Time
Task
Materials
Concrete, Representational, Abstract Teaching Sequence
CRA Article
http://www.mathspecialists.org/dww/Response_t
o_Intervention/Practice_Intentional_Teaching/Se
e/1923_it_mat_cra.pdf
Bradley Witzel Video on CRA
http://www.opi.mt.gov/streamer/profdev/13_1018DWW_WestEd_Box/DWW%20Response%20to%20Interv
ention%20-%20Math/Practice%20%20Intentional%20Teaching/Practice%20Summary%20a
nd%20Learn/1502%20%20Visual%20Representation%20Witzel%20Video/1502_it
_video_witzel2.mov
Graphic Organizer of Strategies:
Ask participants to solve 9 x 4 using more than one strategy.
While participants are solving the problem, walk around and select participants to share who are solving the problem using
the following strategies. As you are selecting strategies, be sure to put them in a pre-determined sequence as suggested
below. You may need to share some of the strategies that were not used.
Participants will draw models as they are discussed on the backside of the graphic organizer. After strategies are discussed
participants will partner to complete the front side of the graphic organizer, filling in such things as vocabulary, core
connections and possible questions at varying DOK levels.
*At this point, don’t tell them whether the strategy is concrete, representational, or abstract.
Model with manipulatives (concrete)
Draw a picture (representational)
Nine groups with 4 objects in each group.
Number line (representational)
Make 9 jumps of 4. Each number represented on the number line is 4 more than the previous number.
Strategy Graphic Organizer
Document Camera, or white board
An array (representational)
9 x 4 or 4 x 9 (Discuss the commutative property)
Use arrays to show that if you turn them they are the same, you are not adding anything or taking anything away.
Repeated addition or skip counting (abstract)
Double and double again (abstract)
Solve
2 x 9 = 18.
Double for 4 x 9 = 18 + 18 = 36.
Solve using a 10 (abstract)
10 x 4 = 40
Remove 1 group of 4. 40 – 4 = 36
To prove it, draw a 10 x 4 array and remove one group of 4 for 9 x 4 (representational).
Doubling and Halving (abstract)
9 x 4 = 18 x 2 = 36 x 1
You double one factor and half the other until you are left with an easier problem to solve
Use the 9s pattern (abstract)
All products of 9 add up to nine. Whatever number you are multiplying by, subtract 1. In this case it is 4. 4 – 1 = 3. Three is the
number in the tens place. To find the number in the ones place, think what plus 3 equals 9. 3 + 6 = 9. So the answer is 36.
Distributive property (abstract)
Break one factor into 2 numbers that are easy to multiply by. Leave the other factor whole.
Show video clip 4:59:
You will need to be aware that there is an error in the video that some participants may notice. In
the beginning when the narrator shows a multiplication fact to be solved, he demonstrates columns
and rows. In fact, if we’re being technical, rows represents the first number and columns represent
the second number. This is not the main point of the video, however. The point is that we can solve a
fact we don't know by using the distributive property. Another important concept of the video is that
one number can be broken into more manageable chunks, while the other number needs to
remain whole. Also because of the commutative property, the fact can be looked at either way.
Distributive Property Video
http://learnzillion.com/lessons/966-use-thedistributive-property-of-multiplication-to-solveunfamiliar-facts
30 min
Graphic Organizer Partner Activity
With a partner participants will now complete the front side of the graphic organizer. They will reference their core, the
progression documents and the math cognitive rigor matrix to complete the graphic organizer. Allow 10 minutes to share
out.
10 min
The Distributive Property Reading
Distribute NCTM article and allow time for reading. Ask participants at their tables to discuss article using the Final Word
Protocol below.
35 min
Final Word Protocol
1. In tables identify a facilitator/time-keeper.
2. Each person needs to have one “most” significant idea from the text underlined or highlighted in the article. It is often
helpful to identify a “back up” quote as well.
3. The first person begins by reading what “struck him or her the most” from the article. Have this person refer to where the
quote is in the text - one thought or quote only. Then, in less than 3 minutes, this person describes why that quote struck him
or her. For example, why does s/he agree/disagree with the quote, what questions does s/he have about that quote, and
what issues does it raise for him or her, what does s/he now wonder about in relation to that quote?
4. Continuing around the circle each person responds to that quote and what the presenter said, briefly, in less than a
minute. The purpose of the response is:
• to expand on the presenter’s thinking about the quote and the issues raised for him or her by the quote,
• to provide a different look at the quote,
• to clarify the presenter’s thinking about the quote, and/or
• to question the presenter’s assumptions about the quote and the issues raised (although at this time there is no
response from the presenter).
5. After going around the circle with each person having responded for less than one minute, the person that began has the
“final word.” In no more than one minute the presenter responds to what has been said. Now what is s/he thinking? What is
his or her reaction to what s/he has heard?
Graph Paper Array Activity
Write each problem one at a time on the board or large post-it pad. Solve each odd number problem by representing it on
graph paper. Solve the even number problems by manipulating the odd number array. See examples on paper copy.
1.
2.
3.
4.
5.
6.
Strategy Graphic Organizer
Core
Progressions
Cognitive Rigor Matrix
NCTM Article “The Distributive Property in Grade
3?”
Final Word Protocol
Graph paper
10 x 9
9x9
6x6
3x6
9x3
6x9
**Now go back and identify each strategy as being concrete, representational, or abstract.
Graphic Organizer of Strategies:
Solve 12 x 23 using more than one strategy.
While participants are solving the problem, walk around and select participants who are solving the problem using the
following strategies. As you are selecting strategies, be sure to put them in a pre-determined sequence as suggested below.
You may need to share some of the strategies if they were not used.
Participants will draw models as they are discussed on the backside of the graphic organizer. After strategies are discussed
participants will partner to complete the front side of the graphic organizer, filling in such things as vocabulary, core
connections and possible questions at varying DOK levels.
Strategy Graphic Organizer
Document Camera, or white board
Note: manipulatives, pictures, and repeated addition are no longer efficient strategies for a 2-digit by 2-digit multiplication
problem.
An Array (representational)
Break into friendlier chunks. This example used tens and ones.
(10 + 2) x (20 + 3)
Partial Products
12
x23
6 (3 x 2)
30 (3 x 10)
40 (20 x 2)
200 (10 x 20)
276
Distributive Property (abstract)
Break one factor into 2 numbers that are easy to multiply by. Leave the other factor whole.
(10 x 23) + (2 x 23) = 12 x 23
230 + 46
= 276
Common mistake: (10 x 20) + (2 x 3). Show the array model to demonstrate that not all parts are represented.
Reason behind error: students are used to breaking one number apart and keeping one number whole.
Compensation (abstract)
Select an easier number to multiply by (in this case 25), and then subtract the number of groups you added on.
Common mistake: only subtract 2 instead of 12 groups of 2.
Lattice: Great opportunity to discuss procedure without understanding.
Often time students love lattice model, however this method is very procedural and lacks place value. Students who prefer
this method are probably lacking foundational place value understanding. These are the students that struggle with the
traditional algorithm because they do not understand the role zero is playing.
Traditional algorithm (abstract)
Connect to the array model. Each problem solved is represented with the corresponding number in the array model.
Graphic Organizer Partner Activity
Strategy Graphic Organizer
Core
Progressions
Cognitive Rigor Matrix
Division Graphic Organizer of Strategies:
Graphic organizer
Graph Paper/ Hundreds Chart
Colored Pencils
Cube Manipulatives or Counters
With a partner participants will now complete the front side of the graphic organizer. They will reference their core, the
progression documents and the math cognitive rigor matrix to complete the graphic organizer. Allow 10 minutes to share
out.
Ask participants to solve 56 ÷ 8 using more than one strategy.
Select participants to share most of the following strategies. You may need to share some of the strategies if they were not
used. As you are selecting strategies, be sure to put them in a predetermined sequence as suggested below, moving from
concrete examples to more abstract examples.
Participants will draw models as they are discussed on the backside of the graphic organizer. After strategies are discussed
participants will partner to complete the front side of the graphic organizer, filling in such things as vocabulary, core
connections and possible questions at varying DOK levels.
Model with Manipulatives (concrete)
Make 8 groups of 7 (or 7 groups of 8) cubes or counters. Share a counting on strategy or partitioning into equal groups.
Draw a picture (representational)
8 groups of 7 objects in each group or 7 groups of 8 objects in each group.
Equal Groups- Partitioning (representational)
8 groups of 7 objects in each group or 7 groups of 8 objects in each group. Similar to modeling with manipulatives and
drawing a picture.
Array (representational)
56 ÷ 8 or 56 ÷ 7
(Discuss the commutative property)
Use arrays to show that if you turn them they are the same, you are not adding anything or taking anything away.
Repeated Addition- Skip Counting (abstract)
7+7+7+7+7+7+7+7
or
8+8+8+8+8+8+8
Repeated Subtraction- Skip Counting (abstract)
56-7-7-7-7-7-7-7-7
or
56-8-8-8-8-8-8-8
Fact Family/ Inverse Operation Multiplication (abstract)
Traditional Algorithm (abstract)
Partial Quotient (abstract)
Answer: Add 5+2=7
Graphic Organizer Partner Activity
With a partner participants will now complete the front side of the graphic organizer. They will reference their core, the
progression documents and the math cognitive rigor matrix to complete the graphic organizer. Allow 10 minutes to share
out.
Using Place Value to Divide (abstract)
Strategy Graphic Organizer
Core
Progressions
Cognitive Rigor Matrix
Show clip: (7 minutes)
https://www.teachingchannel.org/videos/common-core-teaching-division
Questions to Consider:
How does Ms. Simpson encourage mathematical discourse?
Why is it important to discuss and understand multiple strategies?
What can you learn from Ms. Simpson about facilitating discussion?
Discuss value of number strings
Try a number string with participants.
8÷2
16÷2
32÷2
48÷2
48÷4
480÷4
484÷4
480÷40
Use hyperlink to help guide discussion:
http://www.svmimac.org/images/2012CI.NumberStrings4.3-4.pdf
Math Task:
The Teachers’ Lounge LESSON PLAN
Drawing paper
Unifix cubes, tiles, graph paper, poster paper,
markers
LAUNCH:
A teacher notices a serviceman in the teachers’ lounge filling two different vending machines with different beverages. The
first machine only holds water. The second one holds different flavors of juice. The teacher asks the person filling the
machines how many bottles of water the machine can hold. The machine can hold 156 bottles. Next to the machine, the
man has a cart with the water bottles on it. The bottles were in six packs, and the teacher wonders how many six packs the
machine will hold.
The second machine, the juice machine, also holds 156 bottles. The machine doesn’t just hold one kind of juice, there are six
different flavors: apple, cranberry, lemonade, grapefruit, grape, and orange. The machine holds an equal amount of each
flavor. The teacher wonders how many bottles of each flavor fit in the juice machine when it is full.
EXPLORE:
Students work on the problems in pairs. Teacher walks around and takes note of the variety of strategies that they see. They
select two or three strategies to share during the debrief / gallery walk.
Here are some strategies you might see:

The bottles arranged in arrays

Students may not realize that the two problems are related, don’t lead them to the connection, let them make it
on their own

Groups of six

Skip counting, with each multiple of six representing another six pack

Partitive grouping

Counting three times

Drawing all bottles, then grouping them by 6’s and counting the groups

Looking at every tenth multiple in the skip counting sequence

Using ten-times to make partial products
Discuss:
As students finish up their work, have them make posters that depict the important things they want to share the next day in
Math congress. The posters should be clear for others to understand.
Questions to ask students as they work on their posters:
How is this strategy the same? How is it different? Did you try any strategies before this one? How do we know this strategy
worked? Which strategy is the most efficient? Will these strategies work for bigger numbers?
ASSESSMENT:
Collect student work. Sort into three groups: those who’ve got it and are ready to move on, those who need a little more
practice, and those that need extra help. How will you support each learner?
GALLERY WALK DAY TWO:
Conduct a gallery walk to give students a chance to comment on their peers’ posters. Ask
strategies and talk about the relationship between problems.
students to explain their
Pass out sticky notes for students to ask questions or comment on the class posters. Allow time for students to post their notes
on the posters and then time for them to read the comments and questions on their own posters.
Have students meet to discuss the posters you selected from day one. Encourage students to talk about the strategies
shared and to make comparisons between strategies. Could discuss different division notations.
Chris Natale & Fosnot, C. (2007) The Teachers’ Lounge. (pp. 11-20). Portsmouth, NH:
Heinemann.
https://www.teachingchannel.org/videos/common-core-teaching-division