How can I plan my lessons using the Backwards Approach?
Lesson #1
Identify the outcomes to be learned
N1.9 Demonstrate an understanding of addition of numbers with answers to 20 and
the corresponding subtraction facts, concretely, pictorially, physically, and
symbolically by:
• using familiar and mathematical language to describe additive and subtractive
actions from their experience
• creating and solving problems in context that involve addition and subtraction
• modeling addition and subtraction using a variety of concrete and visual
representations, and recording the process symbolically.
N2.2 Demonstrate understanding of addition (limited to 1 and 2-digit numerals) with
answers to 100 and the corresponding subtraction by:
• representing strategies for adding and subtracting concretely, pictorially, and
symbolically
• creating and solving problems involving addition and subtraction
• estimating
• using personal strategies for adding and subtracting with and without the support of
manipulatives
• analyzing the effect of adding or subtracting zero
• analyzing the effect of the ordering of the quantities (addends, minuends, and
subtrahends) in addition and subtraction statements.
N3.2 Demonstrate understanding of addition of whole numbers with answers to
1000 and their corresponding subtractions (limited to 1, 2, and 3-digit numerals)
including:
 representing strategies for adding and subtracting concretely, pictorially, and
symbolically
 solving situational questions involving addition and subtraction
 estimating using personal strategies for adding and subtracting.
Be aware of the processes that should be integrated into your string of lessons for this outcome.
Now that I have listed my outcome:
Determine how the learning will be observed
What will the children do to know that the learning has occurred?
What should children do to demonstrate the understanding of the mathematical
concepts, skills, and big ideas?
What assessment tools will be the most suitable to provide evidence of student
understanding?
How can I document the children’s learning?
Create your assessment tools before you create your lesson task.
Name
Models thinking using
a ten frame
Uses counting
strategies to find
missing part (counting
forward or backward)
Is able to represent
using part-part –whole
mat
Is able to record
number sentences+-
Plan the learning environment and instruction
What learning opportunities and experiences should I provide to promote the
learning outcomes?
What will the learning environment look like?
What strategies do children use to access prior knowledge and continually
communicate and represent understanding?
What teaching strategies and resources will I use?
How can I differentiate the lesson to challenge all students at their learning
ability? How will I integrate technology, communication, mental math, reasoning,
visualization, etc into this lesson? (7 Processes) Look at your outcomes to see
which of the processes you should be including.
Plan your lesson here: What lesson format will you use?
BEFORE-DURING-AFTER? Math PODS? ETC.
I will begin with an oral journal using the sentence starters:
I see..
I think…
This reminds me of…
-------------------------------------------------------------------------------------------After the oral journal I will pull out the Power of Ten cards and students
will tell/ share what they know. How have they used these cards? What are
they learning? What does it look like/ sound like when you play games with
a partner or small group or by yourself?
(Reviewing expectations of behavior)
I will teach a new game called “Salute”. This game requires students to use
either subtraction from ten or adding on in order to find their mystery
number. Their partner says how many empty spaces or white spaces while
the student needs to solve for the unknown. After, we will take this a step
further and represent the number story in a few different ways
(connections).
A ten- frame will be placed on
partner “A”’s forehead.
Partner “B” will say how many empty spaces. Partner “A”
will then think about how many red spaces. A discussion will
follow in full group strategies how they solved for the
unknown number.
Once the students have had playing time and full group
discussion, the activity will be taken one step further.
Students will represent their work in 3 forms (ten frame,
part-part whole and math sentences.
Show and share in full group.
Assess student learning and follow up
What conclusions can be made from assessment information?
How effective have instructional strategies been?
What are the next steps for instruction?
How will the gaps in the development of understanding be addressed?
How will the children extend their learning?
Lesson 2
How can I plan my lessons using the Backwards Approach?
Identify the outcomes to be learned
N1.9 Demonstrate an understanding of addition of numbers with answers to 20 and the corresponding subtraction facts,
concretely, pictorially, physically, and symbolically by:
• using familiar and mathematical language to describe additive and subtractive actions from their experience
• creating and solving problems in context that involve addition and subtraction
• modeling addition and subtraction using a variety of concrete and visual representations, and recording the process
symbolically.
N2.2 Demonstrate understanding of addition (limited to 1 and 2-digit numerals) with
answers to 100 and the corresponding subtraction by:
• representing strategies for adding and subtracting concretely, pictorially, and
symbolically
• creating and solving problems involving addition and subtraction
• estimating
• using personal strategies for adding and subtracting with and without the support of
manipulatives
• analyzing the effect of adding or subtracting zero
• analyzing the effect of the ordering of the quantities (addends, minuends, and
subtrahends) in addition and subtraction statements.
N3.2 Demonstrate understanding of addition of whole numbers with answers to
1000 and their corresponding subtractions (limited to 1, 2, and 3-digit numerals)
including:
 representing strategies for adding and subtracting concretely, pictorially, and
symbolically
 solving situational questions involving addition and subtraction
estimating using personal strategies for adding and subtracting
Be aware of the processes that should be integrated into your string of lessons for this outcome.
Lesson#2 will focus on the relationship between addition and subtraction FACT FAMILIES.
We will also introduce representation on the number-line.
For students that you feel are still not ready to move on please have them practice virtually
(illuminations) and concretely by playing the game below.
Models and Connections
New mathematics is continuously developed by creating new models as well as combining and
expanding existing models. Although the final products of mathematics are most frequently
represented by symbolic models, their meaning and purpose is often found in the concrete,
physical, pictorial, and oral models and the connections between them.
To develop a deep and meaningful understanding of mathematical concepts, students need to
represent their ideas and strategies using a variety of models (concrete, physical, pictorial, oral,
and symbolic). In addition, students need to make connections between the different
representations. These connections are made by having the students try to move from one type
of representation to another (how could you write what you’ve done here using mathematical
symbols?) or by having students compare their representations with others around the class.
In making these connections, students should also be asked to reflect upon the mathematical ideas and
concepts that students already know are being used in their new models (e.g., I know that addition means
to put things together into a group, so I’m going to move the two sets of blocks together to determine the
sum).
If I was not modeling a lesson you could have the two groups as listed above and the third as a
small group to evaluate progress and understanding by watching, and asking questions.
Now that I have listed my outcome:
Determine how the learning will be observed
What will the children do to know that the learning has occurred?
What should children do to demonstrate the understanding of the mathematical
concepts, skills, and big ideas?
What assessment tools will be the most suitable to provide evidence of student
understanding?
How can I document the children’s learning?
Continue using this assessment.
Lesson #2, I want to provide a connection of the representations for the addition and subtraction
to ten. Introduce this rubric. Tell the students that they should think about all of the ways they are
representing their work. How are they connected to each other?
Plan the learning environment and instruction
What learning opportunities and experiences should I provide to promote the
learning outcomes?
What will the learning environment look like?
What strategies do children use to access prior knowledge and continually
communicate and represent understanding?
What teaching strategies and resources will I use?
How can I differentiate the lesson to challenge all students at their learning
ability? How will I integrate technology, communication, mental math, reasoning,
visualization, etc into this lesson? (7 Processes) Look at your outcomes to see
which of the processes you should be including.
Plan your lesson here: What lesson format will you use?
BEFORE-DURING-AFTER? Math PODS? ETC.
Before: Review learning from last lesson through a discussion.
I see… I think… This reminds me of…
Math vocabulary: commutative property, fact family
Can anyone tell a combining or separating story from this picture?
Pass out the number line and two different colors of linking cubes (rods of ten in 2 colors).
How can we show this number story on a number-line?
Pass out number-lines.
During: Work with a partner to solve.
After Show and share the different
representations students used to solve this
problem.
Discuss the connection. How is the ___ problem
similar to this problem? How are the different
representations used in both stories? This is
proving mathematically in a different way.
Assess student learning and follow up
What conclusions can be made from assessment information?
How effective have instructional strategies been?
What are the next steps for instruction?
How will the gaps in the development of understanding be addressed?
How will the children extend their learning?
Where do we go next?
Have a variety of ‘tools’ Photo copy of ten frames, part-part-whole, number-lines
for students to uses as they solve the problem. These tools are ways for all
students to be successful. Some children may not need the tools as they will be
able to represent these graphic organizers themselves. The more they see and use
these tools; eventually they will not need them as they will be able to draw them on
their own.
Use the NCTM rubric to evaluate.