The session, Math Continuum: Multiplication and Division - Session 1, was presented online on Thursday, Oct 24h. This online session was presented by Heather Wark (Lakehead University), Tien Ngo, and Denise Ladd(LKDSB Instructional Coaches).
Children should learn their number facts. However, they would benefit from learning these facts by using an increasingly sophisticated series of strategies rather than by jumping directly to memorization. - (Lawson, 2016, p. 4)
Strategies help students find an answer even if they forget what was memorized. Discussing math fact strategies focuses attention on number sense, operations, patterns, properties, and other critical number concepts. - (O’Connell & SanGiovanni, 2011, p. 5)
Strategies included:
Unitizing
Modelling composite units and counting by 1
Counting Rhythmically
Skip Counting
Using Repeated Addition
Join us for the next sessions that will dig into the strategies:
Math Strand: Number Sense and Numeration Math Strategy: Skip Counting
English video:
Video french:
Materials: One die, whiteboard and whiteboard marker
(Extension: colour tiles)
Overview:
This game is very engaging for students and helps to prepare students to move from using repeated addition to using grouping of numbers when modeling arrays when solving multiplication problems. After completing this lesson you would then begin a lesson using arrays for equal groups.
On each turn, students roll a 1–6 number cube to find out how many circles to draw and then roll the cube again to find out how many stars to draw in each circle. For example, for a first roll of 4 and a second roll of 3, the student would draw this:
How this activity supports student learning?
Playing Circles and Stars helps students understand that multiplication
can be thought of as combining equal groups and is, therefore, related to addition when the same number is added multiple times. From playing, students become familiar with multiplying numbers from 1X1 to 6X6.
As an extension, you could use try using an 8 sided die.
After students know how to play the game, they learn to represent each
of their turns with addition and multiplication equations. This helps them connect the standard symbols to their experience and also strengthens the connection between addition and multiplication.
Where to next?
In the Alex Lawson text, What To Look For there are several other games that can be played to support unitizing, such as: - How Long? How Many?, which uses Cuisenaire rods - Skip Counting Race - Target 75 - Multiplication Challenge
These can all be found on page 185-189.
Share your classroom experiences with us on Instagram and Twitter at @LKelempro #EngageLK!
Strand: Number Sense and Numeration Strategy: Skip Counting
Materials: One 6-sided die with the numbers 2, 2, 3, 3, 4 and 5 (use stickers over the existing numbers) BLM MK13: Skip Counting Race game board
Overview: The students (or teacher) determines the number that the students will skip count by, prior to beginning the game. Students roll the die and skip count by the agreed upon number, based on the number rolled. For example; if a student is skip counting by 2 and rolls a 3 they must skip count 3 times (2, 4, 6). The first student to reach 60 wins.
How This Supports Students Learning: This game provides students with the opportunity to develop efficiency by using the skip counting strategy. Students that are able to skip counting can use this strategy when solving multiplication problems. They are also strengthening their understanding of unitizing; recognizing that each time they skip count the value represents a given quantity or set of numbers.
In this game, students begin by skip counting with a friendly number once they master this skill they can move on to another number. Teachers may want students to begin with 5 or 2, moving forward to 3 or ’s.
For students that are struggling with skip counting, they may rely on counting rhythmically by 1’s, emphasizing every time they have added a unit (1, 2, 3, 4). It may also be helpful to provide the student with an extra copy of the game board and have them use two different colours to record their counting (colour 2 squares red, two squares blue, etc.).
Where To Next: Once students have developed efficiency with the skip counting strategy, they are able to move towards using repeated addition and doubling strategies. As students are introduced to the repeated addition strategy it may be useful for them to first skip count and then to record their addition sentence above this, in order to make the connection between the two strategies visible.
Additional games to reinforce the concept of skip counting from the What to Look For text by Alex Lawson include Target 75 (page 188) and Array Pattern Flashes (page 194).
Math Strand: Number Sense and Numeration
Math Strategy: Skip Counting
Key Idea: Unitizing
Materials:
one die, paper and pencil
Overview:
This activity is found the Alex Lawson’s “What To Look For” on page 188.
After 6 turns, students are trying to have a total close to 75 without going over.
Students will roll a die and select a factor to multiply by: 1, 2, 3, 4, 5 or 10.
The game will encourage students to think about which number to select to multiply by so they are get as close to 75 as possible.
How this activity supports student learning?
Target 75 helps students make sense of quantity and the numbers they are using.
Students will begin to realize that if they select a higher factor in the beginning, they will need to use smaller ones and adjust their choices accordingly as the game goes on. For example, during their final rolls, they will need to decide between multiplying by 1 or 2 to get as close to 75 as possible.
Students may play the game with a partner. This encourages them to communicate with each other their mathematical thinking and reasoning. (e.g. It's our first roll, we should multiply the 5 by 10! That equals 50. We will be close to 75.)
This game supports students understand of unitizing, which is the idea that objects can simultaneously be viewed as individual objects and as a composite unit that can be counted. Through game play, students may use direct modelling to solve the problems, such as making 3 groups of 4 objects, then counting each object. Moving beyond counting by ones, children may begin to rhythmically count the groups or they may repeatedly count groups on their fingers. This will then move to skip counting, then doubling, repeated addition or even a ratio table.
Students can use rekenreks, hundreds charts, number lines, counters or colour tiles to help support their calculations.
Where to next?
Challenge students to work with numbers larger than 75 or change the factors (e.g. 1, 5,6,7,8,9,10 ).
Another game to try is Multiplication Challenge.
If students are not yet ready for all these factors 1,2,3,4,5,6, reduce the number of choices (Alter the Die by using masking tape or switch to number cards)
On Lawson’s Continua:
Skip counting can be found in the Addition and Subtraction Continuum and the Multiplication and Division Continuum.
Overview:
Skip Counting is counting by regular intervals repeatedly to reach a certain value. It is counting by a number other than one.
In early grades, students learn to skip count by 2, 5, and 10s. They learn to skip count forwards and backwards. Then later they learn to skip by 100s, 50s and 25s. Skip counting is different then counting rhythmically. To ensure students connect the values of the counts have them build the amounts with cubes or counters. This will support students as them begin to understand that each skip count, or grouping, is a unit of a certain amount.
(5, 10, 15, 20 is 4 units of 5)
When skip counting, students learn that you can count to a certain number faster instead of counting all numbers individually. Initially, students will solve problems like: 5 groups of 2 by creating the groups by adding 2 counters into each group. They then count all of the counters individually to find the product. As students learn to skip count and grasp the efficiency of skip counting, they will progress to counting by 2’s in this problem rather than 1’s.
From skip counting students learn that groups of equal sizes can be combined to form a whole quantity. For example, when skip counting by 2’s, students will be able to see that 2, 4, 6, 8, 10 ends up being 5 groups of 2 combined to get 10.
Skip counting helps students identify patterns in numbers but also supports their ability to later think multiplicatively. To facilitate this type of thinking, students can be asked to identify all the ways they can skip count to a certain number. This will help them to identify multiples of that number and then the factors as well.
All the ways to skip count to 24:
2, 4, 6. 8, 10, 12, 14, 16, 18, 20, 22, 24 (12 groups of 2)
3, 6, 9, 12, 15, 18, 21, 24 (8 groups of 3)
4, 8, 12, 16, 20, 24 (6 groups of 4)
6, 12, 18, 24 (4 groups of 6)
8, 16, 24 (3 groups of 8)
12, 24, (2 groups of 12)
According to Marian Small in Making Math Meaningful, pg. 149,
“Multiplicative thinking is thinking that focuses on how one number is made up of groups of another “unit” ... For example, thinking of 20 as 4 fives is thinking of the 20 being made up of the units of 5.” Knowledge of skip counting will support this type of thinking.
Supporting Students to Skip Count:
There are many ways to support skip counting, as seen on the video. These include:
- Using 100 charts to mark the numbers they are skip counting by
- Creating amounts with base 10 blocks to skip count by 1000’s, 100’s, 10’s, or 2’s to find their totals
- Money can be used to support skip counting, such as adding up tens, loonies, quarters, dimes, or nickels.
- Number lines can be used to add up in chunks to show equal groups, which may then progress to open number lines where they don’t show all the numbers on the line
- Students may build arrays and skip count along rows or columns to find the total amounts using colour tiles. As the amounts become difficult to skip count they will split the groups into smaller groups they are more comfortable with
Students should be encouraged to learn to skip count on from a variety of numbers. This will support students to later solve problems by skip counting on from an addend. For example, 37 + 40:
An example of Skip Counting from Lawson's What to Look For, pg 86:
Where to next:
Consider using activities that promote Skip Counting in real life contexts in your class. Some suggestions can be found in the text, “What to Look For” by Alex Lawson, on page 157, specifically pages 178, 185, 190 & 195.
Additionally, “Trading Up to 1000” is another activity that supports skip counting. This game, along with other information and activities on skip counting, can be found in the K-3 Guide to Effective Instruction in Math, Number Sense and Numeration.
Share your classroom experiences with Skip Counting with us on Instagram and Twitter at @LKelempro #EngageLK!
Math Strategy:
This game supports the development of skills required for skip counting, splitting, and taking jumps (see Lawson continuum). Also, it helps students to represent, compare, and order whole numbers to 1000.
Math Strand: Number Sense and Numeration
Overview:
Start with two teams and alternate turns between teams. In this game, students start with a 2 digit number and build their amount in base ten blocks on their place value chart. Then they roll one die to see how many times they can skip count on by ones, tens, and thousands. They decide each time which value they will skip count by. The goal is to be the first team to create a cube (of 1000). This game can be played with only 2 players once students know the game.
This game will allow you to see if your students are still at the concrete stage of skip counting on. If they move away from building with the base ten blocks on the place value chart then you know they have grasped (or are beginning to grasp) how to count on without the use of a model to assist them. If they are relying on the base ten then you know more practice, with this game, would be beneficial to them.
How this supports student learning:
Being able to start counting at any number is important because it shows how well a student understands numerical order. Students will have practice skip counting concretely and numerically at the same time. Thus connecting the two representations. It also helps students to learn that you can use friendly numbers to skip count on from any number. This will prepare your students for situations when they are learning to use strategy taking jumps. See the example below taken from Lawson's What To Look For on page 86.
The game will increase students' familiarity with breaking numbers apart into their place value. This will prepare them for using the addition & subtraction strategy of splitting:
(What To Look For, pg 86)
This game can be changed to meet your students' needs by changing the trading up to a different amount (10 000 or 100). If increasing the amount to 10 000 give the students the additional option of skip counting on by 1000.
Where to next?
This activity can be found in the Guide to Effective Instruction in Mathematics, Number Sense and Numeration, on page 143. To allow students to apply the skills they have learned in this game, begin looking at addition and subtraction activities that support skip counting, splitting, and taking jumps in "What to Look For" by Alex Lawson.
Share your classroom experiences with Trading Up To 1000 with us on Instagram and Twitter at @LKelempro #EngageLK!
