New Zealand Curriculum: Level 3
Learning Progression Frameworks: Multiplicative thinking, Signpost 4 to Signpost 5
Target students
These activities are intended for students who understand multiplication as the repeated addition of equal sets, and who have some knowledge of basic multiplication facts. Students should also have an existing understanding of whole number place value, preferably to six places.
The following diagnostic questions indicate students’ understanding of, and ability to apply, multiplication and division by powers of ten to whole numbers. The questions are given in order of complexity. If the student answers a question confidently and with understanding proceed to the next question. If not, then use the supporting activities to build and strengthen their fluency and understanding. Allow access to pencils and paper but not to a calculator unless stated. (show diagnostic questions)
The questions should be presented orally and in a written form so that the student can refer to them. The questions have been posed using various contexts but can be changed to other contexts that are engaging to your students. Ensure students can record on a piece of paper.
- The recipe calls for 6 ripe bananas. To make the same cake ten times the size, how many bananas do you need?
Signs of fluency and understanding:
Anticipates that 60 bananas are needed using 10 x 6 = 60.
What to notice if your student does not solve the problem fluently:
Unable to turn “ten times bigger” into an action. This indicate that the student has yet to be exposed to scaling situations and to the associated mathematical language.
Repeated addition or skip counting in sixes. This indicates that the student understands the meaning of “ten times bigger” but does not see an opportunity to use multiplication. They may be unaware that 10 x 6 has the same answer as 6 x 10 = 60 (6 tens).
Supporting activity:
Multiplying by ten
- The recipe calls for 24 tomatoes. To make ten times as much soup, how many tomatoes do you need?
Signs of fluency and understanding:
Anticipates that 240 tomatoes are needed using 10 x 24 = 240.
What to notice if your student does not solve the problem fluently:
Unable to turn “ten times bigger” into an action. This may indicate that the student has yet to be exposed to scaling situations and to the associated mathematical language.
Repeated addition is unusual but may occur. This indicates that the student understands the meaning of “ten times bigger” but does not see an opportunity to use multiplication. They may be unaware that 10 x 24 has the same answer as 24 x 10 = 240 (24 tens). This may indicate that the student does not understand the nested nature of whole number place value, such as ten tens make 100.
Supporting activity:
More multiplying by ten
- The phone costs $360. You get the money out of the ATM in $10 notes. How many ten-dollar notes do you have?
Explain how you worked out your answer.
Signs of fluency and understanding:
Calculates the number of $10 notes using 36 x 10 = 360 or knows that $360 contains 36 tens (place value knowledge). Explains their answer using the language of multiplication or division, e.g., “I know 36 times ten equals 360.”
What to notice if your student does not solve the problem fluently:
Repeated addition or skip counting in tens and double tracking (possibly as tallies on paper), i.e., “10, 20, 30, …, 360”. This indicates a lack of place value knowledge, such as ten tens make 100. The tens will be counted to get an answer of 36.
Partial build up and tracking (possibly using pencil and paper), such as 10 tens equal 100, 20 tens equal 200, 3 tens equal 300, 310, 320, 330, …, 360. The number of tens will be tracked somehow, e.g. 10, 20 , 30, 31, 32, …, 36. This indicates that the student knows that 10 tens make 100 but has not yet learnt to combine the tens in hundreds and decades in a single calculation.
Explanations of “subtracting a zero” may result in a correct answer but further questioning is needed to find out if the student understands the effect on quantities of dividing by powers of ten.
Supporting activity:
Dividing by ten
The recipe calls for:
- 5 kilograms of oranges
- 750 grams of sugar
- 500 millilitres of water
- 20 grams of butter
To make a batch of marmalade that is 100 times bigger, how much of each ingredient do you need?
Explain how you worked out each amount.
Signs of fluency and understanding:
Anticipates the amounts correctly using multiplication by 100, i.e. 100 x 1.5 = 150, 100 x 750 = 75 000, 100 x 500 = 50 000, 100 x 20 = 2 000
What to notice if your student does not solve the problem fluently:
Unable to turn “100 times bigger” into an action. This may indicate that the student has yet to be exposed to scaling situations and to the associated mathematical language.
Use of multiplication to correctly find the amounts whilst experiencing difficulty reading the numbers correctly. This indicates that more learning is needed about understanding the place value structure of larger numbers.
Explanations of “adding zeros” may result in correct answers but further questioning is needed to find out if the student understands the effect on quantities of multiplying by powers of ten.
Supporting activity:
Multiplying by 100
- A team of 100 people win a prize of $27,080. If the prize is shared equally, how much money does each person get?
Explain how you got your answer.
Signs of fluency and understanding:
Uses an efficient division strategy such as 27 080 ÷ 100 = 270.8 and explains that the answer means $270.80 in context. Uses the language of division, such as “I divided because I needed to share the money equally.”
What to notice if your student does not solve the problem fluently:
Partial build up and tracking (Using pencil and paper), such as 10 hundreds make 1000, so 270 hundreds make 27 000. This indicates that the student has strong place value knowledge and recognises that the problem involves equal sharing. The student needs to develop efficient calculation strategies for division by powers of ten.
Explanations of “subtracting two zeros” or a written algorithm may result in a correct answer, but further questioning is needed to find out if the student understands the effect on quantities of dividing by powers of ten.
Supporting activity:
Dividing by 100
- Here is the number 7 094. Multiply the number by 1000, make it one thousand times larger.
Explain how you worked out your answer.
Now divide 7 094 by 1000, make it one thousandth of what it was.
Explain how you worked out your answer.
Signs of fluency and understanding:
Uses multiplication and division-based strategies, such as, 1000 x 7 094 = 7 094 000 and 7 094 ÷ 1000 = 7.094.
Recording shows that attention has been given to place values up to millions and down to thousandths.
Explains that multiplying by 1000 shifts the digits three places to the left and dividing by 1000 shifts the digits three places to the right. Correctly states the answers.
What to notice if your student does not solve the problem fluently:
Adding zeros to 7 094 may mean the student has an algorithm for multiplying by 10, 100, 1000, etc. Further inquiry is needed to establish if the algorithm is supported by place value understanding.
Writes the correct answer, 7 094 000 but is unable to say the numbers. This indicates that more fluency is needed with connecting numerals to words.
Unable to conduct the operation 7 094 ÷ 1000 = 7.094 or say the answer as “Seven point zero, nine, four”. This indicates that the student needs further support to develop their place value understanding of decimals.
Supporting activity:
Multiplying and dividing by up to 1000