 3: Regroup if Needed

Regroup when there aren’t enough units to subtract.

Now subtract:

• 8 units from 12 → 4 units left
• 1 long from 3 → 2 longs left

Step 4: Count What’s Left

You’re done once you’ve subtracted everything.

• 2 longs (20) and 4 units = 24

Why This Helps:

• It makes subtraction visual and hands-on
• Helps students understand regrouping (borrowing)
• Builds a strong understanding of place value

Addition Alt Algorithms

1. Expanded Form

Break each number into tens and ones, then add those parts.

Example:

47 + 36

→ 40 + 7 + 30 + 6

→ (40 + 30) + (7 + 6)

→ 70 + 13 = 83

Why it helps:

It reinforces place value and shows where each part of the sum comes from.

2. Left to Right

Start by adding the biggest place values first (usually tens), then move right.

Example:

47 + 36

→ 40 + 30 = 70

→ 7 + 6 = 13

→ 70 + 13 = 83

Why it helps:

It’s how we usually read numbers and build mental math skills.

 3. Friendly Numbers

Change one number to a “friendly” number (like a multiple of 10), then adjust.

Example:

47 + 36

→ Think: 47 + 3 = 50 (add 3 to make a friendly number)

→ Then: 36 - 3 = 33

→ Now: 50 + 33 = 83

Why it helps:

It makes numbers easier to work with mentally, especially for rounding or regrouping.

 4. Trade-Off Strategy

Add to one number and subtract from the other so the total stays the same.

Example:

47 + 36

→ Add 3 to 47 (now it’s 50)

→ Subtract 3 from 36 (now it’s 33)

→ 50 + 33 = 83

Why it helps:

It teaches balance in math and is great for quick mental calculations.

Build/Show Addition

1. Know Your Base

• In base five, each long equals 5, and each unit equals 1.
• In base six, each long equals 6, and each unit still equals 1.
• So instead of regrouping when you reach 10 (like in base ten), you regroup when you reach 5 or 6, depending on the base.

2. Build Each Number with Base Blocks

Let’s say you have 24 in base five (24₅). That means:

• 2 longs (2 × 5 = 10)
• 4 units (4 × 1 = 4)
• Together, that’s 14 in base ten, but we’re staying in base five.

Another number, 13₅, would be:

• 1 long = 5
• 3 units = 3
• That’s 8 in base ten, but we’ll still work in base five.

3. Add the Blocks

Put all the longs and units together:

• Count how many total units you have.
• If you have enough units to make one long (like 5 in base five), regroup.
• Example: 7 units in base five = 1 long and 2 units left over.

4. Regroup and Record the Sum

After regrouping, count how many longs and units you have left.

Write the new number in the same base you started with.

24₅ + 13₅

• Build the numbers:
• 2 longs + 4 units
• 1 long + 3 units
• Combine:
• 3 longs, 7 units
• Regroup 5 units → 1 long, 2 units left
• Total: 4 longs, 2 units = 42₅

Week 1

Ploya 4-Step Process

Polya’s 4-Step Problem-Solving Process

1. Understand the Problem

• What is the question asking?
• What information do you have?
• What do you need to find?

1. Make a Plan

• Choose a strategy (draw a picture, make a list, use a number sentence, etc.)
• Think about how to start

1. Carry Out the Plan

• Follow your steps carefully
• Show your math work
• Use tools if needed (counters, number lines, etc.)

1. Check Your Work

• Does your answer make sense?
• Can you solve it a different way?
• Did you answer the question?

Converting Bases

24 ----> l l ....

24 to base six ---> ll ....

Converting to Base 10

14 base five ---> l .... 5+4=9

26 to base 12 ---> ll ....... 12+12+6=30

*Base states the number of units in each shape.

5-10 Frames

A 5-frame is a row of 5 boxes used in math to:

• Count up to 5
• Show parts of 5 (e.g., 2 + 3 = 5)
• Practice addition and subtraction
• Build number sense using visuals

A ten-frame is a row of 10 boxes used in math to:

• Count up to 10
• Show how numbers make 10 (e.g., 6 + 4 = 10)
• Practice addition and subtraction
• Help with place value and number sense

Intro to Building Different Bases

80 base 5 to base 10

llllllll =40