Developing 10-Anchor: 10-Facts with a 10-Frame

Developing 5-Anchor: 5-Facts with 5-Frame

Developing a sense of 5, or 5-anchor is an important concept for young children. Being able to break apart and combine numbers is a skill that aids in later math concepts and operations. For example, adding 7+5 can be done in a number of ways. A child who has developed 5 and 10 anchors may break 7 into 2+5. Since they know 5+5 is 10, 2 more is 12. As children become familiar with 5-and 10-anchors, they unknowingly use both concepts in later concepts such as multi digit addition, subtraction, multiplication, etc. For example, 9 x 8: This is a hard fact for children to memorize but if they think fluidly of numbers they can use those strategies to make sense of the problem and solve it. A child with developed number sense and good understanding of multiplication as repeated addition might think: I know 10 x 8 is 10 sets of 8 instead of 9 sets of 8. I know 10 x 8 is 80 and 80-8 (less a set of 8) is 72. Take another example, 19 x 18: 20 x 18 is 360 and 360-18 (less a set of 18) can be done easier as 360-20+2; the answer is 342.

I know, you're thinking: kids don't think this way--it's way too complicated. But the fact is many of us develop this way of thinking at an early age. As we are exposed to math, we develop these strategies. Some of us are explicitly taught these strategies, others pick it up from their own experience, and yet others are a combination of self-discovery as well as classroom experiences. We need to give children experiences that they can develop these strategies and use them in practical activities inside and outside the classroom.

Part 2: 1,2 More Than and Less Than activities

Part 1: 1,2 More Than and Less Than activities

Children should develop the idea that numbers have relationships to one another. Working with elementary and middle school age children has reinforced that notion. I've noticed that as kids get older and not develop numeric awareness and other relationships, the more likely the child will resort to memorizing facts, require longer time or not fully develop later math concepts. This has nothing to do with capability or intelligence, rather, the child has not developed key number sense areas and this slows, impedes and/or restricts the child's future math growth. Adopting an approach to emphasize these skills early is preferable but these are skills that can be taught to a wide range of ages. Although, I'd stick to basic number recognition and number skills for children 4 and under.

The More Than, Less Than activities help to develop childrens sense of relationships between numbers. Rather than think of one particular number--and only that number--they view it as a range of related numbers. For example, the number 8 is 3 more than 5; 2 less than 10; 1 less than 9; 1 more than 7; 2 more than 6. This is a concept that is important in a wide range of operations; let's look at addition. A child working the problem 8 +13 can solve this problem in a number of ways. A child without a range of strategies or developed number sense will probably have to count out 8 and then add on 13 to arrive at an answer. A child who's developed the 'Counting On' strategy will start with 13 and count on 8 more. A child who's developed an anchor of 10, may think in terms of completing the Ten-Fact for '13'; 7 more completes the Ten-Fact (and gives us 20) and 1 more is 21. Another child who's developed a range of strategies and has some place value sense may know that 10+13 is 23 but 8 is 2 less than 10, so the answer is 21.

It may seem like an overwhelming thought process but children can learn and use a wide range of strategies. Some children may not pick up strategies so easily and need additional activities and modeling to pick up strategies. This is not to say that a child must use one specific strategy. Rather, it's the opposite. It's OK for a child to use a different strategy from another child--or from the adult in the room. I'm more concerned about concepts that a child may be familiar with. If I see a child can not make generalizations about numbers, I work on activities and games that help foster greater number sense.

Developing Math Foundations: Game-Counting On

Developing Math Foundations: Counting On

Counting on is an important strategy that we typically develop in childhood. What is counting on? Counting on is the strategy that if we have two addends, we count on after where the first addend ends.

For example, 17+3. Children who aren't automatic with numbers, haven't developed addition strategies, and haven't developed counting on as a strategy will count out 17 beginning from 1 (using their fingers or a manipulative) and then add 3 more to arrive at the answer, 20.

A child who counts on will start after where the first addend ends. Since the first addend is 17, he/she will count on 3 begining at 18; 18, 19 and 20. A child who develops a count on strategy will have typically developed the strategy after noticing that the sum remains the same using either the shorter, more efficient counting on strategy or the more laborious and time consuming strategy of individually counting out the first addend and then adding the second.

Some children figure it out on their own but others need explicit instruction and modeling to make sense of the strategy. I remember having problems with counting on up until elementary school. My main problem was uncertainty of where I begin counting on from. For example, 17+3. Do I count on beginning with 17 or after it beginning at 18? I eventually figured out the strategy by using logic and easy examples. I would do simple problems for which I knew the answer to--like 5+3. I would then experiment to figure out which was the correct strategy. Since I knew the answer was 8, only one model of the counting on strategy would give me that answer.

This is the kind of experience our kids go through to make sense of numbers and math. We should allow them to explore and make sense of it for themselves but at the same time provide activities, guidance and support for building key concepts.

Using Unifix Cubes Lesson 2: Representing a number in multiple ways

using base ten blocks lesson 5: Addition with regrouping

Many students begin learning addition with regrouping--also referred to as carrying over-- in a mechanical and almost robotic fashion. Take 17+3: We start adding up in the ones column. 7+3 is 10; bring down the 0 and carry the 1 to the tens column. 1+1 is 2; bring down the 2. The answer is 20. While they get the answer correct, there is little understanding of the process that they went through and even less understanding of the concept of place value.

Using Base 10 blocks for addition with regrouping helps children feel, touch, manipulate and see addition problems. They then begin to develop connections between numbers, quantities, place value, and operations. A child with experience with base 10 blocks will start to physically take apart and recombine numbers-- using the manipulatives--and later, do so mentally; they begin to internalize what they've learned from their experiences.

Before beginning children/students on addition with regrouping, it's important they have had sufficient time to develop familiarity with the blocks as well as some prerequisites including:

1. Recognizing relationships between units, rods and flats, i.e., 10 units make up a rod, 10 rods make up a flat.

2. Used blocks with base 10 chart to represent values. Children should have experience creating quantities using base 10 blocks on a chart and writing down their values. They should recognize that the number 113 is made up of one flat, one rod and three units. They should also create numbers using base ten blocks and write down its value.

3. Addition (no regrouping).

Optional but helpful:

4. Race to 20 (or 100--depending on what is appropriate for the child). The game helps develop the "trade-in"rule, which is an important concept in regrouping. It familiarizes students with the rule but doesn't overteach it. When students move from the game to addition with regrouping, the trade-in rule is already second nature or something that's very familiar to the student.

Using the tips and techniques listed above, students progressively develop important math ideas and build upon experiences, from prior activities, to successfully transition into new concepts. More intriguing is watching students develop a slew of ideas and concepts that we didn't have to teach them; rather they learned from their own exploration and experiences.

Using Base 10 blocks isn't the panacea for all math challenges and ills but it does help children develop a deeper understanding of numbers and the operations they're engaging in.

Using base ten blocks lesson 4: Addition (no regrouping)

Using variations of the classic card game 'War' to promote addition and subtraction

Another game I like to have kids play to promote number sense, addition and subtraction is the classic card game War. The original version of War is a great game in that it allows kids to think in terms of quantities: They have to think about which number is greater--so they're thinking in terms of quantities. I suggest even teaching younger children to play this but using unifix cube, counters, or even counting the suits on the cards. I prefer younger kids use manipulatives so they can make sense of numbers as well as see and feel larger quantities.

Double War is a twist on the orginal. Instead of one card, players put down two card each. The player with the greater sum wins the round.

Difference War is another twist. For this version you'll need 50 counters. I use counting chips in the video demo but you can use anything: craft gems, small pebbles, paper clips, etc. Each player put down a card. Players take the difference of the two cards in counters. So if player A puts down a 7 and player B puts down a 10; the difference of 10 and 7 is 3. Player B had the larger number and so picks up 3 counters. Players take back their card and place in a discard pile. After the 50 counters have run out, both players count up their counters; player with the most counters wins.

Enjoy watching your child play and learn!