Learning Basic Single-Digit Combinations (Facts)
Learning Basic Single-Digit Combinations (Facts): Developing Important Ideas in Mathematics; Part III – Number String Options to Build Derived Strategies
submitted by
James Brickwedde, Project for Elementary Mathematics
www.projectmath.net
MCTM VP for Elementary
Previously in the September and October issues of MathBits, I have described a case for developing strategies for students to learn their basic single-digit combinations. Developing these derived strategies is based upon specific developmental research on how to assist students in gaining fluency with these combinations. Rather than a sole emphasis on rote memorization in high stakes timed environments, consciously spending time on these derived strategies helps students develop the very strategies and algebraic relations that are necessary when operating with multidigit numbers. Example: if I know 12 – 2 = 10, and 10 – 3 = 7, then 12 – 7, instead of counting back one-by-one, I can use the strategy of Get back to a Ten to fluently and accurately solve the problem. This strategy holds if the numbers were 72 – 15 as I can quickly increment backwards going 72 – 10 → 62 – 2 → 60 – 3 → 57. Students who are well grounded in such strategies lay down such strong neurological pathways that they determine the result within two to three seconds. This is the time considered to be within the recall range.
The same is true for multiplication single digit combinations. Instead of teaching students the nines finger trick as a quick (albeit limited) mnemonic device, compensation is an effective mathematical strategy that extends to higher multidigit combinations. Example: 9 x 7 is frequently one of the hard combinations for students to commit to memory. Instead of skip counting by sevens starting from zero, 10 x 7 is an easy known combination that is readily recalled. So, 9 x 7 = 10 x 7 – 1 x 7 as in Nine groups of seven is the same as having ten groups of seven minus one less group of seven. This can extend to a combination such as 29 x 12. 29 x 12 = 30 x 12 – 1 x 12. The strategy is transferable. The number sense is strong. And, if the strategy has become itself well ingrained and fluent, the solving of the more complex combinations become fluent and accurate as well as.
Spending intentional and focused time on nurturing these strategies across the grades yields long term benefits. The strategies are grounded in algebraic properties. The strategies are generative – pedagogically sustainable – as they allow the student to grow mathematically in the future. The strategies build towards the future mathematical concepts. The trouble with mnemonic devices and strict rote recall is that they have limited use. They only pertain to that one particular recalled item. This is the general issue with a number of surface tricks and patterns. They may help students get through the short-term test but the tricks and devices limit the students’ capacity to engage in deeper mathematical concepts.
Multiplication Strategies
The following are examples of how some simple number strings can be designed to help students become fluent with certain multiplication single-digit combinations. Those focused on here are what research studies, along with decades of teacher observations, has demonstrated to be remarkably hard for students in gaining quick recall. This particular image below comes from the work of Constance Kamii at the University of Alabama, Birmingham. Notice the one white area that captures 3×6 through 4×9 is also captured in the commuted combinations of 6×3 through 9×4. The third area captures the longest to commit to memory of 6×6 through 9×9. If, as a teacher, you are aware of these combinations being more difficult with which to gain fluency, then designing a series of number strings presented over a period of days helps students build a network of related combinations that increases the capacity through which to quickly recall. I stress again that these strategies are transferable to multidigit combinations. As a result, you are fostering both fluency and foundational algebraic properties and relational thinking.
Intent:
- Anchor 3×6 and 6×3 (Three groups of six and six groups of three), particularly to interrupting a tendency to skip count to find the answer
- Use these two combinations to derive related and more difficult combinations;
- Nurture use of the distributive property of multiplication over addition as well as the commutative property
Part 1: Posing and Asking the Questions
Day 1: [Show each expression one at a time]
3 x 6 What are three groups of six?
6 x 6 Use what you know about three sixes to figure out six sixes.
12 x 6 Use what you know about six sixes to figure out twelve sixes.
3 x 6 So again, what are three sixes?
5 x 6 Use what you know about three sixes to figure out five sixes.
8 x 6 Use these two previous combinations to figure out this new one.
Day 2:
3 x 6 [Use similar language as previously]
4 x 6
7 x 6
3 x 6
4 x 6
8 x 6
Day 3:
3 x 6
6 x 3 Why does 3 x 6 have the same answer as 6 x 3? 3 x 6 = 6 x 3
6 x 6 Use either of the above combinations to figure out six sixes.
6 x 3
7 x 3
8 x 3
Day 4:
3 x 6
3 x 8
3 x 7
6 x 3
3 x 3
8 x 3
4 x 3
Day 5:
3 x 7
6 x 7
9 x 7
6 x 3
6 x 4
6 x 7
There is nothing particularly magical about the above sequences. The examples are to demonstrate how, in an opening warm-up at the beginning of a lesson, substantive conversation can happen around basic fact combinations and to use what you know about one combination to figure out another combination. Yes, you can skip count starting from zero each time – something I remind students frequently – but how can you use one combination to figure out other combinations to save yourself some time. The message that is continually conveyed is that two-pronged: Use relational thinking – Use what you know to figure out what you don’t know – and work efficiently to save yourself time in the future. Recall, by itself, does not build these two key dispositions of relational thinking and making efficient judgements.
Part II: Capturing Student Thinking Mathematically
Before we end, more than asking students for the answer and verbally having them describe the strategy used to derive it, it is how you capture student’s thinking mathematically that will move the conversation towards exploring the underlying algebraic principles. As a student begins explaining his or her thinking, I intentionally capture that information in very specific formats. The product itself may or may not ever be written. The following is one example of how Day 1 and Day 3 might unfold.
Day 1: [Show each expression one at a time]
3 x 6 What are three groups of six?
- If the student builds off of 2 x 6 then I would say and write…
3 x 6 = 2 x 6 + 1 x 6
So, you are telling me that three groups of six is the same
as having two groups of six plus another one group of six?
Did I hear you correctly? [Yes]
And tell me again what the total of three sixes are. [18]
6 x 6 Use what you know about three sixes to figure out six sixes.
- 6 x 6 = 3 x 6 + 3 x 6
= 18 + 18
= 36
— So, you used three groups of six and added another three groups of six,
and you added the two eighteens together and that
gave you a total of 36, is that correct?
— So, you broke the 6 into 3 + 3 to have a total of 6 groups of six.
— 6 x 6
/ \
3 + 3
12 x 6 Use what you know about six sixes to figure out twelve sixes.
- 12 x 6 = 6 x 6 + 6 x 6
= 36 + 36
= 72
- OR – and you need to expect students to recognize alternative strategies…
12 x 6 = 10 x 6 + 2 x 6
= 60 + 12
Notice with the above notation uses a combination of very formal and informal notation. This is done to focus students’ attention to the decomposition of number. The second aspect of the notation is that is the first student called on for 6 x 6 said, “I just added 18 plus 18 and got 36,” I very specifically channel that statement to first write 3 x 6 + 3 x 6, saying, “So, the eighteen came from using 3 groups of six and you then added another. This editorial choice is, again, an intentional decision to focus student’ attention on the algebraic relationships of what eventually can become 6 x 6 = (3 + 3) x 6 = 3 x 6 + 3 x 6.
Day 3:
3 x 6 What are three groups of three?
– Note if more hands go up quickly compared to the previous two days. This will be an indicator that this combination is gravitating to the recall level
6 x 3 What are six threes?
Ask: Why does 3 x 6 have the same answer as 6 x 3? 3 x 6 = 6 x 3; Yes, they have the same product, the same total, but why do they end up being the same? Six bags with three bagels inside each bag looks and feels different than 3 bags with six bagels inside each. Why do they turn out to being the same?
Equal grouping contexts are asymmetrical. As the context of pages and bagels describe above, the commutativity aspect of the scenario is not as readily apparent to students as adults think. Spending time discussing the relationships of the equality, even to the point of working the context out with drawings or manipulatives so students can rearrange the objects. Arrays and area models are symmetrical. The commutative property is more easily seen. However, life in our culture is more dominated with packaging images – asymmetrical contexts – rather than things that come in neat rows and columns – symmetrical contexts. The time spent discussing the commutative property with single-digit combinations such as 3 x 6 and 6 x 3 is worthwhile so that when students encounter a problem such as 25 boxes of 6 (25 x 6) where the emphasis is on skip counting sixes, commuting the scenario to be 6 boxes of 25 (6 x 25) is a lot less work.
There is no one recipe for doing these number strings. Experiment with designing some combination strings. The goal is to build strategies, to assist students in understanding the underlying mathematics, and to think relationally. This will help them transfer this knowledge to other contexts as they advance in their mathematical learning.