Activities to support Multiplication and division
Where are they now?
Students:
- make and identify equal groups
- need visible items to calculate the total number of items when solving multiplication problems, often
- counting by ones
- know the forward number word sequences when counting in composite units
- know the backward number word sequences when counting in composite units
Where to next?
Students:
- solve multiplication problems by counting in composite units, using perceptual markers
- calculate the total number of items in an array pattern without relying on visual representations
- uses rhythmic counting and skip counting
Teaching point
Literature link: Many traditional folk and fairy tales have a theme based on three, such as The Three Little Pigs and Goldilocks and the Three Bears. Grouping based on these tales can be used in activities.
Outcomes
The following activities provide opportunities for students to demonstrate progress towards the following outcomes. A student:
MA1-1WM describes mathematical situations and methods using everyday and some mathematical language, actions, materials, diagrams and symbols
MA1-2WM uses objects, diagrams and technology to explore mathematical problems
MA1-3WM supports conclusions by explaining or demonstrating how answers were obtained
MA1-6NA uses a range of mental strategies and concrete materials for multiplication and division
How?
Units for two
Collect or draw items commonly found in pairs, such as eyes, shoes, socks, or legs. Model the method of counting the items, using rhythmic counting. Model methods for keeping track of the number of groups as well as the total number of items such as using fingers as a perceptual marker. Allow opportunities for the students to practise the modelled methods. Demonstrate how the total number of items in a specified number of “groups of two” can be found by counting the first number in each pair silently and voicing the second number. Allow opportunities for the students to practise this counting method.
Triangle teddies
Provide the students with a collection of popsticks and a collection of plastic teddies. Instruct the students to make a series of triangles using their popsticks. Ask the students to place a teddy on each corner. The students then count and record the number of teddies on each triangle. Students can then use these recordings to practice counting in multiples.
Variations
- Change triangles to squares and pentagons, etc. as well as a straight line to practise counting in multiples of two
- Practise counting back in composite units by starting with the total
- Have students record their thinking by annotating a photograph, recording themselves counting in composite units
- If the students are unable to skip count by threes, encourage rhythmic or stressed counting
How?
Handprints
Make a handprint to represent a group of five. Repeat printing the handprints across a strip of paper. Ask the students to count the number of hands and to find the total number of fingers using rhythmic or skip counting methods.
Why?
Students need to be able to recall the counting sequences for nominated multiples. Multiplication and division strategies will be limited by students’ knowledge of these sequences.
Groupies
Provide the students with a collection of items such as counters, marbles, plastic teddies, etc. Direct the students to form a specified number of equal groups using the items, for example, “make four groups of three”. Use rhythmic or skip counting, or apply a known fact, to find the total number of items in the specified groups. Have students record their work by taking and annotating photographs or by drawing diagrams.
Variation
• Use dot pattern cards in groups or arrays to represent groups. Ask students to find, for example, 3 groups of 7 or 5 groups of 4. Have students record the cards they found and the number words they would say if counting in multiples to work out the total number of dots
Finding and calculating groups
Provide students with images displaying a large number of items. Instruct students to circle groups of items. For example: “Put a circle around groups of five.” Ask the students to determine how many groups were made and to use rhythmic counting or skip counting to find the total.
Why?
Early multiplication and division strategies focus on the structure and use of groups of items. Students need to develop the concept of seeing a group of items as one “unit” and no longer relying on counting each item within the group.
What’s in the square?
Construct multiple copies of the grid (BLM - Whats In The Square) and accompanying picture cards displaying groups of items. Each pair of students will need a copy of the grid, a deck of picture cards and counters of two different colours. Shuffle the picture cards and place them face down before the students. The first player takes a card from the top of the pile and places a counter on the corresponding square on the grid. For example, if a card displaying one item is drawn, the student places his or her counter on the “one group of 1” square at the top left-hand corner of the grid. Players continue to take turns to turn over cards and mark them on the grid. The winner is the first player to make a line of three counters horizontally, vertically or diagonally.
To extend this activity introduce numeral cards which indicate the total number of items on each picture card. After the students place a picture card onto the grid, instruct them to determine the total of the groups and place a corresponding numeral card on top of the picture card.
Why?
Early multiplication and division strategies focus on the structure and use of groups of items. Students need to develop the concept of seeing a group of items as one “unit” and no longer relying on counting each item within the group.
Dice and grid game
Construct a 6x6 grid for each player. Write the numerals 1 to 36, in sequence, onto the grid, placing one numeral in each square. Each player will also need six sets of dot pattern cards representing the numbers one to six, a numeral dice and a dot pattern dice.
In this activity, the “numeral dice” is used to indicate the number of equal groups, and the “dot pattern dice” indicates the number of items in each group.
Instruct the students to roll the two dice and state the size of each group and the number of equal groups. When the student has stated the size and number of equal groups, ask the student to collect the correct number of dot pattern cards to represent the groups. For example, if the student wishes to make five groups of three, he or she would then select five cards showing three dots on each card.
Students then find the total number of dots by rhythmic or skip counting, and place a counter on the corresponding numeral on the grid. In the above example, the student would place the counter on the numeral 15. The winner is the first to have three counters in a row, horizontally, vertically or diagonally.
Variation
- Allow the students to check the answer on a calculator. Observe whether the student uses repeated addition or multiplication with the calculator
Why?
Early multiplication and division strategies focus on the structure and use of groups of items. Students need to develop the concept of seeing a group of items as one “unit” and no longer relying on counting each item within the group.Teaching point
A note about arrays:
An array is an ordered collection of objects. Arranging objects into rows and columns provides a more efficient way of counting as well as a strong visual representation of amount. The structure of arrays allows us to explore skip counting by rows and by columns, counting by 1s, adding rows together (repeated addition), seeing the relationship between multiplication and division, and so on.
Arrays: changing groups
Arrange eight students into four rows with two students in each row. Explain that this way of arranging items, where we have equal rows is called an array.
Ask students:
- How many students are there in each row?
- How many students are there in each column?
- How many students are there altogether?
Discuss the different ways the students could be counted in order to work out how many there are in total (i.e. counting by 1s, 4s and 2s).
Discuss what would happen if another row of students was added, if another column was added, a column was taken away, etc. Continue adding and removing rows and columns, examining how the total changes.
Variation
- Have students go on an “array hunt”. Provide students with iPads or other tablet devices, or learning journals, and have them document the arrays they find at their school. Students should annotate their examples, explaining how many rows and columns there are, how many in each and how they counted to work out the total
Turning arrays
Provide each student with a small sheet of cardboard and a supply of counters. Instruct students to form arrays by placing the counters onto the cardboard following instructions, such as “make three rows of five counters”. Students then turn the card 90° to show a new array of five rows of three. Discuss with the students the number of rows, the number of counters in each row and the total number of counters for each array pattern.
Variation
- Allow the students to form arrays using shape prints,thumb prints or adhesive stickers. Provide students with instruction cards for making the arrays
Why?
The formation of arrays is important in developing concepts of groups and coordinating groups. This coordination of groups can lead to abstract concepts of multiplication and division.
Array flash
Organise students into pairs. Provide the students with a collection of counters. Instruct one student to use their counters to make an array that is no larger than 5 by 5. The student then briefly shows the array to his or her partner before screening the counters with a sheet of cardboard. The other student then attempts to construct the same array pattern with counters. The students should then compare the two arrays. Ask students to find the total number of counters in the array in the most efficient way possible, encouraging skip or rhythmic counting to ascertain the product. Check by counting by ones. Have students record their thinking before swapping roles.
Teaching point
Observe and question the students to determine how they are calculating the total in the array.
Variation
- Once created, students describe the array to the partner by stating the number of rows (how many) and the number of counters in each row (how much). The partner then attempts to make the array. He or she then determines the total number of counters in the array, recording their thinking before revealing the array and counting to check
Why?
Early multiplication and division strategies focus on the structure and use of groups of items. Students need to develop strategies where they see a group of items as one unit and no longer need to count each individual item within the group.