--- title: Lessons — a course of Mova's primary school, with trees and pragmatics description: The mathematics of the primary school: counting on and back, the number between, doubles, making tens, tens and ones, skip counting, odd and even, places in a line, and word problems of joining, separating and comparing. Every problem is read into a world of things and counts, and Mova answers from --- [Mova](https://md.mova.elamur.ai/free/en/) · [this page in HTML](https://mova.elamur.ai/free/en/corpora/school/lessons/) [home](https://md.mova.elamur.ai/free/en/) / [corpora](https://md.mova.elamur.ai/free/en/corpora/) / [school](https://md.mova.elamur.ai/free/en/corpora/school/) / Lessons # Lessons The mathematics of the primary school: counting on and back, the number between, doubles, making tens, tens and ones, skip counting, odd and even, places in a line, and word problems of joining, separating and comparing. Every problem is read into a world of things and counts, and Mova answers from that world. The lessons course of Mova's primary school: 200 lessons, 644 sentences, each sentence with its syntax tree and pragmatics. Licence: [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/) (Mova corpus 0.4). **1** [2](https://md.mova.elamur.ai/free/en/corpora/school/lessons/page/2/) - [61](https://md.mova.elamur.ai/free/en/corpora/school/lessons/61-counting-on-and-back/) — Counting on and counting back — 4 sentences · To count on, start at the number you have and say the next numbers, one for each step forward. - [62](https://md.mova.elamur.ai/free/en/corpora/school/lessons/62-the-number-between/) — The number just before, just after and in between — 3 sentences · The number just after a number is one more; the number just before it is one less. - [63](https://md.mova.elamur.ai/free/en/corpora/school/lessons/63-doubles-and-near-doubles/) — Doubles and near doubles — 3 sentences · A double is a number added to itself: 7 + 7 = 14. - [64](https://md.mova.elamur.ai/free/en/corpora/school/lessons/64-making-tens/) — How many more to make a full ten — 4 sentences · To find how many more make a round number, count on from what you have to that number. - [65](https://md.mova.elamur.ai/free/en/corpora/school/lessons/65-tens-and-ones/) — Bundles of ten and loose ones — 4 sentences · A bag of 10 is one ten. - [66](https://md.mova.elamur.ai/free/en/corpora/school/lessons/66-how-many-more-to-match/) — How many more to have as many — 3 sentences · Take the smaller amount away from the bigger amount. - [67](https://md.mova.elamur.ai/free/en/corpora/school/lessons/67-skip-counting/) — Skip counting by twos, fives and tens — 4 sentences · When every group has the same number, count by that number: by 2s for pairs, by 5s for hands, by 10s for tens. - [68](https://md.mova.elamur.ai/free/en/corpora/school/lessons/68-pairs-and-odd-ones/) — Pairs and the one left alone — 3 sentences · When things are put in pairs, every pair has 2. - [69](https://md.mova.elamur.ai/free/en/corpora/school/lessons/69-places-in-a-line/) — Places in a line — 3 sentences · In a line, first, second, third… tell the place counted from one end. - [70](https://md.mova.elamur.ai/free/en/corpora/school/lessons/70-number-sense-review/) — Number sense review: tens, ones and how many more — 4 sentences · First find each amount: bags of 10 count as tens, loose ones as ones. - [71](https://md.mova.elamur.ai/free/en/corpora/school/lessons/71-join-result-unknown/) — Some more join in: how many now — 3 sentences · A story can start with some things and then more things join them. - [72](https://md.mova.elamur.ai/free/en/corpora/school/lessons/72-join-change-start-unknown/) — Some more join in: how many came, how many at first — 3 sentences · In a joining story we know two of three numbers: how many at first, how many came, how many at the end. - [73](https://md.mova.elamur.ai/free/en/corpora/school/lessons/73-separate-result-unknown/) — Some go away: how many are left — 3 sentences · A story can start with some things and then some of them go away: given away, lost or sold. - [74](https://md.mova.elamur.ai/free/en/corpora/school/lessons/74-separate-change-start-unknown/) — Some go away: how many went, how many at first — 3 sentences · In a story where some things go away we know two of three numbers: how many at first, how many went, how man… - [75](https://md.mova.elamur.ai/free/en/corpora/school/lessons/75-part-part-whole/) — Two parts and the whole — 2 sentences · Sometimes nothing comes or goes: a group is simply made of two parts, like boys and girls in a class. - [76](https://md.mova.elamur.ai/free/en/corpora/school/lessons/76-compare-difference/) — How many more, how many fewer — 3 sentences · To compare two numbers, we ask how many more one has than the other. - [77](https://md.mova.elamur.ai/free/en/corpora/school/lessons/77-compare-bigger-smaller/) — More than, fewer than, less than — 1 sentences · Read carefully who is compared to whom: «Sam has 9, that is 4 fewer than Lily» means Lily has 4 more, 9 + 4… - [78](https://md.mova.elamur.ai/free/en/corpora/school/lessons/78-two-steps/) — Two steps: some come, some go — 3 sentences · Some stories have two changes: first some things come, then some go away, or the other way round. - [79](https://md.mova.elamur.ai/free/en/corpora/school/lessons/79-extra-number/) — A number we do not need — 6 sentences · Some stories tell us a number that the question does not ask about. - [80](https://md.mova.elamur.ai/free/en/corpora/school/lessons/80-mixed-review/) — Review: adding and taking away stories — 5 sentences · Each story asks for one number that is missing: a start, a change, an end, a whole, a part or a difference. - [81](https://md.mova.elamur.ai/free/en/corpora/school/lessons/81-equal-groups/) — Equal groups — 3 sentences · When there are several groups and every group holds the same number, multiply to find how many in all. - [82](https://md.mova.elamur.ai/free/en/corpora/school/lessons/82-arrays/) — Rows and columns — 3 sentences · When things stand in rows and every row is the same length, they make an array. - [83](https://md.mova.elamur.ai/free/en/corpora/school/lessons/83-times-as-many/) — Times as many — 2 sentences · If Tom has 4 marbles and Ann has 3 times as many, Ann has 4 × 3 = 12 marbles. - [84](https://md.mova.elamur.ai/free/en/corpora/school/lessons/84-sharing/) — Sharing equally — 3 sentences · To share things equally means every one gets the same number. - [85](https://md.mova.elamur.ai/free/en/corpora/school/lessons/85-grouping/) — How many groups — 3 sentences · Sometimes we know how many go in each group and must find how many groups we can make. - [86](https://md.mova.elamur.ai/free/en/corpora/school/lessons/86-round-up/) — When the remainder needs one more — 3 sentences · When everyone or everything must fit, the ones left over still need a place, so we need one more group. - [87](https://md.mova.elamur.ai/free/en/corpora/school/lessons/87-full-boxes/) — Only the full ones count — 3 sentences · Sometimes only full groups count, and the ones left over are not enough for another group. - [88](https://md.mova.elamur.ai/free/en/corpora/school/lessons/88-rates/) — So many each day — 2 sentences · To find how much in several days, multiply the amount for one day by the number of days. - [89](https://md.mova.elamur.ai/free/en/corpora/school/lessons/89-multiply-then-add/) — Multiply, then add or take away — 3 sentences · Some problems take two steps: first multiply to find the total of the equal groups, then add or take away. - [90](https://md.mova.elamur.ai/free/en/corpora/school/lessons/90-mixed-review/) — Multiplying and dividing together — 3 sentences · Many problems need both: first multiply to find the whole, then divide to share it. - [91](https://md.mova.elamur.ai/free/en/corpora/school/lessons/91-counting-coins/) — Counting coins — 3 sentences · A penny is worth 1 cent, a nickel 5 cents, a dime 10 cents and a quarter 25 cents. - [92](https://md.mova.elamur.ai/free/en/corpora/school/lessons/92-total-cost/) — Buying several things at one price — 2 sentences · When every thing costs the same, the total cost is the price of one times the number of things. - [93](https://md.mova.elamur.ai/free/en/corpora/school/lessons/93-change/) — The change from a payment — 3 sentences · A dollar is 100 cents. - [94](https://md.mova.elamur.ai/free/en/corpora/school/lessons/94-saving-weeks/) — Saving every week for a price — 2 sentences · If you save the same amount every week, the number of weeks is the price divided by the amount saved each we… - [95](https://md.mova.elamur.ai/free/en/corpora/school/lessons/95-elapsed-hours/) — How many hours have passed — 2 sentences · To find how long something lasts on the same day, count the hours from the start to the end: end minus start. - [96](https://md.mova.elamur.ai/free/en/corpora/school/lessons/96-hours-minutes/) — Hours and minutes — 3 sentences · One hour has 60 minutes. - [97](https://md.mova.elamur.ai/free/en/corpora/school/lessons/97-days-weeks/) — Days and weeks — 3 sentences · One week has 7 days. - [98](https://md.mova.elamur.ai/free/en/corpora/school/lessons/98-metres-centimetres/) — Metres and centimetres — 3 sentences · One metre is 100 centimetres. - [99](https://md.mova.elamur.ai/free/en/corpora/school/lessons/99-weight-capacity/) — Kilograms and grams, litres and millilitres — 3 sentences · One kilogram is 1000 grams, and one litre is 1000 millilitres. - [100](https://md.mova.elamur.ai/free/en/corpora/school/lessons/100-two-step-measures/) — Yards, feet and inches in two steps — 3 sentences · One yard is 3 feet, and one foot is 12 inches. - [101](https://md.mova.elamur.ai/free/en/corpora/school/lessons/101-one-part-of-a-set/) — One equal part of a set — 3 sentences · A fraction like one third means: split the whole set into equal parts and take one part. - [102](https://md.mova.elamur.ai/free/en/corpora/school/lessons/102-several-parts-of-a-set/) — Several equal parts of a set — 3 sentences · Two thirds means two of the three equal parts. - [103](https://md.mova.elamur.ai/free/en/corpora/school/lessons/103-left-after-a-part/) — What is left after a part is used — 2 sentences · Find the part first, then take it away from the whole. - [104](https://md.mova.elamur.ai/free/en/corpora/school/lessons/104-comparing-parts/) — Comparing a half with a smaller part — 2 sentences · A half of a set is the biggest of these parts: a half of 12 is 6, a third of 12 is 4, a fourth of 12 is 3. - [105](https://md.mova.elamur.ai/free/en/corpora/school/lessons/105-perimeter-of-a-rectangle/) — The perimeter of a rectangle — 2 sentences · A rectangle has two long sides and two short sides: a rectangle 7 cm long and 3 cm wide has a perimeter of 7… - [106](https://md.mova.elamur.ai/free/en/corpora/school/lessons/106-area-of-a-rectangle/) — The area of a rectangle in square units — 4 sentences · Area tells how many unit squares cover a shape. - [107](https://md.mova.elamur.ai/free/en/corpora/school/lessons/107-side-from-perimeter-or-area/) — A missing side from the perimeter or the area — 2 sentences · A square has four equal sides, so one side is the perimeter divided by 4: a perimeter of 28 cm gives 28 ÷ 4… - [108](https://md.mova.elamur.ai/free/en/corpora/school/lessons/108-sides-and-corners-of-shapes/) — Sides and corners of many shapes — 2 sentences · A triangle has 3 sides and 3 corners, a square 4, a pentagon 5 and a hexagon 6: a flat shape has as many cor… - [109](https://md.mova.elamur.ai/free/en/corpora/school/lessons/109-faces-edges-corners-of-cubes/) — Faces, edges and corners of cubes — 1 sentences · One cube has 6 + 12 + 8 = 26 faces, edges and corners together, so 3 cubes have 3 × 26 = 78. - [110](https://md.mova.elamur.ai/free/en/corpora/school/lessons/110-fractions-and-shapes-review/) — Review: fractions and shapes together — 3 sentences · Some problems need two steps: first the area of a rectangle, then a fraction of that area. - [111](https://md.mova.elamur.ai/free/en/corpora/school/lessons/111-unit-price/) — The price of one, then the price of many — 3 sentences · When things cost the same each, first find the price of one: divide the total price by how many were bought. - [112](https://md.mova.elamur.ai/free/en/corpora/school/lessons/112-speed/) — Speed: how far in so many hours — 4 sentences · Speed tells how far something goes in one hour: a car going 60 km each hour has a speed of 60 km an hour. - [113](https://md.mova.elamur.ai/free/en/corpora/school/lessons/113-ages/) — Ages: years ago, years from now — 3 sentences · Everyone gets one year older each year, so in 5 years a child of 8 will be 8 + 5 = 13, and 3 years ago was 8… - [114](https://md.mova.elamur.ai/free/en/corpora/school/lessons/114-day-patterns/) — A pattern day by day — 2 sentences · When something grows by the same number each day, the first day has the start, and each later day adds the s… - [115](https://md.mova.elamur.ai/free/en/corpora/school/lessons/115-ratio/) — For every few of one, so many of another — 2 sentences · Find how many groups there are from the colour you know, then multiply for the other colour. - [116](https://md.mova.elamur.ai/free/en/corpora/school/lessons/116-average/) — The average of a few scores — 2 sentences · The average shares the total out evenly: add all the scores, then divide by how many scores there are. - [117](https://md.mova.elamur.ai/free/en/corpora/school/lessons/117-working-backwards/) — Working backwards through two steps — 3 sentences · To find how many there were at the start, begin at the end and undo each step in the opposite order. - [118](https://md.mova.elamur.ai/free/en/corpora/school/lessons/118-distracting-number/) — Two steps and a number you do not need — 6 sentences · Some stories give a number that does not help answer the question. - [119](https://md.mova.elamur.ai/free/en/corpora/school/lessons/119-three-steps/) — A story in three steps — 4 sentences · A long story can be solved one step at a time: find one number, use it to find the next, then finish. - [120](https://md.mova.elamur.ai/free/en/corpora/school/lessons/120-mixed-review/) — Mixed challenge: reasoning stories — 4 sentences · Each story here is one of the kinds met before: a price per thing, a speed, ages, a pattern, a ratio, an ave… - [121](https://md.mova.elamur.ai/free/en/corpora/school/lessons/121-adding-across-ten/) — Adding across ten — 1 sentences · When two numbers together make more than ten, first fill up a ten, then add what is left. - [122](https://md.mova.elamur.ai/free/en/corpora/school/lessons/122-taking-away-across-ten/) — Taking away across ten — 2 sentences · When there are not enough ones to take away, first take away down to ten, then take away the rest from the t… - [123](https://md.mova.elamur.ai/free/en/corpora/school/lessons/123-counting-in-patterns/) — Counting in a repeating pattern — 4 sentences · A pattern repeats the same group again and again. - [124](https://md.mova.elamur.ai/free/en/corpora/school/lessons/124-how-many-more/) — How many more, how many fewer — 2 sentences · To compare two numbers, match them one to one; what has no partner is the difference. - [125](https://md.mova.elamur.ai/free/en/corpora/school/lessons/125-the-missing-number/) — The missing number in a story — 4 sentences · Sometimes a story hides one number: what there was at first, or how many were added or taken. - [126](https://md.mova.elamur.ai/free/en/corpora/school/lessons/126-more-by-fewer-by/) — More by, fewer by — 3 sentences · «7 more than» means add: 7 more than 20 is 20 + 7 = 27. - [127](https://md.mova.elamur.ai/free/en/corpora/school/lessons/127-two-step-stories/) — Stories in two steps — 3 sentences · Some stories change a number twice. - [128](https://md.mova.elamur.ai/free/en/corpora/school/lessons/128-cuts-posts-and-floors/) — Cuts, posts and floors: the gaps between — 4 sentences · To cut a log into 4 pieces you need only 3 cuts: the last piece needs no cut. - [129](https://md.mova.elamur.ai/free/en/corpora/school/lessons/129-sisters-and-brothers/) — Sisters and brothers — 3 sentences · A girl does not count herself among her own sisters. - [130](https://md.mova.elamur.ai/free/en/corpora/school/lessons/130-staircase-and-handshakes/) — A staircase of cubes and handshakes — 3 sentences · A staircase of 4 steps built of cubes has 1, 2, 3 and 4 cubes in its steps: 1 + 2 + 3 + 4 = 10 cubes. - [131](https://md.mova.elamur.ai/free/en/corpora/school/lessons/131-times-table-stories/) — The times table in stories — 3 sentences · When there are several equal groups, multiply the number of groups by the number in one group. - [132](https://md.mova.elamur.ai/free/en/corpora/school/lessons/132-sharing-and-grouping/) — Division: sharing and grouping — 4 sentences · Division answers two questions. - [133](https://md.mova.elamur.ai/free/en/corpora/school/lessons/133-times-more-times-fewer/) — So many times more, so many times fewer — 1 sentences · So if Ann has 5 and that is 3 times fewer than Ben has, Ben has 5 × 3 = 15. - [134](https://md.mova.elamur.ai/free/en/corpora/school/lessons/134-two-units-in-one/) — Metres and centimetres, dollars and cents — 3 sentences · 1 metre is 100 centimetres, and 1 dollar is 100 cents. - [135](https://md.mova.elamur.ai/free/en/corpora/school/lessons/135-hours-and-minutes/) — Hours and minutes — 3 sentences · 1 hour is 60 minutes, so 2 hours are 60 + 60 = 120 minutes. - [136](https://md.mova.elamur.ai/free/en/corpora/school/lessons/136-perimeter-of-a-triangle/) — Perimeter: all the way round a triangle — 4 sentences · The perimeter of a shape is the length all the way round it: add the lengths of all its sides. - [137](https://md.mova.elamur.ai/free/en/corpora/school/lessons/137-brackets-as-stories/) — Brackets in a story: first the group, then the groups — 5 sentences · Work out what is in brackets first. - [138](https://md.mova.elamur.ai/free/en/corpora/school/lessons/138-equations-as-stories/) — The unknown number: equations as stories — 4 sentences · An equation is a story with a missing number. - [139](https://md.mova.elamur.ai/free/en/corpora/school/lessons/139-cuts-and-gaps/) — Cuts and pieces, posts and gaps — 3 sentences · To cut a log into 5 pieces you need only 4 cuts: the last piece needs no cut of its own. - [140](https://md.mova.elamur.ai/free/en/corpora/school/lessons/140-ages-sum-and-difference/) — Ages that grow together: sum and difference — 3 sentences · Each year both children get one year older, so the sum of two ages grows by 2 each year, and the difference… - [141](https://md.mova.elamur.ai/free/en/corpora/school/lessons/141-three-digit-stories/) — Numbers to 1000: adding and taking away in stories — 3 sentences · A number to 1000 has hundreds, tens and ones: 476 is 4 hundreds, 7 tens and 6 ones. - [142](https://md.mova.elamur.ai/free/en/corpora/school/lessons/142-multiply-by-one-digit/) — Multiplying by a one-digit number — 3 sentences · To multiply a big number by a one-digit number, multiply the ones, the tens and the hundreds apart and add:… - [143](https://md.mova.elamur.ai/free/en/corpora/school/lessons/143-how-many-can-be-bought/) — Dividing by a one-digit number: how many can be bought — 3 sentences · To divide a big number by a one-digit number, divide the hundreds, then the tens, then the ones: 576 ÷ 4: 5… - [144](https://md.mova.elamur.ai/free/en/corpora/school/lessons/144-price-from-change/) — The price of one thing from the money paid and the change — 2 sentences · If you paid with more money than the things cost, the change came back: what the things cost is the money gi… - [145](https://md.mova.elamur.ai/free/en/corpora/school/lessons/145-remainder-round-up/) — What the remainder means: one more is needed — 3 sentences · Dividing with a remainder gives full groups and some left over: 50 ÷ 8 = 6 remainder 2. - [146](https://md.mova.elamur.ai/free/en/corpora/school/lessons/146-area-from-perimeter/) — A rectangle's area from its perimeter — 3 sentences · The perimeter of a rectangle is twice the length plus twice the width, so half the perimeter is length + wid… - [147](https://md.mova.elamur.ai/free/en/corpora/school/lessons/147-sum-and-multiple/) — Sum and multiple: two amounts from their total and «times as many» — 3 sentences · If one amount is 3 times another, draw the smaller as 1 part and the bigger as 3 parts: together they are 1… - [148](https://md.mova.elamur.ai/free/en/corpora/school/lessons/148-composite-shapes/) — The area of a shape made of rectangles — 4 sentences · An L-shaped figure can be seen in two ways: as two rectangles side by side, or as a big rectangle with a cor… - [149](https://md.mova.elamur.ai/free/en/corpora/school/lessons/149-sum-and-difference/) — Sum and difference: two amounts from their total and their gap — 3 sentences · If two amounts add up to a total and one is bigger by a difference, add the difference to the total: now bot… - [150](https://md.mova.elamur.ai/free/en/corpora/school/lessons/150-square-cut-into-strips/) — A square cut into equal strips — 3 sentences · When a square is cut by parallel cuts into equal strips, every strip is as long as the square's side, and it… - [151](https://md.mova.elamur.ai/free/en/corpora/school/lessons/151-fraction-of-a-quantity/) — A fraction of a quantity — 3 sentences · To find a fraction of a quantity, first find one equal part: divide by the number below the line. - [152](https://md.mova.elamur.ai/free/en/corpora/school/lessons/152-the-whole-from-a-fraction/) — The whole from its fraction — 3 sentences · If we know how much a fraction of something is, we can find the whole. - [153](https://md.mova.elamur.ai/free/en/corpora/school/lessons/153-time-intervals/) — How long it lasts: from one clock time to another — 3 sentences · To find how long something lasts, count from the start time to the end time. - [154](https://md.mova.elamur.ai/free/en/corpora/school/lessons/154-calendar-across-months/) — The calendar: days from one month into the next — 2 sentences · April, June, September and November have 30 days; the other months have 31, except February. - [155](https://md.mova.elamur.ai/free/en/corpora/school/lessons/155-difference-and-multiple/) — Difference and multiple: how many times as many, and how many more — 3 sentences · The difference is then 3 - 1 = 2 units. - [156](https://md.mova.elamur.ai/free/en/corpora/school/lessons/156-planting-trees-both-ends/) — Planting trees along a road, with a tree at both ends — 2 sentences · When trees stand in a line with one at each end, there is always one more tree than there are gaps. - [157](https://md.mova.elamur.ai/free/en/corpora/school/lessons/157-planting-trees-ends-and-rings/) — Planting trees: both ends, one end, no ends, and around a pond — 7 sentences · First find the number of gaps: length ÷ distance. - [158](https://md.mova.elamur.ai/free/en/corpora/school/lessons/158-ages-how-many-times/) — Ages: in how many years will one be twice or three times as old — 4 sentences · The difference between two ages never changes: every year both people get one year older. - [159](https://md.mova.elamur.ai/free/en/corpora/school/lessons/159-model-method-units/) — The bar model: the whole from the part that is left, and from a difference — 3 sentences · Draw the whole as a bar cut into equal units. - [160](https://md.mova.elamur.ai/free/en/corpora/school/lessons/160-digits-on-the-pages/) — Olympiad: how many digits number the pages of a book — 6 sentences · Pages 1 to 9 use one digit each: 9 digits. **1** [2](https://md.mova.elamur.ai/free/en/corpora/school/lessons/page/2/) - 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