Common Core Standard: 3.OA.B.5
A week’s’ worth of warm ups or a single lesson; the choice is yours. If you need to revisit for conceptual, interleaved practice to build fluency and retention, then use this as a week of warmups. Alternatively, you could use this as a single lesson building conceptual understanding of multiplication and as an introduction of the distributive property.
DAY ONE WARMUP/ STAGE ONE OF A SINGLE LESSON
Multiplication as an area/array. You could use a pure array, but drawing 56 circles is too tedious, and remember Sticky Math is meant to be quick an easy enough to do on a sticky. This may look like too much to draw; however after drawing a rectangle you are simply drawing some lines with a straight edge. These lines extend past the side of the rectangle to help students find the side lengths.
Connections could include:
- 8 as a side length in both stickies
- seeing it horizontally, 8 groups of seven (seven in each row); although, on the right sticky it is 8 groups of five plus two separated, which is still seven
- 7 as a side length being equivalent to the sum of the side lengths on the right sticky
- seeing it vertically, 7 groups of eight (eight in each column) with 5+2 groups of eight on the right sticky
- 56 boxes created in both stickies
- 56 is equivalent to 40 + 16
- rows of 5 and 2 are present in both stickies if you split the left sticky after 5
The blank sticky is a reminder that students could create their own representations; these might include:
- 56
- a different type of picture
- the same picture rotated 90 degrees
- splitting the pictorial representation to create partial products
- changing the numeric expression to 7 x 8 (the commutative property)
- changing the expression to create partial products
These leads me to a general point about using Sticky Math. What if students do not make these type of connections or a specific connection you would like them to make? Try saying, “Someone in my class last year made this representation,” and draw it on a third sticky. Ask students how it connects to the two existing stickies. You do not have to tell them that the person in the other class was you. When you do this, you are building on their existing knowledge (of the two stickies), which is the true definition of scaffolding.
If students are not making any connections at all, then the activity may not yet be appropriate for them. It is good to have any easier activity on hand as a “Plan B” should it not go well. That said, even when an activity does not go well, it is still gives you valuable formative feedback about what students know and are able to do so you can adjust your instruction accordingly.
DAY TWO WARMUP / STAGE TWO OF A SINGLE LESSON
The idea of day two is to explicitly see the decomposition of the 8 x 7 into 8(5) + 8(2). No symbolic representations are given purposefully to keep the activity fully conceptual and to not overwhelm the students’ working memories with an additional form of representation at this stage; however, should students suggest their own symbolic representation during the connection phase, then that is fantastic. This activity sets the stage for understanding what the symbolic notation means the following day.
Connections could include:
- if you combine the right image, then you get the left image
- 8 as a side length in both stickies
- seeing it horizontally, 8 groups of seven (seven in each row); although, on the right sticky it is 8 groups of five plus two separated, which is still seven
- 7 as a side length being equivalent to the sum of the side lengths on the right sticky
- seeing it vertically, 7 groups of eight (eight in each column) with 5+2 groups of eight on the right sticky
- 56 boxes created in both stickies
- 56 is equivalent to 40 + 16
- rows of 5 and 2 are present in both stickies if you separate the left sticky after 5 in a row
The blank sticky is a reminder that students could create their own representations; these might include:
- splitting the left image in a different way like 4 + 4 instead of 5 + 2
- drawing a simple rectangle of the left sticky and labeling the sides, which is another layer of abstraction
- drawing a simple rectangles of the right sticky and labeling the sides, which is another layer of abstraction
- using symbolic representation to create any type of numeric expression like 8 x 7, 56, 8 x (5 + 2), etc.
- reflecting the right sticky to get 2 + 5 instead of 5 + 2 (the commutative property)
- adding another split to create (5 + 3)(5 + 2) creating four partial products
I doubt students would create the last representation; however, I add it for those using this activity with higher grade levels. Changing the problem to (13)(12) = (10 + 3)(10 +2) would be great for higher grade levels leading to area representations with base ten blocks, which could be further connected to (x + 2)(x + 3) in algebra. That sounds like another blog post!
This leads me to a question and another general point about using Sticky Math, do you feel like we are moving too slow in this progression? I hope so. We have to remember that we have fully developed brains while our students do not, unless you are teaching 25+ year olds. Sometimes math is presented too fast (especially when it is focused on procedures only), connections are not made explicitly (leading to a lack of retention), and there is very little depth (leading to a lack of fluency). The goal with each day of the warmup or stage of the lesson is to build a large web of connections so content can be understood and retained leading to fluency and a greater depth of knowledge.
DAY THREE WARMUP / STAGE THREE OF A SINGLE LESSON
The idea of day three is to explicitly see the decomposition of the 8 x 7 into 8(5) + 8(2) symbolically, pictorially, and to make connections between the two. Students need to make connections between the symbols used and their meaning by connecting to the pictorial representation; otherwise, it may only appear to be a bunch of numbers no matter how often we explain it to them verbally.
Connections could include:
- if you add the 5 and the 2 in the left sticky you get the 7 in the right sticky
- all stickies involve multiplying by 8
- 8 x 5 represents the left section of the middle sticky & both are 40
- 8 x 2 represents the right section of the middle sticky & both are 16
- 8 x 7 represents both sections of the middle sticky & both are 56
- all representations are equivalent to 56
Students could create their own representations; these might include:
- splitting the middle image in a different way like 4 + 4 instead of 5 + 2
- drawing a simple rectangles of the middle sticky and labeling the sides, which is another layer of abstraction
- using symbolic representation to create any type of numeric expression like 7 x 8, 56, 8 x (5 + 2), etc.
- reflecting the middle sticky to get 2 + 5 instead of 5 + 2 (the commutative property)
- changing the expression on the left sticky to get 8(2) + 8(5) instead of 8(5) + 8(2) [the commutative property]
- changing the expression on the left sticky or the picture in the middle sticky to create additional partial products such as 8(2) + 8(2) + 8(2) + 8(1)
Once again, I doubt students would create the last representation; however, I include it to point out that you can give students strategies for finding 8(7) with only knowledge of the math fact families of twos and ones. By the way I choose this example specifically, not only because it is written into the standard, but because it is a math fact many of my secondary students struggle with.
The strategy is not just for remediation; it can be used to extend into higher content. The applications of the distributive property are endless. Here is just one adapting the example in the final bullet: 8(2221) = 8(2000) + 8(200) + 8(20) + 8(1). Students may be able to identify place value, but does it inform their mathematics?
Using this activity with other difficult to memorize math facts will give students a strategy that works for any math fact they should know or could know. Better yet, as a formative assessment of the distributive property and to make this idea explicit, ask students to create their own Sticky Math Pair or Sticky Math Triad for any hard to memorize math fact like 8(3), 8(4), 7(6), etc.
DAY FOUR WARMUP / STAGE FOUR OF A SINGLE LESSON
The idea of day four is to explicitly see the decomposition of the 8 x 7 into 8(5 + 2) symbolically, pictorially, and to make connections between the two. The only change from day three is the symbolic representation. In day three, it is helpful to view the area of each piece individually or circle each piece vertically. Here it is helpful to view it as eighth rows of five plus two or circle each row.
Connections could include:
- if you add the 5 and the 2 in the left sticky you get the 7 in the right sticky
- all stickies involve multiplying by 8
- (5 + 2) on the left sticky is equivalent to the 7 on the right sticky
- (5 + 2) represents one row or 5 + 2 groups (of 8 in a column) on the middle sticky
- 8 represents 8 groups or eight in a column
- 8 x 7 represents both sections of the middle sticky & both are 56
- all representations are equivalent to 56
- any connection from previous days
Students could create their own representations; these might include:
splitting the middle image in a different way like 4 + 4 instead of 5 + 2
- drawing a simple rectangles of the middle sticky and labeling the sides, which is another layer of abstraction
- using symbolic representation to create any type of numeric expression like 7 x 8, 56, 8 x (4 + 4), etc.
- reflecting the middle sticky to get 2 + 5 instead of 5 + 2 (the commutative property)
- changing the expression on the left sticky to get 8(2 + 5) instead of 8(5 + 2) [the commutative property]
- changing the expression on the left sticky or the picture in the middle sticky to create additional partial products such as 8(2 + 2 + 2 + 1)
- anything representation from previous days
DAY FIVE WARMUP / STAGE FIVE OF A SINGLE LESSON
Full symbolic! The idea is to realize that all three are equivalent, but refer to different quantities. To help ensure full understanding conceptually, it may be helpful to have the two pictorial representations from day/stage two present for reference. Better yet, to increase the connectivity, conceptual understanding, and formative assessment inherent in the activity, ask students to draw a different picture for each representation.
Connections could include:
- if you add the 5 and the 2 in the middle sticky you get the 7 in the right sticky; they are equivalent expressions; the 5 & 2 are also in the left sticky
- moving left to right, multiplying both by 8 in the left sticky is what the middle sticky means
- moving from right to left, shows the decomposition of the number seven, and its subsequent multiplication by 8
- simplifying the expressions in the left and middle stickies leads to the expression on the right sticky
- all representations are equivalent to 56
- additional connections can be made after encouraging students to draw each expressions pictorially; those include:
- the left sticky represents two partial products, think of two separate rectangles
- the middle sticky represents a single rectangle with a line of separation
- the right sticky represents a single rectangle
Students could create their own representations; these might include:
- drawing a pictorially representation for each sticky as discussed above
- making any of the representations from days/stages one through four above
A closing, formative assessment activity might be complete a Sticky Math Matching activity like the one below:
The images above are different than those in the rest of the activity. The idea is to see if students can demonstrate transfer of the knowledge they have attained by moving to a higher level of abstraction while matching up pairs. The many small squares are unmarked but can still be reporting back as, “The one without a number goes with D.”
Lesson extension, classwork or homework – give each students 8 stickies have them create their own matching game to illustrate a hard to memorize math fact for them, have them give it to a family member to complete, and then they check it. This differentiates the assignment as some could work with 8(4) while others might work with 2(16). Have students exchange them when they are finished or return to class, match the pairs, and check each other’s work as a ticket out the door or an opening activity the following day.
One final word on the usefulness of the distributive property as a strategy to memorize math facts. Our brain is an amazing tool for mathematics. Should a student use this strategy for finding unknown math facts not only will it apply everywhere, if they do it enough times with a single fact, then their brain will look for a shortcut and just memorize the fact without having ever picked up a single flash card! (And, if you do use flash cards, then try the triangular type that you can multiply and divided with.)
If you have read this far, then give yourself a pat on the back! You are a dedicated teacher and your students are lucky to have you!
Common Core Standard: 3.OA.B.5
What connections are your students making? What modifications are you making to use this with students? We would love to hear your feedback; please submit a comment below or consider submitting your own Sticky Math activity here.