August 21, 2013: Sierra Vista Learning Center: Grades 3 and 4: Room 6
Multiplying Multi-Digit Numbers
Common Core State Standards for Mathematics
Introduction
Big Picture
Sign In Clearly: External evaluation team will be contacting you via email to complete online
questionnaires.
Dates of the 7 Wednesday workshops and the proposed topics:
August 21—Multiplying Multi-Digit Numbers
September 18— 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 instruction
Models for multiplication
Contextual word problems for multiplication
The distributive property of multiplication
Double-digit multiplication
Exit Ticket at the end of each session
Problem Based Learning-One component of Mathematics Instruction
Students learn through solving problems
Problem Based Lesson Plan has three parts:
1) Introduce the Problem:
The introduction should help students understand the context of the problem and what
is expected in their solutions (pictures/diagrams, numbers, and words). If students have a lot
of questions when they are working on the problem, it might be because it was not introduced
well and they do not fully understand it. The introduction should not be modeling how to solve
a similar problem.
2) Support students as they work in small groups:
Whole class instruction should not be a part of this time. The instructor walks around
and looks at students’ work, listens actively, provides hints for students who are stuck, and
provides extension for students who solve the problem quickly.
3) Debrief:
Conduct a classroom discussion based on students sharing their work. Promote a
community of learners that includes all students. Listen actively without evaluating.
5 strategies for classroom talk.
 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.
* Comments:
Multiplication Grade 3
3.OA.1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of
objects in 5 groups of 7 objects each. For example, describe a context in which a total number of
objects can be expressed as 5 × 7.
3.OA.3. Use multiplication and division within 100 to solve word problems in situations
involving equal groups, arrays, and measurement quantities, e.g., by using drawings and
equations with a symbol for the unknown number to represent the problem. (See Table 2.)
Students use a variety of representations for creating and solving one-step word
problems, i.e., numbers, words, pictures, physical objects, or equations. They use
multiplication and division of whole numbers up to 10 x10. Students explain their
thinking, show their work by using at least one representation, and verify that their
answer is reasonable.
** Show 3 X 4 Four different ways
Array, Equal Groups, Repeated Addition, Number line (measurement)
*** Write word problems for 3 X 4 for which the following representation align: Array, Equal
Groups, and Measurement.
Distributive Property
Students are introduced to the distributive property of multiplication over addition as a
strategy for using products they know to solve products they don‘t know. For example, if
students are asked to find the product of 7 x 8, they might decompose 7 into 5 and 2 and then
multiply 5 x 8 and 2 x 8 to arrive at 40 + 16 or 56. Students should learn that they can
decompose either of the factors. It is important to note that the students may record their
thinking in different ways.
****Draw a diagram that shows 5 X 6 is the same as 2 X 6 + 3 X 6
3.MD.7. Relate area to the operations of multiplication and addition.
c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side
lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive
property in mathematical reasoning.
*****Joe and John made a poster that was 4‘ by 3‘. Mary and Amir made a poster that was
4‘ by 2‘. They placed their posters on the wall side-by-side so that that there was no space
between them. How much area will the two posters cover?
Use pictures, words, and numbers to explain your understanding of the distributive property in
this context.
Using the above to help students learn their multiplication facts. Handout.
******Represent 16 x 14 and possible representations.
1) Array, 2) Equal Groups, 3) Repeated Addition, 4) Number line (measurement)
******* Which model seems most doable?
Model for 16 times 14.
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)
******** 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
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 twodigit 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).
Division: (Partitive-size of group unknown------Measurement-number of groups unknown)
*** Show 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?
Solve number 1 here
2. I have 12 carrots and I want to put three on each plate. How many plates do I need?
Solve number 2 here
How were the processes different?
Determine the number of objects in each share (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?
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.