Elementary Mathematics

what some of the earliest stages of number sense arehow to apply a problem solving process.

Base Ten Blocks, 2-Color Counters and a Fraction Manipulative

Ploya's 4-step Problem Solving Process (method to teach students): 1) **Understand the problem**   - Reread the problem   - Explain the problem to someone else   - Break down the parts of the problem 2) **Develop a plan**   - Relate the problem to prior knowledge   - Identify similarities/differences to other problems   - Brainstorm ideas 3) **Carry out the plan**   - Try the plan   - Try other plans 4) **Look back to see if your answer makes sense**   - Is it reasonable?   - Solve another way to see if you get the same answer   - Work backward using your answer  Example: - Two men drive past a farm that has pigs and chickens in a field. - One man says, "I see 50 feet in that field," and the other says, "I see 18 animals." - How many pigs and chickens are there? Possible methods to solve: 1) Guess & Check  - Helps students learn how to write equations.   Example:   - Feet: pig (4 feet) + chicken (2 feet)   - Guess: 5 pigs and 13 chickens   - Equation: 20+ 26= 46 feet2) Diagram (needs total 18 animals)   - Helps students see and understand visual processes.   - Example: use visual representations like circles, dots, or illustrations. 3) Lists| Pigs (P) | Chickens (C) | Pig ft (4P) | Chicken ft (2C) | Total | |----------|--------------|-------------|-----------------|-------| | 1    | 17      | 4      | 34       | 38  | | 2    | 16      | 8      | 32       | 40  | | 3    | 15      | 12     | 30       | 42  | | 4    | 14      | 16     | 28       | 44  | Purpose:- Sets patterns to help students reason. - Helps them solve using patterns. 4) Algorithm- P= pigs - C= chickens **Bodies: (P + C = 18) **Feet: (4P + 2C = 56) Solution: Use two equations to solve. 

Building Addition with Base Ten Blocks ( notes do now show images of base blocks, use notebook for visuals)Using base-ten blocks to demonstrate regrouping:Helps students group sizes together to demonstrate place values and add values.ExamplesExample: 4 + 3 = 7 unitsUse 4 units to fill the 10 frame.Then, add 3 units to the 10 frame.Students count the total units out of 10.Example: 6 + 7Use 6 units and then 7 units.Exchange 10 units for 1 "long" (representing tens), leaving 3 units remaining.Combining Shapes to Represent Larger NumbersExample: 23 + 42Break down the numbers into tens and ones:2 tens (20) + 4 tens (40) = 6 tens (60).3 ones + 2 ones = 5 ones.Total: 65Example: 74 + 29Break down into tens and ones:7 tens (70) + 2 tens (20) = 9 tens (90).4 ones + 9 ones = 13 ones. Exchange 10 ones for 1 "long," leaving 3 ones.Total: 103Rules for Using Base Ten Blocks10 units = 1 long (representing tens).10 longs = 1 flat (representing hundreds).Students group, exchange, then add.Advanced ExamplesExample: 247 + 185Break down into hundreds, tens, and ones:Hundreds: 2 flats (200) + 1 flat (100) = 4 flats (400).Tens: 4 longs (40) + 8 longs (80) = 12 longs. Exchange 10 longs for 1 flat, leaving 2 longs.Ones: 7 units + 5 units = 12 units. Exchange 10 units for 1 long, leaving 2 units.Total: 432Example: 143 + 235Break down:Hundreds: 1 flat (100) + 2 flats (200) = 3 flats.Tens: 4 longs (40) + 3 longs (30) = 7 longs.Ones: 3 units + 5 units = 8 units.Total: 378Bases in any other number, not including 10;ex: In base 6, a Flat is only worth 6 by 6; a Long is only worth 6; a Unit is only worth 1.Notes for TeachingUse clear visual aids to show how units are grouped and exchanged.Encourage students to verbalize their thought process while grouping and exchanging.Practice with different numbers to reinforce understanding of place value and regrouping concepts.

Showing Addition (how do we draw)□ = flats | = long • = unit4 + 3• • • • + • • • • = 7 • • • • • • • → make it vertical to match our long instead5 + 8 • • • • • + • • • • • • • •Combine into a 10-frame: | • • • = 1 long, 3 units = 1336 + 17 | | | • • • • • • + | • • • • • • •Combine: | | | | • • • • • • + • • • • (from 7) this creates a long and • • • left over | | | | | • • • = 5 longs, 3 units = 53423 + 159□ □ □ □ | | • • • + □ | | | | | • • • • • • • • •Combine: □ □ □ □ □ | | | | | | | • • • • • • • • • + • (from 3) this creates long and • • left over□ □ □ □ □ | | | | | | | | • • = 5 flats, 8 longs, 2 units = 582286 + 597 □ □ | | | | | | | | • • • • • • + □ □ □ □ □ | | | | | | | | | • • • • • • •Combine: □ □ □ □ □ □ □ | | | | | | | | | + | (from 8) this creates a flat and | | | | | | | left over• • • • • • • + • • • (from 6) this creates a long and • • • left over□ □ □ □ □ □ □ □ | | | | | | | | • • •= 8 flats, 8 longs, 3 units = 883With your students: Transition from building → drawing to demonstrate quicker and concise steps:Colors and drawing help students to see connections of converting and groupingOnce they see connections, they are ready for algorithms(Make sure drawings are colored, unlike these!)

Alternative Addition AlgorithmsUnderstanding Traditional vs. Alternative AlgorithmsTraditional Algorithm Issues:Can be confusing for students due to carrying values.Example:237 + 185Students may struggle with not writing left-to-right.What Makes a Good AlgorithmExpandable: Works for all numbers without changes.Efficient: Quick and easy to use.Based on Math Sense: Reinforces mathematical principles rather than memorizing steps.Alternative Addition Strategies1. Left-to-Right AdditionMaintains place value and minimizes rewriting.Example:495 + 223400 + 200 = 60090 + 20 = 1105 + 3 = 8Total: 6182. Friendly Numbers (Rounding to Nearest Tens)Adjust numbers to make addition easier.Example:32 + 58Round to 30 + 60 = 90Example:364 + 136Round to 360 + 140 = 5003. Trading-Off (Rearrange for Simplicity)Adjust numbers by taking what is needed.Example:26 + 6723 + 70 = 93Example:539 + 285534 + 290 = 8244. Scratch MethodHelps students keep track of numbers mentally by adding up to 10 and marking a scratch.Example:67 + 73 → Scratch and count groups of 10.5. Lattice AdditionAllows numbers to be added in any order.Requires vertical alignment.Example:426 + 159 using a lattice grid.5. Expanded FormBreaks numbers down into place values for easier understanding.Example:36 + 1330 + 6 + 10 + 340 + 9 = 49Example:1368 + 2471000 + 300 + 60 + 8200 + 40 + 7Total: 1615

Building Subtractionwe want to give students something concrete to better understand what they are learning, which is why we provide an action phrase for subtraction. For beginners instead of minus, say take awayuse base blocks to build subtraction visually-place first value amount of base blocks on board and place the subtracting value of base blocks on the side of board (NOT ON BOARD, it is just there to help reinforce visually how much we are taking away)ex: 7-3-place 7 unit blocks on board and 3 unit blocks off the board-as you take one unit block from the 3 take one from the board and do this three times-now you are left with 4 unit blocks on the boardex:23-8 -for numbers like this start the process and then you will convert or borrow a long to convert it into units so you can take away a total of 8 units. the amount of longs and units left on the board is the answer*when subtracting large numbers, take away the largest place value first, so take away flats...then longs...then units if applicable*if you are subtracting with different base using blocks, make sure when you convert from either flat to long or long to units match the base value NOT 10*Showing Subtractionutilize shapes and diagrams to mimic base blocks to subtractsquare= flat, |(line)= long, and . (dot)= unitex: 7-3= . . . . . . . take away . . .-when taking away three dots (units) do not erase but draw a circle around how many you taking away from the 7 dots and draw an arrow-when you get into large numbers or different bases remember to still convert/borrow ( show this by crossing out the shape) and/or utilize the base value NOT 10

Traditional:working from right-to-left (can be confusing for students who have been learning left to right)1) its expandable2)its efficient3) not based in mathExpanded Form:expanding each number by its place values (great for beginning learners)Ex: 46=40 +61) its expandable2)not efficient3) based in mathLeft to right:subtracting left to right using place values (great for intermediate student learners)ex: 96-35= (90-30)+(6-5)=611) its expandable2) unsure if efficient3) based in mathEqual Addends:measurement between two numbers needing to stay the same but you can add or subtract to make friendly numbersex: 64-38 +2 +2= 66-40= 261) its expandable2)its efficient3) based in math

What does multiplication mean?-1st #= # of groups-2nd #= inside of groupsbetter to demonstrate with building rectangles rather than circles and blocks within circles (though it works)To Buildstart out with placing width value at top and length at side of answer board, then place the needed flats, longs, and/or units to fill up the rectangle on the outside of the answer boardmake sure the values being multiplied are not on the answer board because it is hard to see the answer if they aren'tfill w/ least amount of base 10 blocks you canex: add flats if applicable, and longs instead of using all units

To showwith this method still continue to reiterate to your students what the first and second number mean in multiplyingdemonstrate using square as flat, |(line) as long, and . (dot) as unitutilize a ray (diagrams laid in rows) to resemble rectangles the mostex: ----- . .----- . . = 2 groups of 12for larger numbers you can utilize a diagram where you create a width by length brackets.long brackets=10 and shorter brakes= unitsthen you draw out the lines for each bracket line and it creates a answer board filled with possible flats, longs, and/or unitsmake sure to not close of width and length brackets because then they look like they are part of answer board

Multiplication Automaticitythe automatic recognition of answer of 2 numbers working forwards and backwardsAutomaticity is important to set students up for success in math in the futureHow to teach multiplication table order:group 1( learnt by skip counting)start with 1'sthen 10'sthen 2'sthen 5'sgroup 2 ( learnt by skip counting)start with 3'sthen 9'show to teach 9'sin our answers, the 10's digit is 1 less than the number we are multiplying 9 bythen the 1's is the adding of a value our tens value answer to = nineex: 9x7, 1 less than 7 =6 and 9-6=3, so the answer is 63then doubles (4x4, etc.)group 3remaining values that match on both sides, such as 7x6 and 6x7Other methods to reinforce automaticityflash cards: can be great for independent learning but they need to be flash cards that have the groups we created in the multiplication table so you can choose what group they are working on and the frequency of themTimes table/test: are used for automatic understanding but they can stress students because of the time limit, and we must make sure to utilize the test results correctly. Do not make it a class competition and showing of who didn't do well, etc. Instead make a self progress chart that motivates them to keep trying to do betterGames: a memory card game called concentration for example with the problems and answers can be beneficial for students to engage in the problems and use their math skills quickly

Traditional Algorithmright to left solvingits expandableits efficientnot based in math

Expanded Form23x45= 20+3 x 40+5its expandablenot efficientbased in math

Left-to-Rightmultiply starting with furthest place values with each place value in other numberits expandableits efficientbased in math

Area Modeldraw rectangle and place expanded form values of two numbers on top and side of itat each addition sign draw a line through the rectanglethis mimics the model used in base ten blocks where flats are in upper left, etc.its expandableits efficientbased in math

Latticedraw rectangle and place each number with space in-between each value on top and right side (not using expanded form)draw line through rectangle at each spaceadd lattice lines and then add the values within each lattice lineits expandableits efficientnot based in math

2 ways to look at division:the first value is the # of items we havebut second value can be# of groups we will create or# of things inside each groupwith #1 make sure to evenly distribute into each group and if there's a remainder create fractions based on amount of groups you have so if you have five groups and remainder 2 it would be 2/5 with #2 utilize you base blocks and trade off if needed to create as many groups of the second value that you can from the first value, ex: 60 divided by 12 you make as many groups as you can of 12 out if sixty till you run out of base blocks from your group of sixty or till you hit remainders

To show division you create number of groups with smaller numbers, getting into large numbers create groups out of numberssquare= flat, |(line)= long, . (dot)= unitex: 25 divided by 4, create four circles (amount of groups) and count off units into each circle till you reach 25 but maintain even amount of units in each circle. so there are 6 units in each group and 1 dot left over which needs to be divided into each group so 1/4 ths. so the answer is 6 1/4ex: 139 divided by 25, create groups of 25 until you reach 139 or as far as you can to 139, so we created 5 groups of 25 with 14 left which is 14 out of a group of 25 left, so the answer is 5 14/25

Long division/ traditional1) expandable2) efficient3) not based in mathnumber being divided goes inside "house" and number dividing with is outside "house"-students don't always know which number can go where-students will make pattern that only small number goes inside house so you need to present problems like 60/10 and 6/12-the better students are at math facts, the easier long division is-this method requires more memorization to steps

Repeated Subtraction1) expandable2) kinda efficient (depends on student ability)3) based in mathdraw line down side of house and only put numbers in that columnstudent will pick any number that can go into number being divided multiply number in side column by number we are dividing with and subtract it from number being dividedrepeat this process till we are left with 0 or remainderthen add the values in the column together with the remainder-works with what numbers students are strong with

Area Model1) expandable2) kinda efficient (depends on student ability)3) based in mathdraw rectangle and place dividing number on outside of box on left, start with first value in number we are dividing and place it in first box,then add divisible number on top of outside of box and multiply it by dividing number the remainder in that box is added to next place value in next boxrepeat process until left with 0 or remainderadd values on the top outside of box along with remainder-works with what numbers students are strong with

Upwards Division1) expandable2) efficient30 not based in mathwrite the problem as we say itex: 372/9= 372 ----- 9work from the bottom up ex:9 goes into 37 4 times so add 4 to answer then subtract 36 (9x4) from 37, the remainder 1 goes to 2 so its now 12, then do the same process for 12.9 goes into 12 once so 1 is added to answer which is now 41, and subtract 9 from 12=3 which is our remainderso the answer is 41 3/9

Divisibility is useful for students to reduce the amount of work they have to do and helps them recognize relationships better between numbersDivisibility Rules:2- even numbers3- sum of digits is divisible by 34- last two digits are divisible by 45- number ends in a 5 or 06- if divisible by 2 AND divisible by 37- none8- last three digits are divisible by 89- sum of digits are divisible by 910: number ends in 0

-Finding factors of numbers is important for students to know and develop as they further their math journey-A prime # is a number that only has two factors, 1 and itself-non prime # is called a composite, more than two factors-prime factorization is finding factors of numbers that are only prime numberscan do this with a factor tree ex: 48 / \ 4x12 /\ /\ 2x2 3x4 /\ 2x2 Factors are: 2x2x2x2x3= 2 to the power of 4, x 3OR upside down division ex: 5 |40 ___ 2 |8 ___ 2|4 ___ 2 Factors are: 5x2x2x2= 2 to the power of 3, x5Lowest Common Denominator (LCD) -to find LCD, find the blend of the 2 answers from the two given problems (use largest factors), it is the smallest number that two or more denominators can both divide into evenly ex: 2 to the power of 2 times 3 AND 2 to the power of 4 2 to the power of 4 is divisible by both denominators and 3 is necessary for divisibility of one denominator, so the LCD= 2 to the power of 4 times 3= 48

What do the parts of the fraction mean?numerator: the amount we have, number of piecesdenominator: the size of the pieceFractions are tricky for students because this is one of the first times students are working with inverse relationship numbers (the smaller the denominator, the bigger the piece)When and why do we need to find common denominator?when you add and subtract you need the same size pieceswhen you multiply you do not need to find it because the first number acts as the amount of groups and the second number acts as the amount inside each group

Looking at two different fractions and determining which is bigger with reasoningif the denominator is the same, the one with more amount of pieces is biggerif there is a whole number, the one with greater whole number is biggerif a fraction has more than half pieces out of its piece size then it is greater than the one less than halfif you have the same amount of pieces but different size of pieces, the smaller piece size is largerif there are none of the above BUT they are both missing the same amount of pieces to fill up the piece size, the smaller the missing pieces the bigger The more fraction reasoning we can teach students, the more they will understand the size of fractions, how they are less than 1, and the relationships they have

Different Fraction Manipulatives:fraction tiles (everything below the whole is a different fraction, needs context to make sense)fraction circles (most simple, but should not be only one used)fraction squares (comparing shape below to the one above it and the whole)pattern blocks (different color same size rectangles with different size boxes on them, with shading that represents the number of things we have)colored counters (doesn't show whole but size of fraction)fraction rods (everything below whole is a different fraction, needs context to make sense, the width of the whole is same as the length of the wholeParts of a fraction:number on top= number of pieces/ things we havenumber on bottom is the size of pieces

Area Model:the area of the circle, etc. filled or not filledex: 1/4 of the area of the circle is yellowLinear Model:what in the line is shaded, colored, etc.ex: 1/4 of the line is shaded inyou can use different manipulatives that aren't linear and draw a line through themSet Model:can use any manipulative, or mix of manipulatives/ itemswhat out of the set is _____ex: two of the set are shaded, or two out of the set are rectangularTo build equivalent fractions with fraction manipulative:take any fraction you have and see if you can build it with one color To build non equivalent fractions with fraction manipulative:take any fraction you have and see if you can build it with 2 or more different colors

build with one, two, or three colorsbuild the problemthen put together the two manipulativesthen build with one color ( the fewest amount of tiles the better)ex:1/3 + 1/6⬜⬜▢ (2/3 and 1/6) ▫ ▫ ▫ ▫ ▫ ▫ (sixths)=5/6if you are using circle fractions and not tiles make sure the student figures out how many of the color make a whole to find the size of pieces

place the first fraction on the board and place the fraction you are taking away on the side of the boardtake away the same fraction from board and sideif the fractions do not share the same denominator, find a fraction to swap out the fraction on the board that will also match with the fraction on the side of the boardex: 2/3 circle fractions on board and 1/6 on the side of boardchange the 2/3 to 1/6th pieces and take away 1/6=3/6 = 1/2

build the problem and say the problem as blank groups of blankex: 1/2 x 41) make a group of four2) bring context: half a group of 4 is red3) show using color counters 2 red and 2 white =2ex: 3/4 x 1/21) make two groups of 4 based on the denominators2) provide context: 3/4 of a half are red3) show using color counters 3 out of 8 are red=3/8

Use the area model to demonstrate add, sub, and mult model just needs a shape you label as a unitrectangle may be easiest to use

if same base/size use one box to addif not the same base demonstrate each fraction with one box each and combine together into a third box. the highlighted/ filled in is the answer out of the total

if same base/size use one box to take awayif not the same base demonstrate each fraction with one box each and combine together into a third box. the highlighted/ filled in is the answer out of the totaldraw a circle around what you take away and an arrow

if multiplying a whole number by fraction= answer becomes smallerif multiplying fraction by whole number= answer becomes biggerdemonstrate as groupsif two fractions are being multiplied, show first fraction in one rectangle and fill in what needs to be filled in then incorporate the second fraction into the rectangle, overlap is the answer