Multiplying Multi-Digit Numbers and Division
Common Core State Standards for Mathematics
Dates of the Wednesday workshops and the proposed topics: (NOTE for PLL Users: Look for these resources in the PLL too)
September 18— Multiplying Multi-Digit Numbers and Division
October 16 – Fractions
November 20 –Add and Subtract within 1000
December 25 –to be determined
January 15–Perimeter and Area
February 19 Measurement
March 19 –to be determined
Today—
Problem Based Learning (redux) (BEFORE, DURING, AFTER)
Revisit a model for multiplication-The Power of the Array
Strategies for Learning the Basic Facts
Contextual word problems for multiplication and division
Division as Repeated Subtraction
Problem Based Learning- Students learn through solving problems (One component of Mathematics Instruction).
5 strategies for classroom talk. (AFTER)
· Revoicing (So you are saying that what you did in number 1 was like dealing a deck of cards, one to each and repeat.)
· Asking a student to restate someone else’s reasoning. (Can you repeat what ______just said in your own words?)
· Asking students to apply their own reasoning to someone else’s reasoning. (Do you agree or disagree with what _____ said and why?)
· Prompting students for further participation. (Would someone like to add on to that?)
· Using wait time. (Take your time, we will wait.) Give students time to formulate answers in their minds. Also give them time to think about (process) important ideas.
BEFORE: The Setup (To prevent hands going up as soon as the DURING begins.)
Problem: Maya used place value blocks to divide 87. She made groups of 17 with 2 left over. Use drawings of place value blocks to determine how many groups Maya made (enVisionMATH).
1. Key contextual features of the task scenario are explicitly discussed.
2. Key mathematical ideas and relationships, as represented in the task statement are discussed.
3. The expectations are clear.
DURING: Support the work of groups
Monitor progress, offer hints and suggestions for those who need them, offer extensions to those who need them.
AFTER: Debrief Questions
Purposes of teacher questions / Example questions1. Initial eliciting of students’ thinking
2. Probing students’ answers
a. Trying to figure out what a student means or is thinking when you don’t understand what he or she is saying.
b. Checking whether right answers are supported by correct understanding.
c. Probing wrong answers to understand student thinking.
3. Focusing students to listen and respond to others’ ideas.
4. Supporting students to make connections (e.g., between a model and a mathematical idea or a specific notation)
5. Guiding students to reason mathematically (e.g., make conjectures, state definitions, generalize, prove)
6. Extending students’ current thinking and assessing how far they can be stretched
Recommendations for Learning the Basic Facts
The Power of the Array--Row by Column (Array)
Division: (Partitive-size of group unknown------Measurement-number of groups unknown)
Think about how you how would you go about solving the following two problem using counters?
1. I want to divide 16 carrots equally among 4 people. How many carrots would each person get?
Show number 1 here
2. I have 12 carrots and I want to put three on each plate. How many plates do I need?
Show number 2 here
Identify the type of each of the following contextual problems.
- You have 12 apples to share equally among 3 people. How many apples will each person get?
- You want to separate the 24 students in your class into groups of 4. How many groups will there be?
- You have 3 six packs of Dr. Pepper. How many cans of Dr. Pepper are there?
- There are three equal rows of desks with a total of 15 desks. How many desks are in each row?
- A table in MS Word has 3 rows and 4 columns. How many cells are in the table?
- A rectangle has an Area of 20 cm squared. If the width is 4 cm, what is its length?
- One tennis ball cost 3 dollars. A full case of tennis balls costs 5 times as much. How much does a full case cost?
- A rubber band is stretched to 15 inches, 5 times it relaxed length. What is its relaxed length?
- Car A costs 15,000 dollars. Car B costs 45,000 dollars. How many times more does Car B cost than Car A?
Division Based on Partitive Division
Example of partitive division, size of groups unknown.
A bag has 92 hair clips and Laura and her three friends want to share them equally. How many hair clips will each person get?
Explain the diagram below.
Draw a similar diagram using lines for tens and dots for ones to solve this problem.
Snicklefritz has 72 cricket cards. He wants to store them in three empty shoeboxes. If he puts an equal amount in each, how many cards will be in each of the boxes?
Connection to the Standard Algorithm
Division as Repeated Subtraction The link to the standard algorithm
Determine the number of shares (measurement division, number of groups unknown)
Max the monkey loves bananas. Molly, his trainer, has 24 bananas. If she gives Max four bananas each day, how many days will the bananas last?
Explain the diagram below
Harold likes to eat his cookies 6 at a time. He has a box of 48 cookies, how many servings of cookies does he have?
Connection to the Standard Algorithm
a. Represent a variety of counting problems using arrays, charts, and systematic lists, e.g., tree diagram.
b. Analyze relationships among representations and make connections to the multiplication principle of counting. Tree Diagrams, Chart (Array)
******** List all the different two-topping pizzas that a customer can order from a pizza shop that only offers four toppings: pepperoni, sausage, mushrooms, and onion.
Produce a Systematic List
A Chart
A Tree Diagram
4.OA.1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 _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.
A multiplicative comparison is a situation in which one quantity is multiplied by a specified number to get another quantity (e.g., “a is n times as much as b”). Students should be able to identify and verbalize which quantity is being multiplied and which number tells how many times.
“A blue hat costs $6. A red hat costs 3 times as much as the blue hat.
How much does the red hat cost?”
A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat?
4.OA.3. Solve multistep 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.
Chris bought clothes for school. She bought 3 shirts for $12 each and a skirt for $15. How much money did Chris spend on her new school clothes?
Kim is making candy bags. There will be 5 pieces of candy in each bag. She had 53 pieces of candy. She ate 14 pieces of candy. How many candy bags can Kim make now? (7 bags with 4 leftover)
Kim has 28 cookies. She wants to share them equally between herself and 3 friends. How many cookies will each person get? (7 cookies each) 28 ÷ 4 = a
There are 29 students in one class and 28 students in another class going on a field trip. Each car can hold 5 students. How many cars are needed to get all the students to the field trip? (12 cars, one possible explanation is 11 cars holding 5 students and the 12th holding the remaining 2 students) 29 + 28 = 11 x 5 + 2
AZ.4.OA.3.1 Solve a variety of problems based on the multiplication principle of counting.
a. Represent a variety of counting problems using arrays, charts, and systematic lists, e.g., tree diagram.
b. Analyze relationships among representations and make connections to the multiplication principle of counting. Tree Diagrams, Chart (Array)
4.NBT.5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Thirdly, there is the view that mathematics is a useful but unrelated collection of facts, rules and skills (the instrumentalist view).
How were the processes different?
Determine the number of shares (measurement division, number of groups unknown)
Max the monkey loves bananas. Molly, his trainer, has 24 bananas. If she gives Max four bananas each day, how many days will the bananas last?
Explain the diagram below
Equations in the form of a x b = c and c = a x b should be used interchangeably, with the unknown in different positions.
Examples: Solve the equations below:
24 = ? x 6 Rachel has 3 bags. There are 4 marbles in each bag. How many
marbles does Rachel have altogether? 3 x 4 = m
Students may use interactive whiteboards to create digital models to explain and justify their thinking.
3.OA.4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = ÷ 3, 6 × 6 = ?.
Students apply their understanding of the meaning of the equal sign as ‖the same as‖ to interpret an equation with an unknown. When given 4 x ? = 40, they might think:
4 groups of some number is the same as 40, 4 times some number is the same as 40. I know that 4 groups of 10 is 40 so the unknown number is 10 The missing factor is 10 because 4 times 10 equals 40.
Equations in the form of a x b = c and c = a x b should be used interchangeably, with the unknown in different positions.
Examples: Solve the equations below:
24 = ? x 6 Rachel has 3 bags. There are 4 marbles in each bag. How many
marbles does Rachel have altogether? 3 x 4 = m
Students may use interactive whiteboards to create digital models to explain and justify their thinking.
Interpret and existing diagram--What does this show? Be precise.
Create a diagram—Using a diagram and numbers show that 4 x 6 is the as 4 x 4 + 2 x 4