Focus on Math

Helping children become mathematicians!

10 New Year’s Resolutions for the Math Classroom December 31, 2014

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1. praise effort, not correct answers
2. make sure my students know their intelligence is not fixed: hard work pays off
3. make my classroom a safe place for students to take risks
4. encourage students to take risks
5. give my students rich problems that require they engage in problem solving
6. build a class repertoire of strategies
7. have “thinking tools” handy
8. give regular attention to basic facts (for students who do not know them)
9. give students lots of opportunity to talk to each other when solving problems
10. support math vocabulary learning with a word wall chart

Mathematically yours,



Use What You Know to Figure Out What You Don’t Know March 7, 2014

Screen shot 2014-03-07 at 10.36.15 AMI was working with some students this week who were learning their “basic facts” in multiplication. These are generally considered to be those one-digit times one-digit problems that we use when we figure out the products of multi-digit problems. I was going over some different strategies and ways of thinking that can be used to help students learn those facts.

There are a number of strategies that can help in the learning of basic facts, but one phrase sums up many of those individual strategies: “Use what you know to figure out what you don’t know.”

This phrase actually applies to FAR more than just the learning of basic facts. The truth, however, is that often we condition students to NOT think for themselves in mathematics. We have a long tradition of teaching by telling: the “here’s how to do it now go practice 50” method. In reality, that kind of math lesson programs students to think that unless someone has told them “the way” to do something (and, of course, they must remember exactly how to follow the directions of “the way”). If they forget, they are stymied and cannot know how to proceed. They remain in their “stuck” position until someone comes to rescue them with “the way”.

It is far better to regularly encourage students with the idea that when they are stuck, they need to stop and think about the things they DO know that can be applied. We might ask questions (and teach them to ask themselves) such as these:

  • What might be something similar that you do know?
  • If the problem had smaller or simpler numbers, how would you try to solve it?
  • Why did you choose to do it that way?
  • What is important in the question?
  • Is there a pattern?
  • Is there a way to record what you have done so far so you a pattern might be noticed?
  • Can you think of another way to do that?
  • Does this remind you of another problem you have done?

In the case of basic facts, “Use what you know to figure out what you don’t know,” might look like this: a student cannot remember 6 x 8. But 5 x 8 is known. So, knowing that 5 groups of 8 is 40, he need only add one more group of 8 to have the needed 6 groups of 8; thus 40 + 8 = 48 is the solution to the unknown fact.

Students may need practice in doing such strategies, but the important thing is that there ARE strategies to help. It removes the case of having to rely solely on memory and sitting there stuck if memory fails.

What are you doing in your classroom today that encourages students to help themselves when they are stuck? Maybe post the title phrase for them (and model for them how it looks): “Use what you know to figure out what you don’t know.”

Strategies make a difference in student learning!

Mathematically yours,




Math Bowling October 1, 2013

math bowling pic This “Math Bowling” activity is one that students tend to love! It is great for practicing math facts as well as for stretching students’ thinking.

The activity is done as follows (students alone or in pairs):
Roll three dice (your choice whether to use regular six-sided dice or include one or more different dice, such as a ten-sided die). Write the numbers in the boxes marked “strike”. Using all three numbers each time exactly once, students work to write equations to equal each of the numbers 1 to 10 of the “pins” marked on the sheet and thus “knock them down”. Students may use whatever operations they understand: addition, subtraction, multiplication, and division are standard, but students may also use exponents, roots, and factorials if those are in their realm of mathematical knowledge.

For instance, if the numbers rolled are 2, 3, and 6, students might “knock down”
1 = 6 – 2 – 3 OR 1= 6/(2 x 3)
3 = [( 3!)/6] + 2
4 = (2 x 6) ÷ 3
5 = 6 + 2 – 3
6 = (6 ÷ 2) + 3
7 = 3 + 6 – 2
9 = (6 ÷ 2) x 3

If the students did equations for those 7 numbers/pins, that would constitute the first throw of the ball. Since all the pins are not knocked down, the player may roll the dice a second time, record the numbers in the boxes marked “spare” and try to knock down the three remaining pins using that second set of numbers to score a spare. If that is not accomplished, the student scores the number of pins knocked down in the two “throws”.

If you wish, as players take multiple turns, you can calculate scores in the manner that 10-pin bowling is actually scored. As someone who was on a youth bowling league in my younger days, I know the scoring system well. There is some good math in the score keeping, too! If you are not familiar with that scoring system, here is a website which will walk you through the scoring process.

Download the “Math Bowling” sheet here.
Download the score sheet here.

I hope you will give math bowling a try with your class.
Mathematically yours,

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Basic Facts: Mental Math as a Foundation for Multiplication Fact Strategies March 16, 2012

Basic Facts are still very important. Although newer curricula put a greater emphasis on problem solving, communication, reasoning, and representation of numbers, basic facts are still an integral part number sense in students. If a student is a good “memorizer”, then learning the multiplication facts will not be difficult. However, for many students the random bits information we call “facts” don’t stick well in the brain (the brain tends to remember information that is personally meaningful!), and thus it is important that we support those students in their learning by teaching and rehearsing thinking strategies.

Before we look at those particular strategies that are useful for learning the multiplication facts, there are some “prerequisites” to consider. Many of the strategies I will be suggesting use some kind of mental math to help students go from a known fact to an unknown fact.

The mantra for students is this: “Use something you KNOW to get to something you DON’T KNOW!” This is a comforting thing for students, particularly those who have struggled with learning their facts. They tend to feel that there is no hope for them. In some cases they have worked for a very long time, even several years, to memorize these facts, and at this point they feel like it is a hopeless task. We need to offer hope in the notion that they can begin their learning with things they do know, and build from there.

Consider working with your students to build these kinds of skills, remembering to tie them to concrete and/or visuals (such as ten frames or 100 dot arrays):
• Subtracting a single digit number from a multiple of ten (e.g., students use the known fact of 10 – 6 to solve 60 – 6. Tie into ten frames).
• Subtracting a double-digit number from a multiple of ten (e.g., build on the previous skill and have students solve 60 – 16 by subtracting first 10 and then 6.)
• Doubling any 2 digit number, using whole-part-part strategies if necessary. It may be easy to double 12 (think 10 + 10 + 2 + 2), but it will be harder to double 16. Students might consider 16 as 15 + 1, then double each of the two parts, and add back together (think 15 + 15 + 1 + 1 — look for “friendly numbers ending in 5’s or 0’s).
• Adding any single digit number to a double-digit number, particularly when the sum of the one’s place digits is greater than 10. E.g., 35 + 7 can be considered as 35 + 5 + 2; 48 + 6 can be considered as 48 + 2 + 6. (Pull apart the number to be added in a way that makes a group of ten.)
• Subtracting any single digit number from a double-digit number, particularly when “regrouping” would be required. E.g., 54 – 8 can be 54 – 4 – 4. (Again, break apart the number being subtracted into parts that make the work easier.)

It is well worth the time that you invest with students doing mental math. In As well as being a great life-skill, mental math allows students to be flexible with numbers and use powerful thinking strategies.

Mental Math and Basic Facts — don’t skip these important things!
Mathematically yours,


Basic Facts: the Last Addition Facts February 25, 2011

In three previous entries I have discussed the learning of basic facts with the visual tool of ten frames. Using such visual tools can help children (especially the ones who are not good “memorizers”) use strategies to get get from things they know to things they don’t know — which in this case are the basic facts.

There are, of course, the +0 facts. For young children it is not trivial — they must make the connection that you can add zero things to other things, but that the number of other things does not change. So although we often generalize that adding numbers together gets a bigger number, this is not the case with adding zero (that generalization fall short elsewhere, too, such as with adding integers — it is, in fact, a false generalization!)

So,we have looked at strategies for +0, +1, +2, +9, +8, +5, =10, doubles, and near doubles, and, amazingly enough, every fact of the 0 to 9 addition grid has now been addressed with the exception of four facts (see inserted grid): 3 + 6; its pair 6 + 3; 7 + 4; and its pair 4 + 7.

Both of these pairs of facts can be tied to strategies that children already know. Each can be tied to an =10 fact: if 6 + 4 = 10, then 6 + 3 must equal only 9 (if adding one less, the answer must be one less). Similarly, if 7 + 3 = 10, then 7 + 4 must equal 11 (if adding one more, the answer must be one more). The 6 + 3 = 9 may also be tied to the visual of three rows of three dots, as is on a domino: two rows of three is 6, another row of three totals nine.

It is important to remember that all facts can be learned with efficient, mental strategies. Counting on fingers or with pencil taps is NOT an efficient strategy or a mental strategy, and we should strongly discourage children from using these methods. Drill of addition facts should only take place once the facts and/or strategies are in place. Drilling does not help a child learn the facts if he does not already know them.

Again, basic facts are truly basic. It is very important for children to learn them, and a strategy approach is useful. Here’s hoping things all “add up” for you and your children!

Mathematically yours,


Ten Frames for Learning Math: Basic Facts – Doubles and Near Doubles February 22, 2011

The basic facts which are the doubles are those facts which add two of the same: 3 + 3, 7 + 7, and so forth. They have many connections to things in real life, and these should be explored. Many things on the human body come in pairs of 2; some things come pairs of 5. The legs on insects come in pairs of 3, while spider legs come in pairs of 4. Talking about such doubles is a great way to start with young children.

Skip counting by two’s is also a great connection for the doubles. Many children are quite fascinated when they realize that the answers to all of the doubles questions lie in the skip counting sequence.

The doubles can be tied to ten frames, especially those larger than 5 + 5. If a child looks at two six cards, each has a full five on it. Together these can be put together as a full 10, leaving only the two other single dots. Thus 6 + 6 becomes 5 + 5 + 1 + 1 or 10 +2. Similarly, 8 + 8 can become 10 + 3 + 3. Such strategies offer ways for children to eventually close their eyes, see the needed 10 frames, and answer the questions.

Once the doubles are learned, then the near doubles can be addressed. We want children to recognize that 6 + 7 can be thought of as 6 + 6 + 1 (or 7 + 7 – 1, as some children want to double the larger number). Even the “two-aways” can be learned in this matter. 6 + 8 can be 6 + 6 + 2. That fact can also be addressed by compensation: take one from the 8 and move it over to the 6, thus changing 6 + 8 to an actual double 7 + 7.

Although the ten frames provide a visual/pictorial tool, younger children can use actual counter to go through the motions of these kinds of strategies. Egg cartons are a wonderful tool for this! Just cut two of the “cups” off one end of a carton leaving 10 cups in the same formation of a 10 frame. Children can then put 6 counters in one egg carton, 8 in another and then physically move one from the 8 to the 6, revealing the 7 + 7.

Mathematically yours,


Ten Frames for Learning Math: Basic Facts +5, =10 February 18, 2011

Ten frames (click to download), as discussed in the previous post, are powerful visual tools for helping students of all ages learn basic facts. Besides being helpful for facts that are +1, +2, +9, and +8, the ten frames are also great for +5 and =10 facts.

For the +5 facts, start with the 5 card. If you add any other number 1 to 5, when that card is laid beside (or partially on top of) the 5 card, together they look like one of the other cards 6-10. Since these cards are already familiar to the child, the answer to the fact can easily be visualized. If you add any number 6 to 9 to the five card, the cards can be placed so to see the column of 5 dots on the 5 card beside the column on 5 dots on the other card. Of course, the two columns of 5 together make 10, and then the remaining part of the second card is added to the 10. So, 5 + 8 becomes 5 + 5 + 3 or 10 + 3, a much easier fact to learn and remember.

The =10 facts are the ones apparent on every one of the ten frame cards. For instance, when looking at the 6 card, since there are 4 spaces without a dot, 4 must be added to make 10. The corresponding subtraction facts are useful here, as well. 10 – 4 = 6 is also clearly seen on the ten frame card for 6. The cards can be flipped over one at a time, and for each card the addition fact apparent from the dots and spaces can be practiced.
Remember, basic facts are basic! Have fun with them!
Mathematically yours,