Understanding Multiplication and Division in Word Problems |
Posted in division, elementary, multiplication
by Mrs. Kirk
So, the fun and games of learning how to make arrays, skip counting on number lines and using models to solve multiplication problems has lead to the equally exciting task of solving division problems. Everyone seems to be making great progress and making meaning of multiplication and division in real world problems, UNTIL … wait for it … wait for it … We mix the two together!
And here the real “fun” begins … A few students usually have an intuition about the structure of the problems and just “get it” with out much help, but the majority of students need direct, systematic instruction paired with hands on or pictorial examples to really, truly, deeply, understand the difference between multiplication and division in word problems. But where can we start?
Start with The Same Story!
I like to take the same story and write 3 word problems that use the same story, but ask 3 different questions. This really helps my students to see how multiplication and division can be the same and different in a word problem and help them to see to importance of analyzing the question. How many times have you had a student choose multiplication to solve a division problem because they “saw the word each”? Solving similar problems and then discussing the problems in relation to each other can really get students thinking deeper about problem solving.
For example, my 3 word problems might be:
- Jeff had 4 plates of cookies to take to his neighbors. He put 6 cookies on each plate. How many cookies was Jeff giving his neighbors?
- Jeff had 4 plates of cookies to take to his neighbors. He had 24 cookies and put an equal number of cookies on each plate. How many cookies did he put on each plate?
- Jeff had 24 cookies to take to his neighbors. He put the cookies on plates. Each plate had 6 cookies. How many plates did he use?
The three problems have the same story, but different clues are given and different questions are asked for each problem. (Get this FREEBIE here)
As we work through the problems, I ask my students these questions:
- What is the question asking me to find?
- Does the story tell me the total?
- Does the story tell me the number in each group?
- Does the story tell me the number of groups?
Draw pictures or make diagrams!
We write down the question and clues. Then draw a picture …
After discussing how we made our picture we make our number sentence.
Write a number sentence and label the answers.
Compare and Reflect!
Continue to have meaningful discussion as you practice!
As you continue working guided problems with students ask them to explain why they are choosing to multiplication or division. Encourage students to question each other when working in small groups or with partners. Have them write their own examples of word problems using the same story, but different questions.
And, of course, practice, practice, practice! Download this FREE page to practice!
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division, math, math problems, multiplication, problem solving, problem solving skills, word problems
Simple multiplication and division problems
Everything is very
superficial. No depth, no fullness
answer, showing on a specific task. Where
techniques used on each specified
stage? In what order are they entered?
Problems solved by multiplication and division?
It is necessary to show the peculiarity of work on
tasks not only in the traditional, but
and in UMK «Harmony».
assignments is a form of learning
material related to practical
teaching methods. These methods form
practical knowledge and skills. Task
them is to keep for a long time
time gained knowledge.
In the classroom
mathematics system of practical knowledge
can be used at the stages of fixing
or repetition of learned material.
- Olya bought
3 pencils, 4 rubles each. How many
was this purchase worth it?
There are 4 plates left on the table. None
none of them had sausages. How many
all the sausages on those plates?
bottle of 2 liters. lemonade. How many liters
lemonade in three bottles? at five?
- To each rabbit
gave 3 carrots. How many carrots did you give
- 10 oranges
Divided equally among 2 plates. How many
oranges on every plate?
- For lighting
three classes it took 15 lamps, equally
for each class. How many lamps in each
- Make tasks
according to their decision: 1) 10 • 3 = 30 (l) answer: 30l;
2) 2• 6=12(kg) answer: 12 kg; 3) 3 •4=12(cm) answer:
- Make tasks
according to the drawings so that they are solved
multiplication. Write solutions:
- Notebook price
3r. How much does 5 of these notebooks cost?
Includes a large barrel
buckets of water, and in a small one — 3 times less.
How many buckets of water are included in a small
I bought 2 packs of cookies, 200 gr. each. Put
question and solve the problem.
drawing for a problem that would be solved
so: 2•3, 3•3, 5•3.
Thus, when learning
children of any topic after
practical reinforcement. We know that
theory without practice is impossible.
When studying simple
multiplication and division problems
the teacher should pay great attention
attention to tasks related to these
actions. They contribute to the best
assimilation and memorization of the table
multiplication, the ability to apply the received
knowledge for practice in everyday life.
introduction of concepts
Method (from ancient Greek.
«metodos» — a path to something) is a way to achieve
goals, ordered in a certain way
activity. (Pedagogy. S.P. Baranov)
— these are ways of joint activity
teachers and students sent
to achieve the goals of education, upbringing,
development of schoolchildren.
is the assimilation of material in certain
forms of cognitive theoretical
Method- is a system of rules and techniques
approach to the study of phenomena and regularities
nature, society, thinking.
Method- is the way, the way to achieve
certain results of knowledge and
Pedagogy deals with
various methods that are used
in primary school when teaching any
school subject. Thus, bearing in mind the joint
activities of teacher and student
such methods: explanations of the material by the teacher,
conversation, independent work of students.
Depending on the purchase method
knowledge by children distinguish between methods: dogmatic,
heuristic, research. If
consider methods in terms of path,
then they talk about inductive, deductive
methods and analogies. All these methods are used
and when teaching mathematics, taking into account the peculiarities
the subject itself, spoke in
Selection of methods
learning is determined by many factors:
general learning objectives, content
the material being studied, the level of preparedness
children to master the relevant material.
methods best used:
(introduction) concepts most commonly used
verbal method . More difficult material
the teacher explains, i.e. uses the story.
material studied by the teacher and activation
cognitive activity of students.
For example, at the very beginning, when studying
topics «Simple problems for multiplication and division»
give children a little history
about the quantities.
ready-made knowledge. But more efficient
use conversation when introducing concepts.
Conversation . Teacher
through skillfully asked questions encourages
students to reason and analyze
in a certain logical sequence
studied facts and phenomena and independently
approach the relevant theoretical
conclusions and summaries.
During a conversation
children themselves are looking for answers to some questions,
trying to figure it out on their own
what they develop an active mental
Good to use
visual method for children to imagine
the current situation, especially
they have better developed visual
1. Type-setting sheet,
designed to work with subject
pictures, movable numbers.
typesetting canvases (plot
3. Item sets
4. Sets of subject
tasks to create tasks.
1. Cards with
2. Type-setting sheet.
Children take possession
material based on exercises, independent
also explanatory —
divided into 2 parts: 1) theoretical
2) illustrative. Children are given information
in accordance with the curriculum.
material allows you to link explanation
with the student’s personal experience, makes an explanation
accessible. For example, when introducing the concept
«task» the teacher says the task
and illustrates it in the course of the text. Gives
simultaneously the concept of a condition and a question.
Usually all this is illustrated on typesetting
canvas. When fixing the material, also
verbal method is used, most often
conversation. During the conversation, it becomes clear how
children learned the material. Also used
visual method, i.e., various
visual aids for the child to imagine
the situation and showed (when solving problems,
for example, dividing into equal parts)
answer to the question.
and method of practical exercises.
Solving problems with explanation
students. Can be used on these
lessons non-standard types of exercises,
staging a task by roles: one
student — says «condition», the second — «question»,
the third — «explains the decision», the fourth
— “decides”, the fifth — “reads the answer”.
visual aid is used. For example,
task cards, cards with
drawings, according to which children make up
At this stage
the method of practical
exercises. Children composing themselves
problem can be solved, so at the same time
used and independent work
Can be used
introspection and self-control, where is the child
analyzes his work, actions,
which he performs, also a child
must control their mistakes, be able to
find and fix them.
work makes children think it
fosters working capacity, independence,
attention. Most often independent
work is carried out for self-control.
PRINCIPLES OF LEARNING.
Principle (from lat. Principium — beginning, basis):
1) the main starting position of which —
or theories, teachings, sciences, worldviews,
political organization; 2) internal
a person’s belief that defines him
attitude to reality, norms of behavior
and activities; 3) device base or
action of any device, machine
etc. (Sov. Encyclop. Dictionary).
Principle (from Lat.
basis, origin) — leading
idea, basic rule, basic requirement
to activities and behavior arising from
from the laws established by science.
(Pedagogy S. P. Baranov, L. R. Bolotina
— these are the initial provisions that determine
the activities of the teacher and the nature of cognitive
principles are the most important, the main patterns
and laws that form the basis of didactic
process. They reflect the joint action
many regularities of the educational process,
regulate the activities of teachers and students,
retain their general significance in the study
all subjects and at all stages
learning. Therefore, didactic
principles are the basic principles that govern
content, organizational forms and methods
educational process in accordance with its
common goals and patterns.
topics «Simple multiplication problems
and division” implements the basic principles
didactics in the traditional system of education.
scientific nature of teaching.
Even in the elementary course of mathematics at
familiarization with the topic «Simple tasks
on multiplication and division ”students get acquainted
with different concepts: “multiplication, division,
task, solution”, etc. Realize the real
quantitative relationships between different
objects. Install dependency
between the data and the desired, the condition and the question,
draw conclusions, model, obtain
results. Tasks are solved
different ways, use different
visual aids. Distinguish between «form»
problem solution record” and “method of solving
tasks». Gradually work is going from simple
to more difficult.
systematic training. Material
given in a particular system. Children first
get acquainted with arithmetic operations,
with the concept of «task», condition and question,
and then with different kinds of tasks.
visibility. Children using
visual aids get acquainted with the new
material. Since at this age
the child’s visual memory is better developed,
than auditory. The child thinks concretely
images, then reliance on sensory knowledge
is a necessary condition for learning.
It is proposed to use images,
real items as pictures
in the picture, drawing, diagram. Especially
in the early stages of introducing children to various
types of tasks, use visualization.
For example: Problems for dividing by equals
parts and content. For a deeper
comprehension it is better to use some
activity. Students consciously learn
to this subject. When faced with a task,
by its decision, the child must learn the material,
because You can’t do without a task in life.
Often in the classroom we see and notice that
when learning new material (for example:
acquaintance with the task, increasing the number
several times), children who actively
work in the classroom, understand the material
and already at the next steps can decide
such tasks independently and without difficulty.
strength. The teacher must achieve
solid acquisition of knowledge by children. For
to carry out consolidation, repetition,
explanation, generalization, practical exercises,
use visual aids. For completeness
knowledge, you need to highlight the main idea.
Knowledge, skills, skills will be stronger,
if children apply them in practice
activities in their daily lives.
accessibility. Knowledge that students
receive when studying this topic, they
accessible, i.e. understandable, especially since
children, getting acquainted with this topic, relying on
for visual aids. Definitions studied
in this topic should be given by the teacher
language accessible to children. Let’s say
that the teacher only leads the children to the definition
conditions and question, and then gives the opportunity
children to formulate this definition.
Based on this, the teacher in an accessible language
gives an explicit definition of this concept.
a set of principles allows
characterize the entire educational process,
all aspects of the teacher’s activity
and cognitive activity of students.
Each principle has its rationale.
Guiding Principles for Teaching Mathematics
in elementary grades are:
Word problems — online presentation
Elements of combinatorics (grades 9-11)
Application of the derivative in science and in life
Mathematics project “Mathematics is around us. Patterns and ornaments on dishes»
Introducing children to mathematical signs and coins
Mathematics simulator «Harvesting». Count within 10
Methods for processing experimental data
Lecture 6. Correlation and regression analysis
Solving the problems of the mandatory part of the OGE in geometry
Preparation for the exam in mathematics. Basic level Complex tasks
1. Text tasks
The concept of text
2. A text task is a description of some situation in natural language with the requirement to give a quantitative characteristic
of some component of this situation, to establish the presence or absence
of some relationship between the components
or to determine the type of this relationship
– The dog chased the fox, which was 30 m away from it. AT
while the fox makes three jumps, the dog makes
only two. How many jumps must the dog make,
, to catch up with the fox? How far will
Compare tasks. 1. Masha found 3 mushrooms, Petya found 2 mushrooms. How many mushrooms did the children find in total? 2. How many mushrooms did the children bring home,
Masha found 3 mushrooms, and Petya found 2 mushrooms?
3. Masha found 3 mushrooms, and Petya found 2 mushrooms. They
put them in one basket. Find the number of mushrooms
in the basket.
The requirement is presented as a question.
Condition and requirement is given in one sentence
The requirement is formulated in
4. A requirement is an indication of what needs to be found. It can be expressed in an imperative or interrogative form.
Masha found 3 chanterelles and 2 white mushrooms, and Petya-4
chanterelles. How many mushrooms did the children find in total?
Two girls simultaneously ran towards each other
along a sports track, the length of which is
420 m. When they met, the first ran
60 m more than the second. At what speed did each girl run
if they met
after 30 seconds?
5. Classification of problems
Number of actions
Number of conditions
Ways of solving
For the triple rule
For finding the unknown 90 059 Undefined
Elimination of one of the
Percentages and parts
Solved from the end
Practical method is a method in which
the answer is in the process of action with
items or their substitutes.
7. Solve the problem.
is taller than Kolya, Seryozha is lower than Kolya.
Who is taller?
A logical method is a method in which
the answer is found as a result of
Sometimes several methods are used in the course of
solving, in which case
it is considered that the problem has been solved
by the combined method.
8. Modeling in the process of solving problems
Modeling is one of the mathematical methods
knowledge of the surrounding reality, in which
models are built and studied.
The text problem is a verbal model
To solve the problem, it is necessary to build its
mathematical model. to
auxiliary, and then to
2nd stage — intramodel solution
The values of
numerical expressions are found,
equations are solved.
Stage 3 — translation of the obtained solution
into natural language
Using the obtained solution,
the answer to the question is formulated drawing
— short entry
(requires introduction of scale
and ability to use tools)