Elementary Math

80 base 5 to base 10 llllllll =40

A 5-frame is a row of 5 boxes used in math to:Count up to 5Show parts of 5 (e.g., 2 + 3 = 5)Practice addition and subtractionBuild number sense using visualsA ten-frame is a row of 10 boxes used in math to:Count up to 10Show how numbers make 10 (e.g., 6 + 4 = 10)Practice addition and subtractionHelp with place value and number sense

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.

Polya’s 4-Step Problem-Solving Process Understand the ProblemWhat is the question asking?What information do you have?What do you need to find?Make a PlanChoose a strategy (draw a picture, make a list, use a number sentence, etc.)Think about how to startCarry Out the PlanFollow your steps carefullyShow your math workUse tools if needed (counters, number lines, etc.)Check Your WorkDoes your answer make sense?Can you solve it a different way?Did you answer the question?

1. Know Your BaseIn 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 BlocksLet’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 = 53 units = 3That’s 8 in base ten, but we’ll still work in base five.3. Add the BlocksPut 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 SumAfter regrouping, count how many longs and units you have left.Write the new number in the same base you started with.24₅ + 13₅Build the numbers:2 longs + 4 units1 long + 3 unitsCombine:3 longs, 7 unitsRegroup 5 units → 1 long, 2 units leftTotal: 4 longs, 2 units = 42₅

1. Expanded FormBreak each number into tens and ones, then add those parts.Example:47 + 36→ 40 + 7 + 30 + 6→ (40 + 30) + (7 + 6)→ 70 + 13 = 83Why 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 = 83Why 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 = 83Why 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 = 83Why it helps:It teaches balance in math and is great for quick mental calculations.

How to Show Subtraction with Base Ten BlocksWe 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 = 10Units = 1Steps to Build and Subtract:Step 1: Build the First NumberUse base ten blocks to build the number you're starting with.Example: 42 – 18Build 42 → 4 longs (10s) and 2 units (1s)Step 2: Subtract What You CanTry 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 unitsStep 3: Regroup if NeededRegroup when there aren’t enough units to subtract.Now subtract:8 units from 12 → 4 units left1 long from 3 → 2 longs leftStep 4: Count What’s LeftYou’re done once you’ve subtracted everything.2 longs (20) and 4 units = 24Why This Helps:It makes subtraction visual and hands-onHelps students understand regrouping (borrowing)Builds a strong understanding of place value

1. Subtraction in Expanded FormBreak 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 = 22Why it helps:It shows exactly how much is being taken away from each place value.2. Subtraction Left to RightStart with the biggest place value and move left to right.Example:74 – 52→ 70 – 50 = 20→ 4 – 2 = 2→ 20 + 2 = 22Why it helps:It matches how we read numbers and builds mental math confidence.3. Equal Addends StrategyChange both numbers by the same amount to make subtraction easier.Example:74 – 49→ Add 1 to both numbers: 75 – 50→ 75 – 50 = 25Why it helps:It keeps the difference the same while making the math easier to do in your head.

1. Multiplication as Groups This shows multiplication as a certain number of groups with the same number in each group.Example: 3 × 4Think of it as 3 groups of 4You can draw 3 circles and put 4 dots inside eachVisual:Circle 1 → ⚫⚫⚫⚫Circle 2 → ⚫⚫⚫⚫Circle 3 → ⚫⚫⚫⚫Then count all the dots: 4 + 4 + 4 = 12Why this helps:Great for early multiplicationReinforces repeated addition2. Multiplication as an ArrayThis uses rows and columns to represent multiplication.Example: 3 × 4Draw 3 rows with 4 objects in each (or vice versa)Visual:⚫ ⚫ ⚫ ⚫⚫ ⚫ ⚫ ⚫ ⚫ ⚫ ⚫ ⚫ Count all: 3 rows × 4 columns = 12Why this helps:Builds understanding of the area and structureMakes the commutative property (3×4 = 4×3) visual3. Multiplication with Base Ten Blocks (Area Model)This method uses place value to break apart numbers and multiply them in parts.Example: 13 × 14Step 1: Break each number into tens and ones13 = 10 + 314 = 10 + 4Step 2: Set up the box with one number on the top and one on the side:  10  |  3   ------------------10| 100  | 304  |  40 | 12Step 3: Multiply each part10 × 10 = 10010 × 3 = 304 × 10 = 404 × 3 = 12Step 4: Add all the pieces together100 + 30 + 40 + 12 = 182

1. Multiplication in Expanded FormBreak apart the number using place value, then multiply each part.Example: 23 × 4→ 23 = 20 + 3→ (20 × 4) + (3 × 4)→ 80 + 12 = 92Why it helps:Makes it easier to multiply in stepsReinforces place value2. Multiplication Left to RightMultiply the biggest place value first, then move to the right.Example: 23 × 4→ 20 × 4 = 80→ 3 × 4 = 12→ 80 + 12 = 92Why it helps:Supports mental mathMatches how we read numbers3. Area ModelIn 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 visualReinforces place valueHelps break big problems into smaller steps4. Lattice MethodThis 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 organizedReduces errors when multiplying larger numbersVisual support for place value and regrouping

Steps:Step 1: Build the Total Using UnitsStart with the number you're dividing (the dividend) and represent it using individual units.Step 2: Make Equal GroupsGroup the units based on the divisor. Keep grouping until you can't make a full group anymore.Step 3: Count the GroupsThe number of full groups is your whole-number answer.Step 4: Leftover Units Become a FractionIf some units are left over, write them as a fraction over the divisor.Example 1: Small Numbers (Use Circles)8 ÷ 3Use 8 unitsMake groups of 3→ 3, 3 (2 full groups)Leftover: 2 unitsAnswer: 2 2/3Example 2: Larger Numbers (Base Ten)63 ÷ 12Build 63 using longs (for tens) and unitsGroup the units into sets of 12Leftover units that can’t make a full group become the fraction partWhy this helps:Makes division visual and hands-onReinforces equal groups and the meaning of a remainderShows how remainders can be written as fractions

2 – Number ends in 0, 2, 4, 6, or 83 – Sum of the digits is divisible by 34 – Last 2 digits are divisible by 45 – Number ends in 0 or 56 – Number is divisible by both 2 and 39 – Sum of the digits is divisible by 910 – Number ends in 0Factor TreeYou 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 × 3Prime factors: 2 × 2 × 3 × 3

In a fraction:The numerator (top number) tells the number of pieces we haveThe denominator (bottom number) tells the size of the piece1. Same DenominatorIf 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 NumeratorIf 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

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)