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U n t er r i ch t spl a n
M ul t ip l ic at io n wit h Array s up t o 7
by 8
Altersgruppe: 2nd Gr ade , 3 r d Gr ade
Virginia - Mathematics Standards of Learning (2009): 3 .5 , 3 .6
Virginia - Mathematics Standards of Learning (2016): 3 .3 .b, 3 .4 .a,
3 .4 .b, 3 .4 .c , 3 .4 .d
Fairfax County Public Schools Program of Studies: 3 .5 .a.1, 3 .5 .a.2,
3 .6.a.3 , 3 .6.a.4
Online-Ressourcen: S t i c k A r o und
Opening
T eacher
present s
St udent s
pract ice
Class
discussion
12
12
10
6
8
min
min
min
min
min
Closing
M at h Obj e c t i v e s
E x pe r i e nc e a rectangular array as an efficient way to organize
a collection of objects
P r ac t i c e counting the number of objects in an array using
multiplication
L e ar n to visualize multiplication using arrays
De v e l o p a connection between repeated addition, arrays, and
multiplication
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Ope ni ng | 12 min
Find some examples of items that are packaged or arranged in rectangular
arrays, such as eggs or crayons. For this lesson, try to limit the array to no
larger than 7 by 8.
From your students’ previous work with repeated addition and partitioning a
rectangles into squares (tiling), they may already understand the connection
between the area of a rectangular array and multiplication of the number or
rows by the number of columns. The goal for this portion of the lesson will be
to extend the use of rectangular arrays to counting collections of objects.
Hold up the first item.
For this example, suppose the first item is a standard 2 by 6
carton of eggs.
A sk: How many eggs are in the carton?
Of course, if you are using an empty carton, you can ask how many
eggs could be placed in the carton.
Field answers from your students, writing numbers on the board (if
you get more than one answer).
Then, ask your students to explain why.
Since the number of eggs is relatively small, some students may
simply count them. While this is not incorrect, continue to seek
out other answers, particularly ones that use repeated addition or
multiplication.
If no students use more advanced strategies at this stage, you
can draw attention to these methods in the next example, then
refer back to the first.
Show the second item.
For this example, suppose there is a box of 24 crayons (in 3 rows of
8).
A sk: How many crayons are in the box?
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Again, focus on the explanations your students give, helping to
clarify points as needed.
If your class has not yet found a connection to rectangular arrays,
lead them there.
A sk: How many items are in the first row?
A sk: How many rows are there?
Proceed into a brief discussion about repeated addition, then
tying in multiplication, using concepts from area of rectangles
where appropriate.
Specifically, lead your class from the idea of 3 rows of 8 to
multiplying 3 and 8 to find the total number of crayons.
This should be connected to an equation such as 8 + 8 + 8 = 3 x
8.
If you are looking to provide a challenge, remove one row of
crayons from the box.
So that your students don’t already know the answer, consider
using a different box (or another example where a row of objects
can be removed easily).
A sk: How many crayons fit in the whole box?
The purpose of removing a row is to solidify the concept behind
multiplying rows by columns. Your students should be able to use
the existing rows to determine how many would go in the empty
row.
From here, you can use repeated addition, but ultimately tie in
multiplication as it relates to a rectangular array.
T e ac he r pr e se nt s M at h game : S t i c k A r o und - M ul t i pl y up
t o 7 x 8 | 12 min
Present Matific ’s episode S t ic k A r o u n d - M u lt ip ly u p t o 7 x 8 to the
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class, using the projector.
The goal of the episode is to count the number of objects shown using
multiplication. Specifically, each screen will show objects arranged in a
rectangular array, though at least some of the objects will be hidden behind
blocks. In the example below, the top row is hidden.
The block can be moved, so that students are able to see what it behind it.
However, the block cannot be removed, so it will cover some objects at all
times. It may be worthwhile at first to lead your students to the conclusion
that placement of this block does not change the number of objects it hides.
The overall goal will be to relate the provided visuals to the discussion from
the opening about multiplying the number of columns by the number of rows.
Proceed through the screens with this in mind, while also supporting the
unique strategies your students come up with.
Additionally, locate the obstacles your students have in relating the visuals
to multiplication. Where necessary, relate the visual to those from the
opening. For example, the above visual is similar to the box of crayons with
one row of crayons removed.
As the episode progresses, there is an added challenge of multiple blocks
hiding objects, as shown below. Once again, the blocks can be moved but not
removed.
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By this point, your students should feel more comfortable using
multiplication to find the number of objects. If they are still struggling with
the connection to multiplication, revisit repeated addition as it pertains to
these collections of objects, then build the bridge to multiplication.
Ask them how to arrange the blocks in a way that it is easy to count the
number of rows and columns. Note that this is analogous to counting the
number of objects in a column and the number of objects in a row,
respectively.
The blocks in this example are already positioned in a way that allows your
students to count directly, as shown below.
Of course, most examples will require the blocks be moved to allow for this
explicit counting. As your students get comfortable with the setup, they may
wish to count implicitly, which is a reasonable strategy. Encourage
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r e a s o n in g instead of guessing by asking your students to explain how they
know.
S t ude nt s pr ac t i c e M at h game : S t i c k A r o und - M ul t i pl y up
t o 7 x 8 | 10 min
Have the students play S t ic k A r o u n d - M u lt ip ly u p t o 7 x 8 on their
personal devices. Circulate, answering questions as necessary.
While some of the rudimentary counting strategies of old will still generally
yield correct answers, there is greater room for human error as the arrays get
larger and as the orientation of the blocks becomes more challenging.
Encourage your students to leverage the efficiency of multiplication, while
solidifying the concept of why multiplication works.
C l ass di sc ussi o n | 6 min
Take a few minutes to address any lingering concerns your students have
regarding the episode. Revisit an example or two, as needed.
Ask your students to explain what they have learned so far. It is important
that while acquiring the ability to use multiplication in finding a number of
objects, both the parameters and the reasoning are properly understood.
This method cannot be used, in general, if the objects are not set up in an
array. Further, the bridge from repeated addition to multiplication is an
important one. Shore up any gaps, with your students leading the charge, and
you serving as the moderator.
C l o si ng | 8 min
Split your class into small groups. Provide each group with some specific
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number of objects. Choose a number that has at least one non-trivial pair of
factors. Larger collections of objects may become harder to contain or
control. For the example below, the scenario involves 30 objects.
S ay: I want to put these objects in a box, but I want to make sure
they are arranged in a rectangle. Find out with your group how many
rows there will be.
Consider providing a visual such as the one below (left) to give your
students a sense of what this box could look like.
Note that the box is limited to 7 by 8, per the parameters of this
lessons, and that some spaces will obviously not be used.
As shown above (right), one orientation is 6 rows of 5 objects. Another
orientation is 5 rows of 6 objects. The cases of 2 by 15 and 1 by 30 are not
allowed under the constraints of this box, but if your students mention
those, validate their outside-the-box thinking.
A sk: How many objects did I give your group?
For now, simply reach a consensus about the number; do not dive
into why at this point.
A sk: How many rows did your group make? How many objects are
in each row? Some of the groups will have used a 5 by 6 orientation,
while others used 6 by 5.
A sk: Why or why not?
At this point, continue to build the connection between
multiplication and arrays.
In either case, the result is 30.
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Point out that the array can help us confirm very quickly how many
objects were in the original collection.
Close out the lesson by addressing any remaining questions your
students have, including the other possible arrays (2 by 15 and 1 by
30).
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