Math-Multiplication
overgeneralizing multiplication
As students become experienced with shifting across different units, attention can be directed to a fourth sticking point. Students often think that multiplication leads to a product that is larger than the numbers being multiplied. While this is the case with whole number multiplication, it is not true for all numbers, such as fractions
what determines which way you think about it
Depending on if you start with a whole number and multiply by a fraction, or start with a fraction and multiply by a whole number
Dr. Johanning doesn't really like how the teacher in this video models fraction times fraction multiplication. What is it about the way the teacher made his diagram that Dr. Johanning doesn't like and how would she do it?
Dr. Johanning doesn't really like how who the teacher modeled fraction times fraction multiplication because the teacher divided the whole cake into fourths in his second step. For example, in the video the first problem was ¾ x 2/3. He modeled this by drawing a rectangle and splitting it into three qual parts and shaded 3 of these parts because it was how many he hand. He then immediately split the entir whole into fourths, rather than just the 2/3 he had. He then repeated this process in the next fraction multiplication problem. This meant teacher skipped a step that was stressed in this multiplication module. Rather than splitting the whole into fourths right away, Dr. Johanning would have only split the portion of the fraction you have to begin with. For the problem ¾ x 2/3 in the video, for example, the teacher should have only split the 2/3 into fourth horizontally. This is easier to understand if the fractions in the example problem represent, hypothetically brownies. This is needed because if you only have 2/3 of a brownie pan, for example, you don't have the other 1/3 to cut so why would you extend the lines completely over. After doing this, only then should the teacher extended the lines to determine what each of the parts in the picture represents. In the example form the video, for example, each part represents a twelfth. This means that as I teacher I need to model what is really happening in the problem. If I only have 2/3 of a whole, I should only work with that 2/3 of a whole to begin with then, think about it in the context of the whole to determine the size of each part.
communicative property of multiplication
If 6x4=24 is known 4 x 6 =24 is also known
multiplication does not
always lead to a bigger product
associative property multiplication
3 x 5 x 2 can be found by 3x5=12, then 15x2=30
Multiplication as Arrays thinking begins in
In fourth grade (we introduce students to problem contexts where arrays are used and consider how they can be represented with multiplication sentences.)
Scale Factor
Number of Rows or Groups or Number of Groups
Rate
Number per group or number in one group or Number per group or number in one row
decimal multiplication models
There are a lot of models because it is different if you are multiplying decimals <1 times decimals < 1, decimals < 1 time whole numbers, decimals >1 times whole numbers, and decimals >1 times decimals >1.
what could we do to help children think of multiplication as things that come in equal size groups.
We can ask students to generate a list of things that come in groups. You can start with this idea in 4th grade when beginning to work on multiplication. can make up problems about things that come in groups.
questioning framework for teachers to help students make sense of fraction × fraction problems.
What is the problem asking you to do? Tell me about your picture. What does it show? How much are you starting with? What is the problem asking you to find? How much of what you started with do you need? How much of the whole pan do you need? How many pieces are in the whole pan? How many of those pieces do you end up needing? What number sentence could you write?
Without calculating, what happens when you multiply a number by 0.05? Will the number get bigger, smaller, or stay the same? Explain why.
When I multiply a number by 0.05 the number will be smaller. In general, if you multiply a number by a decimal less than 1 the answer will be smaller. This is true because when the number is less than one you are partitioning or taking a part of the whole. When you are portioning or taking a part of the whole you are breaking up what you started with. This means you want a "part of what you are multiplying by," which means the answer or product will automatically be smaller than what you started with or the number you are multiplying it by. For example, if I multiply .05 x 20 I am finding 5 hundredths or 5/100 of 20. As I'm only finding a part of the whole, it must be smaller.
Multiplying by fractions and decimals less than 1 means that you want a
a "part of what you are multiplying by" and so the product is smaller than what you started with. ½ of 12 = 6 which is a way to say ½ x 12 = 6 0.5 of 12 = 6 is also a way to say ½ x 12 = 6 ½ of something will be less that what you started with.
so, we can model the partial product algorithm using
an area/array model and base 10 materials for -single digit times multidigit numbers -two digit times two digit numbers
teaching multiplication with arrays provides the opportunity for children to develop an understanding of what?
area and vice versa
you uses rows and number of rows for
arrays
Why does she prefer to teach fraction multiplication before decimal multiplication?
because the models used with fraction multiplication are helpful.
fraction multiplication
can be complicated!
2.7 x 3.2
can use picture area method use decimal models and area together
ways to develop multiplication algorithm
connecting a visual model with some type of written format.
Multiplication algorithms are based on the concepts of
distribution.
When representing fraction times fraction
divide vertically, then horizontally for the part your taking
foil is
double distribution first, outer, inner, last
move from using base ten blocks to
drawings to represent multiplication problems
expressing the solution
expressing the solution based on what the problem is asking
interpret multiplication as scaling
focuses on number sense thinking if the answer of different multiplication problems are larger, bigger, smaller scaling something up (if it's a fraction bigger then one) staying the same (if you multiply by 1) scaling something down (if it's a fraction less than 1)
main focus of fifth grade and multiplication of fractions
fraction time fractions. on visual pictures/models
Often the study of data is incorporated into work with
fractions
arrays are
gridded arrangements
identifying the unit
identifying the unit at various stages of the problem (whole pan, half pan, half is split into fourths, etc.)
distribution and multiplication
important concepts that are related
3rd grade multiplication standards
interpret products of whole numbers (Ex: 5 x 7 is the total number of objects in 5 groups of 7 objects each) use multiplication within 100 to solve problems involving equal groups, arrays, and measurement apply properties of operation fluently multiply and divide within 100
main focus of fourth grade and multiplication of fractions
is fraction times whole number and whole number times fraction. on visual pictures/models
Sometimes fraction multiplication is replicating or repeatedly adding.
is replicating or repeatedly adding. For example, A recipe for orangey lemonade says to put 2/3 of a cup of lemonade into a glass and then add 1 cup of orange soda. How much lemonade is needed to make 6 glasses of orangey lemonade? You need 6 replications of the 2/3 cup of lemonade
Most of the focus in 4th and 5th grade for multiplication is
is teaching algorithms for multiplying whole number. partial product algorithm and the area/array model are commonly taught before shifting to the more common multiplication method that you know that involves zeros and carrying (later in 5 or 6 the standard alogrthim can be taught_
How to connect multiplication to addition for : 3 + 3 + 3 + 3 = 12
kids can write number sentence to solve a problem As the teacher, you can connect this sentence, where the same number is added repeatedly can be written with a multiplication number sentence: 3 x 4 = 12.
distributive property multiplication
knowing 8 x 5=40 and 8 x 2 =16 one can find 8x7 as 8 x(5+2)=(8x5) + (8x2)
When they multiply by a whole number they get something that is
larger than what the two numbers they were multiplying by.
When you are finding a part of something, or a fraction of something, you will get
less than you started with
.2 x .4
model one vertically and the other horizontally where it overlaps is the answer (just like with fractions)
Students can make sense of this complexity of fraction x fraction problems by
modeling contextual problems that lead to multiplication.
3rd grade thinking about multiplication
multiplication as things that come in equal size groups. (5 x 3 is five groups of three)
With whole number multiplication we were making multiple groups
of the same number For example, 3 times 4 can mean 4 + 4 + 4 = 12. You can think of this as replicating.
At other times with fraction multiplication you are
partitioning
The Week 4 homework problem about the brownies involved
partitioning You were asked to find a part
When you are multiplying by a fraction you are
partitioning and it involved breaking up what you started with.
5th grade multiplication standards
perform operations with multidigit numbers and with decimals to hundredths fluently multiply multidigit whole numbers using a standard algorithm
partial products algorithm focuses on and looks at
place value (which the standard algorithm doesn't mention, but that's why it's efficient because it combines steps) looks at what is going on in the background. It provides a way to focus on math concepts like place value and distribution, which will in turn help develop student's number sense.
teachers need to engage students in
productive mathematical discussions about how they are solving contextual problems and what their visual diagrams represent
what is important for students to engage in?
productive struggle (finding ways to support students as they work through difficult problems and avoiding telling them how to solve those problems )
key parts of a multiplication number sentence
rate scale factor product
multiplication is commonly introduced as another way to think about
repeatedly adding the same number
multiplying by a whole number you are
replicating
hard to know how to
support learners without taking over their work
Teachers can introduce students to fraction × fraction multiplication using contexts that involve
taking a part of a part of a whole. (think brownie problem)
Multiplication and division of whole numbers are often grouped
together in the ohio standards
Product
total
4th grade multiplication standards
use place value understanding and properties of operations to perform multidigit arithmetic with who numbers use standard algorithm illustrate and explain calculation by using equations, rectangular arrays, and/or array models
For example 14x5
using an area model 5 is distributed over 14 where the 14 is expanded and expressed as 10 + 4. 5(14) goes to 5(10+4) which is 5x10 plus 5x4
how do children develop a better conceptual understanding of fraction times fraction multiplication they are explored using
using problem contexts and visual diagrams
three sticking points that the questions help students focus on?
what fraction do I start with? identifying the unit expressing the solution overgeneralizing multiplication
As teachers when should we encourage students to use multiplication?
when it makes sense take a number of years for some of your students to become good at this and do it automatically. So you will have to remind students about this across 4th and 5th grade.
what fraction do I start with?
which fraction students start with and which fraction students multiply by when modeling fraction multiplication (i.e. what fraction they should draw first, one is their model the other is the scale factor)