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!)