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 * Maths Review Questions **:

** REVIEW QUESTIONS - NUMERATION: **

 ﻿ ﻿ 1. A parent brings a 5 year-old child to school on the first day and says to the Prep teacher, “Sara knows her numbers and can count to one hundred. We think she needs to be extended in maths”. What would you say? What would you endeavour to find out over the first few weeks of term and why? Can she understand that the last number is the “count”? Is she accurate, reliable and consistent in her count? She may only know the number naming sequence. 2. What is meant by trusting the count? Why is it so important in developing more efficient counting strategies and mental strategies? Trusting the count has a range of meanings. These include: Believing that if they count the same collection again they will get the same result, that counting is a strategy to determine how many.Trusting the count means that the student has developed a mental object of the numbers 0-9, and can work with these numbers without having to make, count or see these collections physically.. That they understand the part part whole nature of numbers, eg. 8 is one eight or two fours, or four twos, etc. Trusting the count is important in counting strategies as students are able to use small collections as composite units when counting larger collections. 3. A Prep child consistently miscounts as in the following example:              “one…two…three…four…five…six…seven….eight…nine…ten…eleven…thirteen…” Describe what might be happening and why. Indicate what the child needs to know and how you might go about addressing this situation. She is matching syllables to objects intead of one object to one number. The child needs to know the naming sequence better. This problem lies in mapping counts to objects.

4. Why are ten-frames useful in developing part-part-whole understanding of number in the early years? Describe and illustrate what this means for the number 8. Why is this understanding so important?

Ten frames help to subtise numbers in terms of their smaller parts. Part, Part, Whole knowledge. They reinforce how each number relates to 10. Eight is 2 less than 10, 4 and 4, 5 and 3, 3 and 5, double 4, 6 and 2 more.

5. What knowledge, skills, and understandings would you expect a Year 1 student to demonstrate before you introduced 2-digit place-value?

Need to be able to count to 20. Need to be able to count collections using composite units of 2, 5 10. Part, part whole knowledge of numbers 0-10. Trusting the count (can work with numbers to 10 without having to model the count). Need to be able to read, write and interpret numbers 0-10.

6.Understanding that ten ones is 1 ten, that is, appreciating 1 ten as a composite unit, is an important step in the development of place-value ideas. Describe the materials, generic activities and language you would use in developing 2-digit place-value ideas from the introduction of the new unit through to the confident use of all 2-digit numbers. 1. Establish the new unit- 10 ones is 1 ten. Use bundling materials,eg. icypole sticks to make tens into countable units. 2. Introduce the names for multiples of ten- start with Cardinal (six-ty, seven-ty, eight-ty, nine-ty) Then Ordinal, thirty (should be three-ty), fifty (should be five-ty), then Mispelt, Twenty (should be two-ty) forty (should be four-ty). 3. Make name and record tens and ones for 20-99 (regular numbers first). Eg, make 6 tens and 7 ones, name it as sixty-seven, record on a number chart, six in the 10s column and 7 in the ones column. Use unifix blocks to show 6 of these and 7 of those. 4. Make, name and record tens and ones for numbers 10-19 (least irregular first), eg, EIGHTeen, then FIFTeen, then twelve. As with step 3, make name and record using bundling materials and record tens in one column and ones in another (place value chart). 5. Consolidate place value. Compare- compare 2 numbers using different materials, words and symbols. Which is larger/smaller, why? Order (Sequence) more than 2 numbers from smallest to largest, eg place numbers on a rope) Count forwards and backwards in place vale parts starting from anywhere  Rename numbers in more than one way.  Describe the steps in developing a deep understanding of 2-digit place value. Your answer should demonstrate that you know what numbers are likely to be problematic for young children and how place-value knowledge might be consolidated.  See above  You notice that some of your students in Year 2 are recording numbers in unusual ways, for example, 205 (for twenty-five) and 61 (for sixteen). What do you think might be behind this and what would you do about it? Your answer should demonstrate that you know the sequence involved in developing an understanding of 2 digit numbers, what numbers are likely to be problematic for young children and how place-value knowledge might be consolidated. The student is recording 205 for 25 as they do not understand the idea that the 2 in twenty 5 is 2 tens, they are simply recording the number 20 and 5. For the student recording 61 for sixteen, once again they are just recording the name of the number as they hear it, without understanding that 16 is one ten and 6 ones. 7 . Give ‘real-world’ examples of a comparison numeration activity involving 4-digit numbers and an ordering numeration activity involving ones and tenths.

One week the zoo had 3417 visitors, the next week it was 3471. Which week had more visitors? Order these temperatures from hottest to coldest: 1.2, 3.5, 1.8, 7.9

8. Describe how you would explain (justify) the naming of 5 digit numbers and beyond.

See below Describe and illustrate what is meant by the HTO or second place-value pattern and how it might be used to help students develop an understanding of 5-digit numeration and beyond. The second place value pattern is 1 thousand of these is 1 of those.It is an efficient naming system involving the repeated use of hundreds, tens and ones to count certain units, eg. thousands, millions, billions, from 1 to 999, instead of 1 to 9. students will learn that we count thousands from 1to 999, then millions from 1to 999 and so on.

9. What examples would you use to review Year 4 students understanding of fractions? What key ideas/generalisations would you be looking for and why?

Is it a big share or little share? Would you rather have 2 thirds of a pizza or two quarters of a pizza, why? That equal parts are required. That it is how many divided by how much. As the total number of parts increases, the size of the parts decreases.

10. Describe the thinking involved and use diagrams as appropriate to illustrate the thirding and fifthing partitioning strategies. How can these strategies be used to help children construct their own fraction diagrams? Why is this important?

Thirding: Think 3 equal part...2 equal parts... one third is less than one half... estimate a third then fold the remained in half. Fifthing: Think 5 equal parts.. 4 equal parts... 1 fifth is less than 1 quarter, estimate, then halve and halve again. Children need to create their own fraction diagrams so they are not simply counting partitions on someone else's diagram. This does not require an understanding of fractions.

// Partitioning is a powerful strategy for consolidating fraction ideas. Illustrate and explain what is meant by partitioning in this context, how it might be used to ensure a thorough understanding of simple common fractions and why it is important. // // Describe the thinking involved and use diagrams as appropriate to illustrate the fifthing partitioning strategy. Show how partitioning strategies might be used to locate 3.64 on a number line. // // Use elastic to first show 3.6, then on another piece of elastic show how 3.64 falls between 3.60 and 3.70 // 11. Give ‘real-world’ examples of (i) a comparison activity involving 5-digit numbers and (ii) an ordering activity involving ones, tenths, and hundredths.

One week 52789 people attended the footy, the next week it was 52893. Which week had more people? A long jumper jumped 6.345 metres, and another jumped 6.360. Who jumped the furthest?

12. Renaming is an important activity to consolidate place-value understanding, (i) give an example of this using a 5 digit number and (ii) illustrate what this might mean for 207.4.

52789 is 5278 tens and 9 ones. It is 527 hundreds, 8 tens and 9 ones. it is 52 thousand, 78 tens and 9 ones. 207.4 is 20 tens, 7 ones and 4 tenths. It is 2 hundreds and 74 tenths.

13. Use the numbers 483,028 and 61.02 to illustrate what is meant by renaming and explain how it is used to help round numbers to a particular place-value parts, for example, the nearest hundred in 76,845.

483,028 is 4830 hundreds, 2 tens and 8 ones. 61.02 is 6 tens 1 one and 2 hundredths. The nearest 100 in 76845- look at the 2 possibilities. 786 hundreds or 769 hundreds. Next, look to the next place (the tens). 40 tens, so 768 is the nearest hundred.

14. Illustrate how you would use partitioning strategies to show where 4.372 is located in a number line.

Use elastic as per second part of question 10.

15. Use the fraction  to illustrate how would you use the fraction as division idea to rename common fractions as decimals and percents? // Describe how you would use the fraction ////  //// to illustrate the link between fractions, division, decimals and percent. //

16. By the end of Year 6 students need to be able to rename simple common fractions in a meaningful and efficient manner. Describe and illustrate how you would help Year 5/6 students develop this capacity. State the generalisation students need to arrive at in order to be able to rename fractions consistently and easily.

17. List the pre-requisite knowledge needed to add unlike fractions (fractions with different denominators).

18. You notice that a child consistently orders decimal fractions from smallest to largest in the following way: 2.4, 2.7, 2.19, 2.81, 2.023, 2.483, 2.1539. What might be the thinking behind this? What would you do to help address this situation? What is the correct order? // A group of Year 6 students ordered a group of decimal numbers from smallest to largest in the following way: 1.6 4.8 1.26 3.08 4.18 4.83 4.761. What thinking may have led the students to do this? What would you do about it and why? Include a description and/or illustration of any materials or models you might use to do this. //

19. A child consistently orders fractions from smallest to largest in the following way: ,, , ,  What might be the thinking behind this? What would you do to help address this situation? What is the correct order?

** REVIEW QUESTIONS – OPERATIONS: **

20. Write two word problems (action stories) to illustrate the difference between the two concepts for addition. Which idea would be introduced first? Give reasons for your answer.

21. A small group of Year 1 student insists on using the strategy, ‘make-all/count-all’, to deal with simple action stories such as “Five birds were sitting on the fence, eight birds joined them. How many birds are there altogether?” Describe what you might expect these children to be doing and what you would do to help scaffold their learning. 22. Write two word problems that you might use in Year 1 to illustrate the difference between the ‘take-away’ idea and the ‘missing addend’ idea for subtraction. Which idea would be introduced first? Give reasons for your answer.

I have four apples but two are rotten and need throwing away, how many do I have left? I ha I have 5 oranges but I need 7, how many do I need to buy? In trIntroduce take away first as there is a developmental delay of 6 months between them, Missing addend is harder to understand.

23. You want to introduce the ‘difference’ idea to students in Year 1. What action stories might you use to check the students’ readiness to learn this idea? Describe and illustrate the materials, models, and action stories you might use to introduce the difference idea.

Kate has 3 smarties, Belinda has seven, how many more smarties does Belinda have? Her Heree are two pieces of string. What is the difference in length? Measure your own and your friend’s height. How is taller/shorter, by how much?  24. Name the three mental strategies that might be used by Year 1 or 2 children to solve single-digit addition problems. Give two examples of each to illustrate the thinking involved.

3 and 8, think 8... 9, 10,11. 1 and 6, think 6..7. **Doubles and near doubles**. (preps to 10, grade 1 to 20) 4 and 4, think double 4..8 6 and 7, think 6 and 6 is 12 and 1 more, 13. 9 and 8, think double 9 is 18, 1 less, 17. **Make to ten and count on.** 8 and 3, think 8… 10, 11. 9 and 6 think 9…10, 15. ||  ||
 * <span style="color: #0000ff; font-family: Times New Roman; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">**Count on from larger** (count from the larger number so you get to the answer without having to count as many)- 6 and 2 think 6..7, 8

// Describe and illustrate how you might introduce/scaffold the three mental strategies for addition // <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">25. Describe the thinking and mental strategies you might expect a Year 1 or 2 child to use to solve 16 take away 9 <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">26. Asked to use an open number line to show how you might add 24 and 27, a Year 5 child did the following: ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ 24 51  Comment on the child’s strategy and what it suggests about the child’s knowledge of numeration. Describe what you would do about this and why. // Describe how you would use an open-number line to illustrate an efficient mental solution to 26 and 57 with Year 2 or 3 children. // <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">27. Explain and illustrate how you might use an open number line (or a thinking string) to scaffold an efficient mental solution to 83 take 37 // Describe how you would use an open-number line to illustrate an efficient mental solution to 74 take away 47 with Year 2 or 3 children. // <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">28. Use examples and diagrams as appropriate to illustrate the difference between the ‘groups of’, ‘array’ and ‘region’ ideas for multiplication. Why are the ‘array’ and ‘region’ ideas more useful than the ‘groups of’ idea? <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">29. Why is the ‘region’ idea so important for developing a sound understanding of fractions. Use examples to illustrate your answer. <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">30. Write a word problem to illustrate each of the division concepts. Name the ideas and indicate which is more useful for dealing with larger numbers and fractions? Why? // Describe and illustrate the essential difference between the two ideas for division, ‘quotition’ and ‘partition’, using simple whole number examples. Partitioning is often described as the missing link in developing formal fraction ideas. Why is this idea so critical and how can it be used to support student’s understanding of decimal fractions? // <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">31. Write a word problem to illustrate the Cartesian Product or ‘for each’ idea of multiplication that involves at least 3 variables. Show how you might represent this situation. <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">32. A Year 4/5 class appears to have been taught to count and memorise all groups in learning the multiplication facts. For example, “1 three is three, 2 threes are 6, 3 threes are 9, 4 threes are 12, 5 threes are 15” and so on (standard ‘tables’). They are experiencing considerable difficulty with any facts involving numbers larger than 5, for example 7 eights, 6 nines and 8 fours. What might have lead to this and what would you do to turn this situation around? <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">33. Give 2 examples for each of the following mental strategies and illustrate the thinking involved, (i) double and 1 more group for the 3s multiplication facts, and (ii) relate to tens for the 9s multiplication facts // Name and illustrate a mental strategy and describe the thinking involved to determine (i) 7 fours, (ii) 8 nines, and (iii) 23 sixes. // <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">34. Explain how you would use a number expander and renaming to support a written solution to 6005 – 2679. Include a complete description of the language and recoding involved // Show how you would use a number expander to support a written solution to the problem 13012 take away 6678. Include a complete description of the language and recording involved. // <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">35. Describe the thinking and strategies you might expect a Year 4/5 student to use to mentally calculate (i) 46 multiplied by 6, and (ii) 508 divided by 7. // Describe two mental strategies that might be used to calculate (i) 57 multiplied by 8, and (ii) 627 divided by 9. // <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">36. Describe the materials/models, language and recording you would use to support a written solution to (i) 357 multiplied by 8, and (ii) 2803 multiplied by 7. <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">37. Describe a mental strategy for calculating 342 divided by 6. <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">38. Describe the materials, language and recording you would use to develop an understanding of the ‘area’ concept for multiplication using the example, 34 x 52. // Describe and illustrate (using MAB and the area idea for multiplication) the four steps involved in developing a written solution to the problem 23 multiplied by 42. How might you use this result to justify a solution to 2.3 by 4.2? // <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">39. Describe the language and recording that you might use with grade 6 children to develop a written solution to the problem 1875.87 divided by 9 using the partition (sharing) idea for division <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">40. Describe the language and recording you would use in developing a written solution to the problem, “A pizza shop sold pizza by the slice. Just before rush hour the shop had 7 and 2 third pizzas left for sale. If 2 and 3 quarters of the pizzas remained at the end of the rush hour, how much pizza had been sold?” // Indicate the language and recording you would use to develop a written solution to the following problem: A poultry farmer had 8 and 2/3 trays of eggs on one shelf and 5 and 5/8 trays on another. How many trays of eggs did he have altogether? // <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">41. Use examples to illustrate how you might justify the multiplication of a fraction by a whole number to Year 6 students. <span style="font-family: Arial; font-size: 11pt; margin: 0cm 0cm 6pt 32.2pt; tabstops: list 32.2pt; text-indent: -18pt;">42. ‘Helping Bert with division’ is a widely used assessment task for students in Years 4 to 7 (see Beesey et al, 1998). This task is included below. Complete all questions and describe what you believe to be the teachers’ intent in using this task and how you as a teacher might prevent this situation from arising in future.

4 628 || || 1447 <span style="font-family: Times; height: 714px; left: 3353px; margin: 0cm 0cm 0pt; position: absolute; top: 2808px; width: 1087px;">5 7235
 * One of your best friends, Bert, asks you to help him with division. He shows you some division questions which he has done. They look like this: || 157

|| || 28  <span style="font-family: Times; height: 714px; left: 3334px; margin: 0cm 0cm 0pt; position: absolute; top: 2847px; width: 951px;">6 1248

|| || 165  <span style="font-family: Times; height: 714px; left: 3334px; margin: 0cm 0cm 0pt; position: absolute; top: 2847px; width: 951px;">3 4815

|| First check Bert’s answers. If an answer is correct, tick it. If not, write the correct answer underneath. Create a very hard question which you think Bert might be able to answer correctly. Show how he might work out the answer. Write down two questions which you think Bert might get wrong. Give the answers Bert might give to the questions, and then show the correct answers. What would you show or tell Bert to help him when he is doing division questions? Some of Bert’s answers to the problems at the problems he showed you are far too small. Use one of his answers to explain this to him.

I have four apples but two are rotten and need throwing away, how many do I have left? I have 5 oranges but I need 7, how many do I need to buy? Introduce take away first as there is a developmental delay of 6 months between them, Missing addend is harder to understand.