1
U n t er r i ch t spl a n
De t e rmining t he Unkno wn Us ing
Ad d it io n and Sub t rac t io n
Altersgruppe: Gr ade 1, Gr ade 2, Gr ade 3 , Gr ade 4
Online-Ressourcen: W e i ghi ng t he Opt i o ns
Opening
T eacher
present s
St udent s
play
Class
discussion
12
12
10
10
min
min
min
min
ZIE L E :
E x pe r i e nc e solving for unknowns using visual support
P r ac t i c e using a symbol or letter to represent an unknown
quantity within an equation
L e ar n to connect word problems with visuals, then visuals
with equations
De v e l o p strategies for solving multi-step problems involving
addition and subtraction
Ope ni ng | 12 min
Before class, draw numbers on separate index cards, with specific
Copyright 2015 www.matific.com
2
collections grouped together, as below.
Make sure the sums you will work with are no more than 20, at
least in principle.
12 10 7 5
8495
6 5 10 9
11 2 9 18
2383
Place the index cards on the board with sticky tack, so that they can
be rearranged.
Ask students to create an equation using addition and/or
subtraction and at least 3 of the 4 values.
For example, the first line on numbers might yield simpler
equations, such as 5 + 7 = 12, or more involved equations, such as
12 + 5 – 7 = 10.
Consider variations wherein there are sums or differences on
each side of the equality, such as 12 + 5 = 10 + 7.
Encourage the use of subtraction here, as most students will
default to addition.
For example, 12 – 7 = 10 – 5.
For equations where the sum or difference is not one of the
numbers shown, include that value.
For example, 12 – 7 = 5 = 10 – 5.
Record each new equation off to the side before asking the class if
any other equations can be made using the same numbers.
By the end, each line should have at least a couple of equations.
It may be necessary for you to hint at some possible combinations
if your students get stuck.
Copyright 2015 www.matific.com
3
At this early stage, it is reasonable to promote a certain amount of
guess and check work, but as this portion of the lesson progresses,
be sure to touch on number sense.
Specifically, remind the class of mathematical truths, such as
subtracting a small number from a large number (relatively
speaking) still yields a large number.
For these sets of numbers, a suitable example of this is 18 – 2.
After working on the first set of numbers on the board with student
input, have students come up to the board (perhaps in pairs) to work
on the next set of numbers.
Remind the students to include operations and equalities, where
necessary.
Continue working through sets of numbers until your students have
a good grasp on the concept, as well as some flexibility in finding
multiple solutions.
T e ac he r pr e se nt s W e i ghi ng t he Opt i o ns: A dd and S ubt r ac t
| 12 min
Present Matific ’s episode W e i ghi ng t he Opt i o ns: A dd and
S ubt r ac t to the class, using the projector. The goal of this
episode is to solve for unknown quantities using addition and
subtraction within 20 and a visual.
Each question will begin with a scale, three weights, and one or
more fruit or vegetable.
The scale must be balanced, which requires placing weights on the
side opposite the fruit or vegetable of unknown weight.
Note that it is possible to add weights to the side that already
has the fruit or vegetable of unknown weight on it.
It is also possible to use more than one type of fruit or vegetable,
as will be shown later.
Copyright 2015 www.matific.com
4
Accordingly, make sure the students are clear on the purpose of the
scale, which essentially serves as an equalizer of two quantities.
In the example below, as in most, many fruits and vegetables are
shown.
E x a m p le :
The students must first identify and drag the correct item to the
scale. Then, weights can be dragged from the shelf in order to
balance the scale.
Note that often only one or two weights will be needed to
balance the scale, even though three are available.
This prevents students from assuming one specific algorithm
or method is used every time.
E x a m p le :
Copyright 2015 www.matific.com
5
In the above example, two weights were needed to find a balance.
The corresponding equation would look something like 3 + ? = 10,
where the question mark should be noted as the weight of the
squash and the units of weight should be mentioned (here:
pounds).
Ask the class why we can set these weights (or sums of
weights) equal. This will develop a connection between having
a balanced scale and using an equal sign.
Notably, you can describe this equation as: 3 l b w e i ght +
w e i ght o f sq uash = 10l b w e i ght .
Remind the class that the abbreviation “lbs.” is short for
pounds.
When too much weight is placed on one side of the scale, the scale
will tip down.
Use this as a talking point for why it is necessary to add, remove,
or replace some weights.
Similarly, if not enough weight is placed on the scale, the scale will
fail to tip down or balance out.
For some advanced problem solving, it may not be necessary to
actually balance the scale to determine the weight of an item.
However, this thinking should be reserved for when students are
working individually and have already mastered the core
Copyright 2015 www.matific.com
6
concepts.
Additionally, once the weight of one item has been determined, it
can be used in the same fashion as the weights provided, as below.
This strategy will not necessarily be common amongst your
students, but there will undoubtedly be a few who explore this
route. Encourage their inventive thinking, but also look out for any
potential misuse or reliance on this strategy.
E x a m p le :
Here, suppose the cabbage was previously determined to weigh
four pounds.
Accordingly, the weight of the squash can be determined, since we
can still set up an equation with only o ne unknown.
In this case, the corresponding equation would be ? = 3 + 4.
It is key to maintain a connection between the word problem,
the visual, and the equation. Notably, you can describe this
equation as: w e i ght o f sq uash = 3 l b w e i ght + w e i ght
o f c abbage .
Similarly, it is possible that the cabbage is the item with unknown
weight, meaning the scenario becomes a missing-addend problem,
as was seen previously, with corresponding equation 3 + ? = 7.
The most challenging scenarios in this episode involve the use of
all three weights.
Copyright 2015 www.matific.com
7
E x a m p le :
In the example above, the principles remain the same as in previous
examples, with the corresponding equation looking like 10 + 4 = 5 +
?.
Naturally, this can be simplified by summing the left-hand side
and viewing this as a missing-addend problem.
Note that the weight of the watermelon could have been
determined in several ways, some of which are possibly simpler
than the way shown.
However, this is a key point of instruction. Encourage your
students to seek out other ways to determine the item.
Discussing more than one solution with the class will help
develop the requisite flexibility for success with more
complex problem solving.
Copyright 2015 www.matific.com
8
S t ude nt s pl ay W e i ghi ng t he Opt i o ns: A dd and S ubt r ac t |
10 min
Have the students play W e i ghi ng t he Opt i o ns: A dd and
S ubt r ac t their personal devices.
Circulate, answering questions. Continue to connect the visuals
with the appropriate equations involving an unknown quantity, and
support the clever strategies your students come up with.
Encourage the use of letters or symbols in place of the question
mark in solving for the unknown.
Advanced students can move on to play another variant of
Weighing the Options: W e i ghi ng t he Opt i o ns: A dd- S ubt r ac t
P uz z l e s
This episode contains the same concepts as in the first
episode, but offers the added challenge of a fourth possible
weight to use. Again, some scenarios will not require all
weights, but having more possibilities encourages more
efficient problem solving.
As mentioned previously, some scenarios may allow for the weight
of an item to be determined without actually balancing the scale.
For example, you can hint at the use of inequalities (as
represented by a tilted scale).
If the scale is tilted one direction, then a two pound weight is
added and the scale tilts to the other direction, then it can be
reasoned that adding one pound would balance the scale.
This is implied, since all weights are whole numbers. It is not
necessary to find a combination that proves this. Again, reserve
this type of reasoning for students who have showed a mastery
of the basic concepts and are looking for other means of
determining the unknown weight.
C l ass di sc ussi o n | 10 min
Copyright 2015 www.matific.com
9
Discuss any challenges the students faced while working
individually.
If arithmetic issues persist, consider using this time for small
group work and one-on-one conversations about the
fundamentals behind this lesson.
If concerns persist surrounding the scale functionality, use an
actual scale and weights to clarify the ideas of when and why the
scale will tilt or balance.
Using a projector, display W e i ghi ng t he Opt i o ns: A ddS ubt r ac t P uz z l e s , which is an extension of the principles of
W e i ghi ng t he Opt i o ns: A dd and S ubt r ac t , except with a
fourth weight on the shelf.
E x a m p le :
The basic functionalities for this episode are the same, but the
increase in possible combinations to determine the unknown weight
increases the complexity and requisite problem solving skills.
Use this episode as an opportunity to promote flexibility in problem
solving, being sure to ask your students if they have solutions other
Copyright 2015 www.matific.com
10
than the ones discussed in class first.
Additionally, build the logical connections between the scale
readings and what happens when known or unknown weights are
added to the scale.
Copyright 2015 www.matific.com