Post on 27-Dec-2015
TIPM3Grades 4-5November 15, 2011
Dr. Monica HartmanCathy Melody
Gwen Mitchell
Learning Target
• Learn to use models, structure, and math tools to develop students’ persistence in problem solving.
Mathematical Practice #1
Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. —CCSS
Make sense of problems and persevere in solving them.
Persistence in Problem Solving
Discussion
• What was the math?
• What teacher moves did you notice?• What does she do and why does she do it?
• What did the students do?
Problem
A billionaire wishes to share some of her great wealth. She decides to donate $1000 a day to charity until she has given away one million dollars. How much time will it take her to reach her goal?
Solve your problem three different ways.
• Download the PDF of this activity
TI 15 ActivityThe Value of Place Value
Use base-ten materials or drawings and your calculator to explore how many tens, hundreds, and thousands are in a number. Record your observations in the table.What patterns do you see?
59
362
1,000
1,925
10,000
14,398
1,000,000
Use base-ten materials or drawings and your calculator to explore how many tens, hundreds, and thousands are in a number. Record your observations in the table.What patterns do you see?
59
362
1,000
1,925
10,000
14,398
1,000,000
5
36
100
192
1000
1,439
3
10
19
100
143
10,000
1
10
14
1,000 100,000
1
Analyzing DataDrawing Conclusions• Write 5 numbers that have 15 tens.
• Write 5 numbers that have 32 hundreds.
• Write 5 numbers that have 120 tens.
• How can you use the calculator to check?
Analyzing DataDrawing Conclusions• Write 5 numbers that have 15 tens.• 156, 152, 158, 151, 153• Write 5 numbers that have 32 hundreds.• 3213, 3224, 3237, 3248, 3250• Write 5 numbers that have 120 tens.• 1201, 1203, 1204, 1205 1209• How can you use the calculator to check?
Try This
Hazel has picked a secret number. When she rounds it to the nearest ten, she gets 17,550. When she rounds the number to the nearest hundred, she gets 17,500. When she rounds to the nearest 1,ooo, she gets 18,000. What are all the possible numbers Hazel could have chosen?
Operations and Algebraic Thinking 4.OA
1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.
2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to rep- resent the problem, distinguishing multiplicative comparison from additive comparison
3. Solve multi-step word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Operations and Algebraic Thinking 5.OA2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 x (8 +7). Recognize that 3 x (18,932 + 921) is three times as large as 18,932 + 921, without having to calculate the indicated sum or product.
Look for and make use of structure• Situation Diagrams
Part WholeChange Situations
Comparison Situations (Additive)
• Multiplication/Division DiagramsEqual groupsMultiplicative Comparison
Multiplication/Division Diagrams: Equal Groups
Number of Groups Number in Each Group
How Many in All
There are 6 matchbox cars in each package. Shelby bought 4 packages.How many matchbox cars did Shelby buy in all?
Multiplication/Division Diagrams: Equal Groups
Number of Groups Number in Each Group
How Many in All
4 6 ?
Equation 4 x 6 = ?
There are 6 matchbox cars in each package. Shelby bought 4 packages.How many matchbox cars did Shelby buy in all?
Multiplication/Division Diagrams: Equal Groups
Number of Groups Number in Each Group
How Many in All
Val has $20 for buying gifts. She needs to buy gifts for 4 friends.If she spends the same amount for each gift, what is the most she can spend for each?
Multiplication/Division Diagrams: Equal Groups
Number of Groups Number in Each Group
How Many in All
4 ? 20
4 x ? = 20
Val has $20 for buying gifts. She needs to buy gifts for 4 friends.If she spends the same amount for each gift, what is the most she can spend for each?
Multiplication/Division Diagrams: Equal Groups
Number of Groups
Number in Each Group
How Many in All
4 ? 204 x ? = 20
Number of Groups
Number in Each Group
How Many in All
4 6 ?4 x 6 = ?
Multiplication/Division Diagrams: Equal Groups
Number of Groups
Number in Each Group
How Many in All
?
Design a question where the number of groups is unknown.
Multiplication/Division Diagrams
Comparisons
Thinking Blocks
• Video Index
• Modeling Tool
Models for Multiplication
• Area Model• Expanded Notation• Algebraic Notation
Area Model
A model of multiplication that shows each place-value product within a rectangle drawing.
435 = 400 + 30 + 5
9
400 + 30 + 5
9435 x 9
9 x 400 = 3600 9 x 30 = 270 9 x 5 = 45
Expanded Notation Model
376 = 300 + 70 + 6
9
99 x 300 = 2700 9 x 70 = 630 9 x 6 = 54
3 0 0 + 70 + 6 9
3 7 6 =x 9 =
9 x 300 = 27009 x 70 = 6309 x 6 = 54
3,384
Algebraic Notation Model
475 = 400 + 70 + 5
6
66 x 400 = 2400 6 x 70 = 420 6x5= 30
6 475 = 6 (400 + 70 + 5)
= (6 400) + ( 6 70 ) + ( 6 5 )
= 2400 + 420 + 30
= 2,850
Area Model with Base Ten Blocks• Arrange tables so a person from last year is
in each group.• Try these problems:
7 x 154 x 2412 x 1614 x 23
Unknown Addend1. Each person writes a multiplication problem on their
board.2. In your head, add a number less than 10 to your product,
but do not write it. Instead , write a variable to represent this addend which is unknown to all but you.
3. Write an equal sign next to your expression to make an equation. Include the answer.
4. Taking turns, show your problem to the others in the group. The others at your table hold up fingers to show the unknown addend.
5. Then at a hand signal, the others at your table together say the equation with the answer.
6. Repeat so everyone at your table has a chance to share their unknown addend problem.
Division Dash
Partial Quotients
192 ÷4
Rectangle Sections Method
1924
40
-160 32
+ 32
8
-32 0
= 48
3248 ÷ 5
Rectangle Sections Method
Try this with the rectangle sections method on your own!
Expanded Notation Method192 ÷4
4 19240
-160 32
8
- 320
48
Expanded Notation Method18,435 ÷ 5
Try this with the expanded notations method on your own!
Digit-By-Digit Method192 ÷4
4 192 4
-16 3
8
- 3 20
2
Practice
Property Lists for QuadrilateralsWork in groups of 3 or 4.List as many properties as you can that are applicable to all the shapes on their sheet.
•Use an index card to check right angles.•Use rulers to compare side lengths and draw straight lines.•Look for lines of symmetry.•Use tracing paper for angle congruence.•Use the words “at least” to describe how many of something.
Does the property apply to all the shapes in the category?
Closer Activity: Dice GameWhat you are going to do: Review today’s session 1. Presenter: Get groups into 4’s and number off2. Presenter: Tells teams which number goes first3. Teachers: One at a time, roll the dice and respond to the statement that correlates to
the number on your dice.
1. Name one thing I am going to use when I get back to my classroom.
2. List two strategies you can use to solve a problem.3. Name something you learned from someone else.4. Name something I am still struggling with.5. Name a strategy you are excited about.6. Name a change you are going to make in you
math lesson.
Call or email with any questions
Dr. Monica Hartman586-939-3749
mlhartma@umich.edu